Contraction Mappings and Extensions

A complete survey of all that has been written about contraction mappings would appear to be nearly impossible, and perhaps not really useful. In particular the wealth of applications of Banach’s contraction mapping principle is astonishingly diverse. We only attempt to touch on some of the high points of this profound and seminal development in metric fixed point theory.

[1]  A generalization of Brøndsted’s results and its applications , 1990 .

[2]  W. Walter.  Remarks on a paper by F. Browder about contraction , 1981 .

[3]  J. Jachymski,et al.  Equivalence of some contractivity properties over metrical structures , 1997 .

[4]  R. J. Knill.  Fixed points of uniform contractions , 1965 .

[5]  J. Markin.  A fixed point theorem for set valued mappings , 1968 .

[6]  A converse to a contraction mapping theorem in uniform spaces , 1988 .

[7]  Berthold Schweizer,et al.  Contractions on probabilistic metric spaces: examples and counterexamples. , 1988 .

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

[9]  F. Browder.  Remarks on fixed point theorems of contractive type , 1979 .

[10]  J. Jachymski,et al.  An extension of A. Ostrowski's Theorem on the round-off stability of iterations , 1997 .

[11]  Shin Min Kang,et al.  Generalized contraction mapping principle and differential equations in probabilistic metric spaces , 1996 .

[12]  Michael A. Geraghty,et al.  On contractive mappings , 1973 .

[13]  Sehie Park,et al.  ON EXTENSIONS OF THE CARISTI-KIRK FIXED POINT THEOREM , 1983 .

[14]  Robert M. Tardiff,et al.  Contraction maps on probabilistic metric spaces , 1992 .

[15]  An application of combinatorial techniques to a topological problem , 1973 .

[16]  Ludvík Janoš.  A converse of Banach’s contraction theorem , 1967 .

[17]  S. Banach.  Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales , 1922 .

[18]  B. N. Sadovskii.  A fixed-point principle , 1967 .

[19]  A. T. Bharucha-Reid,et al.  Fixed point theorems in probabilistic analysis , 1976 .

[20]  M. A. Krasnoselʹskii,et al.  Geometrical Methods of Nonlinear Analysis , 1984 .

[21]  Set-valued topological contractions , 1995 .

[22]  I. Ekeland.  Nonconvex minimization problems , 1979 .

[23]  K Menger.  Probabilistic Theories of Relations. , 1951, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Guanrong Chen,et al.  Approximate Solutions of Operator Equations , 1997 .

[25]  Jacek R. Jachymski,et al.  Schröder, James D. Stein Jr.: A Connection between Fixed-Point Theorems and Tiling Problems , 1999, J. Comb. Theory, Ser. A.

[26]  T. Burton.  Integral equations, implicit functions, and fixed points , 1996 .

[27]  Generalizations of the Converse of the Contraction Mapping Principle , 1966, Canadian Journal of Mathematics.

[28]  S. Nadler.  Multi-valued contraction mappings. , 1969 .

[29]  Teck-Cheong Lim,et al.  On fixed point stability for set-valued contractive mappings with applications to generalized differential equations , 1985 .

[30]  J. Bakker,et al.  Denotational models for programming lan-guages: Applications of banachs fixed point theorem , 1998 .

[31]  E. Tarafdar.  An approach to fixed-point theorems on uniform spaces , 1974 .

[32]  F. Browder,et al.  A general principle on ordered sets in nonlinear functional analysis , 1976 .

[33]  Edward R. Vrscay,et al.  SOLVING THE INVERSE PROBLEM FOR FUNCTION/IMAGE APPROXIMATION USING ITERATED FUNCTION SYSTEMS I: THEORETICAL BASIS , 1994 .

[34]  A. M. Ostrowski,et al.  The Round‐off Stability of Iterations , 1967 .

[35]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[36]  A CONVERSE TO THE PRINCIPLE OF CONTRACTING MAPS , 1976 .

[37]  William A. Kirk,et al.  A generalization of Caristi’s theorem with applications to nonlinear mapping theory , 1977 .

[38]  A. Brøndsted.  Fixed points and partial orders , 1976 .

[39]  W. A. Kirk,et al.  Local contractions in metric spaces , 1978 .

[40]  Endre Pap,et al.  Fixed Point Theory in Probabilistic Metric Spaces , 2001 .

[41]  Measurability of Fixed Point Sets of Multivalued Random Operators , 1998 .

[42]  Yong-Zhuo Chen.  Inhomogeneous iterates of contraction mappings and nonlinear ergodic theorems , 2000 .

[43]  K. Menger.  Statistical Metrics. , 1942, Proceedings of the National Academy of Sciences of the United States of America.

[44]  E. R. Vrscay,et al.  Continuity of Attractors and Invariant Measures for Iterated Function Systems , 1994, Canadian Mathematical Bulletin.

[45]  Mau-Hsiang Shih,et al.  Fixed point theorems for point-to-point and point-to-set maps , 1979 .

[46]  Jacek Jachymski,et al.  A minimum condition and some realted fixed-point theorems , 1999 .

[47]  A short proof of the converse to the contraction principle and some related results , 2000 .

[48]  Osmo Kaleva.  Fuzzy differential equations , 1987 .

[49]  Simeon Reich,et al.  Some fixed point problems , 1974 .

[50]  F. Browder.  On the convergence of successive applications for nonlinear functional equations , 1968 .

[51]  W. Oettli,et al.  Equivalents of Ekeland's principle , 1993, Bulletin of the Australian Mathematical Society.

[52]  J. Caristi,et al.  Fixed point theorems for mapping satisfying inwardness conditions , 1976 .

[53]  On park's open questions and some fixed-point theorems for general contractive type mappings , 1999 .

[54]  Emmett B. Keeler,et al.  A theorem on contraction mappings , 1969 .

[55]  Yong-Zhuo Chen,et al.  A Variant of the Meir–Keeler-Type Theorem in Ordered Banach Spaces , 1999 .

[56]  S. Kasahara.  On Fixed Points in Partially Ordered Sets and Kirk-Caristi Theorem , 1975 .

[57]  Hong-Kun Xu.  Random fixed point theorems for nonlinear uniformly Lipschitzian mappings , 1996 .

[58]  Phil Diamond,et al.  Absolute retracts and a general fixed point theorem for fuzzy sets , 1997, Fuzzy Sets Syst..

[59]  Sehie Park.  ON GENERAL CONTRACTIVE TYPE CONDITIONS , 1980 .

[60]  W. A. Kirk,et al.  Topics in Metric Fixed Point Theory , 1990 .

[61]  S Rich.  SOME PROBLEMS AND RESULTS IN FIXED POINT THEORY , 1983 .

[62]  Philip R. Meyers.  A Converse to Banach/s Contraction Theorem , 1967 .

[63]  D. Tan.  A CLASSIFICATION OF CONTRACTIVE MAPPINGS IN PROBABILISTIC METRIC SPACES , 1998 .

[64]  H. Sherwood,et al.  Complete probabilistic metric spaces , 1971 .

[65]  K. Deimling.  Nonlinear functional analysis , 1985 .

[66]  J. Jachymski.  An iff fixed point criterion for continuous self-mappings on a complete metric space , 1994 .

[67]  Stanisław Heilpern,et al.  Fuzzy mappings and fixed point theorem , 1981 .

[68]  K Menger,et al.  Probabilistic Geometry. , 1951, Proceedings of the National Academy of Sciences of the United States of America.

[69]  I. Rus.  Weakly Picard mappings , 1993 .

[70]  T. Burton,et al.  A Fixed Point Theorem of Krasnoselskii—Schaefer Type , 1998 .

[71]  R. F. Brown.  Fixed Point Theory and Its Applications , 1988 .

[72]  Anthony Karel Seda.  Quasi-Metrics and the Semantics of Logic Programs , 1997, Fundam. Informaticae.

[73]  Radko Mesiar,et al.  A Fixed Point Theorem in Probabilistic Metric Spaces and an Application , 1996 .

[74]  Marlène Frigon.  Fixed point results for generalized contractions in gauge spaces and applications , 2000 .

[75]  M. Z. Nashed,et al.  Fixed Points and Stability for a Sum of Two Operators in Locally Convex Spaces , 1971 .

[76]  S. Prieß-Crampe.  Der Banachsche Fixpunktsatz für ultrametrische Räume , 1990 .

[77]  Jaime Carvalho e Silva,et al.  A complete comparison of 25 contraction conditions , 1997 .

[78]  E. Rakotch,et al.  A note on contractive mappings , 1962 .

[79]  Evidence of a conspiracy among fixed point theorems , 1975 .

[80]  J. Matkowski.  Integrable solutions of functional equations , 1975 .

[81]  J. S. W. Wong,et al.  On nonlinear contractions , 1969 .

[82]  Jerrold Siegel.  A NEW PROOF OF CARISTI'S FIXED POINT THEOREM , 1977 .

[83]  Solomon Leader.  A topological characterization of Banach contractions , 1977 .

[84]  G. Jungck,et al.  Commuting Mappings and Fixed Points , 1976 .

[85]  M. Tasković.  A monotone principle of fixed points , 1985 .

[86]  C. Bessaga.  On the converse of Banach "fixed-point principle" , 1959 .

[87]  Hideaki Kaneko,et al.  On a conjecture of S. Reich , 1996 .