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Fixed point theorems for nonexpansive mappings on nonconvex sets in UCED Banach spaces. (English) Zbl 1035.47031
This article deals with self-mappings $$T:C\to C$$ where $$C$$ is a finite union of weakly compact convex subsets in a Banach space $$X$$ which is uniformly convex in every direction (the latter means that, for any $$z$$, $$\| z\|=1$$ and $$\varepsilon>0$$, $$\text{inf}\{1-\|(x+y)/2 \|:\| x\|,\| y\|\leq 1$$, $$x-y=\lambda z$$, $$|\lambda |\geq \varepsilon\}>0.)$$
The authors prove the following two results: (I) Each asymptotically regular or $$\lambda$$-firmly nonexpansive mapping $$T:C\to C$$ has a fixed point, and (II) If $$\{T_i\}_{i\in I}$$ is any compatible family of strongly nonexpansive self-mappings of such a $$C$$ and the graphs of $$T_i$$, $$i\in I$$, have a nonempty intersection, then $$T_i$$, $$i\in I$$, have a common fixed point in $$C$$. Recall that a nonexpansive mapping $$T:C\to C$$ is called asymptotically regular if $$\lim_{n\to \infty}\| T^nx-T^{n+1} x\|=0$$, strongly nonexpansive if $$\| x_n-y_n \|-\| Tx_n-Ty_n \|\to 0$$, $$x_n, y_n\in C$$, implies $$(x_n-y_n) -(Tx_n-Ty_n)\to 0$$; $$T:C\to C$$ is said to be $$\lambda$$-firmly nonexpansive $$(\lambda \in(0,1))$$ if $$\| Tx-Ty|\leq \|(1-\lambda)(x-y)+ \lambda(Tx- Ty)\|$$, $$x,y\in C$$.

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