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Multi-valued nonlinear contraction mappings. (English) Zbl 1179.54053

Among three fixed point theorems established in the paper there is the following: Let \((X, d)\) be a complete metric space and let \(\varphi: [0,\infty)\to[a, 1)\), \(0< a< 1\), be such that \(\varlimsup_{r\to t+}(r)< 1\) for all \(t\in[0,\infty)\). If \(T: X\to\text{Cl}(X)\) (= all nonempty closed sets of \(X\)) is such that \(x\mapsto d(x,Tx)\) is lower-semicontinuous and for any \(x\in X\) there is \(y\in Tx\) with \(\sqrt{\varphi(d(x, y))}d(x,y)\leq d (x, Tx)\) and \(d(y, Ty)\leq\varphi(d(x, y))d(x, y)\), then \(z\in Tz\) for some \(z\in\mathbb{Z}\).
This theorem generalizes results of D. Klim and D. Wiatrowski [J. Math. Anal. Appl. 334, No. 1, 132–139 (2007; Zbl 1133.54025)], Y. Feng and S. Liu [ibid., 317, No. 1, 103–112 (2006; Zbl 1094.47049)], N. Mizoguchi and W. Takahashi [ibid., 141, No. 1, 177–188 (1989; Zbl 0688.54028)].

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
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