On set-valued contractions of Nadler type in cone metric spaces

The fixed point theory for cone metric spaces, which was introduced in 2007 by Huang and Zhang in the paper [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive maps, J. Math. Anal. Appl. 332 (2007) 1467–1475] has recently become a subject of interest for many authors. Cone metric spaces are generalizations of metric spaces where the metric is replaced by the mapping d:M×M→Ed:M×M→E, where M≠∅M≠∅, and EE is a real Banach space. In the present paper for a cone metric space (M,d)(M,d) and for the family AA of subsets of MM we establish a new cone metric H:A×A→EH:A×A→E. Next, we introduce the concept of set-valued contraction of Nadler type and prove a fixed point theorem. Examples are provided.