Research Article A Fixed Point Approach to the Stability of Quadratic Functional Equation with Involution

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1 Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 008, Article ID 73086, 11 pages doi: /008/73086 Research Article A Fixed Point Approach to the Stability of Quadratic Functional Equation with Involution Soon-Mo Jung 1 and Zoon-Hee Lee 1 Mathematics Section, College of Science and Technology, Hong-Ik University, Chochiwon, South Korea Department of Mathematics, Chungnam National University, Deajeon, South Korea Correspondence should be addressed to Soon-Mo Jung, smjung@hongik.ac.kr Received 7 September 007; Accepted 6 November 007 Recommended by Tomas Domínguez Benavides Cădariu and Radu applied the fixed point method to the investigation of Cauchy and Jensen functional equations. In this paper, we will adopt the idea of Cădariu and Radu to prove the Hyers- Ulam-Rassias stability of the quadratic functional equation with involution. Copyright q 008 S.-M. Jung and Z.-H. Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In 190, Ulam 1 gave a wide ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms. Let G 1 be a group and let G be a metric group with the metric d,. Givenε>0, does there exist a δ>0such that if a function h : G 1 G satisfies the inequality dhxy,hxhy <δfor all x, y G 1, then there exists a homomorphism H : G 1 G with dhx,hx <εfor all x G 1? The case of approximately additive functions was solved by Hyers under the assumption that G 1 and G are Banach spaces. Indeed, he proved that each solution of the inequality fxy fx fy ε, for all x and y, can be approximated by an exact solution, say an additive function. Rassias 3 attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows: ( fx y fx fy ε x p y p, 1.1 and generalized the result of Hyers. Since then, the stability problems for several functional equations have been extensively investigated.

2 Fixed Point Theory and Applications The terminology Hyers-Ulam-Rassias stability originates from these historical backgrounds. The terminology can also be applied to the case of other functional equations. For more detailed definitions of such terminologies, we can refer to 9. Let E 1 and E be real vector spaces. If an additive function σ : E 1 E 1 satisfies σ x for all x E 1,thenσ is called an involution of E 1 see 10, 11. For a given involution σ : E 1 E 1, the functional equation fx yf σy fxfy 1. is called the quadratic functional equation with involution. According to 11, Corollary 8, a function f : E 1 E is a solution of 1. if and only if there exists an additive function A : E 1 E and a biadditive symmetric function B : E 1 E 1 E such that A Ax, B,y Bx, y and fx Bx, xax for all x E 1. Indeed, if we set σ I in 1.,whereI : E 1 E 1 denotes the identity function, then 1. reduces to the additive functional equation fx y fxfy. 1.3 On the other hand, if σ I in 1., then1. is transformed into the quadratic functional equation fx yfx y fxfy. 1. Recently, Belaid et al. have proved the Hyers-Ulam-Rassias stability of the quadratic functional equation with involution 1.see 10. In this paper, we will apply the fixed point method to prove the Hyers-Ulam-Rassias stability of the functional equation 1. for a large class of functions from a vector space into a complete -normed space. We remark that Isac and Rassias 1 were the first to apply the Hyers-Ulam-Rassias stability approach for the proof of new fixed point theorems.. Preliminaries Let X be a set. A function d : X X 0, is called a generalized metric on X if and only if d satisfies M 1 dx, y 0, if and only if x y; M dx, y dy, x, for all x, y X; M 3 dx, z dx, ydy, z, for all x, y, z X. Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity. We now introduce one of fundamental results of fixed point theory. For the proof, refer to 13. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. 1. Theorem.1. Let X, d be a generalized complete metric space. Assume that Λ : X X is a strictly contractive operator with the Lipschitz constant 0 <L<1. If there exists a nonnegative integer k such that dλ k1 x, Λ k x < for some x X, then the followings are true:

3 S.-M. Jung and Z.-H. Lee 3 a the sequence {Λ n x} converges to a fixed point x of Λ; b x is the unique fixed point of Λ in X { y X : d ( Λ k x, y < } ;.1 c if y X,then d ( y, x 1 dλy, y.. 1 L Throughout this paper, we fix a real number with 0 < 1andletK denote either R or C. Suppose E is a vector space over K. Afunction : E 0, is called a -norm if and only if it satisfies N 1 x 0, if and only if x 0; N λx λ x, for all λ K and all x E; N 3 x y x y, for all x, y E. Recently, Cădariu and Radu 15 applied the fixed point method to the investigation of the Cauchy additive functional equation see 16, 17. Using such a clever idea, they could present a short, simple proof for the Hyers-Ulam stability of Cauchy and Jensen functional equations. 3. Main results In this section, by using an idea of Cădariu and Radu see 15, 16, we will prove the Hyers- Ulam-Rassias stability of the quadratic functional equation with involution 1.. Theorem 3.1. Let E 1 be a vector space over K and let E be a complete -normed space over K,where isafixedrealnumberwith0 < 1. Suppose a function ϕ : E 1 E 1 0, is given and there exists a constant L, 0 <L<1, such that ϕx, y Lϕx, y, ϕ,y σy Lϕx, y for all x, y E 1. Furthermore, let f : E 1 E be a function satisfying the inequality ( fx yf x σy fx fy ϕx, y for all x, y E 1,whereσ : E 1 E 1 is an involution of E 1. Then there exists a unique solution T : E 1 E of 1. such that for all x E 1. fx Tx 1 1 ϕx, x L

4 Fixed Point Theory and Applications Proof. First, let us define X to be the set of all functions h : E 1 E and introduce a generalized metric on X as follows: dg,hinf { C 0, : gx hx Cϕx, x x E 1 }. 3. Let {f n } be a Cauchy sequence in X, d. According to the definition of Cauchy sequences, there exists, for any given ε > 0, a positive integer N ε such that df m,f n ε for all m, n N ε. By considering the definition of the generalized metric d, we see that ε>0 N ε N m, n N ε x E 1 : f m x f n x εϕx, x. 3.5 If x is any given point of E 1, 3.5 implies that {f n x} is a Cauchy sequence in E. Since E is complete, {f n x} converges in E for each x E 1. Hence, we can define a function f : E 1 E by fx lim n f n x 3.6 for any x E 1. If we let m increase to infinity, it follows from 3.5 that for any ε > 0, there exists a positive integer N ε with f n x fx εϕx, x for all n N ε and for all x E 1,thatis, for any ε>0, there exists a positive integer N ε such that df n,f ε for any n N ε.thisfact leads us to a conclusion that {f n } converges in X, d. Hence, X, d is a complete space cf. the proof of 15, Theorem.5. We now define an operator Λ : X X by Λhx 1 [ ( ] hxh x 3.7 for all x E 1. First, we assert that Λ is strictly contractive on X. Giveng,h X, letc 0, be an arbitrary constant with dg,h C,thatis, gx hx Cϕx, x 3.8 for all x E 1.Ifwereplacey by x in 3.,thenweobtain ( fxf x fx ϕx, x 3.9 for every x E 1. It follows from 3.1 and 3.8 that Λgx Λhx 1 gxg hx h 1 gx hx 1 g h 3.10 C ϕx, x C ϕ,x LCϕx, x

5 S.-M. Jung and Z.-H. Lee 5 for all x E 1,thatis,dΛg,Λh LC. We hence conclude that dλg,λh Ldg,h for any g,h X. Therefore, Λ is strictly contractive because L is a constant with 0 <L<1. Next, we assert that dλf, f <. Ifweputy x in 3. and we divide both sides by,thenweget Λfx fx [ 1 ( ] fxf x fx 1 ϕx, x 3.11 for any x E 1,thatis, dλf, f 1 <. 3.1 Now, it follows from Theorem.1a that there exists a function T : E 1 E which is a fixed point of Λ, such that dλ n f, T 0asn. By mathematical induction, we can easily show and hence we can omit to show that ( Λ n f x 1 n [ f ( n x ( n 1 f ( n 1 x n 1 ] 3.13 for each n N. Since dλ n f, T 0asn, there exists a sequence {C n } such that C n 0asn and dλ n f, T C n for every n N. Hence, it follows from the definition of d that ( Λ n f x Tx C n ϕx, x 3.1 for all x E 1. Thus, for each fixed x E 1,wehave lim ( Λ n f x Tx 0. n 3.15 Therefore 1 [ ( n ( n ( n 1 Tx lim f x 1 f x n 1 ] 3.16 n n for all x E 1. It follows from 3.1, 3., and3.16 that Tx yt σy Tx Ty 1 lim f ( n x n y ( n 1 f ( n 1 x y n 1( σy n n f ( n x n σy ( n 1 f ( n 1 σy n 1 ( y f ( n x n 1 f ( n 1 f ( n y n 1 f ( n 1 ( y σy,

6 6 Fixed Point Theory and Applications 1 lim f ( n x n y f ( n x n σy f ( n x f ( n y n n ( n 1 lim f ( n 1 ( x n 1 ( y σy f ( n 1 ( x n 1 ( y σy n n f ( n 1 f ( n 1( y σy 1 lim n ϕ( n x, n y ( n 1 lim ϕ ( n 1, n 1 ( y σy n n n ( 1 n ( n ( 1 n lim n n L ϕx, y lim n n L ϕx, y for all x, y E 1, which implies that T is a solution of 1.. By Theorem.1c and by 3.1,weobtain df, T 1 1 L dλf, f 1 1 L, 3.18 that is, 3.3 is true for all x E 1. Assume that T 1 : E 1 E is another solution of 1. satisfying 3.3. We know that T 1 is a fixed point of Λ. In view of 3.3 and the definition of d, we can conclude that 3.18 is true with T 1 in place of T. DuetoTheorem.1b, wegett T 1. This proves the uniqueness of T. In a similar way, by applying Theorem.1, we can prove the following theorem. Theorem 3.. Let E 1 be a vector space over K and let E be a complete -normed space over K, where is a fixed real number with 0 < 1. Assume that a function ϕ : E 1 E 1 0, is given and there exists a constant L, 0 <L<1, such that ϕx, y L ϕx, y, ϕ,y σy ϕx, y 3.19 for all x, y E 1. Furthermore, let f : E 1 E be a function satisfying 3. for all x, y E 1,where σ : E 1 E 1 is an involution of E 1. Then there exists a unique solution T : E 1 E of 1. such that for all x E 1. fx Tx 1 L ϕx, x L Proof. We use the same definitions for X and d as in the proof of Theorem 3.1. Then, we can similarly prove that X, d is complete. Let us define an operator Λ : X X by [ Λhx h 1 h ] 3.1

7 S.-M. Jung and Z.-H. Lee 7 for all x E 1. By induction, we can prove that ( Λ n f [ ( ( ( ] x 1 x x n f n n 1 f n1 n1 3. for all x E 1 and for every n N. We applythe same argument as in the proofoftheorem 3.1 andprove that Λ is a strictly contractive operator. Given g,h X, letc 0, be an arbitrary constant with dg,h C, that is, gx hx Cϕx, x for all x E 1. It then follows from 3.19 and 3.1 that Λgx Λhx g g ( x 1 g h ( ( x x h g Cϕ, x Cϕ, x 1 h ( x h LCϕx, x for all x E 1,thatis,dΛg,Λh Ldg,h. If we replace x/, respectively, x/ /, for x and y in 3.,thenweobtain fxf 3.3 ( ( x x f ϕ, x, 3. respectively, f ( x f 1 ϕ, x. 3.5 Therefore, it follows from 3.19, 3.1, 3.,and3.5 that fx Λfx [f fx ( x fxf ( x 1 f ] ( x f ϕ, x 1 ϕ, x ( x f 1 Lϕx, x ( x f 3.6 for all x E 1. This means that dλf, f 1 L. 3.7

8 8 Fixed Point Theory and Applications According to Theorem.1a there exists a unique function T : E 1 E, which is a fixed point of Λ, such that [ ( ( ( ] x 1 x Tx lim n f n n n 1 f n1 n1 3.8 for all x E 1. Analogously to the proof of Theorem 3.1, we can show that T is a solution of 1.. Using Theorem.1c and 3.7,weget which implies the validity of 3.0. df, T 1 L 1 L, 3.9 In the following corollaries, we will investigate some special cases of Theorems 3.1 and 3.. Corollary 3.3. Fix a nonnegative number p less than 1 and choose a constant with p 1/ < 1. LetE 1 be a normed space over K and let E be a complete -normed space over K. If a function f : E 1 E satisfies fx yf σy fx fy ε ( x p y p 3.30 and x p p x p for all x E 1 and for some ε>0, then there exists a unique solution T : E 1 E of 1. such that for any x E 1. fx Tx ε p1 x p 3.31 Proof. If we set ϕx, y ε x p y p for all x, y E 1 and if we set L p1 /,thenwehave 0 <L<1and for all x, y E 1. Furthermore, we get ϕ, y p ε ( x p y p Lϕx, y 3.3 ϕ,y σy Lϕx, y 3.33 for any x, y E 1. According to Theorem 3.1, there exists a unique solution T : E 1 E of 1. such that 3.31 holds for every x E 1.

9 S.-M. Jung and Z.-H. Lee 9 Remark 3.. It may be remarked that if we set p 0and 1inCorollary 3.3, then it reduces to 10, Theorem.1. If we set x in Corollary 3.3, then x p 0 p p x p is true for all x E 1. In this case, 3.30 reduces to fx yfx y fx fy ε ( x p y p, 3.3 and the quadratic function T is defined by 1 Tx lim n f( n x n For the case when x and 1, Corollary 3.3 reduces to 10, Corollary 3.3. If we let x in Corollary 3.3, then x p p x p holds for all x E 1, 3.30 reduces to fx y fx fy ε ( x p y p, 3.36 and the additive function T is given by 1 Tx lim n n f( n x If we set x and 1, then the upper bound of 3.31 is smaller than that of 10, Corollary 3.. Corollary 3.5. Fix a number p larger than 1 and choose a constant with 0 <<p 1/. LetE 1 be a normed space over K and let E be a complete -normed space over K. If a function f : E 1 E satisfies 3.30 and x p p x p for all x, y E 1 and for some ε>0, then there exists a unique solution T : E 1 E of 1. such that for any x E 1. fx Tx ε p 1 x p 3.38 Proof. If we set ϕx, y ε x p y p for all x, y E 1 and if we set L / p 1,thenwehave 0 <L<1and ϕx, y ε ( x p y p L ϕ, y 3.39 for all x, y E 1. Furthermore, we get ϕ,y σy ϕ, y 3.0 for any x, y E 1. According to Theorem 3., there exists a unique solution T : E 1 E of 1. such that 3.38 holds for every x E 1.

10 10 Fixed Point Theory and Applications Remark 3.6. If x in Corollary 3.5, then x p 0 p p x p is true for all x E 1. In this case, 3.30 reduces to fx yfx y fx fy ε ( x p y p, 3.1 and the quadratic function T is defined by ( x Tx lim n f n n 3. for all x E 1.Ifwelet x, p>3and 1inCorollary 3.5, then the upper bound of 3.38 is smaller than that of 10, Corollary.3. We cannot expect the Hyers-Ulam-Rassias stability for 3.1 when p and the range space E of the relevant functions f is a Banach space i.e., E is a complete 1-normed space see 18. However, if E is a complete -normed space over K,where is a fixed real number with 0 <<1/, then 3.1 is stable in the sense of Hyers, Ulam, and Rassias in spite of p. If we set x in Corollary 3.5, then x p p x p p x p for all x E 1, 3.30 reduces to fx y fx fy ε ( x p y p, 3.3 and the additive function T is given by ( x Tx lim n f n n. 3. Unfortunately, if we set x, p>3and 1inCorollary 3.5, then the upper bound of 3.38 is larger than that of 10, Corollary.. References 1 S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, D. H. Hyers, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, vol. 7, no., pp., Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, vol. 7, no., pp , G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Mathematicae, vol. 50, no. 1-, pp , D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, vol.3of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Mathematicae, vol., no. -3, pp , S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations, Dynamic Systems and Applications, vol. 6, no., pp , S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematicae, vol. 6, no. 1, pp , B. Belaid, E. Elhoucien, and Th. M. Rassias, On the genaralized Hyers-Ulam stability of the quadratic functional equation with a general involution, Nonlinear Funct. Anal. Appl. 1, pp. 7 6, 007.

11 S.-M. Jung and Z.-H. Lee H. Stetkær, Functional equations on abelian groups with involution, Aequationes Mathematicae, vol. 5, no. 1-, pp. 1 17, G. Isac and Th. M. Rassias, Stability of ψ-additive mappings: applications to nonlinear analysis, International Journal of Mathematics and Mathematical Sciences, vol. 19, no., pp. 19 8, J. B. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bulletin of the American Mathematical Society, vol. 7, no., pp , D. H. Hyers, G. Isac, and Th. M. Rassias, Topics in Nonlinear Analysis and Applications, World Scientific, River Edge, NJ, USA, L. Cădariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Mathematische Berichte, vol. 36, pp. 3 5, L. Cădariu and V. Radu, Fixed points and the stability of Jensen s functional equation, Journal of Inequalities in Pure and Applied Mathematics, vol., no. 1, article, 7 pages, V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory,vol., no. 1, pp , S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 6, pp. 59 6, 199.