Abstract:
we establish the general solution for a mixed type functional equation of aquartic and a quadratic mapping in linear spaces. In addition, we investigate the generalized Hyers-Ulam stability in -Banach spaces. 1. Introduction and Preliminaries The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. Let be a group, and let be a metric group with the metric . Given , does there exist a such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all In other words, under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that for all , and for some . Then, there exists a unique additive mapping such that for all . The result of Hyers was generalized by Aoki [3] for approximate additive function and by Rassias [4] for approximate linear function 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 may be called the Hyers-Ulam-Rassias stability (see [5, 6]). In 1994, a generalization of Rassias theorem was obtained by G vruta [7], who replaced by a general control function . The functional equation is related to a symmetric biadditive function [8–10]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.3) is said to be a quadratic function. It is well known that a function between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function such that for all in the vector space. The biadditive function is given by A Hyers-Ulam stability problem for the quadratic functional equation (1.3) was proved by Skof for functions , where is normed space and is Banach space (see [11]). In the paper [12], Czerwik proved the Hyers-Ulam-Rassias stability of (1.3). Lee et al. [13] considered the following functional equation: In fact, they proved that a function between two real vector spaces and is a solution of (1.5) if and only if there exists a unique symmetric biquadratic function such that for all . The

Abstract:
We establish the general solution of the functional equation for fixed integers with and investigate the generalized Hyers-Ulam stability of this equation in quasi-Banach spaces.

Abstract:
We establish the general solution of the functional equation f(nx+y)+f(nx y)=n2f(x+y)+n2f(x y)+2(f(nx) n2f(x)) 2(n2 1)f(y) for fixed integers n with n≠0,±1 and investigate the generalized Hyers-Ulam stability of this equation in quasi-Banach spaces.

Abstract:
In this paper, we achieve the general solution and the generalized Hyers-Ulam-Rassias stability of the following functional equation $$f(x+ky)+f(x-ky)=k^2f(x+y)+k^2f(x-y)+2(1-k^2)f(x)\eqno {2 cm}$$for fixed integers $k$ with $k\neq0,\pm1$ in the quasi-Banach spaces.

Abstract:
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-quartic functional equation: in fuzzy Banach spaces.

Abstract:
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation in Banach spaces.

Abstract:
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-quartic functional equation: f(x+2y)+f(x 2y)=2f(x+y)+2f( x y)+2f(x y)+2f(y x) 4f( x) 2f(x)+f(2y)+f( 2y) 4f(y) 4f( y) in fuzzy Banach spaces.

Abstract:
We develop an iterative algorithm for computing the approximate solutions of mixed quasi-variational-like inequality problems with skew-symmetric terms in the setting of reflexive Banach spaces. We use Fan-KKM lemma and concept of -cocoercivity of a composition mapping to prove the existence and convergence of approximate solutions to the exact solution of mixed quasi-variational-like inequalities with skew-symmetric terms. Furthermore, we derive the posteriori error estimates for approximate solutions under quite mild conditions.

Abstract:
We consider the following mixed type cubic and quartic functional equation = where is a fixed integer. We establish the general solution of the functional equation when the integer , and then, by using the fixed point alternative, we investigate the generalized Hyers-Ulam-Rassias stability for this functional equation when the integer . 1. Introduction In 1940, Ulam [1] asked the fundamental question for the stability for the group homomorphisms. Let be a group, and let be a metric group with the metric . Given , does there exist a such that if a mapping satisfies the inequality for all , then there is a homomorphism with for all ? In other words, under what conditions, does there exist a homomorphism near an approximately homomorphism? In the next year, Hyers [2] gave the first affirmative answer to the question of Ulam for Cauchy equation in the Banach spaces. Then, Rassias [3] generalized Hyers’ result by considering an unbounded Cauchy difference, and this stability phenomenon is known as generalized Hyers-Ulam-Rassias stability or Hyers-Ulam-Rassias stability. During the last three decades, the stability problem for several functional equations has been extensively investigated by many mathematicians; see, for example, [4–9] and the references therein. We also refer the readers to the books [10–13]. In [14], Jun and Kim introduced the following functional equation It is easy to see that the function satisfies the functional equation (3). Thus, it is natural that (3) is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic function. In [14], Jun and Kim established the general solution and the generalized Hyers-Ulam-Rassias stability for (3). They proved that a function between real vector spaces is a solution of the functional equation (3) if and only if there exists a function such that for all and is symmetric for each fixed one variable and is additive for fixed two variables. In [15], Lee et al. considered the following quartic functional equation Since the function satisfies the functional equation (4), the functional equation (4) is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic function. In [15], the authors solved the functional equation (4) and proved the stability for it. Actually, they obtained that a function between real vector spaces satisfies the functional equation (4) if and only if there exists a symmetric biquadratic function such that for all . A function between real vector spaces is said to be quadratic if