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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 54C60 Set-valued maps in general topology
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