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

and Applied Analysis 3 Clearly, every Menger PN-space is probabilistic metric space having a metrizable uniformity on X if supa<1T a, a 1. Definition 1.3. Let X,Λ, T be a Menger PN-space. i A sequence {xn} in X is said to be convergent to x in X if, for every > 0 and λ > 0, there exists positive integer N such that Λxn−x > 1 − λ whenever n ≥ N. ii A sequence {xn} in X is called Cauchy sequence if, for every > 0 and λ > 0, there exists positive integer N such that Λxn−xm > 1 − λ whenever n ≥ m ≥ N. iii A Menger PN-space X,Λ, T is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X. Theorem 1.4. If X,Λ, T is a Menger PN-space and {xn} is a sequence such that xn → x, then limn→∞Λxn t Λx t . 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 ‖f x y − f x − f y ‖ to be controlled by ε ‖x‖p ‖y‖p . 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 ε ‖x‖p ‖y‖p by a general control function φ x, y . The functional equation, f x1 x2 f x1 − x2 2f x1 2f x2 , 1.5 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, f 2x1 x2 f 2x1 − x2 2f x1 x2 2f x1 − x2 12f x1 , 1.6 is called the cubic functional equation, since the function f x cx3 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 f 0 0, f x1 kx2 f x1 − kx2 k2f x1 x2 k2f x1 − x2 2 ( 1 − k2 ) f x1 1.7 in quasi-Banach spaces, where k is nonzero integer numbers with k / ±1. It is easy to see that the function f x ax bx2 cx3 is a solution of the functional equation 1.7 , see 31, 32 . 4 Abstract and Applied Analysis 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 n-variables as follows: