On Stability of Two New Generalized Set-Valued Functional Equations

Wuttichai Suriyacharoen, Wutiphol Sintunavarat

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abstract_carpathian_2026_42_1_95-114

https://doi.org/10.37193/CJM.2026.01.07

Published on 30 September 2025

Issue no: Vol 42/2026 no. 1 Tags: set-valued functional equations, Stability, Pompeiu-Hausdorff distance

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Abstract.

Denote by $C_c(Y)$ the set of all closed convex subsets of a Banach space $Y$ . For every $A, B \in C_c(Y)$ , we define an operation $\oplus$ by
$A \oplus B := \overline{A + B}$ which is a closure of the set $A + B:=\{a+b:a\in A,\, b\in B\}$ . The present work aims to establish the Hyers-Ulam-Rassias stability of the following generalized set-valued functional equations

$F\left( \frac{x+y}{2} + (\alpha - 1)z \right) \oplus F\left( \frac{x+z}{2} + (\alpha - 1)y \right) \oplus F\left( \frac{y+z}{2} + (\alpha - 1)x \right) = \alpha \left( F(x) \oplus F(y) \oplus F(z) \right)$

and

$F(\beta x+y) \oplus F(\beta x-y) = F(x+y) \oplus F(x-y) \oplus 2(\beta^2 - 1)F(x),$

for all $x,y,z \in X$ , where $F:X\rightarrow C_c(Y)$ is an unknown set-valued function while $X$ is a real vector space, $\alpha \geq 2$ and $\beta \notin \{-1,0,1\}$ are fixed integers. These two equations are respectively related to Cauchy-Jensen type and quadratic type set-valued functional equations.