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On a general class of multi-valued weakly Picard mappings. (English) Zbl 1117.47039
Let $(X,d)$ be a metric space and $T:X\rightarrow\mathcal{P}(X)$ be a multi-valued operator. $T$ is said to be a generalized $(\alpha ,L)$-weak contraction if there exist a constant $L\geq 0$ and a function $\alpha:[0,\infty)\rightarrow[0,1)$, with $\lim \sup_{r\rightarrow t^+} \alpha (r)<1$ for every $t\in [ 0,\infty )$, such that $$H(Tx, Ty)\leq \alpha (d(x,y)) d(x,y)+LD(y,Tx)$$ for all $x,y\in X$. When $\alpha (t)=\theta\in[0,1)$ for all $t\in[0,\infty)$, we say that $T$ is a $(\theta ,L)$-weak contraction. The authors prove that a generalized $(\alpha ,L)$-weak contraction $T$ has a fixed point whenever $X$ is complete and $T$ has nonempty bounded and closed values. Moreover, if $T$ is a $(\theta,L)$-weak contraction, then for any $x_0\in X$ there exists an orbit $\{x_n\}_{n\geq 0}$ converging to a fixed point $u$ of $T$ for which the following estimate holds: $$d(x_n,u)\leq\min\left\{\frac{h^n}{1-h}d(x_1,x_0),\frac{h}{1-h}d(x_{n-1},x_n)\right\}$$ 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)
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##### References:
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