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Fixed point theorems for multi-valued contractions in complete metric spaces. (English) Zbl 1213.54063

Let $\left(X,d\right)$ be a metric space and $\text{Cl}\left(X\right)$ the collection of nonempty closed sets of $X$. The main results of this paper are two fixed point theorems for multivalued functions. The following theorem is a generalization of Theorem 1 of N. Mizoguchi and W. Takahashi [J. Math. Anal. Appl. 141, No. 1, 177–188 (1989; Zbl 0688.54028)].

Theorem A. Let $\left(X,d\right)$ be a complete metric space and $T:X\to \text{Cl}\left(X\right)$ be a mapping of $X$ into itself. If there exists a function $\varphi :\left[0,\infty \right)\to \left(0,1\right)$ satisfying $lim sup\varphi *r\left(<1$ for each $t\in \left[0,\infty \right)$ and $r\to t+$ such that for any $x\in X$ there exists $y\in T\left(x\right)$ satisfying the following two conditions: $d\left(x,y\right)\le \left(2-\varphi \left(d\left(x,y\right)\right)\right)D\left(x,Tx\right)$ and $D\left(y,T\left(y\right)\right)\le \varphi \left(d\left(x,y\right)\right)d\left(x,y\right)$, then $T$ has a fixed point in $X$ provided $f\left(x\right)=D\left(x,T\left(x\right)\right)$ is lower-semicontinuous.

The following theorem is a generalization of Theorem 1 [op. cit.], of Theorem 2 of Y. Feng and S. Liu [J. Math. Anal. Appl. 317, No. 1, 103–112 (2006; Zbl 1094.47049)] and D. Klim and D. Wardowski [J. Math. Anal. Appl. 334, No. 1, 132–139 (2007; Zbl 1133.54025)].

Theorem B. Let $\left(X,d\right)$ be a complete metric space and $T:X\to \text{Cl}\left(X\right)$ be a mapping of $X$ into itself. If there exists a function $\varphi :\left[0,\infty \right)\to \left(0,1\right)$ and a nondecreasing function $b:\left[0,\infty \right)\to \left[b,1\right)$, $b>0$, such that $\varphi \left(t\right) and ${lim sup}_{t\to r+}\phantom{\rule{0.166667em}{0ex}}\varphi \left(t\right)<{lim sup}_{t\to r+}b\left(t\right)$ for all $t\in \left[0,\infty \right)$, and for any $x\in X$ there exists $y\in T\left(x\right)$ satisfying the following conditions: $b\left(d\left(x,y\right)\right)d\left(x,y\right)\le D\left(x,Tx\right)$ and $D\left(y,T\left(y\right)\right)\le \varphi \left(d\left(x,y\right)\right)$ $d\left(x,y\right)$, then $T$ has a fixed point in $X$ provided $f\left(x\right)=D\left(x,T\left(x\right)\right)$ is lower-semicontinuous.

In the last part of the paper the author constructs two examples which show that the results from this paper are genuine generalizations of the results of Mizaguchi and Takahasi, Feng and Liu and Klim and Wardowski.

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