Generalized Hyers-Ulam Stability of the Pexiderized Cauchy Functional Equation in Non-Archimedean Spaces

Abstract

We prove the generalized Hyers-Ulam stability of the Pexiderized Cauchy functional equation in non-Archimedean spaces.

1. Introduction

The stability problem of functional equations was originated from a question of Ulam [1] concerning the stability of group homomorphisms.

Let be a group and let be a metric group with the metric . Given , does there exist a such that, if a function satisfies the inequality for all , then there exists a homomorphism with for all

In other words, we are looking for situations when the homomorphisms are stable, that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it. If we turn our attention to the case of functional equations, we can ask the following question.

When the solutions of an equation differing slightly from a given one must be close to the true solution of the given equation.

For Banach spaces, the Ulam problem was first solved by Hyers [2] in 1941, which states that, if and is a mapping, where are Banach spaces, such that

(1.1)

for all , then there exists a unique additive mapping such that

(1.2)

for all . Rassias [3] succeeded in extending the result of Hyers by weakening the condition for the Cauchy difference to be unbounded. A number of mathematicians were attracted to this result of Rassias and stimulated to investigate the stability problems of functional equations. The stability phenomenon that was introduced and proved by Rassias is called the generalized Hyers-Ulam stability. Forti [4] and Găvruţa [5] have generalized the result of Rassias, which permitted the Cauchy difference to become arbitrary unbounded. 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. A large list of references can be found, for example, in [3, 6–30].

Definition 1.1.

A field equipped with a function (valuation) from into is called a non-Archimedean field if the function satisfies the following conditions:

(1) if and only if ;

(2);

(3) for all .

Clearly, and for all .

Definition 1.2.

Let be a vector space over scaler field with a non-Archimedean nontrivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:

if and only if ;

;

the strong triangle inequality, namely,

(1.3)

for all and .

The pair is called a non-Archimedean space if is non-Archimedean norm on .

It follows from that

(1.4)

for all , where with . Therefore, a sequence is a Cauchy sequence in non-Archimedean space if and only if the sequence converges to zero in . In a complete non-Archimedean space, every Cauchy sequence is convergent.

In 1897, Hensel [31] discovered the -adic number as a number theoretical analogue of power series in complex analysis. Fix a prime number . For any nonzero rational number , there exists a unique integer such that , where and are integers not divisible by . Then defines a non-Archimedean norm on . The completion of with respect to metric , which is denoted by , is called -adic number field. In fact, is the set of all formal series , where are integers. The addition and multiplication between any two elements of are defined naturally. The norm is a non-Archimedean norm on , and it makes a locally compact field (see [32, 33]).

In [34], Arriola and Beyer showed that, if is a continuous mapping for which there exists a fixed such that for all , then there exists a unique additive mapping such that for all . The stability problem of the Cauchy functional equation and quadratic functional equation has been investigated by Moslehian and Rassias [19] in non-Archimedean spaces.

According to Theorem 6 in [16], a mapping satisfying is a solution of the Jensen functional equation

(1.5)

for all if and only if it satisfies the additive Cauchy functional equation .

In this paper, by using the idea of Găvruţa [5], we prove the stability of the Jensen functional equation and the Pexiderized Cauchy functional equation:

(1.6)

2. Generalized Hyers-Ulam Stability of the Jensen Functional Equation

Throughout this section, let be a normed space with norm and a complete non-Archimedean space with norm .

Theorem 2.1.

Let be a function such that

(2.1)

for all and the limit

(2.2)

for all , which is denoted by , exist. Suppose that a mapping with satisfies the inequality

(2.3)

for all . Then the limit

(2.4)

exists for all and is an additive mapping satisfying

(2.5)

for all . Moreover, if

(2.6)

for all , then is a unique additive mapping satisfying (2.5).

Proof.

Letting in (2.3), we get

(2.7)

for all . If we replace in (2.7) by and multiply both sides of (2.7) to , then we have

(2.8)

for all and all nonnegative integers . It follows from (2.1) and (2.8) that the sequence is a Cauchy sequence in for all . Since is complete, the sequence converges for all . Hence one can define the mapping by (2.4).

By induction on , one can conclude that

(2.9)

for all and . By passing the limit in (2.9) and using (2.2), we obtain (2.5).

Now, we show that is additive. It follows from (2.1), (2.3), and (2.4) that

(2.10)

for all . Therefore, the mapping is additive.

To prove the uniqueness of , let be another additive mapping satisfying (2.5). Since

(2.11)

for all , it follows from (2.6) that

(2.12)

for all . So . This completes the proof.

The following theorem is an alternative result of Theorem 2.1, and its proof is similar to the proof of Theorem 2.1.

Theorem 2.2.

Let be a function such that

(2.13)

for all and the limit

(2.14)

for all , denoted by , exist. Suppose that a mapping with satisfies the inequality

(2.15)

for all . Then the limit

(2.16)

exists for all , and is an additive mapping satisfying

(2.17)

for all . Moreover, if

(2.18)

for all , then is a unique additive mapping satisfying (2.17).

3. Generalized Hyers-Ulam Stability of the Pexiderized Cauchy Functional Equation

Throughout this section, let be a normed space with norm and a complete non-Archimedean space with norm .

Theorem 3.1.

Let be a function such that

(3.1)

for all and the limits

(3.2)

(3.3)

(3.4)

exist for all . Suppose that mappings with satisfy the inequality

(3.5)

for all . Then the limits

(3.6)

exist for all and is an additive mapping satisfying

(3.7)

(3.8)

(3.9)

for all . Moreover, if

(3.10)

for all , then is a unique additive mapping satisfying (3.7), (3.8), and (3.9).

Proof.

It follows from (3.5) that

(3.11)

for all . Let

(3.12)

for all . It follows from (3.1) and (3.2) that

(3.13)

for all . By Theorem 2.1, there exists an additive mapping satisfying (3.7) and

(3.14)

for all . From (3.5), we get

(3.15)

for all . Let

(3.16)

for all . By (3.1) and (3.3), we have

(3.17)

for all . By Theorem 2.1, there exists an additive mapping satisfying (3.8) and

(3.18)

for all . Similarly, (3.5) implies that

(3.19)

for all . Let

(3.20)

for all . By (3.1) and (3.4), we have

(3.21)

for all . By Theorem 2.1, there exists an additive mapping satisfying (3.9) and

(3.22)

for all . The uniqueness of , and follows from (3.10).

Now, we show that . Replacing and by and 0 in (3.5), respectively, and dividing both sides of (3.5) by , we get

(3.23)

for all . By passing the limit in (3.23), we conclude that

(3.24)

for all . Similarly, we get for all . Therefore, (3.6) follows from (3.14), (3.18), and (3.22). This completes the proof.

The next theorem is an alternative result of Theorem 3.1.

Theorem 3.2.

Let be a function such that

(3.25)

for all and the limits

(3.26)

exist for all . Suppose that mappings with satisfy the inequality

(3.27)

for all . Then the limits

(3.28)

exist for all and is an additive mapping satisfying

(3.29)

for all . Moreover, if

(3.30)

for all , then is a unique additive mapping satisfying the above inequalities.