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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 $\left(X,d\right)$ is an isometric mapping $c:\left[0,1\right]\to X$ such that $c\left(0\right)=x$ and $c\left(1\right)=y$. The space is said to be geodesic if any two points can be joined by a geodesic path. It is said to be an $ℝ$-tree if for any $x,y\in X$ there is a unique geodesic path from $x$ to $y$ denoted by $\left[x,y\right]$ such that the following condition holds: if $\left[y,x\right]\cap \left[x,z\right]=\left\{x\right\}$ then $\left[y,x\right]\cup \left[x,z\right]=\left[y,z\right]$. Let then $E$ be a nonempty closed convex subset of a complete $ℝ$-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 $f\left(x\right)$ as the nearest point to $u$ in $T\left(x\right)$. For $s\in \left(0,1\right)$ denote by ${t}_{s}\left(x\right)$ the point $z\in \left[u,f\left(x\right)\right]$ that satisfies $d\left(u,z\right)=s$. 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 $\left\{{x}_{s}\right\}$ remains bounded as $s\to 0$. In this case $\left({x}_{s}\right)$ converges to the unique fixed point of $T$ that is nearest to $u$.
##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces
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