Open Access

A generalization on weak contractions in partially ordered b-metric spaces and its application to quadratic integral equations

Journal of Inequalities and Applications20142014:355

https://doi.org/10.1186/1029-242X-2014-355

Received: 23 May 2014

Accepted: 5 September 2014

Published: 24 September 2014

Abstract

We introduce the notion of almost generalized ( ψ , φ , L ) -contractive mappings, and establish the coincidence and common fixed point results for this class of mappings in partially ordered complete b-metric spaces. Our results extend and improve several known results from the context of ordered metric spaces to the setting of ordered b-metric spaces. As an application, we prove the existence of a unique solution to a class of nonlinear quadratic integral equations.

Keywords

fixed point common fixed point coincidence point integral equations b-metric space partially ordered set

1 Introduction

Fixed points theorems in partially ordered metric spaces were firstly obtained in 2004 by Ran and Reurings [1], and then by Nieto and Rodríguez-López [2]. In this direction several authors obtained further results under weak contractive conditions (see, e.g., [38]). Berinde initiated in [9] the concept of almost contractions and obtained several interesting fixed point theorems. This has been a subject of intense study since then; see, e.g., [1020]. Some authors used related notions as ‘condition (B)’ (Babu et al. [21]) and ‘almost generalized contractive condition’ for two maps (Ćirić et al. [22]), and for four maps (Aghajani et al. [23]). See also a note by Pacurar [15]. On the other hand, the concept of b-metric space was introduced by Czerwik in [24]. After that, several interesting results of the existence of fixed point for single-valued and multivalued operators in b-metric spaces have been obtained (see [2540]). Pacurar [41] proved some results on sequences of almost contractions and fixed points in b-metric spaces. Recently, Hussain and Shah [42] obtained results on KKM mappings in cone b-metric spaces. Using the concepts of partially ordered metric spaces, almost generalized contractive condition, and b-metric spaces, we define a new concept of almost generalized ( ψ , φ , L ) -contractive condition. In this paper, some coincidence and common fixed point theorems for mappings satisfying almost generalized ( ψ , φ , L ) -contractive condition in the setup of partially ordered complete b-metric spaces are proved. Consistent with [43] and [[40], p.264], the following definitions and results will be needed in the sequel.

Definition 1.1 [43]

Let X be a (nonempty) set and s 1 be a given real number. A function d : X × X R + is said to be a b-metric space iff for all x , y , z X , the following conditions are satisfied:
  1. (i)

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

     
  2. (ii)

    d ( x , y ) = d ( y , x ) ,

     
  3. (iii)

    d ( x , y ) s [ d ( x , z ) + d ( z , y ) ] .

     

The pair ( X , d ) is called a b-metric space with the parameter s.

It should be noted that the class of b-metric spaces is effectively larger than that of metric spaces, since a b-metric is a metric, when s = 1 .

The following example shows that in general a b-metric does not necessarily need to be a metric (see, also, [40]).

Example 1.1 [44]

Let ( X , d ) be a metric space and ρ ( x , y ) = ( d ( x , y ) ) p , where p > 1 is a real number. Then ρ is a b-metric with s = 2 p 1 . However, if ( X , d ) is a metric space, then ( X , ρ ) is not necessarily a metric space. For example, if X = R is the set of real numbers and d ( x , y ) = | x y | is the usual Euclidean metric, then ρ ( x , y ) = ( x y ) s is a b-metric on with s = 2 , but it is not a metric on .

Also, the following example of a b-metric space is given in [45].

Example 1.2 [45]

Let X be the set of Lebesgue measurable functions on [ 0 , 1 ] such that 0 1 | f ( x ) | 2 d x < . Define D : X × X [ 0 , ) by D ( f , g ) = 0 1 | f ( x ) g ( x ) | 2 d x . As ( 0 1 | f ( x ) g ( x ) | 2 d x ) 1 2 is a metric on X, then, from the previous example, D is a b-metric on X, with s = 2 , where the b-metric D is defined with D ( x , y ) = d ( x , y ) , d is a cone metric (also see [4649]).

Khamsi [50] also showed that each cone metric space over a normal cone has a b-metric structure.

Definition 1.2 [6]

We shall say that the mapping T is g-nondecreasing if
g x g y T x T y .

2 Main results

Throughout the paper, let Ψ be the family of all functions ψ : [ 0 , ) [ 0 , ) satisfying the following conditions:
  1. (a)

    ψ is continuous,

     
  2. (b)

    ψ is nondecreasing,

     
  3. (c)

    ψ ( 0 ) = 0 < ψ ( t ) for every t > 0 .

     
We denote by Φ the set of all functions φ : [ 0 , ) [ 0 , ) satisfying the following conditions:
  1. (i)

    φ is right continuous,

     
  2. (ii)

    φ is nondecreasing,

     
  3. (iii)

    φ ( t ) < t for every t > 0 .

     
Let ( X , d , ) be a partially ordered b-metric space and T : X X and g : X X be two mappings. Set
M ( x , y ) = max { d ( g x , g y ) , d ( g x , T x ) , d ( g y , T y ) , d ( g x , T y ) + d ( g y , T x ) 2 s }
and
N ( x , y ) = min { d ( g x , T x ) , d ( g y , T y ) , d ( g x , T y ) , d ( g y , T x ) } .

Now, we introduce the following definition.

Definition 2.1 Let ( X , d , ) be a partially ordered b-metric space. We say that T : X X is an almost generalized ( ψ , φ , L ) -contractive mapping with respect to g : X X for some ψ Ψ , φ Φ , and L 0 if
ψ ( s 3 d ( T x , T y ) ) φ ( ψ ( M ( x , y ) ) ) + L ψ ( N ( x , y ) )
(2.1)

for all x , y X with g x g y .

Now, we establish some results for the existence of coincidence point and common fixed point of mappings satisfying almost generalized ( ψ , φ , L ) -contractive condition in the setup of partially ordered b-metric spaces. The first result in this paper is the following coincidence point theorem.

Theorem 2.1 Suppose that ( X , d , ) is a partially ordered complete b-metric space. Let T : X X be an almost generalized ( ψ , φ , L ) -contractive mapping with respect to g : X X , and T and g are continuous such that T is a monotone g-nondecreasing mapping, commutative with g and T ( X ) g ( X ) . If there exists x 0 X such that g x 0 T x 0 , then T and g have a coincidence point in X.

Proof By the given assumptions, there exists x 0 X such that g x 0 T x 0 . Since T ( X ) g ( X ) , we can define x 1 X such that g x 1 = T x 0 , then g x 0 T x 0 = g x 1 . Also there exists x 2 X such that g x 2 = T x 1 . Since T is a monotone g-nondecreasing mapping, we have
g x 1 = T x 0 T x 1 = g x 2 .
Continuing in this way, we construct a sequence { x n } in X such that for all n = 0 , 1 , 2 ,  ,
g x n + 1 = T x n
(2.2)
for which
g x 0 g x 1 g x 2 g x n g x n + 1 .
(2.3)
If there exists k 0 N such that g x k 0 + 1 = g x k 0 , then g x k 0 = T x k 0 . This means that x k 0 is a coincidence point of T, g, and the proof is finished. Thus, g x n + 1 g x n for all n N . From (2.2) and (2.3) and the inequality (2.1) with ( x , y ) = ( x n , x n + 1 ) , we have
ψ ( d ( g x n + 1 , g x n + 2 ) ) ψ ( s 3 d ( g x n + 1 , g x n + 2 ) ) = ψ ( s 3 d ( T x n , T x n + 1 ) ) φ ( ψ ( M ( x n , x n + 1 ) ) ) + L ψ ( N ( x n , x n + 1 ) ) ,
(2.4)
where
M ( x n , x n + 1 ) = max { d ( g x n , g x n + 1 ) , d ( g x n , T x n ) , d ( g x n + 1 , T x n + 1 ) , d ( g x n , T x n + 1 ) + d ( g x n + 1 , T x n ) 2 s } = max { d ( g x n , g x n + 1 ) , d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) , d ( g x n , g x n + 2 ) 2 s }
and
N ( x n , x n + 1 ) = min { d ( g x n , T x n ) , d ( g x n + 1 , T x n + 1 ) , d ( g x n , T x n + 1 ) , d ( g x n + 1 , T x n ) } = 0 .
Since
d ( g x n , g x n + 2 ) 2 s d ( g x n , g x n + 1 ) + d ( g x n + 1 , g x n + 2 ) 2 max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) } ,
then we get
M ( x n , x n + 1 ) = max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) } , N ( x n , x n + 1 ) = 0 .
(2.5)
By (2.4) and (2.5), we have
ψ ( d ( g x n + 1 , g x n + 2 ) ) φ ( ψ ( max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) } ) ) .
(2.6)
Suppose that max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) } = d ( g x n + 1 , g x n + 2 ) > 0 for some n N , then by (2.6)
ψ ( d ( g x n + 1 , g x n + 2 ) ) φ ( ψ ( d ( g x n + 1 , g x n + 2 ) ) ) < ψ ( d ( g x n + 1 , g x n + 2 ) ) ;
a contradiction. Hence,
max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) } = d ( g x n , g x n + 1 )
and thus
ψ ( d ( g x n + 1 , g x n + 2 ) ) φ ( ψ ( d ( g x n , g x n + 1 ) ) ) < ψ ( d ( g x n , g x n + 1 ) ) .
Thus, we get
ψ ( d ( g x n + 1 , g x n + 2 ) ) < ψ ( d ( g x n , g x n + 1 ) )
for all n N . Now, from
ψ ( d ( g x n , g x n + 1 ) ) φ ( ψ ( d ( g x n 1 , g x n ) ) ) φ 2 ( ψ ( d ( g x n 2 , g x n 1 ) ) ) φ n ( ψ ( d ( g x 0 , g x 1 ) ) )
and the property of φ, we obtain lim n ψ ( d ( g x n , g x n + 1 ) ) = 0 , and consequently
lim n d ( g x n , g x n + 1 ) = 0 .
(2.7)
Now, we shall prove that { g x n } is a Cauchy sequence in ( X , d ) . Suppose, on the contrary, that { g x n } is not a Cauchy sequence. Then there exist ϵ > 0 and subsequences { g x m ( k ) } , { g x n ( k ) } of { g x n } with m ( k ) > n ( k ) k such that
d ( g x n ( k ) , g x m ( k ) ) ϵ .
(2.8)
Additionally, corresponding to n ( k ) , we may choose m ( k ) such that it is the smallest integer satisfying (2.8) and m ( k ) > n ( k ) k . Thus,
d ( g x n ( k ) , g x m ( k ) 1 ) < ϵ .
(2.9)
Using the triangle inequality in b-metric space and (2.8) and (2.9) we obtain
ϵ d ( g x m ( k ) , g x n ( k ) ) s d ( g x m ( k ) , g x m ( k ) 1 ) + s d ( g x m ( k ) 1 , g x n ( k ) ) < s d ( g x m ( k ) , g x m ( k ) 1 ) + s ϵ .
Taking the upper limit as k and using (2.7) we obtain
ϵ lim sup k d ( g x n ( k ) , g x m ( k ) ) s ϵ .
(2.10)
Also
ϵ d ( g x n ( k ) , g x m ( k ) ) s d ( g x n ( k ) , g x m ( k ) + 1 ) + s d ( g x m ( k ) + 1 , g x m ( k ) ) s 2 d ( g x n ( k ) , g x m ( k ) ) + s 2 d ( g x m ( k ) , g x m ( k ) + 1 ) + s d ( g x m ( k ) + 1 , g x m ( k ) ) s 2 d ( g x n ( k ) , g x m ( k ) ) + ( s 2 + s ) d ( g x m ( k ) , g x m ( k ) + 1 ) .
So from (2.7) and (2.10), we have
ϵ s lim sup k d ( g x n ( k ) , g x m ( k ) + 1 ) s 2 ϵ .
(2.11)
Also
ϵ d ( g x m ( k ) , g x n ( k ) ) s d ( g x m ( k ) , g x n ( k ) + 1 ) + s d ( g x n ( k ) + 1 , g x n ( k ) ) s 2 d ( g x m ( k ) , g x n ( k ) ) + s 2 d ( g x n ( k ) , g x n ( k ) + 1 ) + s d ( g x n ( k ) + 1 , g x n ( k ) ) s 2 d ( g x m ( k ) , g x n ( k ) ) + ( s 2 + s ) d ( g x n ( k ) , g x n ( k ) + 1 ) .
So from (2.7) and (2.10), we have
ϵ s lim sup k d ( g x m ( k ) , g x n ( k ) + 1 ) s 2 ϵ .
(2.12)
Also
d ( g x n ( k ) + 1 , g x m ( k ) ) s d ( g x n ( k ) + 1 , g x m ( k ) + 1 ) + s d ( g x m ( k ) + 1 , g x m ( k ) ) ,
so from (2.7) and (2.12), we have
ϵ s 2 lim sup k d ( g x n ( k ) + 1 , g x m ( k ) + 1 ) .
(2.13)
Linking (2.7), (2.10), (2.11) together with (2.12) we get
lim sup k M ( x n ( k ) , x m ( k ) ) = max { lim sup k d ( g x n ( k ) , g x m ( k ) ) , lim sup k d ( g x n ( k ) , g x n ( k ) + 1 ) , lim sup k d ( g x m ( k ) , g x m ( k ) + 1 ) , lim sup k d ( g x n ( k ) , g x m ( k ) + 1 ) + lim sup k d ( g x m ( k ) , g x n ( k ) + 1 ) 2 s } max { s ϵ , 0 , 0 , s 2 ϵ + s 2 ϵ 2 s } = s ϵ .
So,
lim sup k M ( x n ( k ) , x m ( k ) ) ϵ s .
(2.14)
Similarly, we have
lim sup k N ( x n ( k ) , x m ( k ) ) = 0 .
(2.15)
Since m ( k ) > n ( k ) from (2.2), we have
g x n ( k ) g x m ( k ) .
Thus,
ψ ( s 3 d ( g x n ( k ) + 1 , g x m ( k ) + 1 ) ) = ψ ( s 3 d ( T x n ( k ) , T x m ( k ) ) ) φ ( ψ ( M ( x n ( k ) , x m ( k ) ) ) ) + L ψ ( N ( x n ( k ) , x m ( k ) ) ) .
Passing to the upper limit as k , and using (2.13), (2.14), and (2.15), we get
ψ ( s ϵ ) ψ ( s 3 lim sup k d ( g x n ( k ) + 1 , g x m ( k ) + 1 ) ) = lim sup k ψ ( s 3 d ( g x n ( k ) + 1 , g x m ( k ) + 1 ) ) = lim sup k ψ ( s 3 d ( T x n ( k ) , T x m ( k ) ) ) lim sup k φ ( ψ ( M ( x n ( k ) , x m ( k ) ) ) ) + lim sup k L ψ ( N ( x n ( k ) , x m ( k ) ) ) = φ ( ψ ( lim sup k M ( x n ( k ) , x m ( k ) ) ) ) + L ψ ( lim sup k N ( x n ( k ) , x m ( k ) ) ) φ ( ψ ( ϵ s ) ) < ψ ( s ϵ ) ,
which is a contradiction. Thus, we proved that { g x n } is a Cauchy sequence in ( X , d ) . Since X is a complete b-metric space, there exists x X such that
lim n g x n + 1 = x .
(2.16)
From the commutativity of T and g, we have
g ( g x n + 1 ) = g ( T ( x n ) ) = T ( g x n ) .
(2.17)
Letting n in (2.17) and from the continuity of T and g, we get
g x = lim n g ( g x n + 1 ) = lim n T ( g x n ) = T ( lim n g x n ) = T ( x ) .

This implies that x is a coincidence point of T and g. This completes the proof. □

Now, we will prove the following result.

Theorem 2.2 Suppose that ( X , d , ) is a partially ordered complete b-metric space. Let T : X X be an almost generalized ( ψ , φ , L ) -contractive mapping with respect to g : X X , T is a g-nondecreasing mapping and T ( X ) g ( X ) . Also suppose
if  { g x n } X  is a nondecreasing sequence with  g x n g z  in  g X , then  g x n g z , g z g ( g z ) n  hold .
(2.18)

Also suppose gX is closed. If there exists x 0 X such that g x 0 T x 0 , then T and g have a coincidence. Further, if T and g commute at their coincidence points, then T and g have a common fixed point.

Proof As in the proof of Theorem 2.1, we can show that { g x n } is a Cauchy sequence. Since gX is a closed, there exists x X such that
lim n g x n + 1 = g x .
(2.19)
Now we show that x is a coincidence point of T and g. Since from (2.18) and (2.19) we have g x n g x for all n, then by the triangle inequality in a b-metric space and (2.1), we get
d ( g x , T x ) s d ( g x , g x n + 1 ) + s d ( g x n + 1 , T x ) = s d ( g x , g x n + 1 ) + s d ( T x n , T x ) , ψ ( d ( g x , T x ) ) lim n ψ ( s d ( T x n , T x ) ) lim n ψ ( s 3 d ( T x n , T x ) ) ψ ( d ( g x , T x ) ) lim n [ φ ( ψ ( M ( x n , x ) ) ) + L ψ ( N ( x n , x ) ) ] ψ ( d ( g x , T x ) ) φ ( ψ ( d ( g x , T x ) ) ) < ψ ( d ( g x , T x ) ) .
Indeed,
lim n M ( x n , x ) = lim n max { d ( g x n , g x ) , d ( g x n , T x n ) , d ( g x , T x ) , d ( g x n , T x ) + d ( g x , T x n ) 2 s } = d ( g x , T x )
and
lim n N ( x n , x ) = lim n min { d ( g x n , T x n ) , d ( g x , T x ) , d ( g x n , T x ) , d ( g x , T x n ) } = 0 .
Hence d ( g x , T x ) = 0 , that is, T x = g x . Thus we proved that T and g have a coincidence. Suppose now that T and g commute at x. Set y = T x = g x . Then
T y = T ( g x ) = g ( T x ) = g y .
Since from (2.18) we have g x g ( g x ) = g y and as g x = T x and g y = T y , from (2.1) we obtain
ψ ( d ( T x , T y ) ) ψ ( s 3 d ( T x , T y ) ) φ ( ψ ( M ( x , y ) ) ) + L ψ ( N ( x , y ) ) = φ ( ψ ( max { d ( g x , g y ) , d ( g x , T x ) , d ( g y , T y ) , d ( g x , T y ) + d ( g y , T x ) 2 s } ) ) + L ψ ( min { d ( g x , T x ) , d ( g y , T y ) , d ( g x , T y ) , d ( g y , T x ) } ) = φ ( ψ ( d ( T x , T y ) ) ) < ψ ( d ( T x , T y ) ) .

Hence d ( T x , T y ) = 0 , that is, y = T x = T y . Therefore, T y = g y = y . Thus we proved that T and g have a common fixed point. □

In the following, we deduce some fixed point theorems from our main results given by Theorems 2.1 and 2.2.

Corollary 2.3 Let ( X , d , ) be a partially ordered complete b-metric space and T : X X is a nondecreasing mapping. Suppose there exist ψ Ψ , φ Φ , and L 0 such that
ψ ( s 3 d ( T x , T y ) ) φ ( ψ ( M ( x , y ) ) ) + L ψ ( N ( x , y ) ) ,
where
M ( x , y ) = max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) + d ( y , T x ) 2 s }
and
N ( x , y ) = min { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) }
for all x , y X with x y . Also suppose either
  1. (a)

    if { x n } X is a nondecreasing sequence with x n z in X, then x n z , for all n, holds, or

     
  2. (b)

    T is continuous.

     

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

Example 2.1 Let X be the set of Lebesgue measurable functions on [ 0 , 1 ] such that 0 1 | x ( t ) | d t < . Define D : X × X [ 0 , ) by
D ( x , y ) = ( 0 1 | x ( t ) y ( t ) | d t ) 2 .
Then D is a b-metric on X, with s = 2 . Also, this space can also be equipped with a partial order given by
x , y X , x y x ( t ) y ( t ) for any  t [ a , b ] .
The operator T : X X defined by
T x ( t ) = t n + e t + 2 4 ln ( | x ( t ) | + 1 ) .
(2.20)
Now, we prove that T has a fixed point. For all x , y X with x y , we have
2 3 D ( T x , T y ) = 2 3 ( 0 1 | T x ( t ) T y ( t ) | d t ) 2 2 2 0 1 | 2 4 ln ( | x ( t ) | + 1 ) 2 4 ln ( | y ( t ) | + 1 ) | d t 0 1 | ( ln ( | x ( t ) | + 1 ) ln ( | y ( t ) | + 1 ) ) | d t 0 1 ln ( | x ( t ) | + 1 | y ( t ) | + 1 ) d t 0 1 ln ( 1 + | x ( t ) y ( t ) | | y ( t ) | + 1 ) d t ln ( 1 + 0 1 | x ( t ) y ( t ) | d t ) ln ( 1 + ( 0 1 | x ( t ) y ( t ) | d t ) 2 ) ln ( 1 + D ( x , y ) ) .

Now, if we define φ ( t ) = ln ( 1 + t ) , ψ ( t ) = t , and x 0 = 0 . Thus, by Corollary 2.3 we see that T has a fixed point.

Remark 2.1 Corollary 2.3 extends and generalizes many existing fixed point theorems in the literature [2, 3, 51, 52].

The following result is the immediate consequence of Corollary 2.3.

Corollary 2.4 Let ( X , d , ) be a partially ordered complete b-metric space and T : X X is a nondecreasing mapping. Suppose there exists φ Φ such that
s 3 d ( T x , T y ) φ ( max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) + d ( y , T x ) 2 s } )
(2.21)
for all x , y X with x y . Also suppose either
  1. (a)

    if { x n } X is a nondecreasing sequence with x n z in X, then x n z , for all n, holds, or

     
  2. (b)

    T is continuous.

     

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

Remark 2.2 Corollary 2.4 is a generalization to [[3], Theorem 1.3].

Taking φ ( t ) = λ t , 0 < λ < 1 , in Corollary 2.4 we obtain the following generalization of the results in [1, 53].

Corollary 2.5 Let ( X , d , ) be a partially ordered complete b-metric space and T : X X is a nondecreasing mapping. Suppose there exists φ Φ such that
s 3 d ( T x , T y ) λ max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) + d ( y , T x ) 2 s }
for all x , y X with x y . Also suppose either
  1. (a)

    if { x n } X is a nondecreasing sequence with x n z in X, then x n z , for all n, holds, or

     
  2. (b)

    T is continuous.

     

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

Corollary 2.6 Let ( X , d , ) be a partially ordered complete b-metric space and T : X X is a nondecreasing mapping. Suppose there exist ψ Ψ and 0 λ < 1 such that
ψ ( s 3 d ( T x , T y ) ) λ ψ ( d ( x , y ) )
for all x , y X with x y . Also suppose either
  1. (a)

    if { x n } X is a nondecreasing sequence with x n z in X, then x n z , for all n, holds, or

     
  2. (b)

    T is continuous.

     

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

3 Application to integral equations

Here, in this section, we wish to study the existence of a unique solution to a nonlinear quadratic integral equation, as an application to the our fixed point theorem. Consider the integral equation
x ( t ) = h ( t ) + λ 0 1 k ( t , s ) f ( s , x ( s ) ) d s , t I = [ 0 , 1 ] , λ 0 .
(3.1)

Let Γ denote the class of those functions γ : [ 0 , + ) [ 0 , + ) for which γ Φ and ( γ ( t ) ) p γ ( t p ) , for all p 1 .

For example, γ 1 ( t ) = k t , where 0 k < 1 and γ 2 ( t ) = t t + 1 are in Γ.

We will analyze (3.1) under the following assumptions:

(a1) f : I × R R is continuous monotone nondecreasing in x, f ( t , x ) 0 and there exist constant 0 L < 1 and γ Γ such that for all x , y R and x y
| f ( t , x ) f ( t , y ) | L γ ( x y ) .

(a2) h : I R is a continuous function.

(a3) k : I × I R is continuous in t I for every s I and measurable in s I for all t I such that
0 1 k ( t , s ) d s K

and k ( t , s ) 0 .

(a4) There exists α C ( I ) such that
α ( t ) h ( t ) + λ 0 1 k ( t , s ) f ( s , α ( s ) ) d s .

(a5) L p λ p K p 1 2 3 p 3 .

We consider the space X = C ( I ) of continuous functions defined on I = [ 0 , 1 ] with the standard metric given by
ρ ( x , y ) = sup t I | x ( t ) y ( t ) | for  x , y C ( I ) .
This space can also be equipped with a partial order given by
x , y C ( I ) , x y x ( t ) y ( t ) for any  t I .
Now for p 1 , we define
d ( x , y ) = ( ρ ( x , y ) ) p = ( sup t I | x ( t ) y ( t ) | ) p = sup t I | x ( t ) y ( t ) | p for  x , y C ( I ) .

It is easy to see that ( X , d ) is a complete b-metric space with s = 2 p 1 [44].

For any x , y X and each t I , max { x ( t ) , y ( t ) } and min { x ( t ) , y ( t ) } belong to X and are upper and lower bounds of x, y, respectively. Therefore, for every x , y X , one can take max { x , y } , min { x , y } X which are comparable to x, y. Now, we formulate the main result of this section.

Theorem 3.1 Under assumptions (a1)-(a5), (3.1) has a unique solution in C ( I ) .

Proof We consider the operator T : X X defined by
T ( x ) ( t ) = h ( t ) + λ 0 1 k ( t , s ) f ( s , x ( s ) ) d s for  t I .
By virtue of our assumptions, T is well defined (this means that if x X then T ( x ) X ). For x y , and t I we have
T ( x ) ( t ) T ( y ) ( t ) = h ( t ) + λ 0 1 k ( t , s ) f ( s , x ( s ) ) d s h ( t ) λ 0 1 k ( t , s ) f ( s , y ( s ) ) d s = λ 0 1 k ( t , s ) [ f ( s , x ( s ) ) f ( s , y ( s ) ) ] d s 0 .
Therefore, T has the monotone nondecreasing property. Also, for x y , we have
| T ( x ) ( t ) T ( y ) ( t ) | = | h ( t ) + λ 0 1 k ( t , s ) f ( s , x ( s ) ) d s h ( t ) λ 0 1 k ( t , s ) f ( s , y ( s ) ) d s | λ 0 1 k ( t , s ) | f ( s , x ( s ) ) f ( s , y ( s ) ) | d s λ 0 1 k ( t , s ) L γ ( y ( s ) x ( s ) ) d s .
Since the function γ is nondecreasing and x y , we have
γ ( y ( s ) x ( s ) ) γ ( sup t I | x ( s ) y ( s ) | ) = γ ( ρ ( x , y ) ) ,
hence
| T ( x ) ( t ) T ( y ) ( t ) | λ 0 1 k ( t , s ) L γ ( ρ ( x , y ) ) d s λ K L γ ( ρ ( x , y ) ) .
Then we obtain
d ( T ( x ) , T ( y ) ) = sup t I | T ( x ) ( t ) T ( y ) ( t ) | p { λ K L γ ( ρ ( x , y ) ) } p = λ p K p L p γ ( ρ ( x , y ) ) p λ p K p L p γ ( ρ ( x , y ) p ) = λ p K p L p γ ( d ( x , y ) ) λ p K p L p φ ( max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) + d ( y , T x ) 2 s } ) 1 2 3 p 3 φ ( max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) + d ( y , T x ) 2 s } ) .

This proves that the operator T satisfies the contractive condition (2.21) appearing in Corollary 2.4. Also, let α, β be the functions appearing in assumption (a4); then, by (a4), we get α T ( α ) . So, (3.1) has a solution and the proof is complete. □

Example 3.1 Consider the following functional integral equation:
x ( t ) = t 2 1 + t 4 + 1 27 0 1 e s sin t 2 ( 1 + t ) | x ( s ) | 1 + | x ( s ) | d s
(3.2)
for t [ 0 , 1 ] . Observe that this equation is a special case of (3.1) with
h ( t ) = t 2 1 + t 4 , k ( t , s ) = e s 1 + t , f ( t , x ) = sin t 2 | x | 1 + | x | .
Indeed, by using γ ( t ) = 1 3 t we see that γ Φ and ( γ ( t ) ) p = ( 1 3 t ) p = 1 3 p t p 1 3 t p = γ ( t p ) , for all p 1 . Further, for arbitrarily fixed x , y R such that x y and for t [ 0 , 1 ] we obtain
| f ( t , x ) f ( t , y ) | = | sin t 2 | x | 1 + | x | sin t 2 | y | 1 + | y | | 1 2 | x y | = 1 6 γ ( x y ) .
Thus, the function f satisfies assumption (a1) with L = 1 6 . It is also easily seen that h is a continuous function. Further, notice that the function k is continuous in t I for every s I and measurable in s I for all t I and k ( t , s ) 0 . Moreover, we have
0 1 k ( t , s ) d s = 0 1 e s 1 + t d s = 1 e 1 1 + t 1 e 1 2 3 = K .
If we put α ( t ) = 3 t 2 4 ( 1 + t 4 ) , we have
α ( t ) = 3 t 2 4 ( 1 + t 4 ) t 2 1 + t 4 t 2 1 + t 4 + 1 27 0 1 e s sin t 2 ( 1 + t ) | α ( s ) | 1 + | α ( s ) | d s = h ( t ) + λ 0 1 k ( t , s ) f ( s , α ( s ) ) d s .
This shows that assumption (a4) holds. Taking L = 1 6 , K = 2 3 and λ = 1 27 , then inequality L p λ p K p 1 2 3 p 3 appearing in assumption (a5) has the following form:
1 6 p × 1 27 p × 2 p 3 p 1 2 3 p 3 .

It is easily seen that each number p 1 satisfies the above inequality. Consequently, all the conditions of Theorem 3.1 are satisfied. Hence the integral equation (3.2) has a unique solution in C ( I ) .

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Mashhad Branch, Islamic Azad University
(2)
Department of Mathematics, Sari Branch, Islamic Azad University
(3)
Department of Mathematics, Karaj Branch, Islamic Azad University

References

  1. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some application to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435-1443. 10.1090/S0002-9939-03-07220-4MathSciNetView ArticleMATHGoogle Scholar
  2. 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 ArticleMATHGoogle Scholar
  3. Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008,87(1):109-116. 10.1080/00036810701556151MathSciNetView ArticleMATHGoogle Scholar
  4. Altun I, Durmaz G: Some fixed point theorems on ordered cone metric spaces. Rend. Circ. Mat. Palermo 2009, 58: 319-325. 10.1007/s12215-009-0026-yMathSciNetView ArticleMATHGoogle Scholar
  5. Beg I, Rashid Butt A: Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces. Nonlinear Anal. 2009, 71: 3699-3704. 10.1016/j.na.2009.02.027MathSciNetView ArticleMATHGoogle Scholar
  6. Ćirić L, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008: Article ID 131294, (2008)Google Scholar
  7. Grana 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 ArticleMATHGoogle Scholar
  8. Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 2009, 71: 3402-3410.MathSciNetView ArticleMATHGoogle Scholar
  9. Berinde V: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 2004, 9: 43-53.MathSciNetMATHGoogle Scholar
  10. Berinde V: Some remarks on a fixed point theorem for Ćirić-type almost contractions. Carpath. J. Math. 2009, 25: 157-162.MathSciNetMATHGoogle Scholar
  11. Berinde V: Common fixed points of noncommuting almost contractions in cone metric spaces. Math. Commun. 2010, 15: 229-241.MathSciNetMATHGoogle Scholar
  12. Berinde V: Approximating common fixed points of noncommuting almost contractions in metric spaces. Fixed Point Theory 2010, 11: 179-188.MathSciNetMATHGoogle Scholar
  13. Hussain N, Ðorić D, Kadelburg Z, Radenović S: Suzuki-type fixed point results in metric type spaces. Fixed Point Theory Appl. 2012: Article ID 126, (2012)Google Scholar
  14. Jovanović M, Kadelburg Z, Radenović S: Common fixed point results in metric-type spaces. Fixed Point Theory Appl. 2010: Article ID 978121, (2010)Google Scholar
  15. Pacurar M: Fixed point theory for cyclic Berinde operators. Fixed Point Theory 2012, 11: 419-428.MathSciNetMATHGoogle Scholar
  16. Radenović S, Kadelburg Z: Quasi-contractions on symmetric and cone symmetric spaces. Banach J. Math. Anal. 2011,5(1):38-50. 10.15352/bjma/1313362978MathSciNetView ArticleMATHGoogle Scholar
  17. Radenović S, Kadelburg Z, Jandrlić D, Jandrlić A: Some results on weakly contractive maps. Bull. Iran. Math. Soc. 2012,38(3):625-645.MathSciNetMATHGoogle Scholar
  18. Shah MH, Simić S, Hussain N, Sretenović A, Radenović S: Common fixed points theorems for occasionally weakly compatible pairs on cone metric type spaces. J. Comput. Anal. Appl. 2012,14(2):290-297.MathSciNetMATHGoogle Scholar
  19. Shukla S: Partial rectangular metric spaces and fixed point theorems. Sci. World J. 2014: Article ID 756298, (2014)Google Scholar
  20. Suzuki T: Fixed point theorems for Berinde mappings. Bull. Kyushu Inst. Technol., Pure Appl. Math. 2011, 58: 13-19.MathSciNetMATHGoogle Scholar
  21. Babu GVR, Sandhya ML, Kameswari MVR: A note on a fixed point theorem of Berinde on weak contractions. Carpath. J. Math. 2008, 24: 8-12.MathSciNetMATHGoogle Scholar
  22. Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 2011, 217: 5784-5789. 10.1016/j.amc.2010.12.060MathSciNetView ArticleMATHGoogle Scholar
  23. 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 ArticleMATHGoogle Scholar
  24. Czerwik S: Contraction mappings in b -metric spaces. Acta Math. Inform. Univ. Ostrav. 1993, 1: 5-11.MathSciNetMATHGoogle Scholar
  25. Aghajani A, Arab R: Fixed points of ( ψ , ϕ , θ ) -contractive mappings in partially ordered b -metric spaces and application to quadratic integral equation. Fixed Point Theory Appl. 2013: Article ID 245, (2013)Google Scholar
  26. Azam A, Mehmood N, Ahmad J, Radenović S: Multivalued fixed point theorems in cone b -metric spaces. J. Inequal. Appl. 2013.2013: Article ID 582, (2013)Google Scholar
  27. Boriceanu M: Fixed point theory for multivalued generalized contraction on a set with two b -metrics. Stud. Univ. Babeş-Bolyai, Math. 2009,LIV(3):3-14.MathSciNetMATHGoogle Scholar
  28. Boriceanu M: Strict fixed point theorems for multivalued operators in b -metric spaces. Int. J. Mod. Math. 2009,4(3):285-301.MathSciNetMATHGoogle Scholar
  29. George, R, Radenović, S, Reshma, KP, Shukla, S: Rectangular b-metric spaces and contraction principle. J. Nonlinear Sci. Appl. (2014, in press)Google Scholar
  30. Hussain N, Parvaneh V, Roshan JR, Kadelburg Z: Fixed points of cyclic ( ψ , φ , L , A , B ) -contractive mappings in ordered b -metric spaces with applications. Fixed Point Theory Appl. 2013: Article ID 256, (2013)Google Scholar
  31. Mustafa Z, Roshan JR, Parvaneh V:Coupled coincidence point results for ( ψ , φ ) -weakly contractive mappings in partially ordered G b -metric spaces. Fixed Point Theory Appl. 2013: Article ID 206, (2013)Google Scholar
  32. Mustafa Z, Roshan JR, Parvaneh V, Kadelburg Z: Some common fixed point results in ordered partial b -metric spaces. J. Inequal. Appl. 2013: Article ID 562, (2013)Google Scholar
  33. Popović B, Radenović S, Shukla S: Fixed point results to TVS-cone b -metric spaces. Gulf J. Math. 2013, 1: 51-64.Google Scholar
  34. Roshan JR, Shobkolaei N, Sedhi S, Parvaneh V, Radenović S:Common fixed point theorems for three maps in discontinuous G b -metric spaces. Acta Math. Sci. Ser. B 2014,34(5):1-12.MathSciNetGoogle Scholar
  35. Shukla S: Partial b -metric spaces and fixed point theorems. Mediterr. J. Math. 2014, 11: 703-711. 10.1007/s00009-013-0327-4MathSciNetView ArticleMATHGoogle Scholar
  36. Zabihi F, Razani A: Fixed point theorems for hybrid rational Geraghty contractive mappings in ordered b -metric spaces. J. Appl. Math. 2014: Article ID 929821, 2014Google Scholar
  37. Roshan JR, Parvaneh V, Altun I: Some coincidence point results in ordered b -metric spaces and applications in a system of integral equations. Appl. Math. Comput. 2014, 226: 725-737.MathSciNetView ArticleGoogle Scholar
  38. Roshan, JR, Parvaneh, V, Kadelburg, Z: Common fixed point theorems for weakly isotone increasing mappings in ordered b-metric spaces. J. Nonlinear Sci. Appl. (to appear). www.tjnsa.comGoogle Scholar
  39. 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: Article ID 159, (2013)Google Scholar
  40. Singh SL, Prasad B: Some coincidence theorems and stability of iterative procedures. Comput. Math. Appl. 2008, 55: 2512-2520. 10.1016/j.camwa.2007.10.026MathSciNetView ArticleMATHGoogle Scholar
  41. Pacurar M: Sequences of almost contractions and fixed points in b -metric spaces. An. Univ. Vest. Timiş., Ser. Mat.-Inform. 2010, 48: 125-137.MathSciNetMATHGoogle Scholar
  42. Hussain N, Shah MH: KKM mappings in cone b -metric spaces. Comput. Math. Appl. 2011, 62: 1677-1684. 10.1016/j.camwa.2011.06.004MathSciNetView ArticleMATHGoogle Scholar
  43. Czerwik S: Nonlinear set-valued contraction mappings in b -metric spaces. Atti Semin. Mat. Fis. Univ. Modena 1998,46(2):263-276.MathSciNetMATHGoogle Scholar
  44. Aghajani, A, Abbas, M, Roshan, JR: Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces. Math. Slovaca (in press)Google Scholar
  45. Khamsi MA, Hussain N: KKM mappings in metric type spaces. Nonlinear Anal. 2010,73(9):3123-3129. 10.1016/j.na.2010.06.084MathSciNetView ArticleMATHGoogle Scholar
  46. Ðukić D, Kadelburg Z, Radenović S: Fixed points of Geraghty-type mappings in various generalized metric spaces. Abstr. Appl. Anal. 2011: Article ID 561245, (2011)Google Scholar
  47. Parvaneh V, Roshan JR, Radenović S: Existence of tripled coincidence point in ordered b -metric spaces and application to a system of integral equations. Fixed Point Theory Appl. 2013: Article ID 130, (2013)Google Scholar
  48. 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 ArticleMATHGoogle Scholar
  49. An TV, Dung NV, Kadelburg Z, Radenović S: Various generalizations of metric spaces and fixed point theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2014. 10.1007/s13398-014-0173-7Google Scholar
  50. Khamsi MA: Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010: Article ID 315398 (2010). 10.1155/2010/315398Google Scholar
  51. Altman M: A fixed point theorem in compact metric spaces. Am. Math. Mon. 1975, 82: 827-829. 10.2307/2319801MathSciNetView ArticleGoogle Scholar
  52. Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 1984, 30: 1-9. 10.1017/S0004972700001659MathSciNetView ArticleMATHGoogle Scholar
  53. Ray BK: On Ciric’s fixed point theorem. Fundam. Math. 1977,94(3):221-229.MATHGoogle Scholar

Copyright

© Allahyari 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/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.