Skip to main content

Some coupled fixed point theorems in quasi-partial metric spaces

Abstract

In this paper, we study some coupled fixed point results in a quasi-partial metric space. Also, we introduce some examples to support the useability of our results.

MSC:47H10, 54H25.

1 Introduction and preliminaries

In 1994, Matthews [1] introduced the notion of partial metric spaces and extended the Banach contraction principle from metric spaces to partial metric spaces. After that, many fixed point theorems in partial metric spaces have been given by several authors (for example, see [229]). Very recently, Haghi et al. [30, 31] showed in their interesting paper that some of fixed point theorems in partial metric spaces can be obtained from metric spaces.

Following Matthews [1], the notion of partial metric space is given as follows.

Definition 1.1 [1]

A partial metric on a nonempty set X is a function p:X×X R + such that for all x,y,zX:

( p 1 ) x=yp(x,x)=p(x,y)=p(y,y),

( p 2 ) p(x,x)p(x,y),

( p 3 ) p(x,y)=p(y,x),

( p 4 ) p(x,y)p(x,z)+p(z,y)p(z,z).

A partial metric space is a pair (X,p) such that X is a nonempty set and p is a partial metric on X.

Karapinar et al. [32] introduced the concept of quasi-partial metric spaces and studied some fixed point theorems on quasi-partial metric spaces.

Definition 1.2 [32]

A quasi-partial metric on a nonempty set X is a function q:X×X R + which satisfies:

( QPM 1 ) If q(x,x)=q(x,y)=q(y,y), then x=y,

( QPM 2 ) q(x,x)q(x,y),

( QPM 3 ) q(x,x)q(y,x), and

( QPM 4 ) q(x,y)+q(z,z)q(x,z)+q(z,y)

for all x,y,zX.

A quasi-partial metric space is a pair (X,q) such that X is a nonempty set and q is a quasi-partial metric on X.

Let q be a quasi-partial metric space on the set X. Then

d q (x,y)=q(x,y)+q(y,x)p(x,x)p(y,y)

is a metric on X.

Definition 1.3 [32]

Let (X,q) be a quasi-partial metric space. Then:

  1. (1)

    A sequence ( x n ) converges to a point xX if and only if

    q(x,x)= lim n q(x, x n )= lim n q( x n ,x).
  2. (2)

    A sequence ( x n ) is called a Cauchy sequence if lim n , m p( x n , x m ) and lim n , m p( x n , x m ) exist (and are finite).

  3. (3)

    The quasi-partial metric space (X,q) is said to be complete if every Cauchy sequence ( x n ) in X converges, with respect to τ q , to a point xX such that

    q(x,x)= lim n , m q( x n , x m )= lim n , m q( x m , x n ).

The following lemma is crucial in our work.

Lemma 1.1 [32]

Let (X,q) be a quasi-partial metric space. Then the following statements hold true:

  1. (A)

    If q(x,y)=0, then x=y.

  2. (B)

    If xy, then q(x,y)>0 and q(y,x)>0.

Bhaskar and Lakshmikantham [33] introduced the concept of coupled fixed point and studied some nice coupled fixed point theorems. Later, Lakshmikantham and Ćirić [34] introduced the notion of a coupled coincidence point of mappings. For some works on a coupled fixed point, we refer the reader to [3546].

Definition 1.4 [33]

Let X be a nonempty set. We call an element (x,y)X×X a coupled fixed point of the mapping F:X×XX if

F(x,y)=xandF(y,x)=y.

Definition 1.5 [34]

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

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

Abbas et al. [47] introduced the concept of w-compatible mappings as follows.

Definition 1.6 [47]

Let X be a nonempty set. We say that the mappings F:X×XX and g:XX are w-compatible if gF(x,y)=F(gx,gy) whenever gx=F(x,y) and gy=F(y,x).

In this paper, we study some coupled fixed point theorems in the setting of quasi-partial metric spaces. We introduce some examples to support our results.

2 The main results

We start this section with the following coupled fixed point theorem.

Theorem 2.1 Let (X,q) be a quasi-partial metric space, g:XX and F:X×XX be two mappings. Suppose that there exist k 1 , k 2 and k 3 in [0,1) with k 1 + k 2 + k 3 <1 such that the condition

q ( F ( x , y ) , F ( x , y ) ) + q ( F ( y , x ) , F ( y , x ) ) k 1 ( q ( g x , g x ) + q ( g y , g y ) ) + k 2 ( q ( g x , F ( x , y ) ) + q ( g y , F ( y , x ) ) ) + k 3 ( q ( g x , F ( x , y ) ) + q ( g y , F ( y , x ) ) )
(2.1)

holds for all x,y, x , y X. Also, suppose the following hypotheses:

  1. (1)

    F(X×X)gX.

  2. (2)

    g(X) is a complete subspace of X with respect to the quasi-partial metric q.

Then the mappings F and g have a coupled coincidence point (u,v) satisfying gu=F(u,v)=F(v,u)=gu.

Moreover, if F and g are w-compatible, then F and g have a unique common fixed point of the form (u,u).

Proof Let x 0 , y 0 X. Since F(X×X)gX, we put g x 1 =F( x 0 , y 0 ) and g y 1 =F( y 0 , x 0 ). Again, since F(X×X)gX, we put g x 2 =F( x 1 , y 1 ) and g y 2 =F( y 1 , x 1 ). Continuing this process, we can construct two sequences (g x n ) and (g y n ) in X such that

g x n =F( x n 1 , y n 1 ),nN,

and

g y n =F( y n 1 , x n 1 ),nN.
  • Let nN. Then by inequality (2.1), we obtain

    (2.2)

From (2.2), we have

q(g x n ,g x n + 1 )+q(g y n ,g y n + 1 ) k 1 + k 2 1 k 3 ( q ( g x n 1 , g x n ) + q ( g y n 1 , g y n ) ) .
(2.3)

Put k= k 1 + k 2 1 k 3 . Then k<1. Repeating (2.3) n-times, we get

q(g x n ,g x n + 1 )+q(g y n ,g y n + 1 ) k n ( q ( g x 0 , g x 1 ) + q ( g y 0 , g y 1 ) ) .

Let m and n be natural numbers with m>n. Then

q ( g x n , g x m ) + q ( g y n , g y m ) i = n m 1 q ( g x i , g x i + 1 ) + q ( g y i , g y i + 1 ) i = n m 1 k i ( q ( g x 0 , g x 1 ) + q ( g y 0 , g y 1 ) ) k n 1 k ( q ( g x 0 , g x 1 ) + q ( g y 0 , g y 1 ) ) .
(2.4)

Letting n,m+, we get

lim n , m + q(g x n ,g x m )= lim n , m + q(g y n ,g y m )=0.
(2.5)
  • By similar arguments as above, we can show that

    lim n , m + q(g x m ,g x n )= lim n , m + q(g y m ,g y n )=0.
    (2.6)

Thus the sequences (g x n ) and (g y n ) are Cauchy in (gX,q). Since (gX,q) is complete, there are u and v in X such that g x n gu and g y n gy with respect to τ q , that is,

q ( g u , g u ) = lim n + q ( g u , g x n ) = lim n + q ( g x n , g u ) = lim n , m + q ( g x m , g x n ) = lim n , m + q ( g x n , g x m )

and

q ( g v , g v ) = lim n + q ( g v , g y n ) = lim n + q ( g y n , g v ) = lim n , m + q ( g y m , g y n ) = lim n , m + q ( g y n , g y m ) .

From (2.5) and (2.6), we have

q ( g u , g u ) = lim n + q ( g u , g x n ) = lim n + q ( g x n , g u ) = lim n , m + q ( g x m , g x n ) = lim n , m + q ( g x n , g x m ) = 0
(2.7)

and

q ( g v , g v ) = lim n + q ( g v , g y n ) = lim n + q ( g y n , g v ) = lim n , m + q ( g y m , g y n ) = lim n , m + q ( g y n , g y m ) = 0 .
(2.8)

For n in ℕ, we obtain

q ( g x n + 1 , F ( u , v ) ) q ( g x n + 1 , g u ) + q ( g u , F ( u , v ) ) q ( g u , g u ) q ( g x n + 1 , g u ) + q ( g u , F ( u , v ) ) q ( g x n + 1 , g u ) + q ( g u , g x n + 1 ) + q ( g x n + 1 , F ( u , v ) ) q ( g x n + 1 , g x n + 1 ) q ( g x n + 1 , g u ) + q ( g u , g x n + 1 ) + q ( g x n + 1 , F ( u , v ) ) .

On letting n+ in the above inequalities and using (2.7) and (2.8), we have

lim n + q ( g x n + 1 , F ( u , v ) ) =q ( g u , F ( u , v ) ) .
(2.9)

Similarly, we have

lim n + q ( g y n + 1 , F ( v , u ) ) =q ( g v , F ( v , u ) ) .
(2.10)
  • We show that gu=F(u,v) and gv=F(v,u).

For nN, we have

q ( g x n + 1 , F ( u , v ) ) + q ( g y n + 1 , F ( v , u ) ) = q ( F ( x n , y n ) , F ( u , v ) ) + q ( F ( y n , x n ) , F ( v , u ) ) k 1 ( q ( g x n , g u ) + q ( g y n , g v ) ) + k 2 ( q ( g x n , F ( x n , y n ) ) + q ( g y n , F ( y n , x n ) ) + k 3 ( q ( g u , F ( u , v ) ) + q ( g v , F ( v , u ) ) ) = k 1 ( q ( g x n , g u ) + q ( g y n , g v ) ) + k 1 ( q ( g x n , g x n + 1 ) ) + q ( g y n , g y n + 1 ) ) + k 3 ( q ( g u , F ( u , v ) ) + q ( g v , F ( v , u ) ) ) .

Letting n+ in above inequalities and using (2.9)-(2.10), we get

q ( g u , F ( u , v ) ) +q ( g v , F ( v , u ) ) k 3 ( q ( g u , F ( u , v ) ) + q ( g v , F ( v , u ) ) ) .

Since k 3 <1, we get q(gu,F(u,v))=q(gv,F(v,u))=0. By Lemma 1.1, we get gu=F(u,v) and gv=F(v,u). Next, we will show that gu=gv. Now, from (2.1) we have

q ( g u , g v ) + q ( g v , g u ) = q ( F ( u , v ) , F ( v , u ) ) + q ( F ( v , u ) , F ( u , v ) ) k 1 ( q ( g u , g v ) + q ( g v , g u ) ) + k 2 ( q ( g u , F ( u , v ) ) + q ( g v , F ( v , u ) ) ) + k 3 ( q ( g v , F ( v , u ) ) + q ( g u , F ( u , v ) ) ) = k 1 ( q ( g u , g v ) + q ( g v , g u ) ) + k 2 ( q ( g u , g u ) + q ( g v , g v ) ) + k 3 ( q ( g v , g v ) + q ( g u , g u ) ) .

Using (2.7) and (2.8), we obtain

q(gu,gv)+q(gv,gu) k 1 ( q ( g u , g v ) + q ( g v , g u ) ) .

Since k 1 <1, we have q(gu,gv)=q(gv,gu)=0 By Lemma 1.1, we get that gu=gv. Finally, assume that g and F are w-compatible. Let u 1 =gu and v 1 =gv. Then

g u 1 =ggu=g ( F ( u , v ) ) =F(gu,gv)=F( u 1 , v 1 )
(2.11)

and

g v 1 =ggv=g ( F ( v , u ) ) =F(gv,gu)=F( v 1 , u 1 ).
(2.12)

From (2.11) and (2.12), we can show that

q(g u 1 ,g u 1 )=q(g v 1 ,g v 1 ).
  • We claim that g u 1 =gu and g v 1 =gv.

From (2.1), we have

q ( g u 1 , g u ) + q ( g v 1 , g v ) = q ( F ( u 1 , v 1 ) , F ( u , v ) ) + q ( F ( v 1 , u 1 ) , F ( v , u ) ) k 1 ( q ( g u 1 , g u ) + q ( g v 1 , g v ) ) + k 2 ( q ( g u 1 , F ( u 1 , v 1 ) ) + q ( g v 1 , F ( v 1 , u 1 ) ) ) + k 3 ( q ( g u , F ( u , v ) ) + q ( g v , F ( v , u ) ) ) = k 1 ( q ( g u 1 , g u ) + q ( g v 1 , g v ) ) + k 2 ( q ( g u 1 , g u 1 ) + q ( g v 1 , g v 1 ) ) + k 3 ( q ( g u , g u ) + q ( g v , g v ) ) = k 1 ( q ( g u 1 , g u ) + q ( g v 1 , g v ) ) .

Since k 1 <1, we conclude that q(g u 1 ,gu)=q(g v 1 ,gv)=0. By Lemma 1.1, we get g u 1 =gu and g v 1 =gv. Therefore u 1 =g u 1 and v 1 =g v 1 . Again, since gu=gv, we get u 1 = v 1 . Hence F and g have a unique common coupled fixed point of the form (u,u). □

Corollary 2.1 Let (X,q) be a quasi-partial metric space, g:XX and F:X×XX be two mappings. Suppose that there exist a, b, c, d, e, f in [0,1) with a+b+c+d+e+f<1 such that

q ( F ( x , y ) , F ( x , y ) ) a q ( g x , g x ) + b q ( g y , g y ) + c q ( g x , F ( x , y ) ) + d q ( g y , F ( y , x ) ) + e q ( g x , F ( x , y ) ) + f q ( g y , F ( y , x ) )
(2.13)

holds for all x,y, x , y X. Also, suppose the following hypotheses:

  1. (1)

    F(X×X)gX.

  2. (2)

    g(X) is a complete subspace of X with respect to the quasi-partial metric q.

Then F and g have a coupled coincidence point (u,v) satisfying gu=F(u,v)=F(v,u)=gu.

Moreover, if F and g are w-compatible, then F and g have a unique common fixed point of the form (u,u).

Proof Given x,y, x , y X. From (2.13), we have

q ( F ( x , y ) , F ( x , y ) ) a q ( g x , g x ) + b q ( g y , g y ) + c q ( g x , F ( x , y ) ) + d q ( g y , F ( y , x ) ) + e q ( g x , F ( x , y ) ) + f q ( g y , F ( y , x ) )
(2.14)

and

q ( F ( y , x ) , F ( y , x ) ) a q ( g y , g y ) + b q ( g x , g x ) + c q ( g y , F ( y , x ) ) + d q ( g x , F ( x , y ) ) + e q ( g y , F ( y , x ) ) + f q ( g x , F ( x , y ) ) .
(2.15)

Adding inequality (2.14) to inequality (2.15), we get

q ( F ( x , y ) , F ( x , y ) ) + q ( F ( y , x ) , F ( y , x ) ) ( a + b ) ( q ( g x , g x ) + q ( g y , g y ) ) + ( c + d ) ( q ( g x , F ( x , y ) ) + q ( g y , F ( y , x ) ) ) + ( e + f ) ( q ( g x , F ( x , y ) ) + q ( g y , F ( y , x ) ) ) .

Thus, the result follows from Theorem 2.1. □

Corollary 2.2 Let (X,q) be a quasi-partial metric space, let g:XX and F:X×XX be two mappings. Suppose that there exists k[0,1) with k 1 + k 2 + k 3 <1 such that

q ( F ( x , y ) , F ( x , y ) ) +q ( F ( y , x ) , F ( y , x ) ) k ( q ( g x , g x ) + q ( g y , g y ) )

holds for all x,y, x , y X. Also, suppose the following hypotheses:

  1. (1)

    F(X×X)gX.

  2. (2)

    g(X) is a complete subspace of X with respect to the quasi-partial metric q.

Then F and g have a coupled coincidence point (u,v) satisfying gu=F(u,v)=F(v,u)=gu.

Moreover, if F and g are w-compatible, then F and g have a unique common fixed point of the form (u,u).

Corollary 2.3 Let (X,q) be a quasi-partial metric space, g:X×X and F:X×XX be two mappings. Suppose that there exists k[0,1) with k<1 such that

q ( F ( x , y ) , F ( x , y ) ) +q ( F ( y , x ) , F ( y , x ) ) k ( q ( g x , F ( x , y ) ) + q ( g y , F ( y , x ) ) )

holds for all x,y, x , y X. Also, suppose the following hypotheses:

  1. (1)

    F(X×X)X.

  2. (2)

    g(X) is a complete subspace of X with respect to the quasi-partial metric q.

Then F and g have a coupled coincidence point (u,v) satisfying gu=F(u,v)=F(v,u)=gu.

Moreover, if F and g are w-compatible, then F and g have a unique common fixed point of the form (u,u).

Corollary 2.4 Let (X,q) be a quasi-partial metric space, g:XX and F:X×XX be two mappings. Suppose that there exists k[0,1) with k<1 such that

q ( F ( x , y ) , F ( x , y ) ) +q ( F ( y , x ) , F ( y , x ) ) k ( q ( g x , F ( x , y ) ) + q ( g y , F ( y , x ) ) )

holds for all x,y, x , y X. Also, suppose the following hypotheses:

  1. (1)

    F(X×X)gX.

  2. (2)

    g(X) is a complete subspace of X with respect to the quasi-partial metric q.

Then F and g have a coupled coincidence point (u,v) satisfying gu=F(u,v)=F(v,u)=gu.

Moreover, if F and g are w-compatible, then F and g have a unique common fixed point of the form (u,u).

Let g= I X (the identity mapping) in Theorem 2.2 and Corollaries 2.1-2.4. Then we have the following results.

Corollary 2.5 Let (X,q) be a quasi-partial metric space and let F:X×XX be a mapping. Suppose that there exist k 1 , k 2 , k 3 [0,1) with k 1 + k 2 + k 3 <1 such that

q ( F ( x , y ) , F ( x , y ) ) + g ( F ( y , x ) , F ( y , x ) ) k 1 ( q ( x , x ) + q ( y , y ) ) + k 2 ( q ( x , F ( x , y ) ) + q ( y , F ( y , x ) ) ) + k 3 ( q ( x , F ( x , y ) ) + q ( y , F ( y , x ) ) )

holds for all x,y, x , y X.

Then F has a unique coupled fixed point of the form (u,u).

Corollary 2.6 Let (X,q) be a quasi-partial metric space and let F:X×XX be a mapping. Suppose that there exist a,b,c,d,e,f[0,1) with a+b+c+d+e+f<1 such that

q ( F ( x , y ) , F ( x , y ) ) a q ( x , x ) + b q ( y , y ) + c q ( x , F ( x , y ) ) + d q ( y , F ( y , x ) ) + e q ( x , F ( x , y ) ) + f q ( y , F ( y , x ) )

holds for all x,y, x , y X.

Then F has a unique coupled fixed point of the form (u,u).

Corollary 2.7 Let (X,q) be a complete quasi-partial metric space and let F:X×XX be a mapping. Suppose that there exists k[0,1) such that

q ( F ( x , y ) , F ( x , y ) ) +q ( F ( y , x ) , F ( y , x ) ) k ( q ( x , x ) + q ( y , y ) )

holds for all x,y, x , y X.

Then F has a unique coupled fixed point of the form (u,u).

Corollary 2.8 Let (X,q) be a complete quasi-partial metric space and let F:X×XX be a mapping. Suppose that there exists k[0,1) with k<1 such that

q ( F ( x , y ) , F ( x , y ) ) +q ( F ( y , x ) , F ( y , x ) ) k ( q ( x , F ( x , y ) ) + q ( y , F ( y , x ) ) )

holds for all x,y, x , y X.

Then F has a unique coupled fixed point of the form (u,u).

Corollary 2.9 Let (X,q) be a complete quasi-partial metric space and let F:X×XX be a mapping. Suppose that there exists k[0,1) with k<1 such that

q ( F ( x , y ) , F ( x , y ) ) +q ( F ( y , x ) , F ( y , x ) ) k ( q ( x , F ( x , y ) ) + q ( y , F ( y , x ) ) )

holds for all x,y, x , y X.

Then F has a unique coupled fixed point of the form (u,u).

Theorem 2.2 Let (X,q) be a complete quasi-partial metric space and let F:X×XX, g:XX be two mappings. Suppose that there exists a function ϕ:gX R + such that

q ( g x , F ( x , y ) ) +q ( g y , F ( y , x ) ) ϕ(gx)+ϕ(gy)ϕ ( F ( x , y ) ) ϕ ( F ( y , x ) )

holds for all (x,y)X×X. Also, assume that the following hypotheses are satisfied:

  1. (a)

    F(X×X)gX;

  2. (b)

    if G:X×XR, G(x,y)=q(F(x,y),gx), then for each sequence (g x n ,g y n )(u,v), we have G(u,v)k lim inf n G( x n , y n ) for some k>0.

Then F and g have a coupled coincidence point (u,v). In addition, q(gu,gu)=0 and q(gv,gv)=0.

Proof Consider ( x 0 , y 0 )X×X. As F(X×X)gX, there are x 1 and y 1 from X such that g x 1 =F( x 0 , y 0 ) and g y 1 =F( y 0 , x 0 ). By repeating this process, we construct two sequences, ( x n ) and ( y n ) with g x n + 1 =F( x n , y n ) and g y n + 1 =F( y n , x n ).

The fourth property of the quasi-partial metric space gives us

q ( g x n , g x n + 2 ) + q ( g y n , g y n + 2 ) q ( g x n , g x n + 1 ) + q ( g x n + 1 , g x n + 2 ) q ( g x n + 1 , g x n + 1 ) + q ( g y n , g y n + 1 ) + q ( g y n + 1 , g y n + 2 ) q ( g y n + 1 , g y n + 1 ) q ( g x n , g x n + 1 ) + q ( g x n + 1 , g x n + 2 ) + q ( g y n , g y n + 1 ) + q ( g y n + 1 , g y n + 2 ) .

Based on the above inequality, for m>n, we obtain

q(g x n ,g x m )+q(g y n ,g y m ) k = n m 1 [ q ( g x k , g x k + 1 ) + q ( g y k , g y k + 1 ) ]
(2.16)
= k = n m 1 [ q ( g x k , F ( x k , y k ) ) + q ( g y k , F ( y k , x k ) ) ] k = n m 1 [ ϕ ( g x k ) + ϕ ( g y k ) ϕ ( F ( x k , y k ) ) ϕ ( F ( y k , x k ) ) ] = k = n m 1 [ ϕ ( g x k ) + ϕ ( g y k ) ϕ ( g x k + 1 ) ϕ ( g y k + 1 ) ] = ϕ ( g x n ) + ϕ ( g y n ) ϕ ( g x m ) ϕ ( g y m ) .
(2.17)

Consider S n (x)= k = 0 n [q(g x k ,g x k + 1 )+q(g y k ,g y k + 1 )]. Inequality (2.17) implies that

S n (x)ϕ(g x 0 )+ϕ(g y 0 ),

hence the nondecreasing sequence { S n } is bounded, so it is convergent. Taking the limit as n,m+ in (2.16), we conclude that

lim n , m + q(g x n ,g x m )= lim n , m + q(g y n ,g y m ).

Using similar arguments, it can be proved that

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

As (g x n ) and (g y n ) are Cauchy sequences in the complete quasi-partial metric space (X,q), there are u, v in X such that u= lim n g x n and v= lim n g v n . Having in mind hypothesis (b), the following relations hold true:

0 q ( F ( u , v ) , g u ) = G ( u , v ) k lim inf n G ( x n , y n ) = k lim inf n q ( F ( x n , y n ) , g x n ) = k lim inf n q ( g x n + 1 , g x n ) = 0 .

We get q(F(u,v),gu)=0, and by Lemma 1.1, it follows that F(u,v)=g(u).

Analogously, it can be proved that F(v,u)=gv.

As a conclusion, we have obtained that (u,v) is a coupled coincidence point of the mappings F and g, and q(gu,gu)=0, q(gv,gv)=0. □

Corollary 2.10 Let (X,q) be a complete quasi-partial metric space and let F:X×XX be a mapping. Suppose that there exists a function ϕ:X R + such that

q ( x , F ( x , y ) ) +q ( y , F ( y , x ) ) ϕ(x)+ϕ(y)ϕ ( F ( x , y ) ) ϕ ( F ( y , x ) )

holds for all (x,y)X×X. Also, assume that the following hypotheses are satisfied:

  1. (a)

    F(X×X)X;

  2. (b)

    if G:X×XR, G(x,y)=q(F(x,y),x), then for each sequence ( x n , y n )(u,v), we have G(u,v)k lim inf n G( x n , y n ) for some k>0.

Then F has a coupled coincidence point (u,v). In addition, q(u,u)=0 and q(v,v)=0.

Proof Follows from Theorem 2.2 by taking g= I X (the identity mapping). □

3 Examples

Now, we introduce some examples to support our results.

Example 3.1 On the set X=[0,1], define

q:X×X R + ,q(x,y)=|xy|+x.

Also, define

F:X×XX,F(x,y)={ 1 4 ( x y ) , x y ; 0 , x < y ,

and g:XX by gx= 1 2 x. Then

  1. (1)

    (X,q) is a complete quasi-partial metric space.

  2. (2)

    F(X×X)gX.

  3. (3)

    For any x,y, x , y X, we have

    q ( F ( x , y ) , F ( x , y ) ) +q ( F ( y , x ) , F ( y , x ) ) 1 2 ( q ( g x , g x ) + q ( g y , g y ) ) .

Proof The proofs of (1) and (2) are clear. To prove (3), we consider the following cases.

Case 1: x<y and x < y . Here we have

F(x,y)=0,F ( x , y ) =0,F(y,x)= y x 4 ,F ( y , x ) = y x 4 .

Therefore

q ( F ( x , y ) , F ( x , y ) ) + q ( F ( y , x ) , F ( y , x ) ) = q ( 0 , 0 ) + q ( x y 4 , y x 4 ) = 1 4 | ( y x ) ( y x ) | + 1 4 ( y x ) 1 2 | ( 1 2 y 1 2 x ) ( 1 2 y 1 2 x ) | + 1 2 ( 1 2 y 1 2 x ) 1 2 | ( 1 2 x 1 2 x ) ( 1 2 y 1 2 y ) | + 1 2 ( 1 2 y + 1 2 x ) 1 2 ( | 1 2 x 1 2 x | + 1 2 x + | 1 2 y 1 2 y | + 1 2 y ) = 1 2 ( | g x g x | + g x + | g y g y | + g y ) = 1 2 ( q ( g x , g x ) + q ( g y , g y ) ) .

Case 2: x<y and x y . Here we have

F(x,y)=0,F ( x , y ) = x y 4 ,F(y,x)= y x 4

and F( y , x )=0. Therefore

q ( F ( x , y ) , F ( x , y ) ) + q ( F ( y , x ) , F ( y , x ) ) = q ( 0 , x y 4 ) + q ( y x 4 , 0 ) = 1 4 | 0 ( x y ) | + 1 4 | y x | + 1 4 ( y x ) = 1 4 ( x y ) + 1 4 ( y x ) + 1 4 ( y x ) = 1 2 ( ( 1 2 x 1 2 x ) 1 2 x + ( 1 2 y 1 2 y ) + 1 2 y ) 1 2 ( ( 1 2 x 1 2 x ) + 1 2 x + ( 1 2 y 1 2 y ) + 1 2 y ) 1 2 ( | 1 2 x 1 2 x | + 1 2 x + | 1 2 y 1 2 y | + 1 2 y ) = 1 2 ( | g x g x | + g x + | g y g y | + g y ) = 1 2 ( q ( g x , g x ) + q ( g y , g y ) ) .

Case 3: x>y and x < y . Using similar arguments to those given in Case (2), we can show that

q ( F ( x , y ) , F ( x , y ) ) +q ( F ( y , x ) , F ( y , x ) ) 1 2 ( q ( g x , g x ) + q ( g y , g y ) ) .

Case 4: xy and x y . Using similar arguments to those given in Case (1), we can show that

q ( F ( x , y ) , F ( x , y ) ) +q ( F ( y , x ) , F ( y , x ) ) 1 2 ( q ( g x , g x ) + q ( g y , g y ) ) .

Thus F and g satisfy all the hypotheses of Corollary 2.7. So, F and g have a unique common fixed point. Here (0,0) is the unique common fixed point of F and g. □

We end with an example related to Theorem 2.2.

Example 3.2 Let X=[0,+). Define

q:X×X R + ,q(x,y)=|xy|+x.

Also, define

F : X × X X , F ( x , y ) = x ; g : X X , g x = 2 x ; ϕ : X R + , ϕ ( x ) = 2 x .

Then:

  1. (1)

    (X,q) is a complete quasi-partial metric space.

  2. (2)

    F(X×X)gX.

  3. (3)

    For any x,yX, we have

    q ( g x , F ( x , y ) ) +q ( g y , F ( y , x ) ) ϕ(gx)+ϕ(gy)ϕ ( F ( x , y ) ) ϕ ( F ( y , x ) ) .
  4. (4)

    Let G:X×X R + be defined by G(x,y)=q(F(x,y),gx). If (g x n ) and (g y n ) are two sequences in X with (g x n ,g y n )(u,v), then G(u,v)4 lim inf n + G( x n , y n ).

Proof The proofs of (1) and (2) are clear. To prove (3) given x,yX, gx=2x, gy=2y, F(x,y)=x, F(y,x)=y, ϕ(x)=2x and ϕ(y)=2y. Thus

q ( g x , F ( x , y ) ) + q ( g y , F ( y , x ) ) = q ( 2 x , x ) + q ( 2 y , y ) = 2 x + 2 y 4 x + 4 y 2 x 2 y = ϕ ( 2 x ) + ϕ ( 2 y ) ϕ ( x ) ϕ ( y ) = ϕ ( g x ) + ϕ ( g y ) ϕ ( F ( x , y ) ) ϕ ( F ( y , x ) ) .

To prove (4), let (g x n ) and (g y n ) be two sequences in X such that (g x n ,g y n )(u,v) for some u,vX. Then g x n u and g y n v. Thus

q(g x n ,u)=q(2 x n ,u)q(u,u)

and

q(u,g x n )=q(u,2 x n )q(u,u).

Therefore

|2 x n u|+2 x n u

and

|u2 x n |+uu.

Therefore

|u2 x n |0.

Hence x n 1 2 u in R + . Now

G ( u , v ) = q ( F ( u , v ) , u ) = q ( u , u ) = u 4 ( 1 2 u ) = 4 lim inf n + x n = 4 lim inf n + G ( x n , x n ) = 4 lim inf n + G ( F ( x n , y n ) , x n ) .

So, F and g satisfy all the hypotheses of Theorem 2.2. Hence F and g have a coupled coincidence point. Here (0,0) is the coupled coincidence point of F and g. □

References

  1. 1.

    Matthews SG: Partial metric topology. Ann. N. Y. Acad. Sci. 728. General Topology and Its Applications 1994, 183–197., Proc. 8th Summer Conf., Queen’s College, 1992

    Google Scholar 

  2. 2.

    Abdeljawad T, Karapinar E, Taş K: Existence and uniqueness of a common fixed point on partial metric spaces. Appl. Math. Lett. 2011, 24(11):1900–1904. 10.1016/j.aml.2011.05.014

    MathSciNet  Article  Google Scholar 

  3. 3.

    Abdeljawad T, Karapinar E, Taş K: A generalized contraction principle with control functions on partial metric spaces. Comput. Math. Appl. 2012, 63(3):716–719.

    MathSciNet  Article  Google Scholar 

  4. 4.

    Abdeljawad T: Fixed points for generalized weakly contractive mappings in partial metric spaces. Math. Comput. Model. 2011, 54(11–12):2923–2927. 10.1016/j.mcm.2011.07.013

    MathSciNet  Article  Google Scholar 

  5. 5.

    Altun I, Erduran A: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 508730

    Google Scholar 

  6. 6.

    Altun I, Simsek H: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud. 2008, 1(1–2):1–8.

    MathSciNet  Google Scholar 

  7. 7.

    Altun I, Simsek H: Some fixed point theorems on ordered metric spaces and application. Fixed Point Theory Appl. 2010., 2010: Article ID 6214469

    Google Scholar 

  8. 8.

    Altun I, Sola F, Simsek H: Generalized contractions on partial metric spaces. Topol. Appl. 2010, 157(18):2778–2785. 10.1016/j.topol.2010.08.017

    MathSciNet  Article  Google Scholar 

  9. 9.

    Altun I, Sadarangani K: Corrigendum to “Generalized contractions on partial metric spaces” [Topology Appl. 157 (2010) 2778–2785]. Topol. Appl. 2011, 158(13):1738–1740. 10.1016/j.topol.2011.05.023

    MathSciNet  Article  Google Scholar 

  10. 10.

    Aydi H: Some fixed point results in ordered partial metric spaces. J. Nonlinear Sci. Appl. 2011, 4(2):1–12.

    MathSciNet  Google Scholar 

  11. 11.

    Aydi H: Some coupled fixed point results on partial metric spaces. Int. J. Math. Math. Sci. 2011., 2011: Article ID 647091

    Google Scholar 

  12. 12.

    Aydi H: Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces. J. Nonlinear Anal. Optim. Theory Appl. 2011, 2(2):33–48.

    MathSciNet  Google Scholar 

  13. 13.

    Aydi H, Karapinar E, Shatanawi W:Coupled fixed point results for (ψ,φ)-weakly contractive condition in ordered partial metric spaces. Comput. Math. Appl. 2011, 62: 4449–4460.

    MathSciNet  Article  Google Scholar 

  14. 14.

    Ćirić L, Samet B, Aydi H, Vetro C: Common fixed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput. 2011, 218: 2398–2406. 10.1016/j.amc.2011.07.005

    MathSciNet  Article  Google Scholar 

  15. 15.

    Golubović Z, Kadelburg Z, Radenović S: Coupled coincidence points of mappings in ordered partial metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 192581

    Google Scholar 

  16. 16.

    Karapınar E, Erhan I: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett. 2011, 24: 1894–1899. 10.1016/j.aml.2011.05.013

    MathSciNet  Article  Google Scholar 

  17. 17.

    Nashine HK, Kadelburg Z, Radenović S: Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces. Math. Comput. Model. 2013, 57: 2355–2365. 10.1016/j.mcm.2011.12.019

    Article  Google Scholar 

  18. 18.

    Oltra S, Valero O: Banach’s fixed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste 2004, 36(1–2):17–26.

    MathSciNet  Google Scholar 

  19. 19.

    Romaguera S: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 493298

    Google Scholar 

  20. 20.

    Romaguera S: Fixed point theorems for generalized contractions on partial metric spaces. Topol. Appl. 2012, 159: 194–199. 10.1016/j.topol.2011.08.026

    MathSciNet  Article  Google Scholar 

  21. 21.

    Samet B, Rajović M, Lazović R, Stoiljković R: Common fixed point results for nonlinear contractions in ordered partial metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 71

    Google Scholar 

  22. 22.

    Shatanawi W, Nashine HK: A generalization of Banach’s contraction principle for nonlinear contraction in a partial metric space. J. Nonlinear Sci. Appl. 2012, 5: 37–43.

    MathSciNet  Google Scholar 

  23. 23.

    Shatanawi W, Nashine HK, Tahat N: Generalization of some coupled fixed point results on partial metric spaces. Int. J. Math. Math. Sci. 2012., 2012: Article ID 686801

    Google Scholar 

  24. 24.

    Shatanawi W, Samet B, Abbas M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model. 2012, 55: 680–687. 10.1016/j.mcm.2011.08.042

    MathSciNet  Article  Google Scholar 

  25. 25.

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

    Google Scholar 

  26. 26.

    Radenović S: Remarks on some coupled fixed point results in partial metric spaces. Nonlinear Funct. Anal. Appl. 2013, 18(1):39–50.

    Google Scholar 

  27. 27.

    Nashine HK, Kadelburg Z, Radenović S: Fixed point theorems via various cyclic contractive conditions in partial metric spaces. Publ. Inst. Math. (Belgr.) 2013, 93(107):69–93. 10.2298/PIM1307069N

    Article  Google Scholar 

  28. 28.

    Valero O: On Banach fixed point theorems for partial metric spaces. Appl. Gen. Topol. 2005, 6(2):229–240.

    MathSciNet  Article  Google Scholar 

  29. 29.

    Altun I, Acar Ö: Fixed point theorems for weak contractions in the sense of Berinde on partial metric spaces. Topol. Appl. 2012, 159: 2642–2648. 10.1016/j.topol.2012.04.004

    MathSciNet  Article  Google Scholar 

  30. 30.

    Haghi RH, Rezapour Sh, Shahzad N: Be careful on partial metric fixed point results. Topol. Appl. 2013, 160: 450–454. 10.1016/j.topol.2012.11.004

    MathSciNet  Article  Google Scholar 

  31. 31.

    Haghi RH, Rezapour S, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 2011, 74: 1799–1803. 10.1016/j.na.2010.10.052

    MathSciNet  Article  Google Scholar 

  32. 32.

    Karapinar E, Erhan İ, Öztürk A: Fixed point theorems on quasi-partial metric spaces. Math. Comput. Model. 2012. doi:10.1016/j.mcm.2012.06.036

    Google Scholar 

  33. 33.

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

    MathSciNet  Article  Google Scholar 

  34. 34.

    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.2008.09.020

    MathSciNet  Article  Google Scholar 

  35. 35.

    Cho YJ, Rhoades BE, Saadati R, Samet B, Shatanawi W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type. Fixed Point Theory Appl. 2012., 2012: Article ID 8

    Google Scholar 

  36. 36.

    Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011, 54: 73–79. 10.1016/j.mcm.2011.01.036

    MathSciNet  Article  Google Scholar 

  37. 37.

    Choudhury, BS, Metiya, N, Postolache, M: A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl. (submitted)

  38. 38.

    Karapinar E: Couple fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl. 2010, 59: 3656–3668.

    MathSciNet  Article  Google Scholar 

  39. 39.

    Samet B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 2010, 72: 4508–4517. 10.1016/j.na.2010.02.026

    MathSciNet  Article  Google Scholar 

  40. 40.

    Sedghi S, Altun I, Shobe N: Coupled fixed point theorems for contractions in fuzzy metric spaces. Nonlinear Anal. 2010, 72: 1298–1304. 10.1016/j.na.2009.08.018

    MathSciNet  Article  Google Scholar 

  41. 41.

    Shatanawi W, Samet B, Abbas M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model. 2012, 55: 680–687. 10.1016/j.mcm.2011.08.042

    MathSciNet  Article  Google Scholar 

  42. 42.

    Shatanawi W: On w -compatible mappings and common coupled coincidence point in cone metric spaces. Appl. Math. Lett. 2012, 25: 925–931. 10.1016/j.aml.2011.10.037

    MathSciNet  Article  Google Scholar 

  43. 43.

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

    MathSciNet  Article  Google Scholar 

  44. 44.

    Radenović S: Remarks on some recent coupled coincidence point results in symmetric G -metric spaces. J. Operators 2013., 2013: Article ID 290525. doi:10.1155/2013/290525

    Google Scholar 

  45. 45.

    Radenović S: Remarks on some coupled coincidence point result in partially ordered metric spaces. Arab J. Math. Sci. 2013. doi:10.1016/j.ajmsc.2013.02.003

    Google Scholar 

  46. 46.

    Shatanawi W: Fixed point theorems for nonlinear weakly C -contractive mappings in metric spaces. Math. Comput. Model. 2011. doi:10.1016/j.mcm.2011.06.069

    Google Scholar 

  47. 47.

    Abbas M, Khan MA, Radenović S: Common coupled fixed point theorems in cone metric spaces for w -compatible mapping. Appl. Math. Comput. 2010, 217(1):195–202. 10.1016/j.amc.2010.05.042

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ariana Pitea.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Shatanawi, W., Pitea, A. Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl 2013, 153 (2013). https://doi.org/10.1186/1687-1812-2013-153

Download citation

Keywords

  • partial metric space
  • quasi-partial metric space
  • coupled fixed point