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A discussion on generalized almost contractions via rational expressions in partially ordered metric spaces

Abstract

The main purpose of this paper is to give some fixed point results for mappings involving generalized (ϕ,ψ)-contractions in partially ordered metric spaces. Our results generalize, extend, and unify several well-known comparable results in the literature (Jaggi in Indian J. Pure Appl. Math. 8(2):223-230, 1977, Harjani et al. in Nonlinear Anal. 71:3403-3410, 2009, Luong and Thuan in Fixed Point Theory Appl. 2011:46, 2011). The presented results are supported by three illustrative examples.

MSC: 46N40, 47H10, 54H25, 46T99.

1 Introduction and preliminaries

The Banach contraction mapping principle [1] is one of the pivotal results of analysis. It is widely considered as the source of metric fixed point theory. Also, its significance lies in its application in a vast number of branches of mathematics. Generalizations of this principle have been investigated heavily (see Jaggi [2], Harjani et al. [3], Luong and Thuan [4]). In particular, in 1977, Jaggi [2] proved the following theorem satisfying a contractive condition of a rational type.

Theorem 1 Let (X,d) be a complete metric space. Let T:XX be a continuous mapping such that

d(Tx,Ty)α d ( x , T x ) d ( y , T y ) d ( x , y ) +βd(x,y)
(1.1)

for all distinct points x,yX where α,β[0,1) with α+β<1. Then T has a unique fixed point.

Existence of fixed point in partially ordered sets has been recently studied in [353].

Recently, Harjani et al. [3] proved the ordered version of Theorem 1. Very recently, Luong and Thuan [4] generalized the results of [3] and proved the following.

Theorem 2 Let (X,) be a partially ordered set. Suppose there exists a metric d such that (X,d) is a metric space. Let T:XX be a non-decreasing mapping such that

d(Tx,Ty)M(x,y)ψ ( M ( x , y ) )
(1.2)

for all distinct points x,yX with yx where ψ:[0,)[0,) is a lower semi-continuous function with the property that ψ(t)=0 if and only if t=0, and

M(x,y)=max { d ( x , T x ) d ( y , T y ) d ( x , y ) , d ( x , y ) } .
(1.3)

Also, assume either

  1. (i)

    T is continuous or

  2. (ii)

    if { x n } is a non-decreasing sequence in X such that x n x, then x=sup{ x n }.

If there exists x 0 X such that x 0 T x 0 , then T has a fixed point.

Set Φ={ϕϕ:[0,)[0,) is continuous and non-decreasing with ϕ(t)=0 if andonly if t=0} and Ψ={ψψ:[0,)[0,) is lower semi continuous,ψ(t)>0 for allt>0, and ψ(0)=0}. For some work on the class of Φ or the class of Ψ, we refer the reader to [21, 51, 54].

In 2004, Berinde [55] introduced an almost contraction, a new class of contractive type mappings which exhibits totally different features more than the one of the particular results incorporated [1, 16, 39, 50], i.e., an almost contraction generally does not have a unique fixed point; see Example 1 in [55]. Thereafter, many authors presented several interesting and useful facts about almost contractions; see [42, 5659].

The purpose of this article is to generalize the above results for a mapping T:XX involving a generalized (ϕ,ψ)-almost contraction. Some examples are also presented to show that our results are effective.

2 Main result

Our essential result is given as follows.

Theorem 3 Let (X,) be a partially ordered set. Suppose there exists a metric d such that (X,d) is a complete metric space. Let T:XX be a non-decreasing mapping which satisfies the inequality

ϕ ( d ( T x , T y ) ) ϕ ( M ( x , y ) ) ψ ( M ( x , y ) ) + L min { d ( x , T y ) , d ( y , T x ) , d ( x , T x ) , d ( y , T y ) }
(2.1)

for all distinct points x,yX with yx where ϕΦ, ψΨ, L0 and

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

Also, assume either

  1. (i)

    T is continuous or

  2. (ii)

    if { x n } is a non-decreasing sequence in X such that x n x, then x=sup{ x n }.

If there exists x 0 X such that x 0 T x 0 , then T has a fixed point.

Proof Let x 0 X such that x 0 T x 0 . We define a sequence { x n } in X as follows:

x n =T x n 1 for n1.
(2.2)

Since T is a non-decreasing mapping together with (2.2), we have x 2 =T x 1 . Inductively, we obtain

x 0 x 1 x 2 x n 1 x n x n + 1 .
(2.3)

Assume that there exists n 0 such that x n 0 = x n 0 + 1 . Since x n 0 = x n 0 + 1 =T x n 0 , then T has a fixed point. Suppose that x n x n + 1 for all nN. Thus, by (2.3) we have

x 0 < x 1 < x 2 << x n 1 < x n < x n + 1 <.
(2.4)

Regarding (2.4), the condition (2.1) implies that

ϕ ( d ( x n , x n + 1 ) ) = ϕ ( d ( T x n 1 , T x n ) ) ϕ ( M ( x n 1 , x n ) ) ψ ( M ( x n 1 , x n ) ) + L min { d ( x n 1 , T x n ) , d ( T x n 1 , x n ) , d ( x n 1 , T x n 1 ) , d ( x n , T x n ) } ϕ ( M ( x n 1 , x n ) ) ψ ( M ( x n 1 , x n ) ) + L min { d ( x n 1 , x n + 1 ) , d ( x n , x n ) , d ( x n 1 , x n ) , d ( x n , x n + 1 ) } = ϕ ( M ( x n 1 , x n ) ) ψ ( M ( x n 1 , x n ) ) ,
(2.5)

where

M ( x n 1 , x n ) = max { d ( x n 1 , T x n 1 ) d ( x n , T x n ) d ( x n 1 , x n ) , d ( x n 1 , x n ) } = max { d ( x n , x n + 1 ) , d ( x n 1 , x n ) } .

Suppose that M( x n 1 , x n )=d( x n , x n + 1 ) for some n1. Then the inequality (2.5) turns into

ϕ ( d ( x n , x n + 1 ) ) ϕ ( d ( x n , x n + 1 ) ) ψ ( d ( x n , x n + 1 ) ) .

Regarding (2.4) and the property of ψ, this is a contradiction. Thus, M( x n 1 , x n )=d( x n 1 , x n ) for all n1. Therefore, the inequality (2.5) yields

ϕ ( d ( x n , x n + 1 ) ) ϕ ( d ( x n 1 , x n ) ) ψ ( d ( x n 1 , x n ) ) <ϕ ( d ( x n 1 , x n ) ) .
(2.6)

Since ϕ is non-decreasing, we have d( x n , x n + 1 )d( x n 1 , x n ). Consequently, {d( x n 1 , x n )} is a decreasing sequence of positive real numbers which is bounded below. So, there exists α0 such that lim n d( x n 1 , x n )=α. We claim that α=0. Suppose, to the contrary, that α>0. By taking the limit of the supremum in the relation ϕ(d( x n , x n + 1 ))ϕ(d( x n 1 , x n ))ψ(d( x n 1 , x n )), as n, we get

ϕ(α)ϕ(α)ψ(α)<ϕ(α),

which is a contradiction. Hence, we conclude that α=0, that is,

lim n d( x n 1 , x n )=0.
(2.7)

We prove that the sequence { x n } is Cauchy in X. Suppose, to the contrary, that { x n } is not a Cauchy sequence. So, there exists ε>0 such that

d( x m ( k ) , x n ( k ) )ε,
(2.8)

where { x m ( k ) } and { x n ( k ) } are subsequences of { x n } with

n(k)>m(k)k.
(2.9)

Moreover, n(k) is chosen to be the smallest integer satisfying (2.8). Thus, we have

d( x m ( k ) , x n ( k ) 1 )<ε.
(2.10)

By the triangle inequality, we get

ε d ( x m ( k ) , x n ( k ) ) d ( x m ( k ) , x n ( k ) 1 ) + d ( x n ( k ) 1 , x n ( k ) ) < ε + d ( x n ( k ) 1 , x n ( k ) ) .

Keeping (2.7) in mind and letting n in the above inequality, we get

lim n d( x m ( k ) , x n ( k ) )=ε.
(2.11)

Due to the triangle inequality, we have

d( x m ( k ) , x n ( k ) )d( x m ( k ) , x m ( k ) 1 )+d( x m ( k ) 1 , x n ( k ) 1 )+d( x n ( k ) 1 , x n ( k ) )
(2.12)

and

d( x m ( k ) 1 , x n ( k ) 1 )d( x m ( k ) 1 , x m ( k ) )+d( x m ( k ) , x n ( k ) )+d( x n ( k ) , x n ( k ) 1 ).
(2.13)

By using (2.7), (2.11), and letting n in (2.12) and (2.13), we get

lim n d( x m ( k ) 1 , x n ( k ) 1 )=ε.
(2.14)

Analogously, we derive

lim n d( x m ( k ) , x n ( k ) 1 )=εand lim n d( x m ( k ) 1 , x n ( k ) )=ε.
(2.15)

Since m(k)<n(k) we have x m ( k ) 1 < x n ( k ) 1 . By (2.1) we have

ϕ ( d ( x m ( k ) , x n ( k ) ) ) = ϕ ( d ( T x m ( k ) 1 , T x n ( k ) 1 ) ) ϕ ( M ( x m ( k ) 1 , x n ( k ) 1 ) ) ψ ( M ( x m ( k ) 1 , x n ( k ) 1 ) ) + L min { d ( x n ( k ) 1 , T x m ( k ) 1 ) , d ( x m ( k ) 1 , T x n ( k ) 1 ) , d ( x m ( k ) 1 , T x m ( k ) 1 ) , d ( x n ( k ) 1 , T x n ( k ) 1 ) } ϕ ( M ( x m ( k ) 1 , x n ( k ) 1 ) ) ψ ( M ( x m ( k ) 1 , x n ( k ) 1 ) ) + L min { d ( x n ( k ) 1 , x m ( k ) ) , d ( x m ( k ) 1 , x n ( k ) ) , d ( x m ( k ) 1 , x m ( k ) ) , d ( x n ( k ) 1 , x n ( k ) ) } ,
(2.16)

where

M ( x m ( k ) 1 , x n ( k ) 1 ) = max { d ( x m ( k ) 1 , T x m ( k ) 1 ) d ( x n ( k ) 1 , T x n ( k ) 1 ) d ( x m ( k ) 1 , x n ( k ) 1 ) , d ( x m ( k ) 1 , x n ( k ) 1 ) } = max { d ( x m ( k ) 1 , x m ( k ) ) d ( x n ( k ) 1 , x n ( k ) ) d ( x m ( k ) 1 , x n ( k ) 1 ) , d ( x m ( k ) 1 , x n ( k ) 1 ) } .
(2.17)

Letting n in (2.16) (and hence in (2.17)), and taking (2.7), (2.11), (2.14), and (2.15) into account, we obtain

ϕ(ε)ϕ ( max { 0 , ε } ) ψ ( max { 0 , ε } ) +Lmin{ε,ε,0,0}<ϕ(ε),
(2.18)

which is a contradiction. Thus, { x n } is a Cauchy sequence in X. Since X is a complete metric space, there exists zX such that lim n x n =z.

We will show that z is a fixed point of T. Assume that (i) holds. Then by the continuity of T, we have

z= lim n x n = lim n T x n 1 =T ( lim n x n 1 ) =Tz.

Suppose that (ii) holds. Since { x n } is a non-decreasing sequence and lim n x n =z then z=sup{ x n }. Hence, x n z for all nN. Since T is a non-decreasing mapping, we conclude that T x n Tz, or equivalently,

x n x n + 1 Tzfor all nN.
(2.19)

Then z=sup{ x n }, and we get zTz.

To this end, we construct a new sequence { y n } as follows:

y 0 =zand y n =T y n 1 for all n1.

Since zTz, we have y 0 T y 0 = y 1 . Hence we find that { y n } is a non-decreasing sequence. By repeating the discussion above, one can conclude that { y n } is Cauchy. Thus there exists yX such that lim n y n =y. By (ii), we have y=sup{ y n } and so we have y n y. From (2.19), we get

x n <z= y 0 Tz=T y 0 y n yfor all nN.
(2.20)

If z=y then the proof is finished. Suppose that zy. On account of (2.20), the expression (2.1) implies that

ϕ ( d ( x n + 1 , y n + 1 ) ) = ϕ ( d ( T x n , T y n ) ) ϕ ( M ( x n , y n ) ) ψ ( M ( x n , y n ) ) + L min { d ( x n , T y n ) , d ( y n , T x n ) , d ( x n , T x n ) , d ( y n , T y n ) } ϕ ( M ( x n , y n ) ) ψ ( M ( x n , y n ) ) + L min { d ( x n , y n + 1 ) , d ( y n , x n + 1 ) , d ( x n , x n + 1 ) , d ( y n , y n + 1 ) } ,
(2.21)

where

M ( x n , y n ) = max { d ( x n , T x n ) d ( y n , T y n ) d ( x n , y n ) , d ( x n , y n ) } = max { d ( x n , x n + 1 ) d ( y n , y n + 1 ) d ( x n , y n ) , d ( x n , y n ) } .
(2.22)

Letting n in (2.21) (and hence (2.22)), we obtain

ϕ ( d ( y , z ) ) ϕ ( d ( y , z ) ) ψ ( d ( y , z ) ) <ϕ ( d ( y , z ) )

which is a contradiction. So y=z and we have zTzz, then Tz=z. □

If we take L=0 in Theorem 3 we get the following result.

Theorem 4 Let (X,) be a partially ordered set. Suppose there exists a metric d such that (X,d) is a complete metric space. Let T:XX be a non-decreasing mapping which satisfies the inequality

ϕ ( d ( T x , T y ) ) ϕ ( M ( x , y ) ) ψ ( M ( x , y ) )
(2.23)

for all distinct x,yX with yx where ϕΦ, ψΨ and

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

Also, assume either

  1. (i)

    T is continuous or

  2. (ii)

    if { x n } is a non-decreasing sequence in X such that x n x, then x=sup{ x n }.

If there exists x 0 X such that x 0 T x 0 , then T has a fixed point.

Other corollaries could be derived.

Corollary 5 Let (X,) be a partially ordered set. Suppose there exists a metric d such that (X,d) is a complete metric space. Let T:XX be a non-decreasing mapping such that

d(Tx,Ty)M(x,y)ψ ( M ( x , y ) ) +Lmin { d ( x , T y ) , d ( y , T x ) , d ( x , T x ) , d ( y , T y ) }
(2.24)

for all distinct x,yX with yx where ψΨ, L0 and

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

Also, assume either

  1. (i)

    T is continuous or

  2. (ii)

    if { x n } is a non-decreasing sequence in X such that x n x, then x=sup{ x n }.

If there exists x 0 X such that x 0 T x 0 , then T has a fixed point.

Proof Take ϕ(t)=t in Theorem 3. □

Corollary 6 Let (X,) be a partially ordered set. Suppose there exists a metric d X such that (X,d) is a complete metric space. Let T:XX be a non-decreasing mapping such that

d(Tx,Ty)kM(x,y)+Lmin { d ( x , T y ) , d ( y , T x ) , d ( x , T x ) , d ( y , T y ) } ,
(2.25)

for all distinct x,yX with yx where L0 and

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

Also, assume either

  1. (i)

    T is continuous or

  2. (ii)

    if { x n } is a non-decreasing sequence in X such that x n x, then x=sup{ x n }.

If there exists x 0 X such that x 0 T x 0 , then T has a fixed point.

Proof Take ψ(t)=(1k)ψ(t) for all t[0,) in Corollary 5. □

Corollary 7 Let (X,) be a partially ordered set. Suppose there exists a metric d such that (X,d) is a complete metric space. Let T:XX be a non-decreasing mapping such that

d(Tx,Ty)α d ( x , T x ) d ( y , T y ) d ( x , y ) +βd(x,y)
(2.26)

for all distinct x,yX with yx where α,β[0,1) with α+β<1. Also, assume either

  1. (i)

    T is continuous or

  2. (ii)

    if { x n } is a non-decreasing sequence in X such that x n x, then x=sup{ x n }.

If there exists x 0 X such that x 0 T x 0 , then T has a fixed point.

Proof Take L=0 and k=α+β for all t[0,) in Corollary 6. Indeed,

d ( T x , T y ) α d ( x , T x ) d ( y , T y ) d ( x , y ) + β d ( x , y ) ( α + β ) max { d ( x , T x ) d ( y , T y ) d ( x , y ) , d ( x , y ) } .
(2.27)

 □

Theorem 8 In addition to the hypotheses of Theorem  3, assume that

for every x,yX there exists zX that is comparable to x and y,
(2.28)

then T has a unique fixed point.

Proof Suppose, to the contrary, that x and y are fixed points of T where xy. By (2.28), there exists a point zX which is comparable with x and y. Without loss of generality, we choose zx. We construct a sequence { z n } as follows:

z 0 =zand z n =T z n 1 for all n1.
(2.29)

Since T is non-decreasing, zx implies TzTx=x. By induction, we get z n x.

If x= z N 0 for some N 0 1 then z n =T z n 1 =Tx=x for all n N 0 1. So lim n z n =x. Analogously, we get lim n z n =y, which completes the proof.

Consider the other case, that is, x z n for all n=0,1,2, . Then, by (2.1), we observe that

ϕ ( d ( x , z n ) ) = ϕ ( d ( T x , T z n 1 ) ) ϕ ( M ( x , z n 1 ) ) ψ ( M ( x , z n 1 ) ) + L min { d ( x , T x ) , d ( z n 1 , T z n 1 ) , d ( x , T z n 1 ) , d ( z n 1 , T z n 1 ) } = ϕ ( M ( x , z n 1 ) ) ψ ( M ( x , z n 1 ) )
(2.30)

for all distinct x,yX with yx where ϕΦ, ψΨ and

M ( x , z n 1 ) = max { d ( x , T x ) d ( z n 1 , T z n 1 ) d ( x , z n 1 ) , d ( x , z n 1 ) } = max { d ( x , x ) d ( z n 1 , z n ) d ( x , z n 1 ) , d ( x , z n 1 ) } = d ( x , z n 1 ) .
(2.31)

Thus,

ϕ ( d ( x , z n ) ) ϕ ( d ( x , z n 1 ) ) ψ ( d ( x , z n 1 ) ) <ϕ ( d ( x , z n ) ) ,

which is a contradiction. This ends the proof. □

Remark

  • Corollary 5 is a generalization of Theorem 2.1 of Luong and Thuan [4].

  • Corollary 7 (with L=0) corresponds to Theorem 2.2 and Theorem 2.3 of Harjani, López and Sadarangani [3].

  • Theorem 2.28 generalizes Theorem 2.4 of Luong and Thuan [4].

Now, we give some examples illustrating our results.

Example 9 Let X={4,5,6} be endowed with the usual metric d(x,y)=|xy| for all x,yX, and :={(4,4),(5,5),(6,6),(6,4)}. Consider the mapping

T= ( 4 5 6 4 6 4 ) .

We define the functions ϕ,ψ:[0,+)[0,+) by ϕ(t)=2t and ψ(t)= 3 2 t. Now, we will check that all the hypotheses required by Theorem 4 (Theorem 3 with L=0) are satisfied.

First, X has the property: if { x n } is a non-decreasing sequence in X such that x n x, then x=sup{ x n }. Indeed, let { z n } be a non-decreasing sequence in X with respect to such that z n zX as n+. We have z n z n + 1 for all nN.

  • If z 0 =4, then z 0 =4 z 1 . From the definition of , we have z 1 =4. By induction, we get z n =4 for all nN and z=4. Then z n z for all nN and z=sup{ z n }.

  • If z 0 =5, then z 0 =5 z 1 . From the definition of , we have z 1 =5. By induction, we get z n =5 for all nN and z=5. Then z n z for all nN and z=sup{ z n }.

  • If z 0 =6, then z 0 =6 z 1 . From the definition of , we have z 1 {6,4}. By induction, we get z n {6,4} for all nN. Suppose that there exists p1 such that z p =4. From the definition of , we get z n = z p =4 for all np. Thus, we have z=4 and z n z for all nN. Now, suppose that z n =6 for all nN. In this case, we get z=6 and z n z for all nN and z=sup{ z n }.

Thus, we proved that in all cases, we have z=sup{ z n }.

Let x,yX such that xy and xy, so we have only x=6 and y=4. In particular

d(T6,T4)=0andM(6,4)=2,

so (2.23) holds easily. On the other hand, it is obvious that T is a non-decreasing mapping with respect to and there exists x 0 =6 such that x 0 T x 0 . All the hypotheses of Theorem 4 are verified and u=4 is a fixed point of T.

Note that Theorem 1 is not applicable. Indeed, taking x=4 and y=5

d(T4,T5)=2>β=α d ( 4 , T 4 ) d ( 5 , T 5 ) d ( 4 , 5 ) +βd(4,5),

for any α,β0 such that α+β<1. Also, we could not apply Theorem 2 in this example. Indeed, for x=6 and y=4 (that is, xy and xy), we have

0=d(T6,T4)>M(T6,T4)ψ ( M ( T 6 , T 4 ) ) =1.

Example 10 Let X=[0,) be endowed with the Euclidean metric and the order given as follows:

xy(x=y) or (x,y1,xy).

Define T:XX by Tx=x if 0x<1 and Tx=0 if x1. Define the functions ϕ,ψ:[0,+)[0,+) by ϕ(t)=4t and ψ(t)=3t.

Take xy and xy. It means that 1x<y. In particular, d(Tx,Ty)=0 and M(x,y)=yx. This implies that (2.23) holds. It is easy that X satisfies the property: if { x n } is a non-decreasing sequence in X such that x n x, then x n x for all nN. Also, the other conditions of Theorem 4 are satisfied and u=0 is a fixed point of T.

Notice that we cannot apply Theorem 1 (since T is not continuous) nor Theorem 2 to this example. Indeed, letting xy and xy (that is, 1x<y), we have

d(Tx,Ty)=0>M(x,y)ψ ( M ( x , y ) ) =2(yx).

Example 11 Let X={(0,1),(1,0),(1,1)} R 2 with the Euclidean distance d 2 . (X, d 2 ) is, obviously, a complete metric space. Moreover, we consider the order ≤ in X given by R={(x,x),xX}{((0,1),(1,1))}. We also consider T:XX given by T((0,1))=(0,1), T((1,0))=(1,0) and T((1,1))=(0,1). Take ϕ(t)=3t and ψ(t)=2t. Obviously, T is a continuous and non-decreasing mapping since (0,1)(1,1) and T(0,1)=(0,1)T(1,1)=(0,1). Let xy and xy, then necessarily x=(0,1) and y=(1,1). Then

d 2 (Tx,Ty)= d 2 ( ( 0 , 1 ) , ( 0 , 1 ) ) =0andM(x,y)= 2 ,

so (2.23) holds. Also, (0,1)T((0,1)), therefore all conditions in Theorem 4 hold and there are two fixed points which are (0,1) and (1,0). The non-uniqueness follows from the fact that the partial order ≤ is not total.

Note that Theorem 1 is not applicable. Indeed, taking x=(0,1) and y=(1,0)

d 2 (Tx,Ty)= 2 >(α+β) 2 =α d 2 ( x , T x ) d 2 ( y , T y ) d 2 ( x , y ) +β d 2 (x,y),

for any α,β0 such that α+β<1. Also, we could not apply Theorem 2 in this example. Indeed, for x=(0,1) and y=(1,1) we have

0= d 2 (Tx,Ty)> 2 2 2 =M(x,y)ψ ( M ( x , y ) ) .

References

  1. 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. 2.

    Jaggi DJ: Some unique fixed point theorems. Indian J. Pure Appl. Math. 1977,8(2):223–230.

    MathSciNet  Google Scholar 

  3. 3.

    Harjani J, López B, Sadarangani K: A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space. Abstr. Appl. Anal. 2010., 2010: Article ID 190701

    Google Scholar 

  4. 4.

    Luong NV, Thuan NX: Fixed point theorem for generalized weak contractions satisfying rational expressions in ordered metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 46

    Google Scholar 

  5. 5.

    Agarwal RA, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151

    MathSciNet  Article  Google Scholar 

  6. 6.

    Amini-Harandi A, Emami H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 2010, 72: 2238–2242. 10.1016/j.na.2009.10.023

    MathSciNet  Article  Google Scholar 

  7. 7.

    Altun I, Simsek H: Some fixed point theorems on ordered metric spaces and applications. Fixed Point Theory Appl. 2010., 2010: Article ID 621469

    Google Scholar 

  8. 8.

    Arshad M, Karapınar E, Jamshaid A: Some unique fixed point theorems for rational contractions in partially ordered metric spaces. J. Inequal. Appl. 2013., 2013: Article ID 248

    Google Scholar 

  9. 9.

    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 

  10. 10.

    Aydi H: Common fixed point results for mappings satisfying (ψ,ϕ)-weak contractions in ordered partial metric spaces. Int. J. Math. Stat. 2012,12(2):63–64.

    MathSciNet  Google Scholar 

  11. 11.

    Aydi H, Nashine HK, Samet B, Yazidi H: Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations. Nonlinear Anal. 2011,74(17):6814–6825. 10.1016/j.na.2011.07.006

    MathSciNet  Article  Google Scholar 

  12. 12.

    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 

  13. 13.

    Aydi H, Karapınar E, Postolache M: Tripled coincidence point theorems for weak f -contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 44

    Google Scholar 

  14. 14.

    Aydi H, Postolache M, Shatanawi W: Coupled fixed point results for (ψ,ϕ) -weakly contractive mappings in ordered G -metric spaces. Comput. Math. Appl. 2012,63(1):298–309. 10.1016/j.camwa.2011.11.022

    MathSciNet  Article  Google Scholar 

  15. 15.

    Chandok S, Karapınar E: Common fixed point of generalized rational type contraction mappings in partially ordered metric spaces. Thai J. Math. 2013,11(2):251–260.

    MathSciNet  Google Scholar 

  16. 16.

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

    Google Scholar 

  17. 17.

    Chandok S, Mustafa Z, Postolache M: Coupled common fixed point theorems for mixed g -monotone mappings in partially ordered G -metric spaces. Sci. Bull. ‘Politeh.’ Univ. Buchar., Ser. A, Appl. Math. Phys. 2013,75(4):11–24.

    MathSciNet  Google Scholar 

  18. 18.

    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 

  19. 19.

    Choudhury BS, Metiya N, Postolache M: A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl. 2013., 2013: Article ID 152

    Google Scholar 

  20. 20.

    Dey D, Ganguly A, Saha M: Fixed point theorems for mappings under general contractive condition of integral type. Bull. Math. Anal. Appl. 2011, 3: 27–34.

    MathSciNet  Google Scholar 

  21. 21.

    Ðorić D: Common fixed point for generalized (ψ,ϕ)-weak contractions. Appl. Math. Lett. 2009, 22: 1896–1900. 10.1016/j.aml.2009.08.001

    MathSciNet  Article  Google Scholar 

  22. 22.

    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

    MathSciNet  Article  Google Scholar 

  23. 23.

    Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 2009, 71: 3403–3410. 10.1016/j.na.2009.01.240

    MathSciNet  Article  Google Scholar 

  24. 24.

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

    Google Scholar 

  25. 25.

    Ding H-S, Li L, Radenović S: Coupled coincidence point theorems for generalized nonlinear contraction in partially ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 96

    Google Scholar 

  26. 26.

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

    MathSciNet  Article  Google Scholar 

  27. 27.

    Karapınar E: Weak φ -contraction on partial metric spaces and existence of fixed points in partially ordered sets. Math. Æterna 2011, 1: 237–244.

    Google Scholar 

  28. 28.

    Karapınar E, Marudai M, Pragadeeswarar AV: Fixed point theorems for generalized weak contractions satisfying rational expression on a ordered partial metric space. Lobachevskii J. Math. 2013,34(1):116–123. 10.1134/S1995080213010083

    MathSciNet  Article  Google Scholar 

  29. 29.

    Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5

    MathSciNet  Article  Google Scholar 

  30. 30.

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

    MathSciNet  Google Scholar 

  31. 31.

    Miandaragh MA, Postolache M, Rezapour S: Approximate fixed points of generalized convex contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 255

    Google Scholar 

  32. 32.

    Abbas M, Nazir T, Radenović S: Common fixed points of four maps in partially ordered metric spaces. Appl. Math. Lett. 2011, 24: 1520–1526. 10.1016/j.aml.2011.03.038

    MathSciNet  Article  Google Scholar 

  33. 33.

    Abbas M, Nazir T, Radenović S: Common coupled fixed points of generalized contractive mappings in partially ordered metric spaces. Positivity 2013. 10.1007/s11117-012-0219-z

    Google Scholar 

  34. 34.

    Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equation. Acta Math. Sin. Engl. Ser. 2007,23(12):2205–2212. 10.1007/s10114-005-0769-0

    MathSciNet  Article  Google Scholar 

  35. 35.

    O’Regan D, Petrusel A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 2008, 341: 1241–1252. 10.1016/j.jmaa.2007.11.026

    MathSciNet  Article  Google Scholar 

  36. 36.

    Olatinwo MO, Postolache M: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput. 2012,218(12):6727–6732. 10.1016/j.amc.2011.12.038

    MathSciNet  Article  Google Scholar 

  37. 37.

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

    MathSciNet  Article  Google Scholar 

  38. 38.

    Ran ACM, Reurings MVB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4

    MathSciNet  Article  Google Scholar 

  39. 39.

    Rhoades BE: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 1977, 226: 257–290.

    MathSciNet  Article  Google Scholar 

  40. 40.

    Radenović S, Kadelburg Z, Jandrlić A: Some results on weakly contractive maps. Bull. Iran. Math. Soc. 2012,38(3):625–645.

    Google Scholar 

  41. 41.

    Radenović S, Kadelburg Z: Generalized weak contractions in partially ordered metric spaces. Comput. Math. Appl. 2010, 60: 1776–1783. 10.1016/j.camwa.2010.07.008

    MathSciNet  Article  Google Scholar 

  42. 42.

    Shobkolaei N, Sedghi S, Roshan JR, Altun I:Common fixed point of mappings satisfying almost generalized (S,T)-contractive condition in partially ordered partial metric spaces. Appl. Math. Comput. 2012, 219: 443–452. 10.1016/j.amc.2012.06.063

    MathSciNet  Article  Google Scholar 

  43. 43.

    Shatanawi W, Postolache M: Common fixed point results of mappings for nonlinear contractions of cyclic form in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 60

    Google Scholar 

  44. 44.

    Shatanawi W, Postolache M: Some fixed point results for a G -weak contraction in G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 815870

    Google Scholar 

  45. 45.

    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 

  46. 46.

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

    Google Scholar 

  47. 47.

    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 

  48. 48.

    Shatanawi W, Pitea A: ω -distance and coupled fixed point in G -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 208

    Google Scholar 

  49. 49.

    Shatanawi W, Pitea A: Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 153

    Google Scholar 

  50. 50.

    Zamfirescu T: Fix point theorems in metric spaces. Arch. Math. 1972, 23: 292–298. 10.1007/BF01304884

    MathSciNet  Article  Google Scholar 

  51. 51.

    Zhang Q, Song Y: Fixed point theory for generalized ϕ -weak contraction. Appl. Math. Lett. 2009, 22: 75–78. 10.1016/j.aml.2008.02.007

    MathSciNet  Article  Google Scholar 

  52. 52.

    Golubović Z, Kadelburg Z, Radenović S: Common fixed points of ordered g -quasicontractions and weak contractions in ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 20

    Google Scholar 

  53. 53.

    Kadelburg Z, Radenović S, Pavlović M: Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces. Comput. Math. Appl. 2010, 59: 3148–3159. 10.1016/j.camwa.2010.02.039

    MathSciNet  Article  Google Scholar 

  54. 54.

    Khan KS, Swaleh M, Sessa S: Fixed point theorems for altering distances between the points. Bull. Aust. Math. Soc. 1984,30(1):1–9. 10.1017/S0004972700001659

    MathSciNet  Article  Google Scholar 

  55. 55.

    Berinde V: Approximation fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 2004, 9: 43–53.

    MathSciNet  Google Scholar 

  56. 56.

    Abbas M, Vetro P, Khan SH: On fixed points of Berinde’s contractive mappings in cone metric spaces. Carpath. J. Math. 2010,26(2):121–133.

    MathSciNet  Google Scholar 

  57. 57.

    Aghajani A, Radenović S, Roshan JR:Common fixed point results for four mappings satisfying almost generalized (S,T)-contractive condition in partially ordered metric spaces. Appl. Math. Comput. 2012, 218: 5665–5670. 10.1016/j.amc.2011.11.061

    MathSciNet  Article  Google Scholar 

  58. 58.

    Babu GVR, Sandhya ML, Kameswari MVR: A note on a fixed point theorem of Berinde on weak contraction. Carpath. J. Math. 2008,24(1):8–12.

    MathSciNet  Google Scholar 

  59. 59.

    Berinde V: General constructive fixed point theorem for Ćirić-type almost contractions in metric spaces. Carpath. J. Math. 2008,24(2):10–19.

    MathSciNet  Google Scholar 

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Acknowledgements

The authors express their gratitude to the referees for constructive and useful remarks and suggestions.

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Correspondence to Erdal Karapınar.

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Mustafa, Z., Karapınar, E. & Aydi, H. A discussion on generalized almost contractions via rational expressions in partially ordered metric spaces. J Inequal Appl 2014, 219 (2014). https://doi.org/10.1186/1029-242X-2014-219

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Keywords

  • ordered set
  • metric space
  • fixed point