Cyclic contractions via auxiliary functions on G-metric spaces

  • Nurcan Bilgili1 and

    Affiliated with

    • Erdal Karapınar2Email author

      Affiliated with

      Fixed Point Theory and Applications20132013:49

      DOI: 10.1186/1687-1812-2013-49

      Received: 24 October 2012

      Accepted: 21 February 2013

      Published: 8 March 2013

      Abstract

      In this paper, we prove the existence and uniqueness of fixed points of certain cyclic mappings via auxiliary functions in the context of G-metric spaces, which were introduced by Zead and Sims. In particular, we extend, improve and generalize some earlier results in the literature on this topic.

      MSC: 47H10, 54H25.

      Keywords

      fixed point G-metric space cyclic maps cyclic contractions

      1 Introduction and preliminaries

      It is well established that fixed point theory, which mainly concerns the existence and uniqueness of fixed points, is today’s one of the most investigated research areas as a major subfield of nonlinear functional analysis. Historically, the first outstanding result in this field that guaranteed the existence and uniqueness of fixed points was given by Banach [1]. This result, known as the Banach mapping contraction principle, simply states that every contraction mapping has a unique fixed point in a complete metric space. Since the first appearance of the Banach principle, the ever increasing application potential of the fixed point theory in various research fields, such as physics, chemistry, certain engineering branches, economics and many areas of mathematics, has made this topic more crucial than ever. Consequently, after the Banach celebrated principle, many authors have searched for further fixed point results and reported successfully new fixed point theorems conceived by the use of two very effective techniques, combined or separately.

      The first one of these techniques is to ‘replace’ the notion of a metric space with a more general space. Quasi-metric spaces, partial metric spaces, rectangular metric spaces, fuzzy metric space, b-metric spaces, D-metric spaces, G-metric spaces are generalizations of metric spaces and can be considered as examples of ‘replacements’. Amongst all of these spaces, G-metric spaces, introduced by Zead and Sims [2], are ones of the interesting. Therefore, in the last decade, the notion of a G-metric space has attracted considerable attention from researchers, especially from fixed point theorists [325].

      The second one of these techniques is to modify the conditions on the operator(s). In other words, it entails the examination of certain conditions under which the contraction mapping yields a fixed point. One of the attractive results produced using this approach was given by Kirk et al.[26] in 2003 through the introduction of the concepts of cyclic mappings and best proximity points. After this work, best proximity theorems and, in particular, the fixed point theorems in the context of cyclic mappings have been studied extensively (see, e.g., [2743]).

      The two upper mentioned topics, cyclic mappings and G-metric spaces, have been combined by Aydi in [22] and Karapınar et al. in [36]. In these papers, the existence and uniqueness of fixed points of cyclic mappings are investigated in the framework of G-metric spaces. In this paper, we aim to improve on certain statements proved on these two topics. For the sake of completeness, we will include basic definitions and crucial results that we need in the rest of this work.

      Mustafa and Sims [2] defined the concept of G-metric spaces as follows.

      Definition 1.1 (See [2])

      Let X be a nonempty set, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq1_HTML.gif be a function satisfying the following properties:
      • (G1) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq2_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq3_HTML.gif ,

      • (G2) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq4_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq5_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq6_HTML.gif ,

      • (G3) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq7_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq8_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq9_HTML.gif ,

      • (G4) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq10_HTML.gif (symmetry in all three variables),

      • (G5) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq11_HTML.gif (rectangle inequality) for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq12_HTML.gif .

      Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif is called a G-metric space.

      Note that every G-metric on X induces a metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq14_HTML.gif on X defined by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ1_HTML.gif
      (1)

      For a better understanding of the subject, we give the following examples of G-metrics.

      Example 1.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq15_HTML.gif be a metric space. The function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq16_HTML.gif , defined by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equa_HTML.gif

      for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq17_HTML.gif , is a G-metric on X.

      Example 1.2 (See, e.g., [2])

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq18_HTML.gif . The function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq19_HTML.gif , defined by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equb_HTML.gif

      for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq17_HTML.gif , is a G-metric on X.

      In their initial paper, Mustafa and Sims [2] also defined the basic topological concepts in G-metric spaces as follows.

      Definition 1.2 (See [2])

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq20_HTML.gif be a G-metric space, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif be a sequence of points of X. We say that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif is G-convergent to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq22_HTML.gif if
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equc_HTML.gif

      that is, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq23_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq24_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq25_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq26_HTML.gif . We call x the limit of the sequence and write http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq27_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq28_HTML.gif .

      Proposition 1.1 (See [2])

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq20_HTML.gif be a G-metric space. The following are equivalent:
      1. (1)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif is G-convergent to x,

         
      2. (2)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq29_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq30_HTML.gif ,

         
      3. (3)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq31_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq30_HTML.gif ,

         
      4. (4)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq32_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq33_HTML.gif .

         

      Definition 1.3 (See [2])

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq20_HTML.gif be a G-metric space. A sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif is called a G-Cauchy sequence if, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq23_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq24_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq34_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq35_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq36_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq37_HTML.gif .

      Proposition 1.2 (See [2])

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq20_HTML.gif be a G-metric space. Then the following are equivalent:
      1. (1)

        the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif is G-Cauchy,

         
      2. (2)

        for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq23_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq24_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq38_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq39_HTML.gif .

         

      Definition 1.4 (See [2])

      A G-metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif is called G-complete if every G-Cauchy sequence is G-convergent in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif .

      Definition 1.5 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif be a G-metric space. A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq40_HTML.gif is said to be continuous if for any three G-convergent sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq42_HTML.gif converging to x, y and z respectively, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq43_HTML.gif is G-convergent to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq44_HTML.gif .

      Note that each G-metric on X generates a topology http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq45_HTML.gif on X whose base is a family of open G-balls http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq46_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq47_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq22_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq23_HTML.gif . A nonempty set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq48_HTML.gif is G-closed in the G-metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq49_HTML.gif . Observe that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equd_HTML.gif

      for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq23_HTML.gif . We recall also the following proposition.

      Proposition 1.3 (See, e.g., [36])

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq20_HTML.gif be a G-metric space and A be a nonempty subset of X. The set A is G-closed if for any G-convergent sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif in A with limit x, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq50_HTML.gif .

      Mustafa [5] extended the well-known Banach contraction principle mapping in the framework of G-metric spaces as follows.

      Theorem 1.1 (See [5])

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif be a complete G-metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq51_HTML.gif be a mapping satisfying the following condition for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq52_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ2_HTML.gif
      (2)

      where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq53_HTML.gif . Then T has a unique fixed point.

      Theorem 1.2 (See [5])

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif be a complete G-metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq51_HTML.gif be a mapping satisfying the following condition for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq54_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ3_HTML.gif
      (3)

      where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq53_HTML.gif . Then T has a unique fixed point.

      Remark 1.1 We notice that the condition (2) implies the condition (3). The converse is true only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq55_HTML.gif . For details, see [5].

      Lemma 1.1 ([5])

      By the rectangle inequality (G5) together with the symmetry (G4), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ4_HTML.gif
      (4)
      A map http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq56_HTML.gif on a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq15_HTML.gif is called a weak ϕ-contraction if there exists a strictly increasing function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq57_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq58_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Eque_HTML.gif

      for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq5_HTML.gif . We notice that these types of contractions have also been a subject of extensive research (see, e.g., [4449]). In what follows, we recall the notion of cyclic weak ψ-contractions on G-metric spaces. Let Ψ be the set of continuous functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq57_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq60_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq61_HTML.gif . In [36], the authors concentrated on two types of cyclic contractions: cyclic-type Banach contractions and cyclic weak ϕ-contractions.

      Theorem 1.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif be a G-complete G-metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq63_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq64_HTML.gif be a map satisfying
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ5_HTML.gif
      (5)
      Suppose that there exists a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq65_HTML.gif such that the map T satisfies the inequality
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ6_HTML.gif
      (6)
      for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq66_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq67_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq68_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ7_HTML.gif
      (7)

      Then T has a unique fixed point in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq69_HTML.gif .

      The following result, which can be considered as a corollary of Theorem 1.3, is stated in [36].

      Theorem 1.4 (See [36])

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif be a G-complete G-metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq64_HTML.gif be a map satisfying
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ8_HTML.gif
      (8)
      If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq71_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ9_HTML.gif
      (9)

      holds for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq66_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq67_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq68_HTML.gif , then T has a unique fixed point in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq69_HTML.gif .

      In this paper, we extend, generalize and enrich the results on the topic in the literature.

      2 Main results

      We start this section by defining some sets of auxiliary functions. Let ℱ denote all functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq72_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq73_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq74_HTML.gif . Let Ψ and Φ be the subsets of ℱ such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equf_HTML.gif
      Lemma 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif be a G-complete G-metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif be a sequence in X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq75_HTML.gif is nonincreasing,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ10_HTML.gif
      (10)
      If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif is not a Cauchy sequence, then there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq23_HTML.gif and two sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq76_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq77_HTML.gif of positive integers such that the following sequences tend to ε when http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq78_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ11_HTML.gif
      (11)

      Proof

      Due to Lemma 1.1, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equg_HTML.gif
      Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq79_HTML.gif regarding the assumption of the lemma, we derive that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ12_HTML.gif
      (12)
      If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif is not G-Cauchy, then, due to Proposition 1.2, there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq23_HTML.gif and corresponding subsequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq80_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq81_HTML.gif of ℕ satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq82_HTML.gif for which
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ13_HTML.gif
      (13)
      where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq83_HTML.gif is chosen as the smallest integer satisfying (13), that is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ14_HTML.gif
      (14)
      By (13), (14) and the rectangle inequality (G5), we easily derive that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ15_HTML.gif
      (15)
      Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq84_HTML.gif in (15) and using (10), we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ16_HTML.gif
      (16)
      Further,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ17_HTML.gif
      (17)
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ18_HTML.gif
      (18)
      Passing to the limit when http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq84_HTML.gif and using (10) and (16), we obtain that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ19_HTML.gif
      (19)
      In a similar way,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ20_HTML.gif
      (20)
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ21_HTML.gif
      (21)
      Passing to the limit when http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq84_HTML.gif and using (10) and (16), we obtain that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ22_HTML.gif
      (22)
      Furthermore,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ23_HTML.gif
      (23)
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ24_HTML.gif
      (24)
      Passing to the limit when http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq84_HTML.gif and using (10) and (16), we obtain that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ25_HTML.gif
      (25)
      By regarding the assumptions (G3) and (G5) together with the expression (13), we derive the following:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ26_HTML.gif
      (26)
      Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq84_HTML.gif in the inequality above and using (12) and (16), we conclude that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ27_HTML.gif
      (27)

       □

      Theorem 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif be a G-complete G-metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq85_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq86_HTML.gif be a map satisfying
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ28_HTML.gif
      (28)
      Suppose that there exist functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq88_HTML.gif such that the map T satisfies the inequality
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ29_HTML.gif
      (29)
      for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq90_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq91_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ30_HTML.gif
      (30)

      Then T has a unique fixed point in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq69_HTML.gif .

      Proof First we show the existence of a fixed point of the map T. For this purpose, we take an arbitrary http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq92_HTML.gif and define a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif in the following way:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ31_HTML.gif
      (31)
      We have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq92_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq93_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq94_HTML.gif , … since T is a cyclic mapping. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq95_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq96_HTML.gif , then, clearly, the fixed point of the map T is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq97_HTML.gif . From now on, assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq98_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq99_HTML.gif . Consider the inequality (29) by letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq101_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ32_HTML.gif
      (32)
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ33_HTML.gif
      (33)
      If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq102_HTML.gif , then the expression (32) implies that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ34_HTML.gif
      (34)
      So, the inequality (34) yields http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq103_HTML.gif . Thus, we conclude that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equh_HTML.gif
      This contradicts the assumption http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq104_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq99_HTML.gif . So, we derive that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ35_HTML.gif
      (35)
      Hence the inequality (32) turns into
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ36_HTML.gif
      (36)
      Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq105_HTML.gif is a nonnegative, nonincreasing sequence that converges to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq106_HTML.gif . We will show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq107_HTML.gif . Suppose, on the contrary, that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq108_HTML.gif . Taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq109_HTML.gif in (36), we derive that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ37_HTML.gif
      (37)
      By the continuity of ψ and the lower semi-continuity of ϕ, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ38_HTML.gif
      (38)
      Then it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq110_HTML.gif . Therefore, we get http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq107_HTML.gif , that is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ39_HTML.gif
      (39)
      Lemma 1.1 with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq111_HTML.gif implies that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ40_HTML.gif
      (40)
      So, we get that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ41_HTML.gif
      (41)
      Next, we will show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif is a G-Cauchy sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif . Suppose, on the contrary, that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif is not G-Cauchy. Then, due to Proposition 1.2, there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq23_HTML.gif and corresponding subsequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq80_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq81_HTML.gif of ℕ satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq82_HTML.gif for which
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ42_HTML.gif
      (42)
      where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq83_HTML.gif is chosen as the smallest integer satisfying (42), that is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ43_HTML.gif
      (43)
      By (42), (43) and the rectangle inequality (G5), we easily derive that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ44_HTML.gif
      (44)
      Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq84_HTML.gif in (44) and using (39), we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ45_HTML.gif
      (45)
      Notice that for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq112_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq113_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq114_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ46_HTML.gif
      (46)
      Thus, for large enough values of k, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq115_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq116_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq117_HTML.gif lie in the adjacent sets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq118_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq119_HTML.gif respectively for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq120_HTML.gif . When we substitute http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq121_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq122_HTML.gif in the expression (29), we get that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ47_HTML.gif
      (47)
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ48_HTML.gif
      (48)
      By using Lemma 2.1, we obtain that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ49_HTML.gif
      (49)
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ50_HTML.gif
      (50)
      So, we obtain that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ51_HTML.gif
      (51)
      So, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq123_HTML.gif . We deduce that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq124_HTML.gif . This contradicts the assumption that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq125_HTML.gif is not G-Cauchy. As a result, the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq21_HTML.gif is G-Cauchy. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif is G-complete, it is G-convergent to a limit, say http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq126_HTML.gif . It easy to see that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq127_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq92_HTML.gif , then the subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq128_HTML.gif , the subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq129_HTML.gif and, continuing in this way, the subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq130_HTML.gif . All the m subsequences are G-convergent in the G-closed sets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq118_HTML.gif and hence they all converge to the same limit http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq127_HTML.gif . To show that the limit w is the fixed point of T, that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq131_HTML.gif , we employ (29) with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq100_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq132_HTML.gif . This leads to
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ52_HTML.gif
      (52)
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ53_HTML.gif
      (53)
      Passing to limsup as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq133_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ54_HTML.gif
      (54)

      Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq134_HTML.gif and hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq135_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq131_HTML.gif .

      Finally, we prove that the fixed point is unique. Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq136_HTML.gif is another fixed point of T such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq137_HTML.gif . Then, since both v and w belong to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq69_HTML.gif , we set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq138_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq132_HTML.gif in (29), which yields
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ55_HTML.gif
      (55)
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ56_HTML.gif
      (56)
      On the other hand, by setting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq139_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq140_HTML.gif in (29), we obtain that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ57_HTML.gif
      (57)
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ58_HTML.gif
      (58)
      If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq141_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq142_HTML.gif . Indeed, by definition, we get that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq143_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq142_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq144_HTML.gif , then by (56) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq145_HTML.gif and by (55),
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ59_HTML.gif
      (59)
      and, clearly, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq146_HTML.gif . So, we conclude that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq142_HTML.gif . Otherwise, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq147_HTML.gif . Then by (58), http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq148_HTML.gif and by (57),
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ60_HTML.gif
      (60)

      and, clearly, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq149_HTML.gif . So, we conclude that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq142_HTML.gif . Hence the fixed point of T is unique. □

      Remark 2.1 We notice that some fixed point result in the context of G-metric can be obtained by usual (well-known) fixed point theorems (see, e.g., [50, 51]). In fact, this is not a surprising result due to strong relationship between the usual metric and G-metric space (see, e.g., [2, 3, 5]). Note that a G-metric space tells about the distance of three points instead of two points, which makes it original. We also emphasize that the techniques used in [50, 51] are not applicable to our main theorem.

      To illustrate Theorem 2.1, we give the following example.

      Example 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq150_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq151_HTML.gif be given as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq152_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq153_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq154_HTML.gif . Define the function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq155_HTML.gif as
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ61_HTML.gif
      (61)

      Clearly, the function G is a G-metric on X. Define also http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq156_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq157_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq158_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq159_HTML.gif . Obviously, the map T has a unique fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq160_HTML.gif .

      It can be easily shown that the map T satisfies the condition (29). Indeed,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equi_HTML.gif
      which yields
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ62_HTML.gif
      (62)
      Moreover, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ63_HTML.gif
      (63)
      We derive from (63) that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ64_HTML.gif
      (64)
      On the other hand, we have the following inequality:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ65_HTML.gif
      (65)
      By elementary calculation, we conclude from (65) and (64) that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ66_HTML.gif
      (66)
      Combining the expressions (62) and (65), we obtain that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ67_HTML.gif
      (67)

      Hence, all conditions of Theorem 2.1 are satisfied. Notice that 0 is the unique fixed point of T.

      For particular choices of the functions ϕ, ψ, we obtain the following corollaries.

      Corollary 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif be a G-complete G-metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq85_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq86_HTML.gif be a map satisfying
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ68_HTML.gif
      (68)
      Suppose that there exists a constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq161_HTML.gif such that the map T satisfies
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ69_HTML.gif
      (69)
      for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq90_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq91_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ70_HTML.gif
      (70)

      Then T has a unique fixed point in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq69_HTML.gif .

      Proof The proof is obvious by choosing the functions ϕ, ψ in Theorem 2.1 as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq162_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq163_HTML.gif . □

      Corollary 2.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif be a G-complete G-metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq85_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq86_HTML.gif be a map satisfying
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ71_HTML.gif
      (71)
      Suppose that there exist constants a, b, c, d and e with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq164_HTML.gif and there exists a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq88_HTML.gif such that the map T satisfies the inequality
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ72_HTML.gif
      (72)

      for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq90_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq165_HTML.gif . Then T has a unique fixed point in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq69_HTML.gif .

      Proof

      Clearly, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ73_HTML.gif
      (73)
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ74_HTML.gif
      (74)

      By Corollary 2.1, the map T has a unique fixed point. □

      Corollary 2.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif be a G-complete G-metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq85_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq86_HTML.gif be a map satisfying
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equj_HTML.gif
      Suppose that there exist functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq88_HTML.gif such that the map T satisfies the inequality
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equk_HTML.gif
      for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq90_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq91_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equ75_HTML.gif
      (75)

      Then T has a unique fixed point in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq69_HTML.gif .

      Proof The expression (75) coincides with the expression (30). Following the lines in the proof of Theorem 2.1, by letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq166_HTML.gif , we get the desired result. □

      Cyclic maps satisfying integral type contractive conditions are amongst common applications of fixed point theorems. In this context, we consider the following applications.

      Corollary 2.4 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif be a G-complete G-metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq85_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq86_HTML.gif be a map satisfying
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equl_HTML.gif
      Suppose also that there exist functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq88_HTML.gif such that the map T satisfies
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equm_HTML.gif
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equn_HTML.gif

      for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq90_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq91_HTML.gif . Then T has a unique fixed point in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq69_HTML.gif .

      Corollary 2.5 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq13_HTML.gif be a G-complete G-metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq85_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq86_HTML.gif be a map satisfying
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equo_HTML.gif
      Suppose also that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equp_HTML.gif
      where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq161_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_Equq_HTML.gif

      for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq90_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq91_HTML.gif . Then T has a unique fixed point in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq69_HTML.gif .

      Proof The proof is obvious by choosing the function ϕ, ψ in Corollary 2.4 as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq162_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_389_IEq163_HTML.gif . □

      Declarations

      Authors’ Affiliations

      (1)
      Department of Mathematics, Institute of Science and Technology, Gazi University
      (2)
      Department of Mathematics, Atilim University

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