Bounds on Iterations of Asymptotically Quasi-Nonexpansive Mappings

This paper establishes explicit quantitative bounds on the computation of approximate fixed points of asymptotically (quasi-) nonexpansive mappings f by means of iterative processes. Here f is a selfmapping of a convex subset C of a uniformly convex normed space X. We consider general Krasnoselski-Mann iterations with and without error terms. As a consequence of our quantitative analysis we also get new qualitative results which show that the assumption on the existence of fixed points of f can be replaced by the existence of approximate fixed points only. We explain how the existence of effective uniform bounds in this context can be inferred already a-priorily by a logical metatheorem recently proved by the first author. Our bounds were in fact found with the help of the general logical machinery behind the proof of this metatheorem. The proofs we present here are, however, completely self-contained and do not require any tools from logic.

[1]  J. A. Clarkson.  Uniformly convex spaces , 1936 .

[2]  W. R. Mann,et al.  Mean value methods in iteration , 1953 .

[3]  O. Hanner.  On the uniform convexity ofLp andlp , 1956 .

[4]  Z. Opial.  Weak convergence of the sequence of successive approximations for nonexpansive mappings , 1967 .

[5]  J. B. Diaz,et al.  On the structure of the set of subsequential limit points of successive approximations , 1967 .

[6]  J. B. Díaz,et al.  On the set of subsequential limit points of successive approximations. , 1969 .

[7]  W. G. Dotson,et al.  On the Mann iterative process , 1970 .

[8]  C. W. Groetsch,et al.  A Note on Segmenting Mann Iterates , 1972 .

[9]  William A. Kirk,et al.  A FIXED POINT THEOREM FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS , 1972 .

[10]  S. Singh,et al.  Topological Methods in Nonlinear Functional Analysis , 1983 .

[11]  William A. Kirk,et al.  Approximate fixed points for nonexpansive mappings in uniformly convex spaces , 1990 .

[12]  Jürgen Schu,et al.  Iterative construction of fixed points of asymptotically nonexpansive mappings , 1991 .

[13]  J. Schu,et al.  Weak and strong convergence to fixed points of asymptotically nonexpansive mappings , 1991, Bulletin of the Australian Mathematical Society.

[14]  Jürgen Schu.  Iterative construction of fixed points of strictly pseudocontractive mappings , 1991 .

[15]  B. Rhoades,et al.  Fixed Point Iterations for Certain Nonlinear Mappings , 1994 .

[16]  Hong-Kun Xu,et al.  Fixed point iteration processes for asymptotically nonexpansive mappings , 1994 .

[17]  A. Kartsatos.  Theory and applications of nonlinear operators of accretive and monotone type , 1996 .

[18]  Yuguang Xu,et al.  Ishikawa and Mann Iterative Processes with Errors for Nonlinear Strongly Accretive Operator Equations , 1998 .

[19]  Ulrich Kohlenbach,et al.  On the Computational Content of the Krasnoselski and Ishikawa Fixed Point Theorems , 2000, CCA.

[20]  Liu Qihou.  Iterative Sequences for Asymptotically Quasi-nonexpansive Mappings , 2001 .

[21]  Liu Qihou.  Iterative Sequences for Asymptotically Quasi-nonexpansive Mappings with Error Member☆ , 2001 .

[22]  U. Kohlenbach.  A QUANTITATIVE VERSION OF A THEOREM DUE TO BORWEIN-REICH-SHAFRIR , 2001 .

[23]  Liu Qihou.  Iteration Sequences for Asymptotically Quasi-nonexpansive Mapping with an Error Member of Uniform Convex Banach Space , 2002 .

[24]  Ulrich Kohlenbach.  Uniform Asymptotic Regularity for Mann Iterates , 2002 .

[25]  Paulo Oliva,et al.  Proof Mining: A Systematic Way of Analysing Proofs in Mathematics , 2002 .

[26]  Concise course on fixed point theorems , 2002 .

[27]  Ulrich Kohlenbach,et al.  Mann iterates of directionally nonexpansive mappings in hyperbolic spaces , 2002 .

[28]  Marius Mikucionis,et al.  Online On-the-Fly Testing of Real-time Systems , 2003 .

[29]  Peter Bro Miltersen,et al.  The Cell Probe Complexity of Succinct Data Structures , 2003 .

[30]  Branimir Lambov.  A Two-Layer Approach to the Computability and Complexity of Real Numbers , 2003 .

[31]  Hans Hüttel,et al.  Recursive Ping-Pong Protocols , 2003 .

[32]  Ulrich Kohlenbach.  Some Logical Metatheorems with Applications in Functional Analysis , 2003 .

[33]  Glynn Winskel,et al.  Domain theory for concurrency , 2003, Theor. Comput. Sci..

[34]  Philipp Gerhardy.  The Role of Quantifier Alternations in Cut Elimination , 2005, Notre Dame J. Formal Log..