Notes on Knaster-Tarski Theorem versus Monotone Nonexpansive Mappings

Dedicated to Ibn al-Banna’ al-Marrakushi (c. 1256 c. 1321)

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Abstract

The purpose of this note is to discuss the recent paper of Espínola and Wiśnicki about the fixed point theory of monotone nonexpansive mappings. In their work, it is claimed that most of the fixed point results of this class of mappings boil down to the classical Knaster-Tarski fixed point theorem. We will show that their approach is very restrictive and fail to have any meaningful usefulness in applications.

[1] M. R. Alfuraidan, Topological aspects of weighted graphs with application to fixed point theory, Applied Mathematics and Computation Volume 314, 1 December 2017, Pages 287-292.

[2] M. R. Alfuraidan and M. A. Khamsi, Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph, Fixed Point Theory and Applications (2015) 2015:44 DOI 10.1186/s13663-015-0294-5.

[3] M. R. Alfuraidan and M. A. Khamsi, A fixed point theorem for monotone asymptotically nonexpansive mappings, Proceedings of the American Mathematical Society 146 (6), 2451-2456

[4] M. R. Alfuraidan and M. A. Khamsi, Fibonacci-Mann Iteration for Monotone Asymptotically Nonexpansive Mappings, Bull. Aust. Math. Soc., Volume 96, Issue 2 (2017), 307-316.

[5] M. R. Alfuraidan, M. Bachar and M. A. Khamsi, Almost Monotone Contractions on Weighted Graphs, J. Nonlinear Sci. Appl., Volume: 9 (2016) Issue: 8, Pages: 5189–5195.

[6] M. R. Alfuraidan and S. A. Shukri, Browder and Göhde fixed point theorem for G-nonexpansive mappings, J. Nonlinear Sci. Appl. 9 (2016), 40784083

[7] M. Bachar, M. A. Khamsi, Delay differential equation in metric spaces: A partial ordered sets approach, Fixed Point Theory and Applications 2014, 2014:193. DOI:10.1186/1687-1812-2014-193

[8] M. Bachar and M. A. Khamsi Fixed Points of Monotone Mappings and Application to Integral Equations, Fixed Point Theory and Applications (2015) 2015:110. DOI:10.1186/s13663-015-0362-x.

[9] S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications, Fund. Math. 3(1922), 133-181.

[10] H. Ben-El-Mechaiekh, The Ran-Reurings fixed point theorem without partial order: A simple proof, J. Fixed Point Theory Appl., 16 (2014), 373-383.

[11] B. A. Bin Dehaish and M. A. Khamsi, Browder and Göhde fixed point theorem for monotone nonexpansive mappings, Fixed Point Theory and Applications 2016, 2016:20 DOI 10.1186/s13663-016-0505-8

[12] B. A. Bin Dehaish and M. A. Khamsi, Mann Iteration Process for Monotone Nonexpansive Mappings, Fixed Point Theory and Applications 2015, 2015:177 DOI 10.1186/s13663-015-0416-0

[13] B. A. Bin Dehaish and M. A. Khamsi, Remarks on Monotone Contractive type Mappings in weighted graphs, J. of Nonlinear and Convex Analysis, Volume 19, Number 6, 1021-1027, 2018.

[14] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1041-1044.

[15] S. Carl, S. Heikkilä, Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory, Springer, Berlin, New York, 2011.

[16] R. Diestel, Graph Theory, Springer-Verlag, new York, 2000.

[17] R. Espínola, A. Wiśnicki, The Knaster-Tarski theorem versus monotone nonexpansive mappings, Bulletin of the Polish Academy of Sciences Mathematics (2017) DOI: 10.4064/ba8120-1-2018

[18] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174.

[19] K. Goebel, W.A. Kirk, Iteration processes for nonexpansive mappings, Contemp. Math., 21 (1983), pp. 115-123.

[20] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251-258.

[21] S. Heikkilä and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker 1994.

[22] S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), 65-71.

[23] J. Jachymski, The Contraction Principle for Mappings on a Metric Space with a Graph, Proc. Amer. Math. Soc. 136(2008), 1359–1373.

[24] R. Johnsonbaugh, Discrete Mathematics, Prentice-Hall, Inc., New Jersey, 1997.

[25] M. A. Khamsi, and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, John Wiley, New York, 2001.

[26] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72(1965), 1004-1006.

[27] B. Knaster, Un théorème sur les fonctions densembles, Annales Soc. Polonaise 6 (1928), 133134.

[28] M. A. Krasnoselskii, Two observations about the method of successive approximations, Uspehi Mat. Nauk 10 (1955), 123-127.

[29] J. J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), no. 3, 223–239.

[30] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), pp. 591-597

[31] M. Pˇacurar, Iterative Methods for Fixed Point Approximation, PhD Thesis, Babeş-Bolyai University, Cluj-Napoca, 2009.

[32] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1435–1443.

[33] S. Reich, I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15, 537-558 (1990)

[34] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appli. 158 (1991), 407–413.

[35] A. Tarski, A lattice-theoretical fixpoint theorem and its applications, Pac. J. Math. 5 (1955), 285309.

[36] E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986, 897 pp.

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