Cyclic contractions via auxiliary functions on G-metric spaces

Open Access
Research

DOI: 10.1186/1687-1812-2013-49

Cite this article as:
Bilgili, N. & Karapınar, E. Fixed Point Theory Appl (2013) 2013: 49. doi:10.1186/1687-1812-2013-49

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 [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

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., [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]).

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 + Open image in new window be a function satisfying the following properties:
  1. (G1)

    G ( x , y , z ) = 0 Open image in new window if x = y = z Open image in new window,

     
  2. (G2)

    0 < G ( x , x , y ) Open image in new window for all x , y X Open image in new window with x y Open image in new window,

     
  3. (G3)

    G ( x , x , y ) G ( x , y , z ) Open image in new window for all x , y , z X Open image in new window with y z Open image in new window,

     
  4. (G4)

    G ( x , y , z ) = G ( x , z , y ) = G ( y , z , x ) = Open image in new window (symmetry in all three variables),

     
  5. (G5)

    G ( x , y , z ) G ( x , a , a ) + G ( a , y , z ) Open image in new window (rectangle inequality) for all x , y , z , a X Open image in new window.

     

Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair ( X , G ) Open image in new window is called a G-metric space.

Note that every G-metric on X induces a metric d G Open image in new window on X defined by
d G ( x , y ) = G ( x , y , y ) + G ( y , x , x ) for all  x , y X . Open image in new window
(1)

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

Example 1.1 Let ( X , d ) Open image in new window be a metric space. The function G : X × X × X [ 0 , + ) Open image in new window, defined by
G ( x , y , z ) = max { d ( x , y ) , d ( y , z ) , d ( z , x ) } Open image in new window

for all x , y , z X Open image in new window, is a G-metric on X.

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

Let X = [ 0 , ) Open image in new window. The function G : X × X × X [ 0 , + ) Open image in new window, defined by
G ( x , y , z ) = | x y | + | y z | + | z x | Open image in new window

for all x , y , z X Open image in new window, 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 ) Open image in new window be a G-metric space, and let { x n } Open image in new window be a sequence of points of X. We say that { x n } Open image in new window is G-convergent to x X Open image in new window if
lim n , m + G ( x , x n , x m ) = 0 , Open image in new window

that is, for any ε > 0 Open image in new window, there exists N N Open image in new window such that G ( x , x n , x m ) < ε Open image in new window for all n , m N Open image in new window. We call x the limit of the sequence and write x n x Open image in new window or lim n + x n = x Open image in new window.

Proposition 1.1 (See [2])

Let ( X , G ) Open image in new windowbe aG-metric space. The following are equivalent:
  1. (1)

    { x n } Open image in new windowisG-convergent tox,

     
  2. (2)

    G ( x n , x n , x ) 0 Open image in new windowas n + Open image in new window,

     
  3. (3)

    G ( x n , x , x ) 0 Open image in new windowas n + Open image in new window,

     
  4. (4)

    G ( x n , x m , x ) 0 Open image in new windowas n , m + Open image in new window.

     

Definition 1.3 (See [2])

Let ( X , G ) Open image in new window be a G-metric space. A sequence { x n } Open image in new window is called a G-Cauchy sequence if, for any ε > 0 Open image in new window, there exists N N Open image in new window such that G ( x n , x m , x l ) < ε Open image in new window for all m , n , l N Open image in new window, that is, G ( x n , x m , x l ) 0 Open image in new window as n , m , l + Open image in new window.

Proposition 1.2 (See [2])

Let ( X , G ) Open image in new windowbe aG-metric space. Then the following are equivalent:
  1. (1)

    the sequence { x n } Open image in new windowisG-Cauchy,

     
  2. (2)

    for any ε > 0 Open image in new window, there exists N N Open image in new windowsuch that G ( x n , x m , x m ) < ε Open image in new windowfor all m , n N Open image in new window.

     

Definition 1.4 (See [2])

A G-metric space ( X , G ) Open image in new window is called G-complete if every G-Cauchy sequence is G-convergent in ( X , G ) Open image in new window.

Definition 1.5 Let ( X , G ) Open image in new window be a G-metric space. A mapping F : X × X × X X Open image in new window is said to be continuous if for any three G-convergent sequences { x n } Open image in new window, { y n } Open image in new window and { z n } Open image in new window converging to x, y and z respectively, { F ( x n , y n , z n ) } Open image in new window is G-convergent to F ( x , y , z ) Open image in new window.

Note that each G-metric on X generates a topology τ G Open image in new window on X whose base is a family of open G-balls { B G ( x , ε ) , x X , ε > 0 } Open image in new window, where B G ( x , ε ) = { y X , G ( x , y , y ) < ε } Open image in new window for all x X Open image in new window and ε > 0 Open image in new window. A nonempty set A X Open image in new window is G-closed in the G-metric space ( X , G ) Open image in new window if A ¯ = A Open image in new window. Observe that
x A ¯ B G ( x , ε ) A Open image in new window

for all ε > 0 Open image in new window. We recall also the following proposition.

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

Let ( X , G ) Open image in new windowbe aG-metric space andAbe a nonempty subset ofX. The setAisG-closed if for anyG-convergent sequence { x n } Open image in new windowinAwith limitx, we have x A Open image in new window.

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 ) Open image in new windowbe a completeG-metric space and T : X X Open image in new windowbe a mapping satisfying the following condition for all x , y , z X Open image in new window:
G ( T x , T y , T z ) k G ( x , y , z ) , Open image in new window
(2)

where k [ 0 , 1 ) Open image in new window. ThenThas a unique fixed point.

Theorem 1.2 (See [5])

Let ( X , G ) Open image in new windowbe a completeG-metric space and T : X X Open image in new windowbe a mapping satisfying the following condition for all x , y X Open image in new window:
G ( T x , T y , T y ) k G ( x , y , y ) , Open image in new window
(3)

where k [ 0 , 1 ) Open image in new window. ThenThas 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 ) Open image in new window. 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 ) . Open image in new window
(4)
A map T : X X Open image in new window on a metric space ( X , d ) Open image in new window is called a weak ϕ-contraction if there exists a strictly increasing function ϕ : [ 0 , ) [ 0 , ) Open image in new window with ϕ ( 0 ) = 0 Open image in new window such that
d ( T x , T y ) d ( x , y ) ϕ ( d ( x , y ) ) , Open image in new window

for all x , y X Open image in new window. We notice that these types of contractions have also been a subject of extensive research (see, e.g., [44, 45, 46, 47, 48, 49]). 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 , ) Open image in new window with ϕ ( 0 ) = 0 Open image in new window and ϕ ( t ) > 0 Open image in new window for t > 0 Open image in new window. In [36], the authors concentrated on two types of cyclic contractions: cyclic-type Banach contractions and cyclic weak ϕ-contractions.

Theorem 1.3Let ( X , G ) Open image in new windowbe aG-completeG-metric space and { A j } j = 1 m Open image in new windowbe a family of nonemptyG-closed subsets ofXwith Y = j = 1 m A j Open image in new window. Let T : Y Y Open image in new windowbe a map satisfying
T ( A j ) A j + 1 , j = 1 , , m , where A m + 1 = A 1 . Open image in new window
(5)
Suppose that there exists a function ϕ Ψ Open image in new windowsuch that the mapTsatisfies the inequality
G ( T x , T y , T z ) M ( x , y , z ) ϕ ( M ( x , y , z ) ) Open image in new window
(6)
for all x A j Open image in new windowand y , z A j + 1 Open image in new window, j = 1 , , m Open image in new window, 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 ) } . Open image in new window
(7)

ThenThas a unique fixed point in j = 1 m A j Open image in new window.

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 ) Open image in new windowbe aG-completeG-metric space and { A j } j = 1 m Open image in new windowbe a family of nonemptyG-closed subsets ofX. Let Y = j = 1 m A j Open image in new windowand T : Y Y Open image in new windowbe a map satisfying
T ( A j ) A j + 1 , j = 1 , , m , where A m + 1 = A 1 . Open image in new window
(8)
If there exists k ( 0 , 1 ) Open image in new windowsuch that
G ( T x , T y , T z ) k G ( x , y , z ) Open image in new window
(9)

holds for all x A j Open image in new windowand y , z A j + 1 Open image in new window, j = 1 , , m Open image in new window, thenThas a unique fixed point in j = 1 m A j Open image in new window.

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

2 Main results

We start this section by defining some sets of auxiliary functions. Let ℱ denote all functions f : [ 0 , ) [ 0 , ) Open image in new window such that f ( t ) = 0 Open image in new window if and only if t = 0 Open image in new window. Let Ψ and Φ be the subsets of ℱ such that
Ψ = { ψ F : ψ  is continuous and nondecreasing } , Φ = { ϕ F : ϕ  is lower semi-continuous } . Open image in new window
Lemma 2.1Let ( X , G ) Open image in new windowbe aG-completeG-metric space and { x n } Open image in new windowbe a sequence inXsuch that G ( x n , x n + 1 , x n + 1 ) Open image in new windowis nonincreasing,
lim n G ( x n , x n + 1 , x n + 1 ) = 0 . Open image in new window
(10)
If { x n } Open image in new windowis not a Cauchy sequence, then there exist ε > 0 Open image in new windowand two sequences { n k } Open image in new windowand { k } Open image in new windowof positive integers such that the following sequences tend toεwhen k Open image in new window:
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 ) . Open image in new window
(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 ) . Open image in new window
Letting n Open image in new window regarding the assumption of the lemma, we derive that
lim n G ( x n , x n , x n + 1 ) = 0 . Open image in new window
(12)
If { x n } Open image in new window is not G-Cauchy, then, due to Proposition 1.2, there exist ε > 0 Open image in new window and corresponding subsequences { n ( k ) } Open image in new window and { ( k ) } Open image in new window of ℕ satisfying n ( k ) > ( k ) > k Open image in new window for which
G ( x ( k ) , x n ( k ) , x n ( k ) ) ε , Open image in new window
(13)
where n ( k ) Open image in new window is chosen as the smallest integer satisfying (13), that is,
G ( x ( k ) , x n ( k ) 1 , x n ( k ) 1 ) < ε . Open image in new window
(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 ) ) . Open image in new window
(15)
Letting k Open image in new window in (15) and using (10), we get
lim k G ( x ( k ) , x n ( k ) , x n ( k ) ) = ε . Open image in new window
(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 ) ) , Open image in new window
(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 ) . Open image in new window
(18)
Passing to the limit when k Open image in new window and using (10) and (16), we obtain that
lim k G ( x ( k ) , x n ( k ) + 1 , x n ( k ) + 1 ) = ε . Open image in new window
(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 ) ) , Open image in new window
(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 ) ) . Open image in new window
(21)
Passing to the limit when k Open image in new window and using (10) and (16), we obtain that
lim k G ( x ( k ) 1 , x n ( k ) , x n ( k ) ) = ε . Open image in new window
(22)
Furthermore,
Passing to the limit when k Open image in new window and using (10) and (16), we obtain that
lim k G ( x ( k ) 1 , x n ( k ) + 1 , x n ( k ) + 1 ) = ε . Open image in new window
(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 ) . Open image in new window
(26)
Letting k Open image in new window in the inequality above and using (12) and (16), we conclude that
lim k G ( x n ( k ) , x ( k ) , x ( k ) + 1 ) = ε . Open image in new window
(27)

 □

Theorem 2.1Let ( X , G ) Open image in new windowbe aG-completeG-metric space and { A j } j = 1 m Open image in new windowbe a family of nonemptyG-closed subsets ofXwith Y = j = 1 m A j Open image in new window. Let T : Y Y Open image in new windowbe a map satisfying
T ( A j ) A j + 1 , j = 1 , 2 , , m , where A m + 1 = A 1 . Open image in new window
(28)
Suppose that there exist functions ϕ Φ Open image in new windowand ψ Ψ Open image in new windowsuch that the mapTsatisfies the inequality
ψ ( G ( T x , T y , T y ) ) ψ ( M ( x , y , y ) ) ϕ ( M ( x , y , y ) ) Open image in new window
(29)
for all x A j Open image in new windowand y A j + 1 Open image in new window, j = 1 , 2 , , m Open image in new window, 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 ) ] } . Open image in new window
(30)

ThenThas a unique fixed point in j = 1 m A j Open image in new window.

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 Open image in new window and define a sequence { x n } Open image in new window in the following way:
x n = T x n 1 , n = 1 , 2 , 3 , . Open image in new window
(31)
We have x 0 A 1 Open image in new window, x 1 = T x 0 A 2 Open image in new window, x 2 = T x 1 A 3 Open image in new window, … since T is a cyclic mapping. If x n 0 + 1 = x n 0 Open image in new window for some n 0 N Open image in new window, then, clearly, the fixed point of the map T is x n 0 Open image in new window. From now on, assume that x n + 1 x n Open image in new window for all n N Open image in new window. Consider the inequality (29) by letting x = x n Open image in new window and y = x n + 1 Open image in new window,
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 ) ) , Open image in new window
(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 ) } . Open image in new window
(33)
If M ( x n , x n + 1 , x n + 1 ) = G ( x n + 1 , x n + 2 , x n + 2 ) Open image in new window, 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 ) ) . Open image in new window
(34)
So, the inequality (34) yields ϕ ( G ( x n + 1 , x n + 2 , x n + 2 ) ) = 0 Open image in new window. Thus, we conclude that
G ( x n + 1 , x n + 2 , x n + 2 ) = 0 . Open image in new window
This contradicts the assumption x n x n + 1 Open image in new window for all n N Open image in new window. So, we derive that
M ( x n , x n + 1 , x n + 1 ) = G ( x n , x n + 1 , x n + 1 ) . Open image in new window
(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 ) ) . Open image in new window
(36)
Thus, { G ( x n , x n + 1 , x n + 1 ) } Open image in new window is a nonnegative, nonincreasing sequence that converges to L 0 Open image in new window. We will show that L = 0 Open image in new window. Suppose, on the contrary, that L > 0 Open image in new window. Taking lim sup n + Open image in new window in (36), we derive that
By the continuity of ψ and the lower semi-continuity of ϕ, we get
ψ ( L ) ψ ( L ) ϕ ( L ) . Open image in new window
(38)
Then it follows that ϕ ( L ) = 0 Open image in new window. Therefore, we get L = 0 Open image in new window, that is,
lim n G ( x n , x n + 1 , x n + 1 ) = 0 . Open image in new window
(39)
Lemma 1.1 with x = x n Open image in new window and y = x n 1 Open image in new window implies that
G ( x n , x n 1 , x n 1 ) 2 G ( x n 1 , x n , x n ) . Open image in new window
(40)
So, we get that
lim n G ( x n , x n 1 , x n 1 ) = 0 . Open image in new window
(41)
Next, we will show that { x n } Open image in new window is a G-Cauchy sequence in ( X , G ) Open image in new window. Suppose, on the contrary, that { x n } Open image in new window is not G-Cauchy. Then, due to Proposition 1.2, there exist ε > 0 Open image in new window and corresponding subsequences { n ( k ) } Open image in new window and { ( k ) } Open image in new window of ℕ satisfying n ( k ) > ( k ) > k Open image in new window for which
G ( x ( k ) , x n ( k ) , x n ( k ) ) ε , Open image in new window
(42)
where n ( k ) Open image in new window is chosen as the smallest integer satisfying (42), that is,
G ( x ( k ) , x n ( k ) 1 , x n ( k ) 1 ) < ε . Open image in new window
(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 ) ) . Open image in new window
(44)
Letting k Open image in new window in (44) and using (39), we get
lim k G ( x ( k ) , x n ( k ) , x n ( k ) ) = ε . Open image in new window
(45)
Notice that for every k N Open image in new window there exists s ( k ) Open image in new window satisfying 0 s ( k ) m Open image in new window such that
n ( k ) ( k ) + s ( k ) 1 ( m ) . Open image in new window
(46)
Thus, for large enough values of k, we have r ( k ) = ( k ) s ( k ) > 0 Open image in new window, and x r ( k ) Open image in new window and x n ( k ) Open image in new window lie in the adjacent sets A j Open image in new window and A j + 1 Open image in new window respectively for some 0 j m Open image in new window. When we substitute x = x r ( k ) Open image in new window and y = x n ( k ) Open image in new window 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 ) ) ) , Open image in new window
(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 ) ] } . Open image in new window
(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 , Open image in new window
(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 . Open image in new window
(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 ) . Open image in new window
(51)
So, we have ϕ ( ε ) = 0 Open image in new window. We deduce that ε = 0 Open image in new window. This contradicts the assumption that { x n } Open image in new window is not G-Cauchy. As a result, the sequence { x n } Open image in new window is G-Cauchy. Since ( X , G ) Open image in new window is G-complete, it is G-convergent to a limit, say w X Open image in new window. It easy to see that w j = 1 m A j Open image in new window. Since x 0 A 1 Open image in new window, then the subsequence { x m ( n 1 ) } n = 1 A 1 Open image in new window, the subsequence { x m ( n 1 ) + 1 } n = 1 A 2 Open image in new window and, continuing in this way, the subsequence { x m ( n 1 ) } n = 1 A m Open image in new window. All the m subsequences are G-convergent in the G-closed sets A j Open image in new window and hence they all converge to the same limit w j = 1 m A j Open image in new window. To show that the limit w is the fixed point of T, that is, w = T w Open image in new window, we employ (29) with x = x n Open image in new window, y = w Open image in new window. This leads to
ψ ( G ( T x n , T w , T w ) ) ψ ( M ( x n , w , w ) ) ϕ ( M ( x n , w , w ) ) , Open image in new window
(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 ) ] } . Open image in new window
(53)
Passing to limsup as n Open image in new window, we get
ψ ( G ( w , T w , T w ) ) ψ ( G ( w , T w , T w ) ) ϕ ( G ( w , T w , T w ) ) . Open image in new window
(54)

Thus, ϕ ( G ( w , T w , T w ) ) = 0 Open image in new window and hence G ( w , T w , T w ) = 0 Open image in new window, that is, w = T w Open image in new window.

Finally, we prove that the fixed point is unique. Assume that v X Open image in new window is another fixed point of T such that v w Open image in new window. Then, since both v and w belong to j = 1 m A j Open image in new window, we set x = v Open image in new window and y = w Open image in new window in (29), which yields
ψ ( G ( T v , T w , T w ) ) ψ ( M ( v , w , w ) ) ϕ ( ( M ( v , w , w ) ) ) , Open image in new window
(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 ) ] } . Open image in new window
(56)
On the other hand, by setting x = w Open image in new window and y = v Open image in new window in (29), we obtain that
ψ ( G ( T w , T v , T v ) ) ψ ( M ( w , v , v ) ) ϕ ( ( M ( w , v , v ) ) ) , Open image in new window
(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 ) ] } . Open image in new window
(58)
If G ( v , w , w ) = G ( w , v , v ) Open image in new window, then v = w Open image in new window. Indeed, by definition, we get that d G ( v , w ) = 0 Open image in new window. Hence v = w Open image in new window. If G ( v , w , w ) > G ( w , v , v ) Open image in new window, then by (56) M ( v , w , w ) = G ( v , w , w ) Open image in new window and by (55),
ψ ( G ( v , w , w ) ) ψ ( G ( v , w , w ) ) ϕ ( ( G ( v , w , w ) ) ) , Open image in new window
(59)
and, clearly, G ( v , w , w ) = 0 Open image in new window. So, we conclude that v = w Open image in new window. Otherwise, G ( w , v , v ) > G ( v , w , w ) Open image in new window. Then by (58), M ( w , v , v ) = G ( w , v , v ) Open image in new window and by (57),
ψ ( G ( w , v , v ) ) ψ ( G ( w , v , v ) ) ϕ ( ( G ( w , v , v ) ) ) , Open image in new window
(60)

and, clearly, G ( w , v , v ) = 0 Open image in new window. So, we conclude that v = w Open image in new window. 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 ] Open image in new window and let T : X X Open image in new window be given as T x = x 8 Open image in new window. Let A = [ 1 , 0 ] Open image in new window and B = [ 0 , 1 ] Open image in new window. Define the function G : X × X × X [ 0 , ) Open image in new window as
G ( x , y , z ) = | x y | + | y z | + | z x | . Open image in new window
(61)

Clearly, the function G is a G-metric on X. Define also ϕ : [ 0 , ) [ 0 , ) Open image in new window as ϕ ( t ) = t 8 Open image in new window and ψ : [ 0 , ) [ 0 , ) Open image in new window as ψ = t 2 Open image in new window. Obviously, the map T has a unique fixed point x = 0 A B Open image in new window.

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 , Open image in new window
which yields
ψ ( G ( T x , T y , T y ) ) = | y x | 8 . Open image in new window
(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 | ] } . Open image in new window
(63)
We derive from (63) that
2 | x y | M ( x , y , y ) . Open image in new window
(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 . Open image in new window
(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 ) ) . Open image in new window
(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 ) ) . Open image in new window
(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 2.1Let ( X , G ) Open image in new windowbe aG-completeG-metric space and { A j } j = 1 m Open image in new windowbe a family of nonemptyG-closed subsets ofXwith Y = j = 1 m A j Open image in new window. Let T : Y Y Open image in new windowbe a map satisfying
T ( A j ) A j + 1 , j = 1 , 2 , , m , where A m + 1 = A 1 . Open image in new window
(68)
Suppose that there exists a constant k ( 0 , 1 ) Open image in new windowsuch that the mapTsatisfies
G ( T x , T y , T y ) k M ( x , y , y ) Open image in new window
(69)
for all x A j Open image in new windowand y A j + 1 Open image in new window, j = 1 , 2 , , m Open image in new window, 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 ) ] } . Open image in new window
(70)

ThenThas a unique fixed point in j = 1 m A j Open image in new window.

Proof The proof is obvious by choosing the functions ϕ, ψ in Theorem 2.1 as ϕ ( t ) = ( 1 k ) t Open image in new window and ψ ( t ) = t Open image in new window. □

Corollary 2.2Let ( X , G ) Open image in new windowbe aG-completeG-metric space and { A j } j = 1 m Open image in new windowbe a family of nonemptyG-closed subsets ofXwith Y = j = 1 m A j Open image in new window. Let T : Y Y Open image in new windowbe a map satisfying
T ( A j ) A j + 1 , j = 1 , 2 , , m , where A m + 1 = A 1 . Open image in new window
(71)
Suppose that there exist constantsa, b, c, dandewith 0 < a + b + c + d + e < 1 Open image in new windowand there exists a function ψ Ψ Open image in new windowsuch that the mapTsatisfies 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 ) ] ) Open image in new window
(72)

for all x A j Open image in new windowand y A j + 1 Open image in new window, j = 1 , 2 , , m Open image in new window. ThenThas a unique fixed point in j = 1 m A j Open image in new window.

Proof

Clearly, we have
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 ) ] } . Open image in new window
(74)

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

Corollary 2.3Let ( X , G ) Open image in new windowbe aG-completeG-metric space and { A j } j = 1 m Open image in new windowbe a family of nonemptyG-closed subsets ofXwith Y = j = 1 m A j Open image in new window. Let T : Y Y Open image in new windowbe a map satisfying
T ( A j ) A j + 1 , j = 1 , 2 , , m , where A m + 1 = A 1 . Open image in new window
Suppose that there exist functions ϕ Φ Open image in new windowand ψ Ψ Open image in new windowsuch that the mapTsatisfies the inequality
ψ ( G ( T x , T y , T z ) ) ψ ( M ( x , y , z ) ) ϕ ( M ( x , y , z ) ) Open image in new window
for all x A j Open image in new windowand y A j + 1 Open image in new window, j = 1 , 2 , , m Open image in new window, 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 ) ] } . Open image in new window
(75)

ThenThas a unique fixed point in j = 1 m A j Open image in new window.

Proof The expression (75) coincides with the expression (30). Following the lines in the proof of Theorem 2.1, by letting x = x n Open image in new window and y = z = x n + 1 Open image in new window, 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 2.4Let ( X , G ) Open image in new windowbe aG-completeG-metric space and { A j } j = 1 m Open image in new windowbe a family of nonemptyG-closed subsets ofXwith Y = j = 1 m A j Open image in new window. Let T : Y Y Open image in new windowbe a map satisfying
T ( A j ) A j + 1 , j = 1 , 2 , , m , where A m + 1 = A 1 . Open image in new window
Suppose also that there exist functions ϕ Φ Open image in new windowand ψ Ψ Open image in new windowsuch that the mapTsatisfies
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 ) , Open image in new window
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 ) ] } Open image in new window

for all x A j Open image in new windowand y A j + 1 Open image in new window, j = 1 , 2 , , m Open image in new window. ThenThas a unique fixed point in j = 1 m A j Open image in new window.

Corollary 2.5Let ( X , G ) Open image in new windowbe aG-completeG-metric space and { A j } j = 1 m Open image in new windowbe a family of nonemptyG-closed subsets ofXwith Y = j = 1 m A j Open image in new window. Let T : Y Y Open image in new windowbe a map satisfying
T ( A j ) A j + 1 , j = 1 , 2 , , m , where A m + 1 = A 1 . Open image in new window
Suppose also that
0 G ( T x , T y , T y ) d s k 0 M ( x , y , y ) d s , Open image in new window
where k ( 0 , 1 ) Open image in new windowand
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 ) ] } Open image in new window

for all x A j Open image in new windowand y A j + 1 Open image in new window, j = 1 , 2 , , m Open image in new window. ThenThas a unique fixed point in j = 1 m A j Open image in new window.

Proof The proof is obvious by choosing the function ϕ, ψ in Corollary 2.4 as ϕ ( t ) = ( 1 k ) t Open image in new window and ψ ( t ) = t Open image in new window. □

Copyright information

© Bilgili and Karapınar; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics, Institute of Science and TechnologyGazi UniversityAnkaraTurkey
  2. 2.Department of MathematicsAtilim UniversityİncekTurkey

Personalised recommendations