Skip to main content

G-β-ψ contractive-type mappings and related fixed point theorems

Abstract

In this paper, we introduce the notion of generalized G-β-ψ contractive mappings which is inspired by the concept of α-ψ contractive mappings. We showed the existence and uniqueness of a fixed point for such mappings in the setting of complete G-metric spaces. The main results of this paper extend, generalize and improve some well-known results on the topic in the literature. We state some examples to illustrate our results. We consider also some applications to show the validity of our results.

1 Introduction and preliminaries

In nonlinear functional analysis, the importance of fixed point theory has been increasing rapidly as an interesting research field. One of the most important reasons for this development is the potential of application of fixed point theory not only in various branches of applied and pure mathematics, but also in many other disciplines such as chemistry, biology, physics, economics, computer science, engineering etc. We also emphasize the crucial role of celebrated results of Banach [1], known as a Banach contraction mapping principle or a Banach fixed point theorem, in the growth of this theory. In 1922, Banach proved that every contraction in a complete metric space has a unique fixed point. After this remarkable paper, a number of authors have extended/generalized/improved the Banach contraction mapping principle in various ways in different abstract spaces (see, e.g., [222]). One of the interesting and recent results in this direction was given by Samet et al. [23]. They defined the notion of α-ψ contractive mappings and proved that including the Banach fixed point theorems, some well-known fixed point results turn into corollaries of their results. Another interesting result was given in 2004 by Mustafa and Sims [24] by introducing the notion of a G-metric space as a generalization of the concept of a metric space. The authors characterized the Banach fixed point theorem in the context of a G-metric space. After this result, many authors have paid attention to this space and proved the existence and uniqueness of a fixed point in the context of a G-metric space (see, e.g., [11, 1720, 2448]). In this paper, we combine these two notions by introducing a G-β-ψ contractive mapping which is a characterization α-ψ contractive mappings in the context of G-metric spaces. Our main results generalize, extend and improve the existence results on the topic in the literature.

Let Ψ be a family of functions ψ:[0,)[0,) satisfying the following conditions:

  1. (i)

    ψ is nondecreasing;

  2. (ii)

    there exist k 0 N and a(0,1) and a convergent series of nonnegative terms k = 1 v k such that

    ψ k + 1 (t)a ψ k (t)+ v k

for k k 0 and any t R + , where R + =[0,).

These functions are known in the literature as Bianchini-Grandolfi gauge functions in some sources (see, e.g., [21, 22, 49]) and as (c)-comparison functions in some other sources (see, e.g., [50]).

Lemma 1 (See [50])

If ψΨ, then the following hold:

  1. (i)

    ( ψ n ( t ) ) n N converges to 0 as n for all t R + ;

  2. (ii)

    ψ(t)<t for any t R + ;

  3. (iii)

    ψ is continuous at 0;

  4. (iv)

    the series k = 1 ψ k (t) converges for any t R + .

Very recently, Karapınar and Samet [32] introduced the following concepts.

Definition 2 Let (X,d) be a metric space and T:XX be a given mapping. We say that T is a generalized α-ψ contractive mapping if there exist two functions α:X×X[0,) and ψΨ such that

α(x,y)d(Tx,Ty)ψ ( M ( x , y ) )

for all x,yX, where

M(x,y)=max { d ( x , y ) , ( d ( x , T x ) + d ( y , T y ) ) / 2 , ( d ( x , T y ) + d ( y , T x ) ) / 2 } .

Clearly, since ψ is nondecreasing, every α-ψ contractive mapping, presented in [23], is a generalized α-ψ contractive mapping.

Definition 3 Let T:XX and α:X×X[0,). We say that T is α-admissible if for all x,yX, we have

α(x,y)1α(Tx,Ty)1.

Various examples of such mappings are presented in [23]. The main results in [32] are the following fixed point theorems.

Theorem 4 Let (X,d) be a complete metric space and T:XX be a generalized α-ψ contractive mapping. Suppose that

  1. (i)

    T is α-admissible;

  2. (ii)

    there exists x 0 X such that α( x 0 ,T x 0 )1;

  3. (iii)

    T is continuous.

Then there exists uX such that Tu=u.

Theorem 5 Let (X,d) be a complete metric space and T:XX be a generalized α-ψ contractive mapping. Suppose that

  1. (i)

    T is α-admissible;

  2. (ii)

    there exists x 0 X such that α( x 0 ,T x 0 )1;

  3. (iii)

    if { x n } is a sequence in X such that α( x n , x n + 1 )1 for all n and x n xX as n, then α( x n ,x)1 for all n.

Then there exists uX such that Tu=u.

Theorem 6 Adding to the hypotheses of Theorem 4 (resp. Theorem 5) the condition: For all x,yFix(T), there exists zX such that α(x,z)1 and α(y,z)1, we obtain the uniqueness of the fixed point of T.

Mustafa and Sims [24] introduced the concept of G-metric spaces as follows.

Definition 7 [24]

Let X be a nonempty set and G:X×X×X R + be a function satisfying the following properties:

(G1) G(x,y,z)=0 if x=y=z;

(G2) 0<G(x,x,y) for all x,yX with xy;

(G3) G(x,x,y)G(x,y,z) for all x,y,zX with yz;

(G4) G(x,y,z)=G(x,z,y)=G(y,z,x)= (symmetry in all three variables);

(G5) G(x,y,z)G(x,a,a)+G(a,y,z) for all x,y,z,aX (rectangle inequality).

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

Every G-metric on X defines a metric d G on X by

d G (x,y)=G(x,y,y)+G(y,x,x)for all x,yX.

Example 8 Let (X,d) be a metric space. The function G:X×X×X R + , defined as

G(x,y,z)=max { d ( x , y ) , d ( y , z ) , d ( z , x ) }

or

G(x,y,z)=d(x,y)+d(y,z)+d(z,x)

for all x,y,zX, is a G-metric on X.

Definition 9 [24]

Let (X,G) be a G-metric space, and let { x n } be a sequence of points of X. We say that { x n } is G-convergent to xX if

lim n , m G(x, x n , x m )=0,

that is, for any ε>0, there exists NN such that G(x, x n , x m )<ε for all n,mN. We call x the limit of the sequence and write x n x or lim n x n =x.

Proposition 10 [24]

Let (X,G) be a G-metric space. The following are equivalent:

  1. (1)

    { x n } is G-convergent to x;

  2. (2)

    G( x n , x n ,x)0 as n;

  3. (3)

    G( x n ,x,x)0 as n;

  4. (4)

    G( x n , x m ,x)0 as n,m.

Definition 11 [24]

Let (X,G) be a G-metric space. A sequence { x n } is called a G-Cauchy sequence if for any ε>0, there is NN such that G( x n , x m , x l )<ε for all n,m,lN, that is, G( x n , x m , x l )0 as n,m,l.

Proposition 12 [24]

Let (X,G) be a G-metric space. Then the following are equivalent:

  1. (1)

    the sequence { x n } is G-Cauchy;

  2. (2)

    for any ε>0, there exists NN such that G( x n , x m , x m )<ε for all n,mN.

Definition 13 [24]

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

Lemma 14 [24]

Let (X,G) be a G-metric space. Then, for any x,y,z,aX, it follows that

  1. (i)

    if G(x,y,z)=0, then x=y=z;

  2. (ii)

    G(x,y,z)G(x,x,y)+G(x,x,z);

  3. (iii)

    G(x,y,y)2G(y,x,x);

  4. (iv)

    G(x,y,z)G(x,a,z)+G(a,y,z);

  5. (v)

    G(x,y,z) 2 3 [G(x,y,a)+G(x,a,z)+G(a,y,z)];

  6. (vi)

    G(x,y,z)G(x,a,a)+G(y,a,a)+G(z,a,a).

Definition 15 (See [24])

Let (X,G) be a G-metric space. A mapping T:XX is said to be G-continuous if {T( x n )} is G-convergent to T(x), where { x n } is any G-convergent sequence converging to x.

In [36], Mustafa characterized the well-known Banach contraction principle mapping in the context of G-metric spaces in the following way.

Theorem 16 (See [36])

Let (X,G) be a complete G-metric space and T:XX be a mapping satisfying the following condition for all x,y,zX:

G(Tx,Ty,Tz)kG(x,y,z),
(1)

where k[0,1). Then T has a unique fixed point.

Theorem 17 (See [36])

Let (X,G) be a complete G-metric space and T:XX be a mapping satisfying the following condition for all x,yX:

G(Tx,Ty,Ty)kG(x,y,y),
(2)

where k[0,1). Then T has a unique fixed point.

Remark 18 The condition (1) implies the condition (2). The converse is true only if k[0, 1 2 ). For details, see [36].

From [24, 36], each G-metric G on X generates a topology τ G on X whose base is a family of open G-balls { B G (x,ε):xX,ε>0}, where B G (x,ε)={yX:G(x,y,y)<ε} for all xX and ε>0. Moreover,

x A ¯ B G (x,ε)A,for all ε>0.

Proposition 19 Let (X,G) be a G-metric space and A be a nonempty subset of X. Then A is G-closed if for any G-convergent sequence { x n } in A with limit x, one has xA.

2 Main results

We introduce the concept of generalized α-ψ contractive mappings as follows.

Definition 20 Let (X,G) be a G-metric space and T:XX be a given mapping. We say that T is a generalized G-β-ψ contractive mapping of type I if there exist two functions β:X×X×X[0,) and ψΨ such that for all x,y,zX, we have

β(x,y,z)G(Tx,Ty,Tz)ψ ( M ( x , y , z ) ) ,
(3)

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 z , T z ) + G ( z , T x , T x ) ) } .

Definition 21 Let (X,G) be a G-metric space and T:XX be a given mapping. We say that T is a generalized G-β-ψ contractive mapping of type II if there exist two functions β:X×X×X[0,) and ψΨ such that for all x,yX, we have

β(x,y,y)G(Tx,Ty,Ty)ψ ( M ( x , y , y ) ) ,
(4)

where

M(x,y,y)=max { G ( x , y , y ) , G ( x , T x , T x ) , G ( y , T y , T y ) , 1 3 ( G ( x , T y , T y ) + G ( y , T y , T y ) + G ( y , T x , T x ) ) } .

Remark 22 Clearly, any contractive mapping, that is, a mapping satisfying (1), is a generalized G-β-ψ contractive mapping of type I with β(x,y,z)=1 for all x,y,zX and ψ(t)=kt, k(0,1). Analogously, a mapping satisfying (2) is a generalized G-β-ψ contractive mapping of type II with β(x,y,y)=1 for all x,yX and ψ(t)=kt, where k(0,1).

Definition 23 Let T:XX and β:X×X×X[0,). We say that T is β-admissible if for all x,y,zX, we have

β(x,y,z)1β(Tx,Ty,Tz)1.

Example 24 Let X=[0,) and T:XX. Define β(x,y,z):X×X×X[0,) by Tx=ln(1+x) and

β(x,y,z)={ e if  x y z , 0 if otherwise .

Then T is β-admissible.

Our first result is the following.

Theorem 25 Let (X,G) be a complete G-metric space. Suppose that T:XX is a generalized G-β-ψ contractive mapping of type I and satisfies the following conditions:

(i) a T is β-admissible;

(ii) a there exists x 0 X such that β( x 0 ,T x 0 ,T x 0 )1;

(iii) b T is G-continuous.

Then there exists uX such that Tu=u.

Proof Let x 0 X be such that β( x 0 ,T x 0 ,T x 0 )1 (such a point exists from the condition (ii) a ). Define the sequence { x n } in X by x n + 1 =T x n for all n0. If x n 0 = x n 0 + 1 for some n 0 , then u= x n 0 is a fixed point of T. So, we can assume that x n x n + 1 for all n. Since T is β-admissible, we have

β( x 0 , x 1 , x 1 )=β( x 0 ,T x 0 ,T x 0 )1β(T x 0 ,T x 1 ,T x 1 )=β( x 1 , x 2 , x 2 )1.

Inductively, we have

β( x n , x n + 1 , x n + 1 )1,for all n=0,1,.
(5)

From (3) and (5), it follows that for all n1, we have

G ( x n , x n + 1 , x n + 1 ) = G ( T x n 1 , T x n , T x n ) β ( x n 1 , x n , x n ) G ( T x n 1 , T x n , T x n ) ψ ( M ( x n 1 , x n , x n ) ) .

On the other hand, we have

Thus, we have

G( x n , x n + 1 , x n + 1 )ψ ( max { G ( x n 1 , x n , x n ) , G ( x n , x n + 1 , x n + 1 ) } ) .

We consider the following two cases:

Case 1: If max{G( x n 1 , x n , x n ),G( x n , x n + 1 , x n + 1 )}=G( x n , x n + 1 , x n + 1 ) for some n, then

G( x n , x n + 1 , x n + 1 )ψ ( G ( x n , x n + 1 , x n + 1 ) ) <G( x n , x n + 1 , x n + 1 ),

which is a contradiction.

Case 2: If max{G( x n 1 , x n , x n ),G( x n , x n + 1 , x n + 1 )}=G( x n 1 , x n , x n ), then

G( x n , x n + 1 , x n + 1 )ψ ( G ( x n 1 , x n , x n ) )

for all n1. Since ψ is nondecreasing, by induction, we get

G( x n , x n + 1 , x n + 1 ) ψ n ( G ( x 0 , x 1 , x 1 ) ) for all n1.
(6)

Using (G5) and (6), we have

G ( x n , x m , x m ) G ( x n , x n + 1 , x n + 1 ) + G ( x n + 1 , x n + 2 , x n + 2 ) + G ( x n + 2 , x n + 3 , x n + 3 ) + + G ( x m 1 , x m , x m ) = k = n m 1 G ( x k , x k + 1 , x k + 1 ) k = n m 1 ψ k ( G ( x 0 , x 1 , x 1 ) ) .

Since ψΨ and G( x 0 , x 1 , x 1 )>0, by Lemma 1, we get that

k = 0 ψ k ( G ( x 0 , x 1 , x 1 ) ) <.

Thus, we have

lim n , m 0 G( x n , x m , x m )=0.

By Proposition 12, this implies that { x n } is a G-Cauchy sequence in the G-metric space (X,G). Since (X,G) is complete, there exists uX such that { x n } is G-convergent to u. Since T is G-continuous, it follows that {T x n } is G-convergent to Tu. By the uniqueness of the limit, we get u=Tu, that is, u is a fixed point of T. □

Definition 26 (See [51])

Let (X,G) be a G-metric space and T:XX be a given mapping. We say that T is a G-β-ψ contractive mapping of type I if there exist two functions β:X×X×X[0,) and ψΨ such that for all x,y,zX, we have

β(x,y,z)G(Tx,Ty,Tz)ψ ( G ( x , y , z ) )
(7)

by following the lines of the proof of Theorem 25.

Corollary 27 Let (X,G) be a complete G-metric space. Suppose that T:XX is a G-β-ψ contractive mapping of type I and satisfies the following conditions:

(i) a T is β-admissible;

(ii) a there exists x 0 X such that β( x 0 ,T x 0 ,T x 0 )1;

(iii) b T is G-continuous.

Then there exists uX such that Tu=u.

Example 28 Let X=[0,) be endowed with the G-metric

G(x,y,z)=|xy|+|yz|+|zx|for all x,y,zX.

Define T:XX by Tx=3x for all xX. We define β:X×X×X[0,) in the following way:

β(x,y,z)={ 1 9 if  ( x , y , z ) ( 0 , 0 , 0 ) , 1 otherwise .

One can easily show that

β(x,y,z)G(Tx,Ty,Tz) 1 9 G(x,y,z)for all x,y,zX.

Then T is a G-β-ψ contractive mapping of type I with ψ(t)= 1 9 t for all t[0,). Take x,y,zX such that β(x,y,z)1. By the definition of T, this implies that x=y=z=0. Then we have β(Tx,Ty,Tz)=β(0,0,0)=1, and so T is β-admissible. All the conditions of Corollary 27 are satisfied. Here, 0 is the fixed point of T. Notice also that the Banach contraction mapping principle is not applicable. Indeed, d(x,y)=|xy| for all x,yX. Then we have xy d(Tx,Ty)=3|xy|>k|xy| for all k[0,1).

It is clear that Theorem 16 is not applicable.

The following result can be easily concluded from Theorem 25.

Corollary 29 Let (X,G) be a complete G-metric space. Suppose that T:XX is a generalized G-β-ψ contractive mapping of type II and satisfies the following conditions:

(i) a T is β-admissible;

(ii) a there exists x 0 X such that β( x 0 ,T x 0 ,T x 0 )1;

(iii) b T is G-continuous.

Then there exists uX such that Tu=u.

The next theorem does not require the continuity of T.

Theorem 30 Let (X,G) be a complete G-metric space. Suppose that T:XX is a generalized G-β-ψ contractive mapping of type I such that ψ is continuous and satisfies the following conditions:

(i) b T is β-admissible;

(ii) b there exists x 0 X such that β( x 0 ,T x 0 ,T x 0 )1;

(iii) b if { x n } is a sequence in X such that β( x n , x n + 1 , x n + 1 )1 for all n and { x n } is a G-convergent to xX, then β( x n ,x,x)1 for all n.

Then there exists uX such that Tu=u.

Proof Following the proof of Theorem 25, we know that the sequence { x n } defined by x n + 1 =T x n for all n0, is a G-Cauchy sequence in the complete G-metric space (X,G), that is, G-convergent to uX. From (5) and (iii) b , we have

β( x n ,u,u)1for all n0.
(8)

Using (8), we have

G ( x n + 1 , T u , T u ) = G ( T x n , T u , T u ) β ( x n , u , u ) G ( T x n , T u , T u ) ψ ( M ( x n , u , u ) ) ,

where

M ( x n , u , u ) = max { G ( x n , u , u ) , G ( x n , T x n , T x n ) , G ( u , T u , T u ) , 1 3 ( G ( x n , T u , T u ) + G ( u , T u , T u ) + G ( u , T x n , T x n ) ) } = max { G ( x n , u , u ) , G ( x n , x n + 1 , x n + 1 ) , G ( u , T u , T u ) , 1 3 ( G ( x n , T u , T u ) + G ( u , T u , T u ) + G ( u , x n + 1 , x n + 1 ) ) } .

Letting n in the following inequality:

G( x n + 1 ,Tu,Tu)ψ ( M ( x n , u , u ) ) ,

it follows that

G(u,Tu,Tu)ψ ( G ( u , T u , T u ) ) ,

which is a contradiction. Thus, we obtain G(u,Tu,Tu)=0, that is, by Lemma 14, u=Tu. □

The following corollary can be easily derived from Theorem 30.

Corollary 31 Let (X,G) be a complete G-metric space. Suppose that T:XX is a generalized G-β-ψ contractive mapping of type II such that ψ is continuous and satisfies the following conditions:

(i) b T is β-admissible;

(ii) b there exists x 0 X such that β( x 0 ,T x 0 ,T x 0 )1;

(iii) b if { x n } is a sequence in X such that β( x n , x n + 1 , x n + 1 )1 for all n and { x n } is a G-convergent to xX, then β( x n ,x,x)1 for all n.

Then there exists uX such that Tu=u.

With the following example, we will show that the hypotheses in Theorems 25 and 30 do not guarantee uniqueness.

Example 32 Let X={(1,0),(0,1)} R 2 be endowed with the following G-metric:

G ( ( x , y ) , ( u , v ) , ( z , w ) ) =|xu|+|uz|+|zx|+|yv|+|vw|+|wy|

for all (x,y),(u,v),(z,w)X. Obviously, (X,G) is a complete metric space. The mapping T(x,y)=(x,y) is trivially continuous and satisfies, for any ψΨ,

β ( ( x , y ) , ( u , v ) , ( z , w ) ) G ( T ( x , y ) , T ( u , v ) , T ( z , w ) ) ψ ( M ( ( x , y ) , ( u , v ) , ( z , w ) ) )

for all (x,y),(u,v),(z,w)X, where

β ( ( x , y ) , ( u , v ) , ( z , w ) ) ={ 1 if  ( x , y ) = ( u , v ) = ( z , w ) , 0 otherwise .

Thus T is a generalized G-β-ψ contractive mapping. On the other hand, for all (x,y),(u,v),(z,w)X, we have

β ( ( x , y ) , ( u , v ) , ( z , w ) ) 1(x,y)=(u,v)=(z,w),

which yields that

T(x,y)=T(u,v)=T(z,w)β ( T ( x , y ) , T ( u , v ) , T ( z , w ) ) 1.

Hence T is β-admissible. Moreover, for all (x,y)X, we have β((x,y),T(x,y),T(x,y))1. So, the assumptions of Theorem 25 are satisfied. Note that the assumptions of Theorem 30 are also satisfied, indeed, if {( x n , y n )} is a sequence in X that converges to some point (x,y)X with β(( x n , y n ),( x n + 1 , y n + 1 ),( x n + 1 , y n + 1 ))1 for all n, then from the definition of β, we have ( x n , y n )=(x,y) for all n, which implies that β(( x n , y n ),(x,y),(x,y))=1 for all n. However, in this case, T has two fixed points in X.

Let X be a set and T be a self-mapping on X. The set of all fixed points of T will be denoted by Fix(T).

Theorem 33 Adding the following condition to the hypotheses of Theorem 25 (resp. Theorem 30, Corollary 29, Corollary 31), we obtain the uniqueness of the fixed point of T.

  1. (iv)

    For xFix(T), β(x,z,z)1 for all zX.

Proof Let u,vFix(T) be two fixed points of T. By (iv), we derive

β(u,v,v)1.

Notice that β(Tu,Tv,Tv)=β(u,v,v) since u and v are fixed points of T. Consequently, we have

G ( u , v , v ) = G ( T u , T v , T v ) β ( u , v , v ) G ( T u , T v , T v ) ψ ( M ( u , v , v ) ) ,

where

M ( u , v , v ) = max { G ( u , v , v ) , G ( u , T u , T u ) , G ( v , T v , T v ) , 1 3 ( G ( u , T v , T v ) + G ( v , T v , T v ) + G ( v , T u , T u ) ) } = max { G ( u , v , v ) , 1 3 ( G ( u , v , v ) + G ( v , u , u ) ) } max { G ( u , v , v ) , 1 3 ( G ( u , v , v ) + 2 G ( u , v , v ) ) } = G ( u , v , v ) .

Thus, we get that

G(u,v,v)ψ ( M ( u , v , v ) ) ψ ( G ( u , v , v ) ) <G(u,v,v),

which is a contradiction. Therefore, u=v, i.e., the fixed point of T is unique. □

Corollary 34 Let (X,G) be a complete G-metric space and let T:XX be a given mapping. Suppose that there exists a continuous function ψΨ such that

G(Tx,Ty,Tz)ψ ( M ( x , y , z ) )

for all x,y,zX. Then T has a unique fixed point.

Corollary 35 Let (X,G) be a complete G-metric space and let T:XX be a given mapping. Suppose that there exists a function ψΨ such that

G(Tx,Ty,Tz)ψ ( G ( x , y , z ) )

for all x,y,zX. Then T has a unique fixed point.

Corollary 36 Let (X,G) be a complete G-metric space and let T:XX be a given mapping. Suppose that there exists λ[0,1) such that

G(Tx,Ty,Tz)λmax { 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 z , T z ) + G ( z , T x , T x ) ) }

for all x,y,zX. Then T has a unique fixed point.

Corollary 37 Let (X,G) be a complete G-metric space and let T:XX be a given mapping. Suppose that there exist nonnegative real numbers a, b, c, d and e with a+b+c+d+e<1 such that

G ( T x , T y , T z ) a G ( x , y , z ) + b G ( x , T x , T x ) + c G ( y , T y , T y ) + d G ( z , T z , T z ) + e 3 ( G ( x , T y , T y ) + G ( y , T z , T z ) + G ( z , T x , T x ) )

for all x,y,zX. Then T has a unique fixed point.

Corollary 38 (See [40])

Let (X,G) be a complete G-metric space and let T:XX be a given mapping. Suppose that there exists λ[0,1) such that

G(Tx,Ty,Tz)λG(x,y,z)

for all x,y,zX. Then T has a unique fixed point.

3 Consequences

3.1 Fixed point theorems on metric spaces endowed with a partial order

Definition 39 Let (X,) be a partially ordered set and T:XX be a given mapping. We say that T is nondecreasing with respect to if

x,yX,xyTxTy.

Definition 40 Let (X,) be a partially ordered set. A sequence { x n }X is said to be nondecreasing with respect to if

x n x n + 1 for all n.

Definition 41 Let (X,) be a partially ordered set and G be a G-metric on X. We say that (X,,G) is G-regular if for every nondecreasing sequence { x n }X such that x n xX as n, x n x for all n.

Theorem 42 Let (X,) be a partially ordered set and G be a G-metric on X such that (X,G) is a complete G-metric space. Let T:XX be a nondecreasing mapping with respect to . Suppose that there exists a function ψΨ such that

G(Tx,Ty,Ty)ψ ( M ( x , y , y ) )
(9)

for all x,yX with xy. Suppose also that the following conditions hold:

  1. (i)

    there exists x 0 X such that x 0 T x 0 ;

  2. (ii)

    T is G-continuous or (X,,G) is G-regular and ψ is continuous.

Then there exists uX such that Tu=u. Moreover, if for xFix(T), xz for all zX, one has the uniqueness of the fixed point.

Proof Define the mapping β:X×X×X[0,) by

β(x,y,y)={ 1 if  x y , 0 otherwise .
(10)

From (9), for all x,yX, we have

β(x,y,y)G(Tx,Ty,Ty)ψ ( M ( x , y , y ) ) .

It follows that T is a generalized G-β-ψ contractive mapping of type II. From the condition (i), we have

β( x 0 ,T x 0 ,T x 0 )1.

By the definition of β and since T is a nondecreasing mapping with respect to , we have

β(x,y,y)1xyTxTyβ(Tx,Ty,Ty)1.

Thus T is β-admissible. Moreover, if T is G-continuous, by Theorem 25, T has a fixed point.

Now, suppose that (X,,G) is G-regular. Let { x n } be a sequence in X such that β( x n , x n + 1 , x n + 1 )1 for all n and x n is G-convergent to xX. By Definition 41, x n x for all n, which gives us β( x n ,x,x)1 for all k. Thus, all the hypotheses of Theorem 30 are satisfied and there exists uX such that Tu=u. To prove the uniqueness, since uFix(T), we have, uz for all zX. By the definition of β, we get that β(u,z,z)1 for all zX. Therefore, the hypothesis (iv) of Theorem 33 is satisfied and we deduce the uniqueness of the fixed point. □

Corollary 43 Let (X,) be a partially ordered set and G be a G-metric on X such that (X,G) is a complete G-metric space. Let T:XX be a nondecreasing mapping with respect to . Suppose that there exists a function ψΨ such that

G(Tx,Ty,Ty)ψ ( G ( x , y , y ) )

for all x,yX with xy. Suppose also that the following conditions hold:

  1. (i)

    there exists x 0 X such that x 0 T x 0 ;

  2. (ii)

    T is G-continuous or (X,,G) is G-regular.

Then there exists uX such that Tu=u. Moreover, if for xFix(T), xz for all zX, one has the uniqueness of the fixed point.

Corollary 44 Let (X,) be a partially ordered set and G be a G-metric on X such that (X,G) is a complete G-metric space. Let T:XX be a nondecreasing mapping with respect to . Suppose that there exists λ[0,1) such that

G(Tx,Ty,Ty)λmax { G ( x , y , y ) , G ( x , T x , T x ) , G ( y , T y , T y ) , 1 3 ( G ( x , T y , T y ) + G ( y , T y , T y ) + G ( y , T x , T x ) ) }

for all x,yX with xy. Suppose also that the following conditions hold:

  1. (i)

    there exists x 0 X such that x 0 T x 0 ;

  2. (ii)

    T is G-continuous or (X,,G) is G-regular.

Then there exists uX such that Tu=u. Moreover, if for xFix(T), xz for all zX, one has the uniqueness of the fixed point.

Corollary 45 Let (X,) be a partially ordered set and G be a G-metric on X such that (X,G) is a complete G-metric space. Let T:XX be a nondecreasing mapping with respect to . Suppose that there exist nonnegative real numbers a, b, c and d with a+b+c+d<1 such that

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 3 ( G ( x , T y , T y ) + G ( y , T y , T y ) + G ( y , T x , T x ) )

for all x,yX with xy. Suppose also that the following conditions hold:

  1. (i)

    there exists x 0 X such that x 0 T x 0 ;

  2. (ii)

    T is G-continuous or (X,,G) is G-regular.

Then there exists uX such that Tu=u. Moreover, if for xFix(T), xz for all zX, one has the uniqueness of the fixed point.

Corollary 46 Let (X,) be a partially ordered set and G be a G-metric on X such that (X,G) is a complete G-metric space. Let T:XX be a nondecreasing mapping with respect to . Suppose that there exists a constant λ[0,1) such that

G(Tx,Ty,Ty)λG(x,y,y)

for all x,yX with xy. Suppose also that the following conditions hold:

  1. (i)

    there exists x 0 X such that x 0 T x 0 ;

  2. (ii)

    T is G-continuous or (X,,G) is G-regular.

Then there exists uX such that Tu=u. Moreover, if for xFix(T), xz for all zX, one has the uniqueness of the fixed point.

3.2 Cyclic contraction

Now, we will prove our results for cyclic contractive mappings in a G-metric space.

Theorem 47 (See [30, 33])

Let A, B be a nonempty G-closed subset of a complete G-metric space (X,G). Suppose also that Y=AB and T:YY is a given self-mapping satisfying

T(A)BandT(B)A.
(11)

If there exists a continuous function ψΨ such that

G(Tx,Ty,Ty)ψ ( M ( x , y , y ) ) ,xA,yB,
(12)

then T has a unique fixed point uAB, that is, Tu=u.

Proof Notice that (Y,G) is a complete G-metric space since A, B is a closed subset of a complete G-metric space (X,G). We define β:X×X×X[0,) in the following way:

β(x,y,y)={ 1 if  ( x , y ) ( A × B ) ( B × A ) , 0 otherwise .

Due to the definition of β and the assumption (12), we have

β(x,y,y)G(Tx,Ty,Ty)ψ ( M ( x , y , y ) ) ,x,yY.
(13)

Hence, T is a generalized G-β-ψ contractive mapping.

Let (x,y)Y×Y be such that β(x,y,y)1. If (x,y)A×B then by the assumption (11), (Tx,Ty)B×A, which yields that β(Tx,Ty,Ty)1. If (x,y)B×A, we get again β(Tx,Ty,Ty)1 by analogy. Thus, in any case, we have β(Tx,Ty,Ty)1, that is, T is β-admissible. Notice also that for any zA, we have (z,Tz)A×B, which yields β(z,Tz,Tz)1.

Take a sequence { x n } in X such that β( x n , x n + 1 , x n + 1 )1 for all n and x n uX as n. Regarding the definition of β, we derive that

( x n , x n + 1 )(A×B)(B×A)for all n.
(14)

By assumption, A, B and hence (A×B)(B×A) is a G-closed set. Hence, we get that (u,u)(A×B)(B×A), which implies that uAB. We conclude, by the definition of β, that β( x n ,u,u)1 for all n.

Now, all hypotheses of Theorem 30 are satisfied and we conclude that T has a fixed point. Next, we show the uniqueness of a fixed point u of T. Since uFix(T) and uAB, we get β(u,a,a)1 for all aY. Thus, the condition (iv) of Theorem 33 is satisfied. □

References

  1. Banach S: Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales. Fundam. Math. 1922, 3: 133–181.

    Google Scholar 

  2. Abbas M, Sintunavarat W, Kumam P: Coupled fixed point of generalized contractive mappings on partially ordered G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 31

    Google Scholar 

  3. Agarwal RP, Alghamdi MA, Shahzad N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 40

    Google Scholar 

  4. Aydi H, Vetro C, Sintunavarat W, Kumam P: Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 124

    Google Scholar 

  5. Ćirić L, Lakshmikantham V: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020

    Article  MathSciNet  Google Scholar 

  6. Gnana-Bhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017

    Article  MathSciNet  Google Scholar 

  7. Guo D, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal., Theory Methods Appl. 1987, 11: 623–632. 10.1016/0362-546X(87)90077-0

    Article  MathSciNet  Google Scholar 

  8. Karapınar E: Fixed point theory for cyclic weak ϕ -contraction. Appl. Math. Lett. 2011,24(6):822–825. 10.1016/j.aml.2010.12.016

    Article  MathSciNet  Google Scholar 

  9. Karapınar E, Sadaranagni K: Fixed point theory for cyclic ( ϕ - ψ )-contractions. Fixed Point Theory Appl. 2011., 2011: Article ID 69

    Google Scholar 

  10. Karapınar E: Coupled fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl. 2010, 59: 3656–3668. 10.1016/j.camwa.2010.03.062

    Article  MathSciNet  Google Scholar 

  11. Nashine HK, Sintunavarat W, Kumam P: Cyclic generalized contractions and fixed point results with applications to an integral equation. Fixed Point Theory Appl. 2012., 2012: Article ID 217

    Google Scholar 

  12. Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory Appl. 2003,4(1):79–89.

    MathSciNet  Google Scholar 

  13. Pacurar M, Rus IA: Fixed point theory for cyclic φ -contractions. Nonlinear Anal. 2010, 72: 1181–1187. 10.1016/j.na.2009.08.002

    Article  MathSciNet  Google Scholar 

  14. Petric MA: Some results concerning cyclical contractive mappings. Gen. Math. 2010,18(4):213–226.

    MathSciNet  Google Scholar 

  15. Petruşel A, Rus IA: Fixed point theorems in ordered L -spaces. Proc. Am. Math. Soc. 2006, 134: 411–418.

    Article  MathSciNet  Google Scholar 

  16. Rus IA: Cyclic representations and fixed points. Ann. “Tiberiu Popoviciu” Sem. Funct. Equ. Approx. Convexity 2005, 3: 171–178.

    Google Scholar 

  17. Sintunavarat W, Kumam P: Coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces. Thai J. Math. 2012,10(3):551–563.

    MathSciNet  Google Scholar 

  18. Sintunavarat W, Cho YJ, Kumam P: Coupled fixed-point theorems for contraction mapping induced by cone ball-metric in partially ordered spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 128

    Google Scholar 

  19. Sintunavarat W, Kim JK, Kumam P: Fixed point theorems for a generalized almost (ϕ,φ) -contraction with respect to S in ordered metric spaces. J. Inequal. Appl. 2012., 2012: Article ID 263

    Google Scholar 

  20. Sintunavarat W, Kumam P: Common fixed point theorem for cyclic generalized multi-valued contraction mappings. Appl. Math. Lett. 2012,25(11):1849–1855. 10.1016/j.aml.2012.02.045

    Article  MathSciNet  Google Scholar 

  21. Bianchini RM, Grandolfi M: Transformazioni di tipo contracttivo generalizzato in uno spazio metrico. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 1968, 45: 212–216.

    MathSciNet  Google Scholar 

  22. Proinov PD: New general convergence theory for iterative processes and its applications to Newton Kantorovich type theorems. J. Complex. 2010, 26: 3–42. 10.1016/j.jco.2009.05.001

    Article  MathSciNet  Google Scholar 

  23. Samet B, Vetro C, Vetro P:Fixed point theorem for αψ contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014

    Article  MathSciNet  Google Scholar 

  24. Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006,7(2):289–297.

    MathSciNet  Google Scholar 

  25. Aydi H, Damjanović B, Samet B, Shatanawi W: Coupled fixed point theorems for nonlinear contractions in partially ordered G -metric spaces. Math. Comput. Model. 2011, 54: 2443–2450. 10.1016/j.mcm.2011.05.059

    Article  MathSciNet  Google Scholar 

  26. Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011, 54: 73–79. 10.1016/j.mcm.2011.01.036

    Article  MathSciNet  Google Scholar 

  27. Aydi H, Karapınar E, Shatanawi W: Tripled fixed point results in generalized metric spaces. J. Appl. Math. 2012., 2012: Article ID 314279

    Google Scholar 

  28. Aydi H, Karapınar E, Mustafa Z: On Common Fixed Points in G -Metric Spaces Using (E.A) Property. Comput. Math. Appl. 2012,64(6):1944–1956. 10.1016/j.camwa.2012.03.051

    Article  MathSciNet  Google Scholar 

  29. Aydi H, Karapınar E, Shatanawi W: Tripled common fixed point results for generalized contractions in ordered generalized metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 101

    Google Scholar 

  30. Aydi, H: Generalized cyclic contractions in G-metric spaces. J. Nonlinear Sci. Appl. (in press)

  31. Karapınar E, Kaymakcalan B, Tas K: On coupled fixed point theorems on partially ordered G -metric spaces. J. Inequal. Appl. 2012., 2012: Article ID 200

    Google Scholar 

  32. Karapınar E, Samet B:Generalized (αψ) contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012., 2012: Article ID 793486

    Google Scholar 

  33. Karapınar E, Erhan IM, Ulus AY: Cyclic contractions on G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 182947

    Google Scholar 

  34. Ding HS, Karapınar E: A note on some coupled fixed point theorems on G -metric space. J. Inequal. Appl. 2012., 2012: Article ID 170

    Google Scholar 

  35. Gul U, Karapınar E: On almost contraction in partially ordered metric spaces viz implicit relation. J. Inequal. Appl. 2012., 2012: Article ID 217

    Google Scholar 

  36. Mustafa, Z: A new structure for generalized metric spaces with applications to fixed point theory. Ph.D. Thesis, The University of Newcastle, Australia (2005)

  37. Mustafa Z, Aydi H, Karapınar E: On common fixed points in image-metric spaces using (E.A) property. Comput. Math. Appl. 2012. doi:10.1016/j.camwa.2012.03.051

    Google Scholar 

  38. Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorem for mapping on complete G -metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 189870

    Google Scholar 

  39. Mustafa Z, Khandaqji M, Shatanawi W: Fixed point results on complete G -metric spaces. Studia Sci. Math. Hung. 2011, 48: 304–319.

    MathSciNet  Google Scholar 

  40. Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175

    Google Scholar 

  41. Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed point results in G -metric spaces. Int. J. Math. Math. Sci. 2009., 2009: Article ID 283028

    Google Scholar 

  42. Shatanawi W: Fixed point theory for contractive mappings satisfying Φ-maps in G -metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 181650

    Google Scholar 

  43. Shatanawi W: Some fixed point theorems in ordered G -metric spaces and applications. Abstr. Appl. Anal. 2011., 2011: Article ID 126205

    Google Scholar 

  44. Shatanawi W: Coupled fixed point theorems in generalized metric spaces. Hacet. J. Math. Stat. 2011,40(3):441–447.

    MathSciNet  Google Scholar 

  45. Shatanawi W, Abbas M, Nazir T: Common coupled coincidence and coupled fixed point results in two generalized metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 80

    Google Scholar 

  46. Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c -distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040

    Article  MathSciNet  Google Scholar 

  47. Tahat N, Aydi H, Karapınar E, Shatanawi W: Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 48

    Google Scholar 

  48. Agarwal RP, Karapınar E: Remarks on some coupled fixed point theorems in G -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 2

    Google Scholar 

  49. Proinov PD: A generalization of the Banach contraction principle with high order of convergence of successive approximations. Nonlinear Anal., Theory Methods Appl. 2007, 67: 2361–2369. 10.1016/j.na.2006.09.008

    Article  MathSciNet  Google Scholar 

  50. Berinde V: Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare; 2002.

    Google Scholar 

  51. Alghamdi, MA, Karapınar, E: G-β-ψ contractive type mappings and related fixed point theorems. Preprint

Download references

Acknowledgements

The research of the first author was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Maryam A Alghamdi.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Rights and permissions

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

Reprints and Permissions

About this article

Cite this article

Alghamdi, M.A., Karapınar, E. G-β-ψ contractive-type mappings and related fixed point theorems. J Inequal Appl 2013, 70 (2013). https://doi.org/10.1186/1029-242X-2013-70

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-70

Keywords

  • Fixed Point Theorem
  • Contractive Mapping
  • Nondecreasing Mapping
  • Fixed Point Theory
  • Unique Fixed Point