Abstract and Applied AnalysisVolume 2012 (2012), Article ID 392179, 17 pagesdoi:10.1155/2012/392179

Research Article

Approximation of Mixed-Type Functional Equations in Menger PN-Spaces

M. Eshaghi Gordji,¹ H. Khodaei,¹ Y. W. Lee,² and G. H. Kim³

¹Department of Mathematics, Semnan University, Semnan 35195-363, Iran²Department of Computer Hacking, Information Security, Daejeon University, Youngwoondong Donggu Daejeon 300-716, Republic of Korea³Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea

Received 28 September 2011; Accepted 8 February 2012

Academic Editor: Agacik Zafer

Copyright © 2012 M. Eshaghi Gordji et al. 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.

Let and be vector spaces. We show that a function with satisfies for all , if and only if there exist functions , and such that for all , where the function is symmetric for each fixed one variable and is additive for fixed two variables, is symmetric bi-additive, is additive and (, ) for all . Furthermore, we solve the stability problem for a given function satisfying , in the Menger probabilistic normed spaces.

1. Introduction and Preliminaries

Menger [1] introduced the notion of a probabilistic metric space in 1942 and since then the theory of probabilistic metric spaces has developed in many directions [2]. The idea of Menger was to use distribution functions instead of nonnegative real numbers as values of the metric. The notion of a probabilistic metric space corresponds to situations when we do not know exactly the distance between two points, but we know probabilities of possible values of this distance. A probabilistic generalization of metric spaces appears to be interest in the investigation of physical quantities and physiological thresholds. It is also of fundamental importance in probabilistic functional analysis. Probabilistic normed spaces were introduced by Sherstnev in 1962 [3] by means of a definition that was closely modelled on the theory of (classical) normed spaces and used to study the problem of best approximation in statistics. In the sequel, we will adopt the usual terminology, notation and conventions of the theory of probabilistic normed spaces as in [2, 4–6, 6, 7, 7–18].

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 at the point . The space is partially ordered by the usual point-wise ordering of functions, that is, if and only if for all . The maximal element for in this order is the distribution function given by

Definition 1.1 (see [2]). A mapping     is a continuous triangular norm (briefly, a continuous  -norm) if satisfies the following conditions:(1) is commutative and associative;(2) is continuous;(3) for all ;(4) whenever and for all .

Two typical examples of continuous -norms are , . Recall (see [19, 20]) that if is a -norm and is a given sequence of numbers in , is defined recurrently by   is defined as .

Definition 1.2. A Menger probabilistic normed space (briefly, Menger PN-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: (PN₁) for all ;(PN₂) for all if and only if ;(PN₃) for all , and all ;(PN₄) for all and all .

Clearly, every Menger PN-space is probabilistic metric space having a metrizable uniformity on if .

Definition 1.3. Let be a Menger PN-space. (i)A sequence in is said to be convergent to in if, for every and , there exists positive integer such that whenever .(ii)A sequence in is called Cauchy sequence if, for every and , there exists positive integer such that whenever .(iii)A Menger PN-space is said to be complete if and only if every Cauchy sequence in is convergent to a point in .

Theorem 1.4. If is a Menger PN-space and is a sequence such that , then .

The concept of stability of a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. The first stability problem concerning group homomorphisms was raised by Ulam [21] in 1940 and affirmatively solved by Hyers [22]. The result of Hyers was generalized by Aoki [23] for approximate additive function and by Rassias [24] for approximate linear functions by allowing the difference Cauchy equation to be controlled by . Taking into consideration a lot of influence of Ulam, Hyers, and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias is called the Hyers-Ulam-Rassias stability. In 1994, a generalization of Rassias’ theorem was obtained by Găvruţa [25], who replaced by a general control function . The functional equation, is related to symmetric biadditive function and is called a quadratic functional equation and every solution of the quadratic equation (1.5) is said to be a quadratic function. For more details about the results concerning such problems, the reader is referred to [26–28].

The functional equation, is called the cubic functional equation, since the function is its solution. Every solution of the cubic functional equation is said to be a cubic mapping. The stability results for the cubic functional equation were proved by Jun and Kim [29].

Eshaghi Gordji and Khodaei [30] have established the general solution and investigated the Hyers-Ulam-Rassias stability for a mixed type of cubic, quadratic, and additive functional equations, with , in quasi-Banach spaces, where is nonzero integer numbers with . It is easy to see that the function is a solution of the functional equation (1.7), see [31, 32]. The stability of different functional equations in probabilistic normed spaces, RN-spaces, and fuzzy normed spaces has been studied in [6, 7, 33–37].

Now, we introduce the new mixed type of cubic, quadratic, and additive functional equation in -variables as follows: where and . As a special case, if in (1.8), then (1.8) reduces to that is, The main purpose of this paper is to prove the stability for (1.8), in Menger probabilistic normed spaces.

2. Results in Menger Probabilistic Normed Spaces

We start our work with a general solution for the mixed functional equation (1.8) and then investigate the stability of this equation in Menger PN-space.

Theorem 2.1. Let and be vector spaces. A function with satisfies (1.8) for all if and only if there exist functions , and such that for all , where the function is symmetric for each fixed one variable and is additive for fixed two variables, is symmetric biadditive and is additive.

Proof. If there exists a function that is symmetric for each fixed one variable and is additive for fixed two variables, is biadditive, and is additive, then by a simple computation one can show that the functions , , and satisfy the functional equation (1.8). Therefore, the function satisfies (1.8).Conversely, let with satisfies (1.8). Hence, according to (1.8), we get for all . Setting in (2.1), we have that is, for all . On the other hand, we have the relation: for all . Hence, we obtain from (2.3) and (2.4) that for all . Replacing by in (2.5), one gets for all . Therefore, satisfies (1.7) for . By Theorem  2.3 of [30], there exist an additive function , symmetric biadditive function , and such that for all , where the function is symmetric for each fixed one variable and is additive for fixed two variables.

From now on, let be a real linear space and let be a complete Menger PN-space. For convenience, we use the following abbreviation for a given function : for all .

Theorem 2.2. Let , and is denoted by be a function such that for all , and for all and . Suppose that an even function with satisfies the inequality: for all and . Then, there exists a unique quadratic function such that for all and .

Proof. Setting and in (2.10) and then use , we obtain that for all and . By using evenness of and the relation , we get for all and . So, for all and , which implies that for all , and . It follows from (2.15) and that for all and . Thus, for all and . In order to prove the convergence of the sequence , we replace with in (2.17) to find that for all and . Since the right-hand side of the inequality tends to 1 as and tend to infinity, the sequence is a Cauchy sequence. Therefore, one can define the function by for all . Now, if we replace with in (2.10), respectively, it follows that for all and . By letting in (2.19), we find that for all , which implies that thus satisfies (1.8). Hence, by Theorem 2.1 (see [30, Lemma  2.1]), the function is quadratic.To prove (2.11), take the limit as in (2.17).Finally, to prove the uniqueness of the quadratic function subject to (2.11), let us assume that there exists a quadratic function which satisfies (2.11). Since and for all and , from (2.11), it follows that for all and . By letting in (2.20), we find that .

Theorem 2.3. Let be a function such that for all , and for all and . Suppose that an odd function satisfies (2.10) for all and . Then, there exists a unique additive function such that for all and .

Proof. Setting and in (2.10), we obtain for all and . By using oddness of and the relation , we lead to for all and . Putting , and in (2.10), we have for all and . So for all and . It follows from (2.25), (2.27), and that for all and . Let be a function defined by for all . From (2.28), we conclude that for all and , which implies that for all , and . It follows from (2.30) and that for all and . Thus, for all and . In order to prove the convergence of the sequence , we replace with in (2.32) to find that for all and . Since the right-hand side of the inequality tends to 1 as and tend to infinity, the sequence is a Cauchy sequence. Therefore, one can define the function by for all . Now, if we replace with in (2.10), respectively, it follows that for all and . By letting in (2.34), we find that for all , which implies , thus satisfies (1.8). Hence, by Theorem 2.1 (see [30, Lemma  2.2]) the function is additive.To prove (2.23), take the limit as in (2.32).Finally, to prove the uniqueness of the additive function subject to (2.23), let us assume that there exists a additive function which satisfies (2.23). Since and for all and , from (2.23), it follows that for all and . By letting in (2.35), we find that .

Theorem 2.4. Let be a function such that for all , and for all and . Suppose that an odd function satisfies (2.10) for all and . Then, there exists a unique cubic function such that for all and .

Proof. Similar to proof of Theorem 2.3, we obtain (2.28) for all and . Let be a function defined by for all . Therefore, (2.28) implies that for all and , which implies that for all , , and . It follows from (2.40) and that for all and . In order to prove the convergence of the sequence , we replace with in (2.41) to find that for all and . Since the right-hand side of the inequality tends to 1 as and tend to infinity, the sequence is a Cauchy sequence. Therefore, one can define the function by for all . Now, if we replace with in (2.10), respectively, it follows that for all and . By letting in (2.43), we find that for all , which implies , thus satisfies (1.8). Hence, by Theorem 2.1 (see [30, Lemma  2.2]), the function is cubic. The rest of the proof is similar to the proof of Theorem 2.3.

Theorem 2.5. Let be a function satisfies (2.21) for all , and (2.22) for all and . Suppose that an odd function satisfies (2.10) for all and . Then, there exists a unique additive function and a unique cubic function such that for all and .

Proof. By Theorems 2.3 and 2.4, there exist an additive function and a cubic function such that for all and . It follows from (2.45) that for all and . Thus, we obtain (2.44) by letting and for all .To prove the uniqueness property of and , let be another additive and cubic functions satisfying (2.44). Let and . So, for all and , then (2.47) implies that for all and . Since the right-hand side of the inequality tends to 1 as tends to infinity, thus for all , which implies that . So, from (2.47), we lead to .

Now, we are ready to prove the main theorem concerning the stability results for (1.8), in Menger PN-space.

Theorem 2.6. Let be a function satisfies (2.8) and (2.21) for all , and satisfies (2.9) and (2.22) for all and . Suppose that a function satisfies (2.10) for all and . Furthermore, assume that in (2.10) for the case is even. Then, there exists a unique additive function , a unique quadratic function , and a unique cubic function satisfying for all and .

Proof. Let for all . Then, , and for all and . Hence, in view of Theorem 2.2, there exist a unique quadratic function such that for all and . On the other hand, let for all . Then, , and by using the above method, from Theorem 2.5, there exist a unique additive function and a unique cubic function such that for all and . Hence, (2.49) follows from (2.51) and (2.52).

Acknowledgment

The third author of this work was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021253).

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