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On a general class of multi-valued weakly Picard mappings. (English) Zbl 1117.47039

Let $\left(X,d\right)$ be a metric space and $T:X\to 𝒫\left(X\right)$ be a multi-valued operator. $T$ is said to be a generalized $\left(\alpha ,L\right)$-weak contraction if there exist a constant $L\ge 0$ and a function $\alpha :\left[0,\infty \right)\to \left[0,1\right)$, with $lim{sup}_{r\to {t}^{+}}\alpha \left(r\right)<1$ for every $t\in \left[0,\infty \right)$, such that

$H\left(Tx,Ty\right)\le \alpha \left(d\left(x,y\right)\right)d\left(x,y\right)+LD\left(y,Tx\right)$

for all $x,y\in X$. When $\alpha \left(t\right)=\theta \in \left[0,1\right)$ for all $t\in \left[0,\infty \right)$, we say that $T$ is a $\left(\theta ,L\right)$-weak contraction. The authors prove that a generalized $\left(\alpha ,L\right)$-weak contraction $T$ has a fixed point whenever $X$ is complete and $T$ has nonempty bounded and closed values. Moreover, if $T$ is a $\left(\theta ,L\right)$-weak contraction, then for any ${x}_{0}\in X$ there exists an orbit ${\left\{{x}_{n}\right\}}_{n\ge 0}$ converging to a fixed point $u$ of $T$ for which the following estimate holds:

$d\left({x}_{n},u\right)\le min\left\{\frac{{h}^{n}}{1-h}d\left({x}_{1},{x}_{0}\right),\frac{h}{1-h}d\left({x}_{n-1},{x}_{n}\right)\right\}$

for a certain constant $h<1$.

##### MSC:
 47H04 Set-valued operators 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 54C60 Set-valued maps (general topology)