Document Type: Research Paper


Department of Mathematics, Faculty of Science, University of Bonab, P.O.Box 5551-761167, Bonab, Iran.


In this paper,  we introduce the notion of  generalized multivalued  $F$- weak contraction and we prove some fixed point theorems related to introduced  contraction for multivalued mapping in complete metric spaces.  Our results extend and improve the results announced by many others with less hypothesis. Also, we give some illustrative examples.


Main Subjects

[1] "{O}.  Acar and I.  Altun, A Fixed Point Theorem for Multivalued Mappings with $delta$-Distance. Abstr. Appl. Anal., 2014 (2014), Article ID 497092, 5 pages.

[2] "{O}. Acar,  G. Durmaz and  G Minak,  Generalized multivalued F-contractions on complete metric spaces. Bull. Iranian Math. Soc., 40 (2014)  1469-1478.

[3] R.P. Agarwal, D. O'Regan and N. Shahzad, Fixed point theory for generalized contractive maps of Meir-Keeler type, Math. Nachr., 276 (2004) 3-22.

[4] I. Altun, G. Minak and H. Dau{u}g, Multivalued F-contractions on complete metric space, J. Convex Anal., Accepted.

[5] S. Banach, Sur les op'{e}rations dans les ensembles abstraits et leur application aux equations itegrales, Fund. Math., 3 (1922) 133--181.

[6] V. Berinde, On the approximation of fixed points of weak contractive mappings, Carpathian J. Math., 19 (2003) 7-22.

[7] V. Berinde, Iterative Approximation of Fixed Points, Springer-Verlag, Berlin, 2007.

[8] D.W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc.,  20 (1969) 458-464.

[9] L.B. '{C}iri'{c}, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc.,  45 (1974) 267-273.

[10] W. S. Du, Some new results and generalizations in metric fixed point theory, Nonlinear Anal., 73 (2010) 1439-1446.

[11] G.E. Hardy and T. D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull., 16 (1973) 201-206.

[12] J. Matkowski, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc., 62 (1977) 344-348.

[13] G.  Minak, A.  Helvac  and  I.  Altun,   '{C}iri'{c}  Type Generalized F-contractions on CompleteMetric Spaces and Fixed Point Results., Filomat 28  (2014) 1143-1151.

[14] SB. Nadler, Multivalued contraction mappings, Pac. J. Math., 30 (1969) 475-488.

[15] H. Piri and P. Kumam, Some fixed point theorems concerning F-contraction in complete metric spaces, Fixed Point Theory Appl., 2014, 2014:210 doi:10.1186/1687-1812-2014-210.

[16] S. Reich, Kannan's fixed point theorem, Boll. Un. Mat. Ital., (4) 4 (1971) 1-11.

[17] D. Wardowski and  N. Van Dung, Fixed points of f-weak contractions on complete metric spaces, Demonstratio Math.,  1 (2014) 146-155

[18] D. Wardowski, Fixed point theory of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl., 2012, Article ID 94 (2012).