J ournal of Mathematical I nequalities Volume 10, Number 2 (2016), 511–519 doi:10.7153/jmi-10-40 ON MONOTONE ´ CIRI ´ C QUASI–CONTRACTION MAPPINGS M. BACHAR AND M. A. KHAMSI (Communicated by A. Guessab) Abstract. We prove the existence of fixed points of monotone quasi-contraction mappings in metric and modular metric spaces. This is the extension of Ran and Reurings fixed point theo- rem for monotone contraction mappings in partially ordered metric spaces to the case of quasi- contraction mappings introduced by ´ Ciri´ c. The proofs are based on Lemmas 2.1 and 3.1, which contain two crucial inequalities essential to obtain the main results. 1. Introduction Banach’s Contraction Principle [2] is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis. This is because the con- tractive condition on the mapping is simple and easy to test, because it requires only a complete metric space for its setting, and because it finds almost canonical applications in the theory of differential and integral equations. Over the years, many mathemati- cians tried successfully to extend this fundamental theorem. Recently a version of this theorem was given in partially ordered metric spaces [9, 12] and in metric spaces with a graph [1, 6]. In this work, we discuss the case of quasi-contractive mappings defined in partially ordered metric spaces and modular metric spaces. For more on metric fixed point theory, the reader may consult the book [7]. 2. Monotone quasi-contraction mappings in metric spaces As a generalization to Banach Contraction Principle, ´ Ciri´ c[5] introduced the con- cept of quasi-contraction mappings (see also [10, 11]). In this section, we investigate monotone mappings which are quasi-contraction mappings. Since in this work we dis- cuss the fixed point theory of monotone mappings, we will need to introduce a partial order. Let (X , d ) be a metric space and assume that a partial order  exists in X . Throughout we assume that order intervals are closed. Recall that an order interval is any of the subsets (i) [a, →)= {x ∈ X ; a  x} , (ii) ( ←, a]= {x ∈ X ; x  a} , Mathematics subject classification (2010): Primary 47H09; Secondary 47H10. Keywords and phrases: Fixed point, modular metric space, monotone mappings, quasi-contraction. c  , Zagreb Paper JMI-10-40 511