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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 $\left(X,d\right)$ be a complete metric space and let $\phi :\left[0,\infty \right)\to \left[a,1\right)$, $0, be such that ${\overline{lim}}_{r\to t+}\left(r\right)<1$ for all $t\in \left[0,\infty \right)$. If $T:X\to \text{Cl}\left(X\right)$ (= all nonempty closed sets of $X$) is such that $x↦d\left(x,Tx\right)$ is lower-semicontinuous and for any $x\in X$ there is $y\in Tx$ with $\sqrt{\phi \left(d\left(x,y\right)\right)}d\left(x,y\right)\le d\left(x,Tx\right)$ and $d\left(y,Ty\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right)$, then $z\in Tz$ for some $z\in ℤ$.

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 in topological spaces 54C60 Set-valued maps (general topology)