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Image recovery by convex combinations of sunny nonexpansive retractions. (English) Zbl 0815.47068

Let \(E\) be a Banach space and let \(C\) be a nonempty closed convex subset of \(E\). Let \(D\) be a subset of \(C\) and let \(P\) be a mapping of \(C\) into \(D\). Then \(P\) is said to be sunny if \(P(Px+ t(x- Px))= Px\) whenever \(Px+ t(x- Px)\in C\) for \(x\in C\) and \(t\geq 0\). A mapping \(P\) of \(C\) into \(C\) is said to be a retraction if \(P^ 2=P\). A subset \(D\) of \(C\) is said to be sunny nonexpansive retract on \(C\) if there exists a sunny and nonexpansive retraction of \(C\) onto \(D\).
In the present paper the authors establish, among others, the following theorem: Let \(E\) be a uniformly convex Banach space with a Fréchet differentiable norm and let \(C\) be a nonempty closed convex subset of \(E\). Let \(C_ 1, C_ 2,\dots, C_ r\) be sunny nonexpansive retracts of \(C\) such that \(\bigcap_{i=1}^ r C_ i \neq \emptyset\). Let \(T\) be an operator on \(C\) given by \(T= \sum_{i=1}^ r \alpha_ i T_ i\), \(0<\alpha_ i< 1\), \(i=1,2, \dots,r\), \(\sum_{i=1}^ r \alpha_ i=1\), such that for each \(i\), \(T_ i= (1- \lambda_ i) I+ \lambda_ i P_ i\), \(0<\lambda_ i <1\), where \(P_ i\) is a sunny nonexpansive retraction of \(C\) onto \(C_ i\). Then \(\text{Fix} (T)= \bigcap_{i=1}^ r C_ i\) and further, for each \(x\in C\), \(\{T^ n x\}\) converges weakly to an element of \(\text{Fix} (T)\).
Finally the authors prove a common fixed point theorem for a finite commuting family of nonexpansive mappings in a uniformly convex and uniformly smooth Banach space.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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