Some fixed point iteration procedures

This paper provides a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions. The author does not claim to provide complete coverage of the literature, and admits to certain biases in the theorems that are cited herein. In spite of these shortcomings, however, this paper should be a useful reference for those persons vishing to become better acquainted with the area.

[1]  William A. Kirk,et al.  A Fixed Point Theorem for Mappings which do not Increase Distances , 1965 .

[2]  C. Chidume.  Iterative solution of nonlinear equations of the monotone type in Banach spaces , 1990, Bulletin of the Australian Mathematical Society.

[3]  S. Naimpally,et al.  Extensions of some fixed point theorems of Rhoades , 1983 .

[4]  P. Lions.  Approximation de Points Fixes de Contractions , 1977 .

[5]  Tudor Zamfirescu,et al.  Fix point theorems in metric spaces , 1972 .

[6]  William A. Kirk,et al.  Fixed point theorems for set-valued mappings of contractive type , 1972 .

[7]  On a Theorem of C. E. Chidume Concerning the Iterative Approximation of Fixed Points , 1991 .

[8]  A. A. Potapenko,et al.  Method of Successive Approximations , 1964, Encyclopedia of Evolutionary Psychological Science.

[9]  R. L. Franks,et al.  A theorem on mean-value iterations , 1971 .

[10]  Iterative approximation of the solution of a locally Lipschitzian equation , 1991 .

[11]  B. Rhoades.  Contractive definitions and continuity , 1988 .

[12]  R. Kannan,et al.  Some results on fixed points , 1968 .

[13]  W. A. Kirk.  On successive approximations for nonexpansive mappings in Banach spaces , 1971, Glasgow Mathematical Journal.

[14]  W. L. Bynum,et al.  Weak Parallelogram Laws for Banach Spaces , 1976, Canadian Mathematical Bulletin.

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

[16]  L. Qihou.  The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings , 1990 .

[17]  Xie Ping Ding.  Iteration processes for nonlinear mappings in convex metric spaces , 1988 .

[18]  B. Halpern.  Fixed points of nonexpanding maps , 1967 .

[19]  Iterative construction of fixed points for multivalued operators of the monotone type , 1986 .

[20]  C. E. Chidume,et al.  Fixed point iterations for certain classes of nonlinear mappings , 1988 .

[21]  Helmut Schaefer,et al.  Über die Methode sukzessiver Approximationen. , 1957 .

[22]  Ronald E. Bruck.  On the almost-convergence of iterates of a nonexpansive mapping in Hilbert space and the structure of the weak ω-limit set , 1978 .

[23]  M. Edelstein.  The construction of an asymptotic center with a fixed-point property , 1972 .

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

[25]  Liu Qihou,et al.  On Naimpally and Singh's open questions , 1987 .

[26]  B. Rhoades,et al.  A comparison of various definitions of contractive mappings , 1977 .

[27]  Alberta Marie Harder,et al.  Fixed point theory and stability results for fixed point iteration procedures , 1987 .

[28]  C. Chidume.  An iterative process for nonlinear Lipschitzian strongly accretive mappings in Lp spaces , 1990 .

[29]  Fixed point iterations of generalized nonexpansive mappings , 1988 .

[30]  B. Rhoades,et al.  Fixed point iterations using infinite matrices , 1974 .

[31]  Dietrich Göhde,et al.  Zum Prinzip der kontraktiven Abbildung , 1965 .

[32]  S. Ishikawa.  Fixed points by a new iteration method , 1974 .

[33]  R. Wittmann.  Approximation of fixed points of nonexpansive mappings , 1992 .

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

[35]  C. Morales,et al.  Surjectivity theorems for multi-valued mappings of accretive type , 1985 .

[36]  Ronald E. Bruck.  The iterative solution of the equation $y \in x + Tx$ for a monotone operator $T$ in Hilbert space , 1973 .

[37]  F. Browder,et al.  Nonlinear ergodic theorems , 1976 .

[38]  L. Ciric,et al.  A generalization of Banach’s contraction principle , 1974 .

[39]  C. E. Chidume,et al.  Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings , 1987 .

[40]  K. L. Singh,et al.  FIXED POINT ITERATIONS USING INFINITE MATRICES , 1979 .

[41]  Extensions of fixed point theorems of Rhoades and Ćirić , 1977 .

[42]  Felix E. Browder,et al.  Nonlinear mappings of nonexpansive and accretive type in Banach spaces , 1967 .

[43]  Hong-Kun Xu.  A note on the Ishikawa iteration scheme , 1992 .

[44]  B. Beauzamy,et al.  Théorème de point fixe et d'approximation , 1985 .

[45]  F. Browder,et al.  Construction of fixed points of nonlinear mappings in Hilbert space , 1967 .

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

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

[48]  L. Ciric,et al.  Generalized contractions and fixed-point theorems. , 1971 .

[49]  Klaus Deimling,et al.  Zeros of accretive operators , 1974 .

[50]  Chi Song Wong.  Fixed point theorems for generalized nonexpansive mappings , 1974, Journal of the Australian Mathematical Society.

[51]  Fixed Point Iterations Using Infinite Matrices, III , 1977 .

[52]  Tosio Kato,et al.  Nonlinear semigroups and evolution equations , 1967 .

[53]  Simeon Reich,et al.  An iterative procedure for constructing zeros of accretive sets in Banach spaces , 1978 .

[54]  Hong-Kun Xu.  Inequalities in Banach spaces with applications , 1991 .

[55]  George G. Lorentz,et al.  A contribution to the theory of divergent sequences , 1948 .

[56]  B. Rhoades,et al.  Comments on two fixed point iteration methods , 1976 .

[57]  F. Browder,et al.  The solution by iteration of nonlinear functional equations in Banach spaces , 1966 .

[58]  Wataru Takahashi,et al.  A convexity in metric space and nonexpansive mappings, I , 1970 .

[59]  S. Ishikawa.  Fixed points and iteration of a nonexpansive mapping in a Banach space , 1976 .

[60]  M. Osilike.  Ishikawa and Mann iteration methods for nonlinear strongly accretive mappings , 1992, Bulletin of the Australian Mathematical Society.

[61]  Troy L. Hicks,et al.  On the Mann iteration process in a Hilbert space , 1977 .

[62]  Xinlong Weng,et al.  Fixed point iteration for local strictly pseudo-contractive mapping , 1991 .

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