On the Stability of a General Mixed Additive-Cubic Functional Equation in Random Normed Spaces

Abstract

We prove the generalized Hyers-Ulam stability of the following additive-cubic equation in the setting of random normed spaces.

1. Introduction

A basic question in the theory of functional equations is as follows: when is it true that a function, which approximately satisfies a functional equation, must be close to an exact solution of the equation?

If the problem accepts a unique solution, we say the equation is stable (see [1]). The first stability problem concerning group homomorphisms was raised by Ulam [2] in 1940 and affirmatively solved by Hyers [3]. The result of Hyers was generalized by Rassias [4] for approximate linear mappings by allowing the Cauchy difference operator to be controlled by . In 1994, a generalization of Rassias' theorem was obtained by Gvrua [5], who replaced by a general control function in the spirit of Th. M. Rassias' approach. 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, e.g., [6–12] and references therein). In addition, J. M. Rassias et al. [13–16] generalized the Hyers stability result by introducing two weaker conditions controlled by the Ulam-Gavruta-Rassias (or UGR) product of different powers of norms and the JM Rassias (or JMR) mixed product-sum of powers of norms, respectively.

The theory of random normed spaces (RN-spaces) is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The RN-spaces may also provide us the appropriate tools to study the geometry of nuclear physics and have important application in quantum particle physics (see [17] and the references therein). The generalized Hyers-Ulam stability of different functional equations in random normed spaces, fuzzy normed spaces, and non-Archimedean fuzzy normed spaces has been recently studied in [14–28].

Najati and Eskandani [29] established the general solution and investigated the Ulam-Hyers stability of the following functional equation.

(1.1)

with in the quasi-Banach spaces. It is easy to see that the mapping is a solution of the functional equation (1.1), which is called a mixed additive-cubic functional equation, and every solution of the mixed additive-cubic functional equation is said to be a mixed additive-cubic mapping.

In [14–16], we considered the following general mixed additive-cubic functional equation:

(1.2)

It is easy to show that the function satisfies the functional equation (1.2). We observe that in case (1.2) yields mixed additive-cubic equation (1.1). Therefore, (1.2) is a generalized form of the mixed additive-cubic equation.

In the present paper, we first prove a theorem on stability of equation in random normed spaces and derive from it results on stability of equation . Next, use those results to establish Ulam-Hyers stability for the general mixed additive-cubic functional equation (1.2) in the setting of random normed spaces. In this way some results will be obtained on stability of the linear functional equations also for the random normed spaces, which correspond, for example, to the papers [30–33].

2. Preliminaries

In the sequel we adopt the usual terminology, notations and conventions of the theory of random normed spaces, as in [17, 28]. Throughout this paper, the space of all probability distribution functions is denoted by

(2.1)

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

(2.2)

Definition 2.1 (see [17, 28]).

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

(a) is commutative and associative;

(b) is continuous;

(c) for all ;

(d) whenever and for all .

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

Now, if is a -norm and is a given sequence of numbers in , we define a sequence recursively by and for all . is defined as .

Definition 2.2 (see [17, 28]).

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 in , and all ;

(RN3) for all and all .

Example 2.3.

Let be a normed space. For all and , consider . Then is a random normed space, where is the minimum -norm. This space is called the induced random normed space.

Definition 2.4.

Let be an RN-space.

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

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

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

3. On the Stability of a General Mixed Additive-Cubic Equation in RN-Spaces

Theorem 3.1.

Let with , be a linear space, be a complete RN-space, and be a mapping for which there is such that

(3.1)

for all and . If for some ,

(3.2)

for all and , then there exists a uniquely determined mapping such that and

(3.3)

for all and .

Proof.

Replacing by in (3.1) and using (3.2), we get

(3.4)

for all and . It follows that

(3.5)

for all and all nonnegative integers and with . Hence

(3.6)

for all and with . As and , the right hand side of the inequality tends to as tend to infinity. Then the sequence is a Cauchy sequence in . Since is a complete RN-space, this sequence converges to some point . Therefore, we may define for all . Fix and put in (3.6). Then we obtain

(3.7)

and so, by (RN3), we have

(3.8)

for every . Taking the limit as in (3.8), by , we get (3.3).

To prove the uniqueness of the mapping , assume that there exists another mapping which satisfies (3.3) and for all . Fix . Clearly, and for all . It follows from (3.2) and (3.3) that

(3.9)

Since , we get . Therefore, it follows from (3.9) that for all and so . This completes the proof.

Corollary 3.2.

Let be fixed, be a linear space, be a complete RN-space, and be a mapping for which there is such that

(3.10)

for all and . If for some , for all and , then there exists a uniquely determined mapping such that and

(3.11)

for all and .

Theorem 3.3.

Let be a linear space, be an RN-space, be a complete RN-space, and be a mapping with for which there is such that

(3.12)

for all and . If for some ,

(3.13)

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

(3.14)

for all and , where

(3.15)

Proof.

Letting in (3.12), we get

(3.16)

for all and . Putting in (3.12), we have

(3.17)

for all and . Replacing by in (3.17), we obtain

(3.18)

for all and . Letting in (3.12), we get

(3.19)

for all and . Letting in (3.12), we have

(3.20)

for all and . Letting in (3.12), we have

(3.21)

for all and . Replacing and by and in (3.12), respectively, we get

(3.22)

for all and . Replacing and by and in (3.12), respectively, we get

(3.23)

for all and . Replacing and by and in (3.12), respectively, we have

(3.24)

for all and . Setting in (3.12), we have

(3.25)

for all and . Letting in (3.12), we have

(3.26)

for all and . Letting in (3.12), we have

(3.27)

for all and . By (3.16), (3.17), (3.23), (3.25), and (3.26), we get

(3.28)

for all and . By (3.16), (3.20), and (3.21), we have

(3.29)

for all and . It follows from (3.22) and (3.29) that

(3.30)

for all and . By (3.24) and (3.30), we have

(3.31)

for all and . By (3.16) and (3.25)–(3.27), we have

(3.32)

for all and . It follows from (3.16), (3.18), (3.19), and (3.32) that

(3.33)

for all and . Hence

(3.34)

for all and . By (3.19), we have

(3.35)

for all and . From (3.33) and (3.35), we have

(3.36)

for all and . Also, from (3.28) and (3.36), we get

(3.37)

for all and .

On the other hand, it follows from (3.31) and (3.37) that

(3.38)

for all and . Therefore by (3.34) and (3.38), we get

(3.39)

for all and . By Corollary 3.2, there exists a unique mapping such that and for all and .

It remains to show that is an additive map. Replacing by in (3.12) we get

(3.40)

for all and . Hence

(3.41)

for all and . Taking the limit as in (3.41), we conclude that fulfills (1.2), and so by [16, Lemma ], we see that the mapping is additive, which implies that the mapping is additive. This completes the proof.

Similar to Theorem 3.3, one can prove the following result.

Theorem 3.4.

Let be a linear space, be an RN-space, be a complete RN-space, and be a mapping with for which there is such that

(3.42)

for all and . If for some , for all and , then there exists a unique cubic mapping such that

(3.43)

for all and , where is defined as in Theorem 3.3.

Remark 3.5.

We can also prove Theorems 3.3 and 3.4 for and , respectively.

Theorem 3.6.

Let be a linear space, be an RN-space, be a complete RN-space, and be a mapping with for which there is such that

(3.44)

for all and . If for some ,

(3.45)

for all and , then there exist a unique additive mapping and a unique cubic mapping such that

(3.46)

for all and , where is defined as in Theorem 3.3.

Proof.

By Theorems 3.3 and 3.4, there exist an additive mapping and a cubic mapping such that

(3.47)

(3.48)

for all and . Therefore from (3.47) and (3.48), we get

(3.49)

for all and . Letting and for all , it follows from (3.49) that

(3.50)

for all and . To prove the uniqueness of and , let be another additive and cubic mapping satisfying (3.46). Set and . So

(3.51)

for all and . By , , and (3.51), we get

(3.52)

for all and . Since the right hand side of the inequality tends to as tend to infinity, we find that . Therefore , and then . This completes the proof.

Remark 3.7.

We can formulate similar statements to Theorem 3.6 for .

Corollary 3.8.

Let be a normed space, be an RN-space, and be a complete RN-space. Let be a non-negative real number such that and let . If is a mapping with such that

(3.53)

for all and , then there exist a unique additive mapping and a unique cubic mapping such that

(3.54)

for all and .

Corollary 3.9.

Let be a normed space, be an RN-space, and be a complete RN-space. Let be non-negative real numbers such that and let . If be a mapping with such that

(3.55)

for all and , then there exist a unique additive mapping and a unique cubic mapping such that

(3.56)

for all and .

Now, we give one example to illustrate the main results of Theorem 3.6. This example is a modification of the example of Zhang et al. [34].

Example 3.10.

Let be a Banach algebra, be a unit vector in and is defined as in Example 2.3. It is easy to see that is a complete -space.

Define by for . For , define

(3.57)

Since , the inequality holds when and . A straightforward computation shows that

(3.58)

for all . Therefore, all the conditions of Theorem 3.6 hold, and there exist a unique additive mapping and a unique cubic mapping such that

(3.59)

for all and , where is defined as in Theorem 3.3.