Research Article Stability of Approximate Quadratic Mappings

Transcription

1 Hindawi Publishing Cororation Journal of Inequalities and Alications Volume 2010, Article ID , 13 ages doi: /2010/ Research Article Stability of Aroximate Quadratic Maings Hark-Mahn Kim, Minyoung Kim, and Juri Lee Deartment of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon , South Korea Corresondence should be addressed to Juri Lee, Received 23 October 2009; Revised 21 January 2010; Acceted 1 February 2010 Academic Editor: Yeol Je Cho Coyright q 2010 Hark-Mahn Kim et al. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. We investigate the general solution of the quadratic functional equation f 2x y 3f 2x y 4f x y 12f x, in the class of all functions between quasi--normed saces, and then we rove the generalized Hyers-Ulam stability of the equation by using direct method and fixed oint method. 1. Introduction In 1940, Ulam 1 gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved roblems. Among these was the following question concerning the stability of homomorhisms. Let G 1 be a grou and let G 2 be a metric grou with metric ρ,. Givenɛ>0, does there exist a δ>0such that if f : G 1 G 2 satisfies ρ f xy,f x f y <δfor all x, y G 1,thena homomorhism h : G 1 G 2 exists with ρ f x,h x <ɛfor all x G 1? In 1941, the first result concerning the stability of functional equations was resented by Hyers 2. And then Aoki 3 and Bourgin 4 have investigated the stability theorems of functional equations with unbounded Cauchy differences. In 1978, Th. M. Rassias 5 rovided a generalization of Hyers Theorem which allows the Cauchy difference to be unbounded. It was shown by Gajda 6 as well as by Th. M. Rassias and Šemrl 7 that one cannot rove the Rassias tye theorem when 1. Găvruta 8 obtained generalized result of Th. M. Rassias Theorem which allow the Cauchy difference to be controlled by a general unbounded function. J. M. Rassias 9, 10 established a similar stability theorem linear and nonlinear maings with the unbounded Cauchy difference. Let E 1 and E 2 be real vector saces. A function f : E 1 E 2 is called a quadratic function if and only if f is a solution function of the quadratic functional equation: f ( x y ) f ( x y ) 2f x 2f ( y ). 1.1

2 2 Journal of Inequalities and Alications It is well known that a function f between real vector saces is quadratic if and only if there exists a unique symmetric biadditive function B such that f x B x, x for all x, where the maing B is given by B x, y 1/4 f x y f x y. See 11, 12 for the details. The Hyers-Ulam stability of the quadratic functional 1.1 was first roved by Skof 13 for functions f : E 1 E 2, where E 1 is a normed sace and E 2 is a Banach sace. Cholewa 14 demonstrated that Skof s theorem is also valid if E 1 is relaced by an abelian grou. Czerwik 15 roved the Hyers-Ulam stability of quadratic functional 1.1 by the similar way to Th. M. Rassias control function 5. According to the theorem of Borelli and Forti 16,weobtain the following generalization of stability theorem for the quadratic functional 1.1 : letg be an abelian grou and E a Banach sace; let f : G E be a maing with f 0 0 satisfying the inequality f ( x y ) f ( x y ) 2f x 2f ( y ) ϕ ( x, y ) 1.2 for all x, y G. Assume that one of the following conditions 1 Φ ( x, y ) k 0 2 ϕ( 2 k x, 2 k y ) <, 2 k 1 : ( ) x 2 2k ϕ 2, y < k 1 2 k 1 k holds for all x, y G, then there exists a unique quadratic function Q : G E such that f x Q x Φ x, x 1.4 for all x G. The stability roblems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this roblem In this aer, we consider a new quadratic functional equation f ( 2x y ) 3f ( 2x y ) 4f ( x y ) 12f x, 1.5 for all vectors in quasi--normed saces. First, we note that a function f is a solution of the functional 1.5 in the class of all functions between vector saces if and only if the function f is quadratic. Further, we investigate the generalized Hyers-Ulam stability of 1.5 by using direct method and fixed oint method. As a result of the aer, we have a much better ossible estimation of aroximate quadratic maings by quadratic maings than that of Czerwik 15 and Skof Stability of 1.5 Now, we consider some basic concets concerning quasi--normed saces and some reliminary results. We fix a realnumber with 0 < 1andletK denote either R or C.

3 Journal of Inequalities and Alications 3 Let X be a linear sace over K. Aquasi- -norm is a real-valued function on X satisfying the following. 1 x 0 for all x X and x 0 if and only if x 0. 2 λx λ x for all λ K and all x X. 3 There is a constant K 1 such that x y K x y for all x, y X. The air X, is called a quasi--normed sace if is a quasi--norm on X. The smallest ossible K is called the modulus of concavity of.aquasi--banach sace is a comlete quasi--normed sace. A quasi--norm is called a, -norm 0 < 1 if x y x y 2.1 for all x, y X. In this case, a quasi--banach sace is called a, -Banach sace. We can refer to 24, 25 for the concet of quasinormed saces and -Banach saces. Given a -norm, the formula d x, y : x y gives us a translation invariant metric on X. By the Aoki-Rolewicz theorem 25 see also 24, each quasinorm is equivalent to some -norm. In 26, Tabor has investigated a version of the D. H. Hyers, Th. M. Rassias, and Z. Gajda theorem see 5, 6 in quasibanach saces. Recently, J. M. Rassias and Kim 27 have obtained stability results of general additive equations in quasi--normed saces. From now on, let X be a quasi--normed sace with norm and let Y be a, - Banach sace with norm unless we give any secific reference. Now, we are ready to investigate the generalized Hyers-Ulam stability roblem for the functional 1.5 using direct method. Theorem 2.1. Assume that a function f : X Y satisfies Df x, y : f 2x y 3f 2x y 4f x y 12f x ϕ ( x, y ) 2.2 for all x, y X and that ϕ satisfies the following control conditions Φ 1 x : ϕ ( 3 i x, 3 i x ) ( ϕ 3 n x, 3 n y ) <, lim i n 9 n i 0 for all x, y X. Then there exists a unique quadratic function Q : X Y satisfying f x f 0 Q x 2 1 Φ 9 1 x 2.4 for all x X,where f 0 ϕ 0, 0 /12. The function Q is defined as ( f 3 k x ) Q x lim 2.5 k 3 2k

4 4 Journal of Inequalities and Alications Proof. Putting x, y : 0in 2.2, weget f 0 ϕ 0, 0 /12. Relacing y by x in 2.2, we obtain f 3x 9f x 4f 0 ϕ x, x 2.6 Dividing 2.6 by 9,weget 1 9 f 3x f x 1 ϕ x, x for all x X where f x f x f 0 /2, x X. Now letting x : 3 i x and dividing 3 2i in 2.7, we have i 1 f 3i 1 x 1 3 2i f 3i x 1 ( 9 ϕ 3 i x, 3 x) i i Therefore we rove from the inequality 2.8 that for any integers m, n with m>n m f 3m x 1 3 2n f 3n x m 1 f 3 i 1 x f 3i x 3 2 i 1 3 2i i n m 1 i n 1 ( 9 ϕ i ). 3 i x, 3 i x Since the right-hand side of 2.9 tends to zero as n, the sequence { 1/3 2n f 3 n x } is Cauchy for all x X and thus converges by the comleteness of Y. Define Q : X Y by ( 1 Q x lim f 3 n x f 0 ) n 3 2n 2 lim n f 3 n x 3 2n, x X Letting x : 3 n x, y : 3 n y in 2.2, resectively, and dividing both sides by 3 2n and after then taking the limit in the resulting inequality, we have Q 2x y 3Q 2x y 4Q x y 12Q x lim n f 3 n 2x y 3f 3 n 2x y 4f 3 n x y 12f 3 n x 1 lim n 9 n ϕy 3n x, 3 n 0, 9 n 2.11 and so the function Q is quadratic.

5 Journal of Inequalities and Alications 5 Taking the limit in 2.9 with n 0asm,weobtainthat f x f 0 Q x ϕ ( 3 i x, 3 i x ) i 0 9 i, 2.12 which yields the estimation 2.4. To rove the uniqueness of the quadratic function Q subject to 2.4, let us assume that there exists a quadratic function Q : X Y which satisfies 1.5 and the inequality 2.4. Obviously, we obtain that Q x 3 2n Q 3 n x, Q x 3 2n Q 3 n x 2.13 Hence it follows from 2.4 that Q x Q x 1 ( Q 3 n x f 3 n x f 0 3 2n ( 3 ϕ 2 n i i 0 ) 3 n i x, 3 n i 2 x f 3n x f 0 ) Q 3 n x 2 9 j n 1 ( 3 ϕ 2j ) 3 j x, 3 j x 2.14 for all n N. Therefore letting n, one has Q x Q x 0 for all x X, comleting the roof of uniqueness. Theorem 2.2. Assume that a function f : X Y satisfies Df x, y ϕ ( x, y ) 2.15 for all x, y X and that ϕ satisfies conditions Φ 2 x : ( x 9 i ϕ 3, x ) ( x <, lim 9 n ϕ i 3 i n 3 n, y ) 3 n i 1 for all x, y X. Then there exists a unique quadratic function Q : X Y satisfying f x Q x 1 Φ 9 2 x 2.17 The function Q is given by ( ) x Q x lim 3 2n f n 3 n 2.18

6 6 Journal of Inequalities and Alications Proof. In this case, f 0 0since i 1 1/9i ϕ 0, 0 < and so ϕ 0, 0 0 by assumtion. Relacing x by x/3in 2.6,weobtain ( ) x f x 9f 3 ( x ϕ 3, x ) for x X. Therefore we rove from inequality 2.19 that for any integers m, n with m>n 0 ( ) ( ) x x m 1 9m f 3 m 9 n f 3 n 9i f i n m 1 ( x 9 i ϕ i n i n ( ) ( ) x x 9 i 1 f 3 i 3 i 1 3 i 1, x 3 i 1 ) 1 m 1 ( x 9 i 1 ϕ 9 3, x i 1 3 i 1 ) 2.20 Since the right-hand side of 2.20 tendstozeroasn, the sequence {3 2n f x/3 n } is Cauchy for all x X and thus converges by the comleteness of Y. Define Q : X Y by ( ) x Q x lim 3 2n f n 3 n 2.21 Thereafter, alying the same argument as in the roof of Theorem 2.1, we obtain the desired result. We now introduce a fundamental result of fixed oint theory. We refer to 28 for the roof of it, and the reader is referred to aers Theorem 2.3. Let Ω,d be a generalized comlete metric sace (i.e., d may assume infinite values). Assume that Λ : Ω Ω is a strictly contractive oerator with the Lischitz constant 0 <L<1. Then for a given element x Ω one of the following assertions is true: A 1 d Λ k 1 x, Λ k x for all k 0; A 2 there exists a nonnegative integer n 0 such that A 2.1 d Λ n 1 x, Λ n x < for all n n 0 ; A 2.2 the sequence {Λ n x} converges to a fixed oint x of Λ; A 2.3 x is the unique fixed oint of Λ in the set Δ {y Ω : d Λ n 0 x, y < }; A 2.4 d y, x 1/1 L d y, Λy for all y Δ. For an extensive theory of fixed oint theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. 32. In 1996, Isac and Th. M. Rassias 33 alied the stability theory of functional equations to rove fixed oint theorems and study some new alications in nonlinear analysis. Cădariu and Radu 29, 31 and Radu 34 alied the fixed

7 Journal of Inequalities and Alications 7 oint theorem of alternative to the investigation of Cauchy and Jensen functional equations. Recently, Jung et al and Jung and Rassias 41 have obtained the generalized Hyers- Ulam stability of functional equations via the fixed oint method. Now we are ready to investigate the generalized Hyers-Ulam stability roblem for the functional 1.5 using the fixed oint method. Theorem 2.4. Let f : X Y be a function with f 0 0 for which there exists a function ϕ : X 2 0, such that there exists a constant L, 0 <L<1, satisfying the inequalities Df x, y ϕ ( x, y ), ϕ ( 3x, 3y ) 9 Lϕ ( x, y ) for all x, y X. Then there exists a unique quadratic function Q : X Y defined by lim k f 3 k x /3 2k Q x such that f x Q x 1 ϕ x, x L Proof. Let us define Ω to be the set of all functions g : X Y and introduce a generalized metric d on Ω as follows: d ( g,h ) { inf C 0, : } g x h x Cϕ x, x, x X Then it is easy to show that Ω,d is comlete see 37, Proof of Theorem 3.1.Nowwedefine an oerator Λ : Ω Ω by Λg x g 3x, g Ω First, we assert that Λ is strictly contractive with constant L on Ω.Giveng,h Ω, let C 0, be an arbitrary constant with d g,h C, that is, g x h x Cϕ x, x. Then it follows from 2.23 that Λg x Λh x 1 g 3x h 3x 1 Cϕ 3x, 3x 9 9 LCϕ x, x 2.27 for all x X, that is, d Λg,Λh LC for any C 0, with d g,h C. Thus we see that d Λg,Λh Ld g,h for any g,h Ω and so Λ is strictly contractive with constant L on Ω.

8 8 Journal of Inequalities and Alications Next, if we ut x, y : x, x in 2.22 and we divide both sides by 9, then we get f 3x 9 f x 1 f 3x 9f x ϕ x, x 9 for all x X, which imlies d Λf, f 1/9 <. Thus alying Theorem 2.3 to the comlete generalized metric sace Ω,d with contractive constant L,weseefrom A 2.2 of Theorem 2.3 that there exists a function Q : X Y which is a fixed oint of Λ, thatis,q x ΛQ x Q 3x /9, such that d Λ k f, Q 0as k. By mathematical induction we know that Λ k Q x Q( 3 k x ) 3 2k Q x 2.29 for all k N. Since d Λ k f, Q 0ask by A 2.3 of Theorem 2.3, there exists a sequence {C k } such that C k 0ask, and d Λ k f, Q C k for every k N. Hence, it follows from the definition of d that Λ k f x Q x C k ϕ x, x 2.30 This imlies lim Λ k f x Q x 0, k ( f 3 k x ) i.e., lim Q x 2.31 k 3 2k In turn, it follows from 2.22 and 2.23 that DQ x, y 1 Df 3 lim k x, 3 k y k 3 2k 1 ( ) lim k 3 ϕ 3 k x, 3 k y 2k 0 lim k L k ϕ ( x, y ) 2.32 for all x, y X, which imlies that Q is a solution of 1.5 and so the maing Q is quadratic. By A 2.4 of Theorem 2.3, weobtain d ( f, Q ) which yields the inequality L d( Λf, f ) L, 2.33

9 Journal of Inequalities and Alications 9 To rove the uniqueness of Q, assume now that Q 1 : X Y is another quadratic maing satisfying the inequality Then Q 1 is a fixed oint of Λ with d f, Q 1 < in view of the inequality This imlies that Q 1 Δ {g Ω : d f, g < } and so Q Q 1 by A 2.3 of Theorem 2.3. The roof is comlete. By a similar way, one can rove the following theorem using the fixed oint method. Theorem 2.5. Let f : X Y be a function with f 0 0 for which there exists a function ϕ : X 2 0, such that there exists a constant L, 0 <L<1, satisfying the inequalities Df x, y ϕ ( x, y ), ϕ ( x, y ) L 9 ϕ( 3x, 3y ) for all x, y X. Then there exists a unique quadratic function Q : X Y defined by lim k 3 2k f x/3 k Q x such that f x Q x L ϕ x, x L Proof. We use the same notations for Ω and d as in the roof of Theorem 2.4. Thus Ω,d is a comlete generalized metric sace. Let us define an oerator Λ : Ω Ω by Λg x 9g ( ) x, g Ω Then it follows from 2.35 that ( ) ( ) ( Λg x Λh x x x x 9 g h 9 Cϕ 3 3 3, x ) LCϕ x, x for all x X, that is, d Λg,Λh LC. Thus we see that d Λg,Λh Ld g,h for any g,h Ω and so Λ is strictly contractive with constant L on Ω. Next, if we ut x, y : x/3,x/3 in 2.34 and we divide both sides by 1/9, then we get by virtue of 2.35 ( ) x f x 9f 3 ( x ϕ 3, x ) L ϕ x, x for all x X, which imlies d f, Λf L/9 <. Thereafter, alying the same argument as in the roof of Theorem 2.4, we obtain the desired results.

10 10 Journal of Inequalities and Alications 3. Alications of Main Results In the following corollary, we have a stability result of 1.5 in the sense of Th. M. Rassias. Corollary 3.1. Let r i and ε i be real numbers such that max{r i : i 1, 2} < 2 and ε i 0 for i 1, 2. Assume that a function f : X Y satisfies the inequality Df x, y ε 1 x r 1 ε 2 y r for all x, y X, and for all x, y X \{0} if r 1,r 2 < 0. Then there exists a unique quadratic function Q : X Y which satisfies the inequality f x f 0 Q x 2 [ ε 1 x r r 1 ] ε 2 x r 2 1/ r 1 for all x X, and for all x X \{0} if r 1,r 2 < 0. The function Q is given by Q x lim n f 3 n x 3 2n, 3.3 for all x X,wheref 0 0 if r 1,r 2 > 0. Proof. If r 1,r 2 > 0, then we get f 0 0byuttingx, y : 0in 3.1. Letting ϕ x, y : ε 1 x r 1 ε 2 y r 2 for all x, y X and then alying Theorem 2.1 we obtain easily the desired results. Corollary 3.2. Let r i and ε i be real numbers such that min{r i : i 1, 2} > 2 and ε i 0 for i 1, 2. Assume that a function f : X Y satisfies the inequality Df x, y ε 1 x r 1 ε 2 y r for all x, y X. Then there exists a unique quadratic function Q : X Y which satisfies the inequality f x Q x [ ε 1 x r 1 3 ε ] 2 x r 2 1/ 3.5 r r The function Q is given by ( ) x Q x lim 3 2n f n 3 n 3.6

11 Journal of Inequalities and Alications 11 In the following corollary, we have a stability result of 1.5 in the sense of Hyers. Corollary 3.3. Let δ be a nonnegative real number. Assume that a function f : X Y satisfies the inequality Df x, y δ 3.7 for all x, y X. Then there exists a unique quadratic function Q : X Y, defined by Q x lim n f 3 n x /3 2n, which satisfies the inequality f x f 0 Q x δ In the next corollary, we get a stability result of 1.5 in the sense of J. M. Rassias. Corollary 3.4. Let ε, r 1,r 2 be real numbers such that ε 0 and r / 2, wherer : r 1 r 2. Suose that a function f : X Y satisfies Df x, y ε x r 1 y r for all x, y X, and for all x, y X \{0} if r 1,r 2 < 0. Then there exists a unique quadratic function Q : X Y which satisfies the inequality f x f 0 Q x ε x r 2 3 r for all x X and all x, y X \{0} if r 1,r 2 < 0, wheref 0 0 if r 1,r 2 > 0. and alying Theorems 2.1 and 2.2, we get the re- Proof. Letting ϕ x, y : ε x r 1 y r 2 sults. Acknowledgment This work was suorted by the National Research Foundation of Korea Grant funded by the Korean Government no

12 12 Journal of Inequalities and Alications References 1 S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Alied 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. 27, , T. Aoki, On the stability of the linear transformation in Banach saces, Journal of the Mathematical Society of Jaan, vol. 2, , D. G. Bourgin, Classes of transformations and bordering transformations, Bulletin of the American Mathematical Society, vol. 57, , Th. M. Rassias, On the stability of the linear maing in Banach saces, Proceedings of the American Mathematical Society, vol. 72, no. 2, , Z. Gajda, On stability of additive maings, International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, , Th. M. Rassias and P. Šemrl, On the behavior of maings which do not satisfy Hyers-Ulam stability, Proceedings of the American Mathematical Society, vol. 114, no. 4, , P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of aroximately additive maings, Journal of Mathematical Analysis and Alications, vol. 184, no. 3, , J. M. Rassias, On aroximation of aroximately linear maings by linear maings, Bulletin des Sciences Mathématiques, vol. 108, no. 4, , J. M. Rassias, On the stability of the non-linear Euler-Lagrange functional equation in real normed linear saces, Journal of Mathematical and Physical Sciences, vol. 28, no. 5, , J. Aczél and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encycloedia of Mathematics and Its Alications, Cambridge University Press, Cambridge, UK, D. H. Hyers and Th. M. Rassias, Aroximate homomorhisms, Aequationes Mathematicae, vol. 44, no. 2-3, , F. Skof, Local roerties and aroximation of oerators, Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, , P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Mathematicae, vol. 27, no. 1-2, , St. Czerwik, On the stability of the quadratic maing in normed saces, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, , C. Borelli and G. L. Forti, On a general Hyers-Ulam stability result, International Journal of Mathematics and Mathematical Sciences, vol. 18, no. 2, , I.-S. Chang and H.-M. Kim, On the Hyers-Ulam stability of quadratic functional equations, Journal of Inequalities in Pure and Alied Mathematics, vol. 3, no. 3, article 33, 12 ages, G.-L. Forti, Comments on the core of the direct method for roving Hyers-Ulam stability of functional equations, Journal of Mathematical Analysis and Alications, vol. 295, no. 1, , M. Eshaghi Gordji, T. Karimi, and S. Kaboli Gharetaeh, Aroximately n-jordan homomorhisms on Banach algebras, Journal of Inequalities and Alications, vol. 2009, Article ID , 8 ages, M. Eshaghi Gordji, S. Zolfaghari, J. M. Rassias, and M. B. Savadkouhi, Solution and stability of a mixed tye cubic and quartic functional equation in quasi-banach saces, Abstract and Alied Analysis, vol. 2009, Article ID , 14 ages, K.-W. Jun and H.-M. Kim, Ulam stability roblem for generalized A-quadratic maings, Journal of Mathematical Analysis and Alications, vol. 305, no. 2, , J. M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese Journal of Mathematics, vol. 20, no. 2, , J. M. Rassias, On the stability of the general Euler-Lagrange functional equation, Demonstratio Mathematica, vol. 29, no. 4, , Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, vol. 48 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, S. Rolewicz, Metric Linear Saces, PWN-Polish Scientific, Warsaw, Poland, 2nd edition, J. Tabor, Stability of the Cauchy functional equation in quasi-banach saces, Annales Polonici Mathematici, vol. 83, no. 3, , 2004.

13 Journal of Inequalities and Alications J. M. Rassias and H.-M. Kim, Generalized Hyers-Ulam stability for general additive functional equationsinquasi--normed saces, Journal of Mathematical Analysis and Alications, vol. 356, no. 1, , J. B. Diaz and B. Margolis, A fixed oint theorem of the alternative, for contractions on a generalized comlete metric sace, Bulletin of the American Mathematical Society, vol. 74, no. 2, , L. Cădariu and V. Radu, Fixed oints and the stability of Jensen s functional equation, Journal of Inequalities in Pure and Alied Mathematics, vol. 4, no. 1, article 4, 7 ages, L. Cădariu and V. Radu, Fixed oints and the stability of quadratic functional equations, Analele Universităţii din Timişoara. Seria Matematică-Informatică, vol. 41, no. 1, , L. Cădariu and V. Radu, On the stability of the Cauchy functional equation: a fixed oint aroach, in Iteration Theory (ECIT 02), vol. 346 of Grazer Mathematische Berichte, , Karl-Franzens- Universitaet Graz, Graz, Austria, D. H. Hyers, G. Isac, and Th. M. Rassias, Toics in Nonlinear Analysis and Alications, World Scientific, River Edge, NJ, USA, G. Isac and Th. M. Rassias, Stability of ψ-additive maings: alications to nonlinear analysis, International Journal of Mathematics and Mathematical Sciences, vol. 19, no. 2, , V. Radu, The fixed oint alternative and the stability of functional equations, Fixed Point Theory, vol. 4, no. 1, , S.-M. Jung, Stability of the quadratic equation of Pexider tye, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 70, , S.-M. Jung and T.-S. Kim, A fixed oint aroach to the stability of the cubic functional equation, Boletín de la Sociedad Mátematica Mexicana, vol. 12, no. 1, , S.-M. Jung and Z.-H. Lee, A fixed oint aroach to the stability of quadratic functional equation with involution, Fixed Point Theory and Alications, vol. 2008, Article ID , 11 ages, S.-M. Jung, A fixed oint aroach to the stability of isometries, Journal of Mathematical Analysis and Alications, vol. 329, no. 2, , S.-M. Jung, A fixed oint aroach to the stability of a Volterra integral equation, Fixed Point Theory and Alications, vol. 2007, Article ID 57064, 9 ages, S.-M. Jung, T.-S. Kim, and K.-S. Lee, A fixed oint aroach to the stability of quadratic functional equation, Bulletin of the Korean Mathematical Society, vol. 43, no. 3, , S.-M. Jung and J. M. Rassias, A fixed oint aroach to the stability of a functional equation of the siral of Theodorus, Fixed Point Theory and Alications, vol. 2008, Article ID , 7 ages, 2008.