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Browder’s convergence theorem for multivalued mappings without endpoint condition. (English) Zbl 1245.54042
A geodesic path joining $x,y$ in a metric space $(X,d)$ is an isometric mapping $c:[0,1]\to X$ such that $c(0)=x$ and $c(1)=y$. The space is said to be geodesic if any two points can be joined by a geodesic path. It is said to be an $\Bbb{R}$-tree if for any $x,y\in X$ there is a unique geodesic path from $x$ to $y$ denoted by $[x,y]$ such that the following condition holds: if $[y,x]\cap[x,z]=\{x\}$ then $[y,x]\cup[x,z]=[y,z]$. Let then $E$ be a nonempty closed convex subset of a complete $\Bbb{R}$-tree $X$ and let $T$ be a multivalued nonexpansive mapping from $E$ into the nonempty compact convex subsets of $E$. Choose $u\in E$ and define $f:E\to E$ by defining $f(x)$ as the nearest point to $u$ in $T(x)$. For $s\in(0,1)$ denote by $t_s(x)$ the point $z\in[u,f(x)]$ that satisfies $d(u,z)=s$. Finally, let $x_s$ be the unique fixed point of $t_s$. The authors prove that $T$ has a fixed point if and only if $\{x_s\}$ remains bounded as $s\to0$. In this case $(x_s)$ converges to the unique fixed point of $T$ that is nearest to $u$.
54H25Fixed-point and coincidence theorems in topological spaces
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
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