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


54H25 Fixed-point and coincidence theorems (topological aspects)