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Iterative approximation of fixed points of nonexpansive maps. (English) Zbl 0997.41021
Let $$(E,\tau)$$ be a topological vector space (TVS) whose topology $$\tau$$ is generated by an $$F$$-norm $$q$$ which satisfies (i) $$q(x)\geq 0$$ and $$q(x)=0$$ iff $$x=0$$ $$(x\in E)$$. (ii) $$q(x+y)\leq q(x)+q(y)$$ for all $$x,y\in E$$. (iii) $$q(\lambda x)\leq q(x)$$ for all (real or complex) scalars $$\lambda$$ with $$|\lambda |\leq 1$$. (iv) If $$q(x_n)\to 0$$ then $$q(\lambda x_n)\to 0$$ for all scalars $$\lambda$$. (v) If $$\lambda_n\to 0$$ then $$q(\lambda_nx)\to 0$$ for all $$x\in E$$. The relation $$d(x,y)=q(x-y)$$ defines a metric on $$E$$. The space $$E$$ is said to be uniformly convex if there corresponds to each pair of positive numbers $$(\varepsilon,r)$$ a positive number $$\delta$$ such that if $$x$$ and $$y$$ lie in $$E$$ with $$q(x-y)\geq\varepsilon$$, $$q(x)<r+\delta$$, $$q(y)<1+\delta$$ then $$q({x+y \over 2})<r$$. A map $$T$$ on a subset $$M$$ of $$E$$ is said to be nonexpansive if $$q(Tx-Ty)\leq q(x-y)$$ for all $$x,y\in M$$. A subset $$M$$ of the space $$E$$ is said to be $$T$$-regular if $$T:M\to M$$ and $${x+Tx\over 2}\in M$$ for each $$x\in M$$. The authors use the notion of $$T$$-regular sets to establish the convergence of the iterates of the bisection mapping $$F(x)= {x+Tx\over 2}$$ to a fixed point of non-expansive map $$T$$ in the context of metrizable topological vector spaces. A well known fixed point theorem of P. Veermani [J. Math. Anal. Appl. 167, 160-166 (1992; Zbl 0780.47047) (Theorem 1.1)] is generalized from the uniformly convex Banach space setting to metrizable topological vector spaces. As an application of this result, they prove the convergence of the sequence of iterates of the bisection map $$F$$ to a fixed point of the nonexpansive map $$T$$ thereby extending Theorem 2.3 ofA. R. Khan and A. Q. Siddiqui [Math. Jap. 36, No. 6, 1129—1134 (1991; Zbl 0763.47028)] from a convex set to a $$T$$-regular set. A similar result for the weak convergence of iterates under various conditions is derived. The authors study conditions for an iteration process for a finite family of nonexpansive maps to converge to a common fixed point of the family thereby extending Theorem 1 of P. K. F. Kuhfittig [Pac. J. Math. 97, 137-139 (1981; Zbl 0478.47036)].

##### MSC:
 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) 47H10 Fixed-point theorems