Skip to content

Advertisement

  • Research
  • Open Access

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

Journal of Inequalities and Applications20142014:219

https://doi.org/10.1186/1029-242X-2014-219

  • Received: 9 October 2013
  • Accepted: 15 May 2014
  • Published:

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.

Keywords

  • ordered set
  • metric space
  • fixed point

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 : X X be a continuous mapping such that
d ( T x , T y ) α d ( x , T x ) d ( y , T y ) d ( x , y ) + β d ( x , y )
(1.1)

for all distinct points x , y X 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 : X X be a non-decreasing mapping such that
d ( T x , T y ) M ( x , y ) ψ ( M ( x , y ) )
(1.2)
for all distinct points x , y X with y x 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 = { ϕ ϕ : [ 0 , ) [ 0 , )  is continuous and non-decreasing with  ϕ ( t ) = 0  if and only if  t = 0 } and Ψ = { ψ ψ : [ 0 , ) [ 0 , )  is lower semi continuous , ψ ( t ) > 0  for all t > 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 : X X 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 : X X 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 , y X with y x where ϕ Φ , ψ Ψ , L 0 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  n 1 .
(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 n N . 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 n 1 . 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 n 1 . 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 α 0 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 ) = ε and 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 } ) + L min { 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 z X 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 ) = T z .
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 n N . Since T is a non-decreasing mapping, we conclude that T x n T z , or equivalently,
x n x n + 1 T z for all  n N .
(2.19)

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

To this end, we construct a new sequence { y n } as follows:
y 0 = z and y n = T y n 1 for all  n 1 .
Since z T z , 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 y X 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 T z = T y 0 y n y for all  n N .
(2.20)
If z = y then the proof is finished. Suppose that z y . 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 z T z z , then T z = 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 : X X 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 , y X with y x where ϕ Φ , ψ Ψ 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 : X X be a non-decreasing mapping such that
d ( T x , T y ) 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 ) }
(2.24)
for all distinct x , y X with y x where ψ Ψ , L 0 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 : X X be a non-decreasing mapping such that
d ( T x , T y ) k M ( x , y ) + L min { d ( x , T y ) , d ( y , T x ) , d ( x , T x ) , d ( y , T y ) } ,
(2.25)
for all distinct x , y X with y x where L 0 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 ) = ( 1 k ) 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 : X X be a non-decreasing mapping such that
d ( T x , T y ) α d ( x , T x ) d ( y , T y ) d ( x , y ) + β d ( x , y )
(2.26)
for all distinct x , y X with y x 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 , y X  there exists  z X  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 x y . By (2.28), there exists a point z X which is comparable with x and y. Without loss of generality, we choose z x . We construct a sequence { z n } as follows:
z 0 = z and z n = T z n 1 for all  n 1 .
(2.29)

Since T is non-decreasing, z x implies T z T x = 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 = T x = 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 , y X with y x where ϕ Φ , ψ Ψ 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 ) = | x y | for all x , y X , 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 ) = 2 t 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 z X as n + . We have z n z n + 1 for all n N .

  • 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 n N and z = 4 . Then z n z for all n N 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 n N and z = 5 . Then z n z for all n N 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 n N . Suppose that there exists p 1 such that z p = 4 . From the definition of , we get z n = z p = 4 for all n p . Thus, we have z = 4 and z n z for all n N . Now, suppose that z n = 6 for all n N . In this case, we get z = 6 and z n z for all n N and z = sup { z n } .

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

Let x , y X such that x y and x y , so we have only x = 6 and y = 4 . In particular
d ( T 6 , T 4 ) = 0 and M ( 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 ( T 4 , T 5 ) = 2 > β = α d ( 4 , T 4 ) d ( 5 , T 5 ) d ( 4 , 5 ) + β d ( 4 , 5 ) ,
for any α , β 0 such that α + β < 1 . Also, we could not apply Theorem 2 in this example. Indeed, for x = 6 and y = 4 (that is, x y and x y ), we have
0 = d ( T 6 , T 4 ) > M ( T 6 , T 4 ) ψ ( M ( T 6 , T 4 ) ) = 1 .
Example 10 Let X = [ 0 , ) be endowed with the Euclidean metric and the order given as follows:
x y ( x = y )  or  ( x , y 1 , x y ) .

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

Take x y and x y . It means that 1 x < y . In particular, d ( T x , T y ) = 0 and M ( x , y ) = y x . 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 n N . 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 x y and x y (that is, 1 x < y ), we have
d ( T x , T y ) = 0 > M ( x , y ) ψ ( M ( x , y ) ) = 2 ( y x ) .
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 ) , x X } { ( ( 0 , 1 ) , ( 1 , 1 ) ) } . We also consider T : X X given by T ( ( 0 , 1 ) ) = ( 0 , 1 ) , T ( ( 1 , 0 ) ) = ( 1 , 0 ) and T ( ( 1 , 1 ) ) = ( 0 , 1 ) . Take ϕ ( t ) = 3 t and ψ ( t ) = 2 t . 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 x y and x y , then necessarily x = ( 0 , 1 ) and y = ( 1 , 1 ) . Then
d 2 ( T x , T y ) = d 2 ( ( 0 , 1 ) , ( 0 , 1 ) ) = 0 and M ( 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 ( T x , T y ) = 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 , β 0 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 ( T x , T y ) > 2 2 2 = M ( x , y ) ψ ( M ( x , y ) ) .

Notes

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar
(2)
Department of Mathematics, Atilim University, Incek, Ankara, 06836, Turkey
(3)
Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia
(4)
Institut Supérieur d’Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1-4011, Hammam Sousse, Tunisie
(5)
Department of Mathematics, College of Education of Jubail, P.O. Box 12020, Industrial Jubail, 31961, Saudi Arabia

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. Jaggi DJ: Some unique fixed point theorems. Indian J. Pure Appl. Math. 1977,8(2):223–230.MathSciNetGoogle Scholar
  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 190701Google Scholar
  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 46Google Scholar
  5. Agarwal RA, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151MathSciNetView ArticleGoogle Scholar
  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.2009.10.023MathSciNetView ArticleGoogle Scholar
  7. Altun I, Simsek H: Some fixed point theorems on ordered metric spaces and applications. Fixed Point Theory Appl. 2010., 2010: Article ID 621469Google Scholar
  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 248Google Scholar
  9. Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 2011,4(2):1–12.MathSciNetGoogle Scholar
  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.MathSciNetGoogle Scholar
  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.2011.07.006MathSciNetView ArticleGoogle Scholar
  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 359054Google Scholar
  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 44Google Scholar
  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.2011.11.022MathSciNetView ArticleGoogle Scholar
  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.MathSciNetGoogle Scholar
  16. Chandok S, Postolache M: Fixed point theorem for weakly Chatterjea-type cyclic contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 28Google Scholar
  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.MathSciNetGoogle Scholar
  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.MathSciNetGoogle Scholar
  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 152Google Scholar
  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.MathSciNetGoogle Scholar
  21. Ðorić D: Common fixed point for generalized ( ψ , ϕ ) -weak contractions. Appl. Math. Lett. 2009, 22: 1896–1900. 10.1016/j.aml.2009.08.001MathSciNetView ArticleGoogle Scholar
  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.2005.10.017MathSciNetView ArticleGoogle Scholar
  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.2009.01.240MathSciNetView ArticleGoogle Scholar
  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 658971Google Scholar
  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 96Google Scholar
  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.2010.03.062MathSciNetView ArticleGoogle Scholar
  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. 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/S1995080213010083MathSciNetView ArticleGoogle Scholar
  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/s11083-005-9018-5MathSciNetView ArticleGoogle Scholar
  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.MathSciNetGoogle Scholar
  31. Miandaragh MA, Postolache M, Rezapour S: Approximate fixed points of generalized convex contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 255Google Scholar
  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.2011.03.038MathSciNetView ArticleGoogle Scholar
  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-zGoogle Scholar
  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/s10114-005-0769-0MathSciNetView ArticleGoogle Scholar
  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.2007.11.026MathSciNetView ArticleGoogle Scholar
  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.2011.12.038MathSciNetView ArticleGoogle Scholar
  37. Petrusel A, Rus IA: Fixed point theorems in ordered L -spaces. Proc. Am. Math. Soc. 2006, 134: 411–418.MathSciNetView ArticleGoogle Scholar
  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/S0002-9939-03-07220-4MathSciNetView ArticleGoogle Scholar
  39. Rhoades BE: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 1977, 226: 257–290.MathSciNetView ArticleGoogle Scholar
  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. Radenović S, Kadelburg Z: Generalized weak contractions in partially ordered metric spaces. Comput. Math. Appl. 2010, 60: 1776–1783. 10.1016/j.camwa.2010.07.008MathSciNetView ArticleGoogle Scholar
  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.2012.06.063MathSciNetView ArticleGoogle Scholar
  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 60Google Scholar
  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 815870Google Scholar
  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 271Google Scholar
  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 54Google Scholar
  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 275Google Scholar
  48. Shatanawi W, Pitea A: ω -distance and coupled fixed point in G -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 208Google Scholar
  49. Shatanawi W, Pitea A: Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 153Google Scholar
  50. Zamfirescu T: Fix point theorems in metric spaces. Arch. Math. 1972, 23: 292–298. 10.1007/BF01304884MathSciNetView ArticleGoogle Scholar
  51. Zhang Q, Song Y: Fixed point theory for generalized ϕ -weak contraction. Appl. Math. Lett. 2009, 22: 75–78. 10.1016/j.aml.2008.02.007MathSciNetView ArticleGoogle Scholar
  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 20Google Scholar
  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.2010.02.039MathSciNetView ArticleGoogle Scholar
  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/S0004972700001659MathSciNetView ArticleGoogle Scholar
  55. Berinde V: Approximation fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 2004, 9: 43–53.MathSciNetGoogle Scholar
  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.MathSciNetGoogle Scholar
  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.2011.11.061MathSciNetView ArticleGoogle Scholar
  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.MathSciNetGoogle Scholar
  59. Berinde V: General constructive fixed point theorem for Ćirić-type almost contractions in metric spaces. Carpath. J. Math. 2008,24(2):10–19.MathSciNetGoogle Scholar

Copyright

© Mustafa et al.; licensee Springer. 2014

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

Advertisement