Article

# Stability of Jensen-Type Functional Equations on Restricted Domains in a Group and Their Asymptotic Behaviors

Journal of Applied Mathematics (Impact Factor: 0.72). 01/2012; 2012. DOI: 10.1155/2012/691981

Source: DBLP

**ABSTRACT**

We consider the Hyers-Ulam stability problems for the Jensen-type functional

equations in general restricted domains. The main purpose of this paper is to

find the restricted domains for which the functional inequality satisfied in

those domains extends to the inequality for whole domain. As consequences of the

results we obtain asymptotic behavior of the equations.

equations in general restricted domains. The main purpose of this paper is to

find the restricted domains for which the functional inequality satisfied in

those domains extends to the inequality for whole domain. As consequences of the

results we obtain asymptotic behavior of the equations.

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**ABSTRACT:**In this paper, using an efficient change of variables we refine the Hyers-Ulam stability of the alternative Jensen functional equations of J. M. Rassias and M. J. Rassias and obtain much better bounds and remove some unnecessary conditions imposed in the previous result. Also, viewing the fundamentals of what our method works, we establish an abstract version of the result and consider the functional equations defined in restricted domains of a group and prove their stabilities. - [Show abstract] [Hide abstract]

**ABSTRACT:**Let R02 = R2{set minus}{(0,0)}, R*2 = {(x,y) ∈ R2 : x2 ≠ y2} and f : R02 → R, g : R *2 → R. In this paper we consider the Ulam-Hyers stability of the functional equations f(ux + vy, uy-vx) = f(x, y) + f(u, v), f(ux-vy, uy + vx) = f(x, y) + f(u, v), g(ux-vy, uy-vx) = g(x, y) + g(u, v), g(ux + vy, uy + vx) = g(x, y) + g(u, v) for all (x, y, u, v)∈Γ, where Γ⊂R4 is of 4-dimensional Lebesgue measure zero. The above functional equations are modified versions of the equations in [9,11,14,18,24] which arise from number theory and are in connection with characterizations of determinant and permanent of two-by-two matrices.