On selections of set-valued inclusions in a single variable with applications to several variables (Q370796)

Suppose that \((Y,d)\) is a metric space and \(n(Y)\) denotes the family of all nonempty subsets of \(Y\) and \(\delta(A)=\sup\{d(x,y): x,y \in A\}\) is the diameter of \(A\subset Y\). Suppose that \(K\) is a nonempty set and \(F: Y \to n(Y), \quad \psi:Y\to Y,\quad a:K\to K\) are maps and \(\lambda \in (0, \infty)\) such that \(d(\psi(x), \psi(y))\leq \lambda d(x,y)\) for \(x, y\in Y\), and \(\lim_{n\to \infty} {\lambda}^n\delta (F(a^n(x)))=0\) for each \(x\in K\). The author proves that (1) If \(Y\) is complete and \(\Psi(F(a(x)))\subset F(x)\) for each \(x\in K\), then \(\lim_{n\to \infty} cl \psi^n\circ F \circ a^n(x)=f(x)\) exists and \(f\) is a unique selection of the multifunction \(clF\) such that \(\psi \circ f \circ a=f\). (2) If \( F(x)\subset \psi(F(a(x)))\) for each \(x\in K\), then \(F\) is a single-valued function and \(\psi \circ F \circ a=F\).