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→ 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 (xn)n in Y such that Tnxn⊆ Y, for each n∈ 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), Petruşel et al. (Set-Valued Var. Anal. 23:223–237, 2015) and provides a solution to the problems posed in Petruşel et al. (Set-Valued Var. Anal. 23:223–237, 2015) and Rus and Şerban (Miskolc Math. Notes 17:1021–1031, 2016). © 2018, Springer International Publishing AG, part of Springer Nature.