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Generalized contractive multivalued mappings and their fixed points. (English) Zbl 0647.54038
Let us denote by (X,d) a metric space, by $$P_ x(X)$$ the family of all bounded proximinal subsets of X and by H the Hausdorff metric induced by d on $$P_ x(X)$$. Let T be a multivalued mapping, $$T: X\to P_ x(X)$$, such that $$H(Tx,Ty)\leq h(x,y)d(x,y),\forall x,y\in X$$, where h satisfies $$\sup \{h(x,y): a\leq d(x,y)\leq b\}<1$$ for each finite closed interval $$[a,b]\in (0,+\infty).$$ Moreover assume that if $$(x_ n,y_ n)\in X\times X$$ and $$d(x_ n,y_ n)\to 0,$$ then $$h(x_ n,y_ n)\to k$$ for some $$k\in [0,1].$$ Then T has a fixed point in X. These hypotheses are obviously satisfied if $$h(x,y)=\alpha (d(x,y)),$$ where $$\alpha$$ is a monotone increasing function such that $$0\leq \alpha (t)<1$$ for each $$t\in (0,+\infty).$$ The result is connected with some known theorems in the setting of point-to-point mappings.
Reviewer: D.Roux

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects)
##### Keywords:
contractive multivalued map; proximinal sets