On monotone multivalued transformations

Volume 10, Issue 10, pp 5321--5327 http://dx.doi.org/10.22436/jnsa.010.10.18 Publication Date: October 20, 2017


Buthinah A. Bin Dehaish - Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
Mohamed A. Khamsi - Department of Mathematics \(\&\) Statistics, King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia and Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968, U.S.A.


In this work, we discuss the recently introduced monotone \(\tau\)-Opial condition in Banach spaces which admit a sequence of monotone approximations of the identity. Then we give a fixed point theorem for monotone multivalued nonexpansive mappings in Banach spaces satisfying the monotone \(\tau\)-Opial condition. This result generalizes those of Markin, Browder and Lami Dozo to monotone mappings.



[1] M. R. Alfuraidan, M. A. Khamsi, A fixed point theorem for monotone asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., (to appear).
[2] M. Bachar, M. A. Khamsi, Recent contributions to fixed point theory of monotone mappings, J. Fixed Point Theory Appl., 19 (2017), 1953–1976.
[3] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181.
[4] T. D. Benavides, P. L. Ramírez, Fixed-point theorems for multivalued non-expansive mappings without uniform convexity, Abstr. Appl. Anal., 2003 (2003), 375–386.
[5] F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), Amer. Math. Soc., Providence, R. I., (1976), 1–308.
[6] T. Domínguez Benavides, J. García Falsett, M. A. Japón Pineda, The \(\tau\)-fixed point property for nonexpansive mappings, Abstr. Appl. Anal., 3 (1998), 343–362.
[7] R. M. Dudley, On sequential convergence, Trans. Amer. Math. Soc., 112 (1964), 483–507.
[8] K. Goebel, W. A. Kirk, Iteration processes for nonexpansive mappings, Topological methods in nonlinear functional analysis, Toronto, Ont., (1982), Contemp. Math., Amer. Math. Soc., Providence, RI, 21 (1983), 115–123.
[9] K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, 28. Cambridge University Press, Cambridge, (1990).
[10] S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59 (1976), 65–71.
[11] M. A. Khamsi, On uniform Opial condition and uniform Kadec-Klee property in Banach and metric spaces, Nonlinear Anal., 10 (1996), 1733–1748.
[12] 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).
[13] M. A. Krasnosel’skiı, Two remarks on the method of successive approximations, (Russian) Uspehi Mat. Nauk (N.S.), 10 (1955), 123–127.
[14] E. Lami Dozo, Multivalued nonexpansive mappings and Opial’s condition, Proc. Amer. Math. Soc., 38 (1973), 286–292.
[15] T.C. Lim, Asymptotic centers and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math., 90 (1980), 135– 143.
[16] J. T. Markin, A fixed point theorem for set valued mappings, Bull. Amer. Math. Soc., 74 (1968), 639–640.
[17] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591–597.
[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.