Fixed point results for multivalued maps in metric spaces with generalized inwardness conditions. (English) Zbl 1188.54018

Summary: We establish fixed point theorems for multivalued mappings defined on a closed subset of a complete metric space. We generalize Lim’s result on weakly inward contractions in a Banach space. We also generalize recent results of Azé and Corvellec, Maciejewski, and Uderzo for contractions and directional contractions. Finally, we present local fixed point theorems and continuation principles for generalized inward contractions.


54H25 Fixed-point and coincidence theorems (topological aspects)
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[1] Nadler, SB, Multivalued contraction mappings, Pacific Journal of Mathematics, 30, 475-487, (1969) · Zbl 0187.45002
[2] Lim, T-C, A fixed point theorem for weakly inward multivalued contractions, Journal of Mathematical Analysis and Applications, 247, 323-327, (2000) · Zbl 0957.47040
[3] Azé, D; Corvellec, J-N, A variational method in fixed point results with inwardness conditions, Proceedings of the American Mathematical Society, 134, 3577-3583, (2006) · Zbl 1113.47044
[4] Maciejewski, M, Inward contractions on metric spaces, Journal of Mathematical Analysis and Applications, 330, 1207-1219, (2007) · Zbl 1126.54016
[5] Song, W, A generalization of Clarke’s fixed point theorem, Applied Mathematics: A Journal of Chinese Universities. Series B, 10, 463-466, (1995) · Zbl 0862.47039
[6] Uderzo, A, Fixed points for directional multi-valued [inlineequation not available: see fulltext.] -contractions, Journal of Global Optimization, 31, 455-469, (2005) · Zbl 1081.47058
[7] Caristi, J, Fixed point theorems for mappings satisfying inwardness conditions, Transactions of the American Mathematical Society, 215, 241-251, (1976) · Zbl 0305.47029
[8] Ekeland, I, Sur LES problèmes variationnels, Comptes Rendus de l’Académie des Sciences, Série A-B, 275, 1057-1059, (1972) · Zbl 0249.49004
[9] Ekeland, I, Nonconvex minimization problems, Bulletin of the American Mathematical Society, 1, 443-474, (1979) · Zbl 0441.49011
[10] Dugundji J, Granas A: Fixed Point Theory. I, Mathematical Monographs. Volume 61. Polish Scientific Publishers, Warszawa, Poland; 1982. · Zbl 0483.47038
[11] Granas A, Dugundji J: Fixed Point Theory, Springer Monographs in Mathematics. Springer, New York, NY, USA; 2003. · Zbl 1025.47002
[12] Bishop, E; Phelps, RR; Klee, V (ed.), The support functionals of a convex set, (1963), Providence, RI, USA · Zbl 0149.08601
[13] Phelps, RR, Support cones in Banach spaces and their applications, Advances in Mathematics, 13, 1-19, (1974) · Zbl 0284.46015
[14] Mizoguchi, N; Takahashi, W, Fixed point theorems for multivalued mappings on complete metric spaces, Journal of Mathematical Analysis and Applications, 141, 177-188, (1989) · Zbl 0688.54028
[15] Downing, D; Kirk, WA, Fixed point theorems for set-valued mappings in metric and Banach spaces, Mathematica Japonica, 22, 99-112, (1977) · Zbl 0372.47030
[16] Clarke, F, Pointwise contraction criteria for the existence of fixed points, Canadian Mathematical Bulletin, 21, 7-11, (1978) · Zbl 0414.54030
[17] Frigon, M; Granas, A, Résultats du type de Leray-Schauder pour des contractions multivoques, Topological Methods in Nonlinear Analysis, 4, 197-208, (1994) · Zbl 0829.47047
[18] Browder, FE, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bulletin of the American Mathematical Society, 74, 660-665, (1968) · Zbl 0164.44801
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