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Year 2017, Volume: 1 Issue: 2, 64 - 112, 20.12.2017
https://doi.org/10.31197/atnaa.379089

Abstract

References

  • R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskii. Measures of Noncompactness and Condensing Operators. Birkha¨user Verlag, Basel, 1992.
  • S. Banach. Sur les operations dans les ensembles abstraits et leur applications aux equations integrales. Fund. Math., 3:133-181, 1922.
  • J. Banas and K. Goebel. Measures of Noncompactness in Banach Spaces, volume 60 of Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, New York and Basel, 1980.
  • H. G. Barone. Limit points of sequences and their transforms by methods of summability. Duke Math. J., 5:740-752, 1939.
  • A. Bollenbacher and T.L.Hicks. A xed point theorem revisited. Proc. Amer. Math. Soc., 102:898-900, 1988.
  • D. W. Boyd and J. S. W. Wong. Another proof of contraction mapping theorem. Canad. Math. Bull., 11:605-606, 1968.
  • D. W. Boyd and J. S. W. Wong. On nonlinear contractions. Proc. Amer. Math. Soc., 20:458-464, 1969.
  • H. Brezis and F. E. Browder. A general principle on ordered sets in nonlinear functional analysis. Advances in Mathematics, 21:355-364, 1976.
  • F. E. Browder and W. V. Petryshyn. The solution by iteration of linear functional equations in banach spaces. Bull. Amer. Math. Soc., 72:571-575, 1966.
  • V. W. Bryant. A remark on a xed point theorem for iterated mappings. Amer. Math. Monthly, 75:399-400, 1968.
  • J. Caristi. Fixed point theorems for mappings satisfying inwardness conditions. Trans Amer. Math. Soc., 215:241-251, 1976.
  • S. K. Chatterjea. Fixed point theorems. C. R. Acad. Bulgare Sci., 15:727-730, 1972.
  • Lj. B. Ciric. Generalized contractions and xed point theorems. Publ. Inst. Math., 12(26):19-26, 1971.
  • Lj. B. Ciric. A generalization of Banach's contraction principle. Proc. Amer. Math. Soc., 45:267-273, 1974.
  • Lj.B. Ciric. Some Recent Results in Metrical Fixed Point Theory. University of Belgrade, Belgrade, 2003.
  • Conference on Computing Fixed Points with Applications, Fixed Point Algorithms and Applications. Fixed point iteration using in nite matrices, III, New York, 1977. Academic Press.
  • G. Darbo. Punti uniti in transformazioni a condominio non compatto. Rend. Sem. Math. Univ. Padova, 24:84-92, 1955.
  • W. G. Dotson. On the Mann iterative methods. Trans. Amer. Math. Soc., 149:65-73, 1970.
  • M. Edelstein. An extension of Banach's contraction principle. Proc. Amer. Math. Soc., 12:7-10, 1961.
  • M. Edelstein. On xed and periodic points under contractive mappings. J. London Math. Soc., 37:74-79, 1962.
  • M. Edelstein. A remark on a theorem of M. A. Krasnoselskii. Amer. Math. Monthly, 73:509-510, 1966.
  • J. Eisenfeld and V. Lakshmikantham. Fixed point theorems on closed sets through abstract cones. Appl. Math Comput., 3:155-166, 1977.
  • B. Fisher. A xed point theorem. Mathematics Magazine, 48:223-225, 1975.
  • R. L. Franks and R. P. Marzec. A theorem on mean{value iterations. Proc. Amer. Math. Soc., 30(20):324-326, 1971.
  • G. E. Hardy and T. D. Rogers. A generalization of a xed point theorem of Reich. Canad. Math. Bull., 16:201-206, 1973.
  • T. L. Hicks and B. E. Rhoades. A Banach type xed point theorem. Math. Japon., 24:327-330, 1979.
  • V. Istratesku. On a measure of noncompactness. Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.), 16:195-197, 1972.
  • V. Istratesku. Fixed Point Theory, An Introduction. Reidel Publishing Company, Dordrecht, Boston and London, 1981.
  • J. E. Joseph and M. H. Kwack. Alternative approaches to proofs of contraction mapping xed point theorems. Missouri J. Math Sci., 11:167-175, 1999.
  • R. Kannan. Some results on xed points II. Amer. Math. Monthly, 76:405{408, 1969.
  • W. A. Kirk and L. M. Saliga. The Brezis{Browder order principle and extensions of Caristi's theorem. Nonlinear Analysis, 47:2765-2778, 2001.
  • M. A. Krasnoselski. Two remarks on the method of successive approximations. Uspehi Math. Nauk (N.S.), 10(1):123-127, 1955.
  • K. Kuratowski. Sur les espaces complets. Fund. Math., 15:301-309, 1930.
  • K. Kuratowski. Topologie. Warsaw, 1958.
  • E. Malkowsky and V. Rakocevic. An introduction into the theory of sequence spaces and measures of noncompactness, volume 9(17) of Zbornik radova, Matematcki institut SANU, pages 143{234. Mathematical Institute of SANU, Belgrade, 2000.
  • E. Malkowsky and V. Rakocevic. Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, chapter On some results using measures of noncompactness, pages 127-180. Springer Verlag, 2017.
  • W. R. Mann. Semigroups of operators and measures of noncompactness. Proc. Amer. Math. Soc., 4:506-510, 1953.
  • S. Mazur. Uber die kleinste konvexe Menge, die eine gegebene kompakte Menge enthalt. Studia Mathematica, 2:7-9, 1930.
  • A. Meir and E.Keeler. A theorem on contraction mappings. J. Math. Anal. Appl., 28:326-329, 1969.
  • Z. Opial. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc., 73:591-597, 1966.
  • C. L. Outlaw and C. W. Groetsch. Averaging iteration in a Banach space. Bull. Amer. Math. Soc., 75:430-432, 1969.
  • R. S. Palais. A simple proof of the Banach contraction principle. J. Fixed Point Theory Appl., 2:221-223, 2007.
  • E. Rakotch. A note on contractive mappings. Proc. Amer. Math. Soc., 13:459-465, 1962.
  • S. Reich. Some remarks concerning contraction mappings. Canad. Math. Bull., 14:121-124, 1971.
  • J. Reinermann. Uber Toeplitzsche Iterationsverfahren und einige ihrer Anwendungen in der konstruktiven Fixpunktheorie. Studia Math., 32:209-227, 1969.
  • B. E. Rhoades. Fixed point iterations using in nite matrices. Preliminary report. Notices Amer. Math. Soc., 19, 1972.
  • B. E. Rhoades. Constructive and Computational Methods for Di erential and Integral Equations, volume 430 of Lecture Notes in Mathematics, chapter Fixed point iterations using in nite matrices II, pages 390{395. Springer{Verlag, New York, Berlin, 1974.
  • B. E. Rhoades. Fixed point iterations using in nite matrices. Trans. Amer. Math. Soc., 196:161-176, 1974.
  • B. E. Rhoades. A comparision of various de nitions of contractive mappings. Notices Amer. Math. Soc., 26:257-290, 1977.
  • H. Schafer. Uber die methode sukzessiver approximation. JBer. Deutsch. Math. Verein., 59:131-140, 1957.
  • J. Schauder. Der Fixpunktsatz in Funktionalraumen. Studia Math., 2:171-180, 1930.
  • V. M. Sehgal. A new proof of Caristi's xed point theorem. Proc. Amer. Math. Soc., 66:54-56, 1977.
  • J. M. Ayerbe Toledano, T. Dominguez Benavides, and G. Lopez Acedo. Measures of Noncompactness in Metric Fixed Point Theory, volume 99 of Operator Theory Advances and Applications. Birkhauser Verlag, Basel, Boston, Berlin, 1997.
  • T. Zamfirescu. Fixed point theorems in metric spaces. Archiv der Mathematik, 23:292-298, 1972.

Some results in metric fixed point theory

Year 2017, Volume: 1 Issue: 2, 64 - 112, 20.12.2017
https://doi.org/10.31197/atnaa.379089

Abstract

This is a survey of results mainly in metric fixed point theory, including the Darbo–Sadovski˘i theorem using measures of noncompactness. Various different proofs are presented for some of the most important historical results. Furthermore many examples and remarks are added to illustrate the topics of the paper.

References

  • R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskii. Measures of Noncompactness and Condensing Operators. Birkha¨user Verlag, Basel, 1992.
  • S. Banach. Sur les operations dans les ensembles abstraits et leur applications aux equations integrales. Fund. Math., 3:133-181, 1922.
  • J. Banas and K. Goebel. Measures of Noncompactness in Banach Spaces, volume 60 of Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, New York and Basel, 1980.
  • H. G. Barone. Limit points of sequences and their transforms by methods of summability. Duke Math. J., 5:740-752, 1939.
  • A. Bollenbacher and T.L.Hicks. A xed point theorem revisited. Proc. Amer. Math. Soc., 102:898-900, 1988.
  • D. W. Boyd and J. S. W. Wong. Another proof of contraction mapping theorem. Canad. Math. Bull., 11:605-606, 1968.
  • D. W. Boyd and J. S. W. Wong. On nonlinear contractions. Proc. Amer. Math. Soc., 20:458-464, 1969.
  • H. Brezis and F. E. Browder. A general principle on ordered sets in nonlinear functional analysis. Advances in Mathematics, 21:355-364, 1976.
  • F. E. Browder and W. V. Petryshyn. The solution by iteration of linear functional equations in banach spaces. Bull. Amer. Math. Soc., 72:571-575, 1966.
  • V. W. Bryant. A remark on a xed point theorem for iterated mappings. Amer. Math. Monthly, 75:399-400, 1968.
  • J. Caristi. Fixed point theorems for mappings satisfying inwardness conditions. Trans Amer. Math. Soc., 215:241-251, 1976.
  • S. K. Chatterjea. Fixed point theorems. C. R. Acad. Bulgare Sci., 15:727-730, 1972.
  • Lj. B. Ciric. Generalized contractions and xed point theorems. Publ. Inst. Math., 12(26):19-26, 1971.
  • Lj. B. Ciric. A generalization of Banach's contraction principle. Proc. Amer. Math. Soc., 45:267-273, 1974.
  • Lj.B. Ciric. Some Recent Results in Metrical Fixed Point Theory. University of Belgrade, Belgrade, 2003.
  • Conference on Computing Fixed Points with Applications, Fixed Point Algorithms and Applications. Fixed point iteration using in nite matrices, III, New York, 1977. Academic Press.
  • G. Darbo. Punti uniti in transformazioni a condominio non compatto. Rend. Sem. Math. Univ. Padova, 24:84-92, 1955.
  • W. G. Dotson. On the Mann iterative methods. Trans. Amer. Math. Soc., 149:65-73, 1970.
  • M. Edelstein. An extension of Banach's contraction principle. Proc. Amer. Math. Soc., 12:7-10, 1961.
  • M. Edelstein. On xed and periodic points under contractive mappings. J. London Math. Soc., 37:74-79, 1962.
  • M. Edelstein. A remark on a theorem of M. A. Krasnoselskii. Amer. Math. Monthly, 73:509-510, 1966.
  • J. Eisenfeld and V. Lakshmikantham. Fixed point theorems on closed sets through abstract cones. Appl. Math Comput., 3:155-166, 1977.
  • B. Fisher. A xed point theorem. Mathematics Magazine, 48:223-225, 1975.
  • R. L. Franks and R. P. Marzec. A theorem on mean{value iterations. Proc. Amer. Math. Soc., 30(20):324-326, 1971.
  • G. E. Hardy and T. D. Rogers. A generalization of a xed point theorem of Reich. Canad. Math. Bull., 16:201-206, 1973.
  • T. L. Hicks and B. E. Rhoades. A Banach type xed point theorem. Math. Japon., 24:327-330, 1979.
  • V. Istratesku. On a measure of noncompactness. Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.), 16:195-197, 1972.
  • V. Istratesku. Fixed Point Theory, An Introduction. Reidel Publishing Company, Dordrecht, Boston and London, 1981.
  • J. E. Joseph and M. H. Kwack. Alternative approaches to proofs of contraction mapping xed point theorems. Missouri J. Math Sci., 11:167-175, 1999.
  • R. Kannan. Some results on xed points II. Amer. Math. Monthly, 76:405{408, 1969.
  • W. A. Kirk and L. M. Saliga. The Brezis{Browder order principle and extensions of Caristi's theorem. Nonlinear Analysis, 47:2765-2778, 2001.
  • M. A. Krasnoselski. Two remarks on the method of successive approximations. Uspehi Math. Nauk (N.S.), 10(1):123-127, 1955.
  • K. Kuratowski. Sur les espaces complets. Fund. Math., 15:301-309, 1930.
  • K. Kuratowski. Topologie. Warsaw, 1958.
  • E. Malkowsky and V. Rakocevic. An introduction into the theory of sequence spaces and measures of noncompactness, volume 9(17) of Zbornik radova, Matematcki institut SANU, pages 143{234. Mathematical Institute of SANU, Belgrade, 2000.
  • E. Malkowsky and V. Rakocevic. Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, chapter On some results using measures of noncompactness, pages 127-180. Springer Verlag, 2017.
  • W. R. Mann. Semigroups of operators and measures of noncompactness. Proc. Amer. Math. Soc., 4:506-510, 1953.
  • S. Mazur. Uber die kleinste konvexe Menge, die eine gegebene kompakte Menge enthalt. Studia Mathematica, 2:7-9, 1930.
  • A. Meir and E.Keeler. A theorem on contraction mappings. J. Math. Anal. Appl., 28:326-329, 1969.
  • Z. Opial. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc., 73:591-597, 1966.
  • C. L. Outlaw and C. W. Groetsch. Averaging iteration in a Banach space. Bull. Amer. Math. Soc., 75:430-432, 1969.
  • R. S. Palais. A simple proof of the Banach contraction principle. J. Fixed Point Theory Appl., 2:221-223, 2007.
  • E. Rakotch. A note on contractive mappings. Proc. Amer. Math. Soc., 13:459-465, 1962.
  • S. Reich. Some remarks concerning contraction mappings. Canad. Math. Bull., 14:121-124, 1971.
  • J. Reinermann. Uber Toeplitzsche Iterationsverfahren und einige ihrer Anwendungen in der konstruktiven Fixpunktheorie. Studia Math., 32:209-227, 1969.
  • B. E. Rhoades. Fixed point iterations using in nite matrices. Preliminary report. Notices Amer. Math. Soc., 19, 1972.
  • B. E. Rhoades. Constructive and Computational Methods for Di erential and Integral Equations, volume 430 of Lecture Notes in Mathematics, chapter Fixed point iterations using in nite matrices II, pages 390{395. Springer{Verlag, New York, Berlin, 1974.
  • B. E. Rhoades. Fixed point iterations using in nite matrices. Trans. Amer. Math. Soc., 196:161-176, 1974.
  • B. E. Rhoades. A comparision of various de nitions of contractive mappings. Notices Amer. Math. Soc., 26:257-290, 1977.
  • H. Schafer. Uber die methode sukzessiver approximation. JBer. Deutsch. Math. Verein., 59:131-140, 1957.
  • J. Schauder. Der Fixpunktsatz in Funktionalraumen. Studia Math., 2:171-180, 1930.
  • V. M. Sehgal. A new proof of Caristi's xed point theorem. Proc. Amer. Math. Soc., 66:54-56, 1977.
  • J. M. Ayerbe Toledano, T. Dominguez Benavides, and G. Lopez Acedo. Measures of Noncompactness in Metric Fixed Point Theory, volume 99 of Operator Theory Advances and Applications. Birkhauser Verlag, Basel, Boston, Berlin, 1997.
  • T. Zamfirescu. Fixed point theorems in metric spaces. Archiv der Mathematik, 23:292-298, 1972.
There are 54 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Eberhard Malkowsky

Vladimir Rakocevic

Publication Date December 20, 2017
Published in Issue Year 2017 Volume: 1 Issue: 2

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