Fuzzy Stability of the Pexiderized Quadratic Functional Equation: A Fixed Point Approach

Abstract

The fixed point alternative methods are implemented to give generalized Hyers-Ulam-Rassias stability for the Pexiderized quadratic functional equation in the fuzzy version. This method introduces a metrical context and shows that the stability is related to some fixed point of a suitable operator.

1. Introduction

The aim of this article is to extend the applications of the fixed point alternative method to provide a fuzzy version of Hyers-Ulam-Rassias stability for the functional equation:

(1.1)

which is said to be a Pexiderized quadratic functional equation or called a quadratic functional equation for . During the last two decades, the Hyers-Ulam-Rassias stability of (1.1) has been investigated extensively by several mathematicians for the mapping with more general domains and ranges [1–4]. In view of fuzzy space, Katsaras [5] constructed a fuzzy vector topological structure on the linear space. Later, some other type fuzzy norms and some properties of fuzzy normed linear spaces have been considered by some mathematicians [6–12]. Recently, considerable attention has been increasing to the problem of fuzzy stability of functional equations. Several various fuzzy stability results concerning Cauchy, Jensen, quadratic, and cubic functional equations have been investigated [13–16].

As we see, the powerful method for studying the stability of functional equation was first suggested by Hyers [17] while he was trying to answer the question originated from the problem of Ulam [18], and it is called a direct method because it allows us to construct the additive function directly from the given function . In , Radu [19] proposed the fixed point alternative method for obtaining the existence of exact solutions and error estimations. Subsequently, Miheţ [20] applied the fixed alternative method to study the fuzzy stability of the Jensen functional equation on the fuzzy space which is defined in [14].

Practically, the application of the two methods is successfully extended to obtain a fuzzy approximate solutions to functional equations [14, 20]. A comparison between the direct method and fixed alternative method for functional equations is given in [19]. The fixed alternative method can be considered as an advantage of this method over direct method in the fact that the range of approximate solutions is much more than the latter [14].

2. Preliminaries

Before obtaining the main result, we firstly introduce some useful concepts: a fuzzy normed linear space is a pair , where is a real linear space and is a fuzzy norm on , which is defined as follow.

Definition 2.1 (cf. [6]).

A function (the so-called fuzzy subset) is said to be a fuzzy norm on if for all and all , is left continuous for every and satisfies

(N1) for ;

(N2) if and only if for all ;

(N3) if ;

(N4);

(N5) is a nondecreasing function on and .

Let be a fuzzy normed linear space. A sequence in is said to be convergent if there exists such that . In that case, is called the limit of the sequence and we write .

A sequence in a fuzzy normed space is called Cauchy if for each and , there exists such that . If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space.

We recall the following result by Margolis and Diaz.

Lemma 2.2 (cf. [19, 21]).

Let be a complete generalized metric space and let be a strictly contractive mapping, that is,

(2.1)

for some . Then, for each fixed element , either

(2.2)

or

(2.3)

for some natural number . Moreover, if the second alternative holds, then:

(i)the sequence is convergent to a fixed point of ;

(ii) is the unique fixed point of in the set and .

3. Main Results

We start our works with a fuzzy generalized Hyers-Ulam-Rassias stability theorem for the Pexiderized quadratic functional equation (1.1). Due to some technical reasons, we first examine the stability for odd and even functions and then we apply our results to a general function.

The aim of this section is to give an alternative proof for that result in [15, Section ], based on the fixed point method. Also, our method even provides a better estimation.

Theorem 3.1.

Let be a linear space and let be a fuzzy normed space. Let be a function such that

(3.1)

for some real number with . Let be a fuzzy Banach space and let and be odd functions from to such that

(3.2)

Then there exists a unique additive mapping such that

(3.3)

(3.4)

where .

The next Lemma 3.2 has been proved in [15, Proposition ].

Lemma 3.2.

If , then and ,  .

Proof of Theorem 3.1.

Without loss of generality we may assume that . By changing the roles of and in (3.2), we obtain

(3.5)

It follows from (3.2), (3.5), and (N4) that

(3.6)

Putting in (3.6), we get

(3.7)

Let and introduce the generalized metric define it on by

(3.8)

Then, it is easy to verify that is a complete generalized metric on (see the proof of [22] or [23]). We now define a function by

(3.9)

We assert that is a strictly contractive mapping with the Lipschitz constant . Given , let be an arbitrary constant with . From the definition of , it follows that

(3.10)

Therefore,

(3.11)

Hence, it holds that , that is, , .

Next, from (see Lemma 3.2), it follows that . From the fixed point alternative, we deduce the existence of a fixed point of , that is, the existence of a mapping such that for each . Moreover, we have , which implies

(3.12)

Also, implies the inequality

(3.13)

If is a decreasing sequence converging to , then

(3.14)

Then implies that

(3.15)

that is, (as is left continuous)

(3.16)

The additivity of can be proved in a similar fashion as in the proof of Proposition [15]. It follows from (3.3) and (3.7) that

(3.17)

whence we obtained (3.4).

The uniqueness of follows from the fact that is the unique fixed point of with the property that there exists such that

(3.18)

This completes the proof of the theorem.

Theorem 3.3.

Let be a linear space and let be a fuzzy normed space. Let be a function such that

(3.19)

for some real number with . Let be a fuzzy Banach space and let and be even functions from to such that and

(3.20)

Then there exists a unique quadratic mapping such that

(3.21)

where .

The following Lemma 3.4 has been proved in [15, Proposition ].

Lemma 3.4.

If , then , and ,,, ,, where ,=,,,,,.

Proof of Theorem 3.3.

Without loss of generality we may assume that . By changing the roles of and in (3.20), we obtain

(3.22)

Putting in (3.20), we get

(3.23)

Putting in (3.20), we get

(3.24)

Similarly, put in (3.20) to obtain

(3.25)

Let and introduce the generalized metric define it on by

(3.26)

Then, it is easy to verify that is a complete generalized metric on (see the proof of [22] or [23]). We now define a function by

(3.27)

We assert that is a strictly contractive mapping with the Lipschitz constant . Given , let be an arbitrary constant with . From the definition of , it follows that

(3.28)

Therefore,

(3.29)

Hence, it holds that , that is, , .

Next, from (see Lemma 3.4), it follows that . From the fixed alternative, we deduce the existence of a fixed point of , that is, the existence of a mapping such that for each . Moreover, we have , which implies that

(3.30)

Also, implies the inequality

(3.31)

If is a decreasing sequence converging to , then

(3.32)

Then implies that

(3.33)

that is, (as is left continuous)

(3.34)

The quadratic of can be proved in a similar fashion as in the proof of Proposition [15].

It follows from (3.25) and (3.34) that

(3.35)

whence

(3.36)

A similar inequality holds for . The rest of the proof is similar to the proof of Theorem 3.1.

Theorem 3.5.

Let be a linear space and let be a fuzzy normed space. Let be a function such that

(3.37)

for some real number with . Let be a fuzzy Banach space and let be a mapping from to such that and

(3.38)

Then there exist unique mapping and from to such that is additive, is quadratic, and

(3.39)

where =,, , ,, , ,, , ,, , ,, , ,, .

Proof.

Let for all , then and

(3.40)

Let for all , then and

(3.41)

Using the proofs of Theorems 3.1 and 3.3, we get unique an additive mapping and unique quadratic mapping satisfying

(3.42)

Therefore,

(3.43)

This completes the proof of the theorem.