Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 425784 5 pageshttpdxdoiorg1011552013425784

Research ArticleApproximate Cubic Lie Derivations

Ajda Fošner1 and Maja Fošner2

1 Faculty of Management University of Primorska Cankarjeva 5 6104 Koper Slovenia2 Faculty of Logistics University of Maribor Mariborska Cesta 7 3000 Celje Slovenia

Correspondence should be addressed to Ajda Fosner ajdafosnerfm-kpsi

Received 11 April 2013 Accepted 27 June 2013

Academic Editor Janusz Brzdek

Copyright copy 2013 A Fosner and M Fosner 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 study the stability and hyperstability of cubic Lie derivations on normed algebras At the end we write some additionalobservations about our results

1 Introduction

A classical question in the theory of functional equations isunder what conditions is it true that a mapping which approxi-mately satisfies a functional equationEmust be somehow closeto an exact solution of E We say that a functional equationE is stable if any approximate solution of E is near to a truesolution of E The study of stability problem for functionalequations is related to a question of Ulam [1] concerningthe stability of group homomorphisms This question wasaffirmatively answered for Banach spaces by Hyers [2] Sub-sequently the result of Hyers was generalized by Aoki [3] foradditive mappings and by Rassias [4] for linear mappings byconsidering an unbounded Cauchy difference Rassiasrsquo paperhas provided a lot of influence in the development of whatwe now call the Hyers-Ulam-Rassias stability of functionalequations For further information about the topic we referthe reader to [5 6]

Jun and Kim [7] introduced the following functionalequation

119891 (2119909 + 119910) + 119891 (2119909 minus 119910) = 2119891 (119909 + 119910) + 2119891 (119909 minus 119910)

+ 12119891 (119909) (1)

and established a general solution for it Note that if wereplace 119909 = 119910 = 0 in (1) we get 119891(0) = 0 Therefore setting119909 = 0 in (1) we obtain 119891(minus119910) = minus119891(119910) Moreover writing119910 = 0 in (1) we get 119891(2119909) = 8119891(119909) Now it is easy to see thata function 119891(119909) = 1205821199093 is a solution of (1) where 120582 is a fixedscalar Thus it is natural that (1) is called a cubic functional

equation Moreover every solution of the cubic functionalequation is said to be a cubicmapping In the last few decadesa number of mathematicians worked on the stability of sometype of cubic functional equations (see eg [8ndash13])

In [14] the authors investigated the stability of cubicderivations in a connection with the functional equation (1)Recently Yang et al [15] studied the stability of cubic lowast-deri-vations on Banach lowast-algebras and in [16] the authors provedthe generalizedHyers-Ulam-Rassias stability of ternary cubicderivations on ternary Frechet algebras Motivated by theseresults we investigate the stability of cubic Lie derivations Inparticular we show that such derivations can be generated byfunctions which satisfy some quite natural and simple condi-tions

2 Stability of Cubic Lie Derivations

Before stating our first theorem let us recall some basicdefinitions and known results which we will use in the sequelThroughout the paper A will be a complex normed algebraand M a Banach A-bimodule We will use the same symbol sdot in order to represent the norms on a normed algebra Aand a normed A-bimodule M For all 119909 119910 isin A the symbol[119909 119910] will denote the commutator 119909119910 minus 119910119909 We say that amapping 119891 A rarr M is cubic homogeneous if 119891(120582119909) =1205823119891(119909) for all 119909 isin A and all scalars 120582 isin C In the followingΛ will stand for the set of all complex units that is

Λ = 120582 isin C |120582| = 1 (2)

2 Abstract and Applied Analysis

A cubic homogeneous mapping 119889 A rarr M is called acubic derivation if

119889 (119909119910) = 119889 (119909) 1199103 + 1199093119889 (119910) (3)

holds for all 119909 119910 isin A Following this notion we introducea cubic Lie derivation as a cubic homogeneous mapping 119897 A rarr M satisfying

119897 ([119909 119910]) = [119897 (119909) 1199103] + [1199093 119897 (119910)] (4)

for all 119909 119910 isin AFor a given map 119891 A rarr M we consider

Δ120582119891(119909 119910) = 119891 (2120582119909 + 120582119910) + 119891 (2120582119909 minus 120582119910) minus 21205823119891 (119909 + 119910)

minus 21205823119891 (119909 minus 119910) minus 121205823119891 (119909)

(5)

where 119909 119910 isin A 120582 isin C and

Δ119891(119909 119910) = 119891 ([119909 119910]) minus [119891 (119909) 119910

3]

minus [1199093 119891 (119910)] 119909 119910 isin A(6)

Moreover for a function 120593 A times A rarr [0infin) we use thefollowing abbreviation

120601 (119909 119910) =infin

sum119896=0

120593 (2119896119909 2119896119910)

8119896 119909 119910 isin A (7)

Theorem 1 Suppose that 119897 A rarr M is a mapping for whichthere exists a function 120593 A timesA rarr [0infin) such that

120601 (119909 119910) lt infin 119909 119910 isin A (8)10038171003817100381710038171003817Δ120582

119897(119909 119910)

10038171003817100381710038171003817 le 120593 (119909 119910) 1003817100381710038171003817Δ 119897 (119909 119910)

1003817100381710038171003817 le 120593 (119909 119910) (9)

for all 119909 119910 isin A and 120582 isin Λ If for each fixed 119909 isin A themapping 119903 997891rarr 119897(119903119909) from R to M is continuous then thereexists a unique cubic Lie derivation 119871 A rarr M such that

119897 (119909) minus 119871 (119909) le120601 (119909 0)

16(10)

for all 119909 isin A

Proof Writing 119910 = 0 and 120582 = 1 in the first inequality of (9)we obtain

10038171003817100381710038171003817100381710038171003817

119897 (2119909)

8minus 119897 (119909)

10038171003817100381710038171003817100381710038171003817le120593 (119909 0)

16(11)

for all 119909 isin A Using the induction it is easy to see that

100381710038171003817100381710038171003817100381710038171003817

119897 (2119901119909)

8119901minus119897 (2119902119909)

8119902

100381710038171003817100381710038171003817100381710038171003817le

1

16

119901minus1

sum119896=119902

120593 (2119896119909 0)

8119896(12)

for all 119909 isin A and all 119901 gt 119902 ge 0 By (8) it follows that for all119909 isin A the sequence 119897(2119899119909)8119899infin

119899=0is Cauchy and since M

is complete it is convergent Thus we can define a mapping119871 A rarr M as

119871 (119909) = lim119899rarrinfin

119897 (2119899119909)

8119899 119909 isin A (13)

Writing 119901 = 119899 and 119902 = 0 in (12) we have

100381710038171003817100381710038171003817100381710038171003817

119897 (2119899119909)

8119899minus 119897 (119909)

100381710038171003817100381710038171003817100381710038171003817le

1

16

119899minus1

sum119896=0

120593 (2119896119909 0)

8119896(14)

for 119899 gt 0 Taking the limit as 119899 tends toinfin we get (10)In the following we will show that 119871 is a unique cubic Lie

derivation such that (10) holds true for all 119909 isin A

Claim 1 (119871 is a cubic mapping) Recall that

10038171003817100381710038171003817Δ120582

119871(119909 119910)

10038171003817100381710038171003817 = lim119899rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

Δ120582119897(2119899119909 2119899119910)

8119899

1003817100381710038171003817100381710038171003817100381710038171003817

le lim119899rarrinfin

120593 (2119899119909 2119899119910)

8119899= 0

(15)

for all 119909 119910 isin A and 120582 isin Λ Hence taking 120582 = 1 in (15) itfollows that 119871 is a cubic mapping

Claim 2 (119871 is cubic homogeneous) By (15) we have Δ120582119871(119909 0) =

0 Thus

119871 (2120582119909) = 81205823119871 (119909) 119909 isin A 120582 isin Λ (16)

Let 1199090isin A be any fixed element Firstly we will show that

119871(1199031199090) = 1199033119871(119909

0) for all 119903 isin R We will omit the details since

the arguments are the same as in [17] (see also [18]) So let uschoose any linear functional 120588 isin Mlowast whereMlowast denotes thedual space ofM Then we can define a function 120595 R rarr R

by

120595 (119903) = 120588 (119871 (1199031199090)) 119903 isin R (17)

It is easy to see that 120595 is cubic Set

120595119899 (119903) = 120588(

119897 (21198991199031199090)

8119899) 119899 ge 0 (18)

Note that 120595 is a pointwise limit of a sequence of continuousfunctions 120595

119899 119899 ge 0 Thus 120595 is a continuous cubic function

and

120595 (119903) = 1199033120595 (1) 119903 isin R (19)

Consequently

120588 (119871 (1199031199090)) = 120595 (119903) = 119903

3120595 (1) = 1199033120588 (119871 (119909

0))

= 120588 (1199033119871 (1199090))

(20)

for all 119903 isin R Since 120588 isin Mlowast was an arbitrary linear functionalwe conclude that

119871 (1199031199090) = 1199033119871 (119909

0) 119903 isin R (21)

Abstract and Applied Analysis 3

Let 120582 isin C Then 120582|120582| isin Λ |120582|2 isin R and we have

119871 (1205821199090) = 119871(2

120582

|120582|(|120582|

21199090)) = 8(

120582

|120582|)3

119871(|120582|

21199090)

= 81205823

|120582|3(|120582|

2)3

119871 (1199090) = 1205823119871 (119909

0)

(22)

Since 1199090was any element fromA we conclude that 119871 is cubic

homogeneous

Claim 3 (119871 is a cubic Lie derivation) Using the second ine-quality in (9) we have

1003817100381710038171003817Δ 119871 (119909 119910)1003817100381710038171003817 = lim119899rarrinfin

100381710038171003817100381710038171003817100381710038171003817

Δ119897(2119899119909 2119899119910)

82119899

100381710038171003817100381710038171003817100381710038171003817

le lim119899rarrinfin

120593 (2119899119909 2119899119910)

82119899= 0

(23)

for all 119909 119910 isin A This together with Claim 2 yields that 119871 is acubic Lie derivation

Claim 4 (119871 is unique) Suppose that there exists another cubicLie derivation A rarr M such that

10038171003817100381710038171003817119897 (119909) minus (119909)10038171003817100381710038171003817 le

120601 (119909 0)

16(24)

for all 119909 isin A Then10038171003817100381710038171003817119871 (119909) minus (119909)

10038171003817100381710038171003817 = lim119899rarrinfin

1

811989910038171003817100381710038171003817119897 (2119899119909) minus (2119899119909)

10038171003817100381710038171003817

le lim119899rarrinfin

1

16(120601 (2119899119909 0)

8119899)

= lim119899rarrinfin

1

16

infin

sum119896=0

120593 (2119896+119899119909 0)

8119896+119899

= lim119899rarrinfin

1

16

infin

sum119896=119899

120593 (2119896119909 0)

8119896= 0

(25)

Therefore 119871(119909) = (119909) for all 119909 isin A which proves theuniqueness of 119871 The proof is completed

Let ] 120576 ge 0 and 0 le 119901 lt 3 Applying Theorem 1 for thecase

120593 (119909 119910) = ] + 120576 (119909119901 + 10038171003817100381710038171199101003817100381710038171003817119901) 119909 119910 isin A (26)

we have the next corollary

Corollary 2 Suppose that 119897 A rarr M is a mapping such that(9) holds true for all 119909 119910 isin A and 120582 isin Λ where 120593 is a functiondefined as above If for each fixed119909 isin A themapping 119903 997891rarr 119897(119903119909)fromR toM is continuous then there exists a unique cubic Liederivation 119871 A rarr M such that

119897 (119909) minus 119871 (119909) le]

14+

120576119909119901

(16 minus 2119901+1)(27)

for all 119909 isin A

Proof Note that we have

120601 (119909 119910) =8]

7+8120576 (119909119901 +

10038171003817100381710038171199101003817100381710038171003817119901)

(8 minus 2119901) 119909 119910 isin A (28)

3 Hyperstability of Cubic Lie Derivations

The investigation of the multiplicative Cauchy functionalequation highlighted a new phenomenon which is nowadayscalled superstability (see eg [19]) In this case the so-called stability inequality implies that the observed functionis either bounded or it is a solution of the functional equationBut it can also happen that each function 119891 satisfying thefunctional equation E approximately must actually be asolution of the proposed equationE In this case we say thatthe functional equation E is hyperstable According to ourbest knowledge the first hyperstability result was published in[20] and concerned the ring homomorphisms However theterm hyperstability has been used for the first time probablyin [21] (see also [22 23]) Formore information about the newresults on the hyperstability we refer the reader to [24ndash26]

Theorem 3 Let A be a normed algebra with an element 119890which is not a zero divisor Suppose that 119897 A rarr A is amapping for which there exists a function 120593 AtimesA rarr [0infin)such that

lim119899rarrinfin

120593 (119899119909 119910)

1198993= 0 (29)

10038171003817100381710038171003817119897 (119909) 1199103 minus 1199093119897 (119910)

10038171003817100381710038171003817 le 120593 (119909 119910) (30)

10038171003817100381710038171003817119897 (119911) ([119909 119910])3minus 1199113 ([119897 (119909) 119910

3] + [1199093 119897 (119910)])10038171003817100381710038171003817 le 120593 (119911 [119909 119910])

(31)

for all 119909 119910 119911 isin A Then 119897 is a cubic Lie derivation onA

Proof We divide the proof into several steps

Claim 1 (119897 is a cubic mapping) Let 119909 119910 119911 isin A Then

1003817100381710038171003817100381711989931199113 (119897 (2119909 + 119910) + 119897 (2119909 minus 119910) minus 2119897 (119909 + 119910)

minus2119897 (119909 minus 119910) minus 12119897 (119909))10038171003817100381710038171003817

le1003817100381710038171003817100381711989931199113119897 (2119909 + 119910) minus 119897 (119899119911) (2119909 + 119910)

310038171003817100381710038171003817

+1003817100381710038171003817100381711989931199113119897 (2119909 minus 119910) minus 119897 (119899119911) (2119909 minus 119910)

310038171003817100381710038171003817

+ 210038171003817100381710038171003817119897 (119899119911) (119909 + 119910)

3minus 11989931199113119897 (119909 + 119910)

10038171003817100381710038171003817

+ 210038171003817100381710038171003817119897 (119899119911) (119909 minus 119910)

3minus 11989931199113119897 (119909 minus 119910)

10038171003817100381710038171003817

+ 1210038171003817100381710038171003817119897 (119899119911) 119909

3 minus 11989931199113119897 (119909)10038171003817100381710038171003817

(32)

4 Abstract and Applied Analysis

for all positive integers 119899 By (30) it follows that100381710038171003817100381710038171199113 (119897 (2119909 + 119910) + 119897 (2119909 minus 119910) minus 2119897 (119909 + 119910)

minus2119897 (119909 minus 119910) minus 12119897 (119909))10038171003817100381710038171003817

le 119899minus3 (120593 (119899119911 2119909 + 119910) + 120593 (119899119911 2119909 minus 119910)

+ 2120593 (119899119911 119909 + 119910) + 2120593 (119899119911 119909 minus 119910)

+ 12120593 (119899119911 119909))

(33)

If we let 119899 rarr infin we conclude that

1199113 (119897 (2119909 + 119910) + 119897 (2119909 minus 119910) minus 2119897 (119909 + 119910)

minus2119897 (119909 minus 119910) minus 12119897 (119909)) = 0(34)

for all 119909 119910 119911 isin A Putting 119911 = 119890 in the last equality weconclude that 119897 satisfies a cubic functional equation (1)

Claim 2 (119897 is cubic homogeneous) Let 120582 isin C and 119909 119910 isin AThen1003817100381710038171003817100381711989931199103 (119897 (120582119909) minus 120582

3119897 (119909))10038171003817100381710038171003817 le

1003817100381710038171003817100381711989931199103119897 (120582119909) minus 119897 (119899119910) 120582

3119909310038171003817100381710038171003817

+ |120582|3 10038171003817100381710038171003817119897 (119899119910) 119909

3 minus 11989931199103119897 (119909)10038171003817100381710038171003817

le 120593 (119899119910 120582119909) + |120582|3120593 (119899119910 119909)

(35)

for all positive integers 119899 Thus100381710038171003817100381710038171199103 (119897 (120582119909) minus 120582

3119897 (119909))10038171003817100381710038171003817 le 119899minus3 (120593 (119899119910 120582119909) + |120582|

3120593 (119899119910 119909))

(36)

Taking the limit when 119899 tends toinfin we conclude that

1199103 (119897 (120582119909) minus 1205823119897 (119909)) = 0 119909 119910 isin A 120582 isin C (37)

Writing 119910 = 119890 we get 119897(120582119909) = 1205823119897(119909) for all 119909 isin A and 120582 isin CIn other words 119897 is cubic homogeneous

Claim 3 (119897 is a cubic Lie derivation onA) Let119909 119910 119911 isin AThen1003817100381710038171003817100381711989931199113 (119897 ([119909 119910]) minus [119897 (119909) 119910

3] minus [1199093 119897 (119910)])10038171003817100381710038171003817

le1003817100381710038171003817100381711989931199113119897 ([119909 119910]) minus 119897 (119899119911) ([119909 119910])

310038171003817100381710038171003817

+10038171003817100381710038171003817119897 (119899119911) ([119909 119910])

3minus 11989931199113 ([119897 (119909) 119910

3] + [1199093 119897 (119910)])10038171003817100381710038171003817

le 2120593 (119899119911 [119909 119910])

(38)

for all positive integers 119899 Thus100381710038171003817100381710038171199113 (119897 ([119909 119910]) minus [119897 (119909) 119910

3] minus [1199093 119897 (119910)])10038171003817100381710038171003817

le 119899minus32120593 (119899119911 [119909 119910])(39)

for all 119909 119910 119911 isin 119860 and 119899 isin N Similarly as in Claims 1 and 2 wecan show that 119897([119909 119910]) = [119897(119909) 1199103]+[1199093 119897(119910)] for all 119909 119910 isin AThe proof is completed

Let 120576 ge 0 and 0 le 119901 lt 3 ApplyingTheorem 3 for the case

120593 (119909 119910) = 1205761199091199011003817100381710038171003817119910

1003817100381710038171003817119901 119909 119910 isin A (40)

we have the next corollary

Corollary 4 Let A be a normed algebra with an element 119890which is not a zero divisor Suppose that 119897 A rarr A is amapping such that

10038171003817100381710038171003817119897 (119909) 1199103 minus 1199093119897 (119910)

10038171003817100381710038171003817 le 1205761199091199011003817100381710038171003817119910

1003817100381710038171003817119901

10038171003817100381710038171003817119897 (119911) [119909 119910]3minus 1199113 ([119897 (119909) 119910

3] + [1199093 119897 (119910)])10038171003817100381710038171003817

le 1205761199111199011003817100381710038171003817[119909 119910]

1003817100381710038171003817119901

(41)

for all 119909 119910 119911 isin A Then 119897 is a cubic Lie derivation onA

4 Some Additional Remarks

In this section we will write some additional observationsabout our results We start with the theorem which isanalogue toTheorem 1 Since the proof is similar it is omitted

Theorem 5 Suppose that 119897 A rarr M is a mapping for whichthere exists a function 120593 AtimesA rarr [0infin) such that (9) holdstrue for all 119909 119910 isin A 120582 isin Λ and

infin

sum119896=0

8119896120593(119909

2119896119910

2119896) lt infin 119909 119910 isin A (42)

If for each fixed 119909 isin A the mapping 119903 997891rarr 119897(119903119909) from R to Mis continuous then there exists a unique cubic Lie derivation119871 A rarr M such that (10) is valid for all 119909 isin A

Similarly the next result is analogue to Theorem 3

Theorem 6 Let A be a normed algebra with an element 119890which is not a zero divisor Suppose that 119897 A rarr A is amapping for which there exists a function 120593 AtimesA rarr [0infin)such that (30) and (31) hold true for all 119909 119910 119911 isin A and

lim119899rarrinfin

1198993120593(119909

119899 119910) = 0 119909 119910 isin A (43)

Then 119897 is a cubic Lie derivation onA

Let us end this paper with a remark on 119896-cubic functionalequations In [27] Najati introduced the following functionalequations

119891 (119896119909 + 119910) + 119891 (119896119909 minus 119910) = 119896119891 (119909 + 119910) + 119896119891 (119909 minus 119910)

+ 2 (1198963 minus 119896)119891 (119909) (44)

where 119896 ge 2 is a positive integer For 119896 = 2 we obtain(1) Najati proved that a function 119891 satisfies the functionalequation (1) if and only if 119891 satisfies the functional equation(44)Therefore every solution of the functional equation (44)is also a cubic function Following these results it is easy tosee that we can consider (44) in our theorems instead of (1)

Abstract and Applied Analysis 5

Acknowledgment

The authors would like to thank the referees for their usefulcomments

References

[1] S M Ulam Problems in Modern Mathematics John Wiley ampSons New York NY USA 1964

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T Aoki ldquoOn the stability of the linear transformation in Banachspacesrdquo Journal of the Mathematical Society of Japan vol 2 pp64ndash66 1950

[4] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[5] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[6] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[7] K-W Jun and H-M Kim ldquoThe generalized Hyers-Ulam-Rassias stability of a cubic functional equationrdquo Journal ofMathematical Analysis and Applications vol 274 no 2 pp 267ndash278 2002

[8] A Bodaghi I A Alias andMH Ghahramani ldquoApproximatelycubic functional equations and cubic multipliersrdquo Journal ofInequalities and Applications vol 2011 articlie 53 2011

[9] M E Gordji S K Gharetapeh J M Rassias and S ZolfagharildquoSolution and stability of a mixed type additive quadratic andcubic functional equationrdquo Advances in Difference Equationsvol 2009 pp 1ndash17 2009

[10] M Eshaghi Gordji S Zolfaghari J M Rassias and M BSavadkouhi ldquoSolution and stability of a mixed type cubic andquartic functional equation in quasi-Banach spacesrdquo Abstractand Applied Analysis vol 2009 Article ID 417473 14 pages2009

[11] K-W Jun H-M Kim and I-S Chang ldquoOn the Hyers-Ulamstability of an Euler-Lagrange type cubic functional equationrdquoJournal of Computational Analysis and Applications vol 7 no 1pp 21ndash33 2005

[12] A Najati ldquoHyers-Ulam-Rassias stability of a cubic functionalequationrdquo Bulletin of the Korean Mathematical Society vol 44no 4 pp 825ndash840 2007

[13] T Z Xu J M Rassias and W X Xu ldquoA generalized mixedtype of quartic-cubic-quadratic-additive functional equationsrdquoUkrainian Mathematical Journal vol 63 no 3 pp 461ndash4792011

[14] M Eshaghi Gordji S Kaboli Gharetapeh M B SavadkouhiM Aghaei and T Karimi ldquoOn cubic derivationsrdquo InternationalJournal of Mathematical Analysis vol 4 no 49ndash52 pp 2501ndash2514 2010

[15] S Y Yang A Bodaghi and K A M Atan ldquoApproximate cubiclowast-derivations on Banach lowast-algebrasrdquo Abstract and AppliedAnalysis vol 2012 Article ID 684179 12 pages 2012

[16] B Hayati M E Gordji M B Savadkouhi and M BidkhamldquoStability of ternary cubic derivations on ternary Frechet alge-brasrdquoAustralian Journal of Basic and Applied Sciences vol 5 pp1224ndash1235 2011

[17] St Czerwik ldquoOn the stability of the quadratic mapping innormed spacesrdquo Abhandlungen aus demMathematischen Semi-nar der Universitat Hamburg vol 62 pp 59ndash64 1992

[18] S Kurepa ldquoOn the quadratic functionalrdquo Publications delrsquoInstitut Mathematique de lrsquoAcademie Serbe des Sciences vol 13pp 57ndash72 1961

[19] Z Moszner ldquoSur les definitions differentes de la stabilite desequations fonctionnellesrdquo Aequationes Mathematicae vol 68no 3 pp 260ndash274 2004

[20] D G Bourgin ldquoApproximately isometric and multiplicativetransformations on continuous function ringsrdquo Duke Mathe-matical Journal vol 16 pp 385ndash397 1949

[21] G Maksa and Z Pales ldquoHyperstability of a class of linear func-tional equationsrdquo Acta Mathematica Academiae PaedagogicaeNyıregyhaziensis vol 17 no 2 pp 107ndash112 2001

[22] E Gselmann ldquoHyperstability of a functional equationrdquo ActaMathematica Hungarica vol 124 no 1-2 pp 179ndash188 2009

[23] Gy Maksa K Nikodem and Zs Pales ldquoResults on 119905-Wrightconvexityrdquo LrsquoAcademie des Sciences Comptes RendusMathemat-iques vol 13 no 6 pp 274ndash278 1991

[24] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[25] J Brzdek ldquoHyperstability of the Cauchy equation on restricteddomainsrdquo Acta Mathematica Hungarica 2013

[26] M Piszczek ldquoRemark on hyperstability of the general linearequationrdquo Aequationes Mathematicae 2013

[27] A Najati ldquoThe generalized Hyers-Ulam-Rassias stability of acubic functional equationrdquo Turkish Journal of Mathematics vol31 no 4 pp 395ndash408 2007

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Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

A cubic homogeneous mapping 119889 A rarr M is called acubic derivation if

119889 (119909119910) = 119889 (119909) 1199103 + 1199093119889 (119910) (3)

holds for all 119909 119910 isin A Following this notion we introducea cubic Lie derivation as a cubic homogeneous mapping 119897 A rarr M satisfying

119897 ([119909 119910]) = [119897 (119909) 1199103] + [1199093 119897 (119910)] (4)

for all 119909 119910 isin AFor a given map 119891 A rarr M we consider

Δ120582119891(119909 119910) = 119891 (2120582119909 + 120582119910) + 119891 (2120582119909 minus 120582119910) minus 21205823119891 (119909 + 119910)

minus 21205823119891 (119909 minus 119910) minus 121205823119891 (119909)

(5)

where 119909 119910 isin A 120582 isin C and

Δ119891(119909 119910) = 119891 ([119909 119910]) minus [119891 (119909) 119910

3]

minus [1199093 119891 (119910)] 119909 119910 isin A(6)

Moreover for a function 120593 A times A rarr [0infin) we use thefollowing abbreviation

120601 (119909 119910) =infin

sum119896=0

120593 (2119896119909 2119896119910)

8119896 119909 119910 isin A (7)

Theorem 1 Suppose that 119897 A rarr M is a mapping for whichthere exists a function 120593 A timesA rarr [0infin) such that

120601 (119909 119910) lt infin 119909 119910 isin A (8)10038171003817100381710038171003817Δ120582

119897(119909 119910)

10038171003817100381710038171003817 le 120593 (119909 119910) 1003817100381710038171003817Δ 119897 (119909 119910)

1003817100381710038171003817 le 120593 (119909 119910) (9)

for all 119909 119910 isin A and 120582 isin Λ If for each fixed 119909 isin A themapping 119903 997891rarr 119897(119903119909) from R to M is continuous then thereexists a unique cubic Lie derivation 119871 A rarr M such that

119897 (119909) minus 119871 (119909) le120601 (119909 0)

16(10)

for all 119909 isin A

Proof Writing 119910 = 0 and 120582 = 1 in the first inequality of (9)we obtain

10038171003817100381710038171003817100381710038171003817

119897 (2119909)

8minus 119897 (119909)

10038171003817100381710038171003817100381710038171003817le120593 (119909 0)

16(11)

for all 119909 isin A Using the induction it is easy to see that

100381710038171003817100381710038171003817100381710038171003817

119897 (2119901119909)

8119901minus119897 (2119902119909)

8119902

100381710038171003817100381710038171003817100381710038171003817le

1

16

119901minus1

sum119896=119902

120593 (2119896119909 0)

8119896(12)

for all 119909 isin A and all 119901 gt 119902 ge 0 By (8) it follows that for all119909 isin A the sequence 119897(2119899119909)8119899infin

119899=0is Cauchy and since M

is complete it is convergent Thus we can define a mapping119871 A rarr M as

119871 (119909) = lim119899rarrinfin

119897 (2119899119909)

8119899 119909 isin A (13)

Writing 119901 = 119899 and 119902 = 0 in (12) we have

100381710038171003817100381710038171003817100381710038171003817

119897 (2119899119909)

8119899minus 119897 (119909)

100381710038171003817100381710038171003817100381710038171003817le

1

16

119899minus1

sum119896=0

120593 (2119896119909 0)

8119896(14)

for 119899 gt 0 Taking the limit as 119899 tends toinfin we get (10)In the following we will show that 119871 is a unique cubic Lie

derivation such that (10) holds true for all 119909 isin A

Claim 1 (119871 is a cubic mapping) Recall that

10038171003817100381710038171003817Δ120582

119871(119909 119910)

10038171003817100381710038171003817 = lim119899rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

Δ120582119897(2119899119909 2119899119910)

8119899

1003817100381710038171003817100381710038171003817100381710038171003817

le lim119899rarrinfin

120593 (2119899119909 2119899119910)

8119899= 0

(15)

for all 119909 119910 isin A and 120582 isin Λ Hence taking 120582 = 1 in (15) itfollows that 119871 is a cubic mapping

Claim 2 (119871 is cubic homogeneous) By (15) we have Δ120582119871(119909 0) =

0 Thus

119871 (2120582119909) = 81205823119871 (119909) 119909 isin A 120582 isin Λ (16)

Let 1199090isin A be any fixed element Firstly we will show that

119871(1199031199090) = 1199033119871(119909

0) for all 119903 isin R We will omit the details since

the arguments are the same as in [17] (see also [18]) So let uschoose any linear functional 120588 isin Mlowast whereMlowast denotes thedual space ofM Then we can define a function 120595 R rarr R

by

120595 (119903) = 120588 (119871 (1199031199090)) 119903 isin R (17)

It is easy to see that 120595 is cubic Set

120595119899 (119903) = 120588(

119897 (21198991199031199090)

8119899) 119899 ge 0 (18)

Note that 120595 is a pointwise limit of a sequence of continuousfunctions 120595

119899 119899 ge 0 Thus 120595 is a continuous cubic function

and

120595 (119903) = 1199033120595 (1) 119903 isin R (19)

Consequently

120588 (119871 (1199031199090)) = 120595 (119903) = 119903

3120595 (1) = 1199033120588 (119871 (119909

0))

= 120588 (1199033119871 (1199090))

(20)

for all 119903 isin R Since 120588 isin Mlowast was an arbitrary linear functionalwe conclude that

119871 (1199031199090) = 1199033119871 (119909

0) 119903 isin R (21)

Abstract and Applied Analysis 3

Let 120582 isin C Then 120582|120582| isin Λ |120582|2 isin R and we have

119871 (1205821199090) = 119871(2

120582

|120582|(|120582|

21199090)) = 8(

120582

|120582|)3

119871(|120582|

21199090)

= 81205823

|120582|3(|120582|

2)3

119871 (1199090) = 1205823119871 (119909

0)

(22)

Since 1199090was any element fromA we conclude that 119871 is cubic

homogeneous

Claim 3 (119871 is a cubic Lie derivation) Using the second ine-quality in (9) we have

1003817100381710038171003817Δ 119871 (119909 119910)1003817100381710038171003817 = lim119899rarrinfin

100381710038171003817100381710038171003817100381710038171003817

Δ119897(2119899119909 2119899119910)

82119899

100381710038171003817100381710038171003817100381710038171003817

le lim119899rarrinfin

120593 (2119899119909 2119899119910)

82119899= 0

(23)

for all 119909 119910 isin A This together with Claim 2 yields that 119871 is acubic Lie derivation

Claim 4 (119871 is unique) Suppose that there exists another cubicLie derivation A rarr M such that

10038171003817100381710038171003817119897 (119909) minus (119909)10038171003817100381710038171003817 le

120601 (119909 0)

16(24)

for all 119909 isin A Then10038171003817100381710038171003817119871 (119909) minus (119909)

10038171003817100381710038171003817 = lim119899rarrinfin

1

811989910038171003817100381710038171003817119897 (2119899119909) minus (2119899119909)

10038171003817100381710038171003817

le lim119899rarrinfin

1

16(120601 (2119899119909 0)

8119899)

= lim119899rarrinfin

1

16

infin

sum119896=0

120593 (2119896+119899119909 0)

8119896+119899

= lim119899rarrinfin

1

16

infin

sum119896=119899

120593 (2119896119909 0)

8119896= 0

(25)

Therefore 119871(119909) = (119909) for all 119909 isin A which proves theuniqueness of 119871 The proof is completed

Let ] 120576 ge 0 and 0 le 119901 lt 3 Applying Theorem 1 for thecase

120593 (119909 119910) = ] + 120576 (119909119901 + 10038171003817100381710038171199101003817100381710038171003817119901) 119909 119910 isin A (26)

we have the next corollary

Corollary 2 Suppose that 119897 A rarr M is a mapping such that(9) holds true for all 119909 119910 isin A and 120582 isin Λ where 120593 is a functiondefined as above If for each fixed119909 isin A themapping 119903 997891rarr 119897(119903119909)fromR toM is continuous then there exists a unique cubic Liederivation 119871 A rarr M such that

119897 (119909) minus 119871 (119909) le]

14+

120576119909119901

(16 minus 2119901+1)(27)

for all 119909 isin A

Proof Note that we have

120601 (119909 119910) =8]

7+8120576 (119909119901 +

10038171003817100381710038171199101003817100381710038171003817119901)

(8 minus 2119901) 119909 119910 isin A (28)

3 Hyperstability of Cubic Lie Derivations

The investigation of the multiplicative Cauchy functionalequation highlighted a new phenomenon which is nowadayscalled superstability (see eg [19]) In this case the so-called stability inequality implies that the observed functionis either bounded or it is a solution of the functional equationBut it can also happen that each function 119891 satisfying thefunctional equation E approximately must actually be asolution of the proposed equationE In this case we say thatthe functional equation E is hyperstable According to ourbest knowledge the first hyperstability result was published in[20] and concerned the ring homomorphisms However theterm hyperstability has been used for the first time probablyin [21] (see also [22 23]) Formore information about the newresults on the hyperstability we refer the reader to [24ndash26]

Theorem 3 Let A be a normed algebra with an element 119890which is not a zero divisor Suppose that 119897 A rarr A is amapping for which there exists a function 120593 AtimesA rarr [0infin)such that

lim119899rarrinfin

120593 (119899119909 119910)

1198993= 0 (29)

10038171003817100381710038171003817119897 (119909) 1199103 minus 1199093119897 (119910)

10038171003817100381710038171003817 le 120593 (119909 119910) (30)

10038171003817100381710038171003817119897 (119911) ([119909 119910])3minus 1199113 ([119897 (119909) 119910

3] + [1199093 119897 (119910)])10038171003817100381710038171003817 le 120593 (119911 [119909 119910])

(31)

for all 119909 119910 119911 isin A Then 119897 is a cubic Lie derivation onA

Proof We divide the proof into several steps

Claim 1 (119897 is a cubic mapping) Let 119909 119910 119911 isin A Then

1003817100381710038171003817100381711989931199113 (119897 (2119909 + 119910) + 119897 (2119909 minus 119910) minus 2119897 (119909 + 119910)

minus2119897 (119909 minus 119910) minus 12119897 (119909))10038171003817100381710038171003817

le1003817100381710038171003817100381711989931199113119897 (2119909 + 119910) minus 119897 (119899119911) (2119909 + 119910)

310038171003817100381710038171003817

+1003817100381710038171003817100381711989931199113119897 (2119909 minus 119910) minus 119897 (119899119911) (2119909 minus 119910)

310038171003817100381710038171003817

+ 210038171003817100381710038171003817119897 (119899119911) (119909 + 119910)

3minus 11989931199113119897 (119909 + 119910)

10038171003817100381710038171003817

+ 210038171003817100381710038171003817119897 (119899119911) (119909 minus 119910)

3minus 11989931199113119897 (119909 minus 119910)

10038171003817100381710038171003817

+ 1210038171003817100381710038171003817119897 (119899119911) 119909

3 minus 11989931199113119897 (119909)10038171003817100381710038171003817

(32)

4 Abstract and Applied Analysis

for all positive integers 119899 By (30) it follows that100381710038171003817100381710038171199113 (119897 (2119909 + 119910) + 119897 (2119909 minus 119910) minus 2119897 (119909 + 119910)

minus2119897 (119909 minus 119910) minus 12119897 (119909))10038171003817100381710038171003817

le 119899minus3 (120593 (119899119911 2119909 + 119910) + 120593 (119899119911 2119909 minus 119910)

+ 2120593 (119899119911 119909 + 119910) + 2120593 (119899119911 119909 minus 119910)

+ 12120593 (119899119911 119909))

(33)

If we let 119899 rarr infin we conclude that

1199113 (119897 (2119909 + 119910) + 119897 (2119909 minus 119910) minus 2119897 (119909 + 119910)

minus2119897 (119909 minus 119910) minus 12119897 (119909)) = 0(34)

for all 119909 119910 119911 isin A Putting 119911 = 119890 in the last equality weconclude that 119897 satisfies a cubic functional equation (1)

Claim 2 (119897 is cubic homogeneous) Let 120582 isin C and 119909 119910 isin AThen1003817100381710038171003817100381711989931199103 (119897 (120582119909) minus 120582

3119897 (119909))10038171003817100381710038171003817 le

1003817100381710038171003817100381711989931199103119897 (120582119909) minus 119897 (119899119910) 120582

3119909310038171003817100381710038171003817

+ |120582|3 10038171003817100381710038171003817119897 (119899119910) 119909

3 minus 11989931199103119897 (119909)10038171003817100381710038171003817

le 120593 (119899119910 120582119909) + |120582|3120593 (119899119910 119909)

(35)

for all positive integers 119899 Thus100381710038171003817100381710038171199103 (119897 (120582119909) minus 120582

3119897 (119909))10038171003817100381710038171003817 le 119899minus3 (120593 (119899119910 120582119909) + |120582|

3120593 (119899119910 119909))

(36)

Taking the limit when 119899 tends toinfin we conclude that

1199103 (119897 (120582119909) minus 1205823119897 (119909)) = 0 119909 119910 isin A 120582 isin C (37)

Writing 119910 = 119890 we get 119897(120582119909) = 1205823119897(119909) for all 119909 isin A and 120582 isin CIn other words 119897 is cubic homogeneous

Claim 3 (119897 is a cubic Lie derivation onA) Let119909 119910 119911 isin AThen1003817100381710038171003817100381711989931199113 (119897 ([119909 119910]) minus [119897 (119909) 119910

3] minus [1199093 119897 (119910)])10038171003817100381710038171003817

le1003817100381710038171003817100381711989931199113119897 ([119909 119910]) minus 119897 (119899119911) ([119909 119910])

310038171003817100381710038171003817

+10038171003817100381710038171003817119897 (119899119911) ([119909 119910])

3minus 11989931199113 ([119897 (119909) 119910

3] + [1199093 119897 (119910)])10038171003817100381710038171003817

le 2120593 (119899119911 [119909 119910])

(38)

for all positive integers 119899 Thus100381710038171003817100381710038171199113 (119897 ([119909 119910]) minus [119897 (119909) 119910

3] minus [1199093 119897 (119910)])10038171003817100381710038171003817

le 119899minus32120593 (119899119911 [119909 119910])(39)

for all 119909 119910 119911 isin 119860 and 119899 isin N Similarly as in Claims 1 and 2 wecan show that 119897([119909 119910]) = [119897(119909) 1199103]+[1199093 119897(119910)] for all 119909 119910 isin AThe proof is completed

Let 120576 ge 0 and 0 le 119901 lt 3 ApplyingTheorem 3 for the case

120593 (119909 119910) = 1205761199091199011003817100381710038171003817119910

1003817100381710038171003817119901 119909 119910 isin A (40)

we have the next corollary

Corollary 4 Let A be a normed algebra with an element 119890which is not a zero divisor Suppose that 119897 A rarr A is amapping such that

10038171003817100381710038171003817119897 (119909) 1199103 minus 1199093119897 (119910)

10038171003817100381710038171003817 le 1205761199091199011003817100381710038171003817119910

1003817100381710038171003817119901

10038171003817100381710038171003817119897 (119911) [119909 119910]3minus 1199113 ([119897 (119909) 119910

3] + [1199093 119897 (119910)])10038171003817100381710038171003817

le 1205761199111199011003817100381710038171003817[119909 119910]

1003817100381710038171003817119901

(41)

for all 119909 119910 119911 isin A Then 119897 is a cubic Lie derivation onA

4 Some Additional Remarks

In this section we will write some additional observationsabout our results We start with the theorem which isanalogue toTheorem 1 Since the proof is similar it is omitted

Theorem 5 Suppose that 119897 A rarr M is a mapping for whichthere exists a function 120593 AtimesA rarr [0infin) such that (9) holdstrue for all 119909 119910 isin A 120582 isin Λ and

infin

sum119896=0

8119896120593(119909

2119896119910

2119896) lt infin 119909 119910 isin A (42)

If for each fixed 119909 isin A the mapping 119903 997891rarr 119897(119903119909) from R to Mis continuous then there exists a unique cubic Lie derivation119871 A rarr M such that (10) is valid for all 119909 isin A

Similarly the next result is analogue to Theorem 3

Theorem 6 Let A be a normed algebra with an element 119890which is not a zero divisor Suppose that 119897 A rarr A is amapping for which there exists a function 120593 AtimesA rarr [0infin)such that (30) and (31) hold true for all 119909 119910 119911 isin A and

lim119899rarrinfin

1198993120593(119909

119899 119910) = 0 119909 119910 isin A (43)

Then 119897 is a cubic Lie derivation onA

Let us end this paper with a remark on 119896-cubic functionalequations In [27] Najati introduced the following functionalequations

119891 (119896119909 + 119910) + 119891 (119896119909 minus 119910) = 119896119891 (119909 + 119910) + 119896119891 (119909 minus 119910)

+ 2 (1198963 minus 119896)119891 (119909) (44)

where 119896 ge 2 is a positive integer For 119896 = 2 we obtain(1) Najati proved that a function 119891 satisfies the functionalequation (1) if and only if 119891 satisfies the functional equation(44)Therefore every solution of the functional equation (44)is also a cubic function Following these results it is easy tosee that we can consider (44) in our theorems instead of (1)

Abstract and Applied Analysis 5

Acknowledgment

The authors would like to thank the referees for their usefulcomments

References

[1] S M Ulam Problems in Modern Mathematics John Wiley ampSons New York NY USA 1964

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T Aoki ldquoOn the stability of the linear transformation in Banachspacesrdquo Journal of the Mathematical Society of Japan vol 2 pp64ndash66 1950

[4] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[5] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[6] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[7] K-W Jun and H-M Kim ldquoThe generalized Hyers-Ulam-Rassias stability of a cubic functional equationrdquo Journal ofMathematical Analysis and Applications vol 274 no 2 pp 267ndash278 2002

[8] A Bodaghi I A Alias andMH Ghahramani ldquoApproximatelycubic functional equations and cubic multipliersrdquo Journal ofInequalities and Applications vol 2011 articlie 53 2011

[9] M E Gordji S K Gharetapeh J M Rassias and S ZolfagharildquoSolution and stability of a mixed type additive quadratic andcubic functional equationrdquo Advances in Difference Equationsvol 2009 pp 1ndash17 2009

[10] M Eshaghi Gordji S Zolfaghari J M Rassias and M BSavadkouhi ldquoSolution and stability of a mixed type cubic andquartic functional equation in quasi-Banach spacesrdquo Abstractand Applied Analysis vol 2009 Article ID 417473 14 pages2009

[11] K-W Jun H-M Kim and I-S Chang ldquoOn the Hyers-Ulamstability of an Euler-Lagrange type cubic functional equationrdquoJournal of Computational Analysis and Applications vol 7 no 1pp 21ndash33 2005

[12] A Najati ldquoHyers-Ulam-Rassias stability of a cubic functionalequationrdquo Bulletin of the Korean Mathematical Society vol 44no 4 pp 825ndash840 2007

[13] T Z Xu J M Rassias and W X Xu ldquoA generalized mixedtype of quartic-cubic-quadratic-additive functional equationsrdquoUkrainian Mathematical Journal vol 63 no 3 pp 461ndash4792011

[14] M Eshaghi Gordji S Kaboli Gharetapeh M B SavadkouhiM Aghaei and T Karimi ldquoOn cubic derivationsrdquo InternationalJournal of Mathematical Analysis vol 4 no 49ndash52 pp 2501ndash2514 2010

[15] S Y Yang A Bodaghi and K A M Atan ldquoApproximate cubiclowast-derivations on Banach lowast-algebrasrdquo Abstract and AppliedAnalysis vol 2012 Article ID 684179 12 pages 2012

[16] B Hayati M E Gordji M B Savadkouhi and M BidkhamldquoStability of ternary cubic derivations on ternary Frechet alge-brasrdquoAustralian Journal of Basic and Applied Sciences vol 5 pp1224ndash1235 2011

[17] St Czerwik ldquoOn the stability of the quadratic mapping innormed spacesrdquo Abhandlungen aus demMathematischen Semi-nar der Universitat Hamburg vol 62 pp 59ndash64 1992

[18] S Kurepa ldquoOn the quadratic functionalrdquo Publications delrsquoInstitut Mathematique de lrsquoAcademie Serbe des Sciences vol 13pp 57ndash72 1961

[19] Z Moszner ldquoSur les definitions differentes de la stabilite desequations fonctionnellesrdquo Aequationes Mathematicae vol 68no 3 pp 260ndash274 2004

[20] D G Bourgin ldquoApproximately isometric and multiplicativetransformations on continuous function ringsrdquo Duke Mathe-matical Journal vol 16 pp 385ndash397 1949

[21] G Maksa and Z Pales ldquoHyperstability of a class of linear func-tional equationsrdquo Acta Mathematica Academiae PaedagogicaeNyıregyhaziensis vol 17 no 2 pp 107ndash112 2001

[22] E Gselmann ldquoHyperstability of a functional equationrdquo ActaMathematica Hungarica vol 124 no 1-2 pp 179ndash188 2009

[23] Gy Maksa K Nikodem and Zs Pales ldquoResults on 119905-Wrightconvexityrdquo LrsquoAcademie des Sciences Comptes RendusMathemat-iques vol 13 no 6 pp 274ndash278 1991

[24] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[25] J Brzdek ldquoHyperstability of the Cauchy equation on restricteddomainsrdquo Acta Mathematica Hungarica 2013

[26] M Piszczek ldquoRemark on hyperstability of the general linearequationrdquo Aequationes Mathematicae 2013

[27] A Najati ldquoThe generalized Hyers-Ulam-Rassias stability of acubic functional equationrdquo Turkish Journal of Mathematics vol31 no 4 pp 395ndash408 2007

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

Abstract and Applied Analysis 3

Let 120582 isin C Then 120582|120582| isin Λ |120582|2 isin R and we have

119871 (1205821199090) = 119871(2

120582

|120582|(|120582|

21199090)) = 8(

120582

|120582|)3

119871(|120582|

21199090)

= 81205823

|120582|3(|120582|

2)3

119871 (1199090) = 1205823119871 (119909

0)

(22)

Since 1199090was any element fromA we conclude that 119871 is cubic

homogeneous

Claim 3 (119871 is a cubic Lie derivation) Using the second ine-quality in (9) we have

1003817100381710038171003817Δ 119871 (119909 119910)1003817100381710038171003817 = lim119899rarrinfin

100381710038171003817100381710038171003817100381710038171003817

Δ119897(2119899119909 2119899119910)

82119899

100381710038171003817100381710038171003817100381710038171003817

le lim119899rarrinfin

120593 (2119899119909 2119899119910)

82119899= 0

(23)

for all 119909 119910 isin A This together with Claim 2 yields that 119871 is acubic Lie derivation

Claim 4 (119871 is unique) Suppose that there exists another cubicLie derivation A rarr M such that

10038171003817100381710038171003817119897 (119909) minus (119909)10038171003817100381710038171003817 le

120601 (119909 0)

16(24)

for all 119909 isin A Then10038171003817100381710038171003817119871 (119909) minus (119909)

10038171003817100381710038171003817 = lim119899rarrinfin

1

811989910038171003817100381710038171003817119897 (2119899119909) minus (2119899119909)

10038171003817100381710038171003817

le lim119899rarrinfin

1

16(120601 (2119899119909 0)

8119899)

= lim119899rarrinfin

1

16

infin

sum119896=0

120593 (2119896+119899119909 0)

8119896+119899

= lim119899rarrinfin

1

16

infin

sum119896=119899

120593 (2119896119909 0)

8119896= 0

(25)

Therefore 119871(119909) = (119909) for all 119909 isin A which proves theuniqueness of 119871 The proof is completed

Let ] 120576 ge 0 and 0 le 119901 lt 3 Applying Theorem 1 for thecase

120593 (119909 119910) = ] + 120576 (119909119901 + 10038171003817100381710038171199101003817100381710038171003817119901) 119909 119910 isin A (26)

we have the next corollary

Corollary 2 Suppose that 119897 A rarr M is a mapping such that(9) holds true for all 119909 119910 isin A and 120582 isin Λ where 120593 is a functiondefined as above If for each fixed119909 isin A themapping 119903 997891rarr 119897(119903119909)fromR toM is continuous then there exists a unique cubic Liederivation 119871 A rarr M such that

119897 (119909) minus 119871 (119909) le]

14+

120576119909119901

(16 minus 2119901+1)(27)

for all 119909 isin A

Proof Note that we have

120601 (119909 119910) =8]

7+8120576 (119909119901 +

10038171003817100381710038171199101003817100381710038171003817119901)

(8 minus 2119901) 119909 119910 isin A (28)

3 Hyperstability of Cubic Lie Derivations

The investigation of the multiplicative Cauchy functionalequation highlighted a new phenomenon which is nowadayscalled superstability (see eg [19]) In this case the so-called stability inequality implies that the observed functionis either bounded or it is a solution of the functional equationBut it can also happen that each function 119891 satisfying thefunctional equation E approximately must actually be asolution of the proposed equationE In this case we say thatthe functional equation E is hyperstable According to ourbest knowledge the first hyperstability result was published in[20] and concerned the ring homomorphisms However theterm hyperstability has been used for the first time probablyin [21] (see also [22 23]) Formore information about the newresults on the hyperstability we refer the reader to [24ndash26]

Theorem 3 Let A be a normed algebra with an element 119890which is not a zero divisor Suppose that 119897 A rarr A is amapping for which there exists a function 120593 AtimesA rarr [0infin)such that

lim119899rarrinfin

120593 (119899119909 119910)

1198993= 0 (29)

10038171003817100381710038171003817119897 (119909) 1199103 minus 1199093119897 (119910)

10038171003817100381710038171003817 le 120593 (119909 119910) (30)

10038171003817100381710038171003817119897 (119911) ([119909 119910])3minus 1199113 ([119897 (119909) 119910

3] + [1199093 119897 (119910)])10038171003817100381710038171003817 le 120593 (119911 [119909 119910])

(31)

for all 119909 119910 119911 isin A Then 119897 is a cubic Lie derivation onA

Proof We divide the proof into several steps

Claim 1 (119897 is a cubic mapping) Let 119909 119910 119911 isin A Then

1003817100381710038171003817100381711989931199113 (119897 (2119909 + 119910) + 119897 (2119909 minus 119910) minus 2119897 (119909 + 119910)

minus2119897 (119909 minus 119910) minus 12119897 (119909))10038171003817100381710038171003817

le1003817100381710038171003817100381711989931199113119897 (2119909 + 119910) minus 119897 (119899119911) (2119909 + 119910)

310038171003817100381710038171003817

+1003817100381710038171003817100381711989931199113119897 (2119909 minus 119910) minus 119897 (119899119911) (2119909 minus 119910)

310038171003817100381710038171003817

+ 210038171003817100381710038171003817119897 (119899119911) (119909 + 119910)

3minus 11989931199113119897 (119909 + 119910)

10038171003817100381710038171003817

+ 210038171003817100381710038171003817119897 (119899119911) (119909 minus 119910)

3minus 11989931199113119897 (119909 minus 119910)

10038171003817100381710038171003817

+ 1210038171003817100381710038171003817119897 (119899119911) 119909

3 minus 11989931199113119897 (119909)10038171003817100381710038171003817

(32)

4 Abstract and Applied Analysis

for all positive integers 119899 By (30) it follows that100381710038171003817100381710038171199113 (119897 (2119909 + 119910) + 119897 (2119909 minus 119910) minus 2119897 (119909 + 119910)

minus2119897 (119909 minus 119910) minus 12119897 (119909))10038171003817100381710038171003817

le 119899minus3 (120593 (119899119911 2119909 + 119910) + 120593 (119899119911 2119909 minus 119910)

+ 2120593 (119899119911 119909 + 119910) + 2120593 (119899119911 119909 minus 119910)

+ 12120593 (119899119911 119909))

(33)

If we let 119899 rarr infin we conclude that

1199113 (119897 (2119909 + 119910) + 119897 (2119909 minus 119910) minus 2119897 (119909 + 119910)

minus2119897 (119909 minus 119910) minus 12119897 (119909)) = 0(34)

for all 119909 119910 119911 isin A Putting 119911 = 119890 in the last equality weconclude that 119897 satisfies a cubic functional equation (1)

Claim 2 (119897 is cubic homogeneous) Let 120582 isin C and 119909 119910 isin AThen1003817100381710038171003817100381711989931199103 (119897 (120582119909) minus 120582

3119897 (119909))10038171003817100381710038171003817 le

1003817100381710038171003817100381711989931199103119897 (120582119909) minus 119897 (119899119910) 120582

3119909310038171003817100381710038171003817

+ |120582|3 10038171003817100381710038171003817119897 (119899119910) 119909

3 minus 11989931199103119897 (119909)10038171003817100381710038171003817

le 120593 (119899119910 120582119909) + |120582|3120593 (119899119910 119909)

(35)

for all positive integers 119899 Thus100381710038171003817100381710038171199103 (119897 (120582119909) minus 120582

3119897 (119909))10038171003817100381710038171003817 le 119899minus3 (120593 (119899119910 120582119909) + |120582|

3120593 (119899119910 119909))

(36)

Taking the limit when 119899 tends toinfin we conclude that

1199103 (119897 (120582119909) minus 1205823119897 (119909)) = 0 119909 119910 isin A 120582 isin C (37)

Writing 119910 = 119890 we get 119897(120582119909) = 1205823119897(119909) for all 119909 isin A and 120582 isin CIn other words 119897 is cubic homogeneous

Claim 3 (119897 is a cubic Lie derivation onA) Let119909 119910 119911 isin AThen1003817100381710038171003817100381711989931199113 (119897 ([119909 119910]) minus [119897 (119909) 119910

3] minus [1199093 119897 (119910)])10038171003817100381710038171003817

le1003817100381710038171003817100381711989931199113119897 ([119909 119910]) minus 119897 (119899119911) ([119909 119910])

310038171003817100381710038171003817

+10038171003817100381710038171003817119897 (119899119911) ([119909 119910])

3minus 11989931199113 ([119897 (119909) 119910

3] + [1199093 119897 (119910)])10038171003817100381710038171003817

le 2120593 (119899119911 [119909 119910])

(38)

for all positive integers 119899 Thus100381710038171003817100381710038171199113 (119897 ([119909 119910]) minus [119897 (119909) 119910

3] minus [1199093 119897 (119910)])10038171003817100381710038171003817

le 119899minus32120593 (119899119911 [119909 119910])(39)

for all 119909 119910 119911 isin 119860 and 119899 isin N Similarly as in Claims 1 and 2 wecan show that 119897([119909 119910]) = [119897(119909) 1199103]+[1199093 119897(119910)] for all 119909 119910 isin AThe proof is completed

Let 120576 ge 0 and 0 le 119901 lt 3 ApplyingTheorem 3 for the case

120593 (119909 119910) = 1205761199091199011003817100381710038171003817119910

1003817100381710038171003817119901 119909 119910 isin A (40)

we have the next corollary

Corollary 4 Let A be a normed algebra with an element 119890which is not a zero divisor Suppose that 119897 A rarr A is amapping such that

10038171003817100381710038171003817119897 (119909) 1199103 minus 1199093119897 (119910)

10038171003817100381710038171003817 le 1205761199091199011003817100381710038171003817119910

1003817100381710038171003817119901

10038171003817100381710038171003817119897 (119911) [119909 119910]3minus 1199113 ([119897 (119909) 119910

3] + [1199093 119897 (119910)])10038171003817100381710038171003817

le 1205761199111199011003817100381710038171003817[119909 119910]

1003817100381710038171003817119901

(41)

for all 119909 119910 119911 isin A Then 119897 is a cubic Lie derivation onA

4 Some Additional Remarks

In this section we will write some additional observationsabout our results We start with the theorem which isanalogue toTheorem 1 Since the proof is similar it is omitted

Theorem 5 Suppose that 119897 A rarr M is a mapping for whichthere exists a function 120593 AtimesA rarr [0infin) such that (9) holdstrue for all 119909 119910 isin A 120582 isin Λ and

infin

sum119896=0

8119896120593(119909

2119896119910

2119896) lt infin 119909 119910 isin A (42)

If for each fixed 119909 isin A the mapping 119903 997891rarr 119897(119903119909) from R to Mis continuous then there exists a unique cubic Lie derivation119871 A rarr M such that (10) is valid for all 119909 isin A

Similarly the next result is analogue to Theorem 3

Theorem 6 Let A be a normed algebra with an element 119890which is not a zero divisor Suppose that 119897 A rarr A is amapping for which there exists a function 120593 AtimesA rarr [0infin)such that (30) and (31) hold true for all 119909 119910 119911 isin A and

lim119899rarrinfin

1198993120593(119909

119899 119910) = 0 119909 119910 isin A (43)

Then 119897 is a cubic Lie derivation onA

Let us end this paper with a remark on 119896-cubic functionalequations In [27] Najati introduced the following functionalequations

119891 (119896119909 + 119910) + 119891 (119896119909 minus 119910) = 119896119891 (119909 + 119910) + 119896119891 (119909 minus 119910)

+ 2 (1198963 minus 119896)119891 (119909) (44)

where 119896 ge 2 is a positive integer For 119896 = 2 we obtain(1) Najati proved that a function 119891 satisfies the functionalequation (1) if and only if 119891 satisfies the functional equation(44)Therefore every solution of the functional equation (44)is also a cubic function Following these results it is easy tosee that we can consider (44) in our theorems instead of (1)

Abstract and Applied Analysis 5

Acknowledgment

The authors would like to thank the referees for their usefulcomments

References

[1] S M Ulam Problems in Modern Mathematics John Wiley ampSons New York NY USA 1964

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T Aoki ldquoOn the stability of the linear transformation in Banachspacesrdquo Journal of the Mathematical Society of Japan vol 2 pp64ndash66 1950

[4] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[5] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[6] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[7] K-W Jun and H-M Kim ldquoThe generalized Hyers-Ulam-Rassias stability of a cubic functional equationrdquo Journal ofMathematical Analysis and Applications vol 274 no 2 pp 267ndash278 2002

[8] A Bodaghi I A Alias andMH Ghahramani ldquoApproximatelycubic functional equations and cubic multipliersrdquo Journal ofInequalities and Applications vol 2011 articlie 53 2011

[9] M E Gordji S K Gharetapeh J M Rassias and S ZolfagharildquoSolution and stability of a mixed type additive quadratic andcubic functional equationrdquo Advances in Difference Equationsvol 2009 pp 1ndash17 2009

[10] M Eshaghi Gordji S Zolfaghari J M Rassias and M BSavadkouhi ldquoSolution and stability of a mixed type cubic andquartic functional equation in quasi-Banach spacesrdquo Abstractand Applied Analysis vol 2009 Article ID 417473 14 pages2009

[11] K-W Jun H-M Kim and I-S Chang ldquoOn the Hyers-Ulamstability of an Euler-Lagrange type cubic functional equationrdquoJournal of Computational Analysis and Applications vol 7 no 1pp 21ndash33 2005

[12] A Najati ldquoHyers-Ulam-Rassias stability of a cubic functionalequationrdquo Bulletin of the Korean Mathematical Society vol 44no 4 pp 825ndash840 2007

[13] T Z Xu J M Rassias and W X Xu ldquoA generalized mixedtype of quartic-cubic-quadratic-additive functional equationsrdquoUkrainian Mathematical Journal vol 63 no 3 pp 461ndash4792011

[14] M Eshaghi Gordji S Kaboli Gharetapeh M B SavadkouhiM Aghaei and T Karimi ldquoOn cubic derivationsrdquo InternationalJournal of Mathematical Analysis vol 4 no 49ndash52 pp 2501ndash2514 2010

[15] S Y Yang A Bodaghi and K A M Atan ldquoApproximate cubiclowast-derivations on Banach lowast-algebrasrdquo Abstract and AppliedAnalysis vol 2012 Article ID 684179 12 pages 2012

[16] B Hayati M E Gordji M B Savadkouhi and M BidkhamldquoStability of ternary cubic derivations on ternary Frechet alge-brasrdquoAustralian Journal of Basic and Applied Sciences vol 5 pp1224ndash1235 2011

[17] St Czerwik ldquoOn the stability of the quadratic mapping innormed spacesrdquo Abhandlungen aus demMathematischen Semi-nar der Universitat Hamburg vol 62 pp 59ndash64 1992

[18] S Kurepa ldquoOn the quadratic functionalrdquo Publications delrsquoInstitut Mathematique de lrsquoAcademie Serbe des Sciences vol 13pp 57ndash72 1961

[19] Z Moszner ldquoSur les definitions differentes de la stabilite desequations fonctionnellesrdquo Aequationes Mathematicae vol 68no 3 pp 260ndash274 2004

[20] D G Bourgin ldquoApproximately isometric and multiplicativetransformations on continuous function ringsrdquo Duke Mathe-matical Journal vol 16 pp 385ndash397 1949

[21] G Maksa and Z Pales ldquoHyperstability of a class of linear func-tional equationsrdquo Acta Mathematica Academiae PaedagogicaeNyıregyhaziensis vol 17 no 2 pp 107ndash112 2001

[22] E Gselmann ldquoHyperstability of a functional equationrdquo ActaMathematica Hungarica vol 124 no 1-2 pp 179ndash188 2009

[23] Gy Maksa K Nikodem and Zs Pales ldquoResults on 119905-Wrightconvexityrdquo LrsquoAcademie des Sciences Comptes RendusMathemat-iques vol 13 no 6 pp 274ndash278 1991

[24] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[25] J Brzdek ldquoHyperstability of the Cauchy equation on restricteddomainsrdquo Acta Mathematica Hungarica 2013

[26] M Piszczek ldquoRemark on hyperstability of the general linearequationrdquo Aequationes Mathematicae 2013

[27] A Najati ldquoThe generalized Hyers-Ulam-Rassias stability of acubic functional equationrdquo Turkish Journal of Mathematics vol31 no 4 pp 395ndash408 2007

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 Abstract and Applied Analysis

for all positive integers 119899 By (30) it follows that100381710038171003817100381710038171199113 (119897 (2119909 + 119910) + 119897 (2119909 minus 119910) minus 2119897 (119909 + 119910)

minus2119897 (119909 minus 119910) minus 12119897 (119909))10038171003817100381710038171003817

le 119899minus3 (120593 (119899119911 2119909 + 119910) + 120593 (119899119911 2119909 minus 119910)

+ 2120593 (119899119911 119909 + 119910) + 2120593 (119899119911 119909 minus 119910)

+ 12120593 (119899119911 119909))

(33)

If we let 119899 rarr infin we conclude that

1199113 (119897 (2119909 + 119910) + 119897 (2119909 minus 119910) minus 2119897 (119909 + 119910)

minus2119897 (119909 minus 119910) minus 12119897 (119909)) = 0(34)

for all 119909 119910 119911 isin A Putting 119911 = 119890 in the last equality weconclude that 119897 satisfies a cubic functional equation (1)

Claim 2 (119897 is cubic homogeneous) Let 120582 isin C and 119909 119910 isin AThen1003817100381710038171003817100381711989931199103 (119897 (120582119909) minus 120582

3119897 (119909))10038171003817100381710038171003817 le

1003817100381710038171003817100381711989931199103119897 (120582119909) minus 119897 (119899119910) 120582

3119909310038171003817100381710038171003817

+ |120582|3 10038171003817100381710038171003817119897 (119899119910) 119909

3 minus 11989931199103119897 (119909)10038171003817100381710038171003817

le 120593 (119899119910 120582119909) + |120582|3120593 (119899119910 119909)

(35)

for all positive integers 119899 Thus100381710038171003817100381710038171199103 (119897 (120582119909) minus 120582

3119897 (119909))10038171003817100381710038171003817 le 119899minus3 (120593 (119899119910 120582119909) + |120582|

3120593 (119899119910 119909))

(36)

Taking the limit when 119899 tends toinfin we conclude that

1199103 (119897 (120582119909) minus 1205823119897 (119909)) = 0 119909 119910 isin A 120582 isin C (37)

Writing 119910 = 119890 we get 119897(120582119909) = 1205823119897(119909) for all 119909 isin A and 120582 isin CIn other words 119897 is cubic homogeneous

Claim 3 (119897 is a cubic Lie derivation onA) Let119909 119910 119911 isin AThen1003817100381710038171003817100381711989931199113 (119897 ([119909 119910]) minus [119897 (119909) 119910

3] minus [1199093 119897 (119910)])10038171003817100381710038171003817

le1003817100381710038171003817100381711989931199113119897 ([119909 119910]) minus 119897 (119899119911) ([119909 119910])

310038171003817100381710038171003817

+10038171003817100381710038171003817119897 (119899119911) ([119909 119910])

3minus 11989931199113 ([119897 (119909) 119910

3] + [1199093 119897 (119910)])10038171003817100381710038171003817

le 2120593 (119899119911 [119909 119910])

(38)

for all positive integers 119899 Thus100381710038171003817100381710038171199113 (119897 ([119909 119910]) minus [119897 (119909) 119910

3] minus [1199093 119897 (119910)])10038171003817100381710038171003817

le 119899minus32120593 (119899119911 [119909 119910])(39)

for all 119909 119910 119911 isin 119860 and 119899 isin N Similarly as in Claims 1 and 2 wecan show that 119897([119909 119910]) = [119897(119909) 1199103]+[1199093 119897(119910)] for all 119909 119910 isin AThe proof is completed

Let 120576 ge 0 and 0 le 119901 lt 3 ApplyingTheorem 3 for the case

120593 (119909 119910) = 1205761199091199011003817100381710038171003817119910

1003817100381710038171003817119901 119909 119910 isin A (40)

we have the next corollary

Corollary 4 Let A be a normed algebra with an element 119890which is not a zero divisor Suppose that 119897 A rarr A is amapping such that

10038171003817100381710038171003817119897 (119909) 1199103 minus 1199093119897 (119910)

10038171003817100381710038171003817 le 1205761199091199011003817100381710038171003817119910

1003817100381710038171003817119901

10038171003817100381710038171003817119897 (119911) [119909 119910]3minus 1199113 ([119897 (119909) 119910

3] + [1199093 119897 (119910)])10038171003817100381710038171003817

le 1205761199111199011003817100381710038171003817[119909 119910]

1003817100381710038171003817119901

(41)

for all 119909 119910 119911 isin A Then 119897 is a cubic Lie derivation onA

4 Some Additional Remarks

In this section we will write some additional observationsabout our results We start with the theorem which isanalogue toTheorem 1 Since the proof is similar it is omitted

Theorem 5 Suppose that 119897 A rarr M is a mapping for whichthere exists a function 120593 AtimesA rarr [0infin) such that (9) holdstrue for all 119909 119910 isin A 120582 isin Λ and

infin

sum119896=0

8119896120593(119909

2119896119910

2119896) lt infin 119909 119910 isin A (42)

If for each fixed 119909 isin A the mapping 119903 997891rarr 119897(119903119909) from R to Mis continuous then there exists a unique cubic Lie derivation119871 A rarr M such that (10) is valid for all 119909 isin A

Similarly the next result is analogue to Theorem 3

Theorem 6 Let A be a normed algebra with an element 119890which is not a zero divisor Suppose that 119897 A rarr A is amapping for which there exists a function 120593 AtimesA rarr [0infin)such that (30) and (31) hold true for all 119909 119910 119911 isin A and

lim119899rarrinfin

1198993120593(119909

119899 119910) = 0 119909 119910 isin A (43)

Then 119897 is a cubic Lie derivation onA

Let us end this paper with a remark on 119896-cubic functionalequations In [27] Najati introduced the following functionalequations

119891 (119896119909 + 119910) + 119891 (119896119909 minus 119910) = 119896119891 (119909 + 119910) + 119896119891 (119909 minus 119910)

+ 2 (1198963 minus 119896)119891 (119909) (44)

where 119896 ge 2 is a positive integer For 119896 = 2 we obtain(1) Najati proved that a function 119891 satisfies the functionalequation (1) if and only if 119891 satisfies the functional equation(44)Therefore every solution of the functional equation (44)is also a cubic function Following these results it is easy tosee that we can consider (44) in our theorems instead of (1)

Abstract and Applied Analysis 5

Acknowledgment

The authors would like to thank the referees for their usefulcomments

References

[1] S M Ulam Problems in Modern Mathematics John Wiley ampSons New York NY USA 1964

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T Aoki ldquoOn the stability of the linear transformation in Banachspacesrdquo Journal of the Mathematical Society of Japan vol 2 pp64ndash66 1950

[4] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[5] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[6] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[7] K-W Jun and H-M Kim ldquoThe generalized Hyers-Ulam-Rassias stability of a cubic functional equationrdquo Journal ofMathematical Analysis and Applications vol 274 no 2 pp 267ndash278 2002

[8] A Bodaghi I A Alias andMH Ghahramani ldquoApproximatelycubic functional equations and cubic multipliersrdquo Journal ofInequalities and Applications vol 2011 articlie 53 2011

[9] M E Gordji S K Gharetapeh J M Rassias and S ZolfagharildquoSolution and stability of a mixed type additive quadratic andcubic functional equationrdquo Advances in Difference Equationsvol 2009 pp 1ndash17 2009

[10] M Eshaghi Gordji S Zolfaghari J M Rassias and M BSavadkouhi ldquoSolution and stability of a mixed type cubic andquartic functional equation in quasi-Banach spacesrdquo Abstractand Applied Analysis vol 2009 Article ID 417473 14 pages2009

[11] K-W Jun H-M Kim and I-S Chang ldquoOn the Hyers-Ulamstability of an Euler-Lagrange type cubic functional equationrdquoJournal of Computational Analysis and Applications vol 7 no 1pp 21ndash33 2005

[12] A Najati ldquoHyers-Ulam-Rassias stability of a cubic functionalequationrdquo Bulletin of the Korean Mathematical Society vol 44no 4 pp 825ndash840 2007

[13] T Z Xu J M Rassias and W X Xu ldquoA generalized mixedtype of quartic-cubic-quadratic-additive functional equationsrdquoUkrainian Mathematical Journal vol 63 no 3 pp 461ndash4792011

[14] M Eshaghi Gordji S Kaboli Gharetapeh M B SavadkouhiM Aghaei and T Karimi ldquoOn cubic derivationsrdquo InternationalJournal of Mathematical Analysis vol 4 no 49ndash52 pp 2501ndash2514 2010

[15] S Y Yang A Bodaghi and K A M Atan ldquoApproximate cubiclowast-derivations on Banach lowast-algebrasrdquo Abstract and AppliedAnalysis vol 2012 Article ID 684179 12 pages 2012

[16] B Hayati M E Gordji M B Savadkouhi and M BidkhamldquoStability of ternary cubic derivations on ternary Frechet alge-brasrdquoAustralian Journal of Basic and Applied Sciences vol 5 pp1224ndash1235 2011

[17] St Czerwik ldquoOn the stability of the quadratic mapping innormed spacesrdquo Abhandlungen aus demMathematischen Semi-nar der Universitat Hamburg vol 62 pp 59ndash64 1992

[18] S Kurepa ldquoOn the quadratic functionalrdquo Publications delrsquoInstitut Mathematique de lrsquoAcademie Serbe des Sciences vol 13pp 57ndash72 1961

[19] Z Moszner ldquoSur les definitions differentes de la stabilite desequations fonctionnellesrdquo Aequationes Mathematicae vol 68no 3 pp 260ndash274 2004

[20] D G Bourgin ldquoApproximately isometric and multiplicativetransformations on continuous function ringsrdquo Duke Mathe-matical Journal vol 16 pp 385ndash397 1949

[21] G Maksa and Z Pales ldquoHyperstability of a class of linear func-tional equationsrdquo Acta Mathematica Academiae PaedagogicaeNyıregyhaziensis vol 17 no 2 pp 107ndash112 2001

[22] E Gselmann ldquoHyperstability of a functional equationrdquo ActaMathematica Hungarica vol 124 no 1-2 pp 179ndash188 2009

[23] Gy Maksa K Nikodem and Zs Pales ldquoResults on 119905-Wrightconvexityrdquo LrsquoAcademie des Sciences Comptes RendusMathemat-iques vol 13 no 6 pp 274ndash278 1991

[24] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[25] J Brzdek ldquoHyperstability of the Cauchy equation on restricteddomainsrdquo Acta Mathematica Hungarica 2013

[26] M Piszczek ldquoRemark on hyperstability of the general linearequationrdquo Aequationes Mathematicae 2013

[27] A Najati ldquoThe generalized Hyers-Ulam-Rassias stability of acubic functional equationrdquo Turkish Journal of Mathematics vol31 no 4 pp 395ndash408 2007

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

Abstract and Applied Analysis 5

Acknowledgment

The authors would like to thank the referees for their usefulcomments

References

[1] S M Ulam Problems in Modern Mathematics John Wiley ampSons New York NY USA 1964

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T Aoki ldquoOn the stability of the linear transformation in Banachspacesrdquo Journal of the Mathematical Society of Japan vol 2 pp64ndash66 1950

[4] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[5] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[6] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[7] K-W Jun and H-M Kim ldquoThe generalized Hyers-Ulam-Rassias stability of a cubic functional equationrdquo Journal ofMathematical Analysis and Applications vol 274 no 2 pp 267ndash278 2002

[8] A Bodaghi I A Alias andMH Ghahramani ldquoApproximatelycubic functional equations and cubic multipliersrdquo Journal ofInequalities and Applications vol 2011 articlie 53 2011

[9] M E Gordji S K Gharetapeh J M Rassias and S ZolfagharildquoSolution and stability of a mixed type additive quadratic andcubic functional equationrdquo Advances in Difference Equationsvol 2009 pp 1ndash17 2009

[10] M Eshaghi Gordji S Zolfaghari J M Rassias and M BSavadkouhi ldquoSolution and stability of a mixed type cubic andquartic functional equation in quasi-Banach spacesrdquo Abstractand Applied Analysis vol 2009 Article ID 417473 14 pages2009

[11] K-W Jun H-M Kim and I-S Chang ldquoOn the Hyers-Ulamstability of an Euler-Lagrange type cubic functional equationrdquoJournal of Computational Analysis and Applications vol 7 no 1pp 21ndash33 2005

[12] A Najati ldquoHyers-Ulam-Rassias stability of a cubic functionalequationrdquo Bulletin of the Korean Mathematical Society vol 44no 4 pp 825ndash840 2007

[13] T Z Xu J M Rassias and W X Xu ldquoA generalized mixedtype of quartic-cubic-quadratic-additive functional equationsrdquoUkrainian Mathematical Journal vol 63 no 3 pp 461ndash4792011

[14] M Eshaghi Gordji S Kaboli Gharetapeh M B SavadkouhiM Aghaei and T Karimi ldquoOn cubic derivationsrdquo InternationalJournal of Mathematical Analysis vol 4 no 49ndash52 pp 2501ndash2514 2010

[15] S Y Yang A Bodaghi and K A M Atan ldquoApproximate cubiclowast-derivations on Banach lowast-algebrasrdquo Abstract and AppliedAnalysis vol 2012 Article ID 684179 12 pages 2012

[16] B Hayati M E Gordji M B Savadkouhi and M BidkhamldquoStability of ternary cubic derivations on ternary Frechet alge-brasrdquoAustralian Journal of Basic and Applied Sciences vol 5 pp1224ndash1235 2011

[17] St Czerwik ldquoOn the stability of the quadratic mapping innormed spacesrdquo Abhandlungen aus demMathematischen Semi-nar der Universitat Hamburg vol 62 pp 59ndash64 1992

[18] S Kurepa ldquoOn the quadratic functionalrdquo Publications delrsquoInstitut Mathematique de lrsquoAcademie Serbe des Sciences vol 13pp 57ndash72 1961

[19] Z Moszner ldquoSur les definitions differentes de la stabilite desequations fonctionnellesrdquo Aequationes Mathematicae vol 68no 3 pp 260ndash274 2004

[20] D G Bourgin ldquoApproximately isometric and multiplicativetransformations on continuous function ringsrdquo Duke Mathe-matical Journal vol 16 pp 385ndash397 1949

[21] G Maksa and Z Pales ldquoHyperstability of a class of linear func-tional equationsrdquo Acta Mathematica Academiae PaedagogicaeNyıregyhaziensis vol 17 no 2 pp 107ndash112 2001

[22] E Gselmann ldquoHyperstability of a functional equationrdquo ActaMathematica Hungarica vol 124 no 1-2 pp 179ndash188 2009

[23] Gy Maksa K Nikodem and Zs Pales ldquoResults on 119905-Wrightconvexityrdquo LrsquoAcademie des Sciences Comptes RendusMathemat-iques vol 13 no 6 pp 274ndash278 1991

[24] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[25] J Brzdek ldquoHyperstability of the Cauchy equation on restricteddomainsrdquo Acta Mathematica Hungarica 2013

[26] M Piszczek ldquoRemark on hyperstability of the general linearequationrdquo Aequationes Mathematicae 2013

[27] A Najati ldquoThe generalized Hyers-Ulam-Rassias stability of acubic functional equationrdquo Turkish Journal of Mathematics vol31 no 4 pp 395ndash408 2007

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