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An extension of multi-valued contraction mappings and fixed points. (English) Zbl 0948.47058
Let \(X\) be a complete metric space and \(CB(X)\) the set of all non-empty bounded and closed subsets of \(X\) together with a Hausdorff metric defined on it. In this interesting paper the authors have proved the following generalization of the well-known Banach contraction principle: Let \(F\): \(X\to CB(X)\) be a multivalued and upper semi-continuous mapping, \(x_0\in X\) a given point and \(\sigma\in (0,1]\) a constant. Let \(h:[0,+\infty)\to [0,+\infty)\) be a continuous non-decreasing function satisfying \[ \int^{+\infty}_0 {dr\over 1+ h(r)}=+\infty, \] such that for any \(x\in X\), if \(x\not\in F(x)\), then for any \(y\in F(x)\) \[ \sup_{x\in F(x)} d(z, F(y))\leq \Biggl(1-{\sigma\over 1+ h(d(x_0, x))}\Biggr) d(x,y). \] Then \(F\) has a fixed point. Several consequences of the above result and examples are also given.
Reviewer: Ismat Beg (Kuwait)

47H10 Fixed-point theorems
47H04 Set-valued operators
Full Text: DOI
[1] James Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241 – 251. · Zbl 0305.47029
[2] Sam B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475 – 488. · Zbl 0187.45002
[3] Hong Wei Yi and Yi Chun Zhao, Fixed point theorems for weakly inward multivalued mappings and their randomizations, J. Math. Anal. Appl. 183 (1994), no. 3, 613 – 619. · Zbl 0815.47072 · doi:10.1006/jmaa.1994.1167 · doi.org
[4] V. M. Sehgal and R. E. Smithson, A fixed point theorem for weak directional contraction multifunctions, Math. Japon. 25 (1980), no. 3, 345 – 348. · Zbl 0453.54030
[5] Cheng-Kui Zhong, A generalization of Ekeland’s variational principle and application to the study of the relation between the weak P.S. condition and coercivity, Nonlinear Anal. 29 (1997), no. 12, 1421 – 1431. · Zbl 0912.49021 · doi:10.1016/S0362-546X(96)00180-0 · doi.org
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