Coupled fixed point theorems for α-ψ-contractive type mappings in partially ordered metric spaces

  • Mohammad Mursaleen1,

    Affiliated with

    • Syed Abdul Mohiuddine2Email author and

      Affiliated with

      • Ravi P Agarwal2, 3

        Affiliated with

        Fixed Point Theory and Applications20122012:228

        DOI: 10.1186/1687-1812-2012-228

        Received: 17 May 2012

        Accepted: 29 November 2012

        Published: 18 December 2012

        Abstract

        The object of this paper is to determine some coupled fixed point theorems for nonlinear contractive mappings in the framework of a metric space endowed with partial order. We also prove the uniqueness of a coupled fixed point for such mappings in this setup.

        MSC: 47H10, 54H25, 34B15.

        Keywords

        coupled fixed point contractive mapping partially ordered set metric space

        1 Introduction

        Fixed point theory is a very useful tool in solving a variety of problems in control theory, economic theory, nonlinear analysis and global analysis. The Banach contraction principle [1] is the most famous, simplest and one of the most versatile elementary results in fixed point theory. A huge amount of literature is witnessed on applications, generalizations and extensions of this principle carried out by several authors in different directions, e.g., by weakening the hypothesis, using different setups, considering different mappings.

        Many authors obtained important fixed point theorems, e.g., Abbas et al.[2], Agarwal et al.[3, 4], Bhaskar and Lakshmikantham [5], Choudhury and Kundu [6], Choudhury and Maity [7], Ćirić et al.[8], Luong and Thuan [9], Nieto and López [10, 11], Ran and Reurings [12] and Samet [13] presented some new results for contractions in partially ordered metric spaces. In [14], Ilić and Rakočević determined some common fixed point theorems by considering the maps on cone metric spaces. Recently, Haghi et al.[15] have shown that some coincidence point and common fixed point generalizations in fixed point theory are not real generalizations. For more detail on fixed point theory and related concepts, we refer to [1634] and the references therein.

        In [5], Bhaskar and Lakshmikantham introduced the notions of mixed monotone property and coupled fixed point for the contractive mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq1_HTML.gif , where X is a partially ordered metric space, and proved some coupled fixed point theorems for a mixed monotone operator. As an application of the coupled fixed point theorems, they determined the existence and uniqueness of the solution of a periodic boundary value problem. Recently, Lakshmikantham and Ćirić [35] have proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces. Most recently, Samet et al.[36] have defined α-ψ-contractive and α-admissible mapping and proved fixed point theorems for such mappings in complete metric spaces.

        The aim of this paper is to determine some coupled fixed point theorems for generalized contractive mappings in the framework of partially ordered metric spaces.

        2 Definitions and preliminary results

        We start with the definition of a mixed monotone property and a coupled fixed point and state the related results.

        Definition 2.1 ([5])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq2_HTML.gif be a partially ordered set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq3_HTML.gif be a mapping. Then a map F is said to have the mixed monotone property if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq4_HTML.gif is monotone non-decreasing in x and is monotone non-increasing in y; that is, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq5_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equa_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equb_HTML.gif

        Definition 2.2 ([5])

        An element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq6_HTML.gif is said to be a coupled fixed point of the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq3_HTML.gif if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equc_HTML.gif

        Theorem 2.3 ([5])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq2_HTML.gif be a partially ordered set and suppose there is a metric d on X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq7_HTML.gif is a complete metric space. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq3_HTML.gif be a continuous mapping having the mixed monotone property on X. Assume that there exists a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq8_HTML.gif with
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equd_HTML.gif
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq10_HTML.gif . If there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq11_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Eque_HTML.gif

        then there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq5_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq13_HTML.gif .

        Theorem 2.4 ([5])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq2_HTML.gif be a partially ordered set and suppose there is a metric d on X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq7_HTML.gif is a complete metric space. Assume that X has the following property:
        1. (i)

          if a non-decreasing sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq14_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq15_HTML.gif for all n;

           
        2. (ii)

          if a non-increasing sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq16_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq17_HTML.gif for all n.

           
        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq3_HTML.gif be a mapping having the mixed monotone property on X. Assume that there exists a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq8_HTML.gif with
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equf_HTML.gif
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq10_HTML.gif . If there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq11_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equg_HTML.gif

        then there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq5_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq13_HTML.gif .

        3 Main results

        In this section, we establish some coupled fixed point results by considering maps on metric spaces endowed with partial order.

        Denote by Ψ the family of non-decreasing functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq18_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq19_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq20_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq21_HTML.gif is the nth iterate of ψ satisfying (i) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq22_HTML.gif , (ii) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq23_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq20_HTML.gif and (iii) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq24_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq20_HTML.gif .

        Lemma 3.1 If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq25_HTML.gif is non-decreasing and right continuous, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq26_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq27_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq28_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq23_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq20_HTML.gif .

        Definition 3.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq7_HTML.gif be a partially ordered metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq3_HTML.gif be a mapping. Then a map F is said to be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq29_HTML.gif -contractive if there exist two functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq31_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equh_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq32_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq10_HTML.gif .

        Definition 3.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq3_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq30_HTML.gif be two mappings. Then F is said to be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq33_HTML.gif -admissible if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equi_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq32_HTML.gif .

        Theorem 3.4 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq2_HTML.gif be a partially ordered set and suppose there is a metric d on X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq7_HTML.gif is a complete metric space. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq3_HTML.gif be a mapping having the mixed monotone property of X. Suppose that there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq30_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq32_HTML.gif , the following holds:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ1_HTML.gif
        (3.1)
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq10_HTML.gif . Suppose also that
        1. (i)

          F is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq33_HTML.gif -admissible,

           
        2. (ii)
          there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq11_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equj_HTML.gif
           
        3. (iii)

          F is continuous.

           
        If there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq11_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq35_HTML.gif , then F has a coupled fixed point; that is, there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq5_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equk_HTML.gif
        Proof Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq11_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq37_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq38_HTML.gif (say) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq39_HTML.gif (say). Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq40_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq42_HTML.gif . Continuing this process, we can construct two sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq44_HTML.gif in X as follows:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equl_HTML.gif
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq45_HTML.gif . We will show that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ2_HTML.gif
        (3.2)
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq45_HTML.gif . We will use the mathematical induction. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq46_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq35_HTML.gif and as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq47_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq48_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq49_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq50_HTML.gif . Thus, (3.2) hold for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq46_HTML.gif . Now suppose that (3.2) hold for some fixed n, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq45_HTML.gif . Then, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq51_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq52_HTML.gif and by the mixed monotone property of F, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equm_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equn_HTML.gif
        From above, we conclude that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equo_HTML.gif
        Thus, by the mathematical induction, we conclude that (3.2) hold for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq45_HTML.gif . If for some n we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq53_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq55_HTML.gif ; that is, F has a coupled fixed point. Now, we assumed that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq56_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq45_HTML.gif . Since F is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq33_HTML.gif -admissible, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equp_HTML.gif
        Thus, by the mathematical induction, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ3_HTML.gif
        (3.3)
        and similarly,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ4_HTML.gif
        (3.4)
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq57_HTML.gif . Using (3.1) and (3.3), we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ5_HTML.gif
        (3.5)
        Similarly, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ6_HTML.gif
        (3.6)
        Adding (3.5) and (3.6), we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equq_HTML.gif
        Repeating the above process, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equr_HTML.gif
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq57_HTML.gif . For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq58_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq59_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equs_HTML.gif
        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq60_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq61_HTML.gif . Then, by using the triangle inequality, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equt_HTML.gif
        This implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq62_HTML.gif . Since
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equu_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equv_HTML.gif
        and hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq44_HTML.gif are Cauchy sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq7_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq7_HTML.gif is a complete metric space and hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq44_HTML.gif are convergent in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq7_HTML.gif . Then there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq5_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equw_HTML.gif
        Since F is continuous and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq63_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq64_HTML.gif , taking limit http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq27_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equx_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equy_HTML.gif

        that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq13_HTML.gif and hence F has a coupled fixed point. □

        In the next theorem, we omit the continuity hypothesis of F.

        Theorem 3.5 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq2_HTML.gif be a partially ordered set and suppose there is a metric d on X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq7_HTML.gif is a complete metric space. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq3_HTML.gif be a mapping such that F has the mixed monotone property. Assume that there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq31_HTML.gif and a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq30_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equz_HTML.gif
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq32_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq10_HTML.gif . Suppose that
        1. (i)

          conditions (i) and (ii) of Theorem 3.4 hold,

           
        2. (ii)
          if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq44_HTML.gif are sequences in X such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equaa_HTML.gif
           
        for all n and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq66_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equab_HTML.gif

        If there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq11_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq35_HTML.gif , then there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq5_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq13_HTML.gif ; that is, F has a coupled fixed point in X.

        Proof Proceeding along the same lines as in the proof of Theorem 3.4, we know that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq44_HTML.gif are Cauchy sequences in the complete metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq7_HTML.gif . Then there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq5_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ7_HTML.gif
        (3.7)
        On the other hand, from (3.3) and hypothesis (ii), we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ8_HTML.gif
        (3.8)
        and similarly,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ9_HTML.gif
        (3.9)
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq57_HTML.gif . Using the triangle inequality, (3.8) and the property of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq23_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq20_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equac_HTML.gif
        Similarly, using (3.9), we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equad_HTML.gif
        Taking the limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq27_HTML.gif in the above two inequalities, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equae_HTML.gif

        Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq13_HTML.gif . Thus, F has a coupled fixed point. □

        In the following theorem, we will prove the uniqueness of the coupled fixed point. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq2_HTML.gif is a partially ordered set, then we endow the product http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq67_HTML.gif with the following partial order relation:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equaf_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq68_HTML.gif .

        Theorem 3.6 In addition to the hypothesis of Theorem 3.4, suppose that for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq69_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq70_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq67_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq71_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq67_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equag_HTML.gif

        and also assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq72_HTML.gif is comparable to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq70_HTML.gif . Then F has a unique coupled fixed point.

        Proof From Theorem 3.4, the set of coupled fixed points is nonempty. Suppose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq70_HTML.gif are coupled fixed points of the mappings http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq3_HTML.gif ; that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq73_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq75_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq76_HTML.gif . By assumption, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq72_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq67_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq72_HTML.gif is comparable to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq70_HTML.gif . Put http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq77_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq78_HTML.gif and choose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq79_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq80_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq81_HTML.gif . Thus, we can define two sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq82_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq83_HTML.gif as
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equah_HTML.gif
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq72_HTML.gif is comparable to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq69_HTML.gif , it is easy to show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq85_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq87_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq88_HTML.gif . Since for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq89_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq90_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ10_HTML.gif
        (3.10)
        Since F is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq33_HTML.gif -admissible, so from (3.10), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equai_HTML.gif
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq77_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq78_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equaj_HTML.gif
        Thus,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equak_HTML.gif
        Therefore, by the mathematical induction, we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ11_HTML.gif
        (3.11)
        for all n∈ and similarly, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq91_HTML.gif . From (3.10) and (3.11), we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ12_HTML.gif
        (3.12)
        Similarly, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ13_HTML.gif
        (3.13)
        Adding (3.12) and (3.13), we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equal_HTML.gif
        Thus,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ14_HTML.gif
        (3.14)
        for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq88_HTML.gif . Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq27_HTML.gif in (3.14) and using Lemma 3.1, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equam_HTML.gif
        This implies
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ15_HTML.gif
        (3.15)
        Similarly, one can show that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equ16_HTML.gif
        (3.16)

        From (3.15) and (3.16), we conclude that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq93_HTML.gif . Hence, F has a unique coupled fixed point. □

        Example 3.7 (Linear case)

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq94_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq95_HTML.gif be a standard metric. Define a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq3_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq96_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq5_HTML.gif . Consider a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq30_HTML.gif be such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equan_HTML.gif
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq97_HTML.gif holds for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq32_HTML.gif . Therefore, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equao_HTML.gif
        It follows that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equap_HTML.gif

        Thus (3.1) holds for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq98_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq20_HTML.gif , and we also see that all the hypotheses of Theorem 3.4 are fulfilled. Then there exists a coupled fixed point of F. In this case, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq99_HTML.gif is a coupled fixed point of F.

        Example 3.8 (Nonlinear case)

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq101_HTML.gif be a standard metric. Define a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq1_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq102_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq5_HTML.gif . Consider a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq103_HTML.gif be such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equaq_HTML.gif
        Then we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equar_HTML.gif
        Thus,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_Equas_HTML.gif

        Therefore (3.1) holds for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq104_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq20_HTML.gif , and also the hypothesis of Theorem 3.4 is fulfilled. Then there exists a coupled fixed point of F. In this case, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq99_HTML.gif is a coupled fixed point of F.

        4 Concluding remark

        The author of [33] recently established some coupled fixed point theorems in partially ordered metric spaces shortly by using some usual corresponding fixed point theorems on the metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq105_HTML.gif . Note that if the right-hand side of the α-ψ-contractive type condition (3.1) is replaced by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq106_HTML.gif , then a very short proof similar to what followed in [33] can be provided for a coupled fixed point theorem of Theorem 3.4 type by making just use of the results in [36]. However, since the right-hand side of (3.1) is not of the form http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq106_HTML.gif , specially for nonlinear functions ψ, then it is not possible to apply the method [33]. In this connection, notice that Example 3.7 works for both when the right-hand side is either http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq106_HTML.gif or as in (3.1), but Example 3.8 works only for (3.1). Hence, our results are more interesting and different from the existing results of [33] and [36].

        Declarations

        Acknowledgements

        The work of the second author was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. He acknowledges with thanks DSR technical and financial support.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Aligarh Muslim University
        (2)
        Department of Mathematics, Faculty of Science, King Abdulaziz University
        (3)
        Department of Mathematics, Texas A&M University-Kingsville

        References

        1. Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3:133–181.MATH
        2. Abbas M, Nazir T, Radenović S: Fixed points of four maps in partially ordered metric spaces. Appl. Math. Lett. 2011, 24:1520–1526.MathSciNetMATHView Article
        3. Agarwal RP, Meehan M, O’Regan D: Fixed Point Theory and Applications. Cambridge University Press, Cambridge; 2001.MATHView Article
        4. Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87:109–116.MathSciNetMATHView Article
        5. Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65:1379–1393.MathSciNetMATHView Article
        6. Choudhury BS, Kundu A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal. 2010, 73:2524–2531.MathSciNetMATHView Article
        7. Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011, 54:73–79.MathSciNetMATHView Article
        8. Ćirić L, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 131294
        9. Luong NV, Thuan NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal. 2011, 74:983–992.MathSciNetMATHView Article
        10. Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22:223–239.MathSciNetMATHView Article
        11. Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007,23(12):2205–2212.MathSciNetMATHView Article
        12. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132:1435–1443.MathSciNetMATHView Article
        13. Samet B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 2010, 72:4508–4517.MathSciNetMATHView Article
        14. Ilić D, Rakočević V: Common fixed points for maps on cone metric space. J. Math. Anal. Appl. 2008, 341:876–882.MathSciNetMATHView Article
        15. Haghi RH, Rezapour S, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 2011, 74:1799–1803.MathSciNetMATHView Article
        16. Haghi RH, Rezapour S: Fixed points of multi functions on regular cone metric spaces. Expo. Math. 2010, 28:71–77.MathSciNetMATHView Article
        17. Derafshpour M, Rezapour S, Shahzad N: Best proximity points of cyclic φ -contractions on reflexive Banach space. Topol. Methods Nonlinear Anal. 2011,37(1):193–202.MathSciNetMATH
        18. Aleomraninejad SMA, Rezapour S, Shahzad N: Some fixed point results on a metric space with a graph. Topol. Appl. 2012, 159:659–663.MathSciNetMATHView Article
        19. Ghorbanian V, Rezapour S, Shahzad N: Some ordered fixed point results and the property (P). Comput. Math. Appl. 2012, 63:1361–1368.MathSciNetMATHView Article
        20. Mohiuddine SA, Alotaibi A: On coupled fixed point theorems for nonlinear contractions in partially ordered G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 897198
        21. Mohiuddine SA, Alotaibi A: Some results on tripled fixed point for nonlinear contractions in partially ordered G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 179
        22. Nashine HK, Kadelburg Z, Radenović S:Coupled common fixed point theorems for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_331_IEq107_HTML.gif -compatible mappings in ordered cone metric spaces. Appl. Math. Comput. 2012, 218:5422–5432.MathSciNetMATHView Article
        23. Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c -distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62:1969–1978.MathSciNetMATHView Article
        24. Aydi H, Samet B, Vetro C: Coupled fixed point results in cone metric spaces for W -compatible mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 27
        25. Jleli M, Cojbasic Rajic V, Samet B, Vetro C: Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations. J. Fixed Point Theory Appl. 2012. doi:10.1007/s11784–012–0081–4
        26. Nashine HK, Samet B, Vetro C: Coupled coincidence points for compatible mappings satisfying mixed monotone property. J. Nonlinear Sci. Appl. 2012,5(2):104–114.MathSciNet
        27. Samet B, Vetro C: Coupled fixed point, F -invariant set and fixed point of N -order. Ann. Funct. Anal. 2010,1(2):46–56.MathSciNetMATH
        28. Samet B, Vetro C: Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74:4260–4268.MathSciNetMATHView Article
        29. Sintunavarat W, Kumam P, Cho YJ: Coupled fixed point theorems for nonlinear contractions without mixed monotone property. Fixed Point Theory Appl. 2012., 2012: Article ID 170
        30. Karapinar E, Samet B: Generalized α - ψ -contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012., 2012: Article ID 793486
        31. Abdeljawad T, Karapinar E, Aydi H: A new Meir-Keeler type coupled fixed point on ordered partial metric spaces. Math. Probl. Eng. 2012., 2012: Article ID 327273
        32. Abdeljawad T: Coupled fixed point theorems for partially contractive type mappings. Fixed Point Theory Appl. 2012., 2012: Article ID 148
        33. Amini-Harandi, A: Coupled and tripled fixed point theory in partially ordered metric spaces with applications to initial value problem. Math. Comput. Model. doi:10.1016/j.mcm.2011.12.006 (in press).
        34. Shatanawi W, Samet B, Abbas M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model. 2012, 55:680–687.MathSciNetMATHView Article
        35. Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70:4341–4349.MathSciNetMATHView Article
        36. Samet B, Vetro C, Vetro P: Fixed point theorems for α - ψ -contractive type mappings. Nonlinear Anal. 2012, 75:2154–2165.MathSciNetMATHView Article

        Copyright

        © Mursaleen et al.; licensee Springer 2012

        This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.