Abstract. The purpose of this paper is to present several fixed point theorems for the socalled set-valued Y -contractions. Set-valued Y -contractions in ordered metric spaces, set-valued graphic contractions, set-valued contractions outside a bounded set and set-valued operators on a metric space with cyclic representations are considered.

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The purpose of this paper is to present some existence results for coupled fixed points of contraction type operators in metric spaces endowed with a directed graph. Our results generalize the results obtained by Gnana Bhaskar and Lakshmikantham in (Nonlinear Anal. 65:1379-1393, 2006. As an application, the existence of a continuous solution for a system of Fredholm and Volterra integral equations is obtained. MSC: 47H10; 54H25Keywords: fixed point; coupled fixed point; metric space; connected graph PreliminariesIn fixed point theory, the importance of study of coupled fixed points is due to their applications to a wide variety of problems. Bhaskar and Lakshmikantham [] gave some existence results for coupled fixed point for a mixed monotone type mapping in a metric space endowed with partial order, using a weak contractivity type of assumption.The purpose of this paper is to generalize these results using the context of metric spaces endowed with a graph. This new research direction in the theory of fixed points was initiated by Jachymski Let (X, d) be a metric space and be the diagonal of X × X. Let G be a directed graph, such that the set V (G) of its vertices coincides with X and ⊆

A very interesting approach in the theory of fixed points in some general structures was recently given by Jachymski (Proc. Amer. Math. Soc. 136:1359Soc. 136: -1373Soc. 136: , 2008 and Gwóźdź-Lukawska and Jachymski (J. Math. Anal. Appl. 356:453-463, 2009) by using the context of metric spaces endowed with a graph. The purpose of this article is to present some new fixed point results for graphic contractions and forĆirić-Reich-Rus G-contractions on complete metric spaces endowed with a graph. The particular case of almost contractions is also considered. MSC: 47H10; 54H25