Fixed point theorems for generalized contractions on GP-metric spaces

  • Nurcan Bilgili1,

    Affiliated with

    • Erdal Karapınar2Email author and

      Affiliated with

      • Peyman Salimi3

        Affiliated with

        Journal of Inequalities and Applications20132013:39

        DOI: 10.1186/1029-242X-2013-39

        Received: 8 November 2012

        Accepted: 23 January 2013

        Published: 6 February 2013

        Abstract

        In this paper, we present two fixed point theorems on mappings, defined on GP-complete GP-metric spaces, which satisfy a generalized contraction property determined by certain upper semi-continuous functions. Furthermore, we illustrate applications of our theorems with a number of examples. Inspired by the work of Jachymski, we also establish equivalences of certain auxiliary maps in the context of GP-complete GP-metric spaces.

        MSC:47H10, 54H25.

        Keywords

        fixed point partial metric space GP-metric space

        1 Introduction and preliminaries

        In 1922, Stefan Banach [1] stated his celebrated theorem on the existence and uniqueness of a fixed point of certain self-maps defined on certain metric spaces for the first time. Specifically, this elegant theorem, also known as the Banach contraction mapping principle, can be formulated as follows: any mapping T : ( X , d ) ( X , d ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq1_HTML.gif has a unique point x X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq2_HTML.gif such that T x = x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq3_HTML.gif provided that there exists a constant k ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq4_HTML.gif satisfying the inequality d ( T x , T y ) k d ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq5_HTML.gif for every x , y X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq6_HTML.gif, where ( X , d ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq7_HTML.gif is a complete metric space. A mapping T for which the inequality mentioned above holds is called a contraction.

        Since its first appearance, the Banach contraction mapping principle has become the main tool to study contractions as they appear abundantly in a wide array of quantitative sciences. Its most well-known application is in ordinary differential equations, particularly, in the proof of the Picard-Lindelöf theorem which guarantees the existence and uniqueness of solutions of first-order initial value problems. It is worth emphasizing that the remarkable strength of the Banach principle originates from the constructive process it provides to identify the fixed point. This notable strength further attracted the attention of not only many prominent mathematicians studying in many branches of mathematics related to nonlinear analysis, but also many researchers who are interested in iterative methods to examine the quantitative problems involving certain mappings and space structures required in their work in various areas such as social sciences, biology, economics, and computer sciences.

        Indeed, in 1994, Matthews, a computer scientist who is an expert on semantics, announced in [2] an analog of Banach’s principle in a new space he called a partial metric space. Matthews’s innovative approach was quickly adopted and improved by fixed point theorists (see, e.g., [327]) with the aim of discovering analogs of Banach’s principle in the context of partial metric spaces to broaden its applications and enrich the fixed point theory as a result.

        A closer look to the work of these distinguished mathematicians after Matthews’s studies reveals that their discoveries can be categorized in terms of the techniques implemented to produce the analogs of Banach’s principle. The first technique is to introduce new space structures with certain properties which guarantee the existence and/or uniqueness of fixed points of contractions. In addition to Matthews’s investigations, cone metric spaces, D-metric spaces, and G-metric spaces (see, e.g., [2844]) constitute a few of the examples to the first approach. The second technique is to introduce mappings defined on metric spaces satisfying certain new contractive conditions. For example, cyclic contractions and weak ϕ contractions can be listed as a few.

        As another example to the first approach mentioned above, Zand and Nezhad [43] recently introduced GP-metric spaces which are a combination of the notions of partial metric spaces and G-metric spaces. Then they proved a number of fixed point theorems on these new spaces for certain type of contractions. In this paper, we exercise the second approach by using the space structure they initiated to prove certain fixed point theorems for generalized contractions. First, we review the necessary notation, definitions, and fundamental results produced on GP-metric spaces that we will need in this work.

        Definition 1.1 [43]

        Let X be a non-empty set. A function G p : X × X × X [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq8_HTML.gif is called a GP-metric if the following conditions are satisfied:

        (GP1) x = y = z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq9_HTML.gif if G p ( x , y , z ) = G p ( z , z , z ) = G p ( y , y , y ) = G p ( x , x , x ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq10_HTML.gif;

        (GP2) 0 G p ( x , x , x ) G p ( x , x , y ) G p ( x , y , z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq11_HTML.gif for all x , y , z X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq12_HTML.gif;

        (GP3) G p ( x , y , z ) = G p ( x , z , y ) = G p ( y , z , x ) = http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq13_HTML.gif , symmetry in all three variables;

        (GP4) G p ( x , y , z ) G p ( x , a , a ) + G p ( a , y , z ) G p ( a , a , a ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq14_HTML.gif for any x , y , z , a X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq15_HTML.gif.

        Then the pair ( X , G ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq16_HTML.gif is called a GP-metric space.

        Example 1.1 [43]

        Let X = [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq17_HTML.gif and define G p ( x , y , z ) = max { x , y , z } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq18_HTML.gif for all x , y , z X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq12_HTML.gif. Then ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq19_HTML.gif is a GP-metric space.

        Proposition 1.1 [43]

        Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif be a GP-metric space, then for any x, y, z and a X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq21_HTML.gif, it follows that
        1. (i)

          G p ( x , y , z ) G p ( x , x , y ) + G p ( x , x , z ) G p ( x , x , x ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq22_HTML.gif;

           
        2. (ii)

          G p ( x , y , y ) 2 G p ( x , x , y ) G p ( x , x , x ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq23_HTML.gif;

           
        3. (iii)

          G p ( x , y , z ) G p ( x , a , a ) + G p ( y , a , a ) + G p ( z , a , a ) 2 G p ( a , a , a ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq24_HTML.gif;

           
        4. (iv)

          G p ( x , y , z ) G p ( x , a , z ) + G p ( a , y , z ) G p ( a , a , a ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq25_HTML.gif.

           

        Proposition 1.2 [43]

        Every GP-metric space ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq19_HTML.gif defines a metric space ( X , D G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq26_HTML.gif where
        D G p ( x , y ) = G p ( x , y , y ) + G p ( y , x , x ) G p ( x , x , x ) G p ( y , y , y ) for all x , y X . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equa_HTML.gif

        Definition 1.2 [43]

        Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq19_HTML.gif be a GP-metric space and let { x n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq27_HTML.gif be a sequence of points of X. A point x X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq28_HTML.gif is said to be the limit of the sequence { x n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq27_HTML.gif or x n x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq29_HTML.gif if
        lim n , m G p ( x , x m , x n ) = G p ( x , x , x ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equb_HTML.gif

        Proposition 1.3 [43]

        Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq19_HTML.gif be a GP-metric space. Then, for any sequence { x n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq30_HTML.gif in X and a point x X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq2_HTML.gif, the following are equivalent:
        1. (A)

          { x n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq27_HTML.gif is GP-convergent to x;

           
        2. (B)

          G p ( x n , x n , x ) G p ( x , x , x ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq31_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq32_HTML.gif;

           
        3. (C)

          G p ( x n , x , x ) G p ( x , x , x ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq33_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq34_HTML.gif.

           

        Definition 1.3 [43]

        Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif be a GP-metric space.

        (S1) A sequence { x n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq27_HTML.gif is called a GP-Cauchy if and only if lim m , n G p ( x n , x m , x m ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq35_HTML.gif exists (and is finite);

        (S2) A GP-partial metric space ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif is said to be GP-complete if and only if every GP-Cauchy sequence in X is GP-convergent to x X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq28_HTML.gif such that G p ( x , x , x ) = lim m , n G p ( x n , x m , x m ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq36_HTML.gif.

        Now, we introduce the following.

        Definition 1.4 Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif be a GP-metric space.

        (M1) A sequence { x n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq27_HTML.gif is called 0-GP-Cauchy if and only if lim m , n G p ( x n , x m , x m ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq37_HTML.gif;

        (M2) A GP-metric space ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif is said to be 0-GP-complete if and only if every 0-GP-Cauchy sequence in X GP-converges to a point x X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq28_HTML.gif such that G p ( x , x , x ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq38_HTML.gif.

        Example 1.2 Let X = [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq39_HTML.gif and define G p ( x , y , z ) = max { x , y , z } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq18_HTML.gif for all x , y , z X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq12_HTML.gif. Then ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq19_HTML.gif is a GP-complete GP-metric space. Moreover, if X = Q [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq40_HTML.gif (where ℚ denotes a set of rational numbers), then ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq19_HTML.gif is a 0-GP-complete GP-metric space.

        Lemma 1.1 (See [45])

        Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq19_HTML.gif be a GP-metric space. Then
        1. (A)

          If G p ( x , y , z ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq41_HTML.gif, then x = y = z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq9_HTML.gif;

           
        2. (B)

          If x y http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq42_HTML.gif, then G p ( x , y , y ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq43_HTML.gif.

           

        In the rest of this paper, we will denote the positive natural numbers by N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq44_HTML.gif and the natural numbers by ℕ.

        2 Main results

        In this section, we present our findings on fixed point theorems on 0-GP-complete GP-metric spaces. We first start with the following definition.

        Definition 2.1 Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif be a GP-metric space and T : ( X , G p ) ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq45_HTML.gif be a map. Let M ( x , y , y ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq46_HTML.gif denote the value
        max { G p ( x , y , y ) , G p ( x , T x , T x ) , G p ( y , T y , T y ) , 1 2 [ G p ( x , T y , T y ) + G p ( y , T x , T x ) ] } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ1_HTML.gif
        (1)

        for all x , y X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq47_HTML.gif.

        Lemma 2.1 If ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif is a GP-metric space and T : X X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq48_HTML.gif is a map, then, for each x X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq2_HTML.gif, we have
        M ( x , T x , T x ) = max { G p ( x , T x , T x ) , G p ( T x , T 2 x , T 2 x ) } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ2_HTML.gif
        (2)
        Proof Let x X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq2_HTML.gif. Then
        http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ3_HTML.gif
        (3)

        The proof is complete. □

        Lemma 2.2 Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif be a GP-metric space and let T : X X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq48_HTML.gif be a map such that
        G p ( T x , T y , T y ) ϕ ( M ( x , y , y ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ4_HTML.gif
        (4)
        for all x , y X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq6_HTML.gif, where ϕ : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq49_HTML.gif is a function such that ϕ ( t ) < t http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq50_HTML.gif for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq51_HTML.gif. If x X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq2_HTML.gif satisfies T n x T n + 1 x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq52_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif, then the following hold:
        1. (a)

          M ( T n x , T n + 1 x , T n + 1 x ) = G p ( T n x , T n + 1 x , T n + 1 x ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq54_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif;

           
        2. (b)

          G p ( T n x , T n + 1 x , T n + 1 x ) ϕ ( G p ( T n 1 x , T n x , T n x ) ) < G p ( T n 1 x , T n x , T n x ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq55_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq56_HTML.gif.

           
        Proof
        1. (a)
          From Lemma 2.1, we have
          M ( T n x , T n + 1 x , T n + 1 x ) = max { G p ( T n x , T n + 1 x , T n + 1 x ) , G p ( T n + 1 x , T n + 2 x , T n + 2 x ) } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equc_HTML.gif
           
        Since T n x T n + 1 x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq52_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif, then by Lemma 1.1(B), we get G p ( T n x , T n + 1 x , T n + 1 x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq57_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif. Consequently, M ( T n x , T n + 1 x , T n + 1 x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq58_HTML.gif. Now by condition (4), we deduce that
        G p ( T n + 1 x , T n + 2 x , T n + 2 x ) ϕ ( M ( T n x , T n + 1 x , T n + 1 x ) ) < M ( T n x , T n + 1 x , T n + 1 x ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ5_HTML.gif
        (5)

        that is, G p ( T n + 1 x , T n + 2 x , T n + 2 x ) < G p ( T n x , T n + 1 x , T n + 1 x ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq59_HTML.gif. Hence, (a) holds.

        Clearly, (b) follows from (4), (a), and the fact that G p ( T n 1 x , T n x , T n x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq60_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif. □

        Definition 2.2 A function ϕ : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq49_HTML.gif is called upper semi-continuous from the right if for each t 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq61_HTML.gif and each sequence ( t n ) n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq62_HTML.gif such that t n t http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq63_HTML.gif and lim n t n = t http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq64_HTML.gif, the equality holds lim sup n ϕ ( t n ) ϕ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq65_HTML.gif.

        Theorem 2.1 Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif be a GP-complete GP-metric space and let T : X X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq66_HTML.gif be a map such that
        G p ( T x , T y , T y ) ϕ ( M ( x , y , y ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ6_HTML.gif
        (6)

        for all x , y X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq6_HTML.gif, where ϕ : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq49_HTML.gif is an upper semi-continuous function from the right such that ϕ ( t ) < t http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq50_HTML.gif for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq51_HTML.gif. Then T has a unique fixed point z X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq67_HTML.gif. Moreover, G p ( z , z , z ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq68_HTML.gif.

        Proof Let x X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq2_HTML.gif. If there is n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif such that T n x = T n + 1 x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq69_HTML.gif, then T n x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq70_HTML.gif is a fixed point of T and the uniqueness of T n x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq70_HTML.gif follows as in the last part of the proof below. Hence, we assume that T n x T n + 1 x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq52_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif. Put x 0 = x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq71_HTML.gif and construct the sequence ( x n ) n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq72_HTML.gif, where x n = T n x 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq73_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif. Thus, x n + 1 = T x n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq74_HTML.gif and G p ( x n , x n + 1 , x n + 1 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq75_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq76_HTML.gif. Define the sequence { s n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq77_HTML.gif by s n = G p ( x n , x n + 1 , x n + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq78_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq76_HTML.gif. From Lemma 2.2(b) we know that { s n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq77_HTML.gif is a non-increasing sequence. Hence, there exists c R + http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq79_HTML.gif such that s n c http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq80_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq81_HTML.gif. We will show that c must be equal to 0. Let c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq82_HTML.gif. By taking limitsup as n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq81_HTML.gif in condition (b) of Lemma 2.2, we get that
        c = lim sup n G p ( x n , x n + 1 , x n + 1 ) = lim sup n ϕ ( G p ( x n , x n + 1 , x n + 1 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ7_HTML.gif
        (7)
        and so, by upper semi-continuity from the right of the function ϕ, we deduce
        c = lim sup n ϕ ( G p ( x n , x n + 1 , x n + 1 ) ) ϕ ( c ) < c , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ8_HTML.gif
        (8)
        which is a contradiction. Hence, c = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq83_HTML.gif. Consequently, lim n G p ( x n , x n + 1 , x n + 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq84_HTML.gif. Next we show that lim n , m G p ( x n , x m , x m ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq85_HTML.gif. Assume the contrary. Then there exist ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq86_HTML.gif and sequences ( n k ) k N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq87_HTML.gif, ( m k ) k N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq88_HTML.gif in N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq44_HTML.gif with m k n k k http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq89_HTML.gif and such that G p ( x n k , x m k , x m k ) ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq90_HTML.gif for all k N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq91_HTML.gif. From the fact that lim n G p ( x n , x n + 1 , x n + 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq92_HTML.gif, we can suppose, without loss of generality, that G p ( x n k , x m k 1 , x m k 1 ) < ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq93_HTML.gif. For each k N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq91_HTML.gif, we have
        ε G p ( x n k , x m k , x m k ) G p ( x n k , x m k 1 , x m k 1 ) + G p ( x m k 1 , x m k , x m k ) < ε + G p ( x m k 1 , x m k , x m k ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ9_HTML.gif
        (9)
        and hence lim k G p ( x n k , x m k , x m k ) = ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq94_HTML.gif. Now, let k 0 N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq95_HTML.gif be such that G p ( x n k , x n k + 1 , x n k + 1 ) < ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq96_HTML.gif and G p ( x m k , x m k + 1 , x m k + 1 ) < ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq97_HTML.gif for all k k 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq98_HTML.gif. Then
        G p ( x n k , x m k , x m k ) M ( x n k , x m k , x m k ) = max { G p ( x n k , x m k , x m k ) , G p ( x n k , x n k + 1 , x n k + 1 ) , G p ( x m k , x m k + 1 , x m k + 1 ) , 1 2 [ G p ( x n k , x m k + 1 , x m k + 1 ) + G p ( x m k , x n k + 1 , x n k + 1 ) ] } max { G p ( x n k , x m k , x m k ) , G p ( x n k , x n k + 1 , x n k + 1 ) , G p ( x m k , x m k + 1 , x m k + 1 ) , 1 2 [ G p ( x n k , x n k + 1 , x n k + 1 ) + G p ( x n k + 1 , x n k + 2 , x n k + 2 ) + + G p ( x m k , x m k + 1 , x m k + 1 ) + G p ( x m k , x m k 1 , x m k 1 ) + G p ( x m k 1 , x m k 2 , x m k 2 ) + + G p ( x n k + 2 , x n k + 1 , x n k + 1 ) ] } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ10_HTML.gif
        (10)
        for all k k 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq98_HTML.gif. So, lim k M ( x n k , x m k , x m k ) = ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq99_HTML.gif. Since M ( x n k , x m k , x m k ) ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq100_HTML.gif for all k N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq101_HTML.gif and ϕ is upper semi-continuous from the right, we deduce that lim sup k ϕ ( M ( x n k , x m k , x m k ) ) ϕ ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq102_HTML.gif. On the other hand, for each k N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq91_HTML.gif, we have
        ε G p ( x n k , x m k , x m k ) G p ( x n k , x n k + 1 , x n k + 1 ) + G p ( x n k + 1 , x m k + 1 , x m k + 1 ) + G p ( x m k + 1 , x m k , x m k ) G p ( x n k , x n k + 1 , x n k + 1 ) + ϕ ( M ( x n k , x m k , x m k ) ) + G p ( x m k + 1 , x m k , x m k ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ11_HTML.gif
        (11)
        so ε lim sup k ϕ ( M ( x n k , x m k , x m k ) ) ϕ ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq103_HTML.gif, a contradiction because ϕ ( ε ) < ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq104_HTML.gif. Consequently, lim n , m G p ( x n , x m , x m ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq105_HTML.gif and thus ( x n ) n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq72_HTML.gif is a Cauchy sequence in the GP-complete GP-metric space ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif. Hence, there is z X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq67_HTML.gif such that
        lim n , m G p ( x n , x m , x m ) = lim n G p ( z , x n , x n ) = G p ( z , z , z ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ12_HTML.gif
        (12)
        We show that z is a fixed point of T. To this end, we first note that G p ( z , T z , T z ) = lim n M ( z , x n , x n ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq106_HTML.gif, so lim sup n ϕ ( M ( z , x n , x n ) ) ϕ ( G p ( z , T z , T z ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq107_HTML.gif. On the other hand, since for each n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif, G p ( z , T z , T z ) G p ( z , x n , x n ) + G p ( x n , T z , T z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq108_HTML.gif, it follows that
        G p ( z , T z , T z ) lim sup n ( G p ( z , x n , x n ) + G p ( x n , T z , T z ) ) = lim sup n G p ( x n , T z , T z ) lim sup n ϕ ( M ( x n 1 , z , z ) ) ϕ ( G p ( z , T z , T z ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ13_HTML.gif
        (13)
        Therefore, G p ( z , T z , T z ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq109_HTML.gif and thus z = T z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq110_HTML.gif. Finally, let u X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq111_HTML.gif be such that T u = u http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq112_HTML.gif. Then
        G p ( u , z , z ) = G p ( T u , T z , T z ) ϕ ( M ( u , z , z ) ) = ϕ ( G p ( u , z , z ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ14_HTML.gif
        (14)

        Hence, G p ( u , z , z ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq113_HTML.gif, i.e., u = z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq114_HTML.gif. This concludes the proof. □

        Definition 2.3 Let ψ ( , ) : X × X [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq115_HTML.gif be a given function. Then Q ( x , y , z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq116_HTML.gif will denote the value
        http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ15_HTML.gif
        (15)

        Then we obtain the following statement.

        Corollary 2.1 Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif be a GP-complete GP-metric space and let T : X X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq66_HTML.gif be a map such that
        0 [ ψ ( z , z ) ψ ( y , y ) ] max { G p ( x , x , x ) , G p ( y , y , y ) , G p ( z , z , z ) } + ψ ( x , y ) ψ ( x , z ) [ ϕ ( Q ( x , y , z ) ) G p ( T x , T y , T z ) ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equd_HTML.gif

        for all x , y , z X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq117_HTML.gif, where ϕ : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq49_HTML.gif is an upper semi-continuous from the right function such that ϕ ( t ) < t http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq50_HTML.gif for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq51_HTML.gif. Then T has a unique fixed point z X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq67_HTML.gif. Moreover, G p ( z , z , z ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq68_HTML.gif.

        Proof Clearly, by taking y = z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq118_HTML.gif in the hypothesis, we have
        G p ( T x , T y , T y ) ϕ ( Q ( x , y , y ) ) = ϕ ( M ( x , y , y ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Eque_HTML.gif

        for all x , y X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq6_HTML.gif. Then the conditions of Theorem 2.1 hold. This concludes the proof. □

        Example 2.1 Let X = [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq17_HTML.gif, G p : X × X × X R http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq119_HTML.gif be defined by G p ( x , y , z ) = max { x , y , z } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq120_HTML.gif. Then ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif is a GP-complete GP-metric space. Let f : X X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq121_HTML.gif be defined by T x = x 2 ( x + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq122_HTML.gif and ψ ( t ) = t 1 + t http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq123_HTML.gif for all t [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq124_HTML.gif.

        Proof Without loss of generality, we assume that x y http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq125_HTML.gif. Then
        G p ( T x , T y , T y ) = x 2 ( x + 1 ) x x + 1 = ψ ( x ) = ψ ( G p ( x , y , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equf_HTML.gif

        Then the condition of Theorem 2.1 holds and T has a unique fixed point 0 in [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq126_HTML.gif. Moreover, G p ( 0 , 0 , 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq127_HTML.gif. □

        Example 2.2 Let X = [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq17_HTML.gif, G p : X × X × X R http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq119_HTML.gif be defined by G p ( x , y , z ) = max { x , y , z } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq120_HTML.gif. Then ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif is a GP-complete GP-metric space. Let T : X X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq128_HTML.gif be defined by
        T x = { 1 2 x 2 if  0 x < 1 / 3 , ( 1 x ) / 2 if  1 / 3 x 1 , 1 4 x if  x > 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equg_HTML.gif

        and ψ ( t ) = 1 2 t http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq129_HTML.gif for all t [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq130_HTML.gif.

        Proof To prove this example, we need to consider the following cases:

        • Let 0 x , y < 1 / 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq131_HTML.gif. Then
          G p ( T x , T y , T y ) = 1 2 max { x 2 , y 2 } 1 2 max { x , y } = ψ ( G p ( x , y , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equh_HTML.gif
        • Let 1 / 3 x , y 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq132_HTML.gif. Then
          G p ( T x , T y , T y ) = 1 2 max { 1 x , 1 y } 1 2 max { x , y } = ψ ( G p ( x , y , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equi_HTML.gif
        • Let x , y > 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq133_HTML.gif. Then
          G p ( T x , T y , T y ) = 1 4 max { x , y } 1 2 max { x , y } = ψ ( G p ( x , y , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equj_HTML.gif
        • Let 0 x < 1 / 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq134_HTML.gif and 1 / 3 y 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq135_HTML.gif. Then
          G p ( T x , T y , T y ) = max { 1 2 x 2 , ( 1 y ) / 2 } 1 2 max { x , y } = ψ ( G p ( x , y , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equk_HTML.gif
        • Let 0 x < 1 / 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq134_HTML.gif and y > 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq136_HTML.gif. Then
          G p ( T x , T y , T y ) = max { 1 2 x 2 , 1 4 y } 1 2 max { x , y } = ψ ( G p ( x , y , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equl_HTML.gif
        • Let 1 / 3 x 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq137_HTML.gif and y > 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq136_HTML.gif. Then
          G p ( T x , T y , T y ) = max { ( 1 x ) / 2 , 1 4 y } 1 2 max { x , y } = ψ ( G p ( x , y , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equm_HTML.gif

        Then the condition of Theorem 2.1 holds and T has a unique fixed point 0 in [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq126_HTML.gif. Moreover, G p ( 0 , 0 , 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq127_HTML.gif. □

        Lemma 2.3 Let ϕ : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq49_HTML.gif be nondecreasing and let t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq51_HTML.gif. If lim n ϕ n ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq138_HTML.gif, then ϕ ( t ) < t http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq50_HTML.gif.

        Theorem 2.2 Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif be a GP-complete GP-metric space and T : X X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq66_HTML.gif be a map such that
        G p ( T x , T y , T y ) ϕ ( N ( x , y , y ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ16_HTML.gif
        (16)

        where N ( x , y , y ) = max { G p ( x , y , y ) , G p ( x , T x , T x ) , G p ( y , T y , T y ) } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq139_HTML.gif for all x , y X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq6_HTML.gif, and ϕ : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq140_HTML.gif is a nondecreasing function such that lim n ϕ n ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq138_HTML.gif for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq51_HTML.gif. Then T has a unique fixed point z X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq67_HTML.gif. Moreover, G p ( z , z , z ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq68_HTML.gif.

        Proof Let x X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq2_HTML.gif. If there is n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif such that T n x = T n + 1 x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq69_HTML.gif, then T n x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq70_HTML.gif is a fixed point of T and the uniqueness of T n x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq70_HTML.gif follows as in the last part of the proof below. Hence, we will assume that T n x T n + 1 x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq52_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif. Put x 0 = x http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq71_HTML.gif and construct the sequence ( x n ) n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq72_HTML.gif, where x n = T n x 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq73_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif. Thus, x n + 1 = T x n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq74_HTML.gif and G p ( x n , x n + 1 , x n + 1 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq75_HTML.gif for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif. By Lemma 2.2(b),
        G p ( x n , x n + 1 , x n + 1 ) ϕ ( G p ( x n 1 , x n , x n ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ17_HTML.gif
        (17)
        for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif. Then, since ϕ is nondecreasing, we deduce that
        G p ( x n , x n + 1 , x n + 1 ) ϕ n ( G p ( x 0 , x 1 , x 1 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ18_HTML.gif
        (18)
        for all n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq53_HTML.gif. Hence, lim n G p ( x n , x n + 1 , x n + 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq141_HTML.gif. Now, choose an arbitrary ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq86_HTML.gif. Since lim n ϕ n ( ε ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq142_HTML.gif, it follows from Lemma 2.3 that ϕ ( ε ) < ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq104_HTML.gif, so there is n ε N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq143_HTML.gif such that
        G p ( x n , x n + 1 , x n + 1 ) < ε ϕ ( ε ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ19_HTML.gif
        (19)
        for all n n ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq144_HTML.gif. Therefore,
        G p ( x n , x n + 2 , x n + 2 ) G p ( x n , x n + 1 , x n + 1 ) + G p ( x n + 1 , x n + 2 , x n + 2 ) < ε ϕ ( ε ) + ϕ ( G p ( x n , x n + 1 , x n + 1 ) ) ε ϕ ( ε ) + ϕ ( ε ) = ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ20_HTML.gif
        (20)
        for all n n ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq144_HTML.gif. So,
        G p ( x n , x n + 3 , x n + 3 ) G p ( x n , x n + 1 , x n + 1 ) + G p ( x n + 1 , x n + 3 , x n + 3 ) < ε ϕ ( ε ) + ϕ ( N ( x n , x n + 2 , x n + 2 ) ) ε ϕ ( ε ) + ϕ ( ε ) = ε , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ21_HTML.gif
        (21)
        and following this process,
        G p ( x n , x n + k , x n + k ) < ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ22_HTML.gif
        (22)
        for all n n ε http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq144_HTML.gif and k N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq145_HTML.gif. Consequently,
        lim n , m G p ( x n , x m , x m ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ23_HTML.gif
        (23)
        and thus { x n } n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq146_HTML.gif is a Cauchy sequence in the G p http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq147_HTML.gif-complete G p http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq147_HTML.gif-metric space ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif. Hence, there is z X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq67_HTML.gif such that
        0 = lim n , m G p ( x n , x m , x m ) = lim n G p ( z , x n , x n ) = G p ( z , z , z ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ24_HTML.gif
        (24)
        We show that z is a fixed point of T. Assume the contrary. Then G p ( z , T z , T z ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq148_HTML.gif. For each n N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq149_HTML.gif, we have
        G p ( z , T z , T z ) G p ( z , x n , x n ) + G p ( x n , T z , T z ) G p ( z , x n , x n ) + ϕ ( N ( z , z , x n 1 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ25_HTML.gif
        (25)
        From our assumption that G p ( z , T z , T z ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq148_HTML.gif, it easily follows that there is n 0 N http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq150_HTML.gif such that N ( z , z , x n 1 ) = G p ( z , T z , T z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq151_HTML.gif for all n n 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq152_HTML.gif. So,
        G p ( z , T z , T z ) G p ( z , x n , x n ) + ϕ ( G p ( z , T z , T z ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ26_HTML.gif
        (26)
        for all n n 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq152_HTML.gif. Taking limits as n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq34_HTML.gif, we obtain that
        G p ( z , T z , T z ) ϕ ( G p ( z , T z , T z ) ) < G p ( z , T z , T z ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ27_HTML.gif
        (27)

        a contradiction. Consequently, z = T z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq110_HTML.gif. Finally, the uniqueness of z follows as in Theorem 2.1. □

        Example 2.3 Let X = [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq153_HTML.gif, G p : X × X × X R http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq154_HTML.gif be defined by G p ( x , y , z ) = max { x , y , z } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq120_HTML.gif. Then ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif is a GP-complete GP-metric space. Let T : X X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq128_HTML.gif be defined by
        T x = ( x 2 x 4 ) / 8 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equn_HTML.gif

        and ψ ( t ) = 1 4 t 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq155_HTML.gif for all t [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq130_HTML.gif.

        Proof Without loss of generality, we assume that x y http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq125_HTML.gif. Then
        N ( x , y , y ) = max { G p ( x , y , y ) , G p ( x , T x , T x ) , G p ( y , T y , T y ) } = x . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equo_HTML.gif
        And so
        G p ( T x , T y , T y ) = max { ( x 2 x 4 ) / 8 , ( y 2 y 4 ) / 8 } = max { ( x x 2 ) ( x + x 2 ) 8 , ( y y 2 ) ( y + y 2 ) 8 } max { x ( x x 2 ) 4 , y ( y y 2 ) 4 } 1 4 max { x 2 , y 2 } = 1 4 x 2 = ψ ( N ( x , y , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equp_HTML.gif

        Then the condition of Theorem 2.2 holds and T has a unique fixed point 0 in [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq156_HTML.gif. Moreover, G p ( 0 , 0 , 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq127_HTML.gif. □

        Example 2.4 Let X = [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq17_HTML.gif, G p : X × X × X R http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq119_HTML.gif be defined by G p ( x , y , z ) = max { x , y , z } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq120_HTML.gif. Then ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif is a GP-complete GP-metric space. Let T : X X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq128_HTML.gif be defined by
        T x = { 1 4 ( x 2 + x ) if  0 x < 1 , 1 8 x if  x 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equq_HTML.gif

        and ψ ( t ) = 1 2 t http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq129_HTML.gif for all t [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq130_HTML.gif.

        Proof To prove this example, we need to examine the following cases:

        • Let 0 x , y < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq157_HTML.gif. Then N ( x , y , y ) = max { x , y } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq158_HTML.gif and
          G p ( T x , T y , T y ) = max { 1 4 ( x 2 + x ) , 1 4 ( y 2 + y ) } 1 2 max { x , y } = ψ ( N ( x , y , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equr_HTML.gif
        • Let x , y 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq159_HTML.gif. Then N ( x , y , y ) = max { x , y } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq158_HTML.gif and
          G p ( T x , T y , T y ) = 1 8 max { x , y } 1 2 max { x , y } = ψ ( N ( x , y , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equs_HTML.gif
        • Let 0 x < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq160_HTML.gif and y 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq161_HTML.gif. Then N ( x , y , y ) = y http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq162_HTML.gif and
          G p ( T x , T y , T y ) = max { 1 4 ( x 2 + x ) , 1 8 y } 1 2 max { x , y } = 1 2 y = ψ ( N ( x , y , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equt_HTML.gif

        Then the condition of Theorem 2.2 holds and T has a unique fixed point 0 in [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq156_HTML.gif. Moreover, G p ( 0 , 0 , 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq127_HTML.gif. □

        Corollary 2.2 Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif be a GP-complete GP-metric space and T : X X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq66_HTML.gif be a map such that
        G p ( T x , T y , T z ) ϕ ( N ( x , y , z ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equ28_HTML.gif
        (28)

        where N ( x , y , y ) = max { G p ( x , y , z ) , G p ( x , T x , T x ) , G p ( y , T y , T y ) , G p ( z , T z , T z ) } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq163_HTML.gif for all x , y X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq6_HTML.gif, and ϕ : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq49_HTML.gif is a nondecreasing function such that lim n ϕ n ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq164_HTML.gif for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq51_HTML.gif. Then T has a unique fixed point z X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq67_HTML.gif. Moreover, G p ( z , z , z ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq68_HTML.gif.

        Similarly, we have the corollary below.

        Corollary 2.3 Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif be a GP-complete GP-metric space and T : X X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq66_HTML.gif be a map such that
        0 [ ψ ( z , z ) ψ ( y , y ) ] max { G p ( x , x , x ) , G p ( y , y , y ) , G p ( z , z , z ) } + ψ ( x , y ) ψ ( x , z ) [ ϕ ( R ( x , y , z ) ) G p ( T x , T y , T z ) ] , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equu_HTML.gif

        where R ( x , y , z ) = max { G p ( x , y , z ) , G p ( x , T x , T x ) , G p ( y , T y , T y ) , G p ( z , T z , T z ) } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq165_HTML.gif for all x , y , z X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq117_HTML.gif, and ϕ : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq49_HTML.gif is a nondecreasing function such that lim n ϕ n ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq164_HTML.gif for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq51_HTML.gif. Then T has a unique fixed point z X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq67_HTML.gif. Moreover, G p ( z , z , z ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq68_HTML.gif.

        In [46], Jachymski proved the equivalence of auxiliary functions (see Lemma 1). Inspired by the results from this remarkable paper of Jachymski, we finish this paper by stating the following theorem.

        Theorem 2.3 (See [46])

        Let ( X , G p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq20_HTML.gif be a GP-complete GP-metric space and T : X X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq66_HTML.gif be a self-mapping. Assume that
        M ( x , y , y ) = max { G p ( x , y , y ) , G p ( x , T x , T x ) , G p ( y , T y , T y ) , 1 2 [ G p ( x , T y , T y ) + G p ( y , T x , T x ) ] } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equv_HTML.gif
        Then the following statements are equivalent:
        1. (i)
          there exist functions ψ , η Ψ http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq166_HTML.gif such that
          ψ ( G ( T x , T y , T y ) ) ψ ( M ( x , y , y ) ) η ( M ( x , y , y ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equw_HTML.gif
           
        for any x , y X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq47_HTML.gif;
        1. (ii)
          there exists a function β : [ 0 , ) [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq167_HTML.gif such that for any bounded sequence { t n } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq168_HTML.gif of positive reals, β ( t n ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq169_HTML.gif implies t n 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq170_HTML.gif and
          G ( T x , T y , T y ) β ( M ( x , y , y ) ) ψ ( M ( x , y , y ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equx_HTML.gif
           
        for any x , y X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq47_HTML.gif;
        1. (iii)
          there exists a continuous function η : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq171_HTML.gif such that η 1 ( { 0 } ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq172_HTML.gif and
          G ( T x , T y , T y ) M ( x , y , y ) η ( M ( x , y , y ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equy_HTML.gif
           
        for any x , y X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq47_HTML.gif;
        1. (iv)
          there exist a function ψ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq173_HTML.gif and a nondecreasing, right-continuous function φ : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq174_HTML.gif with φ ( t ) < t http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq175_HTML.gif and for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq51_HTML.gif with
          ψ ( G ( T x , T y , T y ) ) φ ( ψ ( M ( x , y , y ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equz_HTML.gif
           
        for any x , y X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq47_HTML.gif;
        1. (v)
          there exists a continuous and nondecreasing function φ : [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq176_HTML.gif such that φ ( t ) < t http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq175_HTML.gif and for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq51_HTML.gif with
          ψ ( G ( T x , T y , T y ) ) φ ( M ( x , y , y ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_Equaa_HTML.gif
           

        for any x , y X http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-39/MediaObjects/13660_2012_Article_492_IEq47_HTML.gif.

        Declarations

        Authors’ Affiliations

        (1)
        Department of Mathematics, Institute of Science and Technology, Gazi University
        (2)
        Department of Mathematics, Atilim University
        (3)
        Department of Mathematics, Sahand University of Technology

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