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.


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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[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: 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., 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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