Open Access

Cyclic contractions via auxiliary functions on G-metric spaces

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, G : X × X × X R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq1_HTML.gif be a function satisfying the following properties:
  1. (G1)

    G ( x , y , z ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq2_HTML.gif if x = y = z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq3_HTML.gif,

     
  2. (G2)

    0 < G ( x , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq4_HTML.gif for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq5_HTML.gif with x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq6_HTML.gif,

     
  3. (G3)

    G ( x , x , y ) G ( x , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq7_HTML.gif for all x , y , z X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq8_HTML.gif with y z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq9_HTML.gif,

     
  4. (G4)

    G ( x , y , z ) = G ( x , z , y ) = G ( y , z , x ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq10_HTML.gif (symmetry in all three variables),

     
  5. (G5)

    G ( x , y , z ) G ( x , a , a ) + G ( a , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq11_HTML.gif (rectangle inequality) for all x , y , z , a X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq12_HTML.gif.

     

Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif is called a G-metric space.

Note that every G-metric on X induces a metric d G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq14_HTML.gif on X defined by
d G ( x , y ) = G ( x , y , y ) + G ( y , x , x ) for all  x , y X . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ1_HTML.gif
(1)

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

Example 1.1 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq15_HTML.gif be a metric space. The function G : X × X × X [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq16_HTML.gif, defined by
G ( x , y , z ) = max { d ( x , y ) , d ( y , z ) , d ( z , x ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equa_HTML.gif

for all x , y , z X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq17_HTML.gif, is a G-metric on X.

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

Let X = [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq18_HTML.gif. The function G : X × X × X [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq19_HTML.gif, defined by
G ( x , y , z ) = | x y | + | y z | + | z x | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equb_HTML.gif

for all x , y , z X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq20_HTML.gif be a G-metric space, and let { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif be a sequence of points of X. We say that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif is G-convergent to x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq22_HTML.gif if
lim n , m + G ( x , x n , x m ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equc_HTML.gif

that is, for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq23_HTML.gif, there exists N N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq24_HTML.gif such that G ( x , x n , x m ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq25_HTML.gif for all n , m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq26_HTML.gif. We call x the limit of the sequence and write x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq27_HTML.gif or lim n + x n = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq28_HTML.gif.

Proposition 1.1 (See [2])

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq20_HTML.gif be a G-metric space. The following are equivalent:
  1. (1)

    { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif is G-convergent to x,

     
  2. (2)

    G ( x n , x n , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq29_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq30_HTML.gif,

     
  3. (3)

    G ( x n , x , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq31_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq30_HTML.gif,

     
  4. (4)

    G ( x n , x m , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq32_HTML.gif as n , m + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq33_HTML.gif.

     

Definition 1.3 (See [2])

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq20_HTML.gif be a G-metric space. A sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif is called a G-Cauchy sequence if, for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq23_HTML.gif, there exists N N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq24_HTML.gif such that G ( x n , x m , x l ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq34_HTML.gif for all m , n , l N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq35_HTML.gif, that is, G ( x n , x m , x l ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq36_HTML.gif as n , m , l + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq37_HTML.gif.

Proposition 1.2 (See [2])

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq20_HTML.gif be a G-metric space. Then the following are equivalent:
  1. (1)

    the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif is G-Cauchy,

     
  2. (2)

    for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq23_HTML.gif, there exists N N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq24_HTML.gif such that G ( x n , x m , x m ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq38_HTML.gif for all m , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq39_HTML.gif.

     

Definition 1.4 (See [2])

A G-metric space ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif is called G-complete if every G-Cauchy sequence is G-convergent in ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif.

Definition 1.5 Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif be a G-metric space. A mapping F : X × X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq40_HTML.gif is said to be continuous if for any three G-convergent sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif, { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq41_HTML.gif and { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq42_HTML.gif converging to x, y and z respectively, { F ( x n , y n , z n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq43_HTML.gif is G-convergent to F ( x , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq44_HTML.gif.

Note that each G-metric on X generates a topology τ G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq45_HTML.gif on X whose base is a family of open G-balls { B G ( x , ε ) , x X , ε > 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq46_HTML.gif, where B G ( x , ε ) = { y X , G ( x , y , y ) < ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq47_HTML.gif for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq22_HTML.gif and ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq23_HTML.gif. A nonempty set A X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq48_HTML.gif is G-closed in the G-metric space ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif if A ¯ = A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq49_HTML.gif. Observe that
x A ¯ B G ( x , ε ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equd_HTML.gif

for all ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq23_HTML.gif. We recall also the following proposition.

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

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif in A with limit x, we have x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif be a complete G-metric space and T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq51_HTML.gif be a mapping satisfying the following condition for all x , y , z X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq52_HTML.gif:
G ( T x , T y , T z ) k G ( x , y , z ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ2_HTML.gif
(2)

where k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq53_HTML.gif. Then T has a unique fixed point.

Theorem 1.2 (See [5])

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif be a complete G-metric space and T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq51_HTML.gif be a mapping satisfying the following condition for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq54_HTML.gif:
G ( T x , T y , T y ) k G ( x , y , y ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ3_HTML.gif
(3)

where k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 k [ 0 , 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq55_HTML.gif. For details, see [5].

Lemma 1.1 ([5])

By the rectangle inequality (G5) together with the symmetry (G4), we have
G ( x , y , y ) = G ( y , y , x ) G ( y , x , x ) + G ( x , y , x ) = 2 G ( y , x , x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ4_HTML.gif
(4)
A map T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq56_HTML.gif on a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq15_HTML.gif is called a weak ϕ-contraction if there exists a strictly increasing function ϕ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq57_HTML.gif with ϕ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq58_HTML.gif such that
d ( T x , T y ) d ( x , y ) ϕ ( d ( x , y ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Eque_HTML.gif

for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 ϕ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq57_HTML.gif with ϕ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq59_HTML.gif and ϕ ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq60_HTML.gif for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif be a G-complete G-metric space and { A j } j = 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with Y = j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq63_HTML.gif. Let T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq64_HTML.gif be a map satisfying
T ( A j ) A j + 1 , j = 1 , , m , where A m + 1 = A 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ5_HTML.gif
(5)
Suppose that there exists a function ϕ Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq65_HTML.gif such that the map T satisfies the inequality
G ( T x , T y , T z ) M ( x , y , z ) ϕ ( M ( x , y , z ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ6_HTML.gif
(6)
for all x A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq66_HTML.gif and y , z A j + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq67_HTML.gif, j = 1 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq68_HTML.gif, where
M ( x , y , z ) = max { G ( x , y , z ) , G ( x , T x , T x ) , G ( y , T y , T y ) , G ( z , T z , T z ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ7_HTML.gif
(7)

Then T has a unique fixed point in j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif be a G-complete G-metric space and { A j } j = 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X. Let Y = j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq70_HTML.gif and T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq64_HTML.gif be a map satisfying
T ( A j ) A j + 1 , j = 1 , , m , where A m + 1 = A 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ8_HTML.gif
(8)
If there exists k ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq71_HTML.gif such that
G ( T x , T y , T z ) k G ( x , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ9_HTML.gif
(9)

holds for all x A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq66_HTML.gif and y , z A j + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq67_HTML.gif, j = 1 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq68_HTML.gif, then T has a unique fixed point in j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 denote all functions f : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq72_HTML.gif such that f ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq73_HTML.gif if and only if t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq74_HTML.gif. Let Ψ and Φ be the subsets of such that
Ψ = { ψ F : ψ  is continuous and nondecreasing } , Φ = { ϕ F : ϕ  is lower semi-continuous } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equf_HTML.gif
Lemma 2.1 Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif be a G-complete G-metric space and { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif be a sequence in X such that G ( x n , x n + 1 , x n + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq75_HTML.gif is nonincreasing,
lim n G ( x n , x n + 1 , x n + 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ10_HTML.gif
(10)
If { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif is not a Cauchy sequence, then there exist ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq23_HTML.gif and two sequences { n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq76_HTML.gif and { k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq77_HTML.gif of positive integers such that the following sequences tend to ε when k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq78_HTML.gif:
G ( x ( k ) , x n ( k ) , x n ( k ) ) , G ( x ( k ) , x n ( k ) + 1 , x n ( k ) + 1 ) , G ( x ( k ) 1 , x n ( k ) , x n ( k ) ) , G ( x ( k ) 1 , x n ( k ) + 1 , x n ( k ) + 1 ) , G ( x n ( k ) , x ( k ) , x ( k ) + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ11_HTML.gif
(11)

Proof

Due to Lemma 1.1, we have
G ( x n , x n + 1 , x n + 1 ) 2 G ( x n , x n + 1 , x n + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equg_HTML.gif
Letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq79_HTML.gif regarding the assumption of the lemma, we derive that
lim n G ( x n , x n , x n + 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ12_HTML.gif
(12)
If { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif is not G-Cauchy, then, due to Proposition 1.2, there exist ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq23_HTML.gif and corresponding subsequences { n ( k ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq80_HTML.gif and { ( k ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq81_HTML.gif of satisfying n ( k ) > ( k ) > k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq82_HTML.gif for which
G ( x ( k ) , x n ( k ) , x n ( k ) ) ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ13_HTML.gif
(13)
where n ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq83_HTML.gif is chosen as the smallest integer satisfying (13), that is,
G ( x ( k ) , x n ( k ) 1 , x n ( k ) 1 ) < ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ14_HTML.gif
(14)
By (13), (14) and the rectangle inequality (G5), we easily derive that
ε G ( x ( k ) , x n ( k ) , x n ( k ) ) G ( x ( k ) , x n ( k ) 1 , x n ( k ) 1 ) + G ( x n ( k ) 1 , x n ( k ) , x n ( k ) ) < ε + G ( x n ( k ) 1 , x n ( k ) , x n ( k ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ15_HTML.gif
(15)
Letting k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq84_HTML.gif in (15) and using (10), we get
lim k G ( x ( k ) , x n ( k ) , x n ( k ) ) = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ16_HTML.gif
(16)
Further,
G ( x ( k ) , x n ( k ) , x n ( k ) ) G ( x ( k ) , x n ( k ) + 1 , x n ( k ) + 1 ) + G ( x n ( k ) + 1 , x n ( k ) , x n ( k ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ17_HTML.gif
(17)
and
G ( x ( k ) , x n ( k ) + 1 , x n ( k ) + 1 ) G ( x ( k ) , x n ( k ) , x n ( k ) ) + G ( x n ( k ) , x n ( k ) + 1 , x n ( k ) + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ18_HTML.gif
(18)
Passing to the limit when k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq84_HTML.gif and using (10) and (16), we obtain that
lim k G ( x ( k ) , x n ( k ) + 1 , x n ( k ) + 1 ) = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ19_HTML.gif
(19)
In a similar way,
G ( x ( k ) 1 , x n ( k ) , x n ( k ) ) G ( x ( k ) 1 , x ( k ) , x ( k ) ) + G ( x ( k ) , x n ( k ) , x n ( k ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ20_HTML.gif
(20)
and
G ( x ( k ) , x n ( k ) , x n ( k ) ) G ( x ( k ) , x ( k ) 1 , x ( k ) 1 ) + G ( x ( k ) 1 , x n ( k ) , x n ( k ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ21_HTML.gif
(21)
Passing to the limit when k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq84_HTML.gif and using (10) and (16), we obtain that
lim k G ( x ( k ) 1 , x n ( k ) , x n ( k ) ) = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ22_HTML.gif
(22)
Furthermore,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ23_HTML.gif
(23)
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ24_HTML.gif
(24)
Passing to the limit when k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq84_HTML.gif and using (10) and (16), we obtain that
lim k G ( x ( k ) 1 , x n ( k ) + 1 , x n ( k ) + 1 ) = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ25_HTML.gif
(25)
By regarding the assumptions (G3) and (G5) together with the expression (13), we derive the following:
ε G ( x ( k ) , x n ( k ) , x n ( k ) ) G ( x n ( k ) , x ( k ) , x ( k ) + 1 ) G ( x n ( k ) , x ( k ) , x ( k ) ) + G ( x ( k ) , x ( k ) , x ( k ) + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ26_HTML.gif
(26)
Letting k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq84_HTML.gif in the inequality above and using (12) and (16), we conclude that
lim k G ( x n ( k ) , x ( k ) , x ( k ) + 1 ) = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ27_HTML.gif
(27)

 □

Theorem 2.1 Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif be a G-complete G-metric space and { A j } j = 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with Y = j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq85_HTML.gif. Let T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq86_HTML.gif be a map satisfying
T ( A j ) A j + 1 , j = 1 , 2 , , m , where A m + 1 = A 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ28_HTML.gif
(28)
Suppose that there exist functions ϕ Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq87_HTML.gif and ψ Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq88_HTML.gif such that the map T satisfies the inequality
ψ ( G ( T x , T y , T y ) ) ψ ( M ( x , y , y ) ) ϕ ( M ( x , y , y ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ29_HTML.gif
(29)
for all x A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq89_HTML.gif and y A j + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq90_HTML.gif, j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq91_HTML.gif, where
M ( x , y , y ) = max { G ( x , y , y ) , G ( x , T x , T x ) , G ( y , T y , T y ) , G ( x , y , T x ) , 1 3 [ 2 G ( x , T y , T y ) + G ( y , T x , T x ) ] , 1 3 [ G ( x , T y , T y ) + 2 G ( y , T x , T x ) ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ30_HTML.gif
(30)

Then T has a unique fixed point in j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 x 0 A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq92_HTML.gif and define a sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif in the following way:
x n = T x n 1 , n = 1 , 2 , 3 , . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ31_HTML.gif
(31)
We have x 0 A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq92_HTML.gif, x 1 = T x 0 A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq93_HTML.gif, x 2 = T x 1 A 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq94_HTML.gif, … since T is a cyclic mapping. If x n 0 + 1 = x n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq95_HTML.gif for some n 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq96_HTML.gif, then, clearly, the fixed point of the map T is x n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq97_HTML.gif. From now on, assume that x n + 1 x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq98_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq99_HTML.gif. Consider the inequality (29) by letting x = x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq100_HTML.gif and y = x n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq101_HTML.gif,
ψ ( G ( T x n , T x n + 1 , T x n + 1 ) ) = ψ ( G ( x n + 1 , x n + 2 , x n + 2 ) ) ψ ( M ( x n , x n + 1 , x n + 1 ) ) ϕ ( M ( x n , x n + 1 , x n + 1 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ32_HTML.gif
(32)
where
M ( x n , x n + 1 , x n + 1 ) = max { G ( x n , x n + 1 , x n + 1 ) , G ( x n , T x n , T x n ) , G ( x n + 1 , T x n + 1 , T x n + 1 ) , G ( x n , x n + 1 , T x n ) , 1 3 [ 2 G ( x n , T x n + 1 , T x n + 1 ) + G ( x n + 1 , T x n , T x n ) ] , 1 3 [ G ( x n , T x n + 1 , T x n + 1 ) + 2 G ( x n + 1 , T x n , T x n ) ] } = max { G ( x n , x n + 1 , x n + 1 ) , G ( x n , x n + 1 , x n + 1 ) , G ( x n + 1 , x n + 2 , x n + 2 ) , G ( x n , x n + 1 , x n + 1 ) , 1 3 [ 2 G ( x n , x n + 2 , x n + 2 ) + G ( x n + 1 , x n + 1 , x n + 1 ) ] , 1 3 [ G ( x n , x n + 2 , x n + 2 ) + 2 G ( x n + 1 , x n + 1 , x n + 1 ) ] } = max { G ( x n , x n + 1 , x n + 1 ) , G ( x n + 1 , x n + 2 , x n + 2 ) , 2 3 G ( x n , x n + 2 , x n + 2 ) } max { G ( x n , x n + 1 , x n + 1 ) , G ( x n + 1 , x n + 2 , x n + 2 ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ33_HTML.gif
(33)
If M ( x n , x n + 1 , x n + 1 ) = G ( x n + 1 , x n + 2 , x n + 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq102_HTML.gif, then the expression (32) implies that
ψ ( G ( x n + 1 , x n + 2 , x n + 2 ) ) ψ ( G ( x n + 1 , x n + 2 , x n + 2 ) ) ϕ ( G ( x n + 1 , x n + 2 , x n + 2 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ34_HTML.gif
(34)
So, the inequality (34) yields ϕ ( G ( x n + 1 , x n + 2 , x n + 2 ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq103_HTML.gif. Thus, we conclude that
G ( x n + 1 , x n + 2 , x n + 2 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equh_HTML.gif
This contradicts the assumption x n x n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq104_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq99_HTML.gif. So, we derive that
M ( x n , x n + 1 , x n + 1 ) = G ( x n , x n + 1 , x n + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ35_HTML.gif
(35)
Hence the inequality (32) turns into
ψ ( G ( x n + 1 , x n + 2 , x n + 2 ) ) ψ ( G ( x n , x n + 1 , x n + 1 ) ) ϕ ( G ( x n , x n + 1 , x n + 1 ) ) ψ ( G ( x n , x n + 1 , x n + 1 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ36_HTML.gif
(36)
Thus, { G ( x n , x n + 1 , x n + 1 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq105_HTML.gif is a nonnegative, nonincreasing sequence that converges to L 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq106_HTML.gif. We will show that L = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq107_HTML.gif. Suppose, on the contrary, that L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq108_HTML.gif. Taking lim sup n + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq109_HTML.gif in (36), we derive that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ37_HTML.gif
(37)
By the continuity of ψ and the lower semi-continuity of ϕ, we get
ψ ( L ) ψ ( L ) ϕ ( L ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ38_HTML.gif
(38)
Then it follows that ϕ ( L ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq110_HTML.gif. Therefore, we get L = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq107_HTML.gif, that is,
lim n G ( x n , x n + 1 , x n + 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ39_HTML.gif
(39)
Lemma 1.1 with x = x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq100_HTML.gif and y = x n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq111_HTML.gif implies that
G ( x n , x n 1 , x n 1 ) 2 G ( x n 1 , x n , x n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ40_HTML.gif
(40)
So, we get that
lim n G ( x n , x n 1 , x n 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ41_HTML.gif
(41)
Next, we will show that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif is a G-Cauchy sequence in ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif. Suppose, on the contrary, that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif is not G-Cauchy. Then, due to Proposition 1.2, there exist ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq23_HTML.gif and corresponding subsequences { n ( k ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq80_HTML.gif and { ( k ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq81_HTML.gif of satisfying n ( k ) > ( k ) > k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq82_HTML.gif for which
G ( x ( k ) , x n ( k ) , x n ( k ) ) ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ42_HTML.gif
(42)
where n ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq83_HTML.gif is chosen as the smallest integer satisfying (42), that is,
G ( x ( k ) , x n ( k ) 1 , x n ( k ) 1 ) < ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ43_HTML.gif
(43)
By (42), (43) and the rectangle inequality (G5), we easily derive that
ε G ( x ( k ) , x n ( k ) , x n ( k ) ) G ( x ( k ) , x n ( k ) 1 , x n ( k ) 1 ) + G ( x n ( k ) 1 , x n ( k ) , x n ( k ) ) < ε + G ( x n ( k ) 1 , x n ( k ) , x n ( k ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ44_HTML.gif
(44)
Letting k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq84_HTML.gif in (44) and using (39), we get
lim k G ( x ( k ) , x n ( k ) , x n ( k ) ) = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ45_HTML.gif
(45)
Notice that for every k N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq112_HTML.gif there exists s ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq113_HTML.gif satisfying 0 s ( k ) m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq114_HTML.gif such that
n ( k ) ( k ) + s ( k ) 1 ( m ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ46_HTML.gif
(46)
Thus, for large enough values of k, we have r ( k ) = ( k ) s ( k ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq115_HTML.gif, and x r ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq116_HTML.gif and x n ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq117_HTML.gif lie in the adjacent sets A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq118_HTML.gif and A j + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq119_HTML.gif respectively for some 0 j m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq120_HTML.gif. When we substitute x = x r ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq121_HTML.gif and y = x n ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq122_HTML.gif in the expression (29), we get that
ψ ( G ( T x r ( k ) , T x n ( k ) , T x n ( k ) ) ) ψ ( M ( x r ( k ) , x n ( k ) , x n ( k ) ) ) ϕ ( M ( x r ( k ) , x n ( k ) , x n ( k ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ47_HTML.gif
(47)
where
M ( x r ( k ) , x n ( k ) , x n ( k ) ) = max { G ( x r ( k ) , x n ( k ) , x n ( k ) ) , G ( x r ( k ) , x r ( k ) + 1 , x r ( k ) + 1 ) , G ( x n ( k ) , x n ( k ) + 1 , x n ( k ) + 1 ) , G ( x r ( k ) , x n ( k ) , x r ( k ) + 1 ) , 1 3 [ 2 G ( x r ( k ) , x n ( k ) + 1 , x n ( k ) + 1 ) + G ( x n ( k ) , x r ( k ) + 1 , x r ( k ) + 1 ) ] , 1 3 [ G ( x r ( k ) , x n ( k ) + 1 , x n ( k ) + 1 ) + 2 G ( x n ( k ) , x r ( k ) + 1 , x r ( k ) + 1 ) ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ48_HTML.gif
(48)
By using Lemma 2.1, we obtain that
lim k 1 3 [ 2 G ( x r ( k ) , x n ( k ) + 1 , x n ( k ) + 1 ) + G ( x n ( k ) , x r ( k ) + 1 , x r ( k ) + 1 ) ] = ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ49_HTML.gif
(49)
and
lim k 1 3 [ G ( x r ( k ) , x n ( k ) + 1 , x n ( k ) + 1 ) + 2 G ( x n ( k ) , x r ( k ) + 1 , x r ( k ) + 1 ) ] = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ50_HTML.gif
(50)
So, we obtain that
ψ ( ε ) ψ ( max { ε , 0 , 0 , ε , ε , ε } ) ϕ ( max { ε , 0 , 0 , ε , ε , ε } ) = ψ ( ε ) ϕ ( ε ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ51_HTML.gif
(51)
So, we have ϕ ( ε ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq123_HTML.gif. We deduce that ε = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq124_HTML.gif. This contradicts the assumption that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq125_HTML.gif is not G-Cauchy. As a result, the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq21_HTML.gif is G-Cauchy. Since ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif is G-complete, it is G-convergent to a limit, say w X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq126_HTML.gif. It easy to see that w j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq127_HTML.gif. Since x 0 A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq92_HTML.gif, then the subsequence { x m ( n 1 ) } n = 1 A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq128_HTML.gif, the subsequence { x m ( n 1 ) + 1 } n = 1 A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq129_HTML.gif and, continuing in this way, the subsequence { x m ( n 1 ) } n = 1 A m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq130_HTML.gif. All the m subsequences are G-convergent in the G-closed sets A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq118_HTML.gif and hence they all converge to the same limit w j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq127_HTML.gif. To show that the limit w is the fixed point of T, that is, w = T w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq131_HTML.gif, we employ (29) with x = x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq100_HTML.gif, y = w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq132_HTML.gif. This leads to
ψ ( G ( T x n , T w , T w ) ) ψ ( M ( x n , w , w ) ) ϕ ( M ( x n , w , w ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ52_HTML.gif
(52)
where
M ( x n , w , w ) = max { G ( x n , w , w ) , G ( x n , x n + 1 , x n + 1 ) , G ( w , T w , T w ) , G ( x n , w , x n + 1 ) , 1 3 [ 2 G ( x n , T w , T w ) + G ( w , x n + 1 , x n + 1 ) ] , 1 3 [ G ( x n , T w , T w ) + 2 G ( w , x n + 1 , x n + 1 ) ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ53_HTML.gif
(53)
Passing to limsup as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq133_HTML.gif, we get
ψ ( G ( w , T w , T w ) ) ψ ( G ( w , T w , T w ) ) ϕ ( G ( w , T w , T w ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ54_HTML.gif
(54)

Thus, ϕ ( G ( w , T w , T w ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq134_HTML.gif and hence G ( w , T w , T w ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq135_HTML.gif, that is, w = T w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq131_HTML.gif.

Finally, we prove that the fixed point is unique. Assume that v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq136_HTML.gif is another fixed point of T such that v w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq137_HTML.gif. Then, since both v and w belong to j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq69_HTML.gif, we set x = v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq138_HTML.gif and y = w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq132_HTML.gif in (29), which yields
ψ ( G ( T v , T w , T w ) ) ψ ( M ( v , w , w ) ) ϕ ( ( M ( v , w , w ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ55_HTML.gif
(55)
where
M ( v , w , w ) = max { G ( v , w , w ) , G ( v , T v , T v ) , G ( w , T w , T w ) , 1 3 [ 2 G ( v , T w , T w ) + G ( w , T v , T v ) ] , 1 3 [ G ( v , T w , T w ) + 2 G ( w , T v , T v ) ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ56_HTML.gif
(56)
On the other hand, by setting x = w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq139_HTML.gif and y = v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq140_HTML.gif in (29), we obtain that
ψ ( G ( T w , T v , T v ) ) ψ ( M ( w , v , v ) ) ϕ ( ( M ( w , v , v ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ57_HTML.gif
(57)
where
M ( w , v , v ) = max { G ( w , v , v ) , G ( w , T w , T w ) , G ( v , T v , T v ) , G ( w , v , T w ) , 1 3 [ 2 G ( w , T v , T v ) + G ( v , T w , T w ) ] , 1 3 [ G ( w , T v , T v ) + 2 G ( v , T w , T w ) ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ58_HTML.gif
(58)
If G ( v , w , w ) = G ( w , v , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq141_HTML.gif, then v = w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq142_HTML.gif. Indeed, by definition, we get that d G ( v , w ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq143_HTML.gif. Hence v = w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq142_HTML.gif. If G ( v , w , w ) > G ( w , v , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq144_HTML.gif, then by (56) M ( v , w , w ) = G ( v , w , w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq145_HTML.gif and by (55),
ψ ( G ( v , w , w ) ) ψ ( G ( v , w , w ) ) ϕ ( ( G ( v , w , w ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ59_HTML.gif
(59)
and, clearly, G ( v , w , w ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq146_HTML.gif. So, we conclude that v = w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq142_HTML.gif. Otherwise, G ( w , v , v ) > G ( v , w , w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq147_HTML.gif. Then by (58), M ( w , v , v ) = G ( w , v , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq148_HTML.gif and by (57),
ψ ( G ( w , v , v ) ) ψ ( G ( w , v , v ) ) ϕ ( ( G ( w , v , v ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ60_HTML.gif
(60)

and, clearly, G ( w , v , v ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq149_HTML.gif. So, we conclude that v = w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 X = [ 1 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq150_HTML.gif and let T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq151_HTML.gif be given as T x = x 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq152_HTML.gif. Let A = [ 1 , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq153_HTML.gif and B = [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq154_HTML.gif. Define the function G : X × X × X [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq155_HTML.gif as
G ( x , y , z ) = | x y | + | y z | + | z x | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ61_HTML.gif
(61)

Clearly, the function G is a G-metric on X. Define also ϕ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq156_HTML.gif as ϕ ( t ) = t 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq157_HTML.gif and ψ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq158_HTML.gif as ψ = t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq159_HTML.gif. Obviously, the map T has a unique fixed point x = 0 A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq160_HTML.gif.

It can be easily shown that the map T satisfies the condition (29). Indeed,
G ( T x , T y , T y ) = | T x T y | + | T y T y | + | T y T x | = 2 | T x T y | = | y x | 4 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equi_HTML.gif
which yields
ψ ( G ( T x , T y , T y ) ) = | y x | 8 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ62_HTML.gif
(62)
Moreover, we have
M ( x , y , y ) = max { | x y | + | y y | + | y x | , | x T x | + | T x T x | + | T x x | , | y T y | + | T y T y | + | T y y | , | x y | + | T x y | + | T x x | , 1 3 [ 2 ( | x T y | + | T y T y | + | T y x | ) + | y T x | + | T x T x | + | T x y | ] , 1 3 [ | x T y | + | T y T y | + | T y x | + 2 ( | y T x | + | T x T x | + | T x y | ) ] } = max { 2 | x y | , 2 | T x x | , 2 | T y y | , 1 3 [ 4 | T y x | + 2 | T x y | ] , 1 3 [ 2 | T y x | + 4 | T x y | ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ63_HTML.gif
(63)
We derive from (63) that
2 | x y | M ( x , y , y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ64_HTML.gif
(64)
On the other hand, we have the following inequality:
ψ ( M ( x , y , y ) ) ϕ ( M ( x , y , y ) ) = M ( x , y , y ) 2 M ( x , y , y ) 8 = 3 M ( x , y , y ) 8 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ65_HTML.gif
(65)
By elementary calculation, we conclude from (65) and (64) that
3 | x y | 4 3 M ( x , y , y ) 8 = ψ ( M ( x , y , y ) ) ϕ ( M ( x , y , y ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ66_HTML.gif
(66)
Combining the expressions (62) and (65), we obtain that
ψ ( G ( T x , T y , T y ) ) = | y x | 8 3 | x y | 4 3 M ( x , y , y ) 8 = ψ ( M ( x , y , y ) ) ϕ ( M ( x , y , y ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif be a G-complete G-metric space and { A j } j = 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with Y = j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq85_HTML.gif. Let T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq86_HTML.gif be a map satisfying
T ( A j ) A j + 1 , j = 1 , 2 , , m , where A m + 1 = A 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ68_HTML.gif
(68)
Suppose that there exists a constant k ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq161_HTML.gif such that the map T satisfies
G ( T x , T y , T y ) k M ( x , y , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ69_HTML.gif
(69)
for all x A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq89_HTML.gif and y A j + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq90_HTML.gif, j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq91_HTML.gif, where
M ( x , y , y ) = max { G ( x , y , y ) , G ( x , T x , T x ) , G ( y , T y , T y ) , 1 3 [ 2 G ( x , T y , T y ) + G ( y , T x , T x ) ] , 1 3 [ G ( x , T y , T y ) + 2 G ( y , T x , T x ) ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ70_HTML.gif
(70)

Then T has a unique fixed point in j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq69_HTML.gif.

Proof The proof is obvious by choosing the functions ϕ, ψ in Theorem 2.1 as ϕ ( t ) = ( 1 k ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq162_HTML.gif and ψ ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq163_HTML.gif. □

Corollary 2.2 Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif be a G-complete G-metric space and { A j } j = 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with Y = j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq85_HTML.gif. Let T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq86_HTML.gif be a map satisfying
T ( A j ) A j + 1 , j = 1 , 2 , , m , where A m + 1 = A 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ71_HTML.gif
(71)
Suppose that there exist constants a, b, c, d and e with 0 < a + b + c + d + e < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq164_HTML.gif and there exists a function ψ Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq88_HTML.gif such that the map T satisfies the inequality
ψ ( G ( T x , T y , T y ) ) a G ( x , y , y ) + b G ( x , T x , T x ) + c G ( y , T y , T y ) + d ( 1 3 [ 2 G ( x , T y , T y ) + G ( y , T x , T x ) ] ) + e ( 1 3 [ G ( x , T y , T y ) + 2 G ( y , T x , T x ) ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ72_HTML.gif
(72)

for all x A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq89_HTML.gif and y A j + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq90_HTML.gif, j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq165_HTML.gif. Then T has a unique fixed point in j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq69_HTML.gif.

Proof

Clearly, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ73_HTML.gif
(73)
where
M ( x , y , y ) = max { G ( x , y , y ) , G ( x , T x , T x ) , G ( y , T y , T y ) , 1 3 [ 2 G ( x , T y , T y ) + G ( y , T x , T x ) ] , 1 3 [ G ( x , T y , T y ) + 2 G ( y , T x , T x ) ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ74_HTML.gif
(74)

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

Corollary 2.3 Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif be a G-complete G-metric space and { A j } j = 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with Y = j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq85_HTML.gif. Let T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq86_HTML.gif be a map satisfying
T ( A j ) A j + 1 , j = 1 , 2 , , m , where A m + 1 = A 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equj_HTML.gif
Suppose that there exist functions ϕ Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq87_HTML.gif and ψ Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq88_HTML.gif such that the map T satisfies the inequality
ψ ( G ( T x , T y , T z ) ) ψ ( M ( x , y , z ) ) ϕ ( M ( x , y , z ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equk_HTML.gif
for all x A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq89_HTML.gif and y A j + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq90_HTML.gif, j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq91_HTML.gif, where
M ( x , y , z ) = max { G ( x , y , z ) , G ( x , T x , T x ) , G ( y , T y , T y ) , G ( z , T z , T z ) , 1 3 [ G ( x , T y , T y ) + G ( y , T x , T x ) + G ( z , T x , T x ) ] , 1 3 [ G ( x , T z , T z ) + G ( z , T x , T x ) + G ( y , T x , T x ) ] , 1 3 [ G ( y , T x , T x ) + G ( x , T y , T y ) + G ( z , T y , T y ) ] , 1 3 [ G ( y , T z , T z ) + G ( z , T y , T y ) + G ( x , T y , T y ) ] , 1 3 [ G ( z , T x , T x ) + G ( x , T z , T z ) + G ( y , T z , T z ) ] , 1 3 [ G ( z , T y , T y ) + G ( y , T z , T z ) + G ( x , T z , T z ) ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equ75_HTML.gif
(75)

Then T has a unique fixed point in j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 x = x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq100_HTML.gif and y = z = x n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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 ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif be a G-complete G-metric space and { A j } j = 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with Y = j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq85_HTML.gif. Let T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq86_HTML.gif be a map satisfying
T ( A j ) A j + 1 , j = 1 , 2 , , m , where A m + 1 = A 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equl_HTML.gif
Suppose also that there exist functions ϕ Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq87_HTML.gif and ψ Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq88_HTML.gif such that the map T satisfies
ψ ( 0 G ( T x , T y , T y ) d s ) ψ ( 0 M ( x , y , y ) d s ) ϕ ( 0 M ( x , y , y ) d s ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equm_HTML.gif
where
M ( x , y , y ) = max { G ( x , y , y ) , G ( x , T x , T x ) , G ( y , T y , T y ) , 1 3 [ 2 G ( x , T y , T y ) + G ( y , T x , T x ) ] , 1 3 [ G ( x , T y , T y ) + 2 G ( y , T x , T x ) ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equn_HTML.gif

for all x A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq89_HTML.gif and y A j + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq90_HTML.gif, j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq91_HTML.gif. Then T has a unique fixed point in j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq69_HTML.gif.

Corollary 2.5 Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq13_HTML.gif be a G-complete G-metric space and { A j } j = 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq62_HTML.gif be a family of nonempty G-closed subsets of X with Y = j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq85_HTML.gif. Let T : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq86_HTML.gif be a map satisfying
T ( A j ) A j + 1 , j = 1 , 2 , , m , where A m + 1 = A 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equo_HTML.gif
Suppose also that
0 G ( T x , T y , T y ) d s k 0 M ( x , y , y ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equp_HTML.gif
where k ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq161_HTML.gif and
M ( x , y , y ) = max { G ( x , y , y ) , G ( x , T x , T x ) , G ( y , T y , T y ) , 1 3 [ 2 G ( x , T y , T y ) + G ( y , T x , T x ) ] , 1 3 [ G ( x , T y , T y ) + 2 G ( y , T x , T x ) ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_Equq_HTML.gif

for all x A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq89_HTML.gif and y A j + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq90_HTML.gif, j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq91_HTML.gif. Then T has a unique fixed point in j = 1 m A j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq69_HTML.gif.

Proof The proof is obvious by choosing the function ϕ, ψ in Corollary 2.4 as ϕ ( t ) = ( 1 k ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_389_IEq162_HTML.gif and ψ ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-49/MediaObjects/13663_2012_Article_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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