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Zbl 1123.47047
Jung, Jong Soo
Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 66, No. 11, A, 2345-2354 (2007). ISSN 0362-546X

The paper gives convergence theorems for approximating fixed points of multivalued nonexpansive nonself mappings by means of viscosity type methods. The main result (Theorem~1) goes as follows. Let $E$ be a uniformly convex Banach space with uniformly Gâteaux differentiable norm, $C$ be a nonempty closed convex subset of $E$ and $T:C\rightarrow \mathcal{K}(E)$ be a nonself nonexpansive multivalued mapping (here, $\mathcal{K}(E)$ denotes the set of all nonempty compact subsets of $E$). Suppose that $C$ is a nonexpansive retract of $E$ and $T$ has only strict fixed points, that is, $T(y)=\{y\}$ for all fixed points $y$ of $T$. For each $u\in C$ and $t\in (0,1)$, consider the multivalued contraction $G_t:C\rightarrow \mathcal{K}(E)$ defined by $$G_t=tTx+(1-t)u,\ x\in C,$$ and assume that $G_t$ has a fixed point $x_t\in C$. Then $T$ has a fixed point if and only if $x_t$ remains bounded as $t\rightarrow 1$ and, in this case, $x_t$ converges strongly as $t\rightarrow 1$ to a fixed point of $T$. Several other related results are obtained in the same way or as corollaries.
[Vasile Berinde (Baia Mare)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H04 Set-valued operators
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties

Keywords: Banach space; multivalued nonexpansive nonself mapping; fixed point; viscosity method; strong convergence

Cited in: Zbl 1245.54042 Zbl 1135.47054

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