Multivalued pseudo-contractive mappings defined on unbounded sets in Banach spaces. (English) Zbl 0794.47038

Morales, Claudio H.

Commentat. Math. Univ. Carol. 33, No. 4, 625-630 (1992).

Summary: Let \(X\) be a real Banach space. A multivalued operator \(T\) from \(K\) into \(2^ X\) is said to be pseudo-contractive if for every \(x\), \(y\) in \(K\), \(u\in T(x)\), \(v\in T(y)\) and all \(r> 0\), \(\| x-y\|\leq \|(1+ r)(x- y)- r(u- v)\|\). Denote by \(G(z,w)\) the set \(\{u\in K: \| u- w\|\leq \| u- z\|\}\). Suppose every bounded closed and convex subset of \(X\) has the fixed point property with respect to nonexpansive selfmappings. Now if \(T\) is a Lipschitzian and pseudo-contractive mapping from \(K\) into the family of closed and bounded subsets of \(K\) so that the set \(G(x,w)\) is bounded for some \(z\in K\) and some \(w\in T(z)\), then \(T\) has a fixed point in \(K\).

MSC:

47H10	Fixed-point theorems
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

pseudo-contractive mappings; multivalued operator; fixed point property; nonexpansive selfmappings