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

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.