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Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces. (English) Zbl 1123.47047

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.

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.
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