Ulam Stability of Some Functional Inclusions for Multi-valued Mappings

Abstract

We we show that some multifunctions $F:K\to n(Y)$, satisfying functional inclusions of the form
$$
\Psi \big(x,F(\xi_1(x)), \ldots, F(\xi_n(x))\big)\subset F(x)G(x),
$$
admit near-selections $f:K\to G$, fulfilling the functional equation
$$
\Psi \big(x,f(\xi_1(x)), \ldots, f(\xi_n(x))\big)= f(x),
$$
where functions $G:K\to n(Y)$, $\Psi \colon K\times Y^n\to Y$ and $\xi_1,\ldots,\xi_n\in K^K$ are given, $n$ is a fixed positive integer, $K$ is a nonempty set, $(Y,\cdot)$ is a group and $n(Y)$ denotes the family of all nonempty subsets of $Y$.

Our results have been motivated by the notion of Ulam stability and some earlier outcomes. The main tool in the proofs is a very recent fixed point theorem for nonlinear operators, acting on some spaces of multifunctions.