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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 \[ P_T(x)=\{u_x\in Tx: \left\| x-u_x\right\| =d(x,Tx)\}, \] 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.

Citations:

Zbl 1123.47047
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References:

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