Random homomorphisms and random derivations in random normed algebras via fixed point method

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

MSC: 47H10, 39B52, 37H10, 60H25, 17B40, 39B72, 47B47, 54E70.

The stability problem of functional equations originated from the question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [4] has provided a lot of influence on the development of what we call Hyers-Ulam stability of functional equations. A generalization of the Rassias theorem was obtained by Găvruta [5] by replacing the unbounded Cauchy difference with a general control function in the spirit of Rassias’ approach. Important contributions to Hyers-Ulam stability were made by Forti [6]. For Jensen’s functional equation stability, significant generalizations were given by Jung [7] and successively, by Lee and Jun [8] by using the direct method (Hyers-Ulam method).

A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [9] for mappings , where X is a normed space and Y is a Banach space. Cholewa [10] noticed that the theorem of Skof is still true if the relevant domain X is replaced with an Abelian group. 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 [11-27]).

In the sequel, we adopt the usual terminology, notations and conventions of the theory of random normed spaces, as in [28-32]. Throughout this paper, is the space of distribution functions, that is, the space of all mappings such that F is left-continuous and non-decreasing on , and . is a subset of consisting of all functions for which , where denotes the left limit of the function f at the point x, that is, . The space is partially ordered by the usual point-wise ordering of functions, i.e., if and only if for all t in . The maximal element for in this order is the distribution function given by

Definition 1.1 ([31])

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

(a) T is commutative and associative;

(b) T is continuous;

(c) for all ;

(d) whenever and for all .

Typical examples of continuous t-norms are , and (the Lukasiewicz t-norm).

Definition 1.2 ([32])

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

() for all if and only if ;

() for all , ;

() for all and all .

Every normed space defines a random normed space , where

for all , and is the minimum t-norm. This space is called the induced random normed space.

Definition 1.3 A random normed algebra is a random normed space with algebraic structure such that () for all and all .

Example 1.4 Every normed algebra defines a random normed algebra , where

for all . This space is called the induced random normed algebra.

Definition 1.5

(1) Let and be random normed algebras. An -linear mapping is called a random homomorphism if for all .

(2) An -linear mapping is called a random derivation if for all .

Definition 1.6 Let be an RN-space.

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

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

(3) An RN-space is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X.

Theorem 1.7 ([31])

Ifis an RN-space andis a sequence such that, thenalmost everywhere.

Let X be a set. A function is called a generalized metric on X if d satisfies the following:

(1) if and only if ;

(2) for all ;

(3) for all .

We recall a fundamental result in fixed point theory.

Theorem 1.8 ([33-35])

Letbe a complete generalized metric space and letbe a strictly contractive mapping with the Lipschitz constant. Then for each given element, either

for all nonnegative integersnor there exists a positive integersuch that

(1) , ;

(2) the sequenceconverges to a fixed pointofJ;

(3) is the unique fixed point ofJin the set;

(4) for all.

In 1996, Isac and Rassias [36] were the first to provide applications of the stability theory of functional equations for the proof of new fixed point theorems with applications. Starting with 2003, the fixed point alternative was applied to investigate the Hyers-Ulam stability for Jensen’s functional equation in [26,33,37] as well as for the Cauchy functional equation in [38] (see also [39] for quadratic functional equations, [40] for monomial functional equations and [41] for operatorial equations etc.). By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [26,29,33,37-40,42-44]).

Gilányi [45] showed that if f satisfies the functional inequality

then f satisfies the Jordan-von Neumann functional equation

See also [46]. Fechner [47] and Gilányi [48] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [49] investigated the Cauchy additive functional inequality

and the Cauchy-Jensen additive functional inequality

and proved the Hyers-Ulam stability of the functional inequalities (1.2) and (1.3) in Banach spaces.

Throughout this paper, assume that is a random normed algebra and that is a complete random normed algebra.

The Hyers-Ulam stability of different functional equations in random normed and fuzzy normed spaces has been recently studied in [29,30,39,50-53]. They are completed with the recent paper [54], which contains some stability results for functional equations in probabilistic metric and random normed spaces.

This paper is organized as follows. In Section 2, we prove the Hyers-Ulam stability of random homomorphisms in complete random normed algebras associated with the Cauchy additive functional inequality (1.2). In Section 3, we prove the Hyers-Ulam stability of random derivations in complete random normed algebras associated with the Cauchy-Jensen additive functional inequality (1.3).

In this section, using the fixed point method, we prove the Hyers-Ulam stability of random homomorphisms in complete random normed algebras associated with the Cauchy additive functional inequality (1.2).

Theorem 2.1Letbe a function such that there exists anwith

for all. Letbe an odd mapping satisfying

for all, alland all. Thenexists for eachand defines a random homomorphismsuch that

for alland all.

Proof Since f is odd, . So . Letting and and replacing z by in (2.1), we get

for all .

Consider the set

and introduce the generalized metric on S:

where, as usual, . It is easy to show that is complete (see the proof of [30], Lemma 2.1]).

Now we consider the linear mapping such that

for all .

Let be given such that . Then

for all and all . Hence

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

for all .

It follows from (2.4) that

for all and all . So .

By Theorem 1.8, there exists a mapping satisfying the following:

(1) H is a fixed point of J, i.e.,

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

This implies that H is a unique mapping satisfying (2.5) such that there exists a satisfying

for all ;

(2) as . This implies the equality

for all ;

(3) , which implies the inequality

This implies that the inequality (2.3) holds.

Let in (2.1). By (2.1),

for all , all and all . So

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

for all and all . So the mapping is Cauchy additive.

Let and in (2.1). By (2.1),

for all , all , all and all . So

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

for all , all and all . Thus the additive mapping is -linear.

By (2.2),

for all , all and all . So

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

for all and all . Thus the mapping is multiplicative.

Therefore, there exists a unique random homomorphism satisfying (2.3). □

Theorem 2.2Letbe a function such that there exists anwith

for all. Letbe an odd mapping satisfying (2.1) and (2.2). Thenexists for eachand defines a random homomorphismsuch that

for alland all.

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

Consider the linear mapping such that

for all .

It follows from (2.4) that

for all and all . So .

By Theorem 1.8, there exists a mapping satisfying the following:

(1) H is a fixed point of J, i.e.,

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

This implies that H is a unique mapping satisfying (2.7) such that there exists a satisfying

for all ;

(2) as . This implies the equality

for all ;

(3) , which implies the inequality

This implies that the inequality (2.6) holds.

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

In this section, using the fixed point method, we prove the Hyers-Ulam stability of random derivations on complete random normed algebras associated with the Cauchy-Jensen additive functional inequality (1.3).

Theorem 3.1Letbe a function such that there exists anwith

for all. Letbe an odd mapping satisfying

for all, alland all. Thenexists for eachand defines a random derivationsuch that

for alland all.

Note that .

Proof Letting in (3.1), we get

for all .

Consider the set

and introduce the generalized metric on S:

where, as usual, . It is easy to show that is complete (see the proof of [30], Lemma 2.1]).

Now we consider the linear mapping such that

for all .

It follows from (3.4) that

for all and all . So .

By Theorem 1.8, there exists a mapping satisfying the following:

(1) D is a fixed point of J, i.e.,

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

This implies that D is a unique mapping satisfying (3.5) such that there exists a satisfying

for all ;

(2) as . This implies the equality

for all ;

(3) , which implies the inequality

This implies that the inequality (3.3) holds.

Let in (3.1). By (3.1),

for all , all and all . So

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

for all and all . So the mapping is Cauchy additive.

Let , and in (3.1). By (3.1),

for all , all , all and all . So

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

for all , all and all . Thus the additive mapping is -linear.

By (3.2),

for all , all and all . So

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

for all and all . Thus the mapping satisfies for all .

Therefore, there exists a unique random derivation satisfying (3.3). □

Theorem 3.2Letbe a function such that there exists anwith

for all. Letbe an odd mapping satisfying (3.1) and (3.2). Thenexists for eachand defines a random derivationsuch that

for alland all.

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

Consider the linear mapping such that

for all .

The rest of the proof is similar to the proofs of Theorems 2.1 and 3.1. □

The authors declare that they have no competing interests.

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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