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On selections of set-valued inclusions in a single variable with applications to several variables. (English) Zbl 1277.39032
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 $$\psi \circ F \circ a=F$$.

##### MSC:
 39B05 General theory of functional equations and inequalities 39B82 Stability, separation, extension, and related topics for functional equations 54C60 Set-valued maps in general topology 54C65 Selections in general topology
##### Keywords:
set-valued map; selection; inclusion; metric space
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##### References:
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