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Strong convergence results for nonself multimaps in Banach spaces. (English) Zbl 1135.47054
Let $E$ be a uniformly convex Banach space, $D$ be a nonempty closed convex subset of $E$ and $T:D\rightarrow K(E)$ be a multimap, where $K(E)$ is the family of all nonempty compact subsets of $E$. If we denote $$P_T(x)=\{u_x\in Tx: \left\Vert x-u_x\right\Vert =d(x,Tx)\},$$ then $P_T:D\rightarrow K(E)$ is nonempty and compact for every $x\in D$. The first main result of the paper (Theorem 3.1) shows that, if $D$ is a nonexpansive retract of $E$ and if, for each $u\in D$ and $t\in (0,1)$, the multivalued contraction $S_t$ defined by $S_tx=tP_Tx+(1-t)u$ has a fixed point $x_t\in D$, then $T$ has a fixed point if and only if $\{x_t\}$ remains bounded as $t\rightarrow 1$. Moreover, in this case, $\{x_t\}$ converges strongly to a fixed point of $T$ as $t\rightarrow 1$. A similar result (Theorem 3.2) is then obtained for nonself-multimaps satisfying the inwardness condition in the case of reflexive Banach spaces having a uniformly Gâteaux differentiable norm. Several corollaries of these results are also presented.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H04 Set-valued operators 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties
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