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Generalized Meir-Keeler type n-tupled fixed point theorems in ordered partial metric spaces

Abstract

In this paper, we prove n-tupled fixed point theorems (for even n) for mappings satisfying Meir-Keeler type contractive condition besides enjoying mixed monotone property in ordered partial metric spaces. As applications, some results of integral type are also derived. Our results generalize the corresponding results of Erduran and Imdad (J. Nonlinear Anal. Appl. 2012:jnaa-00169, 2012).

1 Introduction

Existence of a fixed point for contraction type mappings in partially ordered metric spaces with possible applications have been considered recently by many authors (e.g. [131]). Recently many researchers have obtained fixed and common fixed point results on partially ordered metric spaces (see [5, 10, 11, 18, 22, 32, 33]). In 2006, Bhaskar and Lakshmikantham [13] initiated the idea of coupled fixed point and proved some interesting coupled fixed point theorems for mappings satisfying a mixed monotone property. In this continuation, Lakshmikantham and Ćirić [17] generalized these results for nonlinear ϕ-contraction mappings by introducing two ideas namely: coupled coincidence point and mixed g-monotone property. Thereafter Samet and Vetro [34] extended the idea of coupled fixed point to higher dimensions by introducing the notion of fixed point of n-order (or n-tupled fixed point, where n is natural number greater than or equal to 2) and presented some n-tupled fixed point results in complete metric spaces, using a new concept of F-invariant set. On the other hand, Imdad et al. [35] generalized the idea of n-tupled fixed point by considering even-tupled coincidence point besides exploiting the idea of mixed g-monotone property on X n and proved an even-tupled coincidence point theorem for nonlinear ϕ-contraction mappings satisfying mixed g-monotone property.

The concept of partial metric space was introduced by Matthews [36] in 1994, which is a generalization of usual metric space. In such spaces, the distance of a point to itself may not be zero. The main motivation behind the idea of a partial metric space is to transfer mathematical techniques into computer science. Following this initial work, Matthews [36] generalized the Banach contraction principle in the context of complete partial metric spaces. For more details, we refer the reader to [4, 79, 20, 2426, 3748].

Samet [27] introduced the concept of generalized Meir-Keeler type contraction function and proved some coupled fixed point theorems in partially ordered metric spaces. In different years, many researchers studied and worked on this contraction condition. Recently, in [12], Berinde and Pǎcurar gave the concept of symmetric Meir-Keeler type condition and generalized several results in the literature. Very recently, Erduran and Imdad [49] generalized the coupled fixed point theorems in the context of partial metric spaces. In this paper, we established some n-tupled fixed point theorems for generalized Meir-Keeler type contraction condition in ordered partial metric spaces enjoying strict mixed monotone property. In this paper, we prove the existence and uniqueness of some Meir-Keeler type n-tupled fixed point theorems in the context of partially ordered partial metric spaces. The presented theorems extend and improve the recent coupled fixed point theorems due to Erduran and Imdad [49].

2 Preliminaries

In this section, we collect some definitions and properties of partial metric space which are relevant to our presentation.

Definition 2.1 A partial metric on a nonempty set X is a function p:X×X R + such that for all x,y,zX,

(p1) x=yp(x,x)=p(x,y)=p(y,y),

(p2) p(x,x)p(x,y),

(p3) p(x,y)=p(y,x),

(p4) 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.

Remark 2.1 It is clear that if p(x,y)=0, then from (p1), (p2) and (p3), x=y. But if x=y, p(x,y) may not be zero.

Each partial metric p on X generates a T 0 topology τ p on X which has as a base the family of open p-balls { B p (x,ϵ),xX,ϵ>0}, where B p (x,ϵ)={yX:p(x,y)<p(x,x)+ϵ} for all xX and ϵ>0.

If p is a partial metric on X, then the function p s :X×X R + given by

p s (x,y)=2p(x,y)p(x,x)p(y,y)

is a metric on X.

Example 2.1 [36, 37, 43]

Consider X= R + with p(x,y)=max{x,y}. Then ( R + ,p) is a partial metric space. It is clear that p is not a (usual) metric. Note that in this case p s (x,y)=|xy|.

Example 2.2 [50]

Let X={[a,b]:a,bR,ab} and define p([a,b],[c,d])=max{b,d}min{a,c}. Then (X,p) is a partial metric space.

Example 2.3 [50]

Let X=[0,1][2,3] and define p:X×X[0,) by

p(x,y)= { max { x , y } , { x , y } [ 2 , 3 ] ; | x y | , { x , y } [ 0 , 1 ] .

Then (X,p) is a complete partial metric space.

Example 2.4 [51]

Let (X,d) and (X,p) be metric space and partial metric space, respectively. Then the mappings ρ i :X×X R + (i{1,2,3}) defined by

ρ 1 ( x , y ) = d ( x , y ) + p ( x , y ) , ρ 2 ( x , y ) = d ( x , y ) + max { ω ( x ) , ω ( y ) } , ρ 3 ( x , y ) = d ( x , y ) + a

induce partial metrics on X, where ω:X R + is an arbitrary function and a0.

Definition 2.2 Let (X,p) be a partial metric space and { x n } be a sequence in X. Then

  1. (i)

    { x n } converges to a point xX if and only if p(x,x)= lim n + p(x, x n ),

  2. (ii)

    { x n } is said to be a Cauchy sequence if lim n , m + p( x n , x m ) exists (and is finite).

Definition 2.3 A partial metric space (X,p) is said to be complete if every Cauchy sequence { x n }X converges with respect to τ p , to a point xX, such that p(x,x)= lim n , m + p( x n , x m ).

Lemma 2.1 Let (X,p) be a partial metric space. Then

  1. (i)

    { x n } is a Cauchy sequence in (X,p) if and only if it is a Cauchy sequence in the metric space (X, p s ),

  2. (ii)

    (X,p) is complete if and only if the metric space (X, p s ) is complete. Furthermore, lim n + p s ( x n ,x)=0 if and only if

    p(x,x)= lim n + p( x n ,x)= lim n , m + p( x n , x m ).

In [52], Meir-Keeler generalized the well-known Banach fixed point theorem by proving the following interesting fixed point theorem.

Theorem 2.1 [52]

Let (X,d) be a complete metric space and T:XX be a mapping. Suppose that for every ϵ>0 there exists δ(ϵ)>0 such that for all

x,yXwith ϵd(x,y)<ϵ+δ(ϵ)d(Tx,Ty)<ϵ.
(2.1)

Then T has a unique fixed point zX and for all xX, the sequence { T n x} converges to z.

In recent years, many authors generalized Meir-Keeler fixed point theorems in various ways in various spaces which include complete metric space as well as ordered metric space. In [27], Samet introduced the concept of generalized Meir-Keeler type contraction function and proved some coupled fixed point results. Samet [27] introduced the definition below to modify the Meir-Keeler contraction and extended its applications.

Definition 2.4 [27]

Let (X,d) be a partially ordered metric space and F:X×XX be a given mapping. Then F is a generalized Meir-Keeler type function if for all ϵ>0 there exists δ(ϵ)>0 such that

ux,yv,ϵ 1 2 [ d ( x , u ) + d ( y , v ) ] <ϵ+δ(ϵ)d ( F ( x , y ) , F ( u , v ) ) <ϵ.
(2.2)

Very recently Erduran and Imdad [49] generalized the results of Samet [27] for ordered partial metric spaces. For more details, see [12, 27, 53, 54].

Erduran and Imdad [49] proved the following result:

Theorem 2.2 [49]

Let (X,) be a partially ordered set and suppose there is a partial metric p on X such that (X,p) is complete partial metric space. Let F:X×XX be mapping satisfying the following hypotheses:

  1. (1)

    F has the mixed strict monotone property,

  2. (2)

    F is a generalized Meir-Keeler type function,

  3. (3)

    F is continuous or X has the following properties:

  4. (a)

    if a nondecreasing sequence { x n }x, then x n x for all n,

  5. (b)

    if a nonincreasing sequence { x n }x, then x x n for all n.

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 there exists (x,y)X×X such that x=F(x,y) and y=F(y,x). Furthermore, p(x,x)=p(y,y)=0.

Note Throughout the paper we consider n to be an even integer.

Let (X,p) be a partial metric. We endow X×X××X, n times (= X n ) with the partial metric η defined for ( x 1 , x 2 ,, x n ),( y 1 , y 2 ,, y n ) X n by

η ( ( x 1 , x 2 , , x n ) , ( y 1 , y 2 , , y n ) ) =max { p ( x 1 , y 1 ) , p ( x 2 , y 2 ) , , p ( x n , y n ) } .

Let F: X n X be a given mapping. Then for all ( x 1 , x 2 ,, x n ) X n and for all mN, m2, we denote

F m ( x 1 , x 2 , , x n ) = F ( F m 1 ( x 1 , x 2 , , x n ) , F m 1 ( x 2 , , x n , x 1 ) , , F m 1 ( x n , x 1 , , x n 1 ) ) .

In this paper, we used the concept of n-tupled fixed point given by Samet and Vetro [34]. We recall some basic concepts.

Definition 2.5 [35]

Let (X,) be a partially ordered set and F: X n X be a mapping. The mapping F is said to have the mixed monotone property if F is nondecreasing in its odd position arguments and nonincreasing in its even position arguments, that is, if,

( i ) for all  x 1 1 , x 2 1 X , x 1 1 x 2 1 F ( x 1 1 , x 2 , x 3 , , x n ) F ( x 2 1 , x 2 , x 3 , , x n ) , ( ii ) for all  x 1 2 , x 2 2 X , x 1 2 x 2 2 F ( x 1 , x 2 2 , x 3 , , x n ) F ( x 1 , x 1 2 , x 3 , , x n ) , ( iii ) for all  x 1 3 , x 2 3 X , x 1 3 x 2 3 F ( x 1 , x 2 , x 1 3 , , x n ) F ( x 1 , x 2 , x 2 3 , , x n ) , ( iii ) ( iii ) for all  x 1 n , x 2 n X , x 1 n x 2 n F ( x 1 , x 2 , x 3 , , x 2 n ) F ( x 1 , x 2 , x 3 , , x 1 n ) .

Definition 2.6 [34]

An element ( x 1 , x 2 ,, x n ) X n is called an n-tupled fixed point of the mapping F: X n X if

{ F ( x 1 , x 2 , x 3 , , x n ) = x 1 , F ( x 2 , x 3 , , x n , x 1 ) = x 2 , F ( x 3 , , x n , x 1 , x 2 ) = x 3 , F ( x n , x 1 , x 2 , , x n 1 ) = x n .

Example 2.5 Let (R,d) be a partially ordered metric space under natural setting and let F: R n R be a mapping defined by F( x 1 , x 2 , x 3 ,, x n )=sin( x 1 x 2 x 3 x n ), for any x 1 , x 2 ,, x n R. Then (0,0,,0) is an n-tupled fixed point of F.

Remark 2.2 Definition 2.6 with n=2,4 respectively yields the definition of coupled fixed point [13] and quadrupled fixed point [55].

3 Main results

We begin this section by defining the following definitions:

Definition 3.1 Let (X,) be a partially ordered set and F: X n X be a mapping. The mapping F is said to have the mixed strict monotone property if F is nondecreasing in its odd position arguments and nonincreasing in its even position arguments, that is, if,

( i ) for all  x 1 1 , x 2 1 X , x 1 1 x 2 1 F ( x 1 1 , x 2 , x 3 , , x n ) F ( x 2 1 , x 2 , x 3 , , x n ) , ( ii ) for all  x 1 2 , x 2 2 X , x 1 2 x 2 2 F ( x 1 , x 2 2 , x 3 , , x n ) F ( x 1 , x 1 2 , x 3 , , x n ) , ( iii ) for all  x 1 3 , x 2 3 X , x 1 3 x 2 3 F ( x 1 , x 2 , x 1 3 , , x n ) F ( x 1 , x 2 , x 2 3 , , x n ) , ( iii ) ( iii ) for all  x 1 n , x 2 n X , x 1 n x 2 n F ( x 1 , x 2 , x 3 , , x 2 n ) F ( x 1 , x 2 , x 3 , , x 1 n ) .

Definition 3.2 Let (X,p) be a partially ordered partial metric space and F: X n X be a given mapping. We say that F is a generalized Meir-Keeler type function if for all ϵ>0 there exists δ(ϵ)>0 such that for ( x 1 , x 2 ,, x n ),( y 1 , y 2 ,, y n ) X n with x 1 y 1 , y 2 x 2 , x 3 y 3 ,, y n x n

{ ϵ max { p ( x 1 , y 1 ) , p ( x 2 , y 2 ) , p ( x 3 , y 3 ) , , p ( x n , y n ) } < ϵ + δ ( ϵ ) p ( F ( x 1 , x 2 , x 3 , , x n ) , F ( y 1 , y 2 , y 3 , , y n ) ) < ϵ .
(3.1)

The aim of this work is to prove the following results:

Lemma 3.1 Let (X,) be a partially ordered set and suppose that there is a partial metric p on X such that (X,p) is a complete partial metric space. Let F: X n X be a given mapping. If F is a generalized Meir-Keeler type function, then for ( x 1 , x 2 ,, x n ),( y 1 , y 2 ,, y n ) X n

p ( F ( x 1 , x 2 , , x n ) , F ( y 1 , y 2 , , y n ) ) <max { p ( x 1 , y 1 ) , p ( x 2 , y 2 ) , , p ( x n , y n ) }

with x 1 y 1 , y 2 x 2 , x 3 y 3 ,, y n x n or x 1 y 1 , y 2 x 2 , x 3 y 3 ,, y n x n .

Proof Let x 1 , x 2 ,, x n , y 1 , y 2 ,, y n X such that

x 1 y 1 , y 2 x 2 , x 3 y 3 , , y n x n or x 1 y 1 , y 2 x 2 , x 3 y 3 , , y n x n .
(3.2)

Then max{p( x 1 , y 1 ),p( x 2 , y 2 ),p( x 3 , y 3 ),,p( x n , y n )}>0. Since F is a generalized Meir-Keeler type function. Therefore for ϵ=max{p( x 1 , y 1 ),p( x 2 , y 2 ),,p( x n , y n )}, there exists δ(ϵ)>0 such that

x 1 y 1 , y 2 x 2 , x 3 y 3 , , y n x n , ϵ max { p ( x 1 , y 1 ) , p ( x 2 , y 2 ) , p ( x 3 , y 3 ) , , p ( x n , y n ) } < ϵ + δ ( ϵ ) p ( F ( x 1 , x 2 , x 3 , , x n ) , F ( y 1 , y 2 , y 3 , , y n ) ) < ϵ .

Putting x 1 = x 1 , x 2 = x 2 ,, x n = x n and y 1 = y 1 , y 2 = y 2 ,, y n = y n , we obtain the desired result. □

Lemma 3.2 Let (X,) be a partially ordered set and suppose that there is a partial metric p on X such that (X,p) is a complete partial metric space. Let F: X n X be a given mapping. Assume that the following hypotheses hold:

  1. (1)

    F has the mixed strict monotone property,

  2. (2)

    F is a generalized Meir-Keeler type function,

  3. (3)

    there exist ( x 1 , x 2 ,, x n ),( y 1 , y 2 ,, y n ) X n with x 1 y 1 , y 2 x 2 , x 3 y 3 ,, y n x n .

Then

η ( ( F m ( x 1 , x 2 , x 3 , , x n ) , F m ( x 2 , x 3 , , x n , x 1 ) , , F m ( x n , x 1 , x 2 , , x n 1 ) ) , ( F m ( y 1 , y 2 , y 3 , , y n ) , F m ( y 2 , y 3 , , y n , y 1 ) , , F m ( y n , y 1 , y 2 , , y n 1 ) ) ) 0 as  m .

Proof We claim that:

{ F m ( x 1 , x 2 , x 3 , , x n ) F m ( y 1 , y 2 , y 3 , , y n ) , F m ( y 2 , y 3 , , y n , y 1 ) F m ( x 2 , x 3 , , x n , x 1 ) , F m ( y n , y 1 , y 2 , , y n 1 ) F m ( x n , x 1 , x 2 , , x n 1 ) ,
(3.3)

with the notation F 1 F. Then by the mixed strict monotone property of F,

{ x 1 y 1 F ( x 1 , x 2 , x 3 , , x n ) F ( y 1 , x 2 , x 3 , , x n ) , y 2 x 2 F ( y 1 , x 2 , x 3 , , x n ) F ( y 1 , y 2 , x 3 , , x n ) , x 3 y 3 F ( y 1 , y 2 , x 3 , x 4 , , x n ) F ( y 1 , y 2 , y 3 , x 4 , , x n ) , y n x n F ( y 1 , y 2 , , y n 1 , x n ) F ( y 1 , y 2 , , y n 1 , y n ) .

Then we have F( x 1 , x 2 , x 3 ,, x n )F( y 1 , y 2 , y 3 ,, y n ). Also

{ y 2 x 2 F ( y 2 , y 3 , , y n , y 1 ) F ( x 2 , y 3 , , y n , y 1 ) , x 3 y 3 F ( x 2 , y 3 , y 4 , , y n , y 1 ) F ( x 2 , x 3 , y 4 , , y n , y 1 ) , y n x n F ( x 2 , x 3 , , y n , y 1 ) F ( x 2 , x 3 , , x n , y 1 ) , x 1 y 1 F ( x 2 , x 3 , , x n , y 1 ) F ( x 2 , x 3 , , x n , x 1 ) .

Therefore F( y 2 , y 3 ,, y n , y 1 )F( x 2 , x 3 ,, x n , x 1 ).

And similarly

{ y n x n F ( y n , y 1 , y 2 , , y n 1 ) F ( x n , y 1 , y 2 , , y n 1 ) , x 1 y 1 F ( x n , y 1 , y 2 , , y n 1 ) F ( x n , x 1 , y 2 , , y n 1 ) , y 2 x 2 F ( x n , x 1 , y 2 , , y n 1 ) F ( x n , x 1 , x 2 , y 3 , , y n 1 ) , x n 1 y n 1 F ( x n , x 1 , , x n 2 , y n 1 ) F ( x n , x 1 , , x n 2 , x n 1 ) .

Therefore F( y n , y 1 , y 2 ,, y n 1 )F( x n , x 1 , x 2 ,, x n 1 ). Thus (3.3) is satisfied for m=1. For m=2, we use the same strategy. We have

F ( x 1 , x 2 , x 3 , , x n ) F ( y 1 , y 2 , y 3 , , y n ) F ( F ( x 1 , x 2 , x 3 , , x n ) , F ( x 2 , x 3 , , x n , x 1 ) , , F ( x n , x 1 , x 2 , , x n 1 ) ) F ( F ( y 1 , y 2 , y 3 , , y n ) , F ( x 2 , x 3 , , x n , x 1 ) , , F ( x n , x 1 , x 2 , , x n 1 ) ) , F ( y 2 , y 3 , , y n , y 1 ) F ( x 2 , x 3 , , x n , x 1 ) F ( F ( y 1 , y 2 , y 3 , , y n ) , F ( x 2 , x 3 , , x n , x 1 ) , , F ( x n , x 1 , x 2 , , x n 1 ) ) F ( F ( y 1 , y 2 , y 3 , , y n ) , F ( y 2 , y 3 , , y n , y 1 ) , , F ( x n , x 1 , x 2 , , x n 1 ) ) , F ( y n , y 1 , y 2 , , , y n 1 ) F ( x n , y 1 , y 2 , , y n 1 ) F ( F ( y 1 , y 2 , y 3 , , y n ) , F ( y 2 , y 3 , , y n , y 1 ) , , F ( x n , x 1 , x 2 , , x n 1 ) ) F ( F ( y 1 , y 2 , y 3 , , y n ) , F ( y 2 , y 3 , , y n , y 1 ) , , F ( y n , y 1 , y 2 , , y n 1 ) ) .

Thus we get

F 2 ( x 1 , x 2 , x 3 , , x n ) F 2 ( y 1 , y 2 , y 3 , , y n ) .

Now,

F ( y 2 , y 3 , , y n , y 1 ) F ( x 2 , x 3 , , x n , x 1 ) F ( F ( y 2 , y 3 , , y n , y 1 ) , , F ( y n , y 1 , y 2 , , y n 1 ) , F ( y 1 , y 2 , y 3 , , y n ) ) F ( F ( x 2 , x 3 , , x n , x 1 ) , , F ( y n , y 1 , y 2 , , y n 1 ) , F ( y 1 , y 2 , y 3 , , y n ) ) , F ( y n , y 1 , y 2 , , y n 1 ) F ( x n , x 1 , x 2 , , x n 1 ) F ( F ( x 2 , x 3 , , x n , x 1 ) , F ( y n , y 1 , y 2 , , y n 1 ) , , F ( y 1 , y 2 , y 3 , , y n ) ) F ( F ( x 2 , x 3 , , x n , x 1 ) , F ( x n , x 1 , x 2 , , x n 1 ) , , F ( y 1 , y 2 , y 3 , , y n ) ) , F ( x 1 , x 2 , x 3 , , x n ) F ( y 1 , y 2 , y 3 , , y n ) F ( F ( x 2 , x 3 , , x n , x 1 ) , , F ( x n , x 1 , x 2 , , x n 1 ) , F ( y 1 , y 2 , y 3 , , y n ) ) F ( F ( x 2 , x 3 , , x n , x 1 ) , , F ( x n , x 1 , x 2 , , x n 1 ) , F ( x 1 , x 2 , x 3 , , x n ) ) .

Therefore we get

F 2 ( y 2 , y 3 , , y n , y 1 ) F 2 ( x 2 , x 3 , , x n , x 1 ) .

In the same way,

F ( y n , y 1 , y 2 , , y n 1 ) F ( x n , x 1 , x 2 , , x n 1 ) F ( F ( y n , y 1 , y 2 , , y n 1 ) , F ( y 1 , y 2 , y 3 , , y n ) , , F ( y n 1 , , y 2 , y 1 , y n ) ) F ( F ( x n , x 1 , x 2 , , x n 1 ) , F ( y 1 , y 2 , y 3 , , y n ) , , F ( y n 1 , , y 2 , y 1 , y n ) ) , F ( x 1 , x 2 , x 3 , , x n ) F ( y 1 , y 2 , y 3 , , y n ) F ( F ( x n , x 1 , x 2 , , x n 1 ) , F ( y 1 , y 2 , y 3 , , y n ) , , F ( x n 1 , , x 2 , x 1 , x n ) ) F ( F ( x n , x 1 , x 2 , , x n 1 ) , F ( x 1 , x 2 , x 3 , , x n ) , , F ( x n 1 , , x 2 , x 1 , x n ) ) , F ( y 2 , y 3 , , y n , y 1 ) F ( x 2 , x 3 , , x n , x 1 ) F ( F ( x n , x 1 , , x n 1 ) , F ( x 1 , x 2 , , x n ) , F ( y 2 , , y n , y 1 ) , , F ( x n 1 , , x 1 , x n ) ) F ( F ( x n , x 1 , , x n 1 ) , F ( x 1 , x 2 , , x n ) , F ( x 2 , , x n , x 1 ) , , F ( x n 1 , , x 1 , x n ) ) .

Thus we have

F 2 ( y n , y 1 , y 2 , , y n 1 ) F 2 ( x n , x 1 , x 2 , , x n 1 ) .

Thus (3.3) is satisfied for m=2. Repeating the same argument for each m, we see that (3.3) holds. Now using Lemma 3.1 and (3.3), we get

p ( F m + 1 ( x 1 , x 2 , x 3 , , x n ) , F m + 1 ( y 1 , y 2 , y 3 , , y n ) ) = p ( F ( F m ( x 1 , x 2 , x 3 , , x n ) , F m ( x 2 , x 3 , , x n , x 1 ) , , F m ( x n , x 1 , x 2 , , x n 1 ) ) , F ( F m ( y 1 , y 2 , y 3 , , y n ) , F m ( y 2 , y 3 , , y n , y 1 ) , , F m ( y n , y 1 , y 2 , , y n 1 ) ) ) < max [ p ( F m ( x 1 , x 2 , x 3 , , x n ) , F m ( y 1 , y 2 , y 3 , , y n ) ) , p ( F m ( x 2 , x 3 , , x n , x 1 ) , F m ( y 2 , y 3 , , y n , y 1 ) ) , , p ( F m ( x n , x 1 , x 2 , , x n 1 ) , F m ( y n , y 1 , y 2 , , y n 1 ) ) ] .
(3.4)

Also we have

p ( F m + 1 ( x 2 , x 3 , , x n , x 1 ) , F m + 1 ( y 2 , y 3 , , y n , y 1 ) ) = p ( F ( F m ( x 2 , x 3 , , x n , x 1 ) , , F m ( x n , x 1 , x 2 , , x n 1 ) , F m ( x 1 , x 2 , x 3 , , x n ) ) , F ( F m ( y 2 , y 3 , , y n , y 1 ) , , F m ( y n , y 1 , y 2 , , y n 1 ) , F m ( y 1 , y 2 , y 3 , , y n ) ) ) < max [ p ( F m ( x 2 , x 3 , , x n , x 1 ) , F m ( y 2 , y 3 , , y n , y 1 ) ) , , p ( F m ( x n , x 1 , x 2 , , x n 1 ) , F m ( y n , y 1 , y 2 , , y n 1 ) ) , p ( F m ( x 1 , x 2 , x 3 , , x n ) , F m ( y 1 , y 2 , y 3 , , y n ) ) ] .
(3.5)

Similarly we have,

p ( F m + 1 ( x n , x 1 , x 2 , , x n 1 ) , F m + 1 ( y n , y 1 , y 2 , , y n 1 ) ) < max [ p ( F m ( x n , x 1 , x 2 , , x n 1 ) , F m ( y n , y 1 , y 2 , , y n 1 ) ) , p ( F m ( x 1 , x 2 , x 3 , , x n ) , F m ( y 1 , y 2 , y 3 , , y n ) ) , , p ( F m ( x n 1 , , x 2 , x 1 , x n ) , F m ( y n 1 , , y 2 , y 1 , y n ) ) ] .
(3.6)

Combining (3.4), (3.5), and (3.6), we get

η ( ( F m + 1 ( x 1 , x 2 , x 3 , , x n ) , F m + 1 ( x 2 , x 3 , , x n , x 1 ) , , F m + 1 ( x n , x 1 , x 2 , , x n 1 ) ) , ( F m + 1 ( y 1 , y 2 , y 3 , , y n ) , F m + 1 ( y 2 , y 3 , , y n , y 1 ) , , F m + 1 ( y n , y 1 , y 2 , , y n 1 ) ) ) < η ( ( F m ( x 1 , x 2 , x 3 , , x n ) , F m ( x 2 , x 3 , , x n , x 1 ) , , F m ( x n , x 1 , x 2 , , x n 1 ) ) , ( F m ( y 1 , y 2 , y 3 , , y n ) , F m ( y 2 , y 3 , , y n , y 1 ) , , F m ( y n , y 1 , y 2 , , y n 1 ) ) ) .

This implies that

{ η ( ( F m ( x 1 , x 2 , x 3 , , x n ) , F m ( x 2 , x 3 , , x n , x 1 ) , , F m ( x n , x 1 , x 2 , , x n 1 ) ) , ( F m ( y 1 , y 2 , y 3 , , y n ) , F m ( y 2 , y 3 , , y n , y 1 ) , , F m ( y n , y 1 , y 2 , , y n 1 ) ) ) }

is a decreasing convergent sequence. Thus there exists ϵ0 such that

lim m [ η ( ( F m ( x 1 , x 2 , x 3 , , x n ) , F m ( x 2 , x 3 , , x n , x 1 ) , , F m ( x n , x 1 , x 2 , , x n 1 ) ) , ( F m ( y 1 , y 2 , y 3 , , y n ) , F m ( y 2 , y 3 , , y n , y 1 ) , , F m ( y n , y 1 , y 2 , , y n 1 ) ) ) ] = ϵ .

Now we show that ϵ=0. Assume that ϵ>0. This implies that there exists m 0 N such that

ϵ < η ( ( F m 0 ( x 1 , x 2 , x 3 , , x n ) , F m 0 ( x 2 , x 3 , , x n , x 1 ) , , F m 0 ( x n , x 1 , x 2 , , x n 1 ) ) , ( F m 0 ( y 1 , y 2 , y 3 , , y n ) , F m 0 ( y 2 , y 3 , , y n , y 1 ) , , F m 0 ( y n , y 1 , y 2 , , y n 1 ) ) ) < ϵ + δ ( ϵ ) .

In this case we have

ϵ max { p ( F m 0 ( x 1 , x 2 , x 3 , , x n ) , F m 0 ( y 1 , y 2 , y 3 , , y n ) ) , p ( F m 0 ( x 2 , x 3 , , x n , x 1 ) , F m 0 ( y 2 , y 3 , , y n , y 1 ) ) , , p ( F m 0 ( x n , x 1 , x 2 , , x n 1 ) , F m 0 ( y n , y 1 , y 2 , , y n 1 ) ) } < ϵ + δ ( ϵ ) .

It follows from (3.3) and hypothesis (2) that

p ( ( F ( F m 0 ( x 1 , x 2 , x 3 , , x n ) , F m 0 ( x 2 , x 3 , , x n , x 1 ) , , F m 0 ( x n , x 1 , x 2 , , x n 1 ) ) ) , ( F ( F m 0 ( y 1 , y 2 , y 3 , , y n ) , F m 0 ( y 2 , y 3 , , y n , y 1 ) , , F m 0 ( y n , y 1 , y 2 , , y n 1 ) ) ) ) < ϵ ,

that is,

p ( F m 0 + 1 ( x 1 , x 2 , x 3 , , x n ) , F m 0 + 1 ( y 1 , y 2 , y 3 , , y n ) ) <ϵ.
(3.7)

On the other hand, we have

ϵ max { p ( F m 0 ( x 2 , x 3 , , x n , x 1 ) , F m 0 ( y 2 , y 3 , , y n , y 1 ) ) , , p ( F m 0 ( x n , x 1 , x 2 , , x n 1 ) , F m 0 ( y n , y 1 , y 2 , , y n 1 ) ) , p ( F m 0 ( x 1 , x 2 , x 3 , , x n ) , F m 0 ( y 1 , y 2 , y 3 , , y n ) ) } < ϵ + δ ( ϵ ) ,

which implies that

p ( F m 0 + 1 ( x 2 , x 3 , , x n , x 1 ) , F m 0 + 1 ( y 2 , y 3 , , y n , y 1 ) ) <ϵ.
(3.8)

Similarly,

p ( F m 0 + 1 ( x n , x 1 , x 2 , , x n 1 ) , F m 0 + 1 ( y n , y 1 , y 2 , , y n 1 ) ) <ϵ.
(3.9)

Combining (3.7), (3.8), and (3.9), we have

η ( ( F m 0 + 1 ( x 1 , x 2 , x 3 , , x n ) , F m 0 + 1 ( x 2 , x 3 , , x n , x 1 ) , , F m 0 + 1 ( x n , x 1 , x 2 , , x n 1 ) ) , ( F m 0 + 1 ( y 1 , y 2 , y 3 , , y n ) , F m 0 + 1 ( y 2 , y 3 , , y n , y 1 ) , , F m 0 + 1 ( y n , y 1 , y 2 , , y n 1 ) ) ) < ϵ ,

which is a contradiction. Therefore, we have necessarily ϵ=0. That is,

η ( ( F m ( x 1 , x 2 , x 3 , , x n ) , F m ( x 2 , x 3 , , x n , x 1 ) , , F m ( x n , x 1 , x 2 , , x n 1 ) ) , ( F m ( y 1 , y 2 , y 3 , , y n ) , F m ( y 2 , y 3 , , y n , y 1 ) , , F m ( y n , y 1 , y 2 , , y n 1 ) ) ) = 0 .

 □

Remark 3.1 Lemma 3.2 also holds if we replace condition (3) by ( x 1 , x 2 , x 3 ,, x n ),( y 1 , y 2 , y 3 ,, y n ) X n such that x 1 y 1 , y 2 x 2 , x 3 y 3 ,, y n x n .

Theorem 3.1 Let (X,) be a partially ordered set and suppose that there is a partial metric p on X such that (X,p) is a complete partial metric space. Let F: X n X be a given mapping satisfying the following hypotheses:

  1. (1)

    F is continuous,

  2. (2)

    F has the mixed strict monotone property,

  3. (3)

    F is a generalized Meir-Keeler type function,

  4. (4)

    there exist x 0 1 , x 0 2 , x 0 3 ,, x 0 n X such that

    { x 0 1 F ( x 0 1 , x 0 2 , x 0 3 , , x 0 n ) , F ( x 0 2 , x 0 3 , , x 0 n , x 0 1 ) x 0 2 , x 0 3 F ( x 0 3 , , x 0 n , x 0 1 , x 0 2 ) , F ( x 0 n , x 0 1 , x 0 2 , , x 0 n 1 ) x 0 n .
    (3.10)

Then there exist ( x 1 , x 2 , x 3 ,, x n ) X n such that x 1 =F( x 1 , x 2 , x 3 ,, x n ), x 2 =F( x 2 , x 3 ,, x n , x 1 ),, x n =F( x n , x 1 , x 2 ,, x n 1 ).

Proof Let us define sequences { x m 1 },{ x m 2 },,{ x m n } in X by

{ x m 1 = F m ( x 0 1 , x 0 2 , x 0 3 , , x 0 n ) , x m 2 = F m ( x 0 2 , x 0 3 , , x 0 n , x 0 1 ) , x m 3 = F m ( x 0 3 , , x 0 n , x 0 1 , x 0 2 ) , x m n = F m ( x 0 n , x 0 1 , x 0 2 , , x 0 n 1 ) , m N .

Since F has mixed monotone property and from (3.3) we have

x 0 1 x 1 1 x 2 1 x m 1 x m + 1 1 x m + 1 2 x m 2 x 2 2 x 1 2 x 0 2 , x 0 3 x 1 3 x 2 3 x m 3 x m + 1 3 x 0 3 x m + 1 n x m n x 2 n x 1 n x 0 n .

Applying Lemma 3.2 by taking x 1 = x 0 1 , x 2 = x 0 2 ,, x n = x 0 n and y 1 = x 1 1 , y 2 = x 1 2 ,, y n = x 1 n , then we get

η ( ( F m ( x 0 1 , x 0 2 , , x 0 n ) , F m ( x 0 2 , , x 0 n , x 0 1 ) , , F m ( x 0 n , x 0 1 , , x 0 n 1 ) ) , ( F m ( x 1 1 , x 1 2 , , x 1 n ) , F m ( x 1 2 , , x 1 n , x 1 1 ) , , F m ( x 1 n , x 1 1 , , x 1 n 1 ) ) ) 0 as  m ,

that is,

η ( ( x m 1 , x m 2 , , x m n ) , ( x m + 1 1 , x m + 1 2 , , x m + 1 n ) ) 0as m.
(3.11)

Denote

η s ( ( x m 1 , x m 2 , , x m n ) , ( x m + 1 1 , x m + 1 2 , , x m + 1 n ) ) = 2 max { p s ( x m 1 , x m + 1 1 ) , p s ( x m 2 , x m + 1 2 ) , , p s ( x m n , x m + 1 n ) } , m N .

From the definition of p s , it is clear that

η s ( ( x m 1 , x m 2 , , x m n ) , ( x m + 1 1 , x m + 1 2 , , x m + 1 n ) ) 2 η ( ( x m 1 , x m 2 , , x m n ) , ( x m + 1 1 , x m + 1 2 , , x m + 1 n ) ) , m N .

Using (3.11), we get

lim m η s ( ( x m 1 , x m 2 , , x m n ) , ( x m + 1 1 , x m + 1 2 , , x m + 1 n ) ) = lim m max { p s ( x m 1 , x m + 1 1 ) , p s ( x m 2 , x m + 1 2 ) , , p s ( x m n , x m + 1 n ) } = 0 .
(3.12)

Let ϵ>0. It follows from (3.12) that there exists kN such that

η s ( ( x k 1 , x k 2 , , x k n ) , ( x k + 1 1 , x k + 1 2 , , x k + 1 n ) ) <δ(ϵ).
(3.13)

Without restriction of generality, we can suppose that δ(ϵ)ϵ. We introduce the set X n defined by

: = { ( x 1 , x 2 , x 3 , , x n ) X n : x k 1 x 1 , x 2 x k 2 , x k 3 x 3 , , x n x k n , η s ( ( x k 1 , x k 2 , x k 3 , , x k n ) , ( x k + 1 1 , x k + 1 2 , x k + 1 3 , , x k + 1 n ) ) < ϵ + δ ( ϵ ) } .

Now we will prove that ( x 1 , x 2 , x 3 ,, x n ),

( F ( x 1 , x 2 , x 3 , , x n ) , F ( x 2 , x 3 , , x n , x 1 ) , , F ( x n , x 1 , x 2 , , x n 1 ) ) .
(3.14)

Let ( x 1 , x 2 , x 3 ,, x n ). We have

η s ( ( x k 1 , x k 2 , , x k n ) , ( F ( x 1 , x 2 , , x n ) , F ( x 2 , , x n , x 1 ) , , F ( x n , x 1 , , x n 1 ) ) ) = max { p s ( x k 1 , F ( x 1 , x 2 , , x n ) ) , p s ( x k 2 , F ( x 2 , , x n , x 1 ) ) , , p s ( x k n , F ( x n , x 1 , , x n 1 ) ) } max { p s ( x k 1 , x k + 1 1 ) + p s ( x k + 1 1 , F ( x 1 , x 2 , , x n ) ) , p s ( x k 2 , x k + 1 2 ) + p s ( x k + 1 2 , F ( x 2 , , x n , x 1 ) ) , , p s ( x k n , x k + 1 n ) + p s ( x k + 1 n , F ( x n , x 1 , , x n 1 ) ) } = max { p s ( x k 1 , x k + 1 1 ) + p s ( F ( x k 1 , x k 2 , , x k n ) , F ( x 1 , x 2 , , x n ) ) , p s ( x k 2 , x k + 1 2 ) + p s ( F ( x k 2 , , x k n , x k 1 ) , F ( x 2 , , x n , x 1 ) ) , , p s ( x k n , x k + 1 n ) + p s ( F ( x k n , x k 1 , , x k n 1 ) , F ( x n , x 1 , , x n 1 ) ) } max { p s ( x k 1 , x k + 1 1 ) , p s ( x k 2 , x k + 1 2 ) , , p s ( x k n , x k + 1 n ) } + max { p s ( F ( x k 1 , x k 2 , , x k n ) , F ( x 1 , x 2 , , x n ) ) , p s ( F ( x k 2 , , x k n , x k 1 ) , F ( x 2 , , x n , x 1 ) ) , , p s ( F ( x k n , x k 1 , , x k n 1 ) , F ( x n , x 1 , , x n 1 ) ) } < δ ( ϵ ) + max { p s ( F ( x k 1 , x k 2 , , x k n ) , F ( x 1 , x 2 , , x n ) ) , p s ( F ( x k 2 , , x k n , x k 1 ) , F ( x 2 , , x n , x 1 ) ) , , p s ( F ( x k n , x k 1 , , x k n 1 ) , F ( x n , x 1 , , x n 1 ) ) } ( by (3.13) ) .

We consider the following two cases.

Case I: η s (( x k 1 , x k 2 ,, x k n ),( x 1 , x 2 ,, x n ))ϵ.

By Lemma 3.1, we have

η s ( ( x k 1 , x k 2 , , x k n ) , ( F ( x 1 , x 2 , , x n ) , F ( x 2 , , x n , x 1 ) , , F ( x n , x 1 , , x n 1 ) ) ) < δ ( ϵ ) + max { p s ( F ( x k 1 , x k 2 , , x k n ) , F ( x 1 , x 2 , , x n ) ) , p s ( F ( x k 2 , , x k n , x k 1 ) , F ( x 2 , , x n , x 1 ) ) , , p s ( F ( x k n , x k 1 , , x k n 1 ) , F ( x n , x 1 , , x n 1 ) ) } < δ ( ϵ ) + max { max [ p s ( x k 1 , x 1 ) , p s ( x k 2 , x 2 ) , , p s ( x k n , x n ) ] , max [ p s ( x k 2 , x 2 ) , , p s ( x k n , x n ) , p s ( x k 1 , x 1 ) ] , , max [ p s ( x k n , x n ) , p s ( x k 1 , x 1 ) , , p s ( x k n 1 , x n 1 ) ] } < δ ( ϵ ) + η s ( ( x k 1 , x k 2 , , x k n ) , ( x 1 , x 2 , , x n ) ) δ ( ϵ ) + ϵ .

Case II: ϵ+ η s (( x k 1 , x k 2 ,, x k n ),( x 1 , x 2 ,, x n ))δ(ϵ)+ϵ.

We have

η s ( ( x k 1 , x k 2 , , x k n ) , ( F ( x 1 , x 2 , , x n ) , F ( x 2 , , x n , x 1 ) , , F ( x n , x 1 , , x n 1 ) ) ) < δ ( ϵ ) + max { p s ( F ( x k 1 , x k 2 , , x k n ) , F ( x 1 , x 2 , , x n ) ) , p s ( F ( x k 2 , , x k n , x k 1 ) , F ( x 2 , , x n , x 1 ) ) , , p s ( F ( x k n , x k 1 , , x k n 1 ) , F ( x n , x 1 , , x n 1 ) ) } .
(3.15)

In this case, we get

ϵ<max { p s ( x k 1 , x 1 ) , p s ( x k 2 , x 2 ) , , p s ( x k n , x n ) } <ϵ+δ(ϵ).

Since x k 1 x 1 , x 2 x k 2 , x k 3 x 3 ,, x n x k n , by (3) we get

p s ( F ( x 1 , x 2 , , x n ) , F ( x k 1 , x k 2 , , x k n ) ) <ϵ.
(3.16)

Also we have

ϵ<max { p s ( x k 2 , x 2 ) , , p s ( x k n , x n ) , p s ( x k 1 , x 1 ) } <ϵ+δ(ϵ).

By (3), this implies that

p s ( F ( x k 2 , , x k n , x k 1 ) , F ( x 2 , , x n , x 1 ) ) <ϵ.
(3.17)

In the same way we have

p s ( F ( x n , x 1 , , x n 1 ) , F ( x k n , x k 1 , , x k n 1 ) ) <ϵ.
(3.18)

Hence combining (3.15)-(3.18), we obtain

η s ( ( x k 1 , x k 2 , , x k n ) , ( F ( x k 1 , x k 2 , , x k n ) , F ( x k 2 , , x k n , x k 1 ) , , F ( x k n , x k 1 , , x k n 1 ) ) ) <ϵ+δ(ϵ).

On the other hand, using (2), we can check easily that

x k 1 F ( x 1 , x 2 , , x n ) , F ( x 2 , , x n , x 1 ) x k 2 , , F ( x n , x 1 , , x n 1 ) x k n .

Hence, we deduce that (3.14) holds. By (3.13), we have ( x k + 1 1 , x k + 1 2 ,, x k + 1 n ). This implies with (3.14) that

( x k + 1 1 , x k + 1 2 , , x k + 1 n ) ( F ( x k + 1 1 , x k + 1 2 , , x k + 1 n ) , F ( x k + 1 2 , , x k + 1 n , x k + 1 1 ) , , F ( x k + 1 n , x k + 1 1 , , x k + 1 n 1 ) ) = ( x k + 2 1 , x k + 2 2 , , x k + 2 n ) ( F ( x k + 2 1 , x k + 2 2 , , x k + 2 n ) , F ( x k + 2 2 , , x k + 2 n , x k + 2 1 ) , , F ( x k + 2 n , x k + 2 1 , , x k + 2 n 1 ) ) = ( x k + 3 1 , x k + 3 2 , , x k + 3 n ) ( x m 1 , x m 2 , , x m n ) .

Thus for all m>k, we have ( x m 1 , x m 2 ,, x m n ). This implies that for all m,l>k, we have

η s ( ( x m 1 , x m 2 , , x m n ) , ( x l 1 , x l 2 , , x l n ) ) = max { p s ( x m 1 , x l 1 ) , p s ( x m 2 , x l 2 ) , , p s ( x m n , x l n ) } max { p s ( x m 1 , x k 1 ) + p s ( x k 1 , x l 1 ) , p s ( x m 2 , x k 2 ) + p s ( x k 2 , x l 2 ) , , p s ( x m n , x k n ) + p s ( x k n , x l n ) } max { p s ( x m 1 , x k 1 ) , p s ( x m 2 , x k 2 ) , , p s ( x m n , x k n ) } + max { p s ( x k 1 , x l 1 ) , p s ( x k 2 , x l 2 ) , , p s ( x k n , x l n ) } = η ( ( x m 1 , x m 2 , , x m n ) , ( x k 1 , x k 2 , , x k n ) ) + η ( ( x k 1 , x k 2 , , x k n ) , ( x m 1 , x m 2 , , x m n ) ) < 2 ( ϵ + δ ( ϵ ) ) < 4 ϵ .

We deduce that {( x m 1 , x m 2 ,, x m n )} is a Cauchy sequence in the metric space ( X n , η s ). Since (X,p) is complete, from Lemma 2.1, (X, p s ) is a complete metric space. Therefore ( X n , η s ) is complete. Hence there exist ( x 1 , x 2 ,, x n ) X n such that

η s ( ( x m 1 , x m 2 , , x m n ) , ( x 1 , x 2 , , x n ) ) 0as m,

which shows that

p s ( F m ( x 0 1 , x 0 2 , , x 0 n ) , x 1 ) 0 as  m , p s ( F m ( x 0 2 , , x 0 n , x 0 1 ) , x 2 ) 0 as  m , p s ( F m ( x 0 n , x 0 1 , , x 0 n 1 ) , x n ) 0 as  m .

Therefore from Lemma 2.1 and using (3.12), we have

{ p ( x 1 , x 1 ) = lim m p ( x m 1 , x 1 ) = lim m p ( x m 1 , x m 1 ) = 0 , p ( x 2 , x 2 ) = lim m p ( x m 2 , x 2 ) = lim m p ( x m 2 , x m 2 ) = 0 , p ( x n , x n ) = lim m p ( x m n , x n ) = lim m p ( x m n , x m n ) = 0 .
(3.19)

We will show that x 1 =F( x 1 , x 2 ,, x n ), x 2 =F( x 2 ,, x n , x 1 ),, x n =F( x n , x 1 ,, x n 1 ). Since F is continuous on X, then F is continuous at ( x 1 , x 2 ,, x n ). Hence for any ϵ>0, there exists δ(ϵ)>0 such that if ( y 1 , y 2 ,, y n ) X n verifying

η ( ( x 1 , x 2 , , x n ) , ( y 1 , y 2 , , y n ) ) <η ( ( x 1 , x 2 , , x n ) , ( x 1 , x 2 , , x n ) ) +δ(ϵ)

means that

max { p ( x 1 , y 1 ) , p ( x 2 , y 2 ) , , p ( x n , y n ) } < max { p ( x 1 , x 1 ) , p ( x 2 , x 2 ) , , p ( x n , x n ) } + δ ( ϵ ) = δ ( ϵ ) ,

because p( x 1 , x 1 )=p( x 2 , x 2 )==p( x n , x n )=0, then we have

p ( F ( x 1 , x 2 , , x n ) , F ( y 1 , y 2 , , y n ) ) <p ( F ( x 1 , x 2 , , x n ) , F ( x 1 , x 2 , , x n ) ) + ϵ 2 .

Since

lim m p ( x m 1 , x 1 ) = lim m p ( x m 2 , x 2 ) == lim m p ( x m n , x n ) =0

for α=min{ δ ( ϵ ) 2 , ϵ 2 }>0, there exists m 0 , l 0 N such that for m> m 0 , l> l 0 , p( x m 1 , x 1 )<α,p( x m 2 , x 2 )<α,,p( x m n , x n )<α. Then for mN, mmax{ m 0 , l 0 }, we have

max { p ( x m 1 , x 1 ) , p ( x m 2 , x 2 ) , , p ( x m n , x n ) } <α< δ ( ϵ ) 2 ,

so we get

p ( F ( x 1 , x 2 , , x n ) , F ( x m 1 , x m 2 , , x m n ) ) <p ( F ( x 1 , x 2 , , x n ) , F ( x 1 , x 2 , , x n ) ) + ϵ 2 .
(3.20)

Now, for any mmax{ m 0 , l 0 },

p ( F ( x 1 , x 2 , , x n ) , x 1 ) p ( F ( x 1 , x 2 , , x n ) , x m + 1 1 ) + p ( x m + 1 1 , x 1 ) = p ( F ( x 1 , x 2 , , x n ) , F ( x m 1 , x m 2 , , x m n ) ) + p ( x m + 1 1 , x 1 ) < p ( F ( x 1 , x 2 , , x n ) , F ( x 1 , x 2 , , x n ) ) + ϵ 2 + α ( by (3.20) ) < p ( F ( x 1 , x 2 , , x n ) , F ( x 1 , x 2 , , x n ) ) + ϵ .

On the other hand, since F is a generalized Meir-Keeler type function, then from Lemma 3.1, we have

p ( F ( x 1 , x 2 , , x n ) , F ( x 1 , x 2 , , x n ) ) <max { p ( x 1 , x 1 ) , p ( x 2 , x 2 ) , , p ( x n , x n ) } =0.

In this case, for any ϵ>0, p(F( x 1 , x 2 ,, x n ), x 1 )<ϵ. This implies that F( x 1 , x 2 ,, x n )= x 1 . Similarly we can show that

F ( x 2 , , x n , x 1 ) = x 2 ,,F ( x n , x 1 , , x n 1 ) = x n .

Thus we have proved that F has an n-tupled fixed point. □

Remark 3.2 Theorem 3.1 still holds if we replace (3.10) by x 0 1 , x 0 2 , x 0 3 ,, x 0 n X such that

{ x 0 1 F ( x 0 1 , x 0 2 , x 0 3 , , x 0 n ) , F ( x 0 2 , x 0 3 , , x 0 n , x 0 1 ) x 0 2 , x 0 3 F ( x 0 3 , , x 0 n , x 0 1 , x 0 2 ) , F ( x 0 n , x 0 1 , x 0 2 , , x 0 n 1 ) x 0 n .

Theorem 3.2 Let (X,) be a partially ordered set and suppose that there is a partial metric p on X such that (X,p) is a complete partial metric space. Assume that X has the following properties:

  1. (a)

    if a nondecreasing sequence x m x then x m x for all m0,

  2. (b)

    if a nonincreasing sequence x m x then x x m for all m0.

Let F: X n X be a given mapping satisfying the following hypotheses:

  1. (1)

    F is continuous,

  2. (2)

    F has the mixed strict monotone property,

  3. (3)

    F is a generalized Meir-Keeler type function,

  4. (4)

    there exist x 0 1 , x 0 2 , x 0 3 ,, x 0 n X such that (3.10) holds.

Then there exists ( x 1 , x 2 , x 3 ,, x n ) X n such that x 1 =F( x 1 , x 2 , x 3 ,, x n ), x 2 =F( x 2 , x 3 ,, x n , x 1 ),, x n =F( x n , x 1 , x 2 ,, x n 1 ). Furthermore, p( x 1 , x 1 )=p( x 2 , x 2 )==p( x n , x n )=0.

Proof Following the proof of Theorem 3.1, we only have to prove that

x 1 = F ( x 1 , x 2 , x 3 , , x n ) , x 2 = F ( x 2 , x 3 , , x n , x 1 ) , , x n = F ( x n , x 1 , x 2 , , x n 1 ) .

Let ϵ>0. Since F m ( x 0 1 , x 0 2 ,, x 0 n ) x 1 , F m ( x 0 2 ,, x 0 n , x 0 1 ) x 2 ,, F m ( x 0 n , x 0 1 ,, x 0 n 1 ) x n . Then there exist m 1 , m 2 ,, m n N such that for all m m 1 ,l m 2 ,,t m n ,

{ p ( F m ( x 0 1 , x 0 2 , , x 0 n ) , x 1 ) < ϵ , p ( F l ( x 0 2 , , x 0 n , x 0 1 ) , x 2 ) < ϵ , p ( F t ( x 0 n , x 0 1 , , x 0 n 1 ) , x n ) < ϵ .
(3.21)

Taking mmax{ m 1 , m 2 ,, m n } and using

F m ( x 0 1 , x 0 2 , , x 0 n ) x 1 , x 2 F m ( x 0 2 , , x 0 n , x 0 1 ) , , x n F m ( x 0 n , x 0 1 , , x 0 n 1 ) ,

by (3.21) and Lemma 3.1, we get

p ( F ( x 1 , x 2 , , x n ) , x 1 ) p ( F ( x 1 , x 2 , , x n ) , F m + 1 ( x 0 1 , x 0 2 , , x 0 n ) ) + p ( F m + 1 ( x 0 1 , x 0 2 , , x 0 n ) , x 1 ) = p ( F ( x 1 , x 2 , , x n ) , F ( F m ( x 0 1 , x 0 2 , , x 0 n ) , F m ( x 0 2 , , x 0 n , x 0 1 ) , , F m ( x 0 n , x 0 1 , , x 0 n 1 ) ) ) + p ( F m + 1 ( x 0 1 , x 0 2 , , x 0 n ) , x 1 ) < max { p ( x 1 , F m ( x 0 1 , x 0 2 , , x 0 n ) ) , p ( x 2 , F m ( x 0 2 , , x 0 n , x 0 1 ) ) , , p ( x n , F m ( x 0 n , x 0 1 , , x 0 n 1 ) ) } + p ( F m + 1 ( x 0 1 , x 0 2 , , x 0 n ) , x 1 ) < 2 ϵ .

This implies that F( x 1 , x 2 ,, x n )= x 1 . Similarly, we can show that

p ( F ( x 2 , , x n , x 1 ) , x 2 ) <2ϵ,,p ( F ( x n , x 1 , , x n 1 ) , x n ) <2ϵ,

which implies that F( x 2 ,, x n , x 1 )= x 2 ,,F( x n , x 1 ,, x n 1 )= x n .

This completes the proof. □

Now we endow the product space X n with the following partial order: for ( x 1 , x 2 ,, x n ),( y 1 , y 2 ,, y n ) X n ,

( y 1 , y 2 , , y n ) ( x 1 , x 2 , , x n ) y 1 x 1 , x 2 y 2 , y 3 x 3 , , x n y n .

One can prove that n-tupled fixed point is in fact unique and the product space X n endow with this partial order has the following property:

  1. (A)

    ( x 1 , x 2 ,, x n ),( z 1 , z 2 ,, z n ) X n , ( t 1 , t 2 ,, t n ) X n that is comparable to ( x 1 , x 2 ,, x n ) and ( z 1 , z 2 ,, z n ).

Theorem 3.3 Adding (A) to the hypotheses of Theorem  3.1 (respectively, Theorem  3.2), we obtain the uniqueness of n-tupled fixed point of F.

Proof Suppose that ( z 1 , z 2 ,, z n ) X n is another n-tupled fixed point of F. We distinguish two cases:

Case I: ( x 1 , x 2 ,, x n ) is comparable to ( z 1 , z 2 ,, z n ) with respect to ordering in X n , where

lim m F m ( x 0 1 , x 0 2 , , x 0 n ) = x 1 , lim m F m ( x 0 2 , , x 0 n , x 0 1 ) = x 2 , , lim m F m ( x 0 n , x 0 1 , , x 0 n 1 ) = x n .

Without restriction of generality, we can suppose that

F ( z 1 , z 2 , , z n ) = z 1 x 1 = F ( x 1 , x 2 , , x n ) , F ( x 2 , , x n , x 1 ) = x 2 z 2 = F ( z 2 , , z n , z 1 ) , F ( x n , x 1 , , x n 1 ) = x n z n = F ( z n , z 1 , , z n 1 ) .

We have

η ( ( x 1 , x 2 , , x n ) , ( z 1 , z 2 , , z n ) ) = max { p ( x 1 , z 1 ) , p ( x 2 , z 2 ) , , p ( x n , z n ) } = max { p ( F ( x 1 , x 2 , , x n ) , F ( z 1 , z 2 , , z n ) ) , p ( F ( x 2 , , x n , x 1 ) , F ( z 2 , , z n , z 1 ) ) , , p ( F ( x n , x 1 , , x n 1 ) , F ( z n , z 1 , , z n 1 ) ) } < max { max [ p ( x 1 , z 1 ) , p ( x 2 , z 2 ) , , p ( x n , z n ) ] , max [ p ( x 1 , z 1 ) , p ( x 2 , z 2 ) , , p ( x n , z n ) ] , , max [ p ( x 1 , z 1 ) , p ( x 2 , z 2 ) , , p ( x n , z n ) ] } = η ( ( x 1 , x 2 , , x n ) , ( z 1 , z 2 , , z n ) ) .

Case II: ( x 1 , x 2 ,, x n ) is not comparable to ( z 1 , z 2 ,, z n ). Then there exists ( t 1 , t 2 ,, t n ) X n that is comparable to ( x 1 , x 2 ,, x n ) and ( z 1 , z 2 ,, z n ). Without restriction of generality, we can assume that

x 1 t 1 , t 2 x 2 , x 3 t 3 , , t n x n and z 1 t 1 , t 2 z 2 , z 3 t 3 , , t n z n .
(3.22)

From (3.22) and Lemma 3.2, we have

η ( ( F m ( x 1 , x 2 , , x n ) , F m ( x 2 , , x n , x 1 ) , , F m ( x n , x 1 , , x n 1 ) ) , ( F m ( t 1 , t 2 , , t n ) , F m ( t 2 , , t n , t 1 ) , , F m ( t n , t 1 , , t n 1 ) ) ) 0 as  m .
(3.23)

Similarly we have

η ( ( F m ( z 1 , z 2 , , z n ) , F m ( z 2 , , z n , z 1 ) , , F m ( z n , z 1 , , z n 1 ) ) , ( F m ( t 1 , t 2 , , t n ) , F m ( t 2 , , t n , t 1 ) , , F m ( t n , t 1 , , t n 1 ) ) ) 0 as  m .
(3.24)

On the other hand, using the triangular inequality, we get

η ( ( x 1 , x 2 , , x n ) , ( z 1 , z 2 , , z n ) ) = η ( ( F m ( x 1 , x 2 , , x n ) , F m ( x 2 , , x n , x 1 ) , , F m ( x n , x 1 , , x n 1 ) ) , ( F m ( z 1 , z 2 , , z n ) , F m ( z 2 , , z n , z 1 ) , , F m ( z n , z 1 , , z n 1 ) ) ) η ( ( F m ( x 1 , x 2 , , x n ) , F m ( x 2 , , x n , x 1 ) , , F m ( x n , x 1 , , x n 1 ) ) , ( F m ( t 1 , t 2 , , t n ) , F m ( t 2 , , t n , t 1 ) , , F m ( t n , t 1 , , t n 1 ) ) ) + η ( ( F m ( t 1 , t 2 , , t n ) , F m ( t 2 , , t n , t 1 ) , , F m ( t n , t 1 , , t n 1 ) ) , ( F m ( z 1 , z 2 , , z n ) , F m ( z 2 , , z n , z 1 ) , , F m ( z n , z 1 , , z n 1 ) ) ) .

By (3.22) and (3.23), we have η(( x 1 , x 2 ,, x n ),( z 1 , z 2 ,, z n ))=0, we get

( x 1 , x 2 , , x n ) = ( z 1 , z 2 , , z n ) .

This completes the proof. □

4 Applications

In this section, using the earlier results proved in the preceding section, we obtain some n-tupled fixed point theorem for mappings satisfying a general contractive condition of integral type in partially ordered complete partial metric spaces.

Theorem 4.1 Let (X,) be a partially ordered set and suppose that there is a partial metric p on X such that (X,p) is a complete partial metric space. Let F: X n X be a given mapping. Assume that there exists a function θ from [0,) into itself satisfying the following:

  1. (1)

    θ(0)=0 and θ(t)>0 for every t>0,

  2. (2)

    θ is nondecreasing and right continuous,

  3. (3)

    for every ϵ>0, there exists δ(ϵ)>0 such that

    ϵ < θ ( max { p ( x 1 , y 1 ) , p ( x 2 , y 2 ) , , p ( x n , y n ) } ) < ϵ + δ ( ϵ ) θ ( p ( F ( x 1 , x 2 , , x n ) , F ( y 1 , y 2 , , y n ) ) ) < ϵ

for all y 1 x 1 , x 2 y 2 , y 3 x 3 ,, x n y n .

Then F is a generalized Meir-Keeler type function.

Proof Fix ϵ>0. Since θ(ϵ)>0, there exists α>0 and ( a 1 , a 2 ,, a n ),( b 1 , b 2 ,, b n ) X n such that

θ ( ϵ ) θ ( max { p ( a 1 , b 1 ) , p ( a 2 , b 2 ) , , p ( a n , b n ) } ) < θ ( ϵ ) + δ ( ϵ ) θ ( p ( F ( a 1 , a 2 , , a n ) , F ( b 1 , b 2 , , b n ) ) ) < ϵ .
(4.1)

From the right continuity of θ, there exists δ>0 such that θ(ϵ+δ)<θ(ϵ)+α. Fix x 1 , x 2 ,, x n , y 1 , y 2 ,, y n X, then

ϵmax { p ( x 1 , y 1 ) , p ( x 2 , y 2 ) , , p ( x n , y n ) } <ϵ+δ.

Since θ is a nondecreasing function, we get

θ(ϵ)θ ( max { p ( x 1 , y 1 ) , p ( x 2 , y 2 ) , , p ( x n , y n ) } ) <θ(ϵ+δ)<θ(ϵ)+α.

By (4.1) we get

θ ( p ( F ( x 1 , x 2 , , x n ) , F ( y 1 , y 2 , , y n ) ) ) <θ(ϵ),

and hence

p ( F ( x 1 , x 2 , , x n ) , F ( y 1 , y 2 , , y n ) ) <ϵ.

 □

The following result is an immediate consequence of Theorems 3.1, 3.2 and 4.1.

Corollary 4.1 Let (X,) be a partially ordered set and suppose that there is a partial metric p on X such that (X,p) is a complete partial metric space. Let F: X n X be a given mapping satisfying the following hypotheses:

  1. (1)

    F is continuous,

  2. (2)

    F has the mixed strict monotone property,

  3. (3)

    for all ϵ>0, there exists δ(ϵ)>0 such that

    ϵ 0 max { p ( x 1 , y 1 ) , p ( x 2 , y 2 ) , , p ( x n , y n ) } φ ( t ) d t < ϵ + δ ( ϵ ) 0 p ( F ( x 1 , , x n ) , F ( y 1 , , y n ) ) φ ( t ) d t < ϵ

for all y 1 x 1 , x 2 y 2 , y 3 x 3 ,, x n y n , where φ is a locally integrable function from [0,) into itself satisfying

0 s φ(t)dt>0,s>0,
  1. (4)

    x 0 1 , x 0 2 , x 0 3 ,, x 0 n X such that (3.10) holds.

Then there exists ( x 1 , x 2 ,, x n ) X n such that

x 1 =F ( x 1 , x 2 , , x n ) , x 2 =F ( x 2 , , x n , x 1 ) ,, x n =F ( x n , x 1 , , x n 1 ) .

Moreover, if property (A) is satisfied, then the n-tupled fixed point of F remains unique.

Remark 4.1 The conclusions of the preceding corollary remain valid if we replace the continuity hypothesis of F by hypotheses (a) and (b) of Theorem 3.2.

Corollary 4.2 Let (X,) be a partially ordered set and suppose that there is a partial metric p on X such that (X,p) is a complete partial metric space. Let F: X n X be a given mapping satisfying the following hypotheses:

  1. (1)

    F is continuous,

  2. (2)

    F has the mixed strict monotone property,

  3. (3)

    for all y 1 x 1 , x 2 y 2 , y 3 x 3 ,, x n y n ,

    0 p ( F ( x 1 , x 2 , , x n ) , F ( y 1 , y 2 , , y n ) ) φ(t)dtk 0 max { p ( x 1 , y 1 ) , p ( x 2 , y 2 ) , , p ( x n , y n ) } φ(t)dt,

where k(0,1) and φ is a locally integrable function from [0,) into itself satisfying

0 s φ(t)dt>0,s>0,
  1. (4)

    x 0 1 , x 0 2 , x 0 3 ,, x 0 n X such that (3.10) holds.

Then there exists ( x 1 , x 2 ,, x n ) X n such that

x 1 =F ( x 1 , x 2 , , x n ) , x 2 =F ( x 2 , , x n , x 1 ) ,, x n =F ( x n , x 1 , , x n 1 ) .

Moreover, if property (A) is satisfied, then the n-tupled fixed point of F remains unique.

Proof For all ϵ>0, take δ(ϵ)=( 1 k 1)ϵ and apply Corollary 4.1. □

Remark 4.2 We replace the continuity hypothesis of F by hypotheses (a) and (b) of Theorem 3.2, then this result also remains true.

5 Example

We give the following example to illustrate our main result.

Example 5.1 Let X=[0,1]. Then (X,) is a partially ordered set under the natural ordering of real numbers. Define p:[0,1]×[0,1] R + by p(x,y)=max{x,y}, x,y[0,1]. Then (X,p) is a complete partial metric space.

Now for any fixed even integer n>1, consider the product space X n =[0,1]×[0,1]××[0,1], n times (in short we write X n = [ 0 , 1 ] n ). Define F: X n X by

F ( x 1 , x 2 , x 3 , , x n ) = x 1 n for  x 1 , x 2 ,, x n [0,1].

Then F has the mixed strict monotone property. Also F is a generalized Meir-Keeler type function. The proof follows in two parts, that is, we prove the following:

For ( x 1 , x 2 ,, x n ),( y 1 , y 2 ,, y n ) X n with x 1 y 1 , y 2 x 2 , x 3 y 3 ,, y n x n ,

( 1 ) p ( F ( x 1 , x 2 , , x n ) , F ( y 1 , y 2 , , y n ) ) < max { p ( x 1 , y 1 ) , p ( x 2 , y 2 ) , , p ( x n , y n ) } , ( 2 ) η ( ( F m ( x 1 , x 2 , x 3 , , x n ) , F m ( x 2 , x 3 , , x n , x 1 ) , , F m ( x n , x 1 , x 2 , , x n 1 ) ) , ( 2 ) ( F m ( y 1 , y 2 , y 3 , , y n ) , F m ( y 2 , y 3 , , y n , y 1 ) , , F m ( y n , y 1 , y 2 , , y n 1 ) ) ) ( 2 ) 0 as  m .

The first part is trivial. For second part, we have

η ( ( F m ( x 1 , x 2 , x 3 , , x n ) , F m ( x 2 , x 3 , , x n , x 1 ) , , F m ( x n , x 1 , x 2 , , x n 1 ) ) , ( F m ( y 1 , y 2 , y 3 , , y n ) , F m ( y 2 , y 3 , , y n , y 1 ) , , F m ( y n , y 1 , y 2 , , y n 1 ) ) ) = η ( ( F ( F m 1 ( x 1 , x 2 , , x n ) , F m 1 ( x 2 , , x n , x 1 ) , , F m 1 ( x n , x 1 , , x n 1 ) ) , F ( F m 1 ( x 2 , , x n , x 1 ) , , F m 1 ( x n , x 1 , , x n 1 ) , F m 1 ( x 1 , x 2 , , x n ) ) , , F ( F m 1 ( x n , x 1 , , x n 1 ) , F m 1 ( x 1 , x 2 , , x n ) , , F m 1 ( x n 1 , , x 1 , x n ) ) ) , ( F ( F m 1 ( y 1 , y 2 , , y n ) , F m 1 ( y 2 , , y n , y 1 ) , , F m 1 ( y n , y 1 , , y n 1 ) ) , F ( F m 1 ( y 2 , , y n , y 1 ) , , F m 1 ( y n , y 1 , , y n 1 ) , F m 1 ( y 1 , y 2 , , y n ) ) , , F ( F m 1 ( y n , y 1 , , y n 1 ) , F m 1 ( y 1 , y 2 , , y n ) , , F m 1 ( y n 1 , , y 1 , y n ) ) ) ) = η ( ( F ( x 1 n m 1 , x 2 n m 1 , x 3 n m 1 , , x n n m 1 ) , F ( x 2 n m 1 , x 3 n m 1 , , x n n m 1 , x 1 n m 1 ) , , F ( x n n m 1 , x 1 n m 1 , x 2 n m 1 , , x n 1 n m 1 ) ) , ( F ( y 1 n m 1 , y 2 n m 1 , y 3 n m 1 , , y n n m 1 ) , F ( y 2 n m 1 , y 3 n m 1 , , y n n m 1 , y 1 n m 1 ) , , F ( y n n m 1 , y 1 n m 1 , y 2 n m 1 , , y n 1 n m 1 ) ) ) = η ( ( x 1 n m , x 2 n m , x 3 n m , , x n n m ) , ( y 1 n m , y 2 n m , y 3 n m , , y n n m ) ) = max { p ( x 1 n m , y 1 n m ) , p ( x 2 n m , y 2 n m ) , p ( x 3 n m , y 3 n m ) , , p ( x n n m , y n n m ) } = max { y 1 n m , x 2 n m , y 3 n m , , x n n m } 0 as  m .

Hence all the hypotheses of Theorem 3.1 are satisfied. Therefore, F has a unique n-tupled fixed point. Here (0,0,,0) is an n-tupled fixed point of F.

References

  1. Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 1–8. 10.1080/00036810701714164

    Article  MathSciNet  Google Scholar 

  2. Altun I, Damjanovic B, Djoric D: Fixed point and common fixed point theorems on ordered cone metric spaces. Appl. Math. Lett. 2010, 23: 310–316. 10.1016/j.aml.2009.09.016

    Article  MathSciNet  MATH  Google Scholar 

  3. Altun I, Simsek H: Some fixed point theorems on ordered metric spaces and application. Fixed Point Theory Appl. 2010., 2010: Article ID 621469

    Google Scholar 

  4. 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

    Article  MathSciNet  MATH  Google Scholar 

  5. Amini-Harandi A: Coupled and tripled fixed point theory in partially ordered metric spaces with application to initial value problem. Math. Comput. Model. 2013, 57: 2343–2348. 10.1016/j.mcm.2011.12.006

    Article  MATH  Google Scholar 

  6. Amini-Harandi A, Emami H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 2010, 72: 2238–2242. 10.1016/j.na.2009.10.023

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  8. Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 2011, 4(2):1–12.

    MathSciNet  MATH  Google Scholar 

  9. Aydi H:Common fixed point results for mappings satisfying (φ,ϕ)-weak contractions in ordered partial metric space. Int. J. Math. Stat. 2012, 12(2):53–64.

    MathSciNet  MATH  Google Scholar 

  10. Berinde V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 7347–7355. 10.1016/j.na.2011.07.053

    Article  MathSciNet  MATH  Google Scholar 

  11. Berinde V: Coupled fixed point theorems for contractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 2012, 75: 3218–3228. 10.1016/j.na.2011.12.021

    Article  MathSciNet  MATH  Google Scholar 

  12. Berinde V, Pǎcurar M: Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 115

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Harjani J, Sadarangani K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 2010, 72: 1188–1197. 10.1016/j.na.2009.08.003

    Article  MathSciNet  MATH  Google Scholar 

  15. Hong S: Fixed points of multivalued operators in ordered metric spaces with applications. Nonlinear Anal. 2010, 72: 3929–3942. 10.1016/j.na.2010.01.013

    Article  MathSciNet  MATH  Google Scholar 

  16. Kadelburg Z, Pavlović M, Radenović S: Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces. Comput. Math. Appl. 2010. 10.1016/j.camwa.2010.02.039

    Google Scholar 

  17. Lakshmikantham V, Ćirić LB: 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

    Article  MathSciNet  MATH  Google Scholar 

  18. Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equation. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5

    Article  MathSciNet  MATH  Google Scholar 

  19. Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23(12):2205–2212. 10.1007/s10114-005-0769-0

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  21. O’Regan D, Petrusel A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 2008, 341: 1231–1252.

    MathSciNet  Google Scholar 

  22. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4

    Article  MathSciNet  MATH  Google Scholar 

  23. Rezapour S, Derafshpour M, Shahzad N: Best proximity points of cyclic φ -contractions in ordered metric spaces. Topol. Methods Nonlinear Anal. 2010, 37(1):193–202.

    MathSciNet  Google Scholar 

  24. Romaguera S, Schellekens M: Partial metric monoids and semivaluation spaces. Topol. Appl. 2005, 153(5–6):948–962. 10.1016/j.topol.2005.01.023

    Article  MathSciNet  MATH  Google Scholar 

  25. Romaguera S, Valero O: A quantitative computational model for complete partialmetric spaces via formal balls. Math. Struct. Comput. Sci. 2009, 19(3):541–563. 10.1017/S0960129509007671

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  27. 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

    Article  MathSciNet  MATH  Google Scholar 

  28. Wolk ES: Continuous convergence in partially ordered sets. Gen. Topol. Appl. 1975, 5: 221–234. 10.1016/0016-660X(75)90022-7

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang X: Fixed point theorems of multivalued monotone mappings in ordered metric spaces. Appl. Math. Lett. 2010, 23: 235–240. 10.1016/j.aml.2009.06.011

    Article  MathSciNet  MATH  Google Scholar 

  30. Ćirić LB, Olatinwo MO, Gopal D, Akinbo G: Coupled fixed point theorems for mappings satisfying a contractive condition of rational type on a partially ordered metric space. Adv. Fixed Point Theory 2012, 2(1):1–8.

    Google Scholar 

  31. Murthy PP, Rashmi : Tripled common fixed point theorems for w -compatible mappings in ordered cone metric spaces. Adv. Fixed Point Theory 2012, 2(2):157–175.

    MathSciNet  MATH  Google Scholar 

  32. Nashine HK, Samet B, Vetro C: Monotone generalized nonlinear contractions type mappings in ordered metric spaces. Math. Comput. Model. 2011, 54: 712–720. 10.1016/j.mcm.2011.03.014

    Article  MathSciNet  MATH  Google Scholar 

  33. Husain S, Sahper H: Generalized n -tupled fixed point theorems in partially ordered metric spaces involving an ICS map. Adv. Fixed Point Theory 2013, 3(3):476–492.

    Google Scholar 

  34. Samet B, Vetro C: Coupled fixed point, f -invariant set and fixed point of N -order. Ann. Funct. Anal. 2010, 1(2):4656–4662.

    Article  MathSciNet  Google Scholar 

  35. Imdad M, Soliman AH, Choudhury BS, Das P: On n -tupled coincidence and common fixed points results in metric spaces. J. Oper. 2013., 2013: Article ID 532867

    Google Scholar 

  36. Matthews SG: Partial metric topology. Ann. New York Acad. Sci. 728. Proc. 8th Summer Conference on General Topology and Applications 1994, 183–197.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Golubović Z, Kadelburg Z, Radenović S: Coupled coincidence points of mappings in ordered partial metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 192581 10.1155/2012/192581

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Heckmann R: Approximation of metric spaces by partial metric spaces. Appl. Categ. Struct. 1999, 7: 71–83. 10.1023/A:1008684018933

    Article  MathSciNet  MATH  Google Scholar 

  41. Jleli M, Karapınar E, Samet B: Further remarks on fixed point theorems in the context of partial metric spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 715456

    Google Scholar 

  42. Kadelburg Z, Nashine HK, Radenović S: Coupled fixed point results in 0-complete ordered partial metric spaces. J. Adv. Math. Stud. 2013, 6(1):159–172.

    MathSciNet  MATH  Google Scholar 

  43. Karapınar E: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 4

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  46. Samet B, Vetro C, Vetro F: From metric spaces to partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 5 10.1186/1687-1812-2013-5

    Google Scholar 

  47. Shatanawi W, Samet B, Abbas M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model. 2011. 10.1016/j.mcm.2011.08.042

    Google Scholar 

  48. Valero O: On Banach fixed point theorems for partial metric spaces. Appl. Gen. Topol. 2005, 6(2):229–240. 10.4995/agt.2005.1957

    Article  MathSciNet  MATH  Google Scholar 

  49. Erduran A, Imdad M: Coupled fixed point theorems for generalized Meir-Keeler contractions in ordered partial metric spaces. J. Nonlinear Anal. Appl. 2012., 2012: Article ID jnaa-00169

    Google Scholar 

  50. Ilic D, Pavlovic V, Rakocecic V: Some new extensions of Banach’s contraction principle to partial metric space. Appl. Math. Lett. 2011, 24(8):1326–1330. 10.1016/j.aml.2011.02.025

    Article  MathSciNet  MATH  Google Scholar 

  51. Shobkolaei N, Vaezpour SM, Sedghi S: A common fixed point theorem on ordered partial metric spaces. J. Basic Appl. Sci. Res. 2011, 1(12):3433–3439.

    Google Scholar 

  52. Meir A, Keeler E: A theorem on contraction mappings. J. Math. Anal. Appl. 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6

    Article  MathSciNet  MATH  Google Scholar 

  53. Suzuki T: Meir-Keeler contraction of integral type are still Meir-Keeler contractions. Int. J. Math. Math. Sci. 2007., 2007: Article ID 39281 10.1155/2007/39281

    Google Scholar 

  54. Suzuki T: A generalized Banach contraction principle which characterizes metric completeness. Proc. Am. Math. Soc. 2008, 136: 1861–1869.

    Article  MATH  Google Scholar 

  55. Karapınar, E: Quartet fixed point for nonlinear contraction. arXiv:1106.5472

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Imdad, M., Sharma, A. & Erduran, A. Generalized Meir-Keeler type n-tupled fixed point theorems in ordered partial metric spaces. Fixed Point Theory Appl 2014, 114 (2014). https://doi.org/10.1186/1687-1812-2014-114

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