Common fixed points of monotone Lipschitzian semigroups in Banach spaces

Volume 11, Issue 1, pp 73--79

Publication Date: 2017-12-24


M. Bachar - Department of Mathematics, College of Science, King Saud University, Saudi Arabia
Mohamed A. Khamsi - Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, U.S.A.
W. M. Kozlowski - School of Mathematics and Statistics, University of New South Wales, Australia
M. Bounkhel - Department of Mathematics, College of Science, King Saud University, Saudi Arabia


In this paper, we investigate the existence of common fixed points of monotone Lipschitzian semigroup in Banach spaces under the natural condition that the images under the action of the semigroup at certain point are comparable to the point. In particular, we prove that if one map in the semigroup is a monotone contraction mapping, then such common fixed point exists. In the case of monotone nonexpansive semigroup we prove the existence of common fixed points if the Banach space is uniformly convex in every direction. This assumption is weaker than uniform convexity.


Common fixed point, fixed point, monotone contraction mappings, monotone nonexpansive mappings, monotone Lipschitzian semigroup


[1] M. R. Alfuraidan, M. A. Khamsi, Fibonacci-Mann iteration for monotone asymptotically nonexpansive mappings, Bull. Aust. Math. Soc., 96 (2017), 307–316.
[2] M. Bachar, M. A. Khamsi, On common approximate fixed points of monotone nonexpansive semigroups in Banach spaces, Fixed Point Theory Appl., 2015 (2015), 11 pages.
[3] M. Bachar, M. A. Khamsi, Recent contributions to fixed point theory of monotone mappings, J. Fixed Point Theory Appl., 19 (2017), 1953–1976.
[4] B. Beauzamy, Introduction to Banach spaces and their geometry, North-Holland Mathematics Studies, Notas de Matemática [Mathematical Notes], North-Holland Publishing Co., Amsterdam-New York, (1982).
[5] L. P. Belluce,W. A. Kirk, Nonexpansive mappings and fixed-points in Banach spaces, Illinois J. Math., 11 (1967), 474–479.
[6] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1041–1044.
[7] R. E. Bruck, Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math. Soc., 179 (1973), 251–262.
[8] A. L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Amer. Math. Soc. Transl. Ser. II, 39 (1964), 111–132.
[9] K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, (1990).
[10] D. Göhde, Zum Prinzip der kontraktiven Abbildung, (German) Math. Nachr., 30 (1965), 251–258.
[11] M. A. Khamsi, W. A. Kirk, An introduction to metric spaces and fixed point theory, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, (2001).
[12] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004–1006.
[13] W. M. Kozlowski, Monotone Lipschitzian semigroups in Banach spaces, J. Aust. Math. Soc., (in press).
[14] T. C. Lim, A fixed point theorem for families on nonexpansive mappings, Pacific J. Math., 53 (1974), 487–493.
[15] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591–597.
[16] N. H. Pavel, Nonlinear evolution operators and semigroups, Applications to partial differential equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, (1987).
[17] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Springer-Verlag, New York, (1983).
[18] 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), 1435–1443.
[19] F. Vetro, On approximating curves associated with nonexpansive mappings, Carpathian J. Math., 27 (2011), 142–147.
[20] F. Vetro, Fixed point results for nonexpansive mappings on metric spaces, Filomat, 29 (2015), 2011–2020.
[21] V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. Rozprawy Mat., 87 (1971), 33 pages.


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