Generalized Ulam—Hyers—Rassias stability of a Cauchy type functional equation

Using the alternative fixed point theorem, we establish the generalized Hyers—Ulam—Rassias stability of a Cauchy type functional equation for functions taking values in arbitrary complete (real or complex) β-normed spaces. Subjclass [2000] : 39B10, 26D20, 39B70, 47H10.


Introduction
In 1940, S. M. Ulam (see [30]) proposed the following problem: Given a group G 1 , a metric group (G 2 , d) and a positive number , does there exist a δ > 0 such that if a function f : G 1 −→ G 2 satisfies the inequality d(f (xy), f(x)f (y)) < δ for all x, y ∈ G 1 , then there exists a homomorphism T : G 1 → G 2 such that d(f (x), T (x)) < for all x ∈ G 1 ?
When this problem has a solution, we say that the homomorphisms from G 1 to G 2 are stable or that the functional equation defining homomorphisms is stable in the sense of Ulam.In 1941, D. H. Hyers [14] gave a partial solution of Ulam , s problem for the case of approximate additive mappings under the assumption that G 1 and G 2 are Banach spaces.Indeed, he proved that each solution of the inequality kf (x + y) − f (x) − f (x)k ≤ , ∀x, y ∈ G 1 can be approximated by an exact solution.That is by an additive mapping.
In 1950, T. Aoki [2] was the second author to study this problem for additive mappings.
In 1978, Th.M. Rassias [21] generalized the result of Hyers by considering the stability problem for unbounded Cauchy differences.This phenomenon of stability introduced by Th.M. Rassias [21] is called the Hyers-Ulam-Rassias stability.
Theorem 1.1.(Th.M. Rassias [21]) Let f : E 1 −→ E 2 be a mapping from a real normed vector space E 1 into a Banach space E 2 satisfying the inequality kf (x + y) − f (x) − f (y)k ≤ (kxk p + kyk p ) ( 1 .1)for all x, y ∈ E 1 , where and p are constants with > 0 and p < 1.Then there exists a unique additive mapping T : (1.2) If p < 0 then inequality (1.1) holds for all x, y 6 = 0, and (1.2) for x 6 = 0. Also, if the function t 7 → f (tx) from R into E 2 is continuous for each fixed x ∈ E, then T is linear.
In [13], Gajda considered also the stability problem with unbounded Cauchy differences.From the papers of Hyers, Rassias and Gajda, we have the following Theorem which completes the results of Theorem 1.1.
Suppose that E 1 is a real normed space, E 2 is a real Banach space, f : E 1 −→ E 2 is a given function, and the following condition holds for some p ∈ [0, +∞) \ {1}.
Then there exists a unique additive function T : It is worth noting that many of the subsequent proofs for Hyers-Ulam-Rassias stability used the Hyers method.Namely, the function T : E 1 −→ E 2 is explicitly constructed, starting from the given function f , by the formulae This method is called a direct method.We observe that the estimate (Est p ) depends on the parameter p.
In [20], V. Radu obtained some stability results via the alternative fixed point theorem.In their paper [6], L. Cǎdariu and V. Radu have used the same fixed point method to establish the stability of functional equations of Jensen type.
It is worth noting that J. A. Baker [3] has started the use of fixed point theorems in the stability theory.In [3], Baker used the Banach principle to study Ulam-Hyers stability of a class of nonlinear functional equations.The results of [3] were extended by M. Akkouchi (see [1]) by using a Ćirić fixed point theorem.
The aim of this paper is to apply the alternative fixed point theorem to deal with functional equations of Cauchy type.That is starting from initial conditions of the form where k is a given positive integer, F is a β-complete normed space over the real or complex field K , G is a linear space over K and ϕ is a control function.
In particular, our results extend both Theorem 1.1 and Theorem 1.2 to the case of functions taking values in arbitrary complete (real or complex) β-normed linear spaces.

Preliminaries
For a nonempty set X, we recall the definition of the generalized metric on We observe that the only one difference of the generalized metric from the usual metric is that the range of the former is allowed to include the infinity.
We now recall one of fundamental results of fixed point theory.For the proof, we refer to [11].
Theorem 2.1.(The alternative of fixed point [11]) Suppose we are given a complete generalized metric space (X, d) and a strictly contractive mapping Λ : X → X, with the Lipschitz constant L.
Then, for each given point x ∈ X, either : there exists a nonnegative integer n 0 such that: Throughout this paper, we fix a real number β with 0 < β ≤ 1 and let K denote either R or C. Suppose F is a vector space over

Main results
Let (G, +) be an abelian group.Let k ≥ 1 be an integer.Let F be a vector space over the (real or complex) field K endowed with a β-norm k.k β .For any function f : G −→ F , we consider the difference defined for all x, y ∈ G In connection with this difference, we have the following Cauchy type functional equation: It is easy to prove the following lemma concerning the solutions of the equation (3.2).Lemma 3.1.Let (G, +) be an abelian group.Let k ≥ 1 be an integer.Let F be a vector space over the (real or complex) field K endowed with a β-norm k.k β .Let a function f : G −→ F be given.
Then the following assertions are equivalent: The first main result of this paper reads as follows.
Theorem 3.2.Let (G, +) be an abelian group.Let k ≥ 1 be an integer.Let (F, k.k β ) be a complete β-normed vector space over the (real or complex) field K endowed with a β-norm k.k β , where 0 < β ≤ 1.Let f : G −→ F be a mapping for which there exists a function ϕ : We suppose also that there exists a constant L, 0 < L < 1 such that and Then there exists a unique additive mapping for all x ∈ G.
Proof.We consider the set For each pair {g, h} of elements of X, we define with the convention inf ∅ := +∞.Then it is easy to see that d ϕ is a generalized distance on the set X.As in [7], one can prove that the generalized metric space (X, d ϕ ) is complete.
We define an operator Λ : X → X by for all x ∈ G. First, we start by proving that Λ is strictly contractive on the (X, d ϕ ).To this end, let g, h ∈ X be given.Without loss of generality, we may suppose that d ϕ (g, h) is finite.In this case, let c ∈ [0, ∞) be any arbitrary constant such that By replacing x in the last inequality by 2kx and making use of (3.4), we have for every x ∈ G, i.e, d ϕ (Λg, Λh) ≤ Lc.This implies that Next, we prove that d ϕ (Λf, f ) is finite.To this end, we set x = y in (3.3).We get which implies that Thus, we may apply Theorem 2.1.It follows that there exists a unique function f * in the set X(f, 0) (see Theorem 2.1) which is fixed by Λ, i.e, Λ(f * ) = f * such that lim n→∞ d ϕ (Λ n g, f * ) = 0 for each g ∈ X(f, 0).In particular, since f ∈ X(f, 0), we have lim n→∞ d ϕ (Λ n f, f * ) = 0, from which we deduce that From (iv) of Theorem 2.1, we obtain which implies that the inequality (3.6) is true for all x ∈ G. Now, we prove that f * is additive.To this respect, we start by substituting 2 n k n x and 2 n k n y for x and y in (3.3), respectively.We obtain which gives after dividing by (2k) nβ the following inequality for all x, y ∈ G.
According to (3.5) and (3.8), by letting n tend to infinity in (3.10), it follows that Thus f * satisfies the functional equation (3.2).By Lemma 3.1, we know that f * must be additive.
Finally, we prove that f * is uniquely determined.Assume that inequality (3.6) is also satisfied with another additive function f : G → F besides f * .As f is an additive function, f satisfies that That is, f is a fixed point of Λ.Since f satisfies (3.6), it follows that By using the triangle inequality, we have Hence f is another fixed point of Λ which belongs to the set Thus, Theorem 3.3 (iv) implies that f = f * .This proves the uniqueness of f * and completes the proof. 2 When the control function φ satisfies the conditions (3.4) and (3.5), the solution f * to the problem of Ulam for the Cauchy type equation (3.2) is given by the formulae If we replace the conditions (3.4) and (3.5) by the conditions (3.12) and (3.13) given below, then we obtain a similar formulae.Precisely, we prove in the next theorem that under these conditions, the solution f * to the problem of Ulam for the equation (3.2) is given by the formulae Theorem 3.3.Let G and (F, k.k β ) be a vecor space over the (real or complex) field K and a complete β-normed sapce over K, respectively with 0 < β ≤ 1.Let k ≥ 1 be a given integer.Let f : G −→ F be a mapping for which there exists a function ϕ : We suppose also that there exists a constant L, 0 < L < 1 such that and lim Then there exists a unique additive mapping for all x ∈ G.
Proof.As in the proof of Theorem 3.2, we consider X the set of functions from G to F .We equipp X with the generalized distance d ϕ .We know that the generalized metric space (X, d ϕ ) is complete.
We define an operator Λ : X → X by By using the same argument as in the proof of Theorem 3.2, one can prove that Λ is a strictly contractive operator.Precisely, we have Moreover, one can prove instead of (3.7).
According to (iii) of Theorem 2.1, there exists a unique function Since the integer n of Theorem 2.1 is 0 and f ∈ X(f, 0), using Theorem 2.1 (iv) and (3.15), we get which implies that the inequality (3.14) is true.
In order to prove that f * is additive, we proceed as in the last proof of Theorem 3.2 by replacing x and y in (3.11) by x 2 n k n and y 2 n k n , respectively.We get , which gives after multiplying by (2k) nβ the following inequality .17) for all x, y ∈ G.
According to (3.13) and (3.16), by letting n tend to infinity in (3.17), it follows that f * (k(x + y)) = kf * (x) + kf * (y), for all x, y ∈ G. Thus f * satisfies the functional equation (3.2) and so, by Lemma 3.1, f * is additive.As in the proof of Theorem 3.2, we prove that f * is unique.This ends the proof. 2 The results obtained in Theorem 3.3 complete those obtained in Theorem 3.2.

Applications
The following result concerns the Ulam-Hyers-Rassias stability of the functional equation (3.2).Corollary 4.1.Let (G, k.k) and (F, k.k β ) be a normed vecor space over the (real or complex) field K and a complete β-normed space over K, respectively, with 0 < β ≤ 1.Let k ≥ 1 be a given integer.Let p be a real number such that 0 for some given number θ > 0.
Then there exists a unique additive mapping f for all x ∈ G.
For every x ∈ G, we have for all x ∈ G.
In addition, we have for some given number δ > 0.
Then there exists a unique additive mapping f * : G → F such that for all x ∈ G.
Corollary 4.2 says that the functional equation (3.2) is stable in the sense of Ulam-Hyers.
In the last result, we deal with the case p > β to complete the study of Ulam-Hyers-Rassias stability for the functional equation (3.2).Corollary 4.3.Let (G, k.k) and (F, k.k β ) be a normed vecor space over the (real or complex) field K and a complete β-normed space over K, respectively, with 0 < β ≤ 1.Let k ≥ 1 be a given integer.Let p be a real number such that p > β. for some given number θ > 0.
In addition, we have as n −→ ∞ for any x, y ∈ G. Thus all the conditions of Theorem 3.2 are satisfied.It follows from Thorem 3.2, that there exists a unique additive function f * : G → F such that the inequality (4.2) holds for all x ∈ G. 2The next result concerns the case p = 0.It is an immediate consequence of Corollary 4.1.Corollary 4.2.Let (G, k.k) and (F, k.k β ) be a normed vecor space over the (real or complex) field K and a complete β-normed space over K, respectively, with 0 < β ≤ 1.Let k ≥ 1 be a given integer.If a function f If a function f : G −→ F satisfies °°°°f (k(x + y)) k − f (x) − f (y) °°°°β ≤ θ(kxk p + kyk p ), ∀x, y ∈ G,(4.5)