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Fuzzy Stability of Additive Functional Inequalities with the Fixed Point Alternative

Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the Cauchy additive functional inequality and of the Cauchy-Jensen additive functional inequality in fuzzy Banach spaces.

1. Introduction and Preliminaries

Katsaras [1] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [24]. In particular, Bag and Samanta [5], following Cheng and Mordeson [6], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [7]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [8].

We use the definition of fuzzy normed spaces given in [5, 9, 10] to investigate a fuzzy version of the generalized Hyers-Ulam stability for the Cauchy additive functional inequality and for the Cauchy-Jensen additive functional inequality in the fuzzy normed vector space setting.

Definition 1.1 (see [5, 911]).

Let be a real vector space. A function is called a m on if for all and all ,

for ;

if and only if for all ;

if ;

;

is a nondecreasing function of and ;

for , is continuous on .

The pair is called a fuzzy normed vector space.

The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [10, 11].

Definition 1.2 (see [5, 911]).

Let be a fuzzy normed vector space. A sequence in is said to be convergent or converge if there exists an such that for all . In this case, is called the limit of the sequence and we denote it by -.

Definition 1.3 (see [5, 9, 10]).

Let be a fuzzy normed vector space. A sequence in is called Cauchy if for each and each there exists an such that for all and all , we have .

It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

We say that a mapping between fuzzy normed vector spaces and is continuous at a point if for each sequence converging to in , then the sequence converges to . If is continuous at each , then is said to be continuous on (see [8]).

The stability problem of functional equations originated from a question of Ulam [12] concerning the stability of group homomorphisms. Hyers [13] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [14] for additive mappings and by Th. M. Rassias [15] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [15] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Gvruta [16] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.

The functional equation

(1.1)

is called a quadratic mapping equation. In particular, every solution of the quadratic functional equation is said to be a quadratic function. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [17] for mappings , where is a normed space and is a Banach space. Cholewa [18] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [19] proved the generalized Hyers-Ulam stability of the quadratic functional equation.

In [20], Jun and Kim considered the following cubic functional equation:

(1.2)

which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping. In [21], Lee et al. considered the following quartic functional equation:

(1.3)

which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping. Quartic functional equations have been investigated in [22, 23].

Surveys of expository results on related advances both in single variables and in multivariables have been given in [24, 25]. 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 [2633]).

Gilányi [34] showed that if satisfies the functional inequality

(1.4)

then satisfies the Jordan-von Neumann functional equation

(1.5)

See also [35]. Fechner [36] and Gilányi [37] proved the generalized Hyers-Ulam stability of the functional inequality (1.4). Park et al. [38] investigated the Cauchy additive functional inequality

(1.6)

and the Cauchy-Jensen additive functional inequality

(1.7)

and proved the generalized Hyers-Ulam stability of the functional inequalities (1.6) and (1.7) in Banach spaces.

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 .

We recall a fundamental result in fixed point theory.

Theorem 1.4 (see [39, 40]).

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

(1.8)

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

(1);

(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 [41] 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 [4247]).

The generalized Hyers-Ulam stability of different functional equations in random normed spaces and in probabilistic normed spaces has been recently studied in [4852].

In [53], Park et al. proved the generalized Hyers-Ulam stability of the functional inequalities (1.6) and (1.7) in fuzzy Banach spaces in the spirit of Hyers, Ulam, and Th. M. Rassias.

This paper is organized as follows. In Section 2, using the fixed point method, we prove the generalized Hyers-Ulam stability of the Cauchy additive functional inequality (1.6) in fuzzy Banach spaces. In Section 3, using fixed point method, we prove the generalized Hyers-Ulam stability of the Cauchy-Jensen additive functional inequality (1.7) in fuzzy Banach spaces.

Throughout this paper, assume that is a vector space and that is a fuzzy Banach space.

2. Fuzzy Stability of the Cauchy Additive Functional Inequality

In this section, using the fixed point method, we prove the generalized Hyers-Ulam stability of the Cauchy additive functional inequality (1.6) in fuzzy Banach spaces.

Theorem 2.1.

Let be a function such that there exists an with

(2.1)

for all . Let be an odd mapping satisfying

(2.2)

for all and all . Then - exists for each and defines an additive mapping such that

(2.3)

for all and all .

Proof.

Since is odd, . So . Letting and replacing by in (2.2), we get

(2.4)

for all .

Consider the set

(2.5)

and introduce the generalized metric on :

(2.6)

where, as usual, . It is easy to show that is complete. (See the proof of Lemma ??2.1 of [49].)

Now we consider the linear mapping such that

(2.7)

for all .

Let be given such that . Then

(2.8)

for all and all . Hence

(2.9)

for all and all . So implies that . This means that

(2.10)

for all .

It follows from (2.4) that

(2.11)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.

  1. (1)

    is a fixed point of , that is,

(2.12)

for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set

(2.13)

This implies that is a unique mapping satisfying (2.12) such that there exists a satisfying

(2.14)

for all .

  1. (2)

    as . This implies the equality

(2.15)

for all .

  1. (3)

    , which implies the inequality

(2.16)

This implies that the inequality (2.3) holds.

By (2.2),

(2.17)

for all , all and all . So

(2.18)

for all , all and all . Since for all and all ,

(2.19)

for all and all . By [53, Lemma ?2.1], the mapping is Cauchy additive, as desired.

Corollary 2.2.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying

(2.20)

for all and all . Then - exists for each and defines an additive mapping such that

(2.21)

for all and all .

Proof.

The proof follows from Theorem 2.1 by taking

(2.22)

for all . Then we can choose and we get the desired result.

Theorem 2.3.

Let be a function such that there exists an with

(2.23)

for all . Let be an odd mapping satisfying (2.2). Then - exists for each and defines an additive mapping such that

(2.24)

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

Consider the linear mapping such that

(2.25)

for all .

Let be given such that . Then

(2.26)

for all and all . Hence

(2.27)

for all and all . So implies that . This means that

(2.28)

for all .

It follows from (2.4) that

(2.29)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.

  1. (1)

    is a fixed point of , that is,

(2.30)

for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set

(2.31)

This implies that is a unique mapping satisfying (2.30) such that there exists a satisfying

(2.32)

for all .

  1. (2)

    as . This implies the equality

(2.33)

for all .

  1. (3)

    , which implies the inequality

(2.34)

This implies that the inequality (2.24) holds.

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.4.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying (2.20). Then - exists for each and defines an additive mapping such that

(2.35)

for all and all .

Proof.

The proof follows from Theorem 2.3 by taking

(2.36)

for all . Then we can choose and we get the desired result.

3. Fuzzy Stability of the Cauchy-Jensen Additive Functional Inequality

In this section, using the fixed point method, we prove the generalized Hyers-Ulam stability of the Cauchy-Jensen additive functional inequality (1.7) in fuzzy Banach spaces.

Theorem 3.1.

Let be a function such that there exists an with

(3.1)

for all . Let be an odd mapping satisfying

(3.2)

for all and all . Then - exists for each and defines an additive mapping such that

(3.3)

for all and all .

Proof.

Letting in (3.2), we get

(3.4)

for all .

Consider the set

(3.5)

and introduce the generalized metric on :

(3.6)

where, as usual, . It is easy to show that is complete. (See the proof of Lemma ?2.1 of [49].)

Now we consider the linear mapping such that

(3.7)

for all .

Let be given such that . Then

(3.8)

for all and all . Hence

(3.9)

for all and all . So implies that . This means that

(3.10)

for all .

It follows from (3.4) that

(3.11)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.

  1. (1)

    is a fixed point of , that is,

(3.12)

for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set

(3.13)

This implies that is a unique mapping satisfying (3.12) such that there exists a satisfying

(3.14)

for all .

  1. (2)

    as . This implies the equality

(3.15)

for all .

  1. (3)

    , which implies the inequality

(3.16)

This implies that the inequality (3.3) holds.

The rest of proof is similar to the proof of Theorem 2.1.

Corollary 3.2.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying

(3.17)

for all and all . Then - exists for each and defines an additive mapping such that

(3.18)

for all and all .

Proof.

The proof follows from Theorem 3.1 by taking

(3.19)

for all . Then we can choose and we get the desired result.

Theorem 3.3.

Let be a function such that there exists an with

(3.20)

for all . Let be an odd mapping satisfying (3.2). Then - exists for each and defines an additive mapping such that

(3.21)

for all and all .

Proof.

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

Consider the linear mapping such that

(3.22)

for all .

Let be given such that . Then

(3.23)

for all and all . Hence

(3.24)

for all and all . So implies that . This means that

(3.25)

for all .

It follows from (3.4) that

(3.26)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.

  1. (1)

    is a fixed point of , that is,

(3.27)

for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set

(3.28)

This implies that is a unique mapping satisfying (3.27) such that there exists a satisfying

(3.29)

for all .

  1. (2)

    as . This implies the equality

(3.30)

for all .

  1. (3)

    , which implies the inequality

(3.31)

This implies that the inequality (3.21) holds.

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 3.4.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying (3.17). Then - exists for each and defines an additive mapping such that

(3.32)

for all and all .

Proof.

The proof follows from Theorem 3.3 by taking

(3.33)

for all . Then we can choose and we get the desired result.

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Acknowledgment

This work was supported by the Hanyang University in 2009.

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Correspondence to Choonkil Park.

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Park, C. Fuzzy Stability of Additive Functional Inequalities with the Fixed Point Alternative. J Inequal Appl 2009, 410576 (2010). https://doi.org/10.1155/2009/410576

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Keywords

  • Additive Mapping
  • Unique Fixed Point
  • Functional Inequality
  • Normed Vector Space
  • Quadratic Functional Equation