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Endpoints of set-valued contractions in metric spaces. (English) Zbl 1226.54042
Summary: Suppose $(X,d)$ to be a complete metric space, and suppose $F:X\to CB(X)$ to be a set-valued map which satisfies $H(Fx,Fy)\le \psi (d(x,y))$, for each $x,y\in X$, where $\psi:[0,\infty)\to [0,\infty)$ is upper semicontinuous, $\psi(t)<t$ for each $t>0$ and satisfies $\liminf_{t\to\infty}(t-\psi(t))>0$. Then $F$ has a unique endpoint if and only if $F$ has the approximate endpoint property.

54H25Fixed-point and coincidence theorems in topological spaces
54C60Set-valued maps (general topology)
54E50Complete metric spaces
Full Text: DOI
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