The Reich–Zaslavski property and fixed points of non-self multivalued mappings

Abstract

Let (X, d) be a metric space, Y be a nonempty subset of X, and let $$T:Y \rightarrow P(X)$$ be a non-self multivalued mapping. In this paper, by a new technique we study the fixed point theory of multivalued mappings under the assumption of the existence of a bounded sequence $$(x_n)_n$$ in Y such that $$T^nx_n\subseteq Y,$$ for each $$n \in \mathbb {N}$$ . Our main result generalizes fixed point theorems due to Matkowski (Diss. Math. 127, 1975), Wȩgrzyk (Diss. Math. (Rozprawy Mat.) 201, 1982), Reich and Zaslavski (Fixed Point Theory 8:303–307, 2007), Petrusel et al. (Set-Valued Var. Anal. 23:223–237, 2015) and provides a solution to the problems posed in Petrusel et al. (Set-Valued Var. Anal. 23:223–237, 2015) and Rus and Şerban (Miskolc Math. Notes 17:1021–1031, 2016).