Skip to main content

Contraction conditions using comparison functions on b-metric spaces

Abstract

In this paper, we consider the setting of b-metric spaces to establish results regarding the common fixed points of two mappings, using a contraction condition defined by means of a comparison function. An example is presented to support our results comparing with existing ones.

MSC:49H09, 47H10.

1 Introduction

The contraction principle of Banach [1], proved in 1922, was followed by diverse works about fixed points theory regarding different classes of contractive conditions on some spaces such as: quasi-metric spaces [2, 3], cone metric spaces [4, 5], partially ordered metric spaces [68], G-metric spaces [9], partial metric spaces [1013], Menger spaces [14], metric-type spaces [15], and fuzzy metric spaces [1618]. Also, there have been developed studies on approximate fixed point or on qualitative aspects of numerical procedures for approximating fixed points see, for example [19, 20].

The concept of b-metric spaces was introduced by Bakhtin [21] in 1989, who used it to prove a generalization of the Banach principle in spaces endowed with such kind of metrics. Since then, this notion has been used by many authors to obtain various fixed point theorems. Aydi et al. in [22] proved common fixed point results for single-valued and multi-valued mappings satisfying a weak ϕ-contraction in b-metric spaces. Roshan et al. in [23] used the notion of almost generalized contractive mappings in ordered complete b-metric spaces and established some fixed and common fixed point results. Starting from the results of Berinde [24], Păcurar [25] proved the existence and uniqueness of fixed points of ϕ-contractions on b-metric spaces. Hussain and Shah in [26] introduced the notion of a cone b-metric space, generalizing both notions of b-metric spaces and cone metric spaces. In this paper they also considered topological properties of cone b-metric spaces and results on KKM mappings in the setting of cone b-metric spaces. Fixed point theorems of contractive mappings in cone b-metric spaces without the assumption of the normality of a corresponding cone are proved by Huang and Xu in [27]. The setting of partially ordered b-metric spaces was used by Hussain et al. in [28] to study tripled coincidence points of mappings which satisfy nonlinear contractive conditions, extending those results of Berinde and Borcut [29] for metric spaces to b-metric spaces. Using the concept of a g-monotone mapping, Shah and Hussain in [30] proved common fixed point theorems involving g-non-decreasing mappings in b-metric spaces, generalizing several results of Agarwal et al. [31] and Ćirić et al. [32]. Some results of Suzuki [33] are extended to the case of metric-type spaces and cone metric-type spaces.

The aim of this paper is to consider and establish results on the setting of b-metric spaces, regarding common fixed points of two mappings, using a contraction condition defined by means of a comparison function. An example is given to support our results.

2 Preliminaries

Definition 1 Let X be a nonempty set and d:X×X[0,+). A function d is called a b-metric with constant (base) s1 if:

  1. (1)

    d(x,y)=0 iff x=y.

  2. (2)

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

  3. (3)

    d(x,y)s(d(x,z)+d(z,y)) for all x,y,zX.

The pair (X,d) is called a b-metric space.

It is obvious that a b-metric space with base s=1 is a metric space. There are examples of b-metric spaces which are not metric spaces (see, e.g., Singh and Prasad [34]).

The notions of a Cauchy sequence and a convergent sequence in b-metric spaces are defined by Boriceanu [35].

Definition 2 Let { x n } be a sequence in a b-metric space (X,d).

  1. (1)

    A sequence { x n } is called convergent if and only if there is xX such that d( x n ,x)0 when n+.

  2. (2)

    { x n } is a Cauchy sequence if and only if d( x n , x m )0, when n,m+.

As usual, a b-metric space is said to be complete if and only if each Cauchy sequence in this space is convergent.

Regarding the properties of a b-metric space, we recall that if the limit of a convergent sequence exists, then it is unique. Also, each convergent sequence is a Cauchy sequence. But note that a b-metric, in the general case, is not continuous (see Roshan et al. [23]).

The continuity of a mapping with respect to a b-metric is defined as follows.

Definition 3 Let (X,d) and ( X , d ) be two b-metric spaces with constant s and s , respectively. A mapping T:X X is called continuous if for each sequence { x n } in X, which converges to xX with respect to d, then T x n converges to Tx with respect to d .

Definition 4 Let s1 be a constant. A mapping φ:[0,+)[0,+) is called comparison function with base s1, if the following two axioms are fulfilled:

  1. (a)

    φ is non-decreasing,

  2. (b)

    lim n + φ n (t)=0 for all t>0.

Clearly, if φ is a comparison function, then φ(t)<t for each t>0.

For different properties and applications of comparison functions on partial metric spaces, we refer the reader to [36].

3 Main results

Now we are ready to prove our main results.

Theorem 1 Let (X,d) be a complete b-metric space with a constant s and T,S:XX two mappings on X. Suppose that there is a constant L< 1 1 + s and a comparison function φ such that the inequality

sd(Tx,Sy)φ ( max { s d ( x , T x ) , s d ( y , S y ) , L [ d ( x , S y ) + d ( T x , y ) ] } )
(3.1)

holds for each x,yX. Suppose that one of the mappings T or S is continuous. Then T and S have a unique common fixed point.

Proof Let x 0 X be arbitrary. We define a sequence { x n } as follows:

x 2 n + 1 =T x 2 n , x 2 n + 2 =S x 2 n + 1 ,nN.
(3.2)

Suppose that there is some nN such that x n = x n + 1 . If n=2k, then x 2 k = x 2 k + 1 and from the contraction condition (3.1) with x= x 2 k and y= x 2 k + 1 we have

s d ( x 2 k + 1 , x 2 k + 2 ) = s d ( T x 2 k , S x 2 k + 1 ) φ ( max { s d ( x 2 k , T x 2 k ) , s d ( x 2 k + 1 , S x 2 k + 1 ) , L [ d ( x 2 k , S x 2 k + 1 ) + d ( T x 2 k , x 2 k + 1 ) ] } ) = φ ( max { s d ( x 2 k , x 2 k + 1 ) , s d ( x 2 k + 1 , x 2 k + 2 ) , L d ( x 2 k , x 2 k + 2 ) } ) .

Hence, as we supposed that x 2 k = x 2 k + 1 and as a comparison function φ is non-decreasing,

s d ( x 2 k + 1 , x 2 k + 2 ) φ ( max { s d ( x 2 k + 1 , x 2 k + 2 ) , L [ s ( d ( x 2 k , x 2 k + 1 ) + d ( x 2 k + 1 , x 2 k + 2 ) ) ] } ) = φ ( max { s d ( x 2 k + 1 , x 2 k + 2 ) , L s d ( x 2 k + 1 , x 2 k + 2 ) } ) = φ ( s d ( x 2 k + 1 , x 2 k + 2 ) ) .

If we assume that d( x 2 k + 1 , x 2 k + 2 )>0, then we have, as φ(t)<t for t>0,

sd( x 2 k + 1 , x 2 k + 2 )φ ( s d ( x 2 k + 1 , x 2 k + 2 ) ) <sd( x 2 k + 1 , x 2 k + 2 ),

a contradiction. Therefore, d( x 2 k + 1 , x 2 k + 2 )=0. Hence x 2 k + 1 = x 2 k + 2 . Thus we have x 2 k = x 2 k + 1 = x 2 k + 2 . By (3.2), it means x 2 k =T x 2 k =S x 2 k , that is, x 2 k is a common fixed point of T and S.

If n=2k+1, then using the same arguments as in the case x 2 k = x 2 k + 1 , it can be shown that x 2 k + 1 is a common fixed point of T and S.

From now on, we suppose that x n x n + 1 for all nN.

Now we shall prove that

sd( x n , x n + 1 )φ ( s d ( x n 1 , x n ) ) for each nN.
(3.3)

There are two cases which we have to consider.

Case I. n=2k, kN.

From the contraction condition (3.1) with x= x 2 k and y= x 2 k 1 we get

s d ( x 2 k + 1 , x 2 k ) = s d ( T x 2 k , S x 2 k 1 ) φ ( max { s d ( x 2 k , T x 2 k ) , s d ( x 2 k 1 , S x 2 k 1 ) , L [ d ( x 2 k , S x 2 k 1 ) + d ( T x 2 k , x 2 k 1 ) ] } ) = φ ( max { s d ( x 2 k , x 2 k + 1 ) , s d ( x 2 k 1 , x 2 k ) , L d ( x 2 k + 1 , x 2 k 1 ) } ) .

Since L<1/2, we get

s d ( x 2 k + 1 , x 2 k ) φ ( max { s d ( x 2 k 1 , x 2 k ) , s d ( x 2 k , x 2 k + 1 ) , s 2 ( d ( x 2 k 1 , x 2 k ) + d ( x 2 k , x 2 k + 1 ) ) } ) = φ ( max { s d ( x 2 k 1 , x 2 k ) , s d ( x 2 k , x 2 k + 1 ) } ) .

Now, if we suppose that max{sd( x 2 k , x 2 k 1 ),sd( x 2 k , x 2 k + 1 )}=sd( x 2 k , x 2 k + 1 ), then by the property (a) of φ in Definition 4 we get

sd( x 2 k , x 2 k + 1 )φ ( s d ( x 2 k , x 2 k + 1 ) ) <sd( x 2 k , x 2 k + 1 ),

a contradiction. Therefore, from the above inequality we have

sd( x 2 k , x 2 k + 1 )φ ( s d ( x 2 k 1 , x 2 k ) ) .
(3.4)

Thus we proved that (3.3) holds for n=2k.

Case II. n=2k+1, kN.

Using the same argument as in the Case I, it can be proved that (3.3) holds for n=2k+1, that is,

sd( x 2 k + 1 , x 2 k + 2 )φ ( s d ( x 2 k , x 2 k + 1 ) ) .
(3.5)

From (3.4) and (3.5) we conclude that the inequality (3.3) holds for all nN.

From (3.3), by the induction it is easy to prove that

sd( x n , x n + 1 ) φ n ( s d ( x 0 , x 1 ) ) for all nN.
(3.6)

Since lim n + φ n (t)=0 for all t>0, from (3.6) it follows that

lim n d( x n , x n + 1 )=0.
(3.7)

Now we shall prove that { x n } is a Cauchy sequence. Let ϵ>0. Since L< 1 1 + s implies s2L>0 and 1L(1+s)>0, from (3.7) we conclude that there exists n 0 N such that

d( x n , x n 1 )< 1 L L s 2 s ϵ
(3.8)

for all n n 0 .

Let m,nN with m>n. By induction on m, we shall prove that

d( x n , x m )<ϵfor all m>n n 0 .
(3.9)

Let n n 0 and m=n+1. Then from (3.3) and (3.8) we get

d( x n , x m )=d( x n , x n + 1 )d( x n , x n 1 )< 1 L L s 2 s ϵ<ϵ.

Thus (3.9) holds for m=n+1.

Assume now that (3.9) holds for some mn+1. We have to prove that (3.9) holds for m+1.

We have to consider four cases.

Case I. n is odd, m+1 is even.

From the contraction condition (3.1) we get

s d ( x n , x m + 1 ) = s d ( T x n 1 , S x m ) φ ( max { s d ( x n 1 , x n ) , s d ( x m , x m + 1 ) , L [ d ( x n 1 , x m + 1 ) + d ( x n , x m ) ] } ) .

Hence we get, as d( x m , x m + 1 )<d( x n 1 , x n ) and φ(t)<t for all t>0,

sd( x n , x m + 1 )<max { s d ( x n 1 , x n ) , L [ d ( x n 1 , x m + 1 ) + d ( x n , x m ) ] } .
(3.10)

If from (3.10) we have sd( x n , x m + 1 )<sd( x n 1 , x n ), then by (3.8),

d( x n , x m + 1 )<d( x n 1 , x n )< 1 L L s 2 s ϵ<ϵ.

If (3.10) implies sd( x n , x m + 1 )<L[d( x n 1 , x m + 1 )+d( x n , x m )], then by the (general) triangle inequality,

sd( x n , x m + 1 )<Lsd( x n 1 , x n )+Lsd( x n , x m + 1 )+Ld( x n , x m ).

Hence we get, as L<1/(1+s) implies L/(1L)<2L<1s,

d ( x n , x m + 1 ) < L 1 L [ d ( x n 1 , x n ) + 1 s d ( x n , x m ) ] < 2 L [ d ( x n 1 , x n ) + 1 s d ( x n , x m ) ] .

Now, by (3.8) and the induction hypothesis (3.9),

d ( x n , x m + 1 ) < 2 L 1 L L s 2 s ϵ + 2 L s ϵ < 1 2 L L ( s 1 ) s ϵ + 2 L s ϵ 1 2 L s ϵ + 2 L s ϵ = 1 s ϵ ϵ .

Thus we proved that in this case (3.9) holds for m+1. Therefore, by induction, we conclude that in Case I the inequality (3.9) holds for all m>n.

Case II. n is even, m+1 is odd. The proof of (3.9) in this case is similar to one given in Case I.

Case III. n is even, m+1 is even.

Using the (general) triangle inequality and the contraction condition (3.1), we obtain

d ( x n , x m + 1 ) s d ( x n , x n + 1 ) + s d ( x n + 1 , x m + 1 ) = s d ( x n , x n + 1 ) + s d ( T x n , S x m ) s d ( x n , x n + 1 ) + φ ( max { s d ( x n , x n + 1 ) , s d ( x m , x m + 1 ) , L [ d ( x n , x m + 1 ) + d ( x n + 1 , x m ) ] } ) = s d ( x n , x n + 1 ) + φ ( max { s d ( x n , x n + 1 ) , L [ d ( x n , x m + 1 ) + d ( x n + 1 , x m ) ] } ) .

Hence we get, as d( x m , x m + 1 )<d( x n 1 , x n ) and φ(t)<t for all t>0,

d( x n , x m + 1 )<sd( x n , x n + 1 )+max { s d ( x n , x n + 1 ) , L [ d ( x n , x m + 1 ) + d ( x n + 1 , x m ) ] } .
(3.11)

If the inequality (3.11) implies d( x n , x m + 1 )<sd( x n , x n + 1 )+sd( x n , x n + 1 ), then from (3.8) we get

d( x n , x m + 1 )<2s 1 L L s 2 s ϵ=ϵ.

If (3.11) implies

d( x n , x m + 1 )<sd( x n , x n + 1 )+L [ d ( x n , x m + 1 ) + d ( x n + 1 , x m ) ] ,

then by the (general) triangle inequality we have

d ( x n , x m + 1 ) < s d ( x n , x n + 1 ) + L d ( x n , x m + 1 ) + L s d ( x n + 1 , x n ) + L s d ( x n , x m ) = ( 1 + L ) s d ( x n , x n + 1 ) + L d ( x n , x m + 1 ) + L s d ( x n , x m ) .

Hence we get

(1L)d( x n , x m + 1 )(1+L)sd( x n , x n + 1 )+Lsd( x n , x m ).

Now, by (3.8) and the induction hypothesis (3.3), we have

(1L)d( x n , x m + 1 )< ( 1 + L ) s [ ( 1 L ) L s ] 2 s ϵ+Lsϵ< [ ( 1 L ) L s ] ϵ+Lsϵ=(1L)ϵ.

Hence

d( x n , x m + 1 )<ϵ.

Thus we proved that (3.9) holds for m+1. Therefore, by induction, we conclude that in Case III the inequality (3.9) holds for all m>n.

Case IV. n is odd, m+1 is odd. The proof of (3.9) in this case is similar to one given in Case III.

Therefore, we proved that in all of four cases the inequality (3.9) holds.

From (3.9) it follows that { x n } is a Cauchy sequence. Since (X,d) is a complete b-metric space, then { x n } converges to some uX as n+.

Now we shall prove that if one of the mappings T or S is continuous, then Tu=Su=u. Without loss of generality, we can suppose that S is continuous. Clearly, as x n u, then by (3.2) we have S x 2 n + 1 = x 2 n + 2 u as n+. Since x 2 n + 1 u and S is continuous, then S x 2 n + 1 Su. Thus, by the uniqueness of the limit in a b-metric space, we have Su=u. Now, from the contraction condition (3.1),

s d ( T u , u ) = s d ( T u , S u ) φ ( max { s d ( u , T u ) , s d ( u , S u ) , L [ d ( T u , u ) + d ( u , S u ) ] } ) = φ ( s d ( u , T u ) ) .

If we suppose that d(u,Tu)>0, then we have

sd(u,Tu)φ ( s d ( u , T u ) ) <sd(u,Tu),

a contradiction. Therefore, d(u,Tu)=0. Hence Tu=u. Thus we proved that u is a common fixed point of T and S.

Suppose now that u and v are different common fixed points of T and S, that is, d(u,v)>0. Then

s d ( u , v ) = s d ( T u , S v ) φ ( max { s d ( u , T u ) , s d ( v , S v ) , L ( d ( u , S v ) + d ( v , T u ) ) } ) = φ ( 2 L d ( u , v ) ) .

Since 2L<1s, then we get sd(u,v)φ(sd(u,v))<sd(u,v), a contradiction. Thus we proved that S and T have a unique common fixed point in X. □

If S=T in Theorem 1, then we have the following result.

Corollary 1 Let (X,d) be a complete b-metric space with a constant s and T:XX two mappings on X. Suppose that there is a constant L< 1 2 and a comparison function φ such that the inequality

sd(Tx,Ty)φ ( max { s d ( x , T x ) , s d ( y , T y ) , L [ d ( x , T y ) + d ( T x , y ) ] } )
(3.12)

holds for each x,yX. Suppose that a mapping T is continuous. Then T has a unique fixed point.

Omitting the continuity assumption of mapping T or S in Theorem 1, modifying the contraction condition (3.1) and imposing on a comparison function φ a corresponding condition, then we can prove the following theorem.

Theorem 2 Let (X,d) be a complete b-metric space with a constant s and T,S:XX two mappings on X. Suppose that there is a constant L< 1 1 + s and a comparison function φ such that the inequality

sd(Tx,Ty)φ ( max { s d ( x , T x ) , d ( y , T y ) , L ( d ( x , T y ) + d ( T x , y ) ) } )
(3.13)

holds for all x,yX. If in addition a comparison function φ satisfies the following condition:

lim sup β α φ(β)<α,α>0,
(3.14)

then T and S have a unique common fixed point.

Proof Since the contraction condition (3.13) implies the contraction condition (3.1) in Theorem 1, then from the proof of Theorem 1 it follows that a sequence { x n }, defined as in (3.3), converges to some uX, that is,

T x 2 n = x 2 n + 1 uandS x 2 n + 1 = x 2 n + 2 uas n+.
(3.15)

Now we prove that Su=u. From the contraction condition (3.13) and by the monotonicity of φ we obtain

s d ( x 2 n + 1 , S u ) = s d ( T x 2 n , S u ) φ ( max { s d ( x 2 n , x 2 n + 1 ) , d ( u , S u ) , L ( d ( x 2 n + 1 , u ) + d ( x 2 n , S u ) ) } ) φ ( max { s d ( x 2 n , x 2 n + 1 ) , s d ( u , x 2 n + 1 ) + s d ( x 2 n + 1 , S u ) , L ( d ( x 2 n + 1 , u ) + s d ( x 2 n , x 2 n + 1 ) + s d ( x 2 n + 1 , S u ) ) } ) .
(3.16)

Since φ is non-decreasing and L<1, from (3.16) we get

sd( x 2 n + 1 ,Su)φ ( s d ( x 2 n , x 2 n + 1 ) + s d ( u , x 2 n + 1 ) + s d ( x 2 n + 1 , S u ) ) .
(3.17)

Set

t n =sd( x 2 n , x 2 n + 1 )+sd(u, x 2 n + 1 )+sd( x 2 n + 1 ,Su).

Then, in virtue of (3.15),

lim sup n t n = lim sup n sd( x 2 n + 1 ,Su)=r,
(3.18)

where r0. Let { t n k } be a subsequence of { t n } such that t n k r as k. For simplicity, denote { t n k } again by { t n }. Then from (3.18),

lim n t n = lim n sd( x 2 n + 1 ,Su)=r.
(3.19)

Suppose that r>0. Then from (3.19), (3.17), and the assumption (3.14) of φ, we have

r= lim n t n = lim n sd( x 2 n + 1 ,Su) lim t n r φ( t n )<r,

a contradiction. Therefore,

lim n t n = lim n sd( x 2 n + 1 ,Su)=0.

Hence we have x 2 n + 1 Su as n. Since by (3.15), x 2 n + 1 u, and as the limit in a b-metric space is unique, it follows that Su=u. Now, by (3.13),

s d ( T u , u ) = s d ( T u , S u ) φ ( max { s d ( u , T u ) , d ( u , S u ) , L ( d ( T u , u ) + d ( u , S u ) ) } ) = φ ( s d ( u , T u ) ) .

If we suppose that d(u,Tu)>0, then we have sd(Tu,u) φ(sd(u,Tu))<sd(u,Tu), a contradiction. Therefore, d(Tu,u)=0, that is, Tu=u. Thus we proved that Tu=Su=u. □

If S=T in Theorem 2, then we get the following result.

Corollary 2 Let (X,d) be a complete b-metric space with a constant s and T:XX a mapping on X. Suppose that there is a constant L< 1 1 + s and a comparison function φ such that the inequality

sd(Tx,Ty)φ ( max { s d ( x , T x ) , d ( y , T y ) , L [ d ( x , T y ) + d ( T x , y ) ] } )

holds for all x,yX. If in addition a comparison function φ satisfies the inequality (3.14), then T has a unique fixed point.

Now we give an example to support our results.

Example 1 Let X=[0,1] endowed with the b-metric

d:X×X[0,+),d(x,y)= ( x y ) 2 ,

with constant s=2. Consider mappings T,S:XX, Tx= 1 4 x, Sx= 1 8 x, and the comparison function φ:[0,+)[0,+), φ(t)= t t + 1 . Clearly, (X,d) is a complete metric space, and S is continuous with respect to d, so we have to verify the contraction condition (3.1). There are three cases to be considered.

Case I. y=2x. Hence Tx=Sy, d(Tx,Sy)=0, and, therefore, the inequality (3.1) holds.

Case II. y>2x. Then 1 8 y> 1 4 x, and

2 d ( T x , S y ) = 2 ( 1 8 y 1 4 x ) 2 1 32 y 2 49 64 + 49 y 2 y 2 = φ ( 49 32 y 2 ) = φ ( 2 d ( y , S y ) ) = φ ( max { 2 d ( x , T x ) , 2 d ( y , S y ) , ( d ( x , S y ) + d ( T x , y ) ) } ) .

Thus in this case the contraction condition (3.1) holds.

Case III. y<2x. Then

2 d ( T x , S y ) = 2 ( 1 4 x 1 8 y ) 2 1 8 x 2 9 8 x 2 = φ ( 2 d ( x , T x ) ) φ ( max { 2 d ( x , T x ) , 2 d ( y , S y ) , ( d ( x , S y ) + d ( T x , y ) ) } ) .

Therefore, we showed that the contraction condition (3.1) is satisfied in all cases. Thus we can apply our Theorem 1, and T and S have a unique common fixed point u=0.

References

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

    Google Scholar 

  2. Caristi J: Fixed point theorems for mapping satisfying inwardness conditions. Trans. Am. Math. Soc. 1976, 215: 241–251.

    Article  MathSciNet  Google Scholar 

  3. Hicks TL: Fixed point theorems for quasi-metric spaces. Math. Jpn. 1988, 33(2):231–236.

    MathSciNet  Google Scholar 

  4. Altun I, Durmaz G: Some fixed point results in cone metric spaces. Rend. Circ. Mat. Palermo 2009, 58: 319–325. 10.1007/s12215-009-0026-y

    Article  MathSciNet  Google Scholar 

  5. Choudhury BS, Metiya N: Coincidence point and fixed point theorems in ordered cone metric spaces. J. Adv. Math. Stud. 2012, 5(2):20–31.

    MathSciNet  Google Scholar 

  6. Aydi H, Shatanawi W, Postolache M, Mustafa Z, Tahat N: Theorems for Boyd-Wong type contractions in ordered metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 359054

    Google Scholar 

  7. Chandok S, Postolache M: Fixed point theorem for weakly Chatterjea-type cyclic contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 28

    Google Scholar 

  8. Shatanawi W, Postolache M: Common fixed point theorems for dominating and weak annihilator mappings in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 271

    Google Scholar 

  9. Shatanawi W, Pitea A: Fixed and coupled fixed point theorems of omega-distance for nonlinear contraction. Fixed Point Theory Appl. 2013., 2013: Article ID 275

    Google Scholar 

  10. Altun I, Simsek H: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud. 2008, 1(1–2):1–8.

    MathSciNet  Google Scholar 

  11. Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 2011, 4(2):1–12.

    MathSciNet  Google Scholar 

  12. Khan AR, Abbas M, Nazir T, Ionescu C: Fixed points of multivalued contractive mappings in partial metric spaces. Abstr. Appl. Anal. 2014., 2014: Article ID 230708

    Google Scholar 

  13. Shatanawi W, Postolache M: Coincidence and fixed point results for generalized weak contractions in the sense of Berinde. Fixed Point Theory Appl. 2013., 2013: Article ID 54

    Google Scholar 

  14. Menger K: Statistical metrics. Proc. Natl. Acad. Sci. USA 1942, 28: 535–537. 10.1073/pnas.28.12.535

    Article  MathSciNet  Google Scholar 

  15. Cosentino M, Salimi P, Vetro P: Fixed point on metric-type spaces. Acta Math. Sci. 2014, 34(4):1–17.

    Article  MathSciNet  Google Scholar 

  16. Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27: 385–389. 10.1016/0165-0114(88)90064-4

    Article  MathSciNet  Google Scholar 

  17. Gregori V, Sapena A: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 2002, 125: 245–252. 10.1016/S0165-0114(00)00088-9

    Article  MathSciNet  Google Scholar 

  18. Ionescu C, Rezapour S, Samei M: Fixed points of some new contractions on intuitionistic fuzzy metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 168

    Google Scholar 

  19. Haghi RH, Postolache M, Rezapour S: On T -stability of the Picard iteration for generalized ϕ -contraction mappings. Abstr. Appl. Anal. 2012., 2012: Article ID 658971

    Google Scholar 

  20. Miandaragh MA, Postolache M, Rezapour S: Some approximate fixed point results for generalized alpha-contractive mappings. Sci. Bull. ‘Politeh.’ Univ. Buchar., Ser. A, Appl. Math. Phys. 2013, 75(2):3–10.

    MathSciNet  Google Scholar 

  21. Bakhtin IA: The contraction mapping principle in almost metric spaces. 30. In Functional Analysis. Ul’yanovsk Gos. Ped. Inst., Ul’yanovsk; 1989:26–37.

    Google Scholar 

  22. Aydi H, Bota MF, Karapinar E, Moradi S: A common fixed point for weak ϕ -contractions on b -metric spaces. Fixed Point Theory 2012, 13(2):337–346.

    MathSciNet  Google Scholar 

  23. Roshan JR, Parvaneh V, Sedghi S, Shobkolaei N, Shatanawi W: Common fixed points of almost generalized ( ψ , φ ) s -contractive mappings in ordered b -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 130

    Google Scholar 

  24. Berinde V: Generalized contractions in quasimetric spaces. Preprint 3. In Seminar on Fixed Point Theory. “Babeş-Bolyai” University, Cluj-Napoca; 1993:3–9.

    Google Scholar 

  25. Păcurar M: A fixed point result for ϕ -contractions and fixed points on b -metric spaces without the boundness assumption. Fasc. Math. 2010, 43(1):127–136.

    Google Scholar 

  26. Hussain N, Shah MH: KKM mappings in cone b -metric spaces. Comput. Math. Appl. 2011, 61(4):1677–1684.

    Article  MathSciNet  Google Scholar 

  27. Huang H, Xu S: Fixed point theorems of contractive mappings in cone b -metric spaces and applications. Fixed Point Theory Appl. 2013., 2013: Article ID 112

    Google Scholar 

  28. Hussain N, Dorić N, Kadelburg Z, Radenović S: Suzuki-type fixed point results in metric type spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 126

    Google Scholar 

  29. Berinde V, Borcut M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74(15):4889–4897. 10.1016/j.na.2011.03.032

    Article  MathSciNet  Google Scholar 

  30. Shah MH, Hussain N: Nonlinear contractions in partially ordered quasi b -metric spaces. Commun. Korean Math. Soc. 2012, 27: 117–128. 10.4134/CKMS.2012.27.1.117

    Article  MathSciNet  Google Scholar 

  31. Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 1–8. 10.1080/00036810701714164

    Article  MathSciNet  Google Scholar 

  32. Ćirić L, Cakić N, Rojović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 131294

    Google Scholar 

  33. Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 2008, 136(5):1861–1869.

    Article  Google Scholar 

  34. Singh SL, Prasad B: Some coincidence theorems and stability of iterative procedures. Comput. Math. Appl. 2008, 55: 2512–2520. 10.1016/j.camwa.2007.10.026

    Article  MathSciNet  Google Scholar 

  35. Boriceanu M: Strict fixed point theorems for multivalued operators in b -metric spaces. Int. J. Mod. Math. 2009, 4(3):285–301.

    MathSciNet  Google Scholar 

  36. Hussain N, Kadelburg Z, Radenović S, Al-Solami F: Comparison functions and fixed point results in partial metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 605781

    Google Scholar 

Download references

Acknowledgements

Rade Lazović was supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ariana Pitea.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. 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

Verify currency and authenticity via CrossMark

Cite this article

Shatanawi, W., Pitea, A. & Lazović, R. Contraction conditions using comparison functions on b-metric spaces. Fixed Point Theory Appl 2014, 135 (2014). https://doi.org/10.1186/1687-1812-2014-135

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1812-2014-135

Keywords

  • b-metric space
  • common fixed point
  • contraction condition
  • comparison function