On the stability of set-valued functional equations with the fixed point alternative

Abstract

Using the fixed point method, we prove the Hyers-Ulam stability of a Cauchy-Jensen type additive set-valued functional equation, a Jensen type additive-quadratic setvalued functional equation, a generalized quadratic set-valued functional equation and a Jensen type cubic set-valued functional equation. Mathematics Subject Classification 2010: 47H10; 54C60; 39B52; 47H04; 91B44.

On the Ulam stability of a quadratic set-valued functional equation

“…It can easily be verified that (S, d) is a complete generalized metric space (see [10]). Now, we define an operator T : S → S by…”

“…In the following, various authors considered the Ulam stability problems of several types of set-valued functional equations by using a similar method [12,13]. Unlike the previous approach, Kenary et al [10] applied the Hausdorff metric defined on all closed convex subsets of a Banach space to characterize the functional inequality and investigated the Ulam stability of several types of setvalued functional equations by using a fixed point technique, which is used to deal with the stability of single-valued functional equations. Recently, Jang et al [8] and Chu et al [3] further studied the Ulam stability problems of some generalized set-valued functional equations in a similar way.…”

On Characterizations of Set-Valued Dynamics

“…In [16], Park et al investigated the stability problems of the Jensen additive, quadratic, cubic and quartic set-valued functional equations. Kenary et al [13] proved the stabilities for various types of the set-valued functional equations. In [2], Brzdek extended and complemented classical results concerning the stability of the additive Cauchy equation.…”

The general solution of a quadratic functional equation and Ulam stability

“…In the following year, Hyers gave an partial answer to the problem [6]. Since then, various generalizations of Ulam's problem and Hyers' theorem have been extensively studied and many elegant results have been obtained [1,18,14,19,13,15,9,11,2]. The theory of nonlinear analysis has become a fast developing field during the past decades.…”