Fixed Points and Random Stability of a Generalized Apollonius Type Quadratic Functional Equation

Abstract

Using the fixed-point method, we prove the generalized Hyers-Ulam stability of a generalized Apollonius type quadratic functional equation in random Banach spaces.

1. Introduction

The stability problem of functional equations was originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of the Th. M. Rassias' approach.

On the other hand, in 1982–1998, J. M. Rassias generalized the Hyers' stability result by presenting a weaker condition controlled by a product of different powers of norms.

Theorem 1.1 (see [6–12]).

Assume that there exist constants and such that , and  :  is a mapping from a normed space into a Banach space , such that the inequality

(11)

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

(12)

for all .

The control function was introduced by Ravi et al. [13] and was used in several papers (see [14–19]).

The functional equation 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 of the quadratic functional equation was proved by Skof [20] for mappings  : , where is a normed space and is a Banach space. Cholewa [21] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [22] 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 [23–44]).

In [45], Park and Th. M. Rassias defined and investigated the following generalized Apollonius type quadratic functional equation:

(13)

in Banach spaces.

2. Preliminaries

We define the notion of a random normed space, which goes back to Šerstnev et al. (see, e.g., [46, 47]).

In the sequel, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [47, 48]. Throughout this paper, let be the space of distribution functions, that is,

(21)

and the subset is the set , where denotes the left limit of the function at the point . The space is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all . The maximal element for in this order is the distribution function given by

(22)

Definition 2.1 (see [48]).

A function  :  is a continuous triangular norm (briefly, a -norm) if satisfies the following conditions:

(TN₁) is commutative and associative;

(TN₂) is continuous;

(TN₃) for all ;

(TN₄) whenever and for all .

Typical examples of continuous -norms are and (the Łukasiewicz -norm). Recall (see [46, 49]) that if is a -norm and is a given sequence of numbers in , is defined recurrently by

(23)

is defined as .

Definition 2.2 (see [47]).

A random normed space (briefly, RN-space) is a triple , where is a vector space, is a continuous -norm, and is a mapping from into , such that the following conditions hold:

(RN₁)   for all if and only if ;

(RN₂)   for all , ;

(RN₃)    for all and all .

Every normed space defines a random normed space , where for all , and is the minimum -norm. This space is called the induced random normed space.

Definition 2.3.

Let be an RN-space.

(1) A sequence in is said to be convergent to in if, for every and , there exists a positive integer , such that whenever .

(2) A sequence in is called Cauchy if, for every and , there exists a positive integer , such that whenever .

(3) An RN-space is said to be complete if every Cauchy sequence in is convergent to a point in . A complete RN-space is said to be a random Banach space.

Theorem 2.4 (see [48]).

If is an RN-space and is a sequence, such that , then almost everywhere.

Starting with the paper [50], the stability of some functional equations in the framework of fuzzy normed spaces or random normed spaces has been investigated in [51–57].

Let be a set. A function  :  is called a generalized metric on if satisfies

(1) if and only if ;

(2) for all ;

(3) for all .

Let be a generalized metric space. An operator  :  satisfies a Lipschitz condition with Lipschitz constant if there exists a constant such that for all . If the Lipschitz constant is less than 1, then the operator is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We recall the following theorem by Diaz and Margolis.

Theorem 2.5 (see [58, 59]).

Let be a complete generalized metric space and let  :  be a strictly contractive mapping with Lipschitz constant , then for each given element , either

(24)

for all nonnegative integers , or there exists a positive integer , such that

(1) ,  for all ;

(2) the sequence converges to a fixed-point of ;

(3) is the unique fixed-point of in the set ;

(4) for all .

In 1996, Isac and Th. M Rassias [60] were the first to provide applications of stability theory of functional equations for the proof of new fixed-point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [61–67]).

In this paper, we prove the generalized Hyers-Ulam stability of the generalized Apollonius type quadratic functional equation (1.3) in random Banach space by using the fixed point method.

Throughout this paper, assume that is a vector spaces and is a complete RN-space.

3. Generalized Hyers-Ulam Stability of the Quadratic Functional Equation (1.3) in RN-Spaces

Let , and for a given mapping  : , consider the mapping  : , defined by

(31)

for all .

Using the fixed-point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation in complete RN-spaces.

Theorem 3.1.

Let  :  be a mapping ( is denoted by ) such that, for some ,

(32)

for all and all . Suppose that an even mapping  :  with satisfies the inequality

(33)

for all and all , then there exists a unique quadratic mapping : , such that

(34)

for all and all .

Proof.

Putting and in (3.3), we get

(35)

for all and all . Therefore,

(36)

for all and all .

Let be the set of all even mappings  :  with and introduce a generalized metric on as follows:

(37)

where, as usual, . It is easy to show that is a generalized complete metric space (see [68, Lemma 2.1]).

Now, we define the mapping  : 

(38)

for all and . Let such that . Therefore,

(39)

that is, if , we have . Hence,

(310)

for all , that is, is a strictly contractive self-mapping on with the Lipschitz constant .

It follows from (3.6) that

(311)

for all and all , which means that .

By Theorem 2.5, there exists a unique mapping  : , such that is a fixed point of , that is, for all .

Also, as , which implies the equality

(312)

for all .

It follows from (3.2) and (3.3) that

(313)

for all and all . Letting in (3.13), we find that for all , which implies . By [45, Lemma 2.1], the mapping,  :  is quadratic.

Since is the unique fixed point of in the set  : , is the unique mapping such that

(314)

for all and all . Using the fixed-point alternative, we obtain that

(315)

which implies the inequality

(316)

for all and all . So

(317)

for all and all . This completes the proof.

Theorem 3.2.

Let  :  be a mapping ( is denoted by ) such that, for some ,

(318)

for all and all . Suppose that an even mapping  :  satisfying and (3.3), then there exists a unique quadratic mapping  : , such that

(319)

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 3.1.

We consider the mapping  :  defined by

(320)

for all and . Let , such that , then

(321)

that is, if , we have . This means that

(322)

for all , that is, is a strictly contractive self-mapping on with the Lipschitz constant .

By Theorem 2.5, there exists a unique mapping  : , such that is a fixed point of , that is, for all .

Also, as , which implies the equality

(323)

for all .

It follows from (3.5) that

(324)

for all and all , which implies that .

Since is the unique fixed point of in the set  : , and is the unique mapping, such that

(325)

for all and all . Using the fixed point alternative, we obtain that

(326)

which implies the inequality

(327)

for all and all . So

(328)

for all and all .

The rest of the proof is similar to the proof of Theorem 3.1.