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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 $\sqrt{\varphi(d(x, y))}d(x,y)\le d (x, Tx)$ and $d(y, Ty)\le\varphi(d(x, y))d(x, y)$, then $z\in Tz$ for some $z\in\bbfZ$. This theorem generalizes results of {\it D. Klim} and {\it D. Wiatrowski} [J. Math. Anal. Appl. 334, No. 1, 132--139 (2007; Zbl 1133.54025)], {\it Y. Feng} and {\it S. Liu} [ibid., 317, No. 1, 103--112 (2006; Zbl 1094.47049)], {\it N. Mizoguchi} and {\it W. Takahashi} [ibid., 141, No. 1, 177--188 (1989; Zbl 0688.54028)].

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54C60 Set-valued maps (general topology)
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##### References:
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