Convergence of approximate sequences for compositions of nonexpansive mappings in banach spaces

Abstract

Let C be a nonempty closed convex subset of a Banach space E and let $T_1, \cdots, T_N$ be nonexpansive mappings from C into itself (recall that a mapping $T : C \longrightarrow C$ is nonexpansive if $\left\$\mid$ Tx - Ty \right\$\mid$ \leq \left\$\mid$ x - y \right\$\mid$$ for all $x, y \in C$). We consider the fixed point problem for nonexpansive mappings : find a common fixed point, i.e., find a point in $\cap_{i=1}^N Fix(T_i)$, where $Fix(T_i) := {x \in C : x = T_i x}$ denotes the set of fixed points of $T_i$.

Keywords

• Nonexpansive mapping;
• fixed points;
• iteration;
• synny and nonexpansive retraction

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References

1. J. Math. Anal. Appl. v.202 The approximation of fixed points of compositions of nonex-pansive mappings in Hilbert space H. H. Bauschke
2. Archs Ration. Mech. Anal. v.24 Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces F. E. Browder
3. Lectures Notes in Math v.485 Geometry of Banach Spaces J. Diestel
4. Israel J. Math. v.22 An example concerning fixed points A. Genel;J. Lindenstrauss
5. Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings K. Goebel;S. Reich
6. Cambridge Studies in Advanced Mathematics v.28 Topics in metric fixed point theory K. Goebel;W. A. Kirk
7. Pacific J. Math. v.40 no.3 Some geometric properties related to the fixed point theory for nonexpansive mappings J. P. Gossez;E. L. Dozo
8. Bull. Amer. Math. Soc. v.73 Fixed points of nonexpansive maps B. Halpern
9. C. R. Acad. Sci. Ser A-B, Paris v.284 Approximation de points fixes de contractions P. L. Lions
10. Bull. Amer. Math. Soc. v.73 Weak convergence of the sequence of successive approximations for nonexpansive mappins Z. Opial
11. J. Math. Anal. Appl. v.75 Strong convergence theorems for resolvents of accretive operations in Banach spaces S. Reich
12. Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces N. Shioji;W. Takahashi
13. Arch. Math. v.59 Approximation of fixed points of nonexpansive mappings R. Wittmann