## 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 $P_T(x)=\{u_x\in Tx: \left\| x-u_x\right\| =d(x,Tx)\},$ 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 involving nonlinear operators 47H04 Set-valued operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.

Zbl 1123.47047
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