Research Article Generalized Uniqueness Theorem for …downloads.hindawi.com/journals/tswj/2014/272479.pdf · Generalized Uniqueness Theorem for Ordinary Differential Equations in

Research ArticleGeneralized Uniqueness Theorem for Ordinary DifferentialEquations in Banach Spaces

Ezzat R Hassan M Sh Alhuthali and M M Al-Ghanmi

Department of Mathematics Faculty of Science King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia

Correspondence should be addressed to Ezzat R Hassan ezzat1611yahoocom

Received 13 November 2013 Accepted 23 December 2013 Published 10 February 2014

Academic Editors F Basar G A Chechkin G Fernandez-Anaya and R Plebaniak

Copyright copy 2014 Ezzat R Hassan et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider nonlinear ordinary differential equations in Banach spaces Uniqueness criterion for the Cauchy problem is givenwhen any of the standard dissipative-type conditions does apply A similar scalar result has been studied byMajorana (1991) Usefulexamples of reflexive Banach spaces whose positive cones have empty interior has been given as well

1 Introduction

Throughout the last century most of the efforts are concen-trated on the study of the classical Cauchy problem alsocalled the initial value problem and denoted by IVP

1199091015840= 119891 (119905 119909) 119909 (0) = 0 (1)

where 119891 [0 1] times 119864 rarr 119864 and 119864 is a real Banach space In thefinite dimensional case the existence is guaranteed by Peanorsquostheorem In order to put our results into context let us startby formulating the classical theorem of Peano

Theorem 1 (see [1]) Let 119864 = R119899 and 119891 isin 119862([0 1] times R119899R119899)Then (1) has a local solution

Such an infinite dimensional Cauchy problem may haveno solutions Dieudonne [2] provided the first example of acontinuous map from an infinitely dimensional nonreflexiveBanach space 119862

0for which there is no solution to the related

Cauchy problem (1) Many counterexamples in various infi-nite dimensional reflexive as well as nonreflexive Banachspaces followed for example [3ndash6] Afterwards Godunov [7]proved that Theorem 1 is false in every infinite dimensionalBanach space It turned out that continuity alone of thefunction119891 is not sufficient to prove a local existence theoremin the case where 119864 is infinite dimensional In order toobtain suitable extensions for the continuity notion on finitelydimensional spaces the ideas were to use different topologies

on 119864 and then the study has taken two directions Onedirection is to impose strong topology assumptions whichcan be found in different works for example [8ndash12] Theother approach is to utilize weak topology assumptions it isobserved that if the Banach space 119864 is reflexive we recoverlocally compactness by endowing it with the weak topologyIn [9 13ndash15] the Cauchy problem (1) has been discussed inreflexive Banach space Astala [16] proved that a Banach space119864 is reflexive if and only if (1) admits a local solution for everyweakly continuous map 119891 Thus there is no hope to extendPeanorsquos theorem in the weak topology setting to nonreflexivespacesThenonreflexive casewas examined by among others[17ndash20] on assuming besides the weak continuity of 119891 somecondition on 119891 involving the measure of weak noncompact-ness to somehow recover the locally compactness lost by thefact that the Banach space we are working on is no longerreflexive There are a lot of works devoted to investigatinguniqueness criteria in which Kamkersquos original hypothesis isreplaced by a dissipative-type condition formulated in termsof a semi-inner product [8 11 20ndash28] Majorana [29] foundout a very close relation between an auxiliary scalar equationASE of the form

119906 = 119905119891 (119905 119906) (2)

and the classical Cauchy problem (1) where 119909 and 0 are reals

Theorem 2 (see [29]) Let the function 119891(119905 119909) be defined in[0 119886]timesR and continuous with respect to 119905 such that119891(119905 0) = 0

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 272479 4 pageshttpdxdoiorg1011552014272479

2 The Scientific World Journal

for every 119905 isin [0 119886] Further let Cauchy problem (1) have twodifferent classical solutions defined in 119905 isin [0 120572]Then for every120576 gt 0 there exists 119905 isin (0 120572] such that (2) has at least two dif-ferent roots 119906 with |119906| lt 120576

An immediate consequence of this latter theorem is thefollowing uniqueness criterion

Theorem 3 (see [29]) Let the function 119891(119905 119909) be defined in[0 119886]timesR and continuous with respect to 119905 such that119891(119905 0) = 0for every 119905 isin [0 119886] Further let there exist 120576 gt 0 such that 119906 = 0is the only root of (2)with |119906| lt 120576 for every 119905 isin [0 120572] Then for(1) 119909(119905) = 0 is the only classical solution defined in 119905 isin [0 120572]

Therefore we have inR a very close relation between (2)and (1) It is one of the goals of this work to retain this relationin a suitable generalized sense However Majoranarsquos resultsare not directly extendable to an arbitrary abstract space asthe following example shows

Example 4 Let us consider the Cauchy problem

1199091015840= 119891 (119905 119909) 119909 (0) = 0 (3)

where119891 (119905 119909)

=

(

2

radic119909

(1199091+ 1199092)

2

radic119909

(1199092minus 1199091)) if 119909 = 0

0 if 119909 = 0(4)

In polar coordinates (3) becomes

1199031015840= 2radic119903 120579

1015840=

minus2

radic119903

(5)

Thus besides the trivial solution 119909 = 0 there is at least oneother given by

119909 =

(1199052 cos ln 1

1199052 1199052 sin ln 1

1199052) if 119905 = 0

0 if 119905 = 0(6)

We conclude that (3) satisfies the assumptions ofTheorem 10however one cannot find more than the trivial root to theauxiliary scalar equation ASE corresponding to (3) namely

1199061= 120582 (119906

1+ 1199062) 119906

2= 120582 (119906

2minus 1199061) (7)

where 120582 is an arbitrary scalar

In his paper [30] the author provided a generalization ofMajoranarsquos theorem in finite dimensional Hilbert spaces anatural question arises can we extend this result to infinitelydimensional Banach spaces The answer is positive as will beshown in the following

A way to provide a version of Majoranarsquos uniqueness the-orem in Banach space consists in replacing (2) with a suitableone Our main concern in this work is the classical Cauchyproblem (1) where 119891 takes values in a real Banach space 119864and 119909 and 0 are in 119864

Before starting the main work we will introduce someconceptsThroughout the following sdot stands for the norm in119864 The underlying idea to provide counterpart to (2) is basedon the following definition

Definition 5 (see [11]) Let 119864 be a real Banach space A subset119875 of 119864 is called a cone if the following are true

(i) 119875 is nonempty and nontrivial (ie 119875 contains anonzero point)

(ii) 120582119875 sub 119875 120582 gt 0(iii) 119875 + 119875 sub 119875(iv) 119875 = 119875 where 119875 denotes the closure of 119864(v) 119875 cap minus119875 = 0 where 0 denotes the zero element of the

Banach space 119864Assume that 1198750 = 120601 1198750 denotes the interior of 119864 The cone 119875of 119864 induces an ordering ldquolerdquo by setting

119909 le 119910 lArrrArr 119910 minus 119909 isin 119875

119909 lt 119910 lArrrArr 119910 minus 119909 isin 1198750

(8)

Let 119875lowast be the set of all continuous linear functionals 119888 on 119864such that 119888(119909) ge 0 for all 119909 isin 119875 and let 119875lowast

0be the set of all

continuous linear functionals 119888 on 119864 such that 119888(119909) gt 0 for all119909 isin 119875

0 The underlying idea to provide counterpart to (2) isbased on the following Lemma which is due to Mazur [31]

Lemma 6 (see [11]) Let 119875 be a cone with nonempty interior1198750 then the following hold

(i) 119909 isin 119875 is equivalent to 119888(119909) ge 0 for all 119888 isin 119875lowast(ii) 119909 isin 120597119875 implies that there exists a 119888 isin 119875

lowast

0such that

119888(119909) = 0 where 120597119875 denotes the boundary of 119875

It is well known that the requirements on the function119891 are dependent on types of solutions since we concentrateourselves to weak solutions then the classical case in the sub-ject is that due to Szep [15] Before giving Szeprsquos Theorem weneed the following definition

Definition 7 (see [11]) A function119891(119905 119909) is said to be weakly-weakly continuous at (119904 119910) if given 120576 gt 0 and 120593 isin 119864

lowast thereexist 120575(120576 120593) gt 0 and ℷ(120576 120593) a weakly open set containing 119910such that |120593(119891(119905 119909) minus 119891(119904 119910))| lt 120576 whenever |119905 minus 119904| lt 120575 and119909 isin ℷ

Definition 8 A function 119909(119905) is said to be weakly differen-tiable at 119905

0if there exists a point in 119864 denoted by 1199091015840(119905

0) such

that 120593(1199091015840(1199050)) = (120593119909)

1015840(1199050) for every 120593 isin 119864lowast

Theorem 9 (see [15]) Let 119864 be a reflexive Banach space andlet 119891 be a weakly-weakly continuous function on119860 0 le 119905 le 119886119909 le 119887 Let 119891(119905 119909) le 119872 on119860Then (1) has at least one weaksolution defined on [0 120572] 120572 = min119886 (119887119872)

2 Main Result

This section contains the main results Throughout thissection we will assume that 119864 is a real reflexive Banach space

The Scientific World Journal 3

endowed with weak topology 119875 sub 119864 is a cone with nonemptyinterior

Theorem 10 Let 119891 [0 1] times 119864 rarr 119864 be a weakly-weakly con-tinuous such that 119891(119905 119909) le 119872 on [0 1]times119864 and let119891(119905 0) = 0for every 119905 isin [0 1] Suppose further that (1) admits two differ-ent weak solutions defined in [0 120572] (120572 lt 119886) Then for every120576 gt 0 and every 119888 isin 119875lowast

0 there exists 119905 isin [0 120572] such that the fol-

lowing scalar equation

119888 (119906) = 119905119888 (119891 (119905 119906)) (9)

has at least two different roots 119906 with 119906 lt 120576

Science119864 is is open there exist 119886 119887 gt 0 such that all pointswhose distance from (0 0) isin [0 1] times 119864 satisfies 0 le 119905 lt 120572 and119909 lt 120573 are contained in [0 1] times 119864 Let 119860 0 le 119905 le 119886 and119909 le 119887 be such that 119860 sube 0 le 119905 lt 120572 and 119909 lt 120573 ThenTheorem 9 applied to (1) guarantees the existence of the solu-tion of (1) on [0 1] times 119864 The idea of the proof comes from[29]

Proof It follows by the assumption 119891(119905 0) = 0 that (1) hasthe zero solution so we assume that (1) has the weak solution119910(119905) = 0

Let 120576 gt 0 and 119888 isin 119875lowast

0be given Since 119906 = 0 is a root of

(9) for every 119905 isin [0 119886] it is sufficient to show that there exists119905 isin [0 120572] for which (9) is satisfied by some 119906 = 0with 119906 lt 120576Let a real-valued function 119860

119888(119905) be defined by setting

119860119888(119905) =

119888 (119910)

119905

119905 = 0

0 119905 = 0

(10)

119905 isin [0 119886] Of course 119860119888(119905) is nonnegative weak continuous

in [0 119886] and weakly differentiable in (0 119886) and for every 119905 in(0 119886) we have

1198601015840

119888(119905) =

1

1199052[119905119888 (119891 (119905 119910 (119905))) minus 119888 (119910)] (11)

Now fix 1199052isin (0 119886) with 119910(119905

2) = 0 such that 119910(119905) lt 120576 for

every 119905 isin [0 1199052] Denote 119905

1= sup119905 isin [0 119905

2] 119860

119888(119905) = 0

Clearly 119910(1199051) = 0 and 119910(119905) = 0 for every 119905 isin (119905

1 1199052] At this

point there are just two possibilitiesP1 If there exists a 119905 isin (119905

1 1199052] such that 1198601015840

119888(119905) = 0 then

from (11) for such a 119905 (9) is satisfied by119906 = 119910(119905) Hencethe proof is accomplished by just taking these 119905 and119906 = 119910(119905)

P2 Otherwise if1198601015840119888(119905) = 0 for every 119905 isin (119905

1 1199052] According

to Darboux property 1198601015840

119888(119905) has a constant sign in

(1199051 1199052] then we take 119906 = 119910(119905

2)( = 0) and define

119866 (119905) = 119905119888 (119891 (119905 119910 (1199052))) minus 119888 (119910 (119905

2)) for every 119905 isin [0 119886]

(12)

Now let us suppose that1198601015840119888(119905) gt 0 for every 119905 isin (119905

1 1199052]119866(0) =

minus119888(119910(1199052)) lt 0 On the other hand we have119866(119905

2) = 1199052

21198601015840

119888(1199052) gt

0 It follows by continuity of 119866 that there exists a 119905 isin (0 1199052]

such that119866(119905) = 0The fact that119860119888(119905) gt 0 for every 119905 isin (119905

1 1199052]

and 119860119888(0) = 0 implies that 1198601015840

119888(119905) lt 0 is impossible and the

proof will thus be accomplished

An immediate consequence ofTheorem 2 is the followinguniqueness criterion

Theorem 11 Let the hypotheses of theTheorem 10 hold and let119891(119905 0) = 0 for every 119905 isin [0 1] Assume further that there exist120576 gt 0 119888 isin 119875

lowast

0 and 119905

0isin (0 1] such that 119906 = 0 is the only

root of the auxiliary scalar equation (9) with 119906 lt 120576 for every119905 isin [0 119905

0] Then (1) admits in the interval [0 119905

0] only the zero

solutionAs it was pointed out byMajorana the crucial point inThe-

orems 10 and 11 is the assumption that (1) has the zero solutionOne follows Majoranarsquos procedure to remove this restriction Ifwe know a weak solution 119910 of (1) then by means of change ofvariables 119909 = 119901 + 119910(119905) (1) becomes

1199011015840= 119865 (119905 119901) 119901 (0) = 0 (13)

where 119865(119905 119901) = 119891(119905 119901 + 119910(119905)) minus 119891(119905 119910(119905)) It is clear thatany two different weak solutions of (1) are mapped to differentweak solutions of (13) Moreover (13) admits the zero solution0 which corresponds to the solution 119910 of (1) One thus gets thecounterpart to (9)

119888 (119906) = 119905119888 (119865 (119905 119906)) (14)

We can restateTheorems 10 and 11 involving the nontrivialsolution 119910 instead of 119909 = 0

Theorem 12 Let 119891 [0 1] times 119864 rarr 119864 be a weakly-weaklycontinuous such that 119891(119905 119909) le 119872 on [0 1] times 119864 and let 119910 bea weak solution of (1) Further let (1) admit two different weaksolutions defined in [0 120572] (120572 lt 119886) Then for every 120576 gt 0 andfor every 119888 isin 119875

lowast

0 there exists 119905 isin [0 120572] such that (14) has at

least two different roots 119906 with 119906 lt 120576

Theorem 13 Let 119891 [0 1] times 119864 rarr 119864 be a weakly-weaklycontinuous such that 119891(119905 119909) le 119872 on [0 1]times119864 and let 119910 be aweak solution of (1) Assume further that there exist 120576 gt 0 119888 isin119875lowast

0 and 119905

0isin (0 1] such that 119906 = 0 is the only root of the auxil-

iary scalar equation (14)with 119906 lt 120576 for every 119905 isin [0 1199050]Then

the (1) admits in the interval [0 1199050] only the weak solution 119910

Remark 14 Another critical point is that as far as we knowin the standard reflexive Banach spaces usually encounteredin differential equations that is 119871119901(Ω) spaces withΩ being ameasurable subset in R119899 and 119901 isin (1infin) the correspondingpositive cones have empty interior Except for the finitedimensional space the only nontrivial example of a positiveconewith nonempty interior is in119862(119870) where119870 is a compacttopological space But119862(119870) is not reflexiveThis motivates usto give the following example

Example 15 We think Sobolev spaces 1198671 and 1198672 are the

required examples Because they are Hilbert spaces 1198671consists of continuous functions and 1198672 consists entirely ofcontinuously differentiable functions See page 382 of [32]


3 Conclusion

We have taken the time interval to be [0 1] and the initialvalue 119909(0) to be 0 only for simplicity of notation our argu-ment would work just as well for any other compact intervaland any other initial value Moreover Szeprsquos assumptions canbe replaced by any set of sufficient conditions that guaranteeexistence of solutions for (1) in reflexive as well as in nonref-lexive Banach spaces

Conflict of Interests

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

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledges with thanks DSR technical and financialsupport

References

[1] G Peano ldquoDemonstration de lrsquointegrabilite des equations differ-entielles ordinairesrdquo Mathematische Annalen vol 37 no 2 pp182ndash228 1890

[2] J Dieudonne ldquoDeux exemples singuliers drsquoequations differ-entiellesrdquoActa ScientiarumMathematicarum vol 12 pp 38ndash401950

[3] A N Godunov ldquoA counterexample to Peanorsquos theorem in aninfinite-dimensional Hilbert spacerdquo Vestnik Moskovskogo Uni-versiteta Serija I Matematika Mehanika vol 27 no 5 pp 31ndash34 1972

[4] E Horst ldquoDifferential equations in Banach spaces five exam-plesrdquo Archiv der Mathematik vol 46 no 5 pp 440ndash444 1986

[5] G E Ladas and V Lakshmikantham Differential Equations inAbstract Spaces Academic Press New York NY USA 1972

[6] J A Yorke ldquoA continuous differential equation in Hilbert spacewithout existencerdquo Funkcialaj Ekvacioj Serio Internacia vol 13pp 19ndash21 1970

[7] A N Godunov ldquoThe Peano theorem in Banach spacesrdquo Func-tional Analysis and Its Applications vol 9 no 1 pp 53ndash55 1975

[8] K Deimling and V Lakshmikantham ldquoOn existence ofextremal solutions of differential equations in Banach spacesrdquoNonlinear AnalysisTheoryMethods and Applications vol 3 no5 pp 563ndash568 1979

[9] E V Teixeira ldquoStrong solutions for differential equations inabstract spacesrdquo Journal of Differential Equations vol 214 no1 pp 65ndash91 2005

[10] V Lakshmikantham ldquoStability and asymptotic behaviour ofsolutions of differential equations in Banach spacerdquo in StabilityProblems vol 65 of CIME Summer Schools pp 38ndash98 2011

[11] V Lakshmikantham and S Leela Nonlinear Differential Equa-tions in Abstract Spaces vol 2 of International Series in Nonlin-ear Mathematics Theory Methods and Applications PergamonPress Oxford UK 1981

[12] R HMartin JrNonlinear Operators andDifferential Equationsin Banach Spaces Robert E KriegerMelbourne VIC Australia1987

[13] S-N Chow and J D Schuur ldquoAn existence theorem forordinary differential equations in Banach spacesrdquo Bulletin of theAmerican Mathematical Society vol 77 pp 1018ndash1020 1971

[14] T Kato ldquoNonlinear semigroups and evolution equationsrdquo Jour-nal of the Mathematical Society of Japan vol 19 pp 508ndash5201967

[15] A Szep ldquoExistence theorem for weak solutions of ordinarydifferential equations in reflexive Banach spacesrdquo Studia Scien-tiarum Mathematicarum Hungarica vol 6 pp 197ndash203 1971

[16] K Astala ldquoOn Peanorsquos theorem in locally convex spacesrdquo StudiaMathematica vol 73 no 3 pp 213ndash223 1982

[17] D Bugajewski ldquoOn the existence of weak solutions of integralequations in Banach spacesrdquo Commentationes MathematicaeUniversitatis Carolinae vol 35 no 1 pp 35ndash41 1994

[18] M Cichon ldquoWeak solutions of differential equations in Banachspacesrdquo Differential Inclusions vol 15 no 1 pp 5ndash14 1995

[19] M Cichon ldquoOn solutions of differential equations in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol60 no 4 pp 651ndash667 2005

[20] E Cramer V Lakshmikantham and A R Mitchell ldquoOn theexistence of weak solutions of differential equations in nonref-lexive Banach spacesrdquo Nonlinear Analysis vol 2 no 2 pp 169ndash177 1978

[21] R P Agarwal andV LakshmikanthamUniqueness andNonuni-queness Criteria for Ordinary Differential Equations WorldScientific 1993

[22] M Arrate ldquoA uniqueness criterion for ordinary differentialequations in Banach spacesrdquo Proceedings of the AmericanMath-ematical Society vol 81 no 3 pp 421ndash424 1981

[23] S Kato ldquoOn existence and uniqueness conditions for nonlinearordinary differential equations in Banach spacesrdquo FunkcialajEkvacioj Serio Internacia vol 19 no 3 pp 239ndash245 1976

[24] M A Krasnoselskiı and S G Kreın ldquoNonlocal existencetheorems and uniqueness theorems for systems of ordinary dif-ferential equationsrdquo Doklady Akademii Nauk SSSR vol 102 pp13ndash16 1955 (Russian)

[25] V Lakshmikantham A R Mitchell and R W MitchellldquoDifferential equations on closed subsets of a Banach spacerdquoTransactions of the AmericanMathematical Society vol 220 pp103ndash113 1976

[26] V Lakshmikantham S Leela and V Moauro ldquoExistence anduniqueness of solutions of delay differential equations on aclosed subset of a Banach spacerdquo Nonlinear Analysis vol 2 no3 pp 311ndash327 1978

[27] R H Martin Jr ldquoDifferential equations on closed subsets ofa Banach spacerdquo Transactions of the American MathematicalSociety vol 179 pp 399ndash414 1973

[28] V BarbuNonlinear Differential Equations of Monotone Types inBanach Spaces Springer Monographs in Mathematics DeLanoScientific LLC 2010

[29] A Majorana ldquoA uniqueness theorem for 1199101015840 = 119891(119909 119910) 119910(1199090) =

1199100rdquo Proceedings of the American Mathematical Society vol 111

no 1 pp 215ndash220 1991[30] E R Hassan ldquoAn extension of the Majorana uniqueness

theorem to Hilbert spacesrdquo Yokohama Mathematical Journalvol 56 no 1-2 pp 1ndash7 2010

[31] S Mazur ldquoUber konvexe mengen in linearen normiertenraumenrdquo Studia Mathematica vol 4 pp 70ndash84 1933

[32] G Chichilnisky and G M Heal ldquoCompetitive equilibrium inSobolev spaces without bounds on short salesrdquo Journal of Eco-nomic Theory vol 59 no 2 pp 364ndash384 1993

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Stochastic AnalysisInternational Journal of


for every 119905 isin [0 119886] Further let Cauchy problem (1) have twodifferent classical solutions defined in 119905 isin [0 120572]Then for every120576 gt 0 there exists 119905 isin (0 120572] such that (2) has at least two dif-ferent roots 119906 with |119906| lt 120576

An immediate consequence of this latter theorem is thefollowing uniqueness criterion

Theorem 3 (see [29]) Let the function 119891(119905 119909) be defined in[0 119886]timesR and continuous with respect to 119905 such that119891(119905 0) = 0for every 119905 isin [0 119886] Further let there exist 120576 gt 0 such that 119906 = 0is the only root of (2)with |119906| lt 120576 for every 119905 isin [0 120572] Then for(1) 119909(119905) = 0 is the only classical solution defined in 119905 isin [0 120572]

Therefore we have inR a very close relation between (2)and (1) It is one of the goals of this work to retain this relationin a suitable generalized sense However Majoranarsquos resultsare not directly extendable to an arbitrary abstract space asthe following example shows

Example 4 Let us consider the Cauchy problem

1199091015840= 119891 (119905 119909) 119909 (0) = 0 (3)

where119891 (119905 119909)

=

(

2

radic119909

(1199091+ 1199092)

2

radic119909

(1199092minus 1199091)) if 119909 = 0

0 if 119909 = 0(4)

In polar coordinates (3) becomes

1199031015840= 2radic119903 120579

1015840=

minus2

radic119903

(5)

Thus besides the trivial solution 119909 = 0 there is at least oneother given by

119909 =

(1199052 cos ln 1

1199052 1199052 sin ln 1

1199052) if 119905 = 0

0 if 119905 = 0(6)

We conclude that (3) satisfies the assumptions ofTheorem 10however one cannot find more than the trivial root to theauxiliary scalar equation ASE corresponding to (3) namely

1199061= 120582 (119906

1+ 1199062) 119906

2= 120582 (119906

2minus 1199061) (7)

where 120582 is an arbitrary scalar

In his paper [30] the author provided a generalization ofMajoranarsquos theorem in finite dimensional Hilbert spaces anatural question arises can we extend this result to infinitelydimensional Banach spaces The answer is positive as will beshown in the following

A way to provide a version of Majoranarsquos uniqueness the-orem in Banach space consists in replacing (2) with a suitableone Our main concern in this work is the classical Cauchyproblem (1) where 119891 takes values in a real Banach space 119864and 119909 and 0 are in 119864

Before starting the main work we will introduce someconceptsThroughout the following sdot stands for the norm in119864 The underlying idea to provide counterpart to (2) is basedon the following definition

Definition 5 (see [11]) Let 119864 be a real Banach space A subset119875 of 119864 is called a cone if the following are true

(i) 119875 is nonempty and nontrivial (ie 119875 contains anonzero point)

(ii) 120582119875 sub 119875 120582 gt 0(iii) 119875 + 119875 sub 119875(iv) 119875 = 119875 where 119875 denotes the closure of 119864(v) 119875 cap minus119875 = 0 where 0 denotes the zero element of the

Banach space 119864Assume that 1198750 = 120601 1198750 denotes the interior of 119864 The cone 119875of 119864 induces an ordering ldquolerdquo by setting

119909 le 119910 lArrrArr 119910 minus 119909 isin 119875

119909 lt 119910 lArrrArr 119910 minus 119909 isin 1198750

(8)

Let 119875lowast be the set of all continuous linear functionals 119888 on 119864such that 119888(119909) ge 0 for all 119909 isin 119875 and let 119875lowast

0be the set of all

continuous linear functionals 119888 on 119864 such that 119888(119909) gt 0 for all119909 isin 119875

0 The underlying idea to provide counterpart to (2) isbased on the following Lemma which is due to Mazur [31]

Lemma 6 (see [11]) Let 119875 be a cone with nonempty interior1198750 then the following hold

(i) 119909 isin 119875 is equivalent to 119888(119909) ge 0 for all 119888 isin 119875lowast(ii) 119909 isin 120597119875 implies that there exists a 119888 isin 119875

lowast

0such that

119888(119909) = 0 where 120597119875 denotes the boundary of 119875

It is well known that the requirements on the function119891 are dependent on types of solutions since we concentrateourselves to weak solutions then the classical case in the sub-ject is that due to Szep [15] Before giving Szeprsquos Theorem weneed the following definition

Definition 7 (see [11]) A function119891(119905 119909) is said to be weakly-weakly continuous at (119904 119910) if given 120576 gt 0 and 120593 isin 119864

lowast thereexist 120575(120576 120593) gt 0 and ℷ(120576 120593) a weakly open set containing 119910such that |120593(119891(119905 119909) minus 119891(119904 119910))| lt 120576 whenever |119905 minus 119904| lt 120575 and119909 isin ℷ

Definition 8 A function 119909(119905) is said to be weakly differen-tiable at 119905

0if there exists a point in 119864 denoted by 1199091015840(119905

0) such

that 120593(1199091015840(1199050)) = (120593119909)

1015840(1199050) for every 120593 isin 119864lowast

Theorem 9 (see [15]) Let 119864 be a reflexive Banach space andlet 119891 be a weakly-weakly continuous function on119860 0 le 119905 le 119886119909 le 119887 Let 119891(119905 119909) le 119872 on119860Then (1) has at least one weaksolution defined on [0 120572] 120572 = min119886 (119887119872)

2 Main Result

This section contains the main results Throughout thissection we will assume that 119864 is a real reflexive Banach space






119888 (119906) = 119905119888 (119891 (119905 119906)) (9)








119860119888(119905) =

119888 (119910)

119905

119905 = 0

0 119905 = 0

(10)



1198601015840

119888(119905) =

1

1199052[119905119888 (119891 (119905 119910 (119905))) minus 119888 (119910)] (11)

Now fix 1199052isin (0 119886) with 119910(119905

2) = 0 such that 119910(119905) lt 120576 for


1= sup119905 isin [0 119905

2] 119860

119888(119905) = 0


1 1199052] At this


1 1199052] such that 1198601015840

119888(119905) = 0 then






(1199051 1199052] then we take 119906 = 119910(119905

2)( = 0) and define

119866 (119905) = 119905119888 (119891 (119905 119910 (1199052))) minus 119888 (119910 (119905

2)) for every 119905 isin [0 119886]

(12)


1 1199052]119866(0) =


2) = 1199052

21198601015840

119888(1199052) gt



1 1199052]

and 119860119888(0) = 0 implies that 1198601015840





lowast

0 and 119905




0] only the zero



1199011015840= 119865 (119905 119901) 119901 (0) = 0 (13)


119888 (119906) = 119905119888 (119865 (119905 119906)) (14)



lowast




0 and 119905








3 Conclusion




Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119888 (119906) = 119905119888 (119891 (119905 119906)) (9)








119860119888(119905) =

119888 (119910)

119905

119905 = 0

0 119905 = 0

(10)



1198601015840

119888(119905) =

1

1199052[119905119888 (119891 (119905 119910 (119905))) minus 119888 (119910)] (11)

Now fix 1199052isin (0 119886) with 119910(119905

2) = 0 such that 119910(119905) lt 120576 for


1= sup119905 isin [0 119905

2] 119860

119888(119905) = 0


1 1199052] At this


1 1199052] such that 1198601015840

119888(119905) = 0 then






(1199051 1199052] then we take 119906 = 119910(119905

2)( = 0) and define

119866 (119905) = 119905119888 (119891 (119905 119910 (1199052))) minus 119888 (119910 (119905

2)) for every 119905 isin [0 119886]

(12)


1 1199052]119866(0) =


2) = 1199052

21198601015840

119888(1199052) gt



1 1199052]

and 119860119888(0) = 0 implies that 1198601015840





lowast

0 and 119905




0] only the zero



1199011015840= 119865 (119905 119901) 119901 (0) = 0 (13)


119888 (119906) = 119905119888 (119865 (119905 119906)) (14)



lowast




0 and 119905








3 Conclusion




Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3 Conclusion




Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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