STABILITY OF DRYGAS TYPE FUNCTIONAL EQUATIONS WITH INVOLUTION IN NON-ARCHIMEDEAN BANACH SPACES BY FIXED POINT METHOD

Abstract

In this paper, we consider the following functional equation with involution f(x + y) + f(x + σ(y)) = 2f(x) + f(y) + f(σ(y)) and prove the generalized Hyers-Ulam stability for it when the target space is a non-Archimedean Banach space.

1. Introduction and Preliminaries

In 1940, Ulam [14] posed the following problem concerning the stability of functional equations: Let G1 be a group and let G2 be a meric group with the metric d(·, ·). Given ϵ > 0, does there exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G1, then there exists a homomorphism H : G1 → G2 with d(h(x),H(x)) < ϵ for all x ∈ G1?

Hyers [9] solved the Ulam’s problem for the case of approximately additive functions in Banach spaces. Since then, the stability of several functional equations have been extensively investigated by several mathematicians [1,4,6,7,8,11,12]. The Hyers-Ulam stability for the quadratic functional equation

was proved by Skof [12] for a function f : E1 → E2, where E1 is a normed space and E2 is a Banach space and later by Jung [11] on unbounded domains.

Let X and Y be real vector spaces. For an additive mapping σ : X → X with σ(σ(x)) = x for all x ∈ X, σ is called an involution of X. For a given involution σ : X → X, the functional equation

is called an additive functional equation with involution and a solution of (2) is called an additive mapping with involution. For a given involution σ : X → X, the functional equation

is called the quadratic functional equation with involution and a solution of (3) is called a quadratic mapping with involution. The functional equation (3) has been studied by Stetkær [13] and the generalized Hyers-Ulam stability for (3) has been obtained by Bouikhalene et al. [2,3,10].

A valuation is a function | · | from a field into [0,∞) such that for any r, s ∈ , the following conditions hold: (i) |r| = 0 if and only if r = 0, (ii) |rs| = |r||s|, and (iii) |r + s| ≤ |r| + |s|.

A field is called a valued field if carries a valuation. The usual absolute values of ℝ and ℂ are examples of valuations. If the triangle inequality is replaced by |r + s| ≤ max{|r|, |s|} for all r, s ∈ , then the valuation | · | is called a non-Archimedean valuation and the field with a non-Archimedean valuation is called a non-Archimedean field. If | · | is a non-Archimedean valuation on , then clearly, |1| = | − 1| and |n| ≤ 1 for all n ∈ ℕ.

Definition 1.1. Let X be a vector space over a scalar field with a non-Archimedean nontrivial valuation | · |. A function ║ · ║ : X → ℝ is called a non-Archimedean norm if it satisfies the following conditions:

(a) ║x║ = 0 if and only if x = 0,

(b) ║rx║ = |r|║x║, and

(c) the strong triangle inequality (ultrametric) holds, that is,

for all x, y ∈ X and all r ∈ . If ║ · ║ is a non-Archimedean norm, then (X, ║ · ║) is called a non-Archimedean normed space.

Let (X, ║ · ║) be a non-Archimedean normed space and {xn} a sequence in X. Then a sequence {xn} is said to be convergent in (X, ║ · ║) if there exists x ∈ X such that limn→∞ ║xn − x║ = 0. In that case, x is called the limit of the sequence {xn}, and one denotes it by limn→∞ xn = x. A sequence {xn} is said to be a Cauchy sequence in (X, ║ · ║) if limn→∞ ║xn+p − xn║ = 0 for all p ∈ ℕ. By (c) of Definion 1.1, we have

and so a sequence {xn} is Cauchy in (X, ║·║) if and only if {xn+1−xn} converges to zero in (X, ║ · ║). By a non-Archimedean Banach space we mean one in which every Cauchy sequence is convergent.

In this paper, using fixed point method, we prove the generalized Hyers-Ulam stability for the following functional equation, called the Drygas type functional equation with involution,

in non-Archimedean Banach spaces. The solution of (4) is called the Drygas type equation with involution. Suppose that f is the Drygas type equation with involution. If f is odd, that is, f(σ(x)) = −f(x) for all x ∈ X, then f satisfies (2) and if f is even, that is, f(σ(x)) = f(x) for all x ∈ X, then f satisfies (3).

Throughout this paper, we assume that X is a non-Archimedean normed space and Y is a non-Archimedean Banach space.

 

2. The Generalized Hyers-Ulam stability for (4)

We will prove the generalized Hyers-Ulam stability for (4) in non-Archimedean Banach spaces. For any mapping f : X → Y , let

We start the following propostion.

Proposition 2.1. A mapping f : X → Y satisfies (4) if and only if fo : X → Y is an additive mapping with involution and fe : X → Y is a quadratic mapping with involution. In case, fo is an additve mapping.

Proof. Cleraly, f(0) = 0. By (4), we have

for all x, y ∈ X and hence fo is an additive mapping with involution. Letting x = 0 in (5), we have

for all y ∈ X. Interchanging x and y in (5), we have

for all x, y ∈ X and by (5), (6), and (7), we obtain

for all x, y ∈ X. Hence fo satisfies (2), that is, fo is an additve mapping. Moreover, fe satisfies (3) and thus one has the result. The converse is trivial. □

We apply the fixed point method to investigate the generalized Hyers-Ulam stability for the functional equation (4) with involution in non-Archimedean Banach spaces. Banachs contraction principle is one of the pivotal results of analysis. It is widely considered as the source of the metric fixed point theory. Also, its significance lies in its vast applicability in a number of branches of mathematics. Many kinds of generalizations of the above principle have been a heavily investigated branch of research. In particular, Diaz and Margolis [5] presented the following definition and the fixed point theorem in a generalized complete metric space.

Definition 2.2. Let X be a non-empty set. Then a mapping d : X2 → [0,∞] is called a generalized metric on X if d satisfies the following conditions:

In case, (X, d) is called a generalized metric space.

Theorem 2.3 ([5]). Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with some Lipschitz constant L with 0 < L < 1. Then for each given element x ∈ X, either d(Jnx, Jn+1x) = ∞ for all nonnegative integer n or there exists a positive integer n0 such that

For any mapping f : X → Y , define the difference operator Df by

for all x, y ∈ X.

Theorem 2.4. Assume that ϕ : X2 → [0,∞) is a mapping and there exists a real number L with 0 < L < 1 such that

for all x, y ∈ X. Suppose that |2| < 1. Let f : X → Y be a mapping satisfying f(0) = 0 and

for all x, y ∈ X. Then there exists a unique mapping F : X → Y such that F is a solution of (4),

for all x ∈ X, and

for all x ∈ X.

Proof. Consider the set S = {g | g : X → Y} and the generalized metric d on S defined by

Then (S, d) is a generalized complete metric space(see [11]). Define a mapping J : S → S by

for all g ∈ S and all x ∈ X. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0,∞). Then by (9), we have

for all x ∈ X, because |3| ≤ 1. Hence we have d(Jg, Jh) ≤ Ld(g, h) for any g, h ∈ S and so J is a strictly contractive mapping.

Putting y = x in (10), we get

for all x ∈ X and putting x = σ(x) and y = σ(x) in (10), we get

for all x ∈ X. By (13) and (14), we obtain

for all x ∈ X and hence we have By Theorem 2.3, there exists a mapping F : X → Y which is a fixed point of J such that d(Jnf, F) → 0 as n → ∞ and by (4) in Theorem 2.3, we have (12). By induction, there are sequences {an}, {bn}, and {cn} in ℝ+ such that

for all x ∈ X and all n ∈ ℕ, where

for all n ∈ ℕ. Hence we have

for all x ∈ X and by (16), we have

for all n ∈ ℕ. Moreover, we have

for all n ∈ ℕ. By (16), we have

Hence we have (11).

Now we claim that F satisfies (4). The strongly triangle inequality implies the triangle inequality and for any two convergent sequences (xn) and (yn) in (Y, ║ · ║), we have

Since F is a fixed point of J, JF(x) = F(x) for all x ∈ X. Hence

By (9) and (10), we have

Since |n| ≤ 1 for all n ∈ ℕ, by the strong triangle inequality,

and letting n → ∞ in the last inequality, we have DF(x, y) = 0 for all x, y ∈ X and so F is a solution of (4).

To prove the uniquness of F, assume that H : X → Y is another solution of (4) satisfying (12). By Proposition 2.1, H is a fixed point of J and so by (4), we have

for all x ∈ X. By (3) of Theorem 2.3, F = H. □

As examples of ϕ(x, y) in Theorem 2.4, we can take ϕ(x, y) = ║x║p+║y║p and for some positive real number p. Then we can formulate the following corollary.

Corollary 2.5. Let δ > 0 and p be a real number with p > 3. Suppose that ║x + σ(x)║ ≤ |2| ║x║ for all x ∈ X and |2| < 1. Let f : X → Y is a mapping satisfying f(0) = 0 and

for all x, y ∈ X. Then there exists a unique additive-quadratic mapping F : X → Y with involution such that

for all x ∈ X.

From Theorem 2.4, we obtain the following corollary concerning the stability of (4).

Corollary 2.6. Let αi : [0,∞) → [0,∞) (i = 1, 2, 3) be an increasing mapping satisfying

Let f : X → Y be a mapping such that f(0) = 0 and

for all x, y ∈ X and some δ > 0. Suppose that ║x+σ(x)║ ≤ |2| ║x║ for all x ∈ X and |2| < 1. Then there exists a unique additive-quadratic mapping F : X → Y with involution such that

for all x ∈ X.

Proof. Let ϕ(x, y) = δ[α1(║x║)α1(║y║) + α2(║x║) + α3(║y║)]. Then by (i) and (ii), we have

for all x, y ∈ X and since ║x + σ(x)║ ≤ |2|║x║ and |8| ≤ |4|, we have

for all x, y ∈ X, because αi is increasing. Letting , by Theorem 2.4, there is a unique additive-quadratic mapping F : X → Y with (18). □

As examples of αi in Corollary 2.6, we can take α1(t) = tp and α2(t) = α3(t) = t2p for some positive real number p. Then we can formulate the following example.

Example 2.7. Let δ > 0 and p be a real number with . Suppose that ║x + σ(x)║ ≤ |2|║x║ for all x ∈ X and |2| < 1. Let f : X → Y is a mapping satisfying f(0) = 0 and

for all x, y ∈ X. Then there exists a unique additive-quadratic mapping F : X → Y with involution such that

for all x ∈ X.