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 Submission Date: April 17, 2017

Download PDF

Download XML

2399 Downloads
4323 Views

Authors

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. - Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968, U.S.A.

Abstract

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.

Share and Cite

ISRP Style

Buthinah A. Bin Dehaish, Mohamed A. Khamsi, On monotone multivalued transformations, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 10, 5321--5327

AMA Style

Bin Dehaish Buthinah A., Khamsi Mohamed A., On monotone multivalued transformations. J. Nonlinear Sci. Appl. (2017); 10(10):5321--5327

Chicago/Turabian Style

Bin Dehaish, Buthinah A., Khamsi, Mohamed A.. "On monotone multivalued transformations." Journal of Nonlinear Sciences and Applications, 10, no. 10 (2017): 5321--5327

Keywords

Fixed point
systems of projections
monotone Opial condition
monotone nonexpansive mappings
multivalued mappings

MSC

47H10
46B99

References

[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.
- Google Scholar

[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.
- MathSciNet
- Google Scholar

[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.
- View Article
- Google Scholar

[7] R. M. Dudley , On sequential convergence, Trans. Amer. Math. Soc., 112 (1964), 483–507.
- Google Scholar

[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.
- Google Scholar

[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.
- MathSciNet
- Google Scholar

[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.
- Google Scholar

[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.