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 → E , where M ≠ ∅ , and E is a real Banach space. In the present paper for a cone metric space ( M , d ) and for the family A of subsets of M we establish a new cone metric H : A × A → E . Next, we introduce the concept of set-valued contraction of Nadler type and prove a fixed point theorem. Examples are provided.