We establish some coupled fixed point theorems for a mapping satisfying some contraction conditions in complete partial metric spaces. Our consequences extend the results of H. Aydi (2011). 1. Introduction and Mathematical Preliminaries The notion of a partial metric space (PMS) was introduced in 1992 by Matthews [1, 2]. Matthews proved a fixed point theorem on this spaces, analogous to the Banach's fixed point theorem. Recently, many authors have focused on partial metric spaces and their topological properties (see, e.g., [3–9]). The definition of a partial metric space is given by Matthews (see [1, 2]) as follows: Definition 1.1. Let be a nonempty set and let satisfies(P1) , for all ,(P2) , for all ,(P3) , for all ,(P4) , for all . Then the pair is called a partial metric space and is called a partial metric on . The function defined by satisfies the conditions of a metric on ; therefore it is a (usual) metric on . Remark 1.2. if , may not be 0.(1)A famous example of partial metric spaces is the pair , where for all . In this case, is the Euclidian metric .(2)Each partial metric on generates a topology on which has a base of open p-balls , where and ( ). The following concepts has been defined as follows on a partial metric space. Definition 1.3 (see e.g., [1, 2]). (i) A sequence in a PMS converges to if and only if . (ii) A sequence in a PMS is called Cauchy if and only if exists (and is finite). (iii) A PMS is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that . The concept of coupled fixed point have been introduced in [10] by Bhaskar and Lakshmikantham as follows. Definition 1.4 (see [10]). An element is called a coupled fixed point of mapping if and . Aydi in [11] has obtained some coupled fixed point results for mappings satisfying different contractive conditions on complete partial metric spaces. Some of these results are the following cases. Theorem 1.5 (see [11, Theorem 2.1]). Let be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition: for all , where are nonnegative constants with . Then, has a unique coupled fixed point. Theorem 1.6 (see [11, Theorem 2.4]). Let (X, p) be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition: for all , where are nonnegative constants with . Then, has a unique coupled fixed point. Theorem 1.7 (see [11, Theorem 2.5]). Let (X, p) be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition: for all , where are
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