Coupled Fixed Point Results For Mappings Involving (α,ψ) - Weak Contractions In Ordered Metric Spaces And Applications

Abstract

In this paper we introduce the notion of , -weak contractions and use the notion to establish the existence and uniqueness of coupled common fixed points for the mixed monotone operators in partially ordered metric spaces. The obtained results extend, improve, complement and unify many recent coupled fixed point results present in the literature. The theoretic results are accompanied with suitable examples. An application to the existence and uniqueness of the solution of the system of integral equations is also presented. 19] proved the following main result. Theorem 1.2 ([19]). Let , ≤ be a partially ordered set and be a metric on such that ( , ) is a complete metric space. Let : × → be a continuous mapping having the mixed monotone property on . Assume that there exists a in [0,1) withthen there exists , in such that = ( , ) and ( , ).In [19, Theorem 2.2], it has also been shown that the continuity assumption of in Theorem 1.2 can be replaced by an alternative condition imposed on convergent non-decreasing and nonincreasing sequences. Assumption 1.1. has the property that (i) if a non-decreasing sequence ⊂ converges to , then ≤ for all , (ii) if a non-increasing sequence ⊂ converges to , then ≥ for all . M. Jain, N. Gupta, S. Kumar / J. Math. Computer Sci. ( ), -25 Berinde [20, Theorem 3], in an alternative way generalized the results of Bhaskar and Lakshmikantham [19] under a weaker contraction given below: , , , + , , , ≤ ( , ) + , , (1.3)with ≥ and ≤ , where ∈ [0,1).On the other hand, Lakshmikantham and C iric [21] extended the notion of mixed monotone property to mixed -monotone property and generalized the results proved in [19] by establishing the existence of coupled coincidence point results using a pair of commutative mappings. This generated new trends in coupled fixed point theory with applications to partially ordered sets. Since then much work has been done in this direction by different authors. For more details the reader may consult ([22-38]). Definition 1. 3 ([21]). Let ( , ≤) be a partially ordered set and : × → and : × → . We say has the mixed -monotone property if is monotone -nondecreasing in its first argument and is monotone -nonincreasing in its second argument; that is, for any