Abstract
In this paper, the concept of multi-valued weak contraction of Berinde and Berinde [8] for the Picard iteration in a complete metric space is extended to the case of multi-valued weak contraction for the Jungck iteration in a complete b-metric space. While our main results generalize the recent results of Berinde and Berinde [8], they also extend, improve and unify several classical results pertainning to single and multi-valued contractive mappings in the fixed point theory. Our results also improve the recent results of Daffer and Kaneko [16].
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References
Agarwal R.P., Meehan M., O’Regan D., Fixed point theory and applications, Cambridge University Press, Cambridge, 2001
Banach S., Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, Fund. Math., 1922, 3, 133–181 (in French)
Berinde V., A priori and a posteriori error estimates for a class of ϕ-contractions, Bulletins for Applied & Computing Math., 1999, 183–192
Berinde V., Iterative approximation of fixed points, Editura Efemeride, Baia Mare, 2002
Berinde V., On the approximation of fixed points of weak ϕ-contractive operators, Fixed Point Theory, 2003, 4, 131–142
Berinde V., On the approximation of fixed points of weak contractive mappings, Carpathian J. Math., 2003, 19, 7–22
Berinde V., Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 2004, 9, 43–53
Berinde M., Berinde V., On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl., 2007, 326, 772–782
Boyd D.W., Wong J.S.W., On nonlinear contractions, Proc. Amer. Math. Soc., 1969, 20, 458–464
Browder F., Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 1965, 54, 1041–1044
Czerwik S., Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1993, 1, 5–11
Czerwik S., Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 1998, 46, 263–276
Ciric L.B., Fixed point theory, contraction mapping principle, FME Press, Beograd, 2003
Ciric L.B., Ume J.S., Common fixed point theorems for multi-valued non-self mappings, Publ. Math. Debrecen, 2002, 60, 359–371
Ciric L.B., Ume J.S., On the convergence of Ishikawa iterates to a common fixed point of multi-valued mappings, Demonstratio Math., 2003, 36, 951–956
Daffer P.Z., Kaneko H., Fixed points of generalized contractive multi-valued mappings, J. Math. Anal. Appl., 1995, 192, 655–666
Geraghty M.A., On contractive mappings, Proc. Amer. Math. Soc., 1973, 40, 604–608
Goffman C., Pedrick G., First course in functional analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1965
Itoh S., Multivalued generalized contractions and fixed point theorems, Comment. Math. Univ. Carolinae, 1977, 18, 247–258
Joshi M.C., Bose R.K., Some topics in nonlinear functional analysis, John Wiley & Sons, Inc., New York, 1985
Kaneko H., A general principle for fixed points of contractive multivalued mappings, Math. Japon., 1986, 31, 407–411
Kaneko H., Generalized contractive multivalued mappings and their fixed points, Math. Japon., 1988, 33, 57–64
Khamsi M.A., Kirk W.A., An introduction to metric spaces and fixed point theory, Wiley-Interscience, New York, 2001
Kubiaczyk I., Ali N.M., On the convergence of the Ishikawa iterates to a common fixed point for a pair of multi-valued mappings, Acta Math. Hungar., 1997, 75, 253–257
Lim T.C., On fixed point stability for set-valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl., 1985, 110, 436–441
Markins J.T., A fixed point theorem for set-valued mappings, Bull. Amer. Math. Soc., 1968, 74, 639–640
Mizoguchi M., Takahashi W., Fixed point theorems for multi-valued mappings on complete metric spaces, J. Math. Anal. Appl., 1989, 141, 177–188
Nadler S.B., Multi-valued contraction mappings, Pacific J. Math., 1969, 30, 475–488
Olatinwo M.O., A generalization of some results on multi-valued weakly Picard mappings in b-metric space, preprint
Picard E., Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J.Math. Pures et Appl., 1890, 6, 145–210 (in French)
Rhoades B.E., A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 1977, 226, 257–290
Rhoades B.E., A fixed point theorem for a multivalued non-self mapping, Comment. Math. Univ. Carolin., 1996, 37, 401–404
Rhoades B.E., Watson B., Fixed points for set-valued mappings on metric spaces, Math. Japon., 1990, 35, 735–743
Rus I.A., Fixed point theorems for multi-valued mappings in complete metric spaces, Math. Japon., 1975, 20, 21–24
Rus I.A., Generalized contractions and applications, Cluj University Press, Cluj Napoca, 2001
Rus I.A., Petrusel A., Petrusel G., Fixed point theory: 1950-2000, Romanian Contributions, House of the Book of Science, Cluj Napoca, 2002
Singh S.L., Bhatnagar C., Hashim A.M., Round-off stability of Picard iterative procedure for multivalued operators, Nonlinear Anal. Forum, 2005, 10, 13–19
Zeidler E., Nonlinear functional analysis and its applications-fixed point theorems, Springer-Verlag, New York, Inc., 1986
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Olatinwo, M.O. Some results on multi-valued weakly jungck mappings in b-metric space. centr.eur.j.math. 6, 610–621 (2008). https://doi.org/10.2478/s11533-008-0047-3
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DOI: https://doi.org/10.2478/s11533-008-0047-3