Research ArticleAlgorithm for Solving a New System of Generalized NonlinearQuasi-Variational-Like Inclusions in Hilbert Spaces

Shamshad Husain Sanjeev Gupta and Huma Sahper

Department of Applied Mathematics Faculty of Engineering amp Technology Aligarh Muslim University Aligarh 202002 India

Correspondence should be addressed to Sanjeev Gupta guptasanmpgmailcom

Received 10 September 2013 Accepted 5 November 2013 Published 5 February 2014

Academic Editors Q Guo and X-G Li

Copyright copy 2014 Shamshad Husain et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We introduce and study a new system of generalized nonlinear quasi-variational-like inclusions with 119867(sdot sdot)-cocoercive operatorin Hilbert spaces We suggest and analyze a class of iterative algorithms for solving the system of generalized nonlinear quasi-variational-like inclusions An existence theorem of solutions for the system of generalized nonlinear quasi-variational-likeinclusions is proved under suitable assumptions which show that the approximate solutions obtained by proposed algorithmsconverge to the exact solutions

1 Introduction

Variational inclusion problems are important generalizationof classical variational inequalities and havewide applicationsto many fields including mechanics physics optimizationand control nonlinear programming economics and engi-neering sciences see for example [1] For these reasonsvarious variational inclusions have been intensively studiedin recent years Many efficient ways have been studied to findsolutions for variational inclusions Those methods includethe projection method and its various forms linear approx-imation descent and Newtonrsquos method and the methodbased on auxiliary principle technique The method basedon the resolvent operator technique is a generalization ofthe projection method and has been widely used to solvevariational inclusions For details we refer to see [2ndash19]

Recently Fang and Huang Kazmi and Khan and Lanet al investigated several resolvent operators for general-ized operators such as 119867-monotone [3 17] 119867-accretive[4] 119860-maximal relaxed accretive [14] (119867 120578)-monotone [5](119860 120578)-accretive [13] (119875 120578)-proximal point [8] and (119875 120578)-accretive [9] operators Very recently Zou and Huang [19]introduced and studied 119867(sdot sdot)-accretive operators Kazmiet al [10ndash12] introduced and studied generalized 119867(sdot sdot)-accretive operators and 119867(sdot sdot)-120578-proximal point mapping

and Xu and Wang [18] introduced and studied (119867(sdot sdot) 120578)-monotone operators Ahmad et al [2 8] introduced andstudied119867(sdot sdot)-cocoercive operators showed some propertiesof the resolvent operator for the119867(sdot sdot)-cocoercive operatorsand obtained an application for solving variational inclusionsin Hilbert spaces They also gave some examples to illustratetheir results

Inspired and motivated by the researches going on in thisarea we introduce and discuss a new system of generalizednonlinear quasi-variational-like inclusions involving 119867(sdot sdot)-cocoercive operators in Hilbert spaces By using the resolventoperators associated with119867(sdot sdot)-cocoercive operators due toAhmad et al [2] we prove that the approximate solutionsobtained by the iterative algorithms converge to the exactsolutions of our system of generalized nonlinear quasi-variational-like inclusions Our results can be viewed as anextension and generalization of some known results in theliterature

2 Preliminaries

Throughout this paper we suppose that 119883 is a real Hilbertspace endowed with a norm sdot and an inner product ⟨sdot sdot⟩119889 is the metric induced by the norm sdot 2119883 (resp 119862119861(119883)) is

Hindawi Publishing CorporationChinese Journal of MathematicsVolume 2014 Article ID 957482 7 pageshttpdxdoiorg1011552014957482

2 Chinese Journal of Mathematics

the family of all the nonempty (resp closed and bounded)subsets of 119883 and D(sdot sdot) is the Hausdorff metric on 119862119861(119883)

defined by

D (119875 119876) = maxsup119909isin119875

119889 (119909 119876) sup119910isin119875

119889 (119875 119910)

forall119875 119876 isin 119862119861 (119883)

(1)

where 119889(119909 119876) = inf119910isin119876

119909 minus 119910 and 119889(119875 119910) = inf119909isin119875

119909 minus 119910In the sequel let us recall some concepts

Definition 1 (see [20]) A mapping 119892 119883 rarr 119883 is said to be

(i) monotone if

⟨119892 (119909) minus 119892 (119910) 119909 minus 119910⟩ ge 0 forall119909 119910 isin 119883 (2)

(ii) 120585-strongly monotone if there exists a constant 120585 gt 0

such that

⟨119892 (119909) minus 119892 (119910) 119909 minus 119910⟩ ge 1205851003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2 forall119909 119910 isin 119883 (3)

(iii) 120583-cocoercive if there exists a constant 120583 gt 0 such that

⟨119892 (119909) minus 119892 (119910) 119909 minus 119910⟩ ge 1205831003817100381710038171003817119892 (119909) minus 119892 (119910)

1003817100381710038171003817

2 forall119909 119910 isin 119883

(4)

(iv) 120574-relaxed cocoercive if there exists a constant 120574 gt 0

such that

⟨119892 (119909) minus 119892 (119910) 119909 minus 119910⟩ ge (minus120574)1003817100381710038171003817119892 (119909) minus 119892 (119910)

1003817100381710038171003817

2 forall119909 119910 isin 119883

(5)

(v) 120582119892-Lipschitz continuous if there exists a constant

120582119892gt 0 such that

1003817100381710038171003817119892 (119909) minus 119892 (119910)1003817100381710038171003817 le 120582119892

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (6)

(vi) 120572-expansive if there exists a constant 120572 gt 0 such that1003817100381710038171003817119892 (119909) minus 119892 (119910)

1003817100381710038171003817 ge 1205721003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin 119883 (7)

if 120572 = 1 then it is expansive

Definition 2 Let119872 119883 rarr 2119883 be set-valued mappingThen

119872 is said to be 1205831015840-cocoercive if there exists a constant 1205831015840 gt 0

such that

⟨119906 minus V 120578 (119909 119910)⟩ ge 1205831015840119906 minus V2

forall119909 119910 isin 119883 119906 isin 119872 (119909) V isin 119872(119910)

(8)

Definition 3 (see [2]) Let119867 119883times119883 rarr 119883 and119860 119861 119883 rarr 119883

be the single-valued mappings Then

(i) 119867(119860 sdot) is said to be 120583-cocoercive with respect to 119860 ifthere exists a constant 120583 gt 0 such that

⟨119867 (119860119909 119906) minus 119867 (119860119910 119906) 119909 minus 119910⟩ ge 1205831003817100381710038171003817119860119909 minus 119860119910

1003817100381710038171003817

2

forall119909 119910 isin 119883

(9)

(ii) 119867(sdot 119861) is said to be 120574-relaxed cocoercive with respectto 119861 if there exists a constant 120574 gt 0 such that

⟨119867 (119906 119861119909) minus 119867 (119906 119861119910) 119909 minus 119910⟩ ge (minus120574)1003817100381710038171003817119861119909 minus 119861119910

1003817100381710038171003817

2

forall119909 119910 isin 119883

(10)

(iii) 119867(119860 sdot) is said to be 1199031-Lipschitz continuous with

respect to 119860 if there exists a constant 1199031gt 0 such that

1003817100381710038171003817119867 (119860119909 sdot) minus 119867 (119860119910 sdot)1003817100381710038171003817 le 1199031

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (11)

(iv) 119867(sdot 119861) is said to be 1199032-Lipschitz continuous with

respect to 119861 if there exists a constant 1199032gt 0 such that

1003817100381710038171003817119867 (sdot 119861119909) minus 119867 (sdot 119861119910)1003817100381710038171003817 le 1199032

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (12)

Definition 4 (see [2]) Let 119867 119883 times 119883 rarr 119883 and119860 119861

119883 rarr 119883 be the single-valued mappingsThen the set-valuedmapping 119872 119883 rarr 2

119883 is said to be 119867(sdot sdot)-cocoercive withrespect to119860 and119861 (or simply119867(sdot sdot)-cocoercive in the sequel)if

(i) 119872 is cocoercive(ii) (119867(119860 119861) + 120582119872)(119883) = 119883 for every 120582 gt 0

Example 5 Let 119883 = R2 with the usual inner product Let119860 119861 R2 rarr R2 be defined by

119860119909 = (21199091minus 21199092 minus21199091+ 41199092)

119861119910 = (minus1199101+ 1199102 minus1199102) forall119909 119910 isin R

2

(13)

Suppose that119867(119860 119861) R2 timesR2 rarr R2 is defined by

119867(119860119909 119861119910) = 119860119909 + 119861119910 forall119909 119910 isin R2 (14)

Then it is easy to check that 119867(119860 119861) is 16-cocoercive withrespect to 119860 and 12-relaxed cocoercive with respect to 119861

Let 119872 = 119868 where 119868 is the identity mapping Then 119872 is119867(sdot sdot)-cocoercive mapping with respect to 119860 and 119861

Example 6 Let 119883 = S2 where S2 denotes the space of all2times2 real symmetric matrices Let119867(119860119909 119861119910) = 119909

2minus119910 for all

119909 119910 isin S2 and119872 = 119868 Then for 120582 = 1 we have

(119867 (119860 119861) +119872) (119909) = 1199092minus 119909 + 119909 = 119909

2 (15)

but

[0 0

0 minus1] notin (119867 (119860 119861) +119872) (S

2) (16)

because [ 0 00 minus1

] is not the square of any 2 times 2 real symmetricmatrix Hence119872 is not 119867(sdot sdot)-cocoercive with respect to 119860and 119861

Proposition 7 (see [2]) Let 119867(119860 119861) be 120583-cocoercive withrespect to 119860 and 120574-relaxed cocoercive with respect to 119861 119860 is120572-expansive 119861 is 120573-Lipschitz continuous and 120583 gt 120574 120572 gt 120573Let 119872 119883 rarr 2

119883 be an 119867(sdot sdot)-cocoercive operator If theinequality

⟨119906 minus V 119909 minus 119910⟩ ge 0 (17)

Chinese Journal of Mathematics 3

holds for all (119910 V) isin 119866119903119886119901ℎ(119872) then 119906 isin 119872119909 where

119866119903119886119901ℎ (119872) = (119906 119909) isin 119883 times 119883 119909 isin 119872 (119906) (18)

Theorem 8 (see [2]) Let119867(119860 119861) be 120583-cocoercive with respectto 119860 and 120574-relaxed cocoercive with respect to 119861 119860 is 120572-expansive 119861 is 120573-Lipschitz continuous and 120583 gt 120574 and 120572 gt 120573Let M be an 119867(sdot sdot)-cocoercive operator with respect to 119860 and119861 Then the operator (119867(119860 119861) + 120582119872)

minus1 is single-valued

Definition 9 (see [2]) Let 119867(119860 119861) be 120583-cocoercive withrespect to 119860 and 120574-relaxed cocoercive with respect to 119861 119860 is120572-expansive 119861 is 120573-Lipschitz continuous and 120583 gt 120574 120572 gt 120573Let119872 be an119867(sdot sdot)-cocoercive operator with respect to119860 and119861 The resolvent operator 119877119867(sdotsdot)

120582119872 119883 rarr 119883 is defined by

119877119867(sdotsdot)

120582119872(119906) = (119867 (119860 119861) + 120582119872)

minus1(119906) forall119906 isin 119883 (19)

Theorem 10 (see [2]) Let 119867(119860 119861) be 120583-cocoercive withrespect to 119860 and 120574-relaxed cocoercive with respect to 119861 119860 is120572-expansive 119861 is 120573-Lipschitz continuous and 120583 gt 120574 120572 gt 120573

with 119903 = 1205831205722minus 1205741205732 Let 119872 be an 119867(sdot sdot)-cocoercive operator

with respect to 119860 and 119861 Then the resolvent operator 119877119867(sdotsdot)120582119872

119883 rarr 119883 is 1(1205831205722 minus 1205741205732)-Lipschitz continuous that is

10038171003817100381710038171003817119877119867(sdotsdot)

120582119872(119906) minus 119877

119867(sdotsdot)

120582119872(V)

10038171003817100381710038171003817le

1

1205831205722 minus 1205741205732119906 minus V forall119906 V isin 119883

(20)

3 The System of Generalized NonlinearQuasi-Variational-Like Inclusions andIterative Algorithm

Let1198831and119883

2be real Hilbert spaces Let 119875 119883

1times1198832rarr 1198831

119876 1198831times1198832rarr 1198832 119892 119860 119861 119883

1rarr 1198831 ℎ 119862119863 119883

2rarr 1198832

119866 1198831times 1198831rarr 1198831 and119867 119883

2times 1198832rarr 1198832be the single-

valued mappings Let 119878 1198831rarr 119862119861(119883

1) and 119879 119883

2rarr

119862119861(1198832) be the set-valued mappings let119872 119883

1times 1198831rarr 21198831

be a set-valued mapping such that for each 119886 isin 1198831 119872(sdot 119886)

is a 119866(sdot sdot)-cocoercive operator with respect to 119860 and 119861 andlet 119873 119883

2times 1198832rarr 21198832 be a set-valued mapping such that

for each 119887 isin 1198832119873(sdot 119887) is an119867(sdot sdot)-cocoercive operator with

respect to 119862 and 119863 Assume that 119892(1198831) cap dom(119872(sdot 119886)) = 0

for each 119886 isin 1198831 and ℎ(119883

2) cap dom(119873(sdot 119887)) = 0 for each 119887 isin

1198832 Then we consider the following system of generalized

nonlinear quasi-variational-like inclusionsFind (119886 119887) isin 119883

1times 1198832with 119901 isin 119878(119886) 119902 isin 119879(119887) such that

0 isin 119875 (119901 119902) +119872(119892 (119886) 119886)

0 isin 119876 (119901 119902) + 119873 (ℎ (119887) 119887)

(21)

Next are some special cases of problem (21)

(1) Let 119878 119879 be identity mappings for each (119886 119887) isin 1198831times

1198832 119872(119892(119886) 119886) = 119872(119909) and 119873(ℎ(119887) 119887) = 119873(119909)

then problem (21) reduces to the following problemconsidered in [15]

0 isin 119875 (119901 119902) +119872 (119886)

0 isin 119876 (119901 119902) + 119873 (119887)

(22)

(2) 1198831= 1198832 119886 = 119887 119878 = 119879 is an identity mapping 119892 = ℎ

119875(119901 119902) = 119876(119901 119902) = 119875(sdot) for each (119886 119887) isin 1198831times 1198832

119872(119892(119886) 119886) = 119873(ℎ(119887) 119887) = 119872(119892(119909)) then problem(21) reduces to the following problem considered in[16]

0 isin 119875 (119909) +119872(119892 (119886)) (23)

For a suitable choice of the mappings 119892 ℎ 119875 119876 119866 119867119860 119861 119862 119863 119878 119879 and the space 119883

1= 1198832 a number of

known systems of quasi-variational inequalities systems ofvariational inequalities systems of quasi-variational inclu-sions and variational inclusions can be obtained as specialcases of the generalized nonlinear quasi-variational inclusionproblem (21) We would like to mention that the problem offinding zero of the sum of two maximal monotone operatorsis also a special case of problem (21) Furthermore these typesof variational inclusion enable us to study many importantproblems arising in mathematical physical and engineeringscience in a general and unified framework

Lemma 11 Let 1198831and 119883

2be real Hilbert spaces Let 119875

1198831times 1198832rarr 119883

1 119876 119883

1times 1198832rarr 119883

2 119892 119860 119861 119883

1rarr

1198831ℎ 119862 119863 119883

2rarr 119883

2be the single-valued mappings

Let 119866 1198831times 1198831

rarr 1198831be a single-valued mapping such

that 119866(119860 119861) is 1205831-cocoercive with respect to 119860 and 120574

1-relaxed

cocoercive with respect to119861119860 is1205721-expansive119861 is120573

1-Lipschitz

continuous and 1205831gt 1205741 1205721gt 1205731 Let 119867 119883

2times 1198832rarr 119883

2

be a single-valued mapping such that119867(119862119863) is 1205832-cocoercive

with respect to 119862 and 1205742-relaxed cocoercive with respect to 119863

119862 is 1205722-expansive 119863 is 120573

2-Lipschitz continuous and 120583

2gt 1205742

1205722gt 1205732 Let 119878 119883

1rarr 119862119861(119883

1) and 119879 119883

2rarr 119862119861(119883

2) be

the set-valued mappings let 119872 1198831times 1198831rarr 21198831 be a set-

valued mapping such that for each 119886 isin 1198831119872(sdot 119886 is a 119866(sdot sdot)-

cocoercive operator and let 119873 1198832times 1198832rarr 21198832 be a set-

valuedmapping such that for each 119887 isin 1198832119873(sdot 119887) is an119867(sdot sdot)-

cocoercive operator Assume that 119892(1198831) capdom(119872(sdot 119886)) = 0 for

each 119886 isin 1198831 and ℎ(119883

2) cap dom(119873(sdot 119887)) = 0 for each 119887 isin 119883

2

Then for any (119886 119887) isin 1198831times 1198832with 119901 isin 119878(119886) 119902 isin 119879(119887) is a

solution of the problem (21) if and only if

119892 (119886) = 119877119866(sdotsdot)

1205821119872(sdot119886)

[119866 (119860 (119892 (119886)) 119861 (119892 (119886))) minus 1205821119875 (119901 119902)]

ℎ (119887) = 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887)) 119863 (ℎ (119887))) minus 1205822119876 (119901 119902)]

(24)

where 119877119866(sdotsdot)

1205821119872(sdot119886)

= (119866(sdot sdot) + 1205821119872(sdot 119886))

minus1 119877119867(sdotsdot)

1205822119873(sdot119887)

=

(119867(sdot sdot) + 1205822119873(sdot 119887))

minus1 and 1205821 1205822gt 0 are constants

Proof By using the definitions of the resolvent operators119877119866(sdotsdot)

1205821119872(sdot119886)

and 119877119867(sdotsdot)1205822119873(sdot119887)

the conclusion follows directly

4 Chinese Journal of Mathematics

The preceding lemma allows us to suggest the followingiterative algorithm for problem (21)

Algorithm 12 For (1198860 1198870) isin 1198831times1198832with119901 isin 119878(119886) 119902 isin 119879(119887)

compute the sequences 119886119899 119887119899 119901119899 119902119899 as follows

119892 (119886119899+1

) = 119877119866(sdotsdot)

1205821119872(sdot119886)

[119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 1205821119875 (119901119899 119902119899) ] 120582

1gt 0

ℎ (119887119899+1

) = 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 1205822119876 (119901119899 119902119899) ] 120582

2gt 0

119901119899isin 119878 (119886

119899)

1003817100381710038171003817119901119899 minus 119901119899+11003817100381710038171003817 le (1 +

1

119899 + 1)

timesD (119878 (119886119899) 119878 (119886

119899+1))

119902119899isin 119879 (119887

119899)

1003817100381710038171003817119902119899 minus 119902119899+11003817100381710038171003817 le (1 +

1

119899 + 1)

timesD (119879 (119887119899) 119879 (119887

119899+1))

(25)

for all 119899 = 0 1 2 and 1205821and 120582

2are constants

Definition 13 Let 119878 1198831rarr 119862119861(119883

1) and 119879 119883

2rarr 119862119861(119883

2)

be two set-valued mappings A single-valued mapping 119875

1198831times 1198832rarr 1198831is said to be

(i) 1205981-Lipschitz continuous in the first argument with

respect to 119878 if there exists a constant 1205981gt 0 such that

1003817100381710038171003817119875 (1199011 sdot) minus 119875 (1199012 sdot)1003817100381710038171003817 le 1205981

10038171003817100381710038171199011 minus 11990121003817100381710038171003817

forall1199011isin 119878 (119886

1) 1199012isin 119878 (119886

2) 1198861 1198862isin 1198831

(26)

(ii) 1205982-Lipschitz continuous in the second argument with

respect to 119879 if there exists a constant 1205982gt 0 such that

1003817100381710038171003817119875 (sdot 1199021) minus 119875 (sdot 1199022)1003817100381710038171003817 le 1205982

10038171003817100381710038171199021 minus 11990221003817100381710038171003817

forall1199021isin 119879 (119887

1) 1199022isin 119879 (119887

2) 1198871 1198872isin 1198832

(27)

Definition 14 Let 119878 1198831rarr 119862119861(119883

1) and 119879 119883

2rarr 119862119861(119883

2)

be two set-valued mappings A single-valued mapping 119876

1198831times 1198832rarr 1198832is said to be

(i) 1205751-Lipschitz continuous in the first argument with

respect to 119878 if there exists a constant 1205751gt 0 such that

1003817100381710038171003817119876 (1199011 sdot) minus 119876 (119901

2 sdot)1003817100381710038171003817 le 1205751

10038171003817100381710038171199011 minus 11990121003817100381710038171003817

forall1199011isin 119878 (119886

1) 1199012isin 119878 (119886

2) 1198861 1198862isin 1198831

(28)

(ii) 1205752-Lipschitz continuous in the second argument with

respect to 119879 if there exists a constant 1205752gt 0 such that

1003817100381710038171003817119876 (sdot 1199021) minus 119876 (sdot 119902

2)1003817100381710038171003817 le 1205752

10038171003817100381710038171199021 minus 11990221003817100381710038171003817

forall1199021isin 119879 (119887

1) 1199022isin 119879 (119887

2) 1198871 1198872isin 1198832

(29)

Definition 15 A set-valuedmapping 119860 119883 rarr 119862119861(119883) is saidto beD-Lipschitz continuous if there exists a constant 120588 gt 0

such that

D (119860 (119909) 119860 (119910)) le 1205881003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin 119883 (30)

Theorem 16 Let 1198831and 119883

2be real Hilbert spaces Let 119875

1198831times 1198832rarr 1198831 119876 119883

1times 1198832rarr 1198832 119892 119860 119861 119883

1rarr 1198831

ℎ 119862 119863 1198832

rarr 1198832 119866 119883

1times 1198831

rarr 1198831 and 119867

1198832times 1198832rarr 1198832be the single-valuedmappings Let 119878 119883

1rarr

119862119861(1198831) and 119879 119883

2rarr 119862119861(119883

2) be the set-valued mappings

let 119872 1198831times 1198831

rarr 21198831 be a set-valued mapping such

that for each 119886 isin 1198831 119872(sdot 119886) is a 119866(sdot sdot)-cocoercive operator

with respect to 119860 and 119861 and let 119873 1198832times 1198832rarr 21198832 be a

set-valued mapping such that for each 119887 isin 1198832 119873(sdot 119887) is an

119867(sdot sdot)-cocoercive operator with respect to 119862 and 119863 Assumethat 119892(119883

1) cap dom(119872(sdot 119886)) = 0 for each 119886 isin 119883

1 and ℎ(119883

2) cap

dom(119873(sdot 119887)) = 0 for each 119887 isin 1198832 Assume that

(i) 119878 119879 are 120588 120591-Lipschitz continuous in the HausdorffmetricD(sdot sdot) respectively

(ii) 119866(119860 119861) is 1205831-cocoercive with respect to 119860 and 120574

1-

relaxed cocoercive with respect to 119861

(iii) 119867(119862119863) is 1205832-cocoercive with respect to 119862 and 120574

2-

relaxed cocoercive with respect to119863

(iv) 119860 119862 is 1205721 1205722-expansive respectively

(v) 119861119863 119894119904 1205731 1205732-Lipschitz continuous respectively

(vi) 119892 is120582119892-Lipschitz continuous and 120585

1-stronglymonotone

(vii) ℎ is 120582ℎ-Lipschitz continuous and 120585

2-stronglymonotone

(viii) 119866(119860 119861) is 1199031-Lipschitz continuouswith respect to119860 and

1199032-Lipschitz continuous with respect to 119861

(ix) H(119862119863) is 1199041-Lipschitz continuous with respect to 119862

and 1199042-Lipschitz continuous with respect to119863

(x) 119875(sdot sdot) is 1205981-Lipschitz continuous in the first argument

and 1205982-Lipschitz continuous in the second argument

(xi) 119876(sdot sdot) is 1205751-Lipschitz continuous in the first argument

and 1205752-Lipschitz continuous in the second argument

(xii)

0 lt

(1199031+ 1199032) 120582119892+ 12058211205981120588

1205851(12058311205722

1minus 12057411205732

1)

+12058221205751 (1 + 1119899) 120588

1205852(12058321205722

2minus 12057421205732

2)lt 1

0 lt(1199041+ 1199042) 120582ℎ+ 12058221205752120591

1205852(12058321205722

2minus 12057421205732

2)

+12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)lt 1

(31)

Then the iterative sequences 119886119899 119887119899 119901119899 and 119902

119899

generated by Algorithm 12 converge strongly to 119886 119887 119901and 119902respectively and (119886 119887 119901 119902) is a solution of problem (21)

Chinese Journal of Mathematics 5

Proof Since 119878 119879 are Lipschitz continuous with constants 120588120591 respectively it follows from Algorithm 12 that

1003817100381710038171003817119901119899 minus 119901119899+11003817100381710038171003817 le (1 +

1

119899)D (119878 (119886

119899) 119878 (119886

119899+1))

le (1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899+11003817100381710038171003817

1003817100381710038171003817119902119899 minus 119902119899+11003817100381710038171003817 le (1 +

1

119899)D (119879 (119887

119899) 119879 (119887

119899+1))

le (1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899+11003817100381710038171003817

(32)

for all 119899 = 0 1 2

Using the 120585-strong monotonicity of 119892 we have1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)

1003817100381710038171003817

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817

ge ⟨119892 (119886119899+1

) minus 119892 (119886119899) 119886119899+1

minus 119886119899⟩

ge 1205851

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817

2

(33)

which implies that

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le

1

1205851

1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)1003817100381710038171003817 (34)

Nowwe estimate 119892(119886119899+1

)minus119892(119886119899) by usingAlgorithm 12 and

the Lipschitz continuity of 119877119866(sdotsdot)1205821119872(sdot119886)

as follows1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)

1003817100381710038171003817

=10038171003817100381710038171003817119877119866(sdotsdot)

1205821119872(sdot119886)

[119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899))) minus 120582

1119875 (119901119899 119902119899)]

minus [119877119866(sdotsdot)

1205821119872(sdot119886)

119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))

minus 1205821119875 (119901119899minus1

119902119899minus1

) ]10038171003817100381710038171003817

le1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899 119902119899) minus 119875 (119901119899minus1 119902119899minus1)1003817100381710038171003817

le1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899)))

1003817100381710038171003817

+1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899 119902119899) minus 119875 (119901119899minus1 119902119899)1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899minus1 119902119899) minus 119875 (119901119899minus1 119902119899minus1)1003817100381710038171003817

(35)

Since119866(119860 119861) is 1199031-Lipschitz continuouswith respect to119860 and

1199032-Lipschitz continuous with respect to 119861 119875 is 120598

1-Lipschitz

continuous in the first argument and 1205982-Lipschitz continuous

in the second argument 119892 is 120582119892-Lipschitz continuous and

using (32) (34) and (35) it becomes

1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)1003817100381710038171003817 le

(1199031+ 1199032) 120582119892

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205981

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119901119899 minus 119901119899minus11003817100381710038171003817

+12058211205982

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119902119899 minus 119902119899minus11003817100381710038171003817

le

(1199031+ 1199032) 120582119892

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205981

12058311205722

1minus 12057411205732

1

(1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205982

12058311205722

1minus 12057411205732

1

(1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le

(1199031+ 1199032) 120582119892+ 12058211205981 (1 + 1119899) 120588

1205851(12058311205722

1minus 12057411205732

1)

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

(36)

Let

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le 1205791119899

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817 + 1205792119899

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817 (37)

where

1205791119899=

(1199031+ 1199032) 120582119892+ 12058211205981 (1 + 1119899) 120588

1205851(12058311205722

1minus 12057411205732

1)

1205792119899=

12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)

(38)

Now we estimate ℎ(119887119899) minus ℎ(119887

119899minus1) by using Algorithm 12 and

the Lipschitz continuity of 119877119867(sdotsdot)1205822119873(sdot119887)

as follows

1003817100381710038171003817ℎ (119887119899+1) minus ℎ (119887119899)1003817100381710038171003817

=10038171003817100381710038171003817119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899))) minus 120582

2119876 (119901119899 119902119899)]

6 Chinese Journal of Mathematics

minus 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))

minus 1205822119876 (119901119899minus1

119902119899minus1

) ]10038171003817100381710038171003817

le1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899 119902119899) minus 119876 (119901

119899minus1 119902119899minus1

)1003817100381710038171003817

le1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899)))

1003817100381710038171003817

+1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899 119902119899) minus 119876 (119901

119899minus1 119902119899)1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899minus1

119902119899) minus 119876 (119901

119899minus1 119902119899minus1

)1003817100381710038171003817

(39)

Since 119867(119862119863) is 1199041-Lipschitz continuous with respect to

119862 and 1199042-Lipschitz continuous with respect to 119863 119876 is 120575

1-

Lipschitz continuous in the first argument and 1205752-Lipschitz

continuous in the second argument ℎ is 120582ℎ-Lipschitz contin-

uous and using (32) (34) and (39) it becomes

1003817100381710038171003817ℎ (119887119899+1) minus ℎ (119887119899)1003817100381710038171003817 le

(1199041+ 1199042) 120582ℎ

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

+12058221205751

12058321205722

2minus 12057421205732

2

(1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058221205752

12058321205722

2minus 12057421205732

2

(1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

(40)

In the light of (34) we have

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le

(1199041+ 1199042) 120582ℎ+ 12058221205752 (1 + 1119899) 120591

1205852(12058321205722

2minus 12057421205732

2)

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

+12058221205751

1205852(12058321205722

2minus 12057421205732

2)(1 +

1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

(41)

Let

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le 1205793119899

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817 + 1205794119899

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817 (42)

where

1205793119899=

12058221205751 (1 + 1119899) 120588

1205852(12058321205722

2minus 12057421205732

2)

1205794119899=(1199041+ 1199042) 120582ℎ+ 12058221205752 (1 + 1119899) 120591

1205852(12058321205722

2minus 12057421205732

2)

(43)

From adding (37) and (42) we get

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 +

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le (120579

1119899+ 1205793119899)1003817100381710038171003817119886119899 minus 119886119899minus1

1003817100381710038171003817

+ (1205792119899+ 1205794119899)1003817100381710038171003817119887119899 minus 119887119899minus1

1003817100381710038171003817

le Θ1198991003817100381710038171003817119886119899 minus 119886119899minus1

1003817100381710038171003817 +1003817100381710038171003817119887119899 minus 119887119899minus1

1003817100381710038171003817

(44)

where Θ119899= max(120579

1119899+ 1205793119899) (1205792119899+ 1205794119899)

Letting 119899 rarr infin we obtain Θ119899rarr 120579 where

120579 = max (1205791+ 1205793) (1205792+ 1205794)

1205791=

(1199031+ 1199032) 120582119892+ 12058211205981120588

1205851(12058311205722

1minus 12057411205732

1)

1205792=

12058211205982120591

1205851(12058311205722

1minus 12057411205732

1)

1205793=

12058221205751120588

1205852(12058321205722

2minus 12057421205732

2)

1205794=(1199041+ 1199042) 120582ℎ+ 12058221205752120591

1205852(12058321205722

2minus 12057421205732

2)

(45)

By (31) 120579 isin (0 1) and (44) 119886119899 and 119887

119899 both are Cauchy

sequencesThus there exists (119886 119887) isin 1198831times1198832such that 119886

119899rarr

119886 119887119899rarr 119887 as 119899 rarr infin From the Lipschitz continuity of 119878

and119879 and (32) 119901119899119902119899 are also Cauchy sequences and thus

there exists (119901 119902) isin 1198831times 1198832such that 119901

119899rarr 119901 119902

119899rarr 119902 as

119899 rarr infinNow we prove that 119901 isin 119878(119886) and 119902 isin 119879(119887) In fact since

119901119899isin 119878(119886119899) and 119902

119899isin 119879(119887119899) we have

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 + 119889 (119901119899 119878 (119886))

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 +D (119878 (119886119899) 119878 (119886))

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 + 1205881003817100381710038171003817119886119899 minus 119886

1003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin

(46)

which implies that 119889(119901 119878(119886)) = 0 Since 119878(119886) isin 119862119861(1198831)

it follows that 119901 isin 119878(119886) Similarly we have 119902 120598 119879(119887) ByLemma 11 it follows that 119886 119887 119901 119902 is a solution of problem(21) and this completes the proof

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Chinese Journal of Mathematics 7

References

[1] J P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquoH(sdotsdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011

[3] Y-P Fang and N-J Huang ldquoH-Monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003

[4] Y-P Fang and N-J Huang ldquoH-accretive operators resolventoperator technique solving variational inclusions in banachspacesrdquo Applied Mathematics Letters vol 17 no 6 pp 647ndash6532004

[5] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (H 120578)-monotone operators inhilbert spacesrdquo Computers and Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005

[6] S Husain and S Gupta ldquoAlgorithm for solving a new system ofgeneralized variational inclusions in Hilbert spacesrdquo Journal ofCalculus of Variations vol 2013 Article ID 461371 8 pages 2013

[7] S Husain S Gupta and V N Mishra ldquoGeneralized H(sdot sdotsdot)120578-cocoercive operators and generalized set-valued variational-like inclusionsrdquo Journal of Mathematics vol 2013 Article ID738491 10 pages 2013

[8] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a P-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007

[9] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involvingP-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008

[10] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalizedimplicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012

[11] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on M-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009

[12] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalizedH(sdot sdot)-accretive mapping in real q-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011

[13] H-Y Lan Y Je Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (A 120578)-accretivemappings in banach spacesrdquo Computers and Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006

[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving A-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009

[15] R U Verma ldquoGeneral system of A-monotone nonlinearvariational inclusion problems with applicationsrdquo Journal of

OptimizationTheory and Applications vol 131 no 1 pp 151ndash1572006

[16] F-Q Xia and N-J Huang ldquoVariational inclusions with ageneral H-monotone operator in Banach spacesrdquo Computersand Mathematics with Applications vol 54 no 1 pp 24ndash302007

[17] F -Q Xia Q -B Zhang and Y -Z Zou ldquoA new iterative algo-rithm for variational inclusions with H-monotone operatorsrdquoThai Journal of Mathematics vol 10 no 3 pp 605ndash616 2012

[18] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010

[19] Y-Z Zou and N-J Huang ldquoH(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008

[20] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquo Mathematical Programming vol 48 no 1ndash3 pp 249ndash263 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Chinese Journal of Mathematics

the family of all the nonempty (resp closed and bounded)subsets of 119883 and D(sdot sdot) is the Hausdorff metric on 119862119861(119883)

defined by

D (119875 119876) = maxsup119909isin119875

119889 (119909 119876) sup119910isin119875

119889 (119875 119910)

forall119875 119876 isin 119862119861 (119883)

(1)

where 119889(119909 119876) = inf119910isin119876

119909 minus 119910 and 119889(119875 119910) = inf119909isin119875

119909 minus 119910In the sequel let us recall some concepts

Definition 1 (see [20]) A mapping 119892 119883 rarr 119883 is said to be

(i) monotone if

⟨119892 (119909) minus 119892 (119910) 119909 minus 119910⟩ ge 0 forall119909 119910 isin 119883 (2)

(ii) 120585-strongly monotone if there exists a constant 120585 gt 0

such that

⟨119892 (119909) minus 119892 (119910) 119909 minus 119910⟩ ge 1205851003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2 forall119909 119910 isin 119883 (3)

(iii) 120583-cocoercive if there exists a constant 120583 gt 0 such that

⟨119892 (119909) minus 119892 (119910) 119909 minus 119910⟩ ge 1205831003817100381710038171003817119892 (119909) minus 119892 (119910)

1003817100381710038171003817

2 forall119909 119910 isin 119883

(4)

(iv) 120574-relaxed cocoercive if there exists a constant 120574 gt 0

such that

⟨119892 (119909) minus 119892 (119910) 119909 minus 119910⟩ ge (minus120574)1003817100381710038171003817119892 (119909) minus 119892 (119910)

1003817100381710038171003817

2 forall119909 119910 isin 119883

(5)

(v) 120582119892-Lipschitz continuous if there exists a constant

120582119892gt 0 such that

1003817100381710038171003817119892 (119909) minus 119892 (119910)1003817100381710038171003817 le 120582119892

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (6)

(vi) 120572-expansive if there exists a constant 120572 gt 0 such that1003817100381710038171003817119892 (119909) minus 119892 (119910)

1003817100381710038171003817 ge 1205721003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin 119883 (7)

if 120572 = 1 then it is expansive

Definition 2 Let119872 119883 rarr 2119883 be set-valued mappingThen

119872 is said to be 1205831015840-cocoercive if there exists a constant 1205831015840 gt 0

such that

⟨119906 minus V 120578 (119909 119910)⟩ ge 1205831015840119906 minus V2

forall119909 119910 isin 119883 119906 isin 119872 (119909) V isin 119872(119910)

(8)

Definition 3 (see [2]) Let119867 119883times119883 rarr 119883 and119860 119861 119883 rarr 119883

be the single-valued mappings Then

(i) 119867(119860 sdot) is said to be 120583-cocoercive with respect to 119860 ifthere exists a constant 120583 gt 0 such that

⟨119867 (119860119909 119906) minus 119867 (119860119910 119906) 119909 minus 119910⟩ ge 1205831003817100381710038171003817119860119909 minus 119860119910

1003817100381710038171003817

2

forall119909 119910 isin 119883

(9)

(ii) 119867(sdot 119861) is said to be 120574-relaxed cocoercive with respectto 119861 if there exists a constant 120574 gt 0 such that

⟨119867 (119906 119861119909) minus 119867 (119906 119861119910) 119909 minus 119910⟩ ge (minus120574)1003817100381710038171003817119861119909 minus 119861119910

1003817100381710038171003817

2

forall119909 119910 isin 119883

(10)

(iii) 119867(119860 sdot) is said to be 1199031-Lipschitz continuous with

respect to 119860 if there exists a constant 1199031gt 0 such that

1003817100381710038171003817119867 (119860119909 sdot) minus 119867 (119860119910 sdot)1003817100381710038171003817 le 1199031

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (11)

(iv) 119867(sdot 119861) is said to be 1199032-Lipschitz continuous with

respect to 119861 if there exists a constant 1199032gt 0 such that

1003817100381710038171003817119867 (sdot 119861119909) minus 119867 (sdot 119861119910)1003817100381710038171003817 le 1199032

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (12)

Definition 4 (see [2]) Let 119867 119883 times 119883 rarr 119883 and119860 119861

119883 rarr 119883 be the single-valued mappingsThen the set-valuedmapping 119872 119883 rarr 2

119883 is said to be 119867(sdot sdot)-cocoercive withrespect to119860 and119861 (or simply119867(sdot sdot)-cocoercive in the sequel)if

(i) 119872 is cocoercive(ii) (119867(119860 119861) + 120582119872)(119883) = 119883 for every 120582 gt 0

Example 5 Let 119883 = R2 with the usual inner product Let119860 119861 R2 rarr R2 be defined by

119860119909 = (21199091minus 21199092 minus21199091+ 41199092)

119861119910 = (minus1199101+ 1199102 minus1199102) forall119909 119910 isin R

2

(13)

Suppose that119867(119860 119861) R2 timesR2 rarr R2 is defined by

119867(119860119909 119861119910) = 119860119909 + 119861119910 forall119909 119910 isin R2 (14)

Then it is easy to check that 119867(119860 119861) is 16-cocoercive withrespect to 119860 and 12-relaxed cocoercive with respect to 119861

Let 119872 = 119868 where 119868 is the identity mapping Then 119872 is119867(sdot sdot)-cocoercive mapping with respect to 119860 and 119861

Example 6 Let 119883 = S2 where S2 denotes the space of all2times2 real symmetric matrices Let119867(119860119909 119861119910) = 119909

2minus119910 for all

119909 119910 isin S2 and119872 = 119868 Then for 120582 = 1 we have

(119867 (119860 119861) +119872) (119909) = 1199092minus 119909 + 119909 = 119909

2 (15)

but

[0 0

0 minus1] notin (119867 (119860 119861) +119872) (S

2) (16)

because [ 0 00 minus1

] is not the square of any 2 times 2 real symmetricmatrix Hence119872 is not 119867(sdot sdot)-cocoercive with respect to 119860and 119861

Proposition 7 (see [2]) Let 119867(119860 119861) be 120583-cocoercive withrespect to 119860 and 120574-relaxed cocoercive with respect to 119861 119860 is120572-expansive 119861 is 120573-Lipschitz continuous and 120583 gt 120574 120572 gt 120573Let 119872 119883 rarr 2

119883 be an 119867(sdot sdot)-cocoercive operator If theinequality

⟨119906 minus V 119909 minus 119910⟩ ge 0 (17)

Chinese Journal of Mathematics 3

holds for all (119910 V) isin 119866119903119886119901ℎ(119872) then 119906 isin 119872119909 where

119866119903119886119901ℎ (119872) = (119906 119909) isin 119883 times 119883 119909 isin 119872 (119906) (18)

Theorem 8 (see [2]) Let119867(119860 119861) be 120583-cocoercive with respectto 119860 and 120574-relaxed cocoercive with respect to 119861 119860 is 120572-expansive 119861 is 120573-Lipschitz continuous and 120583 gt 120574 and 120572 gt 120573Let M be an 119867(sdot sdot)-cocoercive operator with respect to 119860 and119861 Then the operator (119867(119860 119861) + 120582119872)

minus1 is single-valued

Definition 9 (see [2]) Let 119867(119860 119861) be 120583-cocoercive withrespect to 119860 and 120574-relaxed cocoercive with respect to 119861 119860 is120572-expansive 119861 is 120573-Lipschitz continuous and 120583 gt 120574 120572 gt 120573Let119872 be an119867(sdot sdot)-cocoercive operator with respect to119860 and119861 The resolvent operator 119877119867(sdotsdot)

120582119872 119883 rarr 119883 is defined by

119877119867(sdotsdot)

120582119872(119906) = (119867 (119860 119861) + 120582119872)

minus1(119906) forall119906 isin 119883 (19)

Theorem 10 (see [2]) Let 119867(119860 119861) be 120583-cocoercive withrespect to 119860 and 120574-relaxed cocoercive with respect to 119861 119860 is120572-expansive 119861 is 120573-Lipschitz continuous and 120583 gt 120574 120572 gt 120573

with 119903 = 1205831205722minus 1205741205732 Let 119872 be an 119867(sdot sdot)-cocoercive operator

with respect to 119860 and 119861 Then the resolvent operator 119877119867(sdotsdot)120582119872

119883 rarr 119883 is 1(1205831205722 minus 1205741205732)-Lipschitz continuous that is

10038171003817100381710038171003817119877119867(sdotsdot)

120582119872(119906) minus 119877

119867(sdotsdot)

120582119872(V)

10038171003817100381710038171003817le

1

1205831205722 minus 1205741205732119906 minus V forall119906 V isin 119883

(20)

3 The System of Generalized NonlinearQuasi-Variational-Like Inclusions andIterative Algorithm

Let1198831and119883

2be real Hilbert spaces Let 119875 119883

1times1198832rarr 1198831

119876 1198831times1198832rarr 1198832 119892 119860 119861 119883

1rarr 1198831 ℎ 119862119863 119883

2rarr 1198832

119866 1198831times 1198831rarr 1198831 and119867 119883

2times 1198832rarr 1198832be the single-

valued mappings Let 119878 1198831rarr 119862119861(119883

1) and 119879 119883

2rarr

119862119861(1198832) be the set-valued mappings let119872 119883

1times 1198831rarr 21198831

be a set-valued mapping such that for each 119886 isin 1198831 119872(sdot 119886)

is a 119866(sdot sdot)-cocoercive operator with respect to 119860 and 119861 andlet 119873 119883

2times 1198832rarr 21198832 be a set-valued mapping such that

for each 119887 isin 1198832119873(sdot 119887) is an119867(sdot sdot)-cocoercive operator with

respect to 119862 and 119863 Assume that 119892(1198831) cap dom(119872(sdot 119886)) = 0

for each 119886 isin 1198831 and ℎ(119883

2) cap dom(119873(sdot 119887)) = 0 for each 119887 isin

1198832 Then we consider the following system of generalized

nonlinear quasi-variational-like inclusionsFind (119886 119887) isin 119883

1times 1198832with 119901 isin 119878(119886) 119902 isin 119879(119887) such that

0 isin 119875 (119901 119902) +119872(119892 (119886) 119886)

0 isin 119876 (119901 119902) + 119873 (ℎ (119887) 119887)

(21)

Next are some special cases of problem (21)

(1) Let 119878 119879 be identity mappings for each (119886 119887) isin 1198831times

1198832 119872(119892(119886) 119886) = 119872(119909) and 119873(ℎ(119887) 119887) = 119873(119909)

then problem (21) reduces to the following problemconsidered in [15]

0 isin 119875 (119901 119902) +119872 (119886)

0 isin 119876 (119901 119902) + 119873 (119887)

(22)

(2) 1198831= 1198832 119886 = 119887 119878 = 119879 is an identity mapping 119892 = ℎ

119875(119901 119902) = 119876(119901 119902) = 119875(sdot) for each (119886 119887) isin 1198831times 1198832

119872(119892(119886) 119886) = 119873(ℎ(119887) 119887) = 119872(119892(119909)) then problem(21) reduces to the following problem considered in[16]

0 isin 119875 (119909) +119872(119892 (119886)) (23)

For a suitable choice of the mappings 119892 ℎ 119875 119876 119866 119867119860 119861 119862 119863 119878 119879 and the space 119883

1= 1198832 a number of

known systems of quasi-variational inequalities systems ofvariational inequalities systems of quasi-variational inclu-sions and variational inclusions can be obtained as specialcases of the generalized nonlinear quasi-variational inclusionproblem (21) We would like to mention that the problem offinding zero of the sum of two maximal monotone operatorsis also a special case of problem (21) Furthermore these typesof variational inclusion enable us to study many importantproblems arising in mathematical physical and engineeringscience in a general and unified framework

Lemma 11 Let 1198831and 119883

2be real Hilbert spaces Let 119875

1198831times 1198832rarr 119883

1 119876 119883

1times 1198832rarr 119883

2 119892 119860 119861 119883

1rarr

1198831ℎ 119862 119863 119883

2rarr 119883

2be the single-valued mappings

Let 119866 1198831times 1198831

rarr 1198831be a single-valued mapping such

that 119866(119860 119861) is 1205831-cocoercive with respect to 119860 and 120574

1-relaxed

cocoercive with respect to119861119860 is1205721-expansive119861 is120573

1-Lipschitz

continuous and 1205831gt 1205741 1205721gt 1205731 Let 119867 119883

2times 1198832rarr 119883

2

be a single-valued mapping such that119867(119862119863) is 1205832-cocoercive

with respect to 119862 and 1205742-relaxed cocoercive with respect to 119863

119862 is 1205722-expansive 119863 is 120573

2-Lipschitz continuous and 120583

2gt 1205742

1205722gt 1205732 Let 119878 119883

1rarr 119862119861(119883

1) and 119879 119883

2rarr 119862119861(119883

2) be

the set-valued mappings let 119872 1198831times 1198831rarr 21198831 be a set-

valued mapping such that for each 119886 isin 1198831119872(sdot 119886 is a 119866(sdot sdot)-

cocoercive operator and let 119873 1198832times 1198832rarr 21198832 be a set-

valuedmapping such that for each 119887 isin 1198832119873(sdot 119887) is an119867(sdot sdot)-

cocoercive operator Assume that 119892(1198831) capdom(119872(sdot 119886)) = 0 for

each 119886 isin 1198831 and ℎ(119883

2) cap dom(119873(sdot 119887)) = 0 for each 119887 isin 119883

2

Then for any (119886 119887) isin 1198831times 1198832with 119901 isin 119878(119886) 119902 isin 119879(119887) is a

solution of the problem (21) if and only if

119892 (119886) = 119877119866(sdotsdot)

1205821119872(sdot119886)

[119866 (119860 (119892 (119886)) 119861 (119892 (119886))) minus 1205821119875 (119901 119902)]

ℎ (119887) = 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887)) 119863 (ℎ (119887))) minus 1205822119876 (119901 119902)]

(24)

where 119877119866(sdotsdot)

1205821119872(sdot119886)

= (119866(sdot sdot) + 1205821119872(sdot 119886))

minus1 119877119867(sdotsdot)

1205822119873(sdot119887)

=

(119867(sdot sdot) + 1205822119873(sdot 119887))

minus1 and 1205821 1205822gt 0 are constants

Proof By using the definitions of the resolvent operators119877119866(sdotsdot)

1205821119872(sdot119886)

and 119877119867(sdotsdot)1205822119873(sdot119887)

the conclusion follows directly

4 Chinese Journal of Mathematics

The preceding lemma allows us to suggest the followingiterative algorithm for problem (21)

Algorithm 12 For (1198860 1198870) isin 1198831times1198832with119901 isin 119878(119886) 119902 isin 119879(119887)

compute the sequences 119886119899 119887119899 119901119899 119902119899 as follows

119892 (119886119899+1

) = 119877119866(sdotsdot)

1205821119872(sdot119886)

[119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 1205821119875 (119901119899 119902119899) ] 120582

1gt 0

ℎ (119887119899+1

) = 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 1205822119876 (119901119899 119902119899) ] 120582

2gt 0

119901119899isin 119878 (119886

119899)

1003817100381710038171003817119901119899 minus 119901119899+11003817100381710038171003817 le (1 +

1

119899 + 1)

timesD (119878 (119886119899) 119878 (119886

119899+1))

119902119899isin 119879 (119887

119899)

1003817100381710038171003817119902119899 minus 119902119899+11003817100381710038171003817 le (1 +

1

119899 + 1)

timesD (119879 (119887119899) 119879 (119887

119899+1))

(25)

for all 119899 = 0 1 2 and 1205821and 120582

2are constants

Definition 13 Let 119878 1198831rarr 119862119861(119883

1) and 119879 119883

2rarr 119862119861(119883

2)

be two set-valued mappings A single-valued mapping 119875

1198831times 1198832rarr 1198831is said to be

(i) 1205981-Lipschitz continuous in the first argument with

respect to 119878 if there exists a constant 1205981gt 0 such that

1003817100381710038171003817119875 (1199011 sdot) minus 119875 (1199012 sdot)1003817100381710038171003817 le 1205981

10038171003817100381710038171199011 minus 11990121003817100381710038171003817

forall1199011isin 119878 (119886

1) 1199012isin 119878 (119886

2) 1198861 1198862isin 1198831

(26)

(ii) 1205982-Lipschitz continuous in the second argument with

respect to 119879 if there exists a constant 1205982gt 0 such that

1003817100381710038171003817119875 (sdot 1199021) minus 119875 (sdot 1199022)1003817100381710038171003817 le 1205982

10038171003817100381710038171199021 minus 11990221003817100381710038171003817

forall1199021isin 119879 (119887

1) 1199022isin 119879 (119887

2) 1198871 1198872isin 1198832

(27)

Definition 14 Let 119878 1198831rarr 119862119861(119883

1) and 119879 119883

2rarr 119862119861(119883

2)

be two set-valued mappings A single-valued mapping 119876

1198831times 1198832rarr 1198832is said to be

(i) 1205751-Lipschitz continuous in the first argument with

respect to 119878 if there exists a constant 1205751gt 0 such that

1003817100381710038171003817119876 (1199011 sdot) minus 119876 (119901

2 sdot)1003817100381710038171003817 le 1205751

10038171003817100381710038171199011 minus 11990121003817100381710038171003817

forall1199011isin 119878 (119886

1) 1199012isin 119878 (119886

2) 1198861 1198862isin 1198831

(28)

(ii) 1205752-Lipschitz continuous in the second argument with

respect to 119879 if there exists a constant 1205752gt 0 such that

1003817100381710038171003817119876 (sdot 1199021) minus 119876 (sdot 119902

2)1003817100381710038171003817 le 1205752

10038171003817100381710038171199021 minus 11990221003817100381710038171003817

forall1199021isin 119879 (119887

1) 1199022isin 119879 (119887

2) 1198871 1198872isin 1198832

(29)

Definition 15 A set-valuedmapping 119860 119883 rarr 119862119861(119883) is saidto beD-Lipschitz continuous if there exists a constant 120588 gt 0

such that

D (119860 (119909) 119860 (119910)) le 1205881003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin 119883 (30)

Theorem 16 Let 1198831and 119883

2be real Hilbert spaces Let 119875

1198831times 1198832rarr 1198831 119876 119883

1times 1198832rarr 1198832 119892 119860 119861 119883

1rarr 1198831

ℎ 119862 119863 1198832

rarr 1198832 119866 119883

1times 1198831

rarr 1198831 and 119867

1198832times 1198832rarr 1198832be the single-valuedmappings Let 119878 119883

1rarr

119862119861(1198831) and 119879 119883

2rarr 119862119861(119883

2) be the set-valued mappings

let 119872 1198831times 1198831

rarr 21198831 be a set-valued mapping such

that for each 119886 isin 1198831 119872(sdot 119886) is a 119866(sdot sdot)-cocoercive operator

with respect to 119860 and 119861 and let 119873 1198832times 1198832rarr 21198832 be a

set-valued mapping such that for each 119887 isin 1198832 119873(sdot 119887) is an

119867(sdot sdot)-cocoercive operator with respect to 119862 and 119863 Assumethat 119892(119883

1) cap dom(119872(sdot 119886)) = 0 for each 119886 isin 119883

1 and ℎ(119883

2) cap

dom(119873(sdot 119887)) = 0 for each 119887 isin 1198832 Assume that

(i) 119878 119879 are 120588 120591-Lipschitz continuous in the HausdorffmetricD(sdot sdot) respectively

(ii) 119866(119860 119861) is 1205831-cocoercive with respect to 119860 and 120574

1-

relaxed cocoercive with respect to 119861

(iii) 119867(119862119863) is 1205832-cocoercive with respect to 119862 and 120574

2-

relaxed cocoercive with respect to119863

(iv) 119860 119862 is 1205721 1205722-expansive respectively

(v) 119861119863 119894119904 1205731 1205732-Lipschitz continuous respectively

(vi) 119892 is120582119892-Lipschitz continuous and 120585

1-stronglymonotone

(vii) ℎ is 120582ℎ-Lipschitz continuous and 120585

2-stronglymonotone

(viii) 119866(119860 119861) is 1199031-Lipschitz continuouswith respect to119860 and

1199032-Lipschitz continuous with respect to 119861

(ix) H(119862119863) is 1199041-Lipschitz continuous with respect to 119862

and 1199042-Lipschitz continuous with respect to119863

(x) 119875(sdot sdot) is 1205981-Lipschitz continuous in the first argument

and 1205982-Lipschitz continuous in the second argument

(xi) 119876(sdot sdot) is 1205751-Lipschitz continuous in the first argument

and 1205752-Lipschitz continuous in the second argument

(xii)

0 lt

(1199031+ 1199032) 120582119892+ 12058211205981120588

1205851(12058311205722

1minus 12057411205732

1)

+12058221205751 (1 + 1119899) 120588

1205852(12058321205722

2minus 12057421205732

2)lt 1

0 lt(1199041+ 1199042) 120582ℎ+ 12058221205752120591

1205852(12058321205722

2minus 12057421205732

2)

+12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)lt 1

(31)

Then the iterative sequences 119886119899 119887119899 119901119899 and 119902

119899

generated by Algorithm 12 converge strongly to 119886 119887 119901and 119902respectively and (119886 119887 119901 119902) is a solution of problem (21)

Chinese Journal of Mathematics 5

Proof Since 119878 119879 are Lipschitz continuous with constants 120588120591 respectively it follows from Algorithm 12 that

1003817100381710038171003817119901119899 minus 119901119899+11003817100381710038171003817 le (1 +

1

119899)D (119878 (119886

119899) 119878 (119886

119899+1))

le (1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899+11003817100381710038171003817

1003817100381710038171003817119902119899 minus 119902119899+11003817100381710038171003817 le (1 +

1

119899)D (119879 (119887

119899) 119879 (119887

119899+1))

le (1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899+11003817100381710038171003817

(32)

for all 119899 = 0 1 2

Using the 120585-strong monotonicity of 119892 we have1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)

1003817100381710038171003817

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817

ge ⟨119892 (119886119899+1

) minus 119892 (119886119899) 119886119899+1

minus 119886119899⟩

ge 1205851

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817

2

(33)

which implies that

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le

1

1205851

1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)1003817100381710038171003817 (34)

Nowwe estimate 119892(119886119899+1

)minus119892(119886119899) by usingAlgorithm 12 and

the Lipschitz continuity of 119877119866(sdotsdot)1205821119872(sdot119886)

as follows1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)

1003817100381710038171003817

=10038171003817100381710038171003817119877119866(sdotsdot)

1205821119872(sdot119886)

[119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899))) minus 120582

1119875 (119901119899 119902119899)]

minus [119877119866(sdotsdot)

1205821119872(sdot119886)

119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))

minus 1205821119875 (119901119899minus1

119902119899minus1

) ]10038171003817100381710038171003817

le1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899 119902119899) minus 119875 (119901119899minus1 119902119899minus1)1003817100381710038171003817

le1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899)))

1003817100381710038171003817

+1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899 119902119899) minus 119875 (119901119899minus1 119902119899)1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899minus1 119902119899) minus 119875 (119901119899minus1 119902119899minus1)1003817100381710038171003817

(35)

Since119866(119860 119861) is 1199031-Lipschitz continuouswith respect to119860 and

1199032-Lipschitz continuous with respect to 119861 119875 is 120598

1-Lipschitz

continuous in the first argument and 1205982-Lipschitz continuous

in the second argument 119892 is 120582119892-Lipschitz continuous and

using (32) (34) and (35) it becomes

1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)1003817100381710038171003817 le

(1199031+ 1199032) 120582119892

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205981

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119901119899 minus 119901119899minus11003817100381710038171003817

+12058211205982

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119902119899 minus 119902119899minus11003817100381710038171003817

le

(1199031+ 1199032) 120582119892

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205981

12058311205722

1minus 12057411205732

1

(1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205982

12058311205722

1minus 12057411205732

1

(1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le

(1199031+ 1199032) 120582119892+ 12058211205981 (1 + 1119899) 120588

1205851(12058311205722

1minus 12057411205732

1)

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

(36)

Let

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le 1205791119899

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817 + 1205792119899

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817 (37)

where

1205791119899=

(1199031+ 1199032) 120582119892+ 12058211205981 (1 + 1119899) 120588

1205851(12058311205722

1minus 12057411205732

1)

1205792119899=

12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)

(38)

Now we estimate ℎ(119887119899) minus ℎ(119887

119899minus1) by using Algorithm 12 and

the Lipschitz continuity of 119877119867(sdotsdot)1205822119873(sdot119887)

as follows

1003817100381710038171003817ℎ (119887119899+1) minus ℎ (119887119899)1003817100381710038171003817

=10038171003817100381710038171003817119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899))) minus 120582

2119876 (119901119899 119902119899)]

6 Chinese Journal of Mathematics

minus 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))

minus 1205822119876 (119901119899minus1

119902119899minus1

) ]10038171003817100381710038171003817

le1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899 119902119899) minus 119876 (119901

119899minus1 119902119899minus1

)1003817100381710038171003817

le1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899)))

1003817100381710038171003817

+1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899 119902119899) minus 119876 (119901

119899minus1 119902119899)1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899minus1

119902119899) minus 119876 (119901

119899minus1 119902119899minus1

)1003817100381710038171003817

(39)

Since 119867(119862119863) is 1199041-Lipschitz continuous with respect to

119862 and 1199042-Lipschitz continuous with respect to 119863 119876 is 120575

1-

Lipschitz continuous in the first argument and 1205752-Lipschitz

continuous in the second argument ℎ is 120582ℎ-Lipschitz contin-

uous and using (32) (34) and (39) it becomes

1003817100381710038171003817ℎ (119887119899+1) minus ℎ (119887119899)1003817100381710038171003817 le

(1199041+ 1199042) 120582ℎ

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

+12058221205751

12058321205722

2minus 12057421205732

2

(1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058221205752

12058321205722

2minus 12057421205732

2

(1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

(40)

In the light of (34) we have

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le

(1199041+ 1199042) 120582ℎ+ 12058221205752 (1 + 1119899) 120591

1205852(12058321205722

2minus 12057421205732

2)

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

+12058221205751

1205852(12058321205722

2minus 12057421205732

2)(1 +

1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

(41)

Let

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le 1205793119899

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817 + 1205794119899

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817 (42)

where

1205793119899=

12058221205751 (1 + 1119899) 120588

1205852(12058321205722

2minus 12057421205732

2)

1205794119899=(1199041+ 1199042) 120582ℎ+ 12058221205752 (1 + 1119899) 120591

1205852(12058321205722

2minus 12057421205732

2)

(43)

From adding (37) and (42) we get

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 +

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le (120579

1119899+ 1205793119899)1003817100381710038171003817119886119899 minus 119886119899minus1

1003817100381710038171003817

+ (1205792119899+ 1205794119899)1003817100381710038171003817119887119899 minus 119887119899minus1

1003817100381710038171003817

le Θ1198991003817100381710038171003817119886119899 minus 119886119899minus1

1003817100381710038171003817 +1003817100381710038171003817119887119899 minus 119887119899minus1

1003817100381710038171003817

(44)

where Θ119899= max(120579

1119899+ 1205793119899) (1205792119899+ 1205794119899)

Letting 119899 rarr infin we obtain Θ119899rarr 120579 where

120579 = max (1205791+ 1205793) (1205792+ 1205794)

1205791=

(1199031+ 1199032) 120582119892+ 12058211205981120588

1205851(12058311205722

1minus 12057411205732

1)

1205792=

12058211205982120591

1205851(12058311205722

1minus 12057411205732

1)

1205793=

12058221205751120588

1205852(12058321205722

2minus 12057421205732

2)

1205794=(1199041+ 1199042) 120582ℎ+ 12058221205752120591

1205852(12058321205722

2minus 12057421205732

2)

(45)

By (31) 120579 isin (0 1) and (44) 119886119899 and 119887

119899 both are Cauchy

sequencesThus there exists (119886 119887) isin 1198831times1198832such that 119886

119899rarr

119886 119887119899rarr 119887 as 119899 rarr infin From the Lipschitz continuity of 119878

and119879 and (32) 119901119899119902119899 are also Cauchy sequences and thus

there exists (119901 119902) isin 1198831times 1198832such that 119901

119899rarr 119901 119902

119899rarr 119902 as

119899 rarr infinNow we prove that 119901 isin 119878(119886) and 119902 isin 119879(119887) In fact since

119901119899isin 119878(119886119899) and 119902

119899isin 119879(119887119899) we have

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 + 119889 (119901119899 119878 (119886))

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 +D (119878 (119886119899) 119878 (119886))

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 + 1205881003817100381710038171003817119886119899 minus 119886

1003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin

(46)

which implies that 119889(119901 119878(119886)) = 0 Since 119878(119886) isin 119862119861(1198831)

it follows that 119901 isin 119878(119886) Similarly we have 119902 120598 119879(119887) ByLemma 11 it follows that 119886 119887 119901 119902 is a solution of problem(21) and this completes the proof

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Chinese Journal of Mathematics 7

References

[1] J P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquoH(sdotsdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011

[3] Y-P Fang and N-J Huang ldquoH-Monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003

[4] Y-P Fang and N-J Huang ldquoH-accretive operators resolventoperator technique solving variational inclusions in banachspacesrdquo Applied Mathematics Letters vol 17 no 6 pp 647ndash6532004

[5] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (H 120578)-monotone operators inhilbert spacesrdquo Computers and Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005

[6] S Husain and S Gupta ldquoAlgorithm for solving a new system ofgeneralized variational inclusions in Hilbert spacesrdquo Journal ofCalculus of Variations vol 2013 Article ID 461371 8 pages 2013

[7] S Husain S Gupta and V N Mishra ldquoGeneralized H(sdot sdotsdot)120578-cocoercive operators and generalized set-valued variational-like inclusionsrdquo Journal of Mathematics vol 2013 Article ID738491 10 pages 2013

[8] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a P-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007

[9] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involvingP-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008

[10] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalizedimplicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012

[11] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on M-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009

[12] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalizedH(sdot sdot)-accretive mapping in real q-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011

[13] H-Y Lan Y Je Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (A 120578)-accretivemappings in banach spacesrdquo Computers and Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006

[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving A-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009

[15] R U Verma ldquoGeneral system of A-monotone nonlinearvariational inclusion problems with applicationsrdquo Journal of

OptimizationTheory and Applications vol 131 no 1 pp 151ndash1572006

[16] F-Q Xia and N-J Huang ldquoVariational inclusions with ageneral H-monotone operator in Banach spacesrdquo Computersand Mathematics with Applications vol 54 no 1 pp 24ndash302007

[17] F -Q Xia Q -B Zhang and Y -Z Zou ldquoA new iterative algo-rithm for variational inclusions with H-monotone operatorsrdquoThai Journal of Mathematics vol 10 no 3 pp 605ndash616 2012

[18] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010

[19] Y-Z Zou and N-J Huang ldquoH(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008

[20] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquo Mathematical Programming vol 48 no 1ndash3 pp 249ndash263 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Chinese Journal of Mathematics 3

holds for all (119910 V) isin 119866119903119886119901ℎ(119872) then 119906 isin 119872119909 where

119866119903119886119901ℎ (119872) = (119906 119909) isin 119883 times 119883 119909 isin 119872 (119906) (18)

Theorem 8 (see [2]) Let119867(119860 119861) be 120583-cocoercive with respectto 119860 and 120574-relaxed cocoercive with respect to 119861 119860 is 120572-expansive 119861 is 120573-Lipschitz continuous and 120583 gt 120574 and 120572 gt 120573Let M be an 119867(sdot sdot)-cocoercive operator with respect to 119860 and119861 Then the operator (119867(119860 119861) + 120582119872)

minus1 is single-valued

Definition 9 (see [2]) Let 119867(119860 119861) be 120583-cocoercive withrespect to 119860 and 120574-relaxed cocoercive with respect to 119861 119860 is120572-expansive 119861 is 120573-Lipschitz continuous and 120583 gt 120574 120572 gt 120573Let119872 be an119867(sdot sdot)-cocoercive operator with respect to119860 and119861 The resolvent operator 119877119867(sdotsdot)

120582119872 119883 rarr 119883 is defined by

119877119867(sdotsdot)

120582119872(119906) = (119867 (119860 119861) + 120582119872)

minus1(119906) forall119906 isin 119883 (19)

Theorem 10 (see [2]) Let 119867(119860 119861) be 120583-cocoercive withrespect to 119860 and 120574-relaxed cocoercive with respect to 119861 119860 is120572-expansive 119861 is 120573-Lipschitz continuous and 120583 gt 120574 120572 gt 120573

with 119903 = 1205831205722minus 1205741205732 Let 119872 be an 119867(sdot sdot)-cocoercive operator

with respect to 119860 and 119861 Then the resolvent operator 119877119867(sdotsdot)120582119872

119883 rarr 119883 is 1(1205831205722 minus 1205741205732)-Lipschitz continuous that is

10038171003817100381710038171003817119877119867(sdotsdot)

120582119872(119906) minus 119877

119867(sdotsdot)

120582119872(V)

10038171003817100381710038171003817le

1

1205831205722 minus 1205741205732119906 minus V forall119906 V isin 119883

(20)

3 The System of Generalized NonlinearQuasi-Variational-Like Inclusions andIterative Algorithm

Let1198831and119883

2be real Hilbert spaces Let 119875 119883

1times1198832rarr 1198831

119876 1198831times1198832rarr 1198832 119892 119860 119861 119883

1rarr 1198831 ℎ 119862119863 119883

2rarr 1198832

119866 1198831times 1198831rarr 1198831 and119867 119883

2times 1198832rarr 1198832be the single-

valued mappings Let 119878 1198831rarr 119862119861(119883

1) and 119879 119883

2rarr

119862119861(1198832) be the set-valued mappings let119872 119883

1times 1198831rarr 21198831

be a set-valued mapping such that for each 119886 isin 1198831 119872(sdot 119886)

is a 119866(sdot sdot)-cocoercive operator with respect to 119860 and 119861 andlet 119873 119883

2times 1198832rarr 21198832 be a set-valued mapping such that

for each 119887 isin 1198832119873(sdot 119887) is an119867(sdot sdot)-cocoercive operator with

respect to 119862 and 119863 Assume that 119892(1198831) cap dom(119872(sdot 119886)) = 0

for each 119886 isin 1198831 and ℎ(119883

2) cap dom(119873(sdot 119887)) = 0 for each 119887 isin

1198832 Then we consider the following system of generalized

nonlinear quasi-variational-like inclusionsFind (119886 119887) isin 119883

1times 1198832with 119901 isin 119878(119886) 119902 isin 119879(119887) such that

0 isin 119875 (119901 119902) +119872(119892 (119886) 119886)

0 isin 119876 (119901 119902) + 119873 (ℎ (119887) 119887)

(21)

Next are some special cases of problem (21)

(1) Let 119878 119879 be identity mappings for each (119886 119887) isin 1198831times

1198832 119872(119892(119886) 119886) = 119872(119909) and 119873(ℎ(119887) 119887) = 119873(119909)

then problem (21) reduces to the following problemconsidered in [15]

0 isin 119875 (119901 119902) +119872 (119886)

0 isin 119876 (119901 119902) + 119873 (119887)

(22)

(2) 1198831= 1198832 119886 = 119887 119878 = 119879 is an identity mapping 119892 = ℎ

119875(119901 119902) = 119876(119901 119902) = 119875(sdot) for each (119886 119887) isin 1198831times 1198832

119872(119892(119886) 119886) = 119873(ℎ(119887) 119887) = 119872(119892(119909)) then problem(21) reduces to the following problem considered in[16]

0 isin 119875 (119909) +119872(119892 (119886)) (23)

For a suitable choice of the mappings 119892 ℎ 119875 119876 119866 119867119860 119861 119862 119863 119878 119879 and the space 119883

1= 1198832 a number of

known systems of quasi-variational inequalities systems ofvariational inequalities systems of quasi-variational inclu-sions and variational inclusions can be obtained as specialcases of the generalized nonlinear quasi-variational inclusionproblem (21) We would like to mention that the problem offinding zero of the sum of two maximal monotone operatorsis also a special case of problem (21) Furthermore these typesof variational inclusion enable us to study many importantproblems arising in mathematical physical and engineeringscience in a general and unified framework

Lemma 11 Let 1198831and 119883

2be real Hilbert spaces Let 119875

1198831times 1198832rarr 119883

1 119876 119883

1times 1198832rarr 119883

2 119892 119860 119861 119883

1rarr

1198831ℎ 119862 119863 119883

2rarr 119883

2be the single-valued mappings

Let 119866 1198831times 1198831

rarr 1198831be a single-valued mapping such

that 119866(119860 119861) is 1205831-cocoercive with respect to 119860 and 120574

1-relaxed

cocoercive with respect to119861119860 is1205721-expansive119861 is120573

1-Lipschitz

continuous and 1205831gt 1205741 1205721gt 1205731 Let 119867 119883

2times 1198832rarr 119883

2

be a single-valued mapping such that119867(119862119863) is 1205832-cocoercive

with respect to 119862 and 1205742-relaxed cocoercive with respect to 119863

119862 is 1205722-expansive 119863 is 120573

2-Lipschitz continuous and 120583

2gt 1205742

1205722gt 1205732 Let 119878 119883

1rarr 119862119861(119883

1) and 119879 119883

2rarr 119862119861(119883

2) be

the set-valued mappings let 119872 1198831times 1198831rarr 21198831 be a set-

valued mapping such that for each 119886 isin 1198831119872(sdot 119886 is a 119866(sdot sdot)-

cocoercive operator and let 119873 1198832times 1198832rarr 21198832 be a set-

valuedmapping such that for each 119887 isin 1198832119873(sdot 119887) is an119867(sdot sdot)-

cocoercive operator Assume that 119892(1198831) capdom(119872(sdot 119886)) = 0 for

each 119886 isin 1198831 and ℎ(119883

2) cap dom(119873(sdot 119887)) = 0 for each 119887 isin 119883

2

Then for any (119886 119887) isin 1198831times 1198832with 119901 isin 119878(119886) 119902 isin 119879(119887) is a

solution of the problem (21) if and only if

119892 (119886) = 119877119866(sdotsdot)

1205821119872(sdot119886)

[119866 (119860 (119892 (119886)) 119861 (119892 (119886))) minus 1205821119875 (119901 119902)]

ℎ (119887) = 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887)) 119863 (ℎ (119887))) minus 1205822119876 (119901 119902)]

(24)

where 119877119866(sdotsdot)

1205821119872(sdot119886)

= (119866(sdot sdot) + 1205821119872(sdot 119886))

minus1 119877119867(sdotsdot)

1205822119873(sdot119887)

=

(119867(sdot sdot) + 1205822119873(sdot 119887))

minus1 and 1205821 1205822gt 0 are constants

Proof By using the definitions of the resolvent operators119877119866(sdotsdot)

1205821119872(sdot119886)

and 119877119867(sdotsdot)1205822119873(sdot119887)

the conclusion follows directly

4 Chinese Journal of Mathematics

The preceding lemma allows us to suggest the followingiterative algorithm for problem (21)

Algorithm 12 For (1198860 1198870) isin 1198831times1198832with119901 isin 119878(119886) 119902 isin 119879(119887)

compute the sequences 119886119899 119887119899 119901119899 119902119899 as follows

119892 (119886119899+1

) = 119877119866(sdotsdot)

1205821119872(sdot119886)

[119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 1205821119875 (119901119899 119902119899) ] 120582

1gt 0

ℎ (119887119899+1

) = 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 1205822119876 (119901119899 119902119899) ] 120582

2gt 0

119901119899isin 119878 (119886

119899)

1003817100381710038171003817119901119899 minus 119901119899+11003817100381710038171003817 le (1 +

1

119899 + 1)

timesD (119878 (119886119899) 119878 (119886

119899+1))

119902119899isin 119879 (119887

119899)

1003817100381710038171003817119902119899 minus 119902119899+11003817100381710038171003817 le (1 +

1

119899 + 1)

timesD (119879 (119887119899) 119879 (119887

119899+1))

(25)

for all 119899 = 0 1 2 and 1205821and 120582

2are constants

Definition 13 Let 119878 1198831rarr 119862119861(119883

1) and 119879 119883

2rarr 119862119861(119883

2)

be two set-valued mappings A single-valued mapping 119875

1198831times 1198832rarr 1198831is said to be

(i) 1205981-Lipschitz continuous in the first argument with

respect to 119878 if there exists a constant 1205981gt 0 such that

1003817100381710038171003817119875 (1199011 sdot) minus 119875 (1199012 sdot)1003817100381710038171003817 le 1205981

10038171003817100381710038171199011 minus 11990121003817100381710038171003817

forall1199011isin 119878 (119886

1) 1199012isin 119878 (119886

2) 1198861 1198862isin 1198831

(26)

(ii) 1205982-Lipschitz continuous in the second argument with

respect to 119879 if there exists a constant 1205982gt 0 such that

1003817100381710038171003817119875 (sdot 1199021) minus 119875 (sdot 1199022)1003817100381710038171003817 le 1205982

10038171003817100381710038171199021 minus 11990221003817100381710038171003817

forall1199021isin 119879 (119887

1) 1199022isin 119879 (119887

2) 1198871 1198872isin 1198832

(27)

Definition 14 Let 119878 1198831rarr 119862119861(119883

1) and 119879 119883

2rarr 119862119861(119883

2)

be two set-valued mappings A single-valued mapping 119876

1198831times 1198832rarr 1198832is said to be

(i) 1205751-Lipschitz continuous in the first argument with

respect to 119878 if there exists a constant 1205751gt 0 such that

1003817100381710038171003817119876 (1199011 sdot) minus 119876 (119901

2 sdot)1003817100381710038171003817 le 1205751

10038171003817100381710038171199011 minus 11990121003817100381710038171003817

forall1199011isin 119878 (119886

1) 1199012isin 119878 (119886

2) 1198861 1198862isin 1198831

(28)

(ii) 1205752-Lipschitz continuous in the second argument with

respect to 119879 if there exists a constant 1205752gt 0 such that

1003817100381710038171003817119876 (sdot 1199021) minus 119876 (sdot 119902

2)1003817100381710038171003817 le 1205752

10038171003817100381710038171199021 minus 11990221003817100381710038171003817

forall1199021isin 119879 (119887

1) 1199022isin 119879 (119887

2) 1198871 1198872isin 1198832

(29)

Definition 15 A set-valuedmapping 119860 119883 rarr 119862119861(119883) is saidto beD-Lipschitz continuous if there exists a constant 120588 gt 0

such that

D (119860 (119909) 119860 (119910)) le 1205881003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin 119883 (30)

Theorem 16 Let 1198831and 119883

2be real Hilbert spaces Let 119875

1198831times 1198832rarr 1198831 119876 119883

1times 1198832rarr 1198832 119892 119860 119861 119883

1rarr 1198831

ℎ 119862 119863 1198832

rarr 1198832 119866 119883

1times 1198831

rarr 1198831 and 119867

1198832times 1198832rarr 1198832be the single-valuedmappings Let 119878 119883

1rarr

119862119861(1198831) and 119879 119883

2rarr 119862119861(119883

2) be the set-valued mappings

let 119872 1198831times 1198831

rarr 21198831 be a set-valued mapping such

that for each 119886 isin 1198831 119872(sdot 119886) is a 119866(sdot sdot)-cocoercive operator

with respect to 119860 and 119861 and let 119873 1198832times 1198832rarr 21198832 be a

set-valued mapping such that for each 119887 isin 1198832 119873(sdot 119887) is an

119867(sdot sdot)-cocoercive operator with respect to 119862 and 119863 Assumethat 119892(119883

1) cap dom(119872(sdot 119886)) = 0 for each 119886 isin 119883

1 and ℎ(119883

2) cap

dom(119873(sdot 119887)) = 0 for each 119887 isin 1198832 Assume that

(i) 119878 119879 are 120588 120591-Lipschitz continuous in the HausdorffmetricD(sdot sdot) respectively

(ii) 119866(119860 119861) is 1205831-cocoercive with respect to 119860 and 120574

1-

relaxed cocoercive with respect to 119861

(iii) 119867(119862119863) is 1205832-cocoercive with respect to 119862 and 120574

2-

relaxed cocoercive with respect to119863

(iv) 119860 119862 is 1205721 1205722-expansive respectively

(v) 119861119863 119894119904 1205731 1205732-Lipschitz continuous respectively

(vi) 119892 is120582119892-Lipschitz continuous and 120585

1-stronglymonotone

(vii) ℎ is 120582ℎ-Lipschitz continuous and 120585

2-stronglymonotone

(viii) 119866(119860 119861) is 1199031-Lipschitz continuouswith respect to119860 and

1199032-Lipschitz continuous with respect to 119861

(ix) H(119862119863) is 1199041-Lipschitz continuous with respect to 119862

and 1199042-Lipschitz continuous with respect to119863

(x) 119875(sdot sdot) is 1205981-Lipschitz continuous in the first argument

and 1205982-Lipschitz continuous in the second argument

(xi) 119876(sdot sdot) is 1205751-Lipschitz continuous in the first argument

and 1205752-Lipschitz continuous in the second argument

(xii)

0 lt

(1199031+ 1199032) 120582119892+ 12058211205981120588

1205851(12058311205722

1minus 12057411205732

1)

+12058221205751 (1 + 1119899) 120588

1205852(12058321205722

2minus 12057421205732

2)lt 1

0 lt(1199041+ 1199042) 120582ℎ+ 12058221205752120591

1205852(12058321205722

2minus 12057421205732

2)

+12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)lt 1

(31)

Then the iterative sequences 119886119899 119887119899 119901119899 and 119902

119899

generated by Algorithm 12 converge strongly to 119886 119887 119901and 119902respectively and (119886 119887 119901 119902) is a solution of problem (21)

Chinese Journal of Mathematics 5

Proof Since 119878 119879 are Lipschitz continuous with constants 120588120591 respectively it follows from Algorithm 12 that

1003817100381710038171003817119901119899 minus 119901119899+11003817100381710038171003817 le (1 +

1

119899)D (119878 (119886

119899) 119878 (119886

119899+1))

le (1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899+11003817100381710038171003817

1003817100381710038171003817119902119899 minus 119902119899+11003817100381710038171003817 le (1 +

1

119899)D (119879 (119887

119899) 119879 (119887

119899+1))

le (1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899+11003817100381710038171003817

(32)

for all 119899 = 0 1 2

Using the 120585-strong monotonicity of 119892 we have1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)

1003817100381710038171003817

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817

ge ⟨119892 (119886119899+1

) minus 119892 (119886119899) 119886119899+1

minus 119886119899⟩

ge 1205851

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817

2

(33)

which implies that

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le

1

1205851

1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)1003817100381710038171003817 (34)

Nowwe estimate 119892(119886119899+1

)minus119892(119886119899) by usingAlgorithm 12 and

the Lipschitz continuity of 119877119866(sdotsdot)1205821119872(sdot119886)

as follows1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)

1003817100381710038171003817

=10038171003817100381710038171003817119877119866(sdotsdot)

1205821119872(sdot119886)

[119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899))) minus 120582

1119875 (119901119899 119902119899)]

minus [119877119866(sdotsdot)

1205821119872(sdot119886)

119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))

minus 1205821119875 (119901119899minus1

119902119899minus1

) ]10038171003817100381710038171003817

le1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899 119902119899) minus 119875 (119901119899minus1 119902119899minus1)1003817100381710038171003817

le1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899)))

1003817100381710038171003817

+1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899 119902119899) minus 119875 (119901119899minus1 119902119899)1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899minus1 119902119899) minus 119875 (119901119899minus1 119902119899minus1)1003817100381710038171003817

(35)

Since119866(119860 119861) is 1199031-Lipschitz continuouswith respect to119860 and

1199032-Lipschitz continuous with respect to 119861 119875 is 120598

1-Lipschitz

continuous in the first argument and 1205982-Lipschitz continuous

in the second argument 119892 is 120582119892-Lipschitz continuous and

using (32) (34) and (35) it becomes

1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)1003817100381710038171003817 le

(1199031+ 1199032) 120582119892

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205981

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119901119899 minus 119901119899minus11003817100381710038171003817

+12058211205982

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119902119899 minus 119902119899minus11003817100381710038171003817

le

(1199031+ 1199032) 120582119892

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205981

12058311205722

1minus 12057411205732

1

(1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205982

12058311205722

1minus 12057411205732

1

(1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le

(1199031+ 1199032) 120582119892+ 12058211205981 (1 + 1119899) 120588

1205851(12058311205722

1minus 12057411205732

1)

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

(36)

Let

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le 1205791119899

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817 + 1205792119899

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817 (37)

where

1205791119899=

(1199031+ 1199032) 120582119892+ 12058211205981 (1 + 1119899) 120588

1205851(12058311205722

1minus 12057411205732

1)

1205792119899=

12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)

(38)

Now we estimate ℎ(119887119899) minus ℎ(119887

119899minus1) by using Algorithm 12 and

the Lipschitz continuity of 119877119867(sdotsdot)1205822119873(sdot119887)

as follows

1003817100381710038171003817ℎ (119887119899+1) minus ℎ (119887119899)1003817100381710038171003817

=10038171003817100381710038171003817119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899))) minus 120582

2119876 (119901119899 119902119899)]

6 Chinese Journal of Mathematics

minus 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))

minus 1205822119876 (119901119899minus1

119902119899minus1

) ]10038171003817100381710038171003817

le1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899 119902119899) minus 119876 (119901

119899minus1 119902119899minus1

)1003817100381710038171003817

le1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899)))

1003817100381710038171003817

+1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899 119902119899) minus 119876 (119901

119899minus1 119902119899)1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899minus1

119902119899) minus 119876 (119901

119899minus1 119902119899minus1

)1003817100381710038171003817

(39)

Since 119867(119862119863) is 1199041-Lipschitz continuous with respect to

119862 and 1199042-Lipschitz continuous with respect to 119863 119876 is 120575

1-

Lipschitz continuous in the first argument and 1205752-Lipschitz

continuous in the second argument ℎ is 120582ℎ-Lipschitz contin-

uous and using (32) (34) and (39) it becomes

1003817100381710038171003817ℎ (119887119899+1) minus ℎ (119887119899)1003817100381710038171003817 le

(1199041+ 1199042) 120582ℎ

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

+12058221205751

12058321205722

2minus 12057421205732

2

(1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058221205752

12058321205722

2minus 12057421205732

2

(1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

(40)

In the light of (34) we have

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le

(1199041+ 1199042) 120582ℎ+ 12058221205752 (1 + 1119899) 120591

1205852(12058321205722

2minus 12057421205732

2)

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

+12058221205751

1205852(12058321205722

2minus 12057421205732

2)(1 +

1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

(41)

Let

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le 1205793119899

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817 + 1205794119899

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817 (42)

where

1205793119899=

12058221205751 (1 + 1119899) 120588

1205852(12058321205722

2minus 12057421205732

2)

1205794119899=(1199041+ 1199042) 120582ℎ+ 12058221205752 (1 + 1119899) 120591

1205852(12058321205722

2minus 12057421205732

2)

(43)

From adding (37) and (42) we get

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 +

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le (120579

1119899+ 1205793119899)1003817100381710038171003817119886119899 minus 119886119899minus1

1003817100381710038171003817

+ (1205792119899+ 1205794119899)1003817100381710038171003817119887119899 minus 119887119899minus1

1003817100381710038171003817

le Θ1198991003817100381710038171003817119886119899 minus 119886119899minus1

1003817100381710038171003817 +1003817100381710038171003817119887119899 minus 119887119899minus1

1003817100381710038171003817

(44)

where Θ119899= max(120579

1119899+ 1205793119899) (1205792119899+ 1205794119899)

Letting 119899 rarr infin we obtain Θ119899rarr 120579 where

120579 = max (1205791+ 1205793) (1205792+ 1205794)

1205791=

(1199031+ 1199032) 120582119892+ 12058211205981120588

1205851(12058311205722

1minus 12057411205732

1)

1205792=

12058211205982120591

1205851(12058311205722

1minus 12057411205732

1)

1205793=

12058221205751120588

1205852(12058321205722

2minus 12057421205732

2)

1205794=(1199041+ 1199042) 120582ℎ+ 12058221205752120591

1205852(12058321205722

2minus 12057421205732

2)

(45)

By (31) 120579 isin (0 1) and (44) 119886119899 and 119887

119899 both are Cauchy

sequencesThus there exists (119886 119887) isin 1198831times1198832such that 119886

119899rarr

119886 119887119899rarr 119887 as 119899 rarr infin From the Lipschitz continuity of 119878

and119879 and (32) 119901119899119902119899 are also Cauchy sequences and thus

there exists (119901 119902) isin 1198831times 1198832such that 119901

119899rarr 119901 119902

119899rarr 119902 as

119899 rarr infinNow we prove that 119901 isin 119878(119886) and 119902 isin 119879(119887) In fact since

119901119899isin 119878(119886119899) and 119902

119899isin 119879(119887119899) we have

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 + 119889 (119901119899 119878 (119886))

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 +D (119878 (119886119899) 119878 (119886))

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 + 1205881003817100381710038171003817119886119899 minus 119886

1003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin

(46)

which implies that 119889(119901 119878(119886)) = 0 Since 119878(119886) isin 119862119861(1198831)

it follows that 119901 isin 119878(119886) Similarly we have 119902 120598 119879(119887) ByLemma 11 it follows that 119886 119887 119901 119902 is a solution of problem(21) and this completes the proof

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Chinese Journal of Mathematics 7

References

[1] J P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquoH(sdotsdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011

[3] Y-P Fang and N-J Huang ldquoH-Monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003

[4] Y-P Fang and N-J Huang ldquoH-accretive operators resolventoperator technique solving variational inclusions in banachspacesrdquo Applied Mathematics Letters vol 17 no 6 pp 647ndash6532004

[5] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (H 120578)-monotone operators inhilbert spacesrdquo Computers and Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005

[6] S Husain and S Gupta ldquoAlgorithm for solving a new system ofgeneralized variational inclusions in Hilbert spacesrdquo Journal ofCalculus of Variations vol 2013 Article ID 461371 8 pages 2013

[7] S Husain S Gupta and V N Mishra ldquoGeneralized H(sdot sdotsdot)120578-cocoercive operators and generalized set-valued variational-like inclusionsrdquo Journal of Mathematics vol 2013 Article ID738491 10 pages 2013

[8] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a P-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007

[9] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involvingP-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008

[10] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalizedimplicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012

[11] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on M-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009

[12] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalizedH(sdot sdot)-accretive mapping in real q-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011

[13] H-Y Lan Y Je Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (A 120578)-accretivemappings in banach spacesrdquo Computers and Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006

[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving A-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009

[15] R U Verma ldquoGeneral system of A-monotone nonlinearvariational inclusion problems with applicationsrdquo Journal of

OptimizationTheory and Applications vol 131 no 1 pp 151ndash1572006

[16] F-Q Xia and N-J Huang ldquoVariational inclusions with ageneral H-monotone operator in Banach spacesrdquo Computersand Mathematics with Applications vol 54 no 1 pp 24ndash302007

[17] F -Q Xia Q -B Zhang and Y -Z Zou ldquoA new iterative algo-rithm for variational inclusions with H-monotone operatorsrdquoThai Journal of Mathematics vol 10 no 3 pp 605ndash616 2012

[18] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010

[19] Y-Z Zou and N-J Huang ldquoH(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008

[20] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquo Mathematical Programming vol 48 no 1ndash3 pp 249ndash263 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Chinese Journal of Mathematics

The preceding lemma allows us to suggest the followingiterative algorithm for problem (21)

Algorithm 12 For (1198860 1198870) isin 1198831times1198832with119901 isin 119878(119886) 119902 isin 119879(119887)

compute the sequences 119886119899 119887119899 119901119899 119902119899 as follows

119892 (119886119899+1

) = 119877119866(sdotsdot)

1205821119872(sdot119886)

[119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 1205821119875 (119901119899 119902119899) ] 120582

1gt 0

ℎ (119887119899+1

) = 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 1205822119876 (119901119899 119902119899) ] 120582

2gt 0

119901119899isin 119878 (119886

119899)

1003817100381710038171003817119901119899 minus 119901119899+11003817100381710038171003817 le (1 +

1

119899 + 1)

timesD (119878 (119886119899) 119878 (119886

119899+1))

119902119899isin 119879 (119887

119899)

1003817100381710038171003817119902119899 minus 119902119899+11003817100381710038171003817 le (1 +

1

119899 + 1)

timesD (119879 (119887119899) 119879 (119887

119899+1))

(25)

for all 119899 = 0 1 2 and 1205821and 120582

2are constants

Definition 13 Let 119878 1198831rarr 119862119861(119883

1) and 119879 119883

2rarr 119862119861(119883

2)

be two set-valued mappings A single-valued mapping 119875

1198831times 1198832rarr 1198831is said to be

(i) 1205981-Lipschitz continuous in the first argument with

respect to 119878 if there exists a constant 1205981gt 0 such that

1003817100381710038171003817119875 (1199011 sdot) minus 119875 (1199012 sdot)1003817100381710038171003817 le 1205981

10038171003817100381710038171199011 minus 11990121003817100381710038171003817

forall1199011isin 119878 (119886

1) 1199012isin 119878 (119886

2) 1198861 1198862isin 1198831

(26)

(ii) 1205982-Lipschitz continuous in the second argument with

respect to 119879 if there exists a constant 1205982gt 0 such that

1003817100381710038171003817119875 (sdot 1199021) minus 119875 (sdot 1199022)1003817100381710038171003817 le 1205982

10038171003817100381710038171199021 minus 11990221003817100381710038171003817

forall1199021isin 119879 (119887

1) 1199022isin 119879 (119887

2) 1198871 1198872isin 1198832

(27)

Definition 14 Let 119878 1198831rarr 119862119861(119883

1) and 119879 119883

2rarr 119862119861(119883

2)

be two set-valued mappings A single-valued mapping 119876

1198831times 1198832rarr 1198832is said to be

(i) 1205751-Lipschitz continuous in the first argument with

respect to 119878 if there exists a constant 1205751gt 0 such that

1003817100381710038171003817119876 (1199011 sdot) minus 119876 (119901

2 sdot)1003817100381710038171003817 le 1205751

10038171003817100381710038171199011 minus 11990121003817100381710038171003817

forall1199011isin 119878 (119886

1) 1199012isin 119878 (119886

2) 1198861 1198862isin 1198831

(28)

(ii) 1205752-Lipschitz continuous in the second argument with

respect to 119879 if there exists a constant 1205752gt 0 such that

1003817100381710038171003817119876 (sdot 1199021) minus 119876 (sdot 119902

2)1003817100381710038171003817 le 1205752

10038171003817100381710038171199021 minus 11990221003817100381710038171003817

forall1199021isin 119879 (119887

1) 1199022isin 119879 (119887

2) 1198871 1198872isin 1198832

(29)

Definition 15 A set-valuedmapping 119860 119883 rarr 119862119861(119883) is saidto beD-Lipschitz continuous if there exists a constant 120588 gt 0

such that

D (119860 (119909) 119860 (119910)) le 1205881003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin 119883 (30)

Theorem 16 Let 1198831and 119883

2be real Hilbert spaces Let 119875

1198831times 1198832rarr 1198831 119876 119883

1times 1198832rarr 1198832 119892 119860 119861 119883

1rarr 1198831

ℎ 119862 119863 1198832

rarr 1198832 119866 119883

1times 1198831

rarr 1198831 and 119867

1198832times 1198832rarr 1198832be the single-valuedmappings Let 119878 119883

1rarr

119862119861(1198831) and 119879 119883

2rarr 119862119861(119883

2) be the set-valued mappings

let 119872 1198831times 1198831

rarr 21198831 be a set-valued mapping such

that for each 119886 isin 1198831 119872(sdot 119886) is a 119866(sdot sdot)-cocoercive operator

with respect to 119860 and 119861 and let 119873 1198832times 1198832rarr 21198832 be a

set-valued mapping such that for each 119887 isin 1198832 119873(sdot 119887) is an

119867(sdot sdot)-cocoercive operator with respect to 119862 and 119863 Assumethat 119892(119883

1) cap dom(119872(sdot 119886)) = 0 for each 119886 isin 119883

1 and ℎ(119883

2) cap

dom(119873(sdot 119887)) = 0 for each 119887 isin 1198832 Assume that

(i) 119878 119879 are 120588 120591-Lipschitz continuous in the HausdorffmetricD(sdot sdot) respectively

(ii) 119866(119860 119861) is 1205831-cocoercive with respect to 119860 and 120574

1-

relaxed cocoercive with respect to 119861

(iii) 119867(119862119863) is 1205832-cocoercive with respect to 119862 and 120574

2-

relaxed cocoercive with respect to119863

(iv) 119860 119862 is 1205721 1205722-expansive respectively

(v) 119861119863 119894119904 1205731 1205732-Lipschitz continuous respectively

(vi) 119892 is120582119892-Lipschitz continuous and 120585

1-stronglymonotone

(vii) ℎ is 120582ℎ-Lipschitz continuous and 120585

2-stronglymonotone

(viii) 119866(119860 119861) is 1199031-Lipschitz continuouswith respect to119860 and

1199032-Lipschitz continuous with respect to 119861

(ix) H(119862119863) is 1199041-Lipschitz continuous with respect to 119862

and 1199042-Lipschitz continuous with respect to119863

(x) 119875(sdot sdot) is 1205981-Lipschitz continuous in the first argument

and 1205982-Lipschitz continuous in the second argument

(xi) 119876(sdot sdot) is 1205751-Lipschitz continuous in the first argument

and 1205752-Lipschitz continuous in the second argument

(xii)

0 lt

(1199031+ 1199032) 120582119892+ 12058211205981120588

1205851(12058311205722

1minus 12057411205732

1)

+12058221205751 (1 + 1119899) 120588

1205852(12058321205722

2minus 12057421205732

2)lt 1

0 lt(1199041+ 1199042) 120582ℎ+ 12058221205752120591

1205852(12058321205722

2minus 12057421205732

2)

+12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)lt 1

(31)

Then the iterative sequences 119886119899 119887119899 119901119899 and 119902

119899

generated by Algorithm 12 converge strongly to 119886 119887 119901and 119902respectively and (119886 119887 119901 119902) is a solution of problem (21)

Chinese Journal of Mathematics 5

Proof Since 119878 119879 are Lipschitz continuous with constants 120588120591 respectively it follows from Algorithm 12 that

1003817100381710038171003817119901119899 minus 119901119899+11003817100381710038171003817 le (1 +

1

119899)D (119878 (119886

119899) 119878 (119886

119899+1))

le (1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899+11003817100381710038171003817

1003817100381710038171003817119902119899 minus 119902119899+11003817100381710038171003817 le (1 +

1

119899)D (119879 (119887

119899) 119879 (119887

119899+1))

le (1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899+11003817100381710038171003817

(32)

for all 119899 = 0 1 2

Using the 120585-strong monotonicity of 119892 we have1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)

1003817100381710038171003817

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817

ge ⟨119892 (119886119899+1

) minus 119892 (119886119899) 119886119899+1

minus 119886119899⟩

ge 1205851

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817

2

(33)

which implies that

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le

1

1205851

1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)1003817100381710038171003817 (34)

Nowwe estimate 119892(119886119899+1

)minus119892(119886119899) by usingAlgorithm 12 and

the Lipschitz continuity of 119877119866(sdotsdot)1205821119872(sdot119886)

as follows1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)

1003817100381710038171003817

=10038171003817100381710038171003817119877119866(sdotsdot)

1205821119872(sdot119886)

[119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899))) minus 120582

1119875 (119901119899 119902119899)]

minus [119877119866(sdotsdot)

1205821119872(sdot119886)

119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))

minus 1205821119875 (119901119899minus1

119902119899minus1

) ]10038171003817100381710038171003817

le1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899 119902119899) minus 119875 (119901119899minus1 119902119899minus1)1003817100381710038171003817

le1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899)))

1003817100381710038171003817

+1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899 119902119899) minus 119875 (119901119899minus1 119902119899)1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899minus1 119902119899) minus 119875 (119901119899minus1 119902119899minus1)1003817100381710038171003817

(35)

Since119866(119860 119861) is 1199031-Lipschitz continuouswith respect to119860 and

1199032-Lipschitz continuous with respect to 119861 119875 is 120598

1-Lipschitz

continuous in the first argument and 1205982-Lipschitz continuous

in the second argument 119892 is 120582119892-Lipschitz continuous and

using (32) (34) and (35) it becomes

1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)1003817100381710038171003817 le

(1199031+ 1199032) 120582119892

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205981

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119901119899 minus 119901119899minus11003817100381710038171003817

+12058211205982

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119902119899 minus 119902119899minus11003817100381710038171003817

le

(1199031+ 1199032) 120582119892

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205981

12058311205722

1minus 12057411205732

1

(1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205982

12058311205722

1minus 12057411205732

1

(1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le

(1199031+ 1199032) 120582119892+ 12058211205981 (1 + 1119899) 120588

1205851(12058311205722

1minus 12057411205732

1)

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

(36)

Let

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le 1205791119899

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817 + 1205792119899

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817 (37)

where

1205791119899=

(1199031+ 1199032) 120582119892+ 12058211205981 (1 + 1119899) 120588

1205851(12058311205722

1minus 12057411205732

1)

1205792119899=

12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)

(38)

Now we estimate ℎ(119887119899) minus ℎ(119887

119899minus1) by using Algorithm 12 and

the Lipschitz continuity of 119877119867(sdotsdot)1205822119873(sdot119887)

as follows

1003817100381710038171003817ℎ (119887119899+1) minus ℎ (119887119899)1003817100381710038171003817

=10038171003817100381710038171003817119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899))) minus 120582

2119876 (119901119899 119902119899)]

6 Chinese Journal of Mathematics

minus 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))

minus 1205822119876 (119901119899minus1

119902119899minus1

) ]10038171003817100381710038171003817

le1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899 119902119899) minus 119876 (119901

119899minus1 119902119899minus1

)1003817100381710038171003817

le1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899)))

1003817100381710038171003817

+1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899 119902119899) minus 119876 (119901

119899minus1 119902119899)1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899minus1

119902119899) minus 119876 (119901

119899minus1 119902119899minus1

)1003817100381710038171003817

(39)

Since 119867(119862119863) is 1199041-Lipschitz continuous with respect to

119862 and 1199042-Lipschitz continuous with respect to 119863 119876 is 120575

1-

Lipschitz continuous in the first argument and 1205752-Lipschitz

continuous in the second argument ℎ is 120582ℎ-Lipschitz contin-

uous and using (32) (34) and (39) it becomes

1003817100381710038171003817ℎ (119887119899+1) minus ℎ (119887119899)1003817100381710038171003817 le

(1199041+ 1199042) 120582ℎ

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

+12058221205751

12058321205722

2minus 12057421205732

2

(1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058221205752

12058321205722

2minus 12057421205732

2

(1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

(40)

In the light of (34) we have

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le

(1199041+ 1199042) 120582ℎ+ 12058221205752 (1 + 1119899) 120591

1205852(12058321205722

2minus 12057421205732

2)

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

+12058221205751

1205852(12058321205722

2minus 12057421205732

2)(1 +

1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

(41)

Let

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le 1205793119899

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817 + 1205794119899

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817 (42)

where

1205793119899=

12058221205751 (1 + 1119899) 120588

1205852(12058321205722

2minus 12057421205732

2)

1205794119899=(1199041+ 1199042) 120582ℎ+ 12058221205752 (1 + 1119899) 120591

1205852(12058321205722

2minus 12057421205732

2)

(43)

From adding (37) and (42) we get

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 +

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le (120579

1119899+ 1205793119899)1003817100381710038171003817119886119899 minus 119886119899minus1

1003817100381710038171003817

+ (1205792119899+ 1205794119899)1003817100381710038171003817119887119899 minus 119887119899minus1

1003817100381710038171003817

le Θ1198991003817100381710038171003817119886119899 minus 119886119899minus1

1003817100381710038171003817 +1003817100381710038171003817119887119899 minus 119887119899minus1

1003817100381710038171003817

(44)

where Θ119899= max(120579

1119899+ 1205793119899) (1205792119899+ 1205794119899)

Letting 119899 rarr infin we obtain Θ119899rarr 120579 where

120579 = max (1205791+ 1205793) (1205792+ 1205794)

1205791=

(1199031+ 1199032) 120582119892+ 12058211205981120588

1205851(12058311205722

1minus 12057411205732

1)

1205792=

12058211205982120591

1205851(12058311205722

1minus 12057411205732

1)

1205793=

12058221205751120588

1205852(12058321205722

2minus 12057421205732

2)

1205794=(1199041+ 1199042) 120582ℎ+ 12058221205752120591

1205852(12058321205722

2minus 12057421205732

2)

(45)

By (31) 120579 isin (0 1) and (44) 119886119899 and 119887

119899 both are Cauchy

sequencesThus there exists (119886 119887) isin 1198831times1198832such that 119886

119899rarr

119886 119887119899rarr 119887 as 119899 rarr infin From the Lipschitz continuity of 119878

and119879 and (32) 119901119899119902119899 are also Cauchy sequences and thus

there exists (119901 119902) isin 1198831times 1198832such that 119901

119899rarr 119901 119902

119899rarr 119902 as

119899 rarr infinNow we prove that 119901 isin 119878(119886) and 119902 isin 119879(119887) In fact since

119901119899isin 119878(119886119899) and 119902

119899isin 119879(119887119899) we have

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 + 119889 (119901119899 119878 (119886))

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 +D (119878 (119886119899) 119878 (119886))

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 + 1205881003817100381710038171003817119886119899 minus 119886

1003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin

(46)

which implies that 119889(119901 119878(119886)) = 0 Since 119878(119886) isin 119862119861(1198831)

it follows that 119901 isin 119878(119886) Similarly we have 119902 120598 119879(119887) ByLemma 11 it follows that 119886 119887 119901 119902 is a solution of problem(21) and this completes the proof

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Chinese Journal of Mathematics 7

References

[1] J P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquoH(sdotsdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011

[3] Y-P Fang and N-J Huang ldquoH-Monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003

[4] Y-P Fang and N-J Huang ldquoH-accretive operators resolventoperator technique solving variational inclusions in banachspacesrdquo Applied Mathematics Letters vol 17 no 6 pp 647ndash6532004

[5] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (H 120578)-monotone operators inhilbert spacesrdquo Computers and Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005

[6] S Husain and S Gupta ldquoAlgorithm for solving a new system ofgeneralized variational inclusions in Hilbert spacesrdquo Journal ofCalculus of Variations vol 2013 Article ID 461371 8 pages 2013

[7] S Husain S Gupta and V N Mishra ldquoGeneralized H(sdot sdotsdot)120578-cocoercive operators and generalized set-valued variational-like inclusionsrdquo Journal of Mathematics vol 2013 Article ID738491 10 pages 2013

[8] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a P-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007

[9] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involvingP-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008

[10] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalizedimplicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012

[11] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on M-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009

[12] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalizedH(sdot sdot)-accretive mapping in real q-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011

[13] H-Y Lan Y Je Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (A 120578)-accretivemappings in banach spacesrdquo Computers and Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006

[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving A-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009

[15] R U Verma ldquoGeneral system of A-monotone nonlinearvariational inclusion problems with applicationsrdquo Journal of

OptimizationTheory and Applications vol 131 no 1 pp 151ndash1572006

[16] F-Q Xia and N-J Huang ldquoVariational inclusions with ageneral H-monotone operator in Banach spacesrdquo Computersand Mathematics with Applications vol 54 no 1 pp 24ndash302007

[17] F -Q Xia Q -B Zhang and Y -Z Zou ldquoA new iterative algo-rithm for variational inclusions with H-monotone operatorsrdquoThai Journal of Mathematics vol 10 no 3 pp 605ndash616 2012

[18] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010

[19] Y-Z Zou and N-J Huang ldquoH(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008

[20] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquo Mathematical Programming vol 48 no 1ndash3 pp 249ndash263 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Chinese Journal of Mathematics 5

Proof Since 119878 119879 are Lipschitz continuous with constants 120588120591 respectively it follows from Algorithm 12 that

1003817100381710038171003817119901119899 minus 119901119899+11003817100381710038171003817 le (1 +

1

119899)D (119878 (119886

119899) 119878 (119886

119899+1))

le (1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899+11003817100381710038171003817

1003817100381710038171003817119902119899 minus 119902119899+11003817100381710038171003817 le (1 +

1

119899)D (119879 (119887

119899) 119879 (119887

119899+1))

le (1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899+11003817100381710038171003817

(32)

for all 119899 = 0 1 2

Using the 120585-strong monotonicity of 119892 we have1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)

1003817100381710038171003817

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817

ge ⟨119892 (119886119899+1

) minus 119892 (119886119899) 119886119899+1

minus 119886119899⟩

ge 1205851

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817

2

(33)

which implies that

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le

1

1205851

1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)1003817100381710038171003817 (34)

Nowwe estimate 119892(119886119899+1

)minus119892(119886119899) by usingAlgorithm 12 and

the Lipschitz continuity of 119877119866(sdotsdot)1205821119872(sdot119886)

as follows1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)

1003817100381710038171003817

=10038171003817100381710038171003817119877119866(sdotsdot)

1205821119872(sdot119886)

[119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899))) minus 120582

1119875 (119901119899 119902119899)]

minus [119877119866(sdotsdot)

1205821119872(sdot119886)

119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))

minus 1205821119875 (119901119899minus1

119902119899minus1

) ]10038171003817100381710038171003817

le1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899 119902119899) minus 119875 (119901119899minus1 119902119899minus1)1003817100381710038171003817

le1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899)) 119861 (119892 (119886

119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899)))

1003817100381710038171003817

+1

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899)))

minus 119866 (119860 (119892 (119886119899minus1

)) 119861 (119892 (119886119899minus1

)))1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899 119902119899) minus 119875 (119901119899minus1 119902119899)1003817100381710038171003817

+1205821

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119875 (119901119899minus1 119902119899) minus 119875 (119901119899minus1 119902119899minus1)1003817100381710038171003817

(35)

Since119866(119860 119861) is 1199031-Lipschitz continuouswith respect to119860 and

1199032-Lipschitz continuous with respect to 119861 119875 is 120598

1-Lipschitz

continuous in the first argument and 1205982-Lipschitz continuous

in the second argument 119892 is 120582119892-Lipschitz continuous and

using (32) (34) and (35) it becomes

1003817100381710038171003817119892 (119886119899+1) minus 119892 (119886119899)1003817100381710038171003817 le

(1199031+ 1199032) 120582119892

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205981

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119901119899 minus 119901119899minus11003817100381710038171003817

+12058211205982

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119902119899 minus 119902119899minus11003817100381710038171003817

le

(1199031+ 1199032) 120582119892

12058311205722

1minus 12057411205732

1

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205981

12058311205722

1minus 12057411205732

1

(1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205982

12058311205722

1minus 12057411205732

1

(1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le

(1199031+ 1199032) 120582119892+ 12058211205981 (1 + 1119899) 120588

1205851(12058311205722

1minus 12057411205732

1)

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

(36)

Let

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 le 1205791119899

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817 + 1205792119899

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817 (37)

where

1205791119899=

(1199031+ 1199032) 120582119892+ 12058211205981 (1 + 1119899) 120588

1205851(12058311205722

1minus 12057411205732

1)

1205792119899=

12058211205982 (1 + 1119899) 120591

1205851(12058311205722

1minus 12057411205732

1)

(38)

Now we estimate ℎ(119887119899) minus ℎ(119887

119899minus1) by using Algorithm 12 and

the Lipschitz continuity of 119877119867(sdotsdot)1205822119873(sdot119887)

as follows

1003817100381710038171003817ℎ (119887119899+1) minus ℎ (119887119899)1003817100381710038171003817

=10038171003817100381710038171003817119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899))) minus 120582

2119876 (119901119899 119902119899)]

6 Chinese Journal of Mathematics

minus 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))

minus 1205822119876 (119901119899minus1

119902119899minus1

) ]10038171003817100381710038171003817

le1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899 119902119899) minus 119876 (119901

119899minus1 119902119899minus1

)1003817100381710038171003817

le1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899)))

1003817100381710038171003817

+1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899 119902119899) minus 119876 (119901

119899minus1 119902119899)1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899minus1

119902119899) minus 119876 (119901

119899minus1 119902119899minus1

)1003817100381710038171003817

(39)

Since 119867(119862119863) is 1199041-Lipschitz continuous with respect to

119862 and 1199042-Lipschitz continuous with respect to 119863 119876 is 120575

1-

Lipschitz continuous in the first argument and 1205752-Lipschitz

continuous in the second argument ℎ is 120582ℎ-Lipschitz contin-

uous and using (32) (34) and (39) it becomes

1003817100381710038171003817ℎ (119887119899+1) minus ℎ (119887119899)1003817100381710038171003817 le

(1199041+ 1199042) 120582ℎ

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

+12058221205751

12058321205722

2minus 12057421205732

2

(1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058221205752

12058321205722

2minus 12057421205732

2

(1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

(40)

In the light of (34) we have

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le

(1199041+ 1199042) 120582ℎ+ 12058221205752 (1 + 1119899) 120591

1205852(12058321205722

2minus 12057421205732

2)

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

+12058221205751

1205852(12058321205722

2minus 12057421205732

2)(1 +

1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

(41)

Let

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le 1205793119899

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817 + 1205794119899

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817 (42)

where

1205793119899=

12058221205751 (1 + 1119899) 120588

1205852(12058321205722

2minus 12057421205732

2)

1205794119899=(1199041+ 1199042) 120582ℎ+ 12058221205752 (1 + 1119899) 120591

1205852(12058321205722

2minus 12057421205732

2)

(43)

From adding (37) and (42) we get

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 +

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le (120579

1119899+ 1205793119899)1003817100381710038171003817119886119899 minus 119886119899minus1

1003817100381710038171003817

+ (1205792119899+ 1205794119899)1003817100381710038171003817119887119899 minus 119887119899minus1

1003817100381710038171003817

le Θ1198991003817100381710038171003817119886119899 minus 119886119899minus1

1003817100381710038171003817 +1003817100381710038171003817119887119899 minus 119887119899minus1

1003817100381710038171003817

(44)

where Θ119899= max(120579

1119899+ 1205793119899) (1205792119899+ 1205794119899)

Letting 119899 rarr infin we obtain Θ119899rarr 120579 where

120579 = max (1205791+ 1205793) (1205792+ 1205794)

1205791=

(1199031+ 1199032) 120582119892+ 12058211205981120588

1205851(12058311205722

1minus 12057411205732

1)

1205792=

12058211205982120591

1205851(12058311205722

1minus 12057411205732

1)

1205793=

12058221205751120588

1205852(12058321205722

2minus 12057421205732

2)

1205794=(1199041+ 1199042) 120582ℎ+ 12058221205752120591

1205852(12058321205722

2minus 12057421205732

2)

(45)

By (31) 120579 isin (0 1) and (44) 119886119899 and 119887

119899 both are Cauchy

sequencesThus there exists (119886 119887) isin 1198831times1198832such that 119886

119899rarr

119886 119887119899rarr 119887 as 119899 rarr infin From the Lipschitz continuity of 119878

and119879 and (32) 119901119899119902119899 are also Cauchy sequences and thus

there exists (119901 119902) isin 1198831times 1198832such that 119901

119899rarr 119901 119902

119899rarr 119902 as

119899 rarr infinNow we prove that 119901 isin 119878(119886) and 119902 isin 119879(119887) In fact since

119901119899isin 119878(119886119899) and 119902

119899isin 119879(119887119899) we have

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 + 119889 (119901119899 119878 (119886))

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 +D (119878 (119886119899) 119878 (119886))

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 + 1205881003817100381710038171003817119886119899 minus 119886

1003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin

(46)

which implies that 119889(119901 119878(119886)) = 0 Since 119878(119886) isin 119862119861(1198831)

it follows that 119901 isin 119878(119886) Similarly we have 119902 120598 119879(119887) ByLemma 11 it follows that 119886 119887 119901 119902 is a solution of problem(21) and this completes the proof

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Chinese Journal of Mathematics 7

References

[1] J P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquoH(sdotsdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011

[3] Y-P Fang and N-J Huang ldquoH-Monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003

[4] Y-P Fang and N-J Huang ldquoH-accretive operators resolventoperator technique solving variational inclusions in banachspacesrdquo Applied Mathematics Letters vol 17 no 6 pp 647ndash6532004

[5] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (H 120578)-monotone operators inhilbert spacesrdquo Computers and Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005

[6] S Husain and S Gupta ldquoAlgorithm for solving a new system ofgeneralized variational inclusions in Hilbert spacesrdquo Journal ofCalculus of Variations vol 2013 Article ID 461371 8 pages 2013

[7] S Husain S Gupta and V N Mishra ldquoGeneralized H(sdot sdotsdot)120578-cocoercive operators and generalized set-valued variational-like inclusionsrdquo Journal of Mathematics vol 2013 Article ID738491 10 pages 2013

[8] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a P-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007

[9] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involvingP-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008

[10] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalizedimplicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012

[11] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on M-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009

[12] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalizedH(sdot sdot)-accretive mapping in real q-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011

[13] H-Y Lan Y Je Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (A 120578)-accretivemappings in banach spacesrdquo Computers and Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006

[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving A-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009

[15] R U Verma ldquoGeneral system of A-monotone nonlinearvariational inclusion problems with applicationsrdquo Journal of

OptimizationTheory and Applications vol 131 no 1 pp 151ndash1572006

[16] F-Q Xia and N-J Huang ldquoVariational inclusions with ageneral H-monotone operator in Banach spacesrdquo Computersand Mathematics with Applications vol 54 no 1 pp 24ndash302007

[17] F -Q Xia Q -B Zhang and Y -Z Zou ldquoA new iterative algo-rithm for variational inclusions with H-monotone operatorsrdquoThai Journal of Mathematics vol 10 no 3 pp 605ndash616 2012

[18] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010

[19] Y-Z Zou and N-J Huang ldquoH(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008

[20] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquo Mathematical Programming vol 48 no 1ndash3 pp 249ndash263 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Chinese Journal of Mathematics

minus 119877119867(sdotsdot)

1205822119873(sdot119887)

[119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))

minus 1205822119876 (119901119899minus1

119902119899minus1

) ]10038171003817100381710038171003817

le1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899 119902119899) minus 119876 (119901

119899minus1 119902119899minus1

)1003817100381710038171003817

le1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899)) 119863 (ℎ (119887

119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899)))

1003817100381710038171003817

+1

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899)))

minus 119867 (119862 (ℎ (119887119899minus1

)) 119863 (ℎ (119887119899minus1

)))1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899 119902119899) minus 119876 (119901

119899minus1 119902119899)1003817100381710038171003817

+1205822

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119876 (119901119899minus1

119902119899) minus 119876 (119901

119899minus1 119902119899minus1

)1003817100381710038171003817

(39)

Since 119867(119862119863) is 1199041-Lipschitz continuous with respect to

119862 and 1199042-Lipschitz continuous with respect to 119863 119876 is 120575

1-

Lipschitz continuous in the first argument and 1205752-Lipschitz

continuous in the second argument ℎ is 120582ℎ-Lipschitz contin-

uous and using (32) (34) and (39) it becomes

1003817100381710038171003817ℎ (119887119899+1) minus ℎ (119887119899)1003817100381710038171003817 le

(1199041+ 1199042) 120582ℎ

12058321205722

2minus 12057421205732

2

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

+12058221205751

12058321205722

2minus 12057421205732

2

(1 +1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

+12058221205752

12058321205722

2minus 12057421205732

2

(1 +1

119899) 120591

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

(40)

In the light of (34) we have

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le

(1199041+ 1199042) 120582ℎ+ 12058221205752 (1 + 1119899) 120591

1205852(12058321205722

2minus 12057421205732

2)

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817

+12058221205751

1205852(12058321205722

2minus 12057421205732

2)(1 +

1

119899) 120588

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817

(41)

Let

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le 1205793119899

1003817100381710038171003817119886119899 minus 119886119899minus11003817100381710038171003817 + 1205794119899

1003817100381710038171003817119887119899 minus 119887119899minus11003817100381710038171003817 (42)

where

1205793119899=

12058221205751 (1 + 1119899) 120588

1205852(12058321205722

2minus 12057421205732

2)

1205794119899=(1199041+ 1199042) 120582ℎ+ 12058221205752 (1 + 1119899) 120591

1205852(12058321205722

2minus 12057421205732

2)

(43)

From adding (37) and (42) we get

1003817100381710038171003817119886119899+1 minus 1198861198991003817100381710038171003817 +

1003817100381710038171003817119887119899+1 minus 1198871198991003817100381710038171003817 le (120579

1119899+ 1205793119899)1003817100381710038171003817119886119899 minus 119886119899minus1

1003817100381710038171003817

+ (1205792119899+ 1205794119899)1003817100381710038171003817119887119899 minus 119887119899minus1

1003817100381710038171003817

le Θ1198991003817100381710038171003817119886119899 minus 119886119899minus1

1003817100381710038171003817 +1003817100381710038171003817119887119899 minus 119887119899minus1

1003817100381710038171003817

(44)

where Θ119899= max(120579

1119899+ 1205793119899) (1205792119899+ 1205794119899)

Letting 119899 rarr infin we obtain Θ119899rarr 120579 where

120579 = max (1205791+ 1205793) (1205792+ 1205794)

1205791=

(1199031+ 1199032) 120582119892+ 12058211205981120588

1205851(12058311205722

1minus 12057411205732

1)

1205792=

12058211205982120591

1205851(12058311205722

1minus 12057411205732

1)

1205793=

12058221205751120588

1205852(12058321205722

2minus 12057421205732

2)

1205794=(1199041+ 1199042) 120582ℎ+ 12058221205752120591

1205852(12058321205722

2minus 12057421205732

2)

(45)

By (31) 120579 isin (0 1) and (44) 119886119899 and 119887

119899 both are Cauchy

sequencesThus there exists (119886 119887) isin 1198831times1198832such that 119886

119899rarr

119886 119887119899rarr 119887 as 119899 rarr infin From the Lipschitz continuity of 119878

and119879 and (32) 119901119899119902119899 are also Cauchy sequences and thus

there exists (119901 119902) isin 1198831times 1198832such that 119901

119899rarr 119901 119902

119899rarr 119902 as

119899 rarr infinNow we prove that 119901 isin 119878(119886) and 119902 isin 119879(119887) In fact since

119901119899isin 119878(119886119899) and 119902

119899isin 119879(119887119899) we have

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 + 119889 (119901119899 119878 (119886))

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 +D (119878 (119886119899) 119878 (119886))

119889 (119901 119878 (119886)) le1003817100381710038171003817119901 minus 119901119899

1003817100381710038171003817 + 1205881003817100381710038171003817119886119899 minus 119886

1003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin

(46)

which implies that 119889(119901 119878(119886)) = 0 Since 119878(119886) isin 119862119861(1198831)

it follows that 119901 isin 119878(119886) Similarly we have 119902 120598 119879(119887) ByLemma 11 it follows that 119886 119887 119901 119902 is a solution of problem(21) and this completes the proof

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Chinese Journal of Mathematics 7

References

[1] J P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquoH(sdotsdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011

[3] Y-P Fang and N-J Huang ldquoH-Monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003

[4] Y-P Fang and N-J Huang ldquoH-accretive operators resolventoperator technique solving variational inclusions in banachspacesrdquo Applied Mathematics Letters vol 17 no 6 pp 647ndash6532004

[5] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (H 120578)-monotone operators inhilbert spacesrdquo Computers and Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005

[6] S Husain and S Gupta ldquoAlgorithm for solving a new system ofgeneralized variational inclusions in Hilbert spacesrdquo Journal ofCalculus of Variations vol 2013 Article ID 461371 8 pages 2013

[7] S Husain S Gupta and V N Mishra ldquoGeneralized H(sdot sdotsdot)120578-cocoercive operators and generalized set-valued variational-like inclusionsrdquo Journal of Mathematics vol 2013 Article ID738491 10 pages 2013

[8] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a P-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007

[9] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involvingP-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008

[10] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalizedimplicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012

[11] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on M-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009

[12] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalizedH(sdot sdot)-accretive mapping in real q-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011

[13] H-Y Lan Y Je Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (A 120578)-accretivemappings in banach spacesrdquo Computers and Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006

[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving A-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009

[15] R U Verma ldquoGeneral system of A-monotone nonlinearvariational inclusion problems with applicationsrdquo Journal of

OptimizationTheory and Applications vol 131 no 1 pp 151ndash1572006

[16] F-Q Xia and N-J Huang ldquoVariational inclusions with ageneral H-monotone operator in Banach spacesrdquo Computersand Mathematics with Applications vol 54 no 1 pp 24ndash302007

[17] F -Q Xia Q -B Zhang and Y -Z Zou ldquoA new iterative algo-rithm for variational inclusions with H-monotone operatorsrdquoThai Journal of Mathematics vol 10 no 3 pp 605ndash616 2012

[18] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010

[19] Y-Z Zou and N-J Huang ldquoH(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008

[20] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquo Mathematical Programming vol 48 no 1ndash3 pp 249ndash263 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Chinese Journal of Mathematics 7

References

[1] J P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquoH(sdotsdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011

[3] Y-P Fang and N-J Huang ldquoH-Monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003

[4] Y-P Fang and N-J Huang ldquoH-accretive operators resolventoperator technique solving variational inclusions in banachspacesrdquo Applied Mathematics Letters vol 17 no 6 pp 647ndash6532004

[5] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (H 120578)-monotone operators inhilbert spacesrdquo Computers and Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005

[6] S Husain and S Gupta ldquoAlgorithm for solving a new system ofgeneralized variational inclusions in Hilbert spacesrdquo Journal ofCalculus of Variations vol 2013 Article ID 461371 8 pages 2013

[7] S Husain S Gupta and V N Mishra ldquoGeneralized H(sdot sdotsdot)120578-cocoercive operators and generalized set-valued variational-like inclusionsrdquo Journal of Mathematics vol 2013 Article ID738491 10 pages 2013

[8] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a P-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007

[9] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involvingP-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008

[10] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalizedimplicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012

[11] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on M-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009

[12] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalizedH(sdot sdot)-accretive mapping in real q-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011

[13] H-Y Lan Y Je Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (A 120578)-accretivemappings in banach spacesrdquo Computers and Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006

[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving A-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009

[15] R U Verma ldquoGeneral system of A-monotone nonlinearvariational inclusion problems with applicationsrdquo Journal of

OptimizationTheory and Applications vol 131 no 1 pp 151ndash1572006

[16] F-Q Xia and N-J Huang ldquoVariational inclusions with ageneral H-monotone operator in Banach spacesrdquo Computersand Mathematics with Applications vol 54 no 1 pp 24ndash302007

[17] F -Q Xia Q -B Zhang and Y -Z Zou ldquoA new iterative algo-rithm for variational inclusions with H-monotone operatorsrdquoThai Journal of Mathematics vol 10 no 3 pp 605ndash616 2012

[18] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010

[19] Y-Z Zou and N-J Huang ldquoH(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008

[20] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquo Mathematical Programming vol 48 no 1ndash3 pp 249ndash263 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of