Approximate homomorphisms and derivations on random Banach algebras

The motivation of this paper is to investigate the stability of homomorphisms and derivations on random Banach algebras.

MSC: 39B82, 39B52.

The study of stability problems originated from a famous talk Under what condition does there exist a homomorphism near an approximate homomorphism? given by S. M. Ulam [38] in 1940. Next year, in 1941, D. H. Hyers [15] answered affirmatively the question of Ulam for additive mappings between Banach spaces.

Aoki [3] and Rassias [26] provided a generalization of the Hyers theorem for additive and linear functions respectively, by allowing the Cauchy difference to be unbounded.

Theorem 1.1 (Th. M. Rassias)

LetXbe a normed space, Ybe a Banach space andbe a function such that

for all, whereεandpare constants withand. Then there exists a unique additive functionsatisfying

for all. If, then the inequality (1.1) holds forand (1.2) for. Also, if for each fixedthe functionis continuous in, thenAis linear.

The above theorem had a lot of influence on the development of the generalization of the Hyers-Ulam stability concept during the last three decades. This new concept is known as generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations (see [7,16]). Furthermore, Gǎvruta [13] provided a generalization of Rassias’ theorem which allows the Cauchy difference to be controlled by a general unbounded function.

During the last three decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [4,6,8,9,12,17-19,24] and [27-34]). We also refer the readers to the books [1,7,10,16,20,21,28].

Recently, Khodaei and Rassias [22] introduced the generalized additive functional equation

where with , and they established a general solution and the generalized Hyers-Ulam stability for the functional equation (1.3) in various spaces. They proved that a function f between real vector spaces X and Y is a solution of (1.3) if and only if f is additive.

In the sequel we adopt the usual terminology, notations and conventions of the theory of random normed spaces, as in [36,37]. Throughout this paper, let be the space of distribution functions, that is,

and the subset is the set

where, denotes the left limit of the function f at the point x. The space is partially ordered by the usual point-wise ordering of functions, i.e., if and only if for all . The maximal element for in this order is the distribution function given by

Definition 1.2 ([36])

A function is a continuous triangular norm (briefly, a 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 Łukasiewicz t-norm).

Recall (see [14]) that if T is a t-norm and is a given sequence of numbers in is defined recurrently by

is defined as .

It is known [14] that for the Łukasiewicz t-norm the following implication holds:

Definition 1.3 ([37])

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 function from X into such that, the following conditions hold:

(RN1) for all if and only if ;

(RN2) for all , ;

(RN3) for all and .

Definition 1.4 Let be a 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 Cauchy if, for every and , there exists a positive integer N such that whenever .

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

Theorem 1.5 ([36])

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

Definition 1.6 A random normed algebra is a random normed space with algebraic structure such that (RN4) for all and all .

Definition 1.7 Let and be random normed algebras:

(i) An additive mapping is called a random homomorphism if for all .

(ii) An additive mapping is called a random derivation if for all .

The theory of random normed spaces is important as a generalization of the deterministic result of linear normed spaces and also in the study of random operator equations. The random normed spaces may also provide us with the appropriate tools to study the geometry of nuclear physics and have an important application in quantum particle physics. The generalized Hyers-Ulam stability of different functional equations in random and fuzzy normed spaces and random and fuzzy normed algebras has been recently studied in Alsina [2], Miheţ et al. [23], Baktash et al. [5], Saadati et al. [35], Gordji et al. [11], and Park et al. [25].

In this paper, we prove the generalized Hyers-Ulam stability of random homomorphisms and random derivations associated with the generalized additive functional equation (1.3) in random Banach algebras.

We use the following abbreviation for a given function f:

Theorem 2.1LetXbe a real algebra, be a random Banach algebra and (, anddenoted by) be a function such that

for all, and

for alland all. Suppose thatis a function satisfying

for alland. Then there exists a unique homomorphismsuch that

for alland.

Proof Putting and () in (2.3), we obtain that

for all and , that is,

for all and . It follows from the last inequality that

for all and ; hence by using the relation , we have

for all and . So we have

for all and . We can show that the sequence is convergent. Therefore, one can define the function by

for all . Now, if we put , and replace with in (2.3) respectively, it follows that

for all and all . By letting in (2.5), we have ; thus H satisfies (1.3). Hence the function is additive (see also [22]). For the uniqueness property of H, see paper [22].

Finally, we show that H is multiplicative. Since for all and , from (2.1) it follows that

for all and all . Therefore, there exists a unique random homomorphism satisfying (2.4). □

In the following theorem, we establish the stability of derivations on random Banach algebras. We use the following abbreviation for a given function f:

Theorem 2.2Letbe a random Banach algebra andbe a function such that (2.1) and (2.2) hold for alland all. Suppose thatis a function satisfying

for alland. Then there exists a unique derivationsuch that

for alland.

Proof By the same reasoning as in the proof of Theorem 2.1, the sequence is convergent for all , and the function defined by

for all , is a unique additive function which satisfies (2.7). We have to show that is a derivation.

Since for all and , from (2.1) it follows that

for all and all . This means that D is a derivation on X. □

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

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