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Fixed points of (ψ,ϕ,θ)-contractive mappings in partially ordered b-metric spaces and application to quadratic integral equations

Abstract

We prove some coupled coincidence and coupled common fixed point theorems for mappings satisfying (ψ,ϕ,θ)-contractive conditions in partially ordered complete b-metric spaces. The obtained results extend and improve many existing results from the literature. As an application, we prove the existence of a unique solution to a class of nonlinear quadratic integral equations.

MSC:47H10, 54H25.

1 Introduction and preliminaries

In [1, 2], Czerwik introduced the notion of a b-metric space, which is a generalization of the usual metric space, and generalized the Banach contraction principle in the context of complete b-metric spaces. After that, many authors have carried out further studies on b-metric spaces and their topological properties (see, e.g., [114]). In this paper, some coupled coincidence and coupled common fixed point theorems for mappings satisfying (ψ,ϕ,θ)-contractive conditions in partially ordered complete b-metric spaces are proved. Also, we apply our results to study the existence of a unique solution to a large class of nonlinear quadratic integral equations. There are many papers in the literature concerning coupled fixed points introduced by Bhaskar and Lakshmikantham [15] and their applications in the existence and uniqueness of solutions for boundary value problems. A number of articles on this topic have been dedicated to the improvement and generalization; see [1620] and references therein. Also, to see some results on common fixed points for generalized contraction mappings, we refer the reader to [2123]. For the sake of convenience, some definitions and notations are recalled from [1, 3, 24] and [25].

Definition 1.1 [1]

Let X be a (nonempty) set and s1 be a given real number. A function d:X×X R + is said to be a b-metric space iff for all x,y,zX, 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 need not necessarily be a metric (see also [14]).

Example 1.2 [3]

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)=|xy| is the usual Euclidean metric, then ρ(x,y)= ( x y ) s is a b-metric on with s=2, but is not a metric on .

Also, the following example of a b-metric space is given in [26].

Example 1.3 [26]

Let X be the set of Lebesgue measurable functions on [0,1] such that 0 1 | f ( x ) | 2 dx<. Define D:X×X[0,) by D(f,g)= 0 1 | f ( x ) g ( x ) | 2 dx. 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.

Khamsi [27] also showed that each cone metric space over a normal cone has a b-metric structure.

Since, in general, a b-metric is not continuous, we need the following simple lemma about the b-convergent sequences in the proof of our main result.

Lemma 1.4 [3]

Let (X,d) be a b-metric space with s1, and suppose that { x n } and { y n } are b-convergent to x, y, respectively. Then we have

1 s 2 d(x,y)lim infd( x n , y n )lim supd( x n , y n ) s 2 d(x,y).

In particular, if x=y, then we have limd( x n , y n )=0. Moreover, for each zX, we have

1 s d(x,z)lim infd( x n ,z)lim supd( x n ,z)sd(x,z).

In [25], Lakshmikantham and Ćirić introduced the concept of mixed g-monotone property as follows.

Definition 1.5 [25]

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

x 1 , x 2 X,g x 1 g x 2 F( x 1 ,y)F( x 2 ,y)

and

y 1 , y 2 X,g y 1 g y 2 F(x, y 1 )F(x, y 2 ).

Note that if g is an identity mapping, then F is said to have the mixed monotone property (see also [15]).

Definition 1.6 [25]

An element (x,y)X×X is called a coupled coincidence point of a mapping F:X×XX and a mapping g:XX if

F(x,y)=gx,F(y,x)=gy.

Similarly, note that if g is an identity mapping, then (x,y) is called a coupled fixed point of the mapping F (see also [15]).

Definition 1.7 [24]

An element xX is called a common fixed point of a mapping F:X×XX and g:XX if

F(x,x)=gx=x.
(1.1)

Definition 1.8 [25]

Let X be a nonempty set and F:X×XX and g:XX. One says that F and g are commutative if for all x,yX,

F(gx,gy)=g ( F ( x , y ) ) .

Definition 1.9 [28]

The mappings F and g, where F:X×XX and g:XX, are said to be compatible if

lim n d ( g ( F ( x n , y n ) ) , F ( g x n , g y n ) ) =0

and

lim n d ( g ( F ( y n , x n ) ) , F ( g y n , g x n ) ) =0,

whenever { x n } and { y n } are sequences in X such that lim n F( x n , y n )= lim n g x n =x and lim n F( y n , x n )= lim n g y n =y for all x,yX.

2 Main results

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

  1. (a)

    ψ is continuous,

  2. (b)

    ψ non-decreasing,

  3. (c)

    ψ(t)=0 if and only if t=0.

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

  1. (a)

    ϕ is lower semi-continuous,

  2. (b)

    ϕ(t)=0 if and only if t=0,

and Θ the set of all continuous functions θ:[0,)[0,) with θ(t)=0 if and only if t=0.

Let (X,d,) be a partially ordered b-metric space, and let T:X×XX and g:XX be two mappings. Set

M s , T , g ( x , y , u , v ) = max { d ( g x , g u ) , d ( g y , g v ) , d ( g x , T ( x , y ) ) , 1 2 s d ( g u , T ( u , v ) ) , d ( g y , T ( y , x ) ) , 1 2 s d ( g v , T ( v , u ) ) , d ( g x , T ( u , v ) ) + d ( g u , T ( x , y ) ) 2 s , d ( g y , T ( v , u ) ) + d ( g v , T ( y , x ) ) 2 s }

and

N T , g (x,y,u,v)=min { d ( g x , T ( x , y ) ) , d ( g u , T ( u , v ) ) , d ( g u , T ( x , y ) ) , d ( g x , T ( u , v ) ) } .

Now, we introduce the following definition.

Definition 2.1 Let (X,d,) be a partially ordered b-metric space and ψΨ, ϕΦ and θΘ. We say that T:X×XX is an almost generalized (ψ,ϕ,θ)-contractive mapping with respect to g:XX if there exists L0 such that

ψ ( s 3 d ( T ( x , y ) , T ( u , v ) ) ) ψ ( M s , T , g ( x , y , u , v ) ) ϕ ( M s , T , g ( x , y , u , v ) ) + L θ ( N T , g ( x , y , u , v ) )
(2.1)

for all x,y,u,vX with gxgu and gygv.

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

Theorem 2.2 Suppose that (X,d,) is a partially ordered complete b-metric space. Let T:X×XX be an almost generalized (ψ,ϕ,θ)-contractive mapping with respect to g:XX, and T and g are continuous such that T has the mixed g-monotone property and commutes with g. Also, suppose T(X×X)g(X). If there exists ( x 0 , y 0 )X×X such that g x 0 T( x 0 , y 0 ) and g y 0 T( y 0 , x 0 ), then T and g have coupled coincidence point in X.

Proof By the given assumptions, there exists ( x 0 , y 0 )X×X such that g x 0 T( x 0 , y 0 ) and g y 0 T( y 0 , x 0 ). Since T(X×X)g(X), we can define ( x 1 , y 1 )X×X such that g x 1 =T( x 0 , y 0 ) and g y 1 =T( y 0 , x 0 ), then g x 0 T( x 0 , y 0 )=g x 1 and g y 0 T( y 0 , x 0 )=g y 1 . Also, there exists ( x 2 , y 2 )X×X such that g x 2 =T( x 1 , y 1 ) and g y 2 =T( y 1 , x 1 ). Since T has the mixed g-monotone property, we have

g x 1 =T( x 0 , y 0 )T( x 0 , y 1 )T( x 1 , y 1 )=g x 2

and

g y 2 =T( y 1 , x 1 )T( y 0 , x 1 )T( y 0 , x 0 )=g y 1 .

Continuing in this way, we construct two sequences { x n } and { y n } in X such that

g x n + 1 =T( x n , y n )andg y n + 1 =T( y n , x n )for all n=0,1,2,
(2.2)

for which

g x 0 g x 1 g x 2 g x n g x n + 1 , g y 0 g y 1 g y 2 g y n g y n + 1 .
(2.3)

From (2.2) and (2.3) and inequality (2.1) with (x,y)=( x n , y n ) and (u,v)=( x n + 1 , y n + 1 ), we obtain

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 , y n ) , T ( x n + 1 , y n + 1 ) ) ) ψ ( M s , T , g ( x n , y n , x n + 1 , y n + 1 ) ) ϕ ( M s , T , g ( x n , y n , x n + 1 , y n + 1 ) ) + L θ ( N T , g ( x n , y n , x n + 1 , y n + 1 ) ) ,
(2.4)

where

M s , T , g ( x n , y n , x n + 1 , y n + 1 ) = max { d ( g x n , g x n + 1 ) , d ( g y n , g y n + 1 ) , d ( g x n , T ( x n , y n ) ) , 1 2 s d ( g x n + 1 , T ( x n + 1 , y n + 1 ) ) , d ( g y n , T ( y n , x n ) ) , 1 2 s d ( g y n + 1 , T ( y n + 1 , x n + 1 ) ) , d ( g x n , T ( x n + 1 , y n + 1 ) ) + d ( g x n + 1 , T ( x n , y n ) ) 2 s , d ( g y n , T ( y n + 1 , x n + 1 ) ) + d ( g y n + 1 , T ( y n , x n ) ) 2 s } = max { d ( g x n , g x n + 1 ) , d ( g y n , g y n + 1 ) , 1 2 s d ( g x n + 1 , g x n + 2 ) , 1 2 s d ( g y n + 1 , g y n + 2 ) , d ( g x n , g x n + 2 ) 2 s , d ( g y n , g y n + 2 ) 2 s }

and

N T , g ( x n , y n , x n + 1 , y n + 1 ) = min { d ( g x n , T ( x n , y n ) ) , d ( g x n + 1 , T ( x n + 1 , y n + 1 ) ) , d ( g x n + 1 , T ( x n , y n ) ) , d ( g x n + 1 , T ( x n + 1 , y n + 1 ) ) } = 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 ) }

and

d ( g y n , g y n + 2 ) 2 s d ( g y n , g y n + 1 ) + d ( g y n + 1 , g y n + 2 ) 2 max { d ( g y n , g y n + 1 ) , d ( g y n + 1 , g y n + 2 ) } ,

then we get

M s , T , g ( x n , y n , x n + 1 , y n + 1 ) max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) , M s , T , g ( x n , y n , x n + 1 , y n + 1 ) d ( g y n , g y n + 1 ) , d ( g y n + 1 , g y n + 2 ) } , N T , g ( x n , y n , x n + 1 , y n + 1 ) = 0 .
(2.5)

By (2.4) and (2.5), we have

ψ ( d ( g x n + 1 , g x n + 2 ) ) ψ ( max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) , d ( g y n , g y n + 1 ) , d ( g y n + 1 , g y n + 2 ) } ) ϕ ( max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) , d ( g y n , g y n + 1 ) , d ( g y n + 1 , g y n + 2 ) } ) .
(2.6)

Similarly, we can show that

ψ ( d ( g y n + 1 , g y n + 2 ) ) ψ ( max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) , d ( g y n , g y n + 1 ) , d ( g y n + 1 , g y n + 2 ) } ) ϕ ( max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) , d ( g y n , g y n + 1 ) , d ( g y n + 1 , g y n + 2 ) } ) .
(2.7)

Now, denote

δ n =max { d ( g x n , g x n + 1 ) , d ( g y n , g y n + 1 ) } .
(2.8)

Combining (2.6), (2.7) and the fact that max{ψ(a),ψ(b)}=ψ(max{a,b}) for a,b[0,+), we have

ψ( δ n + 1 )=max { ψ ( d ( g x n + 1 , g x n + 2 ) ) , ψ ( d ( g y n + 1 , g y n + 2 ) ) } .
(2.9)

So, using (2.6), (2.7), (2.8) together with (2.9), we obtain

ψ ( δ n + 1 ) ψ ( max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) , d ( g y n , g y n + 1 ) , d ( g y n + 1 , g y n + 2 ) } ) ϕ ( max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) , d ( g y n , g y n + 1 ) , d ( g y n + 1 , g y n + 2 ) } ) .
(2.10)

Now we prove that for all nN,

max { d ( g x n , g x n + 1 ) , d ( g x n + 1 , g x n + 2 ) , d ( g y n , g y n + 1 ) , d ( g y n + 1 , g y n + 2 ) } = δ n and δ n + 1 δ n .
(2.11)

For this purpose, consider the following three cases.

Case 1. If max{d(g x n ,g x n + 1 ),d(g x n + 1 ,g x n + 2 ),d(g y n ,g y n + 1 ),d(g y n + 1 ,g y n + 2 )}= δ n , then by (2.10) we have

ψ( δ n + 1 )ψ( δ n )ϕ( δ n )<ψ( δ n ),
(2.12)

so (2.11) obviously holds.

Case 2. If max{d(g x n ,g x n + 1 ),d(g x n + 1 ,g x n + 2 ),d(g y n ,g y n + 1 ),d(g y n + 1 ,g y n + 2 )}=d(g x n + 1 ,g x n + 2 )>0, then by (2.6) we have

ψ ( d ( g x n + 1 , g x n + 2 ) ) ψ ( d ( g x n + 1 , g x n + 2 ) ) ϕ ( d ( g x n + 1 , g x n + 2 ) ) <ψ ( d ( g x n + 1 , g x n + 2 ) ) ,

which is a contradiction.

Case 3. If max{d(g x n ,g x n + 1 ),d(g x n + 1 ,g x n + 2 ),d(g y n ,g y n + 1 ),d(g y n + 1 ,g y n + 2 )}=d(g y n + 1 ,g y n + 2 )>0, then from (2.7) we have

ψ ( d ( g y n + 1 , g y n + 2 ) ) ψ ( d ( g y n + 1 , g y n + 2 ) ) ϕ ( d ( g y n + 1 , g y n + 2 ) ) < ψ ( d ( g y n + 1 , g y n + 2 ) ) ,

which is again a contradiction.

Thus, in all the cases, (2.11) holds for each nN. It follows that the sequence { δ n } is a monotone decreasing sequence of nonnegative real numbers and, consequently, there exists δ0 such that

lim n δ n =δ.
(2.13)

We show that δ=0. Suppose, on the contrary, that δ>0. Taking the limit as n in (2.12) and using the properties of the function ϕ, we get

ψ(δ)ψ(δ)ϕ(δ)<ψ(δ),

which is a contradiction. Therefore δ=0, that is,

lim n δ n = lim n max { d ( g x n , g x n + 1 ) , d ( g y n , g y n + 1 ) } =0,

which implies that

lim n d(g x n ,g x n + 1 )=0and lim n d(g y n ,g y n + 1 )=0.
(2.14)

Now, we claim that

lim n , m max { d ( g x n , g x m ) , d ( g y n , g y m ) } =0.
(2.15)

Assume, on the contrary, that there exist ϵ>0 and subsequences {g x m ( k ) }, {g x n ( k ) } of {g x n } and {g y m ( k ) }, {g y n ( k ) } of {g y n } with m(k)>n(k)k such that

max { d ( g x n ( k ) , g x m ( k ) ) , d ( g y n ( k ) , g y m ( k ) ) } ϵ.
(2.16)

Additionally, corresponding to n(k), we may choose m(k) such that it is the smallest integer satisfying (2.16) and m(k)>n(k)k. Thus,

max { d ( g x n ( k ) , g x m ( k ) 1 ) , d ( g y n ( k ) , g y m ( k ) 1 ) } <ϵ.
(2.17)

Using the triangle inequality in a b-metric space and (2.16) and (2.17), we obtain that

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 (2.14), we obtain

ϵ lim sup k d(g x n ( k ) ,g x m ( k ) )sϵ.
(2.18)

Similarly, we obtain

ϵ lim sup k d(g y n ( k ) ,g y m ( k ) )sϵ.
(2.19)

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 (2.14) and (2.18), we have

ϵ s lim sup k d(g x n ( k ) ,g x m ( k ) + 1 ) s 2 ϵ.
(2.20)

Similarly, we obtain

ϵ s lim sup k d(g y n ( k ) ,g y m ( k ) + 1 ) s 2 ϵ.
(2.21)

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 (2.14) and (2.18), we have

ϵ s lim sup k d(g x m ( k ) ,g x n ( k ) + 1 ) s 2 ϵ.
(2.22)

In a similar way, we obtain

ϵ s lim sup k d(g y m ( k ) ,g y n ( k ) + 1 ) s 2 ϵ.
(2.23)

Also,

d(g x n ( k ) + 1 ,g x m ( k ) )sd(g x n ( k ) + 1 ,g x m ( k ) + 1 )+sd(g x m ( k ) + 1 ,g x m ( k ) ).

So, from (2.14) and (2.22), we have

ϵ s 2 lim sup k d(g x n ( k ) + 1 ,g x m ( k ) + 1 ).
(2.24)

Similarly, we obtain

ϵ s 2 lim sup k d(g y n ( k ) + 1 ,g y m ( k ) + 1 ).
(2.25)

Linking (2.14), (2.18), (2.19), (2.20), (2.21), (2.22) together with (2.23), we get

ϵ s 2 = min { ϵ , ϵ , ϵ s + ϵ s 2 s , ϵ s + ϵ s 2 s } max { lim sup k d ( g x n ( k ) , g x m ( k ) ) , lim sup k d ( g y n ( k ) , g y m ( k ) ) , 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 , lim sup k d ( g y n ( k ) , g y m ( k ) + 1 ) + lim sup k d ( g y m ( k ) , g y n ( k ) + 1 ) 2 s } max { s ϵ , s ϵ , s 2 ϵ + s 2 ϵ 2 s , s 2 ϵ + s 2 ϵ 2 s } = s ϵ .

So,

ϵ s 2 lim sup k M s , T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) )ϵs.
(2.26)

Similarly, we have

ϵ s 2 lim inf k M s , T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) )ϵs
(2.27)

and

lim k N T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) )=0.
(2.28)

Since m(k)>n(k), from (2.2) we have

g x n ( k ) g x m ( k ) ,g y n ( k ) g y 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 ) , y n ( k ) ) , T ( x m ( k ) , y m ( k ) ) ) ) ψ ( s 3 d ( g x n ( k ) + 1 , g x m ( k ) + 1 ) ) ψ ( M s , T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) ) ) ψ ( s 3 d ( g x n ( k ) + 1 , g x m ( k ) + 1 ) ) = ϕ ( M s , T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) ) ) ψ ( s 3 d ( g x n ( k ) + 1 , g x m ( k ) + 1 ) ) = + L θ ( N T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) ) ) , ψ ( s 3 d ( g y n ( k ) + 1 , g y m ( k ) + 1 ) ) = ψ ( s 3 d ( T ( y n ( k ) , x n ( k ) ) , T ( y m ( k ) , x m ( k ) ) ) ) ψ ( s 3 d ( g y n ( k ) + 1 , g y m ( k ) + 1 ) ) ψ ( M s , T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) ) ) ψ ( s 3 d ( g y n ( k ) + 1 , g y m ( k ) + 1 ) ) = ϕ ( M s , T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) ) ) ψ ( s 3 d ( g y n ( k ) + 1 , g y m ( k ) + 1 ) ) = + L θ ( N T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) ) ) .

Since ψ is a non-decreasing function, we have

max { ψ ( s 3 d ( g x n ( k ) + 1 , g x m ( k ) + 1 ) ) , ψ ( s 3 d ( g y n ( k ) + 1 , g y m ( k ) + 1 ) ) } = ψ ( s 3 max { d ( g x n ( k ) + 1 , g x m ( k ) + 1 ) , d ( g y n ( k ) + 1 , g y m ( k ) + 1 ) } ) .

Taking the upper limit as k and using (2.25) and (2.26), we get

ψ ( s ϵ ) ψ ( s 3 max { lim sup k d ( g x n ( k ) + 1 , g x m ( k ) + 1 ) , lim sup k d ( g y n ( k ) + 1 , g y m ( k ) + 1 ) } ) ψ ( lim sup k M s , T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) ) ) ϕ ( lim inf k M s , T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) ) ) + L θ ( lim sup k N T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) ) ) ψ ( s ϵ ) ϕ ( lim inf k M s , T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) ) ) ,

which implies that

ϕ ( lim inf k M s , T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) ) ) =0,

so

lim inf k M s , T , g ( x n ( k ) , y n ( k ) , x m ( k ) , y m ( k ) )=0,

a contradiction to (2.27). Therefore, (2.15) holds and we have

lim n , m d(g x n ,g x m )=0and lim n , m d(g y n ,g y m )=0.

Since X is a complete b-metric space, there exist x,yX such that

lim n g x n + 1 =xand lim n g y n + 1 =y.
(2.29)

From the commutativity of T and g, we have

g(g x n + 1 )=g ( T ( x n , y n ) ) =T(g x n ,g y n ),g(g y n + 1 )=g ( T ( y n , x n ) ) =T(g y n ,g x n ).
(2.30)

Now, we shall show that

gx=T(x,y)andgy=T(y,x).

Letting n in (2.30), from the continuity of T and g, we get

g x = lim n g ( g x n + 1 ) = lim n T ( g x n , g y n ) = T ( lim n g x n , lim n g y n ) = T ( x , y ) , g y = lim n g ( g y n + 1 ) = lim n T ( g y n , g x n ) = T ( lim n g y n , lim n g x n ) = T ( y , x ) .

This implies that (x,y) is a coupled coincidence point of T and g. This completes the proof. □

Corollary 2.3 Let (X,d,) be a partially ordered complete b-metric space, and let T:X×XX be a continuous mapping such that T has the mixed monotone property. Suppose that there exist ψΨ, ϕΦ, θΘ and L0 such that

ψ ( s 3 d ( T ( x , y ) , T ( u , v ) ) ) ψ ( M s ( x , y , u , v ) ) ϕ ( M s ( x , y , u , v ) ) +Lθ ( N ( x , y , u , v ) ) ,

where

M s ( x , y , u , v ) = max { d ( x , u ) , d ( y , v ) , d ( x , T ( x , y ) ) , 1 2 s d ( u , T ( u , v ) ) , d ( y , T ( y , x ) ) , 1 2 s d ( v , T ( v , u ) ) , d ( x , T ( u , v ) ) + d ( u , T ( x , y ) ) 2 s , d ( y , T ( v , u ) ) + d ( v , T ( y , x ) ) 2 s }

and

N(x,y,u,v)=min { d ( x , T ( x , y ) ) , d ( u , T ( u , v ) ) , d ( u , T ( x , y ) ) , d ( x , T ( u , v ) ) }

for all x,y,u,vX with xu and yv. If there exists ( x 0 , y 0 )X×X such that x 0 T( x 0 , y 0 ) and y 0 T( y 0 , x 0 ), then T has a coupled fixed point in X.

Proof Take g= I X and apply Theorem 2.2. □

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. Let T:X×XX be a continuous mapping such that T has the mixed monotone property. Suppose that there exists ϕΦ such that

d ( T ( x , y ) , T ( u , v ) ) 1 s 3 M s (x,y,u,v) 1 s 3 ϕ ( M s ( x , y , u , v ) ) ,
(2.31)

where

M s ( x , y , u , v ) = max { d ( x , u ) , d ( y , v ) , d ( x , T ( x , y ) ) , 1 2 s d ( u , T ( u , v ) ) , d ( y , T ( y , x ) ) , 1 2 s d ( v , T ( v , u ) ) , d ( x , T ( u , v ) ) + d ( u , T ( x , y ) ) 2 s , d ( y , T ( v , u ) ) + d ( v , T ( y , x ) ) 2 s }

for all x,y,u,vX with xu and yv. If there exists ( x 0 , y 0 )X×X such that x 0 T( x 0 , y 0 ) and y 0 T( y 0 , x 0 ), then T has a coupled fixed point in X.

3 Uniqueness of a common fixed point

In this section we shall provide some sufficient conditions under which T and g have a unique common fixed point. Note that if (X,) is a partially ordered set, then we endow the product X×X with the following partial order relation, for all (x,y),(z,t)X×X,

(x,y)(z,t)xz,yt.

From Theorem 2.2, it follows that the set C(T,g) of coupled coincidences is nonempty.

Theorem 3.1 By adding to the hypotheses of Theorem  2.2, the condition: for every (x,y) and (z,t) in X×X, there exists (u,v)X×X such that (T(u,v),T(v,u)) is comparable to (T(x,y),T(y,x)) and to (T(z,t),T(t,z)), then T and g have a unique coupled common fixed point; that is, there exists a unique (x,y)X×X such that

x=gx=T(x,y),y=gy=T(y,x).

Proof We know, from Theorem 2.2, that there exists at least a coupled coincidence point. Suppose that (x,y) and (z,t) are coupled coincidence points of T and g, that is, T(x,y)=gx, T(y,x)=gy, T(z,t)=gz and T(t,z)=gt. We shall show that gx=gz and gy=gt. By the assumptions, there exists (u,v)X×X such that (T(u,v),T(v,u)) is comparable to (T(x,y),T(y,x)) and to (T(z,t),T(t,z)). Without any restriction of the generality, we can assume that

( T ( x , y ) , T ( y , x ) ) ( T ( u , v ) , T ( v , u ) ) and ( T ( z , t ) , T ( t , z ) ) ( T ( u , v ) , T ( v , u ) ) .

Put u 0 =u, v 0 =v and choose ( u 1 , v 1 )X×X such that

g u 1 =T( u 0 , v 0 ),g v 1 =T( v 0 , u 0 ).

For n1, continuing this process, we can construct sequences {g u n } and {g v n } such that

g u n + 1 =T( u n , v n ),g v n + 1 =T( v n , u n )for all n.

Further, set x 0 =x, y 0 =y and z 0 =z, t 0 =t and in the same way define sequences {g x n }, {g y n } and {g z n }, {g t n }. Then it is easy to see that

g x n T(x,y),g y n T(y,x)andg z n T(z,t),g t n T(t,z)
(3.1)

for all n1. Since (T(x,y),T(y,x))=(gx,gy)=(g x 1 ,g y 1 ) is comparable to (T(u,v),T(v,u))=(gu,gv)=(g u 1 ,g v 1 ), then it is easy to show (gx,gy)(gu,gv). Recursively, we get that

(g x n ,g y n )(g u n ,g v n )for all n.
(3.2)

Thus from (2.1) we have

ψ ( d ( g x , g u n + 1 ) ) ψ ( s 3 d ( g x , g u n + 1 ) ) = ψ ( s 3 d ( T ( x , y ) , T ( u n , v n ) ) ) ψ ( M s , T , g ( x , y , u n , v n ) ) ϕ ( M s , T , g ( x , y , u n , v n ) ) + L θ ( N T , g ( x , y , u n , v n ) ) ,

where

M s , T , g ( x , y , u n , v n ) = max { d ( g x , g u n ) , d ( g y , g v n ) , d ( g x , T ( x , y ) ) , 1 2 s d ( g u n , T ( u n , v n ) ) , d ( g y , T ( y , x ) ) , 1 2 s d ( g v n , T ( v n , u n ) ) , d ( g x , T ( u n , v n ) ) + d ( g u n , T ( x , y ) ) 2 s , d ( g y , T ( v n , u n ) ) + d ( g v n , T ( y , x ) ) 2 s } max { d ( g x , g u n ) , d ( g y , g v n ) , d ( g y , g v n + 1 ) , d ( g x , g u n + 1 ) } .

It is easy to show that

M s , T , g (x,y, u n , v n )max { d ( g x , g u n ) , d ( g y , g v n ) }

and

N T , g (x,y, u n , v n )=0.

Hence,

ψ ( d ( g x , g u n + 1 ) ) ψ ( max { d ( g x , g u n ) , d ( g y , g v n ) } ) ϕ ( max { d ( g x , g u n ) , d ( g y , g v n ) } ) .
(3.3)

Similarly, one can prove that

ψ ( d ( g y , g v n + 1 ) ) ψ ( max { d ( g x , g u n ) , d ( g y , g v n ) } ) ϕ ( max { d ( g x , g u n ) , d ( g y , g v n ) } ) .
(3.4)

Combining (3.3), (3.4) and the fact that max{ψ(a),ψ(b)}=ψ(max{a,b}) for a,b[0,+), we have

ψ ( max { d ( g x , g u n + 1 ) , d ( g y , g v n + 1 ) } ) = max { ψ ( d ( g x , g u n + 1 ) ) , ψ ( d ( g y , g v n + 1 ) ) } ψ ( max { d ( g x , g u n ) , d ( g y , g v n ) } ) ϕ ( max { d ( g x , g u n ) , d ( g y , g v n ) } ) ψ ( max { d ( g x , g u n ) , d ( g y , g v n ) } ) .
(3.5)

Using the non-decreasing property of ψ, we get that

max { d ( g x , g u n + 1 ) , d ( g y , g v n + 1 ) } max { d ( g x , g u n ) , d ( g y , g v n ) }

implies that max{d(gx,g u n ),d(gy,g v n )} is a non-increasing sequence. Hence, there exists r0 such that

lim n max { d ( g x , g u n ) , d ( g y , g v n ) } =r.

Passing the upper limit in (3.5) as n, we obtain

ψ(r)ψ(r)ϕ(r),

which implies that ϕ(r)=0, and then r=0. We deduce that

lim n max { d ( g x , g u n ) , d ( g y , g v n ) } =0,

which concludes

lim n d(gx,g u n )= lim n d(gy,g v n )=0.
(3.6)

Similarly, one can prove that

lim n d(gz,g u n )= lim n d(gt,g v n )=0.
(3.7)

From (3.6) and (3.7), we have gx=gz and gy=gt. Since gx=T(x,y) and gy=T(y,x), by the commutativity of T and g, we have

g(gx)=g ( T ( x , y ) ) =T(gx,gy),g(gy)=g ( T ( y , x ) ) =T(gy,gx).
(3.8)

Denote gx=a and gy=b. Then from (3.8) we have

g(a)=T(a,b),g(b)=T(b,a).
(3.9)

Thus, (a,b) is a coupled coincidence point. It follows that ga=gz and gb=gy, that is,

g(a)=a,g(b)=b.
(3.10)

From (3.9) and (3.10), we obtain

a=g(a)=T(a,b),b=g(b)=T(b,a).
(3.11)

Therefore, (a,b) is a coupled common fixed point of T and g. To prove the uniqueness of the point (a,b), assume that (c,d) is another coupled common fixed point of T and g. Then we have

c=gc=T(c,d),d=gd=T(d,c).

Since (c,d) is a coupled coincidence point of T and g, we have gc=gx=a and gd=gy=b. Thus c=gc=ga=a and d=gd=gb=b, which is the desired result. □

Theorem 3.2 In addition to the hypotheses of Theorem  3.1, if g x 0 and g y 0 are comparable, then T and g have a unique common fixed point, that is, there exists xX such that x=gx=T(x,x).

Proof Following the proof of Theorem 3.1, T and g have a unique coupled common fixed point (x,y). We only have to show that x=y. Since g x 0 and g y 0 are comparable, we may assume that g x 0 g y 0 . By using the mathematical induction, one can show that

g x n g y n for all n0,
(3.12)

where {g x n } and {g y n } are defined by (2.2). From (2.29) and Lemma 1.4, we have

ψ ( s d ( x , y ) ) = ψ ( s 3 1 s 2 d ( x , y ) ) lim sup n ψ ( s 3 d ( g x n + 1 , g y n + 1 ) ) = lim sup n ψ ( s 3 d ( T ( x n , y n ) , T ( y n , x n ) ) ) lim sup n ψ ( M s , T , g ( x n , y n , y n , x n ) ) lim inf n ϕ ( M s , T , g ( x n , y n , y n , x n ) ) + lim sup n L θ ( N T , g ( x n , y n , y n , x n ) ) ψ ( d ( x , y ) ) lim inf n ϕ ( M s ( x n , y n , y n , x n ) ) < ψ ( d ( x , y ) ) ,

a contradiction. Therefore, x=y, that is, T and g have a common fixed point. □

Remark 3.3 Since a b-metric is a metric when s=1, from the results of Jachymski [29], the condition

ψ ( d ( F ( x , y ) , F ( u , v ) ) ) ψ ( max { d ( g x , g u ) , d ( g y , g v ) } ) ϕ ( max { d ( g x , g u ) , d ( g y , g v ) } )

is equivalent to

d ( F ( x , y ) , F ( u , v ) ) φ ( max { d ( g x , g u ) , d ( g y , g v ) } ) ,

where ψΨ, ϕΦ and φ:[0,)[0,) is continuous, φ(t)<t for all t>0 and φ(t)=0 if and only if t=0. So, our results can be viewed as a generalization and extension of the corresponding results in [15, 25, 3032] and several other comparable results.

4 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 our coupled fixed point theorem. Consider the nonlinear quadratic integral equation

x(t)=h(t)+λ 0 1 k 1 (t,s) f 1 ( s , x ( s ) ) ds 0 1 k 2 (t,s) f 2 ( s , x ( s ) ) ds,tI=[0,1],λ0.
(4.1)

Let Γ denote the class of those functions γ:[0,+)[0,+) which satisfy the following conditions:

  1. (i)

    γ is non-decreasing and ( γ ( t ) ) p γ( t p ) for all p1.

  2. (ii)

    There exists ϕΦ such that γ(t)=tϕ(t) for all t[0,+).

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

We will analyze Eq. (4.1) under the following assumptions:

(a1) f i :I×RR (i=1,2) are continuous functions, f i (t,x)0 and there exist two functions m i L 1 (I) such that f i (t,x) m i (t) (i=1,2).

(a2) f 1 (t,x) is monotone non-decreasing in x and f 2 (t,y) is monotone non-increasing in y for all x,yR and tI.

(a3) h:IR is a continuous function.

(a4) k i :I×IR (i=1,2) are continuous in tI for every sI and measurable in sI for all tI such that

0 1 k i (t,s) m i (s)dsK,i=1,2,

and k i (t,x)0.

(a5) There exist constants 0 L i <1 (i=1,2) and γΓ such that for all x,yR and xy,

| f i ( t , x ) f i ( t , y ) | L i γ(xy)(i=1,2).

(a6) There exist α,βC(I) such that

α ( t ) h ( t ) + λ 0 1 k 1 ( t , s ) f 1 ( s , α ( s ) ) d s 0 1 k 2 ( t , s ) f 2 ( s , β ( s ) ) d s h ( t ) + λ 0 1 k 1 ( t , s ) f 1 ( s , β ( s ) ) d s 0 1 k 2 ( t , s ) f 2 ( s , α ( s ) ) d s β ( t ) .

(a7) max{ L 1 p , L 2 p } λ p K 2 p 1 2 4 p 3 .

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,yC(I).

This space can also be equipped with a partial order given by

x,yC(I),xyx(t)y(t)for any tI.

Now, for p1, 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,yC(I).

It is easy to see that (X,d) is a complete b-metric space with s= 2 p 1 [3].

Also, X×X=C(I)×C(I) is a partially ordered set if we define the following order relation:

(x,y),(u,v)X×X,(x,y)(u,v)xuandyv.

For any x,yX and each tI, 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),(u,v)X×X, one can take (max{x,u},min{y,v})X×X which is comparable to (x,y) and (u,v). Now, we formulate the main result of this section.

Theorem 4.1 Under assumptions (a1)-(a7), Eq. (4.1) has a unique solution in C(I).

Proof We consider the operator T:X×XX defined by

T(x,y)(t)=h(t)+λ 0 1 k 1 (t,s) f 1 ( s , x ( s ) ) ds 0 1 k 2 (t,s) f 2 ( s , y ( s ) ) dsfor tI.

By virtue of our assumptions, T is well defined (this means that if x,yX, then T(x,y)X). Firstly, we prove that T has the mixed monotone property. In fact, for x 1 x 2 and tI, we have

T ( x 1 , y ) ( t ) T ( x 2 , y ) ( t ) = h ( t ) + λ 0 1 k 1 ( t , s ) f 1 ( s , x 1 ( s ) ) d s 0 1 k 2 ( t , s ) f 2 ( s , y ( s ) ) d s h ( t ) λ 0 1 k 1 ( t , s ) f 1 ( s , x 2 ( s ) ) d s 0 1 k 2 ( t , s ) f 2 ( s , y ( s ) ) d s = λ 0 1 k 1 ( t , s ) [ f 1 ( s , x 1 ( s ) ) f 1 ( s , x 2 ( s ) ) ] d s 0 1 k 2 ( t , s ) f 2 ( s , y ( s ) ) d s 0 .

Similarly, if y 1 y 2 and tI, then T(x, y 1 )(t)T(x, y 2 )(t). Therefore, T has the mixed monotone property. Also, for (x,y)(u,v), that is, xu and yv, we have

| T ( x , y ) ( t ) T ( u , v ) ( t ) | | λ 0 1 k 1 ( t , s ) f 1 ( s , x ( s ) ) d s 0 1 k 2 ( t , s ) [ f 2 ( s , y ( s ) ) f 2 ( s , v ( s ) ) ] d s + λ 0 1 k 2 ( t , s ) f 2 ( s , v ( s ) ) d s 0 1 k 1 ( t , s ) [ f 1 ( s , x ( s ) ) f 1 ( s , u ( s ) ) ] d s | λ 0 1 k 1 ( t , s ) f 1 ( s , x ( s ) ) d s 0 1 k 2 ( t , s ) | f 2 ( s , y ( s ) ) f 2 ( s , v ( s ) ) | d s + λ 0 1 k 2 ( t , s ) f 2 ( s , v ( s ) ) d s 0 1 k 1 ( t , s ) | f 1 ( s , x ( s ) ) f 1 ( s , u ( s ) ) | d s λ 0 1 k 1 ( t , s ) m 1 ( s ) d s 0 1 k 2 ( t , s ) L 2 γ ( y ( s ) v ( s ) ) d s + λ 0 1 k 2 ( t , s ) m 2 ( s ) d s 0 1 k 1 ( t , s ) L 1 γ ( u ( s ) x ( s ) ) d s .

Since the function γ is non-decreasing and xu and yv, we have

γ ( u ( s ) x ( s ) ) γ ( sup t I | x ( s ) u ( s ) | ) =γ ( ρ ( x , u ) )

and

γ ( y ( s ) v ( s ) ) γ ( sup t I | y ( s ) v ( s ) | ) =γ ( ρ ( y , v ) ) ,

hence

| T ( x , y ) ( t ) T ( u , v ) ( t ) | λ K 0 1 k 2 ( t , s ) L 2 γ ( ρ ( y , v ) ) d s + λ K 0 1 k 1 ( t , s ) L 1 γ ( ρ ( u , x ) ) d s λ K 2 max { L 1 , L 2 } [ γ ( ρ ( u , x ) ) + γ ( ρ ( y , v ) ) ] .

Then we can obtain

d ( T ( x , y ) , T ( u , v ) ) = sup t I | T ( x , y ) ( t ) T ( u , v ) ( t ) | p { λ K 2 max { L 1 , L 2 } [ γ ( ρ ( u , x ) ) + γ ( ρ ( y , v ) ) ] } p = λ p K 2 p max { L 1 p , L 2 p } [ γ ( ρ ( u , x ) ) + γ ( ρ ( y , v ) ) ] p ,

and using the fact that ( a + b ) p 2 p 1 ( a p + b p ) for a,b(0,+) and p>1, we have

d ( T ( x , y ) , T ( u , v ) ) 2 p 1 λ p K 2 p max { L 1 p , L 2 p } [ ( γ ( ρ ( u , x ) ) ) p + ( γ ( ρ ( y , v ) ) ) p ] 2 p 1 λ p K 2 p max { L 1 p , L 2 p } [ γ ( d ( u , x ) ) + γ ( d ( y , v ) ) ] 2 p λ p K 2 p max { L 1 p , L 2 p } [ γ ( M s ( x , y , u , v ) ) ] 2 p λ p K 2 p max { L 1 p , L 2 p } [ M s ( x , y , u , v ) ϕ ( M s ( x , y , u , v ) ) ] 1 2 3 p 3 M s ( x , y , u , v ) 1 2 3 p 3 ϕ ( M s ( x , y , u , v ) ) .

This proves that the operator T satisfies the contractive condition (2.31) appearing in Corollary 2.4.

Finally, let α, β be the functions appearing in assumption (a6); then, by (a6), we get

αT(α,β)T(β,α)β.

Theorem 3.1 gives us that T has a unique coupled fixed point ( x , y )X×X. Since αβ, Theorem 3.2 says that x = y and this implies x =T( x , x ). So, x C(I) is the unique solution of Eq. (4.1) and the proof is complete. □

References

  1. Czerwik S: Nonlinear set-valued contraction mappings in b -metric spaces. Atti Semin. Mat. Fis. Univ. Modena 1998, 46(2):263–276.

    MathSciNet  Google Scholar 

  2. Czerwik S: Contraction mappings in b -metric spaces. Acta Math. Inf. Univ. Ostrav. 1993, 1: 5–11.

    MathSciNet  Google Scholar 

  3. Aghajani, A, Abbas, M, Roshan, JR: Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces. Math. Slovaca (to appear)

  4. Akkouchi M: Common fixed point theorems for two selfmappings of a b -metric space under an implicit relation. Hacet. J. Math. Stat. 2011, 40(6):805–810.

    MathSciNet  Google Scholar 

  5. Aydi H, Bota M, Karapınar E, Mitrovic S: A fixed point theorem for set-valued quasi-contractions in b -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 88

    Google Scholar 

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

    MathSciNet  Google Scholar 

  7. 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.

    MathSciNet  Google Scholar 

  8. Boriceanu M, Bota M, Petrusel A: Multivalued fractals in b -metric spaces. Cent. Eur. J. Math. 2010, 8(2):367–377. 10.2478/s11533-010-0009-4

    Article  MathSciNet  Google Scholar 

  9. Czerwik S, Dlutek K, Singh SL: Round-off stability of iteration procedures for set-valued operators in b -metric spaces. J. Nat. Phys. Sci. 2007, 11: 87–94.

    MathSciNet  Google Scholar 

  10. Huang H, Xu S: Fixed point theorems of contractive mappings in cone b -metric spaces and applications. Fixed Point Theory Appl. 2012., 2012: Article ID 220

    Google Scholar 

  11. Hussain N, Shah MH: KKM mappings in cone b -metric spaces. Comput. Math. Appl. 2011, 62: 1677–1684. 10.1016/j.camwa.2011.06.004

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  13. Pacurar M: Sequences of almost contractions and fixed points in b -metric spaces. An. Univ. Vest. Timiş., Ser. Mat.-Inf. 2010, XLVIII(3):125–137.

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Bhaskar TG, Lakshmikantham V: Fixed point theory in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017

    Article  MathSciNet  Google Scholar 

  16. Abbas M, Sintunavarat W, Kumam P: Coupled fixed point of generalized contractive mappings on partially ordered G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 31

    Google Scholar 

  17. Agarwal RP, Sintunavarat W, Kumam P: Coupled coincidence point and common coupled fixed point theorems lacking the mixed monotone property. Fixed Point Theory Appl. 2013., 2013: Article ID 22

    Google Scholar 

  18. Chandok S, Sintunavarat W, Kumam P: Some coupled common fixed points for a pair of mappings in partially ordered G -metric spaces. Math. Sci. 2013., 7: Article ID 24

    Google Scholar 

  19. Karapınar E, Kumam P, Sintunavarat W: Coupled fixed point theorems in cone metric spaces with a c -distance and applications. Fixed Point Theory Appl. 2012., 2012: Article ID 194

    Google Scholar 

  20. 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 81

    Google Scholar 

  21. Sintunavarat W, Kumam P:Weak condition for generalized multivalued (f,α,β)-weak contraction mappings. Appl. Math. Lett. 2011, 24: 460–465. 10.1016/j.aml.2010.10.042

    Article  MathSciNet  Google Scholar 

  22. Sintunavarat W, Kumam P: Common fixed point theorems for hybrid generalized multi-valued contraction mappings. Appl. Math. Lett. 2012, 25(1):52–57. 10.1016/j.aml.2011.05.047

    Article  MathSciNet  Google Scholar 

  23. Sintunavarat W, Kumam P: Common fixed point theorem for cyclic generalized multivalued contraction mappings. Appl. Math. Lett. 2012, 25(11):1849–1855. 10.1016/j.aml.2012.02.045

    Article  MathSciNet  Google Scholar 

  24. Abbas M, Ali Khan M, Radenović S: Common coupled fixed point theorems in cone metric spaces for w -compatible mappings. Appl. Math. Comput. 2010, 217: 195–202. 10.1016/j.amc.2010.05.042

    Article  MathSciNet  Google Scholar 

  25. Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal., Theory Methods Appl. 2009, 70(12):4341–4349. 10.1016/j.na.2008.09.020

    Article  Google Scholar 

  26. Khamsi MA, Hussain N: KKM mappings in metric type spaces. Nonlinear Anal. 2010, 73(9):3123–3129. 10.1016/j.na.2010.06.084

    Article  MathSciNet  Google Scholar 

  27. Khamsi MA: Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 315398 10.1155/2010/315398

    Google Scholar 

  28. 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.2010.06.025

    Article  MathSciNet  Google Scholar 

  29. Jachymski J: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal. 2011, 74: 768–774. 10.1016/j.na.2010.09.025

    Article  MathSciNet  Google Scholar 

  30. Choudhury BS, Metiya N, Kundu A: Coupled coincidence point theorems in ordered metric spaces. Ann. Univ. Ferrara 2011, 57: 1–16. 10.1007/s11565-011-0117-5

    Article  MathSciNet  Google Scholar 

  31. Harjani J, López B, Sadarangani K: Fixed point theorems for mixed monotone operators and applications to integral equations. Nonlinear Anal. 2011, 74: 1749–1760. 10.1016/j.na.2010.10.047

    Article  MathSciNet  Google Scholar 

  32. Luong NV, Thuan NX: Coupled fixed point theorems in partially ordered metric spaces. Bull. Math. Anal. Appl. 2010, 4: 16–24.

    Google Scholar 

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Correspondence to Asadollah Aghajani.

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Aghajani, A., Arab, R. Fixed points of (ψ,ϕ,θ)-contractive mappings in partially ordered b-metric spaces and application to quadratic integral equations. Fixed Point Theory Appl 2013, 245 (2013). https://doi.org/10.1186/1687-1812-2013-245

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