Journal of Function Spaces
Volume 2015 (2015), Article ID 828967, 7 pages
http://dx.doi.org/10.1155/2015/828967

Research Article

Stability of Pexiderized Quadratic Functional Equation in Random 2-Normed Spaces

Mohammed A. Alghamdi

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 28 November 2014; Accepted 27 March 2015

Academic Editor: Mikail Et

Copyright © 2015 Mohammed A. Alghamdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The aim of this paper is to investigate the stability of Hyers-Ulam-Rassias type theorems by considering the pexiderized quadratic functional equation in the setting of random 2-normed spaces (RTNS), while the concept of random 2-normed space has been recently studied by Goleţ (2005).

1. Introduction and Preliminaries

In 1940, Ulam [1] proposed the famous “Ulam stability problem,” which was solved by Hyers [2], in 1941, for additive mappings. In 1950, Aoki [3] solved this Ulam problem for weaker additive mappings; for some historical comments regarding the work of Aoki we refer to [4]. In 1978, Rassias [5] generalized the theorem of Hyers for linear mappings in which the Cauchy difference is allowed to be unbounded by replacing with a function depending on and in the Hyers theorem. The generalization of Hyers theorem was also presented by Rassias [6–9] in 1982–1989. Some important Ulam stability problems on Cauchy equation on semigroups, approximately additive mappings, and Jensen equation have been investigated by Gajda [10], Găvruta [11], and Jung [12], respectively. Until now, the stability problems for different types of functional equations in various spaces have been extensively studied, for instance, by Mirmostafaee and Moslehian [13, 14], Rassias [15], Chang et al. [16, 17], Xu et al. [18], Jun and Kim [19], Mursaleen et al. [20–22], and many others. Also very interesting results on additive, quadratic, and cubic functional equations have been achieved by Mohiuddine et al. [23–29]. This paper is inspired from the work of Alotaibi and Mohiuddine [30] in which they solved stability problem for cubic functional equation in random 2-normed spaces.

The pexiderized quadratic functional equation is of the form . For , it is called the quadratic functional equation.

The terminology and notations used below are standard as in [31–33].

A function is called a distribution function if it is nondecreasing and is left continuous with and . By , we denote the set of all distribution functions such that .

If , then , where

It is obvious that for all .

A -norm is a continuous mapping such that is abelian monoid with unit one and if and for all . A triangle function is a binary operation on which is commutative and associative and for every .

Gähler [34] presented the following notion of 2-normed space.

Let be a linear space of a dimension (). A function is called 2-normed on if it satisfied the following conditions: for every , (i) if and only if and are linearly dependent; (ii) ; (iii) for every ; and (iv) for every . In this case, is called a 2-norm space.

Goleţ [35] defined and studied the notion of random 2-normed space with the help of 2-norm of Gähler [34]. Recently, the notion of statistical convergence and lacunary statistical convergence have been studied by Mursaleen [36] and Mohiuddine and Aiyub [37], respectively, in random 2-normed spaces.

Let be a linear space of a dimension greater than one and let be a triangle function. A function is called a probabilistic 2-norm on if it satisfies the following conditions:

(i) () if and are linearly dependent,(ii) if and are linearly independent,(iii),(iv) for all , and ,(v) whenever ,

where denotes the value of at and the triple is called a probabilistic 2-normed space. If we replaced (v) by

(v′), for all and ,

then triple is called a random 2-normed space (RTNS).

Example A. Let be a 2-normed space with , , , and for . For all , , and nonzero , consider

Then is a RTNS.

We remark that every 2-normed space can be made RTNS by considering , for every , , and , where .

The notions of convergence and Cauchy sequences have been recently studied by Alotaibi and Mohiuddine [30] in the setting of RTNS.

Let be a RTNS. Then, a sequence is said to be

(i)convergent in (-convergent) to if for every and there exists such that whenever and nonzero . In this case we write -;(ii)Cauchy sequence in (-Cauchy) if for every , , and nonzero there exists a number such that for all . We say that RTNS is if every -Cauchy sequence is -convergent. A complete RTNS is called random 2-Banach space.

2. Main Results

Throughout the paper, by , , and , we denote linear space, random 2-normed space, and random 2-Banach space, respectively. Firstly, we prove the stability of the pexiderized quadratic functional equation in RTNS for an odd case.

Let be a function from to . A mapping is said to be -approximately pexiderized quadratic function if there exist mappings such that

for all , , and nonzero .

Theorem 1. Suppose that and are odd functions from to satisfying (3). If for some real number with

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

where

for all , , and nonzero .

Proof. Replacing by and by in (3), we obtain

for all , , and nonzero . It follows from (3) and (7) that

Substituting in (8), we get

From (8) and (9), we conclude that

for every and nonzero . Then, by our assumption,

Taking in (10), for all , , and nonzero , we get

Putting in (12), we have

Thus,

Therefore, for each ,

where . Let and be given. With the help of the definition of RTNS, we have and, therefore, we can find some such that . The convergence of the series gives some such that for each , . Therefore,

It follows that is a Cauchy sequence in . Since is complete RTNS, this sequence converges to some point in ; that is, . Therefore, a mapping from to is defined by -. Fix and . From (10), we get that

for all . Moreover,

for all . From (17) and (18), we obtain

Thus, . Now by taking (15) with , we get

It follows from (9) and (20) that

Thus, we obtained (5). Now we will prove the uniqueness of . For this, we assume that is another additive mapping from into , which satisfies the required inequality. Since, for each , and , then

We obtain with the help of the definition of RTNS that

Therefore, , for all , , and nonzero . Hence, for all .

Now, we are going to prove the stability of the pexiderized quadratic functional equation in RTNS for an even case.

Theorem 2. If (4) holds for , let , , and be three even functions from to such that and satisfies (3). Then there is a unique quadratic mapping such that, for every , , and nonzero ,

where is defined by (6).

Proof. Substitute by and by in (3). Then, for all , , and nonzero , we obtain

Again substituting in (3), we get

Putting in (3), we get

For , (3) becomes

It follows from (25), (27), and (28) that

By substituting in (29), we get

From (4), we obtain

for every , nonzero and for each . It follows from (30) and (31) that

From (32), we obtain

or, equivalently,

Therefore, for all , , and nonzero and for each

where is the same as Theorem 1. Given and , since , there is some such that . By the convergence of , we can find some such that for each . This gives that

We see that is a Cauchy sequence in and so it is convergent to some point . Therefore, a mapping from to is defined by -. Fix and . Thus, (29) gives that

for all . Furthermore,

Equations (37) and (38) give that

for all , , and nonzero . Thus, . Using (35) with , we get

for sufficiently large . From (28) and (40), we conclude that

Thus,

Similarly, one can show that the above inequality also holds for . We obtain the uniqueness assertion of this theorem by proceeding the same lines as in Theorem 1.

Theorem 3. Suppose that (4) holds with . If a map satisfies

for all , , and nonzero with . Then, there are unique mappings such that is additive, is quadratic, and

for all , , and nonzero , where

Proof. Passing to the odd part and even part of , we deduce from (43) that

On the other hand,

With the help of the proofs of Theorems 1 and 2, we obtain unique additive and quadratic mappings and , respectively, satisfying

Therefore,

for all , , and nonzero .

Remark 4. Let be a 2-inner product space. We can define a 2-norm on by for all . In this case, parallelogram law is given by

for all (for more details of 2-inner product space we refer to [38]).

Now we give the following illustrative example.

Example 5. Let be a 2-inner product space. Let be a 2-normed space such that , where and . Suppose that for all . Suppose that and are two random 2-norms on and , respectively, which are given by Example A. Suppose that the random 2-norm makes into an random 2-Banach space. Fixing and , we define

for each . Using parallelogram law, one can easily verify that

for all . Therefore,

for each , , and nonzero . Moreover, for each . We can see that the conditions of Theorems 1 and 2 for and are satisfied. It follows that odd and even parts of can be approximated by linear and quadratic functions, respectively. In fact , the odd part of and , is linear. The even part of is , and contains a quadratic . Also

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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