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On set-valued contractions of Nadler type in cone metric spaces. (English) Zbl 1206.54067
Summary: The fixed point theory for cone metric spaces, which was introduced in 2007 by {\it L.-G. Huang} and {\it X. Zhang} [in J. Math. Anal. Appl. 332, No. 2, 1467--1475 (2007; Zbl 1118.54022)] has recently become a subject of interest for many authors. Cone metric spaces are generalizations of metric spaces where the metric is replaced by a mapping $d:M\times M\to E$, where $M\ne \emptyset$, and $E$ is a real Banach space. In the present paper for a cone metric space $(M,d)$ and for a family $\cal A$ of subsets of $M$ we establish a new cone metric $H:{\cal A}\times{\cal A}\to E$. Next, we introduce the concept of set-valued contraction of Nadler type and prove a fixed point theorem. Examples are provided.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
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