Open Access

Coupled fixed point results on quasi-Banach spaces with application to a system of integral equations

Fixed Point Theory and Applications20132013:261

https://doi.org/10.1186/1687-1812-2013-261

Received: 4 June 2013

Accepted: 27 August 2013

Published: 7 November 2013

Abstract

The aim of this paper is to obtain coupled fixed point theorems for self-mappings defined on an ordered closed and convex subset of a quasi-Banach space. Our method of proof is different and constructive in nature. As an application of our coupled fixed point results, we establish corresponding coupled coincidence point results without any type of commutativity of underlying maps. Moreover, an application to integral equations is given to illustrate the usability of the obtained results.

MSC:47H10, 54H25, 55M20.

Keywords

quasi-Banach spacemetric-type spacecoupled fixed pointmixed monotone propertyordered setintegral equation

Dedication

Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday

1 Introduction

It is well known that the Banach contraction principle is one of the most important results in classical functional analysis. It is widely considered as the source of metric fixed point theory. Also, its significance lies in its vast applicability in a number of branches of mathematics (see [115]). The study of coupled fixed points in partially ordered metric spaces was first investigated in 1987 by Guo and Lakshmikantham [16], and then it attracted many researchers; see, for example, [3, 5] and references therein. Recently, Bhaskar and Lakshmikantham [12] presented coupled fixed point theorems for contractions in partially ordered metric spaces. Luong and Thuan [17] presented nice generalizations of these results. Alsulami et al. [3] further extended the work of Luong and Thuan to coupled coincidences in partial metric spaces. For more related work on coupled fixed points and coincidences, we refer the readers to recent results in [1330].

In recent years, several authors have obtained coupled fixed point results for various classes of mappings on the setting of many generalized metric spaces. The concept of metric-type space appeared in some works, such as Czerwik [8], Khamsi [9] and Khamsi and Hussain [10]. Metric-type space is a symmetric space with some special properties. A metric-type space can also be regarded as a triplet ( X , d , K ) , where ( X , d ) is a symmetric space and K 1 is a real number such that
d ( x , z ) K ( d ( x , y ) + d ( y , z ) )

for any x , y , z X . In this paper, we adopt a different and constructive method to prove some coupled fixed and coincidence point results for contractive mappings defined on an ordered closed convex subset of a quasi-Banach space. Moreover, an application to integral equations is given to illustrate the usability of the obtained results.

2 Preliminaries

The aim of this section is to present some notions and results used in the paper.

Definition 2.1 Let X be a non-empty set and d : X × X [ 0 , + ) . ( X , d ) is a symmetric space (also called E-space) if and only if it satisfies the following conditions:
  1. (W1)

    d ( x , y ) = 0 if and only if x = y ;

     
  2. (W2)

    d ( x , y ) = d ( y , x ) for any x , y X .

     
Symmetric spaces differ from more convenient metric spaces in the absence of triangle inequality. Nevertheless, many notions can be defined similar to those in metric spaces. For instance, in a symmetric space ( X , d ) , the limit point of a sequence { x n } is defined by
lim n + d ( x n , x ) = 0 if and only if lim n + x n = x .

Also, a sequence { x n } X is said to be a Cauchy sequence if, for every given ε > 0 , there exists a positive integer n ( ε ) such that d ( x m , x n ) < ε for all m , n n ( ε ) . A symmetric space ( X , d ) is said to be complete if and only if each of its Cauchy sequences converges to some x X .

Definition 2.2 Let X be a nonempty set and let K 1 be a given real number. A function d : X × X R + is said to be of metric type if and only if for all x , y , z X the following conditions are satisfied:
  1. (1)

    d ( x , y ) = 0 if and only if x = y ;

     
  2. (2)

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

     
  3. (3)

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

     

A triplet ( X , d , K ) is called a metric-type space.

We observe that a metric-type space is included in the class of symmetric spaces. So the notions of convergent sequence, Cauchy sequence and complete space are defined as in symmetric spaces.

Next, we give some examples of metric-type spaces.

Example 2.3 [4]

The space l p with ( 0 < p < 1 )
l p = { { x n } R : n = 1 + | x n | p < + } ,
together with the function d : l p × l p R , defined by
d ( x , y ) = ( n = 1 + | x n y n | p ) 1 / p ,

where x = { x n } , y = { y n } l p , is a metric-type space. By an elementary calculation, we obtain d ( x , y ) 2 1 / p [ d ( x , y ) + d ( y , z ) ] .

Example 2.4 [4]

The space L p ( 0 < p < 1 ) of all real functions x : [ 0 , 1 ] R such that
0 1 | x ( t ) | p d t < +
is a metric-type space if we take
d ( x , y ) = ( 0 1 | x ( t ) y ( t ) | p d t ) 1 / p

for each x , y L p . The constant K is again equal to 2 1 / p .

For more examples of metric-type (or b-metric) spaces, we refer to [8, 10, 11]. We recall that a quasi-norm defined on a real vector space X is a mapping X R + such that:
  1. (1)

    x > 0 if and only if x 0 ;

     
  2. (2)

    λ x = | λ | x for λ R and x X ;

     
  3. (3)

    x + y K [ x + y ] for all x , y X , where K 1 is a constant independent of x, y.

     

A triplet ( X , , K ) is called a quasi-Banach space.

What makes quasi-Banach spaces different from the more classical Banach spaces is the triangle inequality. In quasi-Banach spaces, the triangle inequality is allowed to hold approximately: x + y K ( x + y ) for some constant K 1 . This relaxation leads to a broad class of spaces. Lebesgue spaces L p are Banach spaces for 1 p + and only quasi-Banach spaces for 0 < p < 1 .

Remark 2.5 Let ( X , , K ) be a quasi-Banach space, then the mapping d : X × X R + defined by d ( x , y ) = x y for all x , y X is a metric-type (b-metric). In general, a metric-type (b-metric) function d is not continuous (see [23, 26]).

The following result is useful for some of the proofs in the paper.

Lemma 2.6 Let ( X , d , K ) be a metric-type space and let { x k } k = 0 n X . Then
d ( x n , x 0 ) K d ( x 0 , x 1 ) + + K n 1 d ( x n 2 , x n 1 ) + K n 1 d ( x n 1 , x n ) .

From Lemma 2.6, we deduce the following lemma.

Lemma 2.7 Let { y n } be a sequence in a metric-type space ( X , d , K ) such that
d ( x n , x n + 1 ) + d ( y n , y n + 1 ) λ [ d ( x n 1 , x n ) + d ( y n 1 , y n ) ]

for some λ, 0 < λ < 1 / K , and each n N . Then { x n } and { y n } are two Cauchy sequences in X.

Definition 2.8 (Mixed monotone property)

Let ( X , ) be a partially ordered set and F : X × X X . We say that the mapping F has the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument. That is, for any x , y X ,
x 1 , x 2 X , x 1 x 2 F ( x 1 , y ) F ( x 2 , y )
(2.1)
and
y 1 , y 2 X , y 1 y 2 F ( x , y 1 ) F ( x , y 2 ) .
(2.2)

Definition 2.9 [13]

Let F : X × X X . We say that ( x , y ) X × X is a coupled fixed point of F if F ( x , y ) = x and F ( y , x ) = y .

Lakshmikantham and Ćirić [13] introduced the following concept of a mixed g-monotone mapping.

Definition 2.10 [13]

Let ( X , ) be a partially ordered set, F : X × X X and g : X X . We say that F has the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument, that is, for any x , y X ,
x 1 , x 2 X , g ( x 1 ) g ( x 2 ) implies F ( x 1 , y ) F ( x 2 , y )
and
y 1 , y 2 X , g ( y 1 ) g ( y 2 ) implies F ( x , y 1 ) F ( x , y 2 ) .

Note that if g is the identity mapping, then this definition reduces to Definition 2.8.

Definition 2.11 [13]

An element ( x , y ) X × X is called a coupled coincidence point of the mappings F : X × X X and g : X X if
F ( x , y ) = g ( x ) , F ( y , x ) = g ( y ) .

3 Main results

Let ( C , ) be an ordered subset of a quasi-Banach space ( X , , K ) . Throughout this paper, we assume that the partial order have the following properties:
  1. (A)

    If x y and λ R + , then λ x λ y ;

     
  2. (B)

    If x y and z C , then x + z y + z .

     

The following theorem is our first main result.

Theorem 3.1 Let ( C , ) be an ordered closed and convex subset of a quasi-Banach space ( X , , K ) where 1 K < 2 and d : X × X R + is such that d ( x , y ) = x y . Assume that F : C × C C is a mapping with the mixed monotone property on C and suppose that there exist non-negative real numbers α, β and γ with 0 γ + 2 α + 2 β + 1 < 2 / K 2 such that
d ( F ( x , y ) , F ( u , v ) ) α d ( x , F ( x , y ) ) + β d ( y , F ( y , x ) ) + γ 2 [ d ( x , u ) + d ( y , v ) ]
(3.1)
for all x , y , u , v C , for which u x and y v . Also suppose that either
  1. (a)

    F is continuous, or

     
  2. (b)
    C has the following property:
    if a non-decreasing sequence  { x n } x ,  then  x n x  for all  n 0 , if a non-increasing sequence  { y n } y ,  then  y y n  for all  n 0 .
     

If there exist x 0 , y 0 C such that x 0 F ( x 0 , y 0 ) and F ( y 0 , x 0 ) y 0 , then there exist x , y C such that x = F ( x , y ) and y = F ( y , x ) , that is, F has a coupled fixed point in C.

Proof Let x 0 , y 0 C be such that x 0 F ( x 0 , y 0 ) and y 0 F ( y 0 , x 0 ) . Then
x 0 = x 0 + x 0 2 x 0 + F ( x 0 , y 0 ) 2 and y 0 = y 0 + y 0 2 y 0 + F ( y 0 , x 0 ) 2 .
Define x 1 , y 1 X such that x 1 = x 0 + F ( x 0 , y 0 ) 2 and y 1 = y 0 + F ( y 0 , x 0 ) 2 . Similarly, x 2 = x 1 + F ( x 1 , y 1 ) 2 and y 2 = y 1 + F ( y 1 , x 1 ) 2 . We construct two sequences { x n } and { y n } such that
y n + 1 = y n + F ( y n , x n ) 2 for all  n 0 ,
(3.2)
and
x n + 1 = x n + F ( x n , y n ) 2 for all  n 0 .
(3.3)
Let us prove that
x n x n + 1 and y n y n + 1 for all  n 0 .
(3.4)
Since
x 0 x 0 + F ( x 0 , y 0 ) 2 = x 1 and y 0 y 0 + F ( y 0 , x 0 ) 2 = y 1 ,
then (3.4) hold for n = 0 . Suppose that (3.4) hold for n 1 . Since F has the mixed monotone property, so we have
x n + 1 = x n + F ( x n , y n ) 2 x n + F ( x n + 1 , y n ) 2 x n + 1 + F ( x n + 1 , y n ) 2 x n + 1 + F ( x n + 1 , y n + 1 ) 2 = x n + 2
and
y n + 1 = y n + F ( y n , x n ) 2 y n + 1 + F ( y n , x n ) 2 y n + 1 + F ( y n + 1 , x n ) 2 y n + 1 + F ( y n + 1 , x n + 1 ) 2 = y n + 2 .

Then, by mathematical induction, it follows that (3.4) hold for all n 0 .

By (3.2) and (3.3) we have
x n + 1 x n = x n x n 1 + [ F ( x n , y n ) F ( x n 1 , y n 1 ) ] 2
and
y n + 1 y n = y n y n 1 + [ F ( y n , x n ) F ( y n 1 , x n 1 ) ] 2 .
Thus
x n + 1 x n K x n x n 1 + K F ( x n , y n ) F ( x n 1 , y n 1 ) 2
and
y n + 1 y n K y n y n 1 + K F ( y n , x n ) F ( y n 1 , x n 1 ) 2 .
Therefore
2 K d ( x n + 1 , x n ) d ( x n , x n 1 ) d ( F ( x n , y n ) , F ( x n 1 , y n 1 ) )
(3.5)
and
2 K d ( y n + 1 , y n ) d ( y n , y n 1 ) d ( F ( y n , x n ) , F ( y n 1 , x n 1 ) ) .
(3.6)
Also,
x n 1 F ( x n 1 , y n 1 ) = 2 ( x n 1 x n 1 + F ( x n 1 , y n 1 ) 2 ) = 2 ( x n 1 x n ) .
Then
d ( x n 1 , F ( x n 1 , y n 1 ) ) = 2 d ( x n 1 , x n ) .
(3.7)
Similarly,
d ( y n 1 , F ( y n 1 , x n 1 ) ) = 2 d ( y n 1 , y n ) .
(3.8)
On the other hand, by (3.1) and (3.4), we have
d ( F ( x n 1 , y n 1 ) , F ( x n , y n ) ) α d ( x n 1 , F ( x n 1 , y n 1 ) ) + β d ( y n 1 , F ( y n 1 , x n 1 ) ) + γ 2 [ d ( x n 1 , x n ) + d ( y n 1 , y n ) ]
and
d ( F ( y n 1 , x n 1 ) , F ( y n , x n ) ) α d ( y n 1 , F ( y n 1 , x n 1 ) ) + β d ( x n 1 , F ( x n 1 , y n 1 ) ) + γ 2 [ d ( y n 1 , y n ) + d ( x n 1 , x n ) ] .
Hence, by (3.5), (3.6), (3.7) and (3.8), we have
2 K d ( x n + 1 , x n ) d ( x n , x n 1 ) 2 α d ( x n 1 , x n ) + 2 β d ( y n 1 , y n ) + γ 2 [ ( d ( x n 1 , x n ) + d ( y n 1 , y n ) ) ]
and
2 K d ( y n + 1 , y n ) d ( y n , y n 1 ) 2 α d ( y n 1 , y n ) + 2 β d ( x n 1 , x n ) + γ 2 [ ( d ( y n 1 , y n ) + d ( x n 1 , x n ) ) ] .
Thus
d ( x n + 1 , x n ) + d ( y n + 1 , y n ) K 2 ( γ + 2 α + 2 β + 1 ) ( d ( x n 1 , x n ) + d ( y n 1 , y n ) ) .

By Lemma 2.7 we conclude that { x n } and { y n } are two Cauchy sequences. Then there exist x , y C such that x n x and y n y .

At first, we assume that F is continuous. Hence
x = lim n x n = lim n F ( x n 1 , y n 1 ) = F ( x , y ) .
Similarly,
y = F ( y , x ) .

That is, F has a coupled fixed point.

Now we assume that (b) holds. Since x n x and y n y as n , then (b) implies that x n x and y y n for all n 0 . Now, by (3.1) with x = x n , y = y n , u = x , v = y , we have
d ( F ( x n , y n ) , F ( x , y ) ) α d ( x n , F ( x n , y n ) ) + β d ( y n , F ( y n , x n ) ) + γ 2 [ d ( x n , x ) + d ( y n , y ) ] ,
which implies
d ( F ( x n , y n ) , F ( x , y ) ) 2 α d ( x n , x n + 1 ) + 2 β d ( y n , y n + 1 ) + γ 2 [ d ( x n , x ) + d ( y n , y ) ] .
Taking the limit as n in the above inequality, we have
lim n d ( F ( x n , y n ) , F ( x , y ) ) = 0 .
Also, by (3.3) we get lim n F ( x n , y n ) = x . Now we can write
d ( x , F ( x , y ) ) K lim n d ( x , F ( x n , y n ) ) + K lim n d ( F ( x n , y n ) , F ( x , y ) ) ,

and hence d ( x , F ( x , y ) ) = 0 . That is, x = F ( x , y ) . Similarly, y = F ( y , x ) as required. □

If in Theorem 3.1 we take α = β = 0 , we obtain following result.

Corollary 3.2 Let ( C , ) be an ordered closed and convex subset of a quasi-Banach space ( X , , K ) , where 1 K < 2 , and let d : X × X R + be such that d ( x , y ) = x y . Assume that F : C × C C is a mapping such that F has the mixed monotone property on X and there exists a non-negative real number γ with 0 γ + 1 < 2 / K 2 such that
d ( F ( x , y ) , F ( u , v ) ) γ 2 [ d ( x , u ) + d ( y , v ) ]
for all x , y , u , v X , for which u x and y v . Also suppose that either
  1. (a)

    F is continuous, or

     
  2. (b)
    C has the following property:
    if a non-decreasing sequence  { x n } x ,  then  x n x  for all  n 0 , if a non-increasing sequence  { y n } y ,  then  y y n  for all  n 0 .
     

If there exist x 0 , y 0 C such that x 0 F ( x 0 , y 0 ) and F ( y 0 , x 0 ) y 0 , then there exist x , y C such that x = F ( x , y ) and y = F ( y , x ) , that is, F has a coupled fixed point.

The following theorem is our second main result.

Theorem 3.3 Let ( C , ) be an ordered closed and convex subset of a quasi-Banach space ( X , , K ) , where 1 K < 1 + 17 4 , and let d : X × X R + be such that d ( x , y ) = x y . Assume that F : C × C C is a mapping such that F has the mixed monotone property on X and there exists a non-negative real number α with 0 α + 2 K 1 < 2 / K such that
d ( F ( x , y ) , F ( u , v ) ) α 2 d ( x , u ) 2 + d ( y , v ) 2 + 1 2 K 2 d ( x , F ( x , y ) ) d ( y , F ( y , x ) )
(3.9)
for all x , y , u , v C , for which u x and y v . Also suppose that either
  1. (a)

    F is continuous, or

     
  2. (b)
    C has the following property:
    if a non-decreasing sequence  { x n } x ,  then  x n x  for all  n 0 , if a non-increasing sequence  { y n } y ,  then  y y n  for all  n 0 .
     

If there exist x 0 , y 0 X such that x 0 F ( x 0 , y 0 ) and F ( y 0 , x 0 ) y 0 , then F has a coupled fixed point.

Proof Let x 0 , y 0 X be such that x 0 F ( x 0 , y 0 ) and y 0 F ( y 0 , x 0 ) . Then
x 0 = ( 2 K 1 ) x 0 + x 0 2 K ( 2 K 1 ) x 0 + F ( x 0 , y 0 ) 2 K
and
y 0 = ( 2 K 1 ) y 0 + y 0 2 K ( 2 K 1 ) y 0 + F ( y 0 , x 0 ) 2 K .
Define x 1 , y 1 X such that x 1 = ( 2 K 1 ) x 0 + F ( x 0 , y 0 ) 2 and y 1 = ( 2 K 1 ) y 0 + F ( y 0 , x 0 ) 2 K . Similarly, x 2 = x 1 + F ( x 1 , y 1 ) 2 and y 2 = ( 2 K 1 ) y 1 + F ( y 1 , x 1 ) 2 K . We construct two sequences { x n } and { y n } such that
y n + 1 = ( 2 K 1 ) y n + F ( y n , x n ) 2 K for all  n 0
(3.10)
and
x n + 1 = ( 2 K 1 ) x n + F ( x n , y n ) 2 K for all  n 0 .
(3.11)
Let us prove that
x n x n + 1 and y n y n + 1 for all  n 0 .
(3.12)
As
x 0 ( 2 K 1 ) x 0 + F ( x 0 , y 0 ) 2 K = x 1 and y 0 ( 2 K 1 ) y 0 + F ( y 0 , x 0 ) 2 K = y 1 ,
so (3.12) hold for n = 0 . Suppose that (3.12) hold for n 1 . As F has the mixed monotone property, so
x n + 1 = ( 2 K 1 ) x n + F ( x n , y n ) 2 K ( 2 K 1 ) x n + F ( x n + 1 , y n ) 2 K ( 2 K 1 ) x n + 1 + F ( x n + 1 , y n ) 2 K ( 2 K 1 ) x n + 1 + F ( x n + 1 , y n + 1 ) 2 K = x n + 2
and
y n + 2 = ( 2 K 1 ) y n + 1 + F ( y n + 1 , x n + 1 ) 2 K ( 2 K 1 ) y n + 1 + F ( y n + 1 , x n ) 2 K ( 2 K 1 ) y n + 1 + F ( y n , x n ) 2 K ( 2 K 1 ) y n + F ( y n , x n ) 2 K = y n + 1 .

Then, by mathematical induction, it follows that (3.12) holds for all n 0 .

By (3.10) and (3.11), we have
x n + 1 x n = ( 2 K 1 ) ( x n x n 1 ) + [ F ( x n , y n ) F ( x n 1 , y n 1 ) ] 2 K
and
y n + 1 y n = ( 2 K 1 ) ( y n y n 1 ) + [ F ( y n , x n ) F ( y n 1 , x n 1 ) ] 2 K .
Thus
x n + 1 x n ( 2 K 1 ) x n x n 1 + F ( x n , y n ) F ( x n 1 , y n 1 ) 2
and
y n + 1 y n ( 2 K 1 ) y n y n 1 + F ( y n , x n ) F ( y n 1 , x n 1 ) 2 .
Therefore
2 d ( x n + 1 , x n ) ( 2 K 1 ) d ( x n , x n 1 ) d ( F ( x n , y n ) , F ( x n 1 , y n 1 ) )
(3.13)
and
2 d ( y n + 1 , y n ) ( 2 K 1 ) d ( y n , y n 1 ) d ( F ( y n , x n ) , F ( y n 1 , x n 1 ) ) .
(3.14)
Also, we have
x n 1 F ( x n 1 , y n 1 ) = 2 K ( x n 1 ( 2 K 1 ) x n 1 + F ( x n 1 , y n 1 ) 2 K ) = 2 K ( x n 1 x n ) ,
which implies
d ( x n 1 , F ( x n 1 , y n 1 ) ) = 2 K d ( x n 1 , x n ) .
(3.15)
Similarly, we have
d ( y n 1 , F ( y n 1 , x n 1 ) ) = 2 K d ( y n 1 , y n ) .
(3.16)
Now, by (3.9), (3.13) and (3.15), (3.16), we have
2 d ( x n + 1 , x n ) ( 2 K 1 ) d ( x n , x n 1 ) α 2 d ( x n 1 , x n ) 2 + d ( y n 1 , y n ) 2 + 2 d ( x n 1 , x n ) d ( y n 1 , y n ) = α 2 [ ( d ( x n 1 , x n ) + d ( y n 1 , y n ) ) ] .
Similarly,
2 d ( y n + 1 , y n ) ( 2 K 1 ) d ( y n , y n 1 ) α 2 [ ( d ( y n 1 , y n ) + d ( x n 1 , x n ) ] .
Thus
d ( x n + 1 , x n ) + d ( y n + 1 , y n ) 1 2 ( α + 2 K 1 ) ( d ( x n 1 , x n ) + d ( y n 1 , y n ) ) .

By Lemma 2.7, we conclude that { x n } and { y n } are Cauchy sequences. Thus, there exist x , y C such that x n x and y n y .

Now, proceeding as in the proof of Theorem 3.1, we can prove that ( x , y ) is a coupled fixed point of F. □

Since
α 2 1 2 K 2 d ( x , F ( x , y ) ) d ( y , F ( y , x ) ) α 2 d ( x , u ) 2 + d ( y , v ) 2 + 1 2 K 2 d ( x , F ( x , y ) ) d ( y , F ( y , x ) ) ,

so by Theorem 3.3 we obtain the following result.

Corollary 3.4 Let ( C , ) be an ordered closed and convex subset of a quasi-Banach space ( X , , K ) , where 1 K < 1 + 17 4 , and let d : X × X R + be such that d ( x , y ) = x y . Assume that F : C × C C , F has the mixed monotone property on C and for a non-negative real number α with 0 α + 2 K 1 < 2 / K , F satisfies following inequality:
d ( F ( x , y ) , F ( u , v ) ) α 2 4 K d ( x , F ( x , y ) ) d ( y , F ( y , x ) )
for all x , y , u , v X , for which u x and y v . Also suppose that either
  1. (a)

    F is continuous, or

     
  2. (b)
    C has the following property:
    if a non-decreasing sequence  { x n } x ,  then  x n x  for all  n 0 , if a non-increasing sequence  { y n } y , then  y y n  for all  n 0 .
     

If there exist x 0 , y 0 C such that x 0 F ( x 0 , y 0 ) and F ( y 0 , x 0 ) y 0 , then F has a coupled fixed point.

By taking K = 1 in the above proved results, we can obtain the following couple fixed results in Banach spaces.

Corollary 3.5 Let ( C , ) be an ordered closed and convex subset of a Banach space ( X , ) , and let F : C × C C be a mapping such that F has the mixed monotone property on C. Suppose that there exist non-negative real numbers α, β and γ with 0 γ + 2 α + 2 β < 1 such that
d ( F ( x , y ) , F ( u , v ) ) α d ( x , F ( x , y ) ) + β d ( y , F ( y , x ) ) + γ 2 [ d ( x , u ) + d ( y , v ) ]
for all x , y , u , v C with u x and y v . Also suppose that either
  1. (a)

    F is continuous, or

     
  2. (b)
    C has the following property:
    if a non-decreasing sequence  { x n } x ,  then  x n x  for all  n 0 , if a non-increasing sequence  { y n } y ,  then  y y n  for all  n 0 .
     

If there exist x 0 , y 0 C such that x 0 F ( x 0 , y 0 ) and F ( y 0 , x 0 ) y 0 , then F has a coupled fixed point.

Corollary 3.6 Let ( C , ) be an ordered closed and convex subset of a Banach space ( X , ) , and let F : C × C C be a mapping such that F has the mixed monotone property on C. Suppose that there exists a non-negative real number γ with 0 γ < 1 such that
d ( F ( x , y ) , F ( u , v ) ) γ 2 [ d ( x , u ) + d ( y , v ) ]
for all x , y , u , v C with u x and y v . Also suppose that either
  1. (a)

    F is continuous, or

     
  2. (b)
    C has the following property:
    if a non-decreasing sequence  { x n } x ,  then  x n x  for all  n 0 , if a non-increasing sequence  { y n } y ,  then  y y n  for all  n 0 .
     

If there exist x 0 , y 0 C such that x 0 F ( x 0 , y 0 ) and F ( y 0 , x 0 ) y 0 , then F has a coupled fixed point.

Corollary 3.7 Let ( C , ) be an ordered closed and convex subset of a Banach space ( X , ) , and let F : C × C C be a mapping such that F has the mixed monotone property on C. Suppose that there exists a non-negative real number α with 0 α < 1 such that
d ( F ( x , y ) , F ( u , v ) ) α 2 d ( x , u ) 2 + d ( y , v ) 2 + 1 2 d ( x , F ( x , y ) ) d ( y , F ( y , x ) )
for all x , y , u , v C with u x and y v . Also suppose that either
  1. (a)

    F is continuous, or

     
  2. (b)
    C has the following property:
    if a non-decreasing sequence  { x n } x ,  then  x n x  for all  n 0 ,
    (3.17)
    if a non-increasing sequence  { y n } y ,  then  y y n  for all  n 0 .
    (3.18)
     

If there exist x 0 , y 0 C such that x 0 F ( x 0 , y 0 ) and F ( y 0 , x 0 ) y 0 , then F has a coupled fixed point.

Corollary 3.8 Let ( C , ) be an ordered closed and convex subset of a Banach space ( X , ) , and let F : C × C C be a mapping such that F has the mixed monotone property on C. Suppose that there exists a non-negative real number α with 0 α < 1 such that
d ( F ( x , y ) , F ( u , v ) ) α 2 4 d ( x , F ( x , y ) ) d ( y , F ( y , x ) )
for all x , y , u , v C with u x and y v . Also suppose that either
  1. (a)

    F is continuous, or

     
  2. (b)
    C has the following property:
    if a non-decreasing sequence  { x n } x ,  then  x n x  for all  n 0 , if a non-increasing sequence  { y n } y ,  then  y y n  for all  n 0 .
     

If there exist x 0 , y 0 C such that x 0 F ( x 0 , y 0 ) and F ( y 0 , x 0 ) y 0 , then F has a coupled fixed point.

The following lemma is an easy consequence of the axiom of choice (see p.5 [25], AC5: For every function f : X X , there is a function g such that D ( g ) = R ( f ) and for every x D ( g ) , f ( g x ) = x ).

Lemma 3.9 Let X be a nonempty set and g : X X be a mapping. Then there exists a subset E X such that g ( E ) = g ( X ) and g : E X is one-to-one.

As an application of Theorem 3.1, we now establish a coupled coincidence point result.

Theorem 3.10 Let ( C , ) be a nonempty ordered subset of a quasi-Banach space ( X , , K ) , where 1 K < 2 , and let d : X × X R + be such that d ( x , y ) = x y . Assume that g : C C and F : C × C C are mappings where F has the mixed g-monotone property on C, g ( C ) is closed and convex and F ( C × C ) g ( C ) . Suppose that there exist non-negative real numbers α, β and a real number γ with 0 γ + 2 α + 2 β + 1 < 2 / K 2 such that
d ( F ( x , y ) , F ( u , v ) ) α d ( g x , F ( x , y ) ) + β d ( g y , F ( y , x ) ) + γ 2 [ d ( g x , g u ) + d ( g y , g v ) ]
(3.19)
for all x , y , u , v C , for which g u g x and g y g v . Also suppose that either
  1. (a)

    F is continuous, or

     
  2. (b)
    C has the following property:
    if a non-decreasing sequence  { g x n } g x ,  then  g x n g x  for all  n 0 , if a non-increasing sequence  { g y n } g y ,  then  g y g y n  for all  n 0 .
     

If there exist x 0 , y 0 C such that g ( x 0 ) F ( x 0 , y 0 ) and F ( y 0 , x 0 ) g ( y 0 ) , then there exist x , y C such that g x = F ( x , y ) and g y = F ( y , x ) , that is, F and g have a coupled coincidence point in C.

Proof Using Lemma 3.9, there exists E C such that g ( E ) = g ( C ) and g : E C is one-to-one. We define a mapping G : g ( E ) × g ( E ) g ( E ) by
G ( g x , g y ) = F ( x , y ) ,
(3.20)
for all g x , g y g ( E ) . As g is one-to-one on g ( E ) and F ( C × C ) g ( C ) , so G is well defined. Thus, it follows from (3.19) and (3.20) that
d ( G ( g x , g y ) , F ( g u , g v ) ) + α d ( g x , G ( g x , g y ) ) + β d ( g y , G ( g y , g x ) ) γ 2 [ d ( g x , g u ) + d ( g y , g v ) ]
(3.21)
for all g x , g y , g u , g v g ( C ) , for which g ( x ) g ( u ) and g ( y ) g ( v ) . Since F has the mixed g-monotone property, for all g x , g y g ( C ) ,
g x 1 , g x 2 g ( C ) , g ( x 1 ) g ( x 2 ) implies G ( g x 1 , g y ) G ( g x 2 , g y )
(3.22)
and
g y 1 , g y 2 g ( C ) , g ( y 1 ) g ( y 2 ) implies G ( g x , g y 1 ) G ( g x , g y 2 ) ,
(3.23)
which imply that G has the mixed monotone property. Also, there exist x 0 , y 0 C such that
g ( x 0 ) F ( x 0 , y 0 ) and g ( y 0 ) F ( y 0 , x 0 ) .
This implies that there exist g x 0 , g y 0 g ( C ) such that
g ( x 0 ) G ( g x 0 , g y 0 ) and g ( y 0 ) G ( g y 0 , g x 0 ) .

Suppose that assumption (a) holds. Since F is continuous, G is also continuous. Using Theorem 3.1 to the mapping G, it follows that G has a coupled fixed point ( u , v ) g ( C ) × g ( C ) .

Suppose that assumption (b) holds. We conclude similarly that the mapping G has a coupled fixed point ( u , v ) g ( C ) × g ( C ) . Finally, we prove that F and g have a coupled coincidence point. Since ( u , v ) is a coupled fixed point of G, we get
u = G ( u , v ) and v = G ( v , u ) .
(3.24)
Since ( u , v ) g ( C ) × g ( C ) , there exists a point ( u 0 , v 0 ) C × C such that
u = g u 0 and v = g v 0 .
(3.25)
It follows from (3.24) and (3.25) that
g u 0 = G ( g u 0 , g v 0 ) and g v 0 = G ( g v 0 , g u 0 ) .
(3.26)
Combining (3.20) and (3.26), we get
g u 0 = F ( u 0 , v 0 ) and g v 0 = F ( v 0 , u 0 ) .

Thus, ( u 0 , v 0 ) is a required coupled coincidence point of F and g. This completes the proof. □

Similarly, as an application of Theorem 3.3, we can prove the following coupled coincidence point result.

Theorem 3.11 Let ( C , ) be a nonempty ordered subset of a quasi-Banach space ( X , , K ) , where 1 K < 1 + 17 4 , and let d : X × X R + be such that d ( x , y ) = x y . Assume that g : C C and F : C × C C are mappings where F has the mixed g-monotone property on C, g ( C ) is closed and convex and F ( C × C ) g ( C ) . Suppose that there exists a real number α with 0 α + 2 K 1 < 2 / K such that
d ( F ( x , y ) , F ( u , v ) ) α 2 d ( g x , g u ) 2 + d ( g y , g v ) 2 + 1 2 K 2 d ( g x , F ( x , y ) ) d ( g y , F ( y , x ) )
(3.27)
for all x , y , u , v C , for which g u g x and g y g v . Also suppose that either
  1. (a)

    F is continuous, or

     
  2. (b)
    C has the following property:
    if a non-decreasing sequence  { g x n } g x ,  then  g x n g x  for all  n 0 , if a non-increasing sequence  { g y n } g y ,  then  g y g y n  for all  n 0 .
     

If there exist x 0 , y 0 C such that g ( x 0 ) F ( x 0 , y 0 ) and F ( y 0 , x 0 ) g ( y 0 ) , then F and g have a coupled coincidence point in C.

4 Existence of a solution for a system of integral equations

We consider the space X = C ( [ 0 , T ] , R ) of continuous functions defined on I = [ 0 , T ] endowed with the structure ( X , ) given by
u = sup t [ 0 , T ] | u ( t ) |
for all u X . We endow X with the partial order given by
x y x ( t ) y ( t ) for all  t [ 0 , T ] .
Clearly, the partial order satisfies conditions A and B. Further, it is known that ( X , d , ) is regular [24], that is,
if a non-decreasing sequence  { x n } x ,  then  x n x  for all  n 0 , if a non-increasing sequence  { y n } y , then  y y n  for all  n 0 .

Motivated by the work in [1, 17, 18, 26], we study the existence of solutions for a system of nonlinear integral equations using the results proved in the previous section.

Consider the integral equations in the following system.
x ( t ) = P ( t ) + 0 T S ( t , r ) [ f ( r , x ( r ) ) + k ( r , y ( r ) ) ] d r , y ( t ) = P ( t ) + 0 T S ( t , r ) [ f ( r , y ( r ) ) + k ( r , x ( r ) ) ] d r .
(4.1)
We will consider system (4.1) under the following assumptions:
  1. (i)

    f , k : [ 0 , T ] × R R are continuous;

     
  2. (ii)

    P : [ 0 , T ] R is continuous;

     
  3. (iii)

    S : [ 0 , T ] × R [ 0 , ) is continuous;

     
  4. (iv)
    there exist a , b , c > 0 with 0 2 a + 2 b + c < 1 such that for all r [ 0 , T ] and all x ( r ) , y ( r ) , u ( r ) , v ( r ) X with u ( r ) x ( r ) y ( r ) v ( r ) , we have
    0 f ( r , y ( r ) ) f ( r , x ( r ) ) a | x ( r ) F ( x ( r ) , y ( r ) ) | + b | y ( r ) F ( y ( r ) , x ( r ) ) | + c 2 [ ( x ( r ) u ( r ) ) + ( y ( r ) v ( r ) ) ] , 0 k ( r , x ( r ) ) k ( r , y ( r ) ) a | x ( r ) F ( x ( r ) , y ( r ) ) | + b | y ( r ) F ( y ( r ) , x ( r ) ) | + c 2 [ ( x ( r ) u ( r ) ) + ( y ( r ) v ( r ) ) ] ,
     
where
F ( x , y ) ( t ) = P ( t ) + 0 T S ( t , r ) [ f ( r , x ( r ) ) + k ( r , y ( r ) ) ] d r ;
  1. (v)
    there exist continuous functions α , γ : [ 0 , T ] R such that
    α ( t ) P ( t ) + 0 T S ( t , r ) [ f ( r , α ( r ) ) + k ( r , γ ( r ) ) ] d r , γ ( t ) P ( t ) + 0 T S ( t , r ) [ f ( r , γ ( r ) ) + k ( r , α ( r ) ) ] d r ;
     
  2. (vi)
    assume that
    sup t [ 0 , T ] 0 T S ( t , r ) d r 1 / 2 .
     

Theorem 4.1 Under assumptions (i)-(vi), system (4.1) has a solution in X 2 , where X = ( C ( [ 0 , T ] , R ) ) is defined above.

Proof We consider the operator F : X 2 X defined by
F ( x 1 , x 2 ) ( t ) = P ( t ) + 0 T S ( t , r ) [ f ( r , x 1 ( r ) ) + k ( r , x 2 ( r ) ) ] d r

for all t , r [ 0 , T ] , x 1 , x 2 X .

Clearly, F has the mixed monotone property [26].

Let x , y , u , v X with u x y v . Since F has the mixed monotone property, we have
F ( u , v ) F ( x , y ) .
Notice that
| F ( x , y ) ( t ) F ( u , v ) ( t ) | = | 0 T S ( t , r ) [ f ( r , x ( r ) ) f ( r , u ( r ) ) ] d r + 0 T S ( t , r ) [ k ( r , y ( r ) ) k ( r , v ( r ) ) ] d r | 0 T S ( t , r ) [ | f ( r , x ( r ) ) f ( r , u ( r ) ) | ] d r + 0 T S ( t , r ) [ | k ( r , y ( r ) ) k ( r , v ( r ) ) | ] d r 0 T S ( t , r ) [ a | x ( r ) F ( x ( r ) , y ( r ) ) | + b | y ( r ) F ( y ( r ) , x ( r ) ) | + c 2 [ | x ( r ) u ( r ) | + | y ( r ) v ( r ) | ] ] d r + 0 T S ( t , r ) [ a | x ( r ) F ( x ( r ) , y ( r ) ) | + b | y ( r ) F ( y ( r ) , x ( r ) ) | + c 2 [ | x ( r ) u ( r ) | + | y ( r ) v ( r ) | ] ] d r = 2 0 T S ( t , r ) [ a | x ( r ) F ( x ( r ) , y ( r ) ) | + b | y ( r ) F ( y ( r ) , x ( r ) ) | + c 2 [ | x ( r ) u ( r ) | + | y ( r ) v ( r ) | ] ] d r 2 sup t [ 0 , T ] 0 T S ( t , r ) d r [ a x F ( x , y ) + b y F ( y , x ) + c 2 ( x u + y v ) ] a x F ( x , y ) + b y F ( y , x ) + c 2 ( x u + y v ) .
Thus,
F ( x , y ) F ( u , v ) a x F ( x , y ) + b y F ( y , x ) + c 2 ( x u + y v ) .
Further, by (v), we get
α F ( α , γ ) , γ F ( γ , α ) .

All of the conditions of Corollary 3.5 are satisfied, so we deduce the existence of x 1 , x 2 X such that x 1 = F ( x 1 , x 2 ) and x 2 = F ( x 2 , x 1 ) . □

Declarations

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first and third authors acknowledge with thanks DSR, KAU for financial support. The second author is thankful for the support of Rasht Branch, Islamic Azad University. The authors would like to express their thanks to the referees for their helpful comments and suggestions.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University
(2)
Young Researchers and Elite Club, Rasht Branch, Islamic Azad University

References

  1. Agarwal RP, Hussain N, Taoudi MA: Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations. Abstr. Appl. Anal. 2012., 2012: Article ID 245872Google Scholar
  2. Akbar F, Khan AR: Common fixed point and approximation results for noncommuting maps on locally convex spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 207503Google Scholar
  3. Alsulami S, Hussain N, Alotaibi A: Coupled fixed and coincidence points for monotone operators in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 173Google Scholar
  4. Berinde V: Generalized contractions in quasi-metric spaces. Preprint 93. Seminar on Fixed Point Theory 1993, 3–9.Google Scholar
  5. Berinde V, Borcut M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 4889–4897. 10.1016/j.na.2011.03.032MathSciNetView ArticleMATHGoogle Scholar
  6. Ćirić L, Hussain N, Cakic N: Common fixed points for Ciric type f -weak contraction with applications. Publ. Math. (Debr.) 2010, 76(1–2):31–49.MathSciNetMATHGoogle Scholar
  7. 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
  8. Czerwik S: Contraction mappings in b -metric spaces. Acta Math. Inform. Univ. Ostrav. 1993, 1: 5–11.MathSciNetMATHGoogle Scholar
  9. Khamsi MA: Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 315398Google Scholar
  10. Khamsi MA, Hussain N: KKM mappings in metric type spaces. Nonlinear Anal. 2010, 73: 3123–3129. 10.1016/j.na.2010.06.084MathSciNetView ArticleMATHGoogle Scholar
  11. Jovanović M, Kadelburg Z, Radenović S: Common fixed point results in metric type spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 978121Google Scholar
  12. Bhaskar TG, 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
  13. Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020MathSciNetView ArticleMATHGoogle Scholar
  14. Choudhury BS, Kundu A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal. 2010, 73: 2524–2531. 10.1016/j.na.2010.06.025MathSciNetView ArticleMATHGoogle Scholar
  15. Cho YJ, Shah MH, Hussain N: Coupled fixed points of weakly F -contractive mappings in topological spaces. Appl. Math. Lett. 2011, 24: 1185–1190. 10.1016/j.aml.2011.02.004MathSciNetView ArticleMATHGoogle Scholar
  16. Guo D, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 1987, 11: 623–632. 10.1016/0362-546X(87)90077-0MathSciNetView ArticleMATHGoogle Scholar
  17. Luong NV, Thuan NX: Coupled fixed point in partially ordered metric spaces and applications. Nonlinear Anal. 2011, 74: 983–992. 10.1016/j.na.2010.09.055MathSciNetView ArticleMATHGoogle Scholar
  18. Hussain N, Khan AR, Agarwal RP: Krasnosel’skii and Ky Fan type fixed point theorems in ordered Banach spaces. J. Nonlinear Convex Anal. 2010, 11(3):475–489.MathSciNetMATHGoogle Scholar
  19. Hussain N, Đorić D, Kadelburg Z, Radenović S: Suzuki-type fixed point results in metric type spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 126Google Scholar
  20. Hussain N, Shah MH, Kutbi MA: Coupled coincidence point theorems for nonlinear contractions in partially ordered quasi-metric spaces with a Q -function. Fixed Point Theory Appl. 2011., 2011: Article ID 703938Google Scholar
  21. Hussain N, Alotaibi A: Coupled coincidences for multi-valued nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 82Google Scholar
  22. Hussain N, Latif A, Shah MH: Coupled and tripled coincidence point results without compatibility. Fixed Point Theory Appl. 2012., 2011: Article ID 77Google Scholar
  23. 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
  24. Nieto JJ, Rodriguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Order 2005, 22(3):223–239. 10.1007/s11083-005-9018-5MathSciNetView ArticleMATHGoogle Scholar
  25. Herman R, Jean ER: Equivalents of the Axiom of Choice. North-Holland, Amsterdam; 1970.MATHGoogle Scholar
  26. 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., 2013: Article ID 130Google Scholar
  27. Salimi, P, Vetro, P: Common fixed point results on quasi-Banach spaces and integral equations. Georgian Math. J. (in press)Google Scholar
  28. Sintunavarat W, Cho YJ, Kumam P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 81Google Scholar
  29. Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 2008, 136: 1861–1869.View ArticleMathSciNetMATHGoogle Scholar
  30. Takahashi W, Lin L-J, Wang SY: Fixed point theorems for contractively generalized hybrid mappings in complete metric spaces. J. Nonlinear Convex Anal. 2012, 13(2):195–206.MathSciNetMATHGoogle Scholar

Copyright

© Hussain et al.; licensee Springer. 2013

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 cited.