Coupled fixed point theorems on partially ordered G-metric spaces

  • Erdal Karapınar1,

    Affiliated with

    • Poom Kumam2, 3 and

      Affiliated with

      • Inci M Erhan1Email author

        Affiliated with

        Fixed Point Theory and Applications20122012:174

        DOI: 10.1186/1687-1812-2012-174

        Received: 7 July 2012

        Accepted: 26 September 2012

        Published: 11 October 2012

        Abstract

        The purpose of this paper is to extend some recent coupled fixed point theorems in the context of partially ordered G-metric spaces in a virtually different and more natural way.

        MSC: 46N40, 47H10, 54H25, 46T99.

        Keywords

        coupled fixed point coupled coincidence point mixed g-monotone property ordered set G-metric space

        1 Introduction and preliminaries

        The notion of metric space was introduced by Fréchet [1] in 1906. In almost all fields of quantitative sciences which require the use of analysis, metric spaces play a major role. Internet search engines, image classification, protein classification (see, e.g., [2]) can be listed as examples in which metric spaces have been extensively used to solve major problems. It is conceivable that metric spaces will be needed to explore new problems that will arise in quantitative sciences in the future. Therefore, it is necessary to consider various generalizations of metrics and metric spaces to broaden the scope of applied sciences. In this respect, cone metric spaces, fuzzy metric spaces, partial metric spaces, quasi-metric spaces and b-metric spaces can be given as the main examples. Applications of these different approaches to metrics and metric spaces make it evident that fixed point theorems are important not only for the branches of mainstream mathematics, but also for many divisions of applied sciences.

        Inspired by this motivation Mustafa and Sims [3] introduced the notion of a G-metric space in 2004 (see also [47]). In their introductory paper, the authors investigated versions of the celebrated theorems of the fixed point theory such as the Banach contraction mapping principle [8] from the point of view of G-metrics. Another fundamental aspect in the theory of existence and uniqueness of fixed points was considered by Ran and Reurings [9] in partially ordered metric spaces. After Ran and Reurings’ pioneering work, several authors have focused on the fixed points in ordered metric spaces and have used the obtained results to discuss the existence and uniqueness of solutions of differential equations, more precisely, of boundary value problems (see, e.g., [1020]). Upon the introduction of the notion of coupled fixed points by Guo and Laksmikantham [14], Gnana-Bhaskar and Lakshmikantham [15] obtained interesting results related to differential equations with periodic boundary conditions by developing the mixed monotone property in the context of partially ordered metric spaces. As a continuation of this trend, many authors conducted research on the coupled fixed point theory and many results in this direction were published (see, for example, [2135]).

        In this paper, we prove the theorem that amalgamates these three seminal approaches in the study of fixed point theory, the so called G-metrics, coupled fixed points and partially ordered spaces.

        We shall start with some necessary definitions and a detailed overview of the fundamental results developed in the remarkable works mentioned above. Throughout this paper, ℕ and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq1_HTML.gif denote the set of non-negative integers and the set of positive integers respectively.

        Definition 1 (See [3])

        Let X be a non-empty set, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq2_HTML.gif be a function satisfying the following properties:

        (G1) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq3_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq4_HTML.gif ,

        (G2) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq5_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq6_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq7_HTML.gif ,

        (G3) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq8_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq9_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq10_HTML.gif ,

        (G4) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq11_HTML.gif (symmetry in all three variables),

        (G5) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq12_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq13_HTML.gif (rectangle inequality).

        Then the function G is called a generalized metric or, more specially, a G-metric on X, and the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq14_HTML.gif is called a G-metric space.

        It can be easily verified that every G-metric on X induces a metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq15_HTML.gif on X given by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ1_HTML.gif
        (1.1)

        Trivial examples of G-metric are as follows.

        Example 2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq16_HTML.gif be a metric space. The function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq17_HTML.gif , defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equa_HTML.gif
        or
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equb_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq18_HTML.gif , is a G-metric on X.

        The concepts of convergence, continuity, completeness and Cauchy sequence have also been defined in [3].

        Definition 3 (See [3])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq19_HTML.gif be a G-metric space, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq20_HTML.gif be a sequence of points of X. We say that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq20_HTML.gif is G-convergent to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq21_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq22_HTML.gif , that is, if for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq23_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq24_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq25_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq26_HTML.gif . We call x the limit of the sequence and write http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq27_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq28_HTML.gif .

        Proposition 4 (See [3])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq19_HTML.gif be a G-metric space. The following statements are equivalent:
        1. (1)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq20_HTML.gif is G-convergent to x,

           
        2. (2)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq29_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq30_HTML.gif ,

           
        3. (3)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq31_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq30_HTML.gif ,

           
        4. (4)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq32_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq33_HTML.gif .

           

        Definition 5 (See [3])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq19_HTML.gif be a G-metric space. A sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq20_HTML.gif is called G-Cauchy sequence if for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq23_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq34_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq35_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq36_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq37_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq38_HTML.gif .

        Proposition 6 (See [3])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq19_HTML.gif be a G-metric space. The following statements are equivalent:
        1. (1)

          The sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq20_HTML.gif is G-Cauchy.

           
        2. (2)

          For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq23_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq34_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq39_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq40_HTML.gif .

           

        Definition 7 (See [3])

        A G-metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq14_HTML.gif is called G-complete if every G-Cauchy sequence is G-convergent in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq14_HTML.gif .

        Definition 8 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq14_HTML.gif be a G-metric space. A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq41_HTML.gif is said to be continuous if for any three G-convergent sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq20_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq42_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq43_HTML.gif converging to x, y and z respectively, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq44_HTML.gif is G-convergent to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq45_HTML.gif .

        We define below g-ordered complete G-metric spaces.

        Definition 9 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq46_HTML.gif be a partially ordered set, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq19_HTML.gif be a G-metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq47_HTML.gif be a mapping. A partially ordered G-metric space, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq48_HTML.gif , is called g-ordered complete if for each G-convergent sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq49_HTML.gif , the following conditions hold:

        ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq50_HTML.gif ) If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq20_HTML.gif is a non-increasing sequence in X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq51_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq52_HTML.gif http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq53_HTML.gif .

        ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq54_HTML.gif ) If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq20_HTML.gif is a non-decreasing sequence in X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq51_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq55_HTML.gif http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq53_HTML.gif .

        In particular, if g is the identity mapping in ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq50_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq54_HTML.gif ), the partially ordered G-metric space, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq48_HTML.gif , is called ordered complete.

        We next recall some basic notions from the coupled fixed point theory. In 1987 Guo and Lakshmikantham [14] defined the concept of a coupled fixed point. In 2006, in order to prove the existence and uniqueness of the coupled fixed point of an operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq56_HTML.gif on a partially ordered metric space, Gnana-Bhaskar and Lakshmikantham [15] reconsidered the notion of a coupled fixed point via the mixed monotone property.

        Definition 10 ([15])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq57_HTML.gif be a partially ordered set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq58_HTML.gif . The mapping F is said to have the mixed monotone property if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq59_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%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq6_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equc_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equd_HTML.gif

        Definition 11 ([15])

        An element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq60_HTML.gif is called a coupled fixed point of the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq61_HTML.gif if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Eque_HTML.gif

        The results in [15] were extended by Lakshmikantham and Ćirić in [16] by defining the mixed g-monotone property.

        Definition 12 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq46_HTML.gif be a partially ordered set, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq62_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq63_HTML.gif . The function F is said to have mixed g-monotone property if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq59_HTML.gif is monotone g-non-decreasing in x and is monotone g-non-increasing in y, that is, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq64_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ2_HTML.gif
        (1.2)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ3_HTML.gif
        (1.3)

        It is clear that Definition 12 reduces to Definition 10 when g is the identity mapping.

        Definition 13 An element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq65_HTML.gif is called a coupled coincidence point of the mappings http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq66_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq63_HTML.gif if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equf_HTML.gif
        and a common coupled fixed point of F and g if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equg_HTML.gif
        Definition 14 The mappings http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq66_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq63_HTML.gif are said to commute if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equh_HTML.gif

        Throughout the rest of the paper, we shall use the notation gx instead of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq67_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq68_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq21_HTML.gif , for brevity. In [35], Nashine proved the following theorems.

        Theorem 15 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq69_HTML.gif be a partially ordered G-metric space. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq47_HTML.gif be mappings such that F has the mixed g-monotone property, and let there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq71_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq73_HTML.gif . Suppose that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq74_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq75_HTML.gif the following holds:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ4_HTML.gif
        (1.4)
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq76_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq77_HTML.gif , where either http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq78_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq79_HTML.gif . Assume the following hypotheses:
        1. (i)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq80_HTML.gif ,

           
        2. (ii)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq81_HTML.gif is G-complete,

           
        3. (iii)

          g is G-continuous and commutes with F.

           

        Then F and g have a coupled coincidence point, that is, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq82_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq84_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq85_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq86_HTML.gif , then F and g have a common fixed point, that is, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq21_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq87_HTML.gif .

        Theorem 16 If in the above theorem, we replace the condition (ii) by the assumption that X is g-ordered complete, then we have the conclusions of Theorem 15.

        We next give the definition of G-compatible mappings inspired by the definition of compatible mappings in [13].

        Definition 17 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq14_HTML.gif be a G-metric space. The mappings http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq88_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq68_HTML.gif are said to be G-compatible if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equi_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equj_HTML.gif

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq89_HTML.gif are sequences in X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq90_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq91_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq6_HTML.gif are satisfied.

        In this paper, we aim to extend the results on coupled fixed points mentioned above. Our results improve, enrich and extend some existing theorems in the literature. We also give examples to illustrate our results. This paper can also be considered as a continuation of the works of Berinde [11, 12].

        2 Main results

        We start with an example which shows the weakness of Theorem 15.

        Example 18 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq92_HTML.gif . Define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq93_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equk_HTML.gif
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq94_HTML.gif . Let ⪯ be usual order. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq14_HTML.gif is a G-metric space. Define a map http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq61_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq95_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq47_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq96_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq64_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq97_HTML.gif . Then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ5_HTML.gif
        (2.1)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ6_HTML.gif
        (2.2)

        It is clear that there is no http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq74_HTML.gif for which the statement (1.4) of Theorem 15 holds. Notice, however, that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq98_HTML.gif is the unique coupled coincidence point of F and g. In fact, it is a common fixed point of F and g, that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq99_HTML.gif .

        We now state our first result which successively guarantees the existence of a coupled coincidence point.

        Theorem 19 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq46_HTML.gif be a partially ordered set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq19_HTML.gif be a G-complete G-metric space. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq101_HTML.gif be two mappings such that F has the mixed g-monotone property on X and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ7_HTML.gif
        (2.3)

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq102_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq103_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq77_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq104_HTML.gif , g is G-continuous and that F and g are G-compatible mappings. Suppose further that either

        1. (a)

          F is continuous or

           
        2. (b)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq69_HTML.gif is g-ordered complete.

           

        Suppose also that there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq105_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq106_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq107_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq108_HTML.gif , then F and g have a coupled coincidence point, that is, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq109_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq110_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq111_HTML.gif .

        Proof Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq112_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq113_HTML.gif . Using the fact that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq104_HTML.gif , we can construct two sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq89_HTML.gif in X in the following way:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ8_HTML.gif
        (2.4)
        We shall prove that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq114_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ9_HTML.gif
        (2.5)
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq113_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq115_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq116_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq117_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq118_HTML.gif , that is, (2.5) holds for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq119_HTML.gif . Assume that (2.5) holds for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq120_HTML.gif . Since F has the mixed g-monotone property, from (2.4), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ10_HTML.gif
        (2.6)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ11_HTML.gif
        (2.7)
        By mathematical induction, it follows that (2.5) holds for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq121_HTML.gif , that is,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ12_HTML.gif
        (2.8)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ13_HTML.gif
        (2.9)
        If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq122_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq123_HTML.gif , then F and g have a coupled coincidence point. Indeed, in that case we would have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equl_HTML.gif

        We suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq124_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq125_HTML.gif . More precisely, we assume that either http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq126_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq127_HTML.gif .

        For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq128_HTML.gif , we set
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equm_HTML.gif
        Then by using (2.3) and (2.6), for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq128_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equn_HTML.gif
        which yields that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ14_HTML.gif
        (2.10)
        Now, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq129_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq130_HTML.gif , by using rectangle inequality (G5) of G-metric and (2.10), we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equo_HTML.gif
        which yields that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equp_HTML.gif

        Then by Proposition 6, we conclude that the sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq131_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq132_HTML.gif are G-Cauchy.

        Noting that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq81_HTML.gif is G-complete, there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq133_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq132_HTML.gif are G-convergent to x and y respectively, i.e.,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ15_HTML.gif
        (2.11)
        Since F and g are G-compatible mappings, by (2.11), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ16_HTML.gif
        (2.12)
        Suppose that the condition (a) holds. For all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq120_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ17_HTML.gif
        (2.13)
        Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq135_HTML.gif in the above inequality, using (2.11), (2.12) and the continuities of F and g, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equq_HTML.gif
        Hence, we derive that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq84_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq136_HTML.gif is a coupled coincidence point of F and g. Suppose that the condition (b) holds. By (2.8), (2.9) and (2.11), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ18_HTML.gif
        (2.14)
        Due to the fact that F and g are G-compatible mappings and g is continuous, by (2.11) and (2.12), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ19_HTML.gif
        (2.15)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ20_HTML.gif
        (2.16)
        Keeping (2.15) and (2.16) in mind, we consider now
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ21_HTML.gif
        (2.17)
        Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq135_HTML.gif in the above inequality, by using (2.15), (2.16) and the continuity of g, we conclude that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ22_HTML.gif
        (2.18)

        By (G1), we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq84_HTML.gif . Consequently, the element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq137_HTML.gif is a coupled coincidence point of the mappings F and g. □

        Corollary 20 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq46_HTML.gif be a partially ordered set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq19_HTML.gif be a G-metric space such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq14_HTML.gif is G-complete. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq101_HTML.gif be two mappings such that F has the mixed g-monotone property on X and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ23_HTML.gif
        (2.19)

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq138_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq139_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq140_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq104_HTML.gif , the self-mapping g is G-continuous and F and g are G-compatible mappings. Suppose that either

        1. (a)

          F is continuous or

           
        2. (b)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq69_HTML.gif is g-ordered complete.

           

        Suppose also that there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq105_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq106_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq141_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq108_HTML.gif , then F and g have a coupled coincidence point.

        Proof It is sufficient to take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq142_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq143_HTML.gif in Theorem 19. □

        Corollary 21 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq46_HTML.gif be a partially ordered set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq19_HTML.gif be a G-metric space such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq14_HTML.gif is G-complete. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq101_HTML.gif be two mappings such that F has the mixed g-monotone property on X and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ24_HTML.gif
        (2.20)

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq138_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq103_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq144_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq104_HTML.gif and that the self-mapping g is G-continuous and commutes with F. Suppose that either

        1. (a)

          F is continuous or

           
        2. (b)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq69_HTML.gif is g-ordered complete.

           

        Suppose further that there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq105_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq106_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq141_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq108_HTML.gif , then F and g have a coupled coincidence point.

        Proof Since g commutes with F, then F and g are G-compatible mappings. Thus, the result follows from Theorem 19. □

        Corollary 22 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq46_HTML.gif be a partially ordered set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq19_HTML.gif be a G-metric space such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq14_HTML.gif is G-complete. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq101_HTML.gif be two mappings such that F has the mixed g-monotone property on X and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ25_HTML.gif
        (2.21)

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq138_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq139_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq140_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq104_HTML.gif and that g is G-continuous and commutes with F. Suppose that either

        1. (a)

          F is continuous or

           
        2. (b)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq69_HTML.gif is g-ordered complete.

           

        Assume also that there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq105_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq106_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq141_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq108_HTML.gif , then F and g have a coupled coincidence point.

        Proof Since g commutes with F, then F and g are G-compatible mappings. Thus, the result follows from Corollary 20. □

        Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq145_HTML.gif in Theorem 19 and in Corollary 20, we get the following results.

        Corollary 23 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq46_HTML.gif be a partially ordered set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq19_HTML.gif be a G-metric space such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq14_HTML.gif is G-complete. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq100_HTML.gif be a mapping having the mixed monotone property on X and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ26_HTML.gif
        (2.22)

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq102_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq146_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq147_HTML.gif . Suppose that either

        1. (a)

          F is continuous or

           
        2. (b)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq69_HTML.gif is ordered complete.

           

        Suppose also that there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq105_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq148_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq149_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq108_HTML.gif , then F has a coupled fixed point.

        Corollary 24 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq46_HTML.gif be a partially ordered set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq19_HTML.gif be a G-metric space such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq14_HTML.gif is G-complete. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq100_HTML.gif be a mapping having the mixed monotone property on X and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ27_HTML.gif
        (2.23)

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq138_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq150_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq151_HTML.gif . Suppose that either

        1. (a)

          F is continuous or

           
        2. (b)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq69_HTML.gif is ordered complete.

           

        Suppose further that there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq105_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq148_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq149_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq108_HTML.gif , then F has a coupled fixed point.

        Example 25 Let us recall Example 18. We have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ28_HTML.gif
        (2.24)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ29_HTML.gif
        (2.25)

        It is clear that there any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq152_HTML.gif provides the statement (2.3) of Theorem 19.

        Notice that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq98_HTML.gif is the unique coupled coincidence point of F and g which is also common coupled fixed point, that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq99_HTML.gif .

        Example 26 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq92_HTML.gif . Define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq93_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equr_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq94_HTML.gif . Let ⪯ be usual order. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq14_HTML.gif is a G-metric space.

        Define a map http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq61_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equs_HTML.gif
        and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq47_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq153_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq64_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq154_HTML.gif . We observe that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equt_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ30_HTML.gif
        (2.26)

        then the statement (2.3) of Theorem 19 is satisfied for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq155_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq98_HTML.gif .

        Notice that if we replace the condition (2.3) of Theorem 19 with the condition (1.4) of Theorem 15 [21], that is,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ31_HTML.gif
        (2.27)

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq156_HTML.gif , then the coupled coincidence point exists even though the contractive condition is not satisfied.

        More precisely, consider http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq97_HTML.gif . Then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ32_HTML.gif
        (2.28)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_Equ33_HTML.gif
        (2.29)

        It is clear that the condition (2.27) holds for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_282_IEq157_HTML.gif .

        Declarations

        Acknowledgements

        The second author gratefully acknowledges the support provided by the Department of Mathematics and Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT) during his stay at the Department of Mathematical and Statistical Sciences, University of Alberta as a visitor for the short term research.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Atilim University
        (2)
        Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)
        (3)
        Department of Mathematical and Statistical Sciences, University of Alberta

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        Copyright

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        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.