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Fixed point theorems for set-valued contractions in complete metric spaces. (English) Zbl 1133.54025
Let $(M,d)$ be a metric space and let $H(A,B)$ denote the Pompeiu-Hausdorff distance between the sets $A,B\subset M$. The main results of this paper are fixed point theorems for set-valued contractions in complete metric spaces which are obtained by considering, instead of the classical contraction conditions of the form $$H(Tx,Ty)\leq \varphi(d(x,y))d(x,y),\,x,y\in M,$$ a more general condition: for each $x\in M$, there exists $$y\in I_{b}^{x}:=\left\{y\in Tx:\,bd(x,y)\leq d(x,Tx)\right\},$$ for a certain $b\in (0,1]$, such that $$ d(y,Ty)\leq \varphi(d(x,y))d(x,y).$$ Several related results in literature are thus extended or generalized.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
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