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Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. (English) Zbl 1094.47049
In this paper, the famous Banach contraction principle and Caristi’s fixed point theorem are generalized to the case of multi-valued mappings. The results are extensions of Nadler’s fixed point theorem and some Caristi type theorems for multi-valued operators.

##### MSC:
 47H10 Fixed-point theorems 47H04 Set-valued operators 54H25 Fixed-point and coincidence theorems (topological aspects) 54C60 Set-valued maps in general topology
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##### References:
 [1] Nadler, S.B., Multi-valued contraction mappings, Pacific J. math., 30, 475-487, (1969) · Zbl 0187.45002 [2] Caristi, J., Fixed point theorems for mappings satisfying inwardness conditions, Trans. amer. math. soc., 215, 241-251, (1976) · Zbl 0305.47029 [3] Jachymski, J.R., Caristi’s fixed point theorem and selections of set-valued contractions, J. math. anal. appl., 227, 55-67, (1998) · Zbl 0916.47044 [4] Zhong, C.K.; Zhu, J.; Zhao, P.H., An extension of multi-valued contraction mappings and fixed points, Proc. amer. math. soc., 128, 2439-2444, (2000) · Zbl 0948.47058 [5] Branciari, A., A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. math. math. sci., 29, 531-536, (2002) · Zbl 0993.54040 [6] Naidu, S.V.R., Fixed-point theorems for a broad class of multimaps, Nonlinear anal., 52, 961-969, (2003) · Zbl 1029.54049 [7] Kirk, W.A., Caristi’s fixed-point theorem and metric convexity, Colloq. math., 36, 81-86, (1976) · Zbl 0353.53041 [8] Mizoguchi, N.; Takahashi, W., Fixed point theorems for multivalued mappings on complete metric spaces, J. math. anal. appl., 141, 177-188, (1989) · Zbl 0688.54028 [9] Aubin, J.P., Optima and equilibria. an introduction to nonlinear analysis, Grad. texts in math., (1998), Springer-Verlag Berlin [10] Wang, T., Fixed point theorems and fixed point stability for multivalued mappings on metric spaces, J. Nanjing univ. math. baq., 6, 16-23, (1989) · Zbl 0715.54034 [11] Aubin, J.P.; Siegel, J., Fixed point and stationary points of dissipative multi-valued maps, Proc. amer. math. soc., 78, 391-398, (1980) · Zbl 0446.47049 [12] Covitz, H.; Nadler, S.B., Multi-valued contraction mappings in generalized metric space, Israel J. math., 8, 5-11, (1970) · Zbl 0192.59802 [13] Zhang, S.S.; Luo, Q., Set-valued Caristi fixed point theorem and Ekeland’s variational principle, Appl. math. mech., Appl. math. mech. (English ed.), 10, 2, 119-121, (1989), (in Chinese), English translation: · Zbl 0738.49009 [14] Petruşel, A.; Sîntămărian, A., Single-valued and multi-valued Caristi type operators, Publ. math. debrecen, 60, 167-177, (2002) [15] Van Hot, L., Fixed point theorems for multi-valued mapping, Comment. math. univ. carolin., 23, 137-145, (1982) · Zbl 0492.47035 [16] Agarawl, R.P.; O’Regan, D.O.; Shahzad, N., Fixed point theorem for generalized contractive maps of meir – keeler type, Math. nachr., 276, 3-22, (2004) · Zbl 1086.47016 [17] Jachymski, J.R., Converses to fixed point theorems of zeremlo and Caristi, Nonlinear anal., 52, 1455-1463, (2003) · Zbl 1030.54033 [18] Vijayaraju, P.; Rhoades, B.E.; Mohanraj, R., A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type, Int. J. math. math. sci., 15, 2359-2364, (2005) · Zbl 1113.54027 [19] Feng, Y.; Liu, S., Fixed point theorems for multi-valued operators in partial ordered spaces, Soochow J. math., 30, 461-469, (2004) · Zbl 1084.47046
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