hyers ulam rassias rns approximation of euler lagrange type additive mappings

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2012, Article ID 672531, 15 pages doi:10.1155/2012/672531 Research Article Hyers-Ulam-Rassias RNS Approximation of Euler-Lagrange-Type Additive Mappings H Azadi Kenary,1 H Rezaei,1 A Ebadian,2 and A R Zohdi3 Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran Department of Mathematics, Payame Noor University, Tehran, Iran Department of Mathematics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran Correspondence should be addressed to A Ebadian, ebadian.ali@gmail.com Received 24 December 2011; Revised March 2012; Accepted 19 March 2012 Academic Editor: Tadeusz Kaczorek Copyright q 2012 H Azadi Kenary et al 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 Recently the generalized Hyers-Ulam or Hyers-Ulam-Rassias stability of the following functional equation m m mf m 1≤i≤m,i / j ri xi j f −rj xj i ri f x i i ri xi where r1 , , rm ∈ R, proved ∗ in Banach modules over a unital C -algebra It was shown that if m i ri / 0, ri , rj / for some ≤ i < j ≤ m and a mapping f : X → Y satisfies the above mentioned functional equation then the mapping f : X → Y is Cauchy additive In this paper we prove the Hyers-Ulam-Rassias stability of the above mentioned functional equation in random normed spaces briefly RNS Introduction and Preliminaries The stability problem of functional equations originated from a question of Ulam concerning the stability of group homomorphisms Hyers gave a first affirmative partial answer to the question of Ulam for Banach spaces Hyers’ Theorem was generalized by Aoki for additive mappings and by Rassias for linear mappings by considering an unbounded Cauchy difference The paper of Rassias has provided a lot of influence in the development of what we call the generalized Hyers-Ulam stability of functional equations In 1994, a generalization of y p by a Rassias’ theorem was obtained by G˘avrut¸a by replacing the bound x p general control function ϕ x, y The functional equation: f x y f x−y 2f x 2f y , 1.1 is called a quadratic functional equation In particular, every solution of the quadratic functional equation is said to be a quadratic mapping The generalized Hyers-Ulam stability problem for Mathematical Problems in Engineering the quadratic functional equation was proved by Skof for mappings f : X → Y , where X is a normed space and Y is a Banach space Cholewa noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group Czerwik proved the generalized Hyers-Ulam stability of the quadratic functional equation The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem see 2, 4, 5, 9–28 In the sequel, we will adopt the usual terminology, notions, and conventions of the theory of random normed spaces as in 29 Throughout this paper, the spaces of all probability distribution functions are denoted by Δ Elements of Δ are functions F : R ∪ {−∞, ∞} → 0, , such that F is left continuous and nondecreasing on R, F 0 {F ∈ Δ : l− F ∞ 1}, where l− f x and F ∞ It’s clear that the subset D limt → x− f t , is a subset of Δ The space Δ is partially ordered by the usual point-wise ordering of functions, that is, for all t ∈ R, F ≤ G if and only if F t ≤ G t For every a ≥ 0, Ha t is the element of D defined by Ha t ⎧ ⎨0 if t ≤ a ⎩1 if t > a 1.2 One can easily show that the maximal element for Δ in this order is the distribution function H0 t Definition 1.1 A function T : 0, → 0, is a continuous triangular norm briefly a tnorm if T satisfies the following conditions: i T is commutative and associative; ii T is continuous; iii T x, x for all x ∈ 0, ; iv T x, y ≤ T z, w whenever x ≤ z and y ≤ w for all x, y, z, w ∈ 0, xy, Tmax x, y max{a Three typical examples of continuous t-norms are TP x, y a, b Recall that, if T is a t-norm and {xn } is a given of numbers b − 1, 0}, and TM x, y in 0, , Tin xi is defined recursively by Ti1 x1 and Tin xi T Tin−1 xi , xn for n ≥ Definition 1.2 A random normed space briefly RNS is a triple X, μ , T , where X is a vector space, T is a continuous t-norm, and μ : X → D is a mapping such that the following conditions hold i μx t H0 t for all t > if and only if x ii μαx t iii μx y μx t/|α| for all α ∈ R, α / 0, x ∈ X and t ≥ t s ≥ T μx t , μy s , for all x, y ∈ X and t, s ≥ Definition 1.3 Let X, μ , T be an RNS i A sequence {xn } in X is said to be convergent to x ∈ X in X if for all t > 0, limn → ∞ μxn −x t Mathematical Problems in Engineering ii A sequence {xn } in X is said to be Cauchy sequence in X if for all t > 0, limn → ∞ μxn −xm t iii The RN-space X, μ , T is said to be complete if every Cauchy sequence in X is convergent Theorem 1.4 If X, μ , T is RNS and {xn } is a sequence such that xn → x, then limn → ∞ μxn t μx t In this paper, we investigate the generalized Hyers-Ulam stability of the following additive functional equation of Euler-Lagrange type: m ⎛ ⎞ f ⎝−rj xj j ri xi ⎠ 1≤i≤m, i / j m ri f xi mf i m ri xi , 1.3 i where r1 , , rn ∈ R, m i ri / 0, and ri , rj / for some ≤ i < j ≤ m, in random normed spaces Every solution of the functional equation 1.3 is said to be a generalized Euler-Lagrange type additive mapping RNS Approximation of Functional Equation 1.3 Remark 2.1 Throughout this paper, r1 , , rm will be real numbers such that ri , rj / for fixed ≤ i < j ≤ m Theorem 2.2 Let X be a real linear space, Z, μ , be an RN space, ϕ : X n → Z be a function such that for some < α < 2, μϕ 2x1 , ,2xm t ≥ μαϕ x1 , ,xm t f ∀xi ∈ X, t > 2.1 t 2.2 and for all xi ∈ X and t > lim μ n→∞ ϕ 2n x1 , ,2n xm /2n Let Y, μ, be a complete RN space If f : x → Y is a mapping such that for all xi , xj ∈ X and t > μ m j f −rj xj 1≤i≤m, i / j ri xi m i ri f xi −mf m i ri xi t ≥ μ ϕ x1 , ,xm t 2.3 Mathematical Problems in Engineering then there is a unique generalized Euler-Lagrange-type additive mapping EL : X → Y such that, for all x ∈ X and all t > μEL x −f x t ≥ TM TM μ ϕi,j 2−α t , μ ϕi,j x/2ri ,− x/2rj μ ϕi,j 2−α t 0,− x/2rj x/2ri ,0 , TM μ ϕi,j 2−α t , 2−α t , x/ri , x/rj μ ϕi,j 2−α t , x/ri ,0 μ ϕi,j 2−α t 0,x/rj 2.4 Proof For each ≤ k ≤ m with k / i, j, let xk in 2.3 Then we get the following inequality: μλ xi ,xj t ≥ μ ϕi,j xi ,xj t , 2.5 for all xi , xj ∈ X, where ⎞ ⎛ ⎜ ϕi,j x, y : ϕ⎜ ⎝0, , 0, x , 0, , 0, ⎟ y , 0, , 0⎟ ⎠, ith jth 2.6 for all x, y ∈ X and all ≤ i < j ≤ m, and λ xi , xj f −ri xi Letting xi f ri xi − rj xj − 2f ri xi rj xj rj xj 2ri f xi 2rj f xj 2.7 in 2.5 , we get μf −rj xj −f rj xj for all xj ∈ X Similarly, letting xj μf 2rj f xj t ≥ μ ϕi,j 0,xj t , 2.8 xi ,0 t , 2.9 in 2.5 , we get −ri xi −f ri xi 2ri f xi t ≥ μ ϕi,j for all xi ∈ X It follows from 2.5 , 2.8 , and 2.9 that for all xi , xj ∈ X μλ xi ,xj − f −ri xi −f ri xi ≥ TM μ ϕi,j xi ,xj 2ri f xi − f −rj xj −f rj xj t , μ ϕi,j xi , 2rj f xj t t , μ ϕi,j 0, xj t 2.10 Mathematical Problems in Engineering Replacing xi and xj by x/ri and y/rj in 2.10 , we get that μf −x y f x−y −2f x y ≥ TM μ ϕi,j for all x, y ∈ X Putting y μ2f x −2f −x −2f 2x f x x/ri ,y/rj t f y −f −x −f −y t , μ ϕi,j t , μ ϕi,j x/ri ,0 0,y/rj t 2.11 , x in 2.11 , we get t ≥ TM μ ϕi,j t , μ ϕi,j x/ri , x/rj x/ri ,0 t , μ ϕi,j 0,x/rj t , 2.12 for all x ∈ X Replacing x and y by x/2 and − x/2 in 2.11 , respectively, we get μf x f −x t ≥ TM μ ϕi,j x/2ri ,− x/2rj t , μ ϕi,j t , μ ϕi,j x/2ri ,0 t 0,− x/2rj , 2.13 for all x ∈ X It follows from 2.12 and 2.13 that μf 2x −2f x t μf x f −x f −x t , μ2f ≥ TM TM μ ϕi,j x/2ri ,− x/2rj ≥ TM μf x t 2f x −2f −x −2f 2x /2 TM μ ϕi,j x/ri ,x/rj x −2f −x −2f 2x t , μ ϕi,j t , μ ϕi,j t t , μ ϕi,j x/2ri ,0 x/ri ,0 t , μ ϕi,j 0,x/rj t 0,− x/2rj t , , 2.14 for all x ∈ X So μf 2x /2 −f x t ≥ TM TM μ ϕi,j x/2ri ,− x/2rj μ ϕi,j 0,− x/2rj TM μ ϕi,j x/ri ,x/rj t t , μ ϕi,j x/2ri ,0 t , , 2t , μ ϕi,j x/ri ,0 2t , μ ϕi,j 0,x/rj 2t 2.15 Mathematical Problems in Engineering Replacing x by 2n x in 2.15 and using 2.1 , we get μf 2n x /2n t − f 2n x /2n ≥ TM TM μ ϕi,j T M μ ϕi,j 2n t 2n x/ri , 2n x/rj μ ϕi,j 2n x/ri ,0 2n t , μ ϕi,j 3αn x/2ri ,− x/2rj TM μ ϕi,j , μ ϕi,j 2n t , μ ϕi,j 2n x/2ri ,0 2n t 0,− 2n x/2rj 2n t , , 2n t 0, 2n x/rj ≥ TM TM μ ϕi,j 2n t , μ ϕi,j 2n x/2ri ,− 2n x/2rj 2n t 3αn x/ri ,x/rj , μ ϕi,j x/2ri ,0 2n t , μ ϕi,j 3αn 2n t 3αn x/ri ,0 , μ ϕi,j 2n t 3αn 0,− x/2rj 0,x/rj , 2n t 3αn , 2.16 for all x ∈ X and all n ∈ N Therefore, we have μf n−1 2n x /2n −f x k μ αk t k n−1 n−1 k f ≥ Tkn−10 μ 2k x /2k 2k x − f f 2k x /2k TM TM μ ϕi,j TM μ ϕi,j k − f 2k x /2k ≥ Tkn−10 TM TM μ ϕi,j TM μ ϕi,j /2k αk t 2k x/2ri ,− x/2rj t , μ ϕi,j 2t , μ ϕi,j x/ri ,x/rj x/2ri ,− x/2rj x/ri ,x/rj αk t k x/ri ,0 t , μ ϕi,j 2t , μ ϕi,j t , μ ϕi,j 2t , μ ϕi,j x/2ri ,0 x/ri ,0 x/2ri ,0 0,x/rj t , μ ϕi,j 2t , μ ϕi,j 2t 0,− x/2rj 0,x/rj t 0,− x/2rj 2t t , , , 2.17 Mathematical Problems in Engineering for all x ∈ X This implies that μf t 2n x /2n −f x ≥ TM TM μ ϕi,j μ ϕi,j TM μ ϕi,j μ ϕi,j t x/2ri ,− x/2rj n−1 k t 0,− x/2rj n−1 k n−1 k 3 n−1 k n−1 k αk /2k , 2.18 2t , μ ϕi,j αk /2k 2t 0,x/rj t x/2ri ,0 , αk /2k 2t x/ri ,x/rj , μ ϕi,j αk /2k x/ri ,0 n−1 k , αk /2k αk /2k Replacing x by 2p x in 2.18 , we obtain μf 2n p x /2n ⎛ p − f 2p x /2p t ⎛ ≥ TM ⎝TM ⎝μ ϕi,j ⎛ x/2ri ,− x/2rj ⎝ 0,− x/2rj ⎛ T M ⎝μ ϕi,j ⎝ μ ϕi,j x/ri ,0 p n−1 k p αk /2k p n−1 k p p n−1 k p ⎠, μ ϕi,j x/2ri ,0 ⎞ t ⎝ p n−1 k p αk /2k ⎠, ⎠⎠, ⎞ αk /2k 2t ⎝ ⎛ ⎞⎞ 2t ⎝ ⎛ αk /2k t ⎛ x/ri ,x/rj p n−1 k p ⎛ μ ϕi,j ⎞ t αk /2k ⎠, ⎞ ⎞⎞⎞ ⎛ ⎠μ ϕi,j 0,x/rj 2t ⎝ p n−1 k p αk /2k ⎠⎠⎠ 2.19 Since the right-hand side of the above inequality tends to 1, when p, n → ∞, then ∞ the sequence {f 2k x /2k }n is a Cauchy sequence in complete RN space Y, μ, , so there exists some point EL x ∈ Y such that EL x for all x ∈ X lim n→∞ f 2k x 2k , 2.20 Mathematical Problems in Engineering Fix x ∈ X and put P μf 2n x /2n −f x in 2.19 Then we obtain t ≥ TM TM μ ϕi,j μ ϕi,j 0,− x/2rj TM μ ϕi,j x/ri ,x/rj μ ϕi,j t x/2ri ,− x/2rj n−1 k αk /2k t n−1 k n−1 k αk /2k 2t αk /2k 2t 0,x/rj n−1 k αk /2k , μ ϕi,j t x/2ri ,0 n−1 k , αk /2k , , μ ϕi,j 2t x/ri ,0 n−1 k , αk /2k , 2.21 and so, for every μEL x −f x t > 0, we have ≥ T μEL x − f ≥T 2n x /2n μEL x − f 2n x /2n ,μ f 2n x /2n −f x t , TM TM μ ϕi,j μ ϕi,j μ ϕi,j TM μ ϕi,j μ ϕi,j μ ϕi,j t x/2ri ,− x/2rj n−1 k t x/2ri ,0 n−1 k αk /2k t 0,− x/2rj n−1 k n−1 k 3 αk /2k 2t 0,x/rj n−1 k , αk /2k 2t n−1 k , αk /2k 2t x/ri ,x/rj x/ri ,0 , αk /2k αk /2k , 2.22 Mathematical Problems in Engineering Taking the limit as n → ∞ and using 2.22 , we get μEL x −f t x ≥ TM TM μ ϕi,j 2−α t , μ ϕi,j x/2ri ,− x/2rj μ ϕi,j 2−α t 0,− x/2rj 2−α t , x/2ri ,0 , TM μ ϕi,j μ ϕi,j μ ϕi,j x t ≥ TM TM μ ϕi,j 2−α t , μ ϕi,j x/2ri ,− x/2rj μ ϕi,j TM μ ϕi,j 2−α t 0,− x/2rj x/ri ,x/rj μ ϕi,j 2−α t , x/ri ,0 0,x/rj 2.23 2−α t → in 2.23 , we get Since was arbitrary by taking μEL x −f 2−α t , x/ri ,x/rj 2−α t , , 2−α t , μ ϕi,j 2−α t 0,x/rj x/2ri ,0 x/ri ,0 2.24 2−α t , Replacing xi by 2n xi for all ≤ i ≤ m, in 2.3 , we get for all xi , xj ∈ X and for all t > 0, μ m j f −2n rj xj 1≤i≤m,i / j 2n ri xi m i ri f 2n xi −mf m i 2n ri xi /2n t ≥ μ ϕ 2n x1 , ,2n xm /2n t 2.25 since lim μ n n n n → ∞ ϕ x1 , ,2 xm /2 t 1, 2.26 We conclude that m j ⎛ ⎞ EL⎝−rj xj ri xi ⎠ 1≤i≤m,i / j m i ri EL xi − mEL m i ri xi 2.27 10 Mathematical Problems in Engineering To prove the uniqueness of mapping EL, assume that there exists another mapping 2n EL x and A 2n x 2n A x , A : X → Y which satisfies 2.4 Fix x ∈ X, clearly EL 2n x limn → ∞ μ EL 2n x /2n − A 2n x /2n t , so for all n ∈ N Since μEL x −A x t μ EL 2n x /2n − A 2n x /2n t ≥ μ EL 2n x /2n ≥ TM TM μ ϕi,j 2n x /2n − A 2n x /2n 2n − α t , μ ϕi,j 12αn x/2ri ,− x/2rj 2n − α t 12αn μ ϕi,j 0,− x/2rj T M μ ϕi,j x/ri ,x/rj μ ϕi,j t ,μ f t − f 2n x /2n t 2n − α t , 12αn , 2n − α t , μ ϕi,j 6αn 2n − α t 6αn 0,x/rj x/2ri ,0 t 2n − α t , 6αn x/ri ,0 2.28 Since the right-hand side of the above inequality tends to 1, when n → ∞, therefore, it follows that for all t > 0, μEL x −A x t and so EL x A x This completes the proof Corollary 2.3 Let X be a real linear space, Z, μ , be an RN space, and Y, μ, a complete and satisfying RN space Let < p < 1, z0 ∈ Z and f : X → Y be a mapping with f μ m j f −rj xj 1≤i≤m, i / j ri xi m i ri f m i ri xi xi −mf t ≥μ m k xk p z0 t , 2.29 limn → ∞ f 2n x /2n exists for all x ∈ X and for all xi , xj ∈ X and t > Then the limit EL x defines a unique Euler-Lagrange additive mapping EL : X → Y such that μEL x −f x t ≥ TM TM μ μ TM μ μ x p z0 p 2p ri rj x p z0 − 2p t |ri |p p 2rj ri rj p |ri |p rj p rj − 2p t rj p − 2p t , μ p − 2p t x z0 x p z0 p x p z0 |2ri |p − 2p t , , , μ x p z0 |ri |p − 2p t , , for all x ∈ X and t > Proof Let α 2p and ϕ : X m → Z be defined as ϕ x1 , , xm m k xi p z0 2.30 Mathematical Problems in Engineering 11 Corollary 2.4 Let X be a real linear space, Z, μ , be an RN space, and Y, μ, a complete and satisfying RN space Let z0 ∈ Z and f : X → Y be a mapping with f μ m j f −rj xj 1≤i≤m, i / j ri xi m i ri f xi −mf m i ri xi t ≥ μ δz0 t , 2.31 limn → ∞ f 2n x /2n exists for for all xi ∈ X for all ≤ i ≤ m and all t > Then, the limit C x all x ∈ X and defines a unique Euler-Lagrange additive mapping EL : X → Y such that μEL x −f x t ≥ TM μ δz0 t t , μ δz0 , 2.32 for all x ∈ X and t > and ϕ : X m → Z be defined as ϕ x1 , , xm Proof Let α δz0 Theorem 2.5 Let X be a real linear space, Z, μ , be an RN space, ϕ : X m → Z be a function such that for some < α < 1/2, μ ϕ x1 /2, ,xm /2 t ≥ μ αϕ x1 , ,xm t ∀xi ∈ X, t > 0, 2.33 Let Y, μ, be a f 0 and for all xi ∈ X and t > 0, limn → ∞ μ 2n ϕ x1 /2n , ,xm /2n t complete RN space If f : X → Y is a mapping satisfying 2.3 , then there is a unique generalized Euler-Lagrange-type additive mapping EL : X → Y such that, for all x ∈ X μEL x −f x t ≥ TM TM μ ϕi,j x/ri ,−x/rj μ ϕi,j TM μ ϕi,j 0,− x/rj x/ri ,x/rj μ ϕi,j for all x ∈ X and all t > x/ri ,0 1−2α t , μ ϕi,j 6α 1−2α t 6α x/rj ,0 1−2α t , 6α , 2.34 1−2α t , 3α 1−2α t , μ ϕi,j 3α 0,x/rj 1−2α t 3α , 12 Mathematical Problems in Engineering Proof Replacing x by x/2n μ2n f x/2n −2n f x/2n TM μ ϕi,j t ·6 0,− x/2n rj t , μ ϕi,j ·3 2n t x/2ri ,− x/2rj αn 2n αn 2n αn 2n ·3 2n t x/2ri ,0 αn 2n ·6 t ·3 0,x/2n rj 2n , , ·6 t x/ri ,x/j ·6 t , μ ϕi,j ·3 x/2n ri ,0 , μ ϕi,j t 0,− x/2rj t , 2n · x/2n ri ,0 , 2n x/2n ri ,x/2n rj μ ϕi,j TM μ ϕi,j t , μ ϕi,j 2n · x/2n ri ,− x/2n rj μ ϕi,j ≥ TM TM μ ϕi,j in 2.14 and using 2.33 , we obtain t ≥ TM TM μ ϕi,j , μ ϕi,j t x/ri ,0 αn 2n , μ ϕi,j ·3 t αn 2n 0,x/rj ·3 2.35 So μ2n f n−1 x/2n −f x 2k αk t i ≥ TM TM μ ϕi,j TM μ ϕi,j t , μ ϕi,j x/ri ,− x/rj x/ri ,x/rj t , μ ϕi,j x/ri ,0 t , μ ϕi,j t , μ ϕi,j x/ri ,0 t 0,− x/rj 0,x/rj t 2.36 , , for all x ∈ X This implies that μ2n f x/2n −f x t ≥ TM TM μ ϕi,j x/ri ,− x/rj μ ϕi,j TM μ ϕi,j 0,− x/rj x/ri ,x/rj μ ϕi,j x/ri ,0 t 6α n−1 k , μ ϕi,j 2k αk t 6α n−1 k 3α n−1 k 3α n−1 k 2k αk t 2k αk t 6α n−1 k 2k αk , , 2k αk t x/ri ,0 2.37 , , μ ϕi,j t 0,x/rj 3α n−1 k 2k αk Mathematical Problems in Engineering 13 Proceeding as in the proof of Theorem 2.2, one can easily show that the sequence {2n f x/2n }n∞1 is a Cauchy sequence in complete RN space Y, μ, , so there exists some point EL x ∈ Y such that x , 2n lim 2n f EL x n→∞ 2.38 for all x ∈ X Taking the limit n → ∞ from both sides of the above inequality, we obtain 2.34 The rest of the proof is similar to the proof of Theorem 2.2 Corollary 2.6 Let X be a real linear space, Z, μ , be an RN space and Y, μ, a complete and satisfying RN space Let p > 1, z0 ∈ Z and f : X → Y be a mapping with f μ m j f −rj xj 1≤i≤m, i / j ri xi m i ri f t ≥μ m i ri xi xi −mf m k xi p t , z0 2.39 limn → ∞ 2n f x/2n exists for all for all xi ∈ X for all ≤ i ≤ m and all t > Then the limit EL x x ∈ X and defines a unique Euler-Lagrange additive mapping EL : X → Y such that μEL x −f x t ≥ TM TM μ x z0 μ p 2p ri rj p |ri |p 2rj p p x p z0 μ rj p ri rj 2p − t |ri |p p rj x z0 p p 2p − t x z0 TM μ 2p − t rj p 2p − t , μ p x z0 |2ri |p 2p − t , , , μ x p z0 |ri |p 2p − t , 2.40 , for all x ∈ X and t > Proof Let α m k 2−p and ϕ : X m → Z be defined as ϕ x1 , , xm xi p z0 Corollary 2.7 Let X be a real linear space, Z, μ , be an RN space and Y, μ, a complete and satisfying RN space Let z0 ∈ Z and f : X → Y be a mapping with f μ m j f −rj xj 1≤i≤m, i / j ri xi m i ri f xi −mf m i ri xi t ≥ μδz0 t , 2.41 limn → ∞ 2n f x/2n exists for all x ∈ X and for all xi , xj ∈ X and t > Then, the limit EL x defines a unique Euler-Lagrange additive mapping EL : X → Y such that μEL x −f for all x ∈ X and t > x t ≥ TM μ δz0 4t 2t , μ δz0 3 , 2.42 14 Proof Let α Mathematical Problems in Engineering 1/4 and ϕ : X m → Z be defined as ϕ x1 , , xm δz0 References S M Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1964 D H Hyers, “On the stability of the linear functional 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