## Limit theorems of operators by convex combinations of nonexpansive retractions in Banach spaces.(English)Zbl 0904.47045

The paper deals with the so-called convex feasibility problem in Banach space setting. Roughly speaking this problem is formulated as follows: Let $$C$$ be a nonempty convex closed subset of a Banach space $$E$$ and let $$C_1,C_2,\dots, C_r$$ be nonexpansive retracts of $$C$$ such that $$\bigcap^r_{i= 1}C_i\neq\emptyset$$. Assume that $$T$$ is a mapping on $$C$$ given by the formula $$T= \sum^r_{i= 1}\alpha_i T_i$$, where $$\alpha_i\in(0, 1)$$, $$\sum^r_{i= 1}\alpha_i= 1$$ and $$T_i= (1-\lambda_i)I+ \lambda_i P_i$$, where $$\lambda_i\in(0, 1)$$ and $$P_i$$ is a nonexpansive retraction of $$C$$ onto $$C_i$$ $$(i= 1,2,\dots,r)$$. One can find assumptions concerning the space $$E$$ which guarantee that the set $$F(T)$$ of fixed points of the mapping $$T$$ can be represented as $$F(T)= \bigcap^r_{i= 1}C_i$$ and for every $$x\in C$$ the sequence $$\{T^nx\}$$ converges weakly to an element of $$F(T)$$. The authors prove that if $$E$$ is a uniformly convex Banach space with a Fréchet differentiable norm (or a reflexive and strictly convex Banach space satisfying the Opial condition) then the convex feasibility problem has a positive solution. Apart from that the problem of finding a common fixed point for a finite commuting family of nonexpansive mappings in a strictly convex and reflexive Banach space is also considered.

### MSC:

 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems
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### References:

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