Additive functional inequalities in Banach spaces

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1 Lu and Park Journal o Inequalities and Applications 01, 01:94 R E S E A R C H Open Access Additive unctional inequalities in Banach spaces Gang Lu 1 and Choonkil Park * * Correspondence: baak@hanyang.ac.kr Department o Mathematics, Research Institute or Natural Sciences, Hanyang University, Seoul, , orea Full list o author inormation is available at the end o the article Abstract In this paper, we prove the Hyers-Ulam stability o the ollowing unction inequalities: ()(y)(z) ( y z ) (0< <3), ()(y)(z) ( y z) (0 < ) in Banach spaces. MSC: Primary 39B6; 39B5; 46B5 eywords: Hyers-Ulam stability; additive unctional inequality; additive mapping 1 Introduction and preliminaries The stability problem o unctional equations originated rom the question o Ulam [1] in 1940 concerning the stability o group homomorphisms. Let (G 1, ) beagroupandlet (G, ) be a metric group with the metric d(, ). Given ɛ >0,doesthereeistaδ0suchthat i a mapping h : G 1 G satisies the inequality d(h( y), h() h(y)) < δ or all, y G 1, then there eists a homomorphism H : G 1 G with d(h(), H()) < ɛ or all G 1?In other words, under what condition does there eist a homomorphism near an approimate homomorphism? The concept o stability or a unctional equation arises when we replace the unctional equation by an inequality which acts as a perturbation o the equation. In 1941, Hyers []gave theirstairmative answertothequestionoulamorbanachspaces. Let : E E be a mapping between Banach spaces such that ( y) () (y) δ or all, y E and or some δ > 0. Then there eists a unique additive mapping T : E E such that () T() δ or all E. Moreover,i (t) is continuous in t R or each ied E, thent is R- linear. In 1978, Th.M. Rassias [3] proved the ollowing theorem. 01 Lu and Park; licensee Springer. This is an Open Access article distributed under the terms o the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Lu and Park Journal o Inequalities and Applications 01, 01:94 Page o 10 Theorem 1.1 Let : E E beamappingromanormedvectorspaceeintoabanach space E subject to the inequality ( y) () (y) ɛ ( p y p) (1.1) or all, y E, where ɛ and p are constants with ɛ >0and p <1.Then there eists a unique additive mapping T : E E such that () T() ɛ p p (1.) or all E. I p <0,then inequality (1.1) holds or all, y 0,and (1.) or 0.Also, i the unction t (t) rom R into E is continuous in t R or each ied E, then T is R-linear. In 1991, Gajda [4] answered the question or the case p > 1, which was raised by Th.M. Rassias. On the other hand, J.M.Rassias [5] generalized the Hyers-Ulam stability result by presenting a weaker condition controlled by a product o dierent powers o norms. Theorem 1. [6, 7] I it is assumed that there eist constants 0 and p 1, p R such that p = p 1 p 1,and : E E is a mapping rom a norm space E into a Banach space E such that the inequality ( y) () (y) ɛ p 1 y p holds or all, y E, then there eists a unique additive mapping T : E E such that () T() p p or all E. I, in addition, or every E, (t) is continuous in t R or each ied E, then T is R-linear. More generalizations and applications o the Hyers-Ulam stability to a number o unctional equations and mappings can be ound in [8 11]. In [1], Park et al. investigated the ollowing inequalities: ()(y)(z) ( ) y z, ()(y)(z) ( y z), ( z) ()(y)(z) y in Banach spaces. Recently, Cho et al. [13] investigated the ollowing unctional inequality: ()(y)(z) ( ) y z ( ) 0< < 3

3 Lu and Park Journal o Inequalities and Applications 01, 01:94 Page 3 o 10 in non-archimedean Banach spaces. Lu and Park [14] investigated the ollowing unctional inequality: N ( ( i ) N i=1 ( ) i) ( ) 0< N i=1 in Fréchet spaces. In this paper, we investigate the ollowing unctional inequalities: ()(y)(z) ( ) y z ( ) 0< <3, (1.3) ()(y) (z) ( ) y z (0 < ) (1.4) and prove the Hyers-Ulam stability o unctional inequalities (1.3) and(1.4) in Banach spaces. Throughout this paper, assume that X is a normed vector space and that (Y, )isa Banach space. Hyers-Ulam stability o unctional inequality (1.3) Throughout this section, assume that is a real number with 0 < <3. Proposition.1 Let : X Y be a mapping such that ()(y)(z) ( ) y z (.1) or all, y, z X. Then the mapping : X Y is additive. Proo Letting = y = z =0in(.1), we get 3 (0) (0). So, (0) = 0. Letting z =0andy = in (.1), we get () ( ) (0) =0 So, ( )= () Letting z = y in (.1), we get () (y) ( y) = () (y) ( y) (0) =0 or all, y X.Thus, ( y)= () (y) or all, y X,asdesired.

4 Lu and Park Journal o Inequalities and Applications 01, 01:94 Page 4 o 10 Theorem. Assume that a mapping : X Ysatisiestheinequality ( ) ()(y)(z) y z (, y, z), (.) where : X 3 [0, ) satisies (, y, z):= j=1 ( j, y j, z ) < (.3) j j or all, y, z X. Then there eists a unique additive mapping A : X Ysuchthat A() () 1 (,,) (,,0) (.4) Proo It ollows rom (.3)that(0,0,0)=0.Letting = y = z =0in(.), we get 3 (0) (0) (0,0,0)= (0).So,(0) = 0. Letting y =, z = in (.), we get () ( ) (,, ) So, ( ) ( ( ), ), (.5) Letting y = and z =0in(.), we get () ( ) (,,0) (.6) It ollows rom (.5)and(.6)that ( ) ( ) l m l m m 1 ( ) ( ) j j1 j j1 m 1 ( j j m 1[ ( j m 1 ) ( ) ( ) ( ) j1 j1 j1 j1 j1 j1 j ) ( j1 j1 ( ) 1 j1, j1, j1 j1 ) ( j1 m 1 j1 ( j1 ) ( )] j1 j1, j1 ),0 j1

5 Lu and Park Journal o Inequalities and Applications 01, 01:94 Page 5 o 10 or all nonnegative integers m and l with m > l and all X. It means that the sequence { n ( n )} is a Cauchy sequence SinceY is complete, the sequence { n ( n )} converges. We deine the mapping A : X Y by A() =lim n n ( n ) Moreover, letting l = 0 and passing the limit m,weget(.4). Net, we show that A() is an additive mapping. A()A( ) = lim n n ( ) ( ) n n ( 1 n n1 and so A( )= A(), n1 n1,0 ) =0 A()A(y) A( y) = A()A(y)A( y) ( ) ( ) ( ) = lim y y n n n 1 n n1 n (, y ( y), n1 n1 n1 n ) =0 or all, y X. Thus, the mapping A : X Y is additive. Now, we prove the uniqueness o A. Assume that T : X Y is another additive mapping satisying (.4). Then we obtain A() T() = lim n n n n n ( ) ( ) A T n n [ ( ) ( ) ( ) ( )] A n n T n n [ (, n, ) ( n n, )] n,0 =0 n ThenwecanconcludethatA() =T() Thiscompletesthe proo. Corollary.3 Let p and θ be positive real numbers with p >1.Let : X Y be a mapping satisying ()(y)(z) ( ) y z θ ( p y p z p) or all, y, z X. Then there eists a unique additive mapping A : X Ysuchthat () A() p 6 p θ p 3 Hyers-Ulam stability o unctional inequality (1.4) Throughout this section, assume that is a real number with 0 <.

6 Lu and Park Journal o Inequalities and Applications 01, 01:94 Page 6 o 10 Proposition 3.1 Let : X Y be a mapping such that ( ) ()(y) (z) y z (3.1) or all, y, z X. Then the mapping : X Y is additive. Proo Letting = y = z =0in(3.1), we get ( ) (0) (0). So, (0) = 0. Letting z =0andy = in (3.1), we get () ( ) (0) =0 So, ( )= () Letting z = y in (3.1), we get ( ) ( ) y ()(y) = y ()(y) (0) =0 or all, y X.Thus, ( ) y = ()(y) (3.) or all, y X. Letting y =0in(3.), we get ( )=() So, ( ) y ( y)= = ()(y) or all, y X,asdesired. Theorem 3. Let be a positive real number with <.Assume that a mapping : X Ysatisiestheinequality ( ) ()(y) (z) y z (, y, z), (3.3) where : X 3 [0, ) satisies (, y, z):= j=1 ( ((, ( ) j ( ) j y, z) < (3.4) or all, y, z X. Then there eists a unique additive mapping A : X Ysuchthat A() () 1 (,, ) (,,0) (3.5)

7 Lu and Park Journal o Inequalities and Applications 01, 01:94 Page 7 o 10 Proo It ollows rom (3.4) that(0,0,0) = 0. Letting = y = z =0in(3.3), we get ( ) (0) (0) (0,0,0)= (0).So,(0) = 0. Letting y =, z =0in(3.3), we get ()( ) (,,0) (3.6) Letting = y =, z = in (3.3), we obtain () ()( ) (,, ) So, ( ) ( ) 1 (, ), (3.7) It ollows rom (3.6)and(3.7)that ( ) l (( ) l ( ) m (( ) m ) ) m 1 ( ) j (( ) j ( ) j1 (( ) j1 ) ) m 1[ ( [( ) j ] ( ) j1 [( ) j1 ] ( ) ( ) j1 [( ) j1 ( ) j1 [( ) j1 ] ( )] ] m 1[ ( ) 1 j ( ) j1 ( ) j1 ( ) j (,, ) ( 1 (( 1, ( ) j1 ( ), 0)] or all nonnegative integers m and l with m > l and all X. It means that the sequence {( )n (( )n )} is a Cauchy sequence SinceY is complete, the sequence {( )n (( )n )} converges. So, we may deine the mapping A : X Y by A() = lim n (( )n (( )n )) Moreover, by letting l = 0 and passing the limit m,weget(3.5). Net, we claim that A() is an additive mapping. It ollows rom (3.6)that ( A()A( ) = lim n n = lim n =0 and so A( )= A() ) n (( ) n ( ( ) ( ) n (( ) n ( ) n,,0) ( ) n1 (( ) n1 ( ), ) n ) ( ) n1 ( ) ),0

8 Lu and Park Journal o Inequalities and Applications 01, 01:94 Page 8 o 10 It ollows rom (3.3)that () () = () (0) ( ) (,0, ) Hence, A()A(y) A( y) = A()A(y)A( y) ( ) ( ) = y y A()A(y)A A A( y) ( ) ( ) y A()A(y)A y A( y) A ( ) n [ (( ) n (( ) n (( ) n ) y = lim ) y) n (( ) n ) (( ) n )] y ( y) ( ) n (( ) n ( ) n ( ) n ) y, y, n ( ) n (( ) n ( ) n ) y lim ( y), 0, =0 n or all, y X. So, the mapping A : X Y is an additive mapping. Now, we show the uniqueness o A.AssumethatT : X Y is another additive mapping satisying (3.5). Then we get A() T() ( ) n = lim A n (( ) n (( ) n ) ) T ( ) n [ (( ) n (( ) n ) A ) (( ) n ) (( ) n ] T ) ( ) (( ) n ( ) n n ( ), ( ), ( ) (( ) n ( ) n lim n,,0) =0 n n n ( ) n ) Thus,wemayconcludethatA()=T() This proves the uniqueness o A. So, the mapping A : X Y is a unique additive mapping satisying (3.5). Corollary 3.3 Let p, θ and be positive real numbers with p >1and <.Let : X Y be a mapping satisying ()(y) (z) ( y ) z θ ( p y p z p) (3.8)

9 Lu and Park Journal o Inequalities and Applications 01, 01:94 Page 9 o 10 or all, y, z X. Then there eists a unique additive mapping A : X Ysuchthat 1 () A() ( )p 6 ( )p θ p Theorem 3.4 Let be a real number with >.Assume that a mapping : X Y satisies inequality (3.3), where : X 3 [0, ) satisies (, y, z):= j=0 ( ((, ( ) j ( ) j y, z) < (3.9) or all, y, z X. Then there eists a unique additive mapping A : X Ysuchthat A() () 1 (,, ) ( ),,0 (3.10) Proo It ollows rom (3.9) that(0,0,0) = 0. Letting = y = z =0in(3.3), we get ( ) (0) (0) (0,0,0)= (0).So,(0) = 0. Replacing by in (3.7), we get ( () ) 1 (,, ) (3.11) It ollows rom (3.6)and(3.11)that ( ) l (( m 1 ( ) j m 1[ ( ) l ) (( ( ) j1 [( m 1[ ( ) 1 j ( 1 ( ) m (( ) [( ) j ] (( (( ( ) j1 1 ( )], 1, ( ) j1 ( ) j, ) m ) (( ) j1 ) [( ) j1 ] ( ) ( ) j1 [( ) j1 ] ] ( ) j1 ) ( ) j1 ( ), 0)] or all nonnegative integers m and l with m > l and all X. It means that the sequence {( )n (( )n )} is a Cauchy sequence SinceY is complete, the sequence {( )n (( )n )} converges. So, we may deine the mapping A : X Y by A() = lim n (( )n (( )n ))

10 Lu and Park Journal o Inequalities and Applications 01, 01:94 Page 10 o 10 Moreover, by letting l = 0 and passing the limit m,weget(3.10). The rest o the proo is similar to the proo o Theorem 3.. Corollary 3.5 Let p, θ and be positive real numbers with p >1and >.Let : X Y be a mapping satisying (3.8). Then there eists a unique additive mapping A : X Ysuch that 1 () A() ( )p 6 θ p ( )p Competing interests The authors declare that they have no competing interests. Authors contributions All authors conceived o the study, participated in its design and coordination, drated the manuscript, participated in the sequence alignment, and read and approved the inal manuscript. Author details 1 Department o Mathematics, School o Science, ShenYang University o Technology, Shenyang, , P.R. China. Department o Mathematics, Research Institute or Natural Sciences, Hanyang University, Seoul, , orea. Received: 30 December 011 Accepted: 6 November 01 Published: 1 December 01 Reerences 1. Ulam, SM: A Collection o the Mathematical Problems. Interscience, New York (1960). Hyers, DH: On the stability o the linear unctional equation. Proc. Natl. Acad. Sci. USA 7, -4 (1941) 3. Rassias, TM: On the stability o the linear mapping in Banach spaces. Proc. Am. Math. Soc. 7, (1978) 4. Gajda, Z: On stability o additive mappings. Int. J. Math. Math. Sci. 14, (1991) 5. Rassias, JM: On approimation o approimately linear mappings by linear mappings. Bull. Sci. Math. 108, (1984) 6. Rassias, JM: On approimation o approimately linear mappings by linear mappings. J. Funct. Anal. 46, (198) 7. Rassias, JM: On a new approimation o approimately linear mappings by linear mappings. Discuss. Math. 7, (1985) 8. Jung, S: Hyers-Ulam-Rassias Stability o Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (001) 9. Lu, G, Park, C: Hyers-Ulam stability o additive set-valued unctional equations. Appl. Math. Lett. 4, (011) 10. Park, C: Homomorphisms between Poisson JC * -algebra. Bull. Braz. Math. Soc. 36, (005) 11. Park, C: Hyers-Ulam-Rassias stability o homomorphisms in quasi-banach algebras. Bull. Sci. Math. 13, (008) 1. Park, C, Cho, YS, Han, M: Functional inequalities associated with Jordan-von Neumann type additive unctional equations. J. Inequal. Appl. 007, Article ID 4180 (007) 13. Cho, YJ, Park, C, Saadati, R: Functional inequalities in non-archimedean Banach spaces. Appl. Math. Lett. 3, (010) 14. Lu, G, Park, C: Functional inequality in Fréchet spaces. Preprint doi: /109-4x Cite this article as: Lu and Park: Additive unctional inequalities in Banach spaces. Journal o Inequalities and Applications 01 01:94.