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Coincidence point and common fixed point theorems in the product spaces of mixed-monotonically complete quasi-ordered metric spaces and their applications to the systems of integral equations and ordinary differential equations

Journal of Inequalities and Applications20142014:518

https://doi.org/10.1186/1029-242X-2014-518

Received: 4 October 2014

Accepted: 9 December 2014

Published: 23 December 2014

Abstract

The main purpose of this paper is to study the coincidence point and common fixed point theorems in the product spaces of mixed-monotonically complete quasi-ordered metric spaces based on some new types of contractive inequalities. In order to investigate the existence and chain-uniqueness of solutions for the systems of integral equations and ordinary differential equations, we shall also study the fixed point theorems for the functions having mixed monotone property or comparable property in the product space of quasi-ordered metric space.

MSC:47H10, 54H25.

Keywords

function of contractive factorcoincidence pointcommon fixed pointschain-uniquenesssystem of integral equationssystem of ordinary differential equations

1 Introduction

The existence of coincidence point has been studied in [14] and the references therein. Also, the existence of common fixed point has been studied in [515] and the references therein. In this paper, we shall introduce the concepts of mixed-monotonically complete quasi-ordered metric space and monotonically complete quasi-ordered metric space. Based on this completeness, we shall establish some new coincidence point and common fixed point theorems in the product spaces of mixed-monotonically complete quasi-ordered metric spaces in which the fixed points of functions having mixed monotone property or mixed comparable property that are defined in the product space of quasi-ordered metric space can be subsequently obtained. We shall also present the interesting applications to the existence and chain-uniqueness of solutions for the systems of integral equations and ordinary differential equations according to the fixed points of functions having mixed monotone property.

In Section 2, we shall derive the coincidence point theorems in the product space of mixed-monotonically complete quasi-ordered metric space. Also, in Section 3, the coincidence point theorems in the product space of monotonically complete quasi-ordered metric space will be studied. On the other hand, in Section 4, we shall study the fixed point theorems for the functions having mixed monotone property in the product space of monotonically complete quasi-ordered metric space. Also, in Section 5, the fixed point theorems for the functions having mixed comparable property in the product space of mixed-monotonically complete quasi-ordered metric space will be derived. In Section 6, we shall present the interesting application to investigate the existence and chain-uniqueness of solutions for the system of integral equations. Finally, in Section 7, we shall also present the interesting application to investigate the existence and chain-uniqueness of solutions for the system of ordinary differential equations.

2 Coincidence point theorems in the mixed-monotonically complete quasi-ordered metric space

Let X be a nonempty set. We consider the product set
The element of X m is represented by the vectorial notation x = ( x ( 1 ) , , x ( m ) ) , where x ( i ) X for i = 1 , , m . We also consider the function F : X m X m defined by
F ( x ) = ( F 1 ( x ) , F 2 ( x ) , , F m ( x ) ) ,
where F k : X m X for all k = 1 , 2 , , m . The vectorial element x ˆ = ( x ˆ ( 1 ) , x ˆ ( 2 ) , , x ˆ ( m ) ) X m is a fixed point of F if and only if F ( x ˆ ) = x ˆ ; that is,
F k ( x ˆ ( 1 ) , x ˆ ( 2 ) , , x ˆ ( m ) ) = x ˆ ( k )

for all k = 1 , 2 , , m .

Definition 2.1 Let X be a nonempty set. Consider the functions F : X m X m and f : X m X m by F = ( F 1 , F 2 , , F k ) and f = ( f 1 , f 2 , , f k ) , where F k : X m X and f k : X m X for k = 1 , 2 , , m .

  • ♦ The element x ˆ X m is a coincidence point of F and f if and only if F ( x ˆ ) = f ( x ˆ ) , i.e., F k ( x ˆ ) = f k ( x ˆ ) for all k = 1 , 2 , , m .

  • ♦ The element x ˆ is a common fixed point of F and f if and only if F ( x ˆ ) = f ( x ˆ ) = x ˆ , i.e., F k ( x ˆ ) = f k ( x ˆ ) = x ˆ ( k ) for all k = 1 , 2 , , m .

  • ♦ The functions F and f are said to be commutative if and only if f ( F ( x ) ) = F ( f ( x ) ) for all x X m .

Let ‘’ be a binary relation defined on X. We say that the binary relation ‘’ is a quasi-order (pre-order or pseudo-order) if and only if it is reflexive and transitive. In this case, ( X , ) is called a quasi-ordered set.

For any x , y X m , we say that x and y are -mixed comparable if and only if, for each k = 1 , , m , one has either x ( k ) y ( k ) or y ( k ) x ( k ) . Let I be a subset of { 1 , 2 , , m } and J = { 1 , 2 , , m } I . In this case, we say that I and J are the disjoint pair of { 1 , 2 , , m } . We can define a binary relation on X m as follows:
x I y if and only if x ( k ) y ( k ) for  k I and y ( k ) x ( k ) for  k J .
(1)
It is obvious that ( X m , I ) is a quasi-ordered set that depends on I. We also have
x I y if and only if y J x .
(2)

We need to mention that I or J is allowed to be an empty set.

Remark 2.2 For any x , y X m , we have the following observations.
  1. (a)

    If x I y for some disjoint pair I and J of { 1 , , m } , then x and y are -mixed comparable.

     
  2. (b)

    If x and y are -mixed comparable, then there exists a disjoint pair I and J of { 1 , , m } such that x I y .

     

Definition 2.3 Let I and J be a disjoint pair of { 1 , 2 , , m } . Given a quasi-ordered set ( X , ) , we consider the quasi-ordered set ( X m , I ) defined in (1).

  • ♦ The sequence { x n } n N in X is said to be a mixed -monotone sequence if and only if x n x n + 1 or x n + 1 x n (i.e., x n and x n + 1 are comparable with respect to ‘’) for all n N .

  • ♦ The sequence { x n } n N in X m is said to be a mixed -monotone sequence if and only if each sequence { x n ( k ) } n N in X is a mixed -monotone sequence for all k = 1 , , m .

  • ♦ The sequence { x n } n N in X m is said to be a mixed I -monotone sequence if and only if x n I x n + 1 or x n + 1 I x n (i.e., x n and x n + 1 are comparable with respect to ‘ I ’) for all n N .

Remark 2.4 Let I and J be a disjoint pair of { 1 , 2 , , m } . We have the following observations.
  1. (a)

    { x n } n N in X m is a mixed I -monotone sequence if and only if it is a mixed J -monotone sequence.

     
  2. (b)

    If { x n } n N in X m is a mixed I -monotone sequence, then it is also a mixed -monotone sequence; that is, each sequence { x n ( k ) } n N in X is a mixed -monotone sequence for all k = 1 , , m .

     
  3. (c)

    If { x n } n N in X m is a mixed -monotone sequence, then given any n N , there exists a disjoint pair of I n and J n (which depends on n) of { 1 , , m } such that x n I n x n + 1 or x n + 1 I n x n .

     
  4. (d)

    { x n } n N in X m is a mixed -monotone sequence if and only if, for each n N , x n and x n + 1 are -mixed comparable

     

Definition 2.5 Let I and J be a disjoint pair of { 1 , 2 , , m } . Given a quasi-ordered set ( X , ) , we also consider the quasi-ordered set ( X m , I ) defined in (1) and the function f : ( X m , d ) ( X m , d ) .

  • The function f is said to have the sequentially mixed -monotone property if and only if, given any mixed -monotone sequence { x n } n N in X m , { f ( x n ) } n N is also a mixed -monotone sequence.

  • The function f is said to have the sequentially mixed I -monotone property if and only if, given any mixed I -monotone sequence { x n } n N in X m , { f ( x n ) } n N is also a mixed I -monotone sequence.

It is obvious that the identity function on X m has the sequentially mixed I -monotone and -monotone property.

Let X be a nonempty set. We consider the functions F : X m X m and f : X m X m satisfying F p ( X m ) f ( X m ) for some p N , where F p ( x ) = F ( F p 1 ( x ) ) for any x X m . Therefore, we have F k p ( x ) = F k ( F p 1 ( x ) ) for k = 1 , , m . Given an initial element x 0 = ( x 0 ( 1 ) , x 0 ( 2 ) , , x 0 ( m ) ) X m , where x 0 ( k ) X for k = 1 , , m , since F p ( X m ) f ( X m ) , there exists x 1 X m such that f ( x 1 ) = F p ( x 0 ) . Similarly, there also exists x 2 X m such that f ( x 2 ) = F p ( x 1 ) . Continuing this process, we can construct a sequence { x n } n N such that
f ( x n ) = F p ( x n 1 )
(3)
for all n N ; that is,
f k ( x n ) = f k ( x n ( 1 ) , , x n ( k ) , , x n ( m ) ) = F k p ( x n 1 ( 1 ) , , x n 1 ( k ) , , x n 1 ( m ) ) = F k p ( x n 1 )
for all k = 1 , , m . We introduce the concepts of mixed monotone seed element as follows.
  1. (A)

    We say that the initial element x 0 is a mixed -monotone seed element of X m if and only if the sequence { x n } n N constructed from (3) is a mixed -monotone sequence; that is, each sequence { x n ( k ) } n N in X is a mixed -monotone sequence for k = 1 , , m .

     
  2. (B)

    Given a disjoint pair I and J of { 1 , 2 , , m } , we say that the initial element x 0 is a mixed I -monotone seed element of X m if and only if the sequence { x n } n N constructed from (3) is a mixed I -monotone sequence.

     

From observation (b) of Remark 2.4, it follows that if x 0 is a mixed I -monotone seed element, then it is also a mixed -monotone seed element.

Example 2.6 Suppose that the initial element x 0 can generate a sequence { x n } n N such that, for each k = 1 , , m , the generated sequence { x n ( k ) } n N is either -increasing or -decreasing. In this case, we define the disjoint pair I and J of { 1 , 2 , , m } as follows:
I = { k : the sequence  { x n ( k ) } n N  is  -increasing } and J = { 1 , 2 , , m } I .
(4)

It means that if k J , then the sequence { x n ( k ) } n N is -decreasing. Therefore, the sequence { x n } n N satisfies x n I x n + 1 for any n N . In this case, the initial element x 0 is a mixed I -monotone seed element with the disjoint pair I and J defined in (4).

Definition 2.7 Let ( X , d , ) be a metric space endowed with a quasi-order ‘’. We say that ( X , d , ) is mixed-monotonically complete if and only if each mixed -monotone Cauchy sequence { x n } n N in X is convergent.

It is obvious that if the quasi-ordered metric space ( X , d , ) is complete, then it is also mixed-monotonically complete. However, the converse is not necessarily true.

For the metric space ( X , d ) , we can consider a product metric space ( X m , d ) in which the metric d is induced by the original metric d. For example, the following distance functions
d ( x , y ) = max k = 1 , , m { d ( x ( k ) , y ( k ) ) }
(5)
and
d ( x , y ) = k = 1 m d ( x ( k ) , y ( k ) )
(6)

make ( X m , d ) to be the product metric spaces. For the general product metric d , we consider the following concepts.

  • We say that the metrics d and d are compatible in the sense of preserving convergence if and only if, given a sequence { x n } n N in X m , the following statement holds:
    d ( x n , x ˆ ) 0 if and only if d ( x n ( k ) , x ˆ ( k ) ) 0 for all  k = 1 , , m .
  • We say that the metrics d and d are compatible in the sense of preserving continuity if and only if, given any ϵ > 0 , there exists a positive constant k > 0 (which depends on ϵ) such that the following statement holds:
    d ( x , y ) < ϵ if and only if d ( x ( k ) , y ( k ) ) < k ϵ for all  k = 1 , , m .

We can check that the product metric d defined in (5) or (6) is compatible with d in the sense of preserving convergence and continuity.

Proposition 2.8 If d and d are compatible in the sense of preserving continuity, then d and d are compatible in the sense of preserving convergence.

Proof Suppose that d ( x n , x ˆ ) 0 . By definition, given any ϵ > 0 , there exists n 0 N such that d ( x n , x ˆ ) < ϵ / k for all n n 0 , i.e., d ( x n ( k ) , x ˆ ( k ) ) < ϵ for all k = 1 , , m and n n 0 . For the converse, given any ϵ > 0 , there exist n 0 ( k ) N such that d ( x n ( k ) , x ˆ ( k ) ) < k ϵ for all n n 0 ( k ) , where k = 1 , , m . Let
n 0 = max k = 1 , , m n 0 ( k ) .

It follows that d ( x n ( k ) , x ˆ ( k ) ) < k ϵ for all n n 0 and all k = 1 , , m , i.e., d ( x n , x ˆ ) < ϵ for all n n 0 . This completes the proof. □

Mizoguchi and Takahashi [16, 17] considered the mapping φ : [ 0 , ) [ 0 , 1 ) that satisfies the following condition:
lim sup x c + φ ( x ) < 1 for all  c [ 0 , )
(7)

in the contractive inequality, and generalized Nadler’s fixed point theorem as shown in [18]. Suzuki [19] also gave a simple proof of the theorem obtained by Mizoguchi and Takahashi [16]. In this paper, we consider the following definition.

Definition 2.9 We say that φ : [ 0 , ) [ 0 , 1 ) is a function of contractive factor if and only if, for any strictly decreasing sequence { x n } n N in [ 0 , ) , we have
0 sup n φ ( x n ) < 1 .
(8)

Using the routine arguments, we can show that the function φ : [ 0 , ) [ 0 , 1 ) satisfies (7) if and only if φ is a function of contractive factor. Throughout this paper, we shall assume that the mapping φ satisfies (8) in order to prove the various types of coincidence point theorems in the product space. The following lemma is obvious and useful for further discussion.

Lemma 2.10 Let φ be a function of contractive factor. We define
κ ( t ) = 1 + φ ( t ) 2 .
Then, for any strictly decreasing sequence { x n } n N in [ 0 , ) , we have
0 φ ( t ) < κ ( t ) < 1 for all  t [ 0 , ) and 0 < sup n κ ( x n ) < 1 .

Let ( X , d ) be a metric space, and let F : ( X m , d ) ( X m , d ) be a function defined on ( X m , d ) into itself. If F is continuous at x ˆ X m , then, given ϵ > 0 , there exists δ > 0 such that x X m with d ( x ˆ , x ) < δ implies d ( F ( x ˆ ) , F ( x ) ) < ϵ .

Suppose that d and d are compatible in the sense of preserving continuity. Then F is continuous at x ˆ X m if and only if each F k is continuous at x ˆ for k = 1 , , m . Indeed, it is obvious that if F is continuous at x ˆ X m , then each F k is continuous at x ˆ for k = 1 , , m . For the converse, given any ϵ > 0 , there exists δ k > 0 such that d ( x ˆ , x ) < δ k implies d ( F k ( x ˆ ) , F k ( x ) ) < k ϵ , where k = 1 , , m . Let
δ = min k = 1 , , m δ k .

It follows that d ( x ˆ , x ) < δ implies d ( F k ( x ˆ ) , F k ( x ) ) < k ϵ for all k = 1 , , m , i.e., d ( F ( x ˆ ) , F ( x ) ) < ϵ . Next, we propose another concept of continuity.

Definition 2.11 Let ( X , d ) be a metric space, and let ( X m , d ) be the corresponding product metric space. Let F : ( X m , d ) ( X m , d ) and f : ( X m , d ) ( X m , d ) be functions defined on ( X m , d ) into itself. We say that F is continuous with respect to f at x ˆ X m if and only if, given any ϵ > 0 , there exists δ > 0 such that x X m with d ( x ˆ , f ( x ) ) < δ implies d ( F ( x ˆ ) , F ( x ) ) < ϵ . We say that F is continuous with respect to f on X m if and only if it is continuous with respect to f at each x ˆ X m .

It is obvious that if the function F is continuous at x ˆ with respect to the identity function, then it is also continuous at x ˆ .

Proposition 2.12 Let ( X , d ) be a metric space, and let F : ( X m , d ) ( X m , d ) and f : ( X m , d ) ( X m , d ) be functions defined on ( X m , d ) into itself. Suppose that d and d are compatible in the sense of preserving continuity. Then F is continuous with respect to f at x ˆ X m if and only if, given any ϵ > 0 , there exists δ > 0 such that x X m with d ( x ˆ ( k ) , f k ( x ) ) < δ for all k = 1 , , m implies d ( F k ( x ˆ ) , F k ( x ) ) < ϵ for all k = 1 , , m .

Proof Suppose that F is continuous with respect to f at x ˆ . By definition, given any ϵ > 0 , there exists δ ˆ > 0 such that x X m with d ( x ˆ , f ( x ) ) < δ ˆ implies d ( F ( x ˆ ) , F ( x ) ) < ϵ / k . Let δ = k δ ˆ . It follows that d ( x ˆ ( k ) , f k ( x ) ) < δ for all k = 1 , , m if and only if d ( x ˆ , f ( x ) ) < δ ˆ , which implies d ( F ( x ˆ ) , F ( x ) ) < ϵ / k , i.e., d ( F k ( x ˆ ) , F k ( x ) ) < ϵ for all k = 1 , , m . For the converse, given any ϵ > 0 , there exists δ > 0 such that d ( x ˆ ( k ) , f k ( x ) ) < δ for all k = 1 , , m implies d ( F k ( x ˆ ) , F k ( x ) ) < k ϵ for all k = 1 , , m . Let δ = δ / k . It follows that d ( x ˆ , f ( x ) ) < δ if and only if d ( x ˆ ( k ) , f k ( x ) ) < δ for all k = 1 , , m , which implies d ( F k ( x ˆ ) , F k ( x ) ) < k ϵ for all k = 1 , , m , i.e., d ( F ( x ˆ ) , F ( x ) ) < ϵ . This completes the proof. □

Lemma 2.13 Let ( X , d ) be a metric space. If x n x as n with respect to the metric d, then, given any fixed y X , d ( x n , y ) d ( x , y ) as n .

Theorem 2.14 Suppose that the quasi-ordered metric space ( X , d , ) is mixed-monotonically complete, and that the metrics d and d are compatible in the sense of preserving continuity. Consider the functions F : ( X m , d ) ( X m , d ) and f : ( X m , d ) ( X m , d ) satisfying F p ( X m ) f ( X m ) for some p N . Let x 0 be a mixed -monotone seed element in X m . Assume that the functions F and f satisfy the following conditions:

  • F and f are commutative;

  • f has the sequentially mixed -monotone property;

  • F p is continuous with respect to f on X m ;

  • each f k is continuous on X m for k = 1 , , m .

Suppose that there exist a function ρ : X m × X m R + and a function of contractive factor φ : [ 0 , ) [ 0 , 1 ) such that, for any two -mixed comparable elements x and y in X m , the following inequalities are satisfied:
ρ ( x , y ) k = 1 m d ( x ( k ) , y ( k ) )
(9)
and, for each k = 1 , , m ,
d ( F k p ( x ) , F k p ( y ) ) 1 m φ ( ρ ( f ( x ) , f ( y ) ) ) ρ ( f ( x ) , f ( y ) ) .
(10)

Then F p has a fixed point x ˆ such that each component x ˆ ( k ) of x ˆ is the limit of the sequence { f k ( x n ) } n N constructed in (3) for all k = 1 , , m .

Proof We consider the sequence { x n } n N constructed from (3). Since x 0 is a mixed -monotone seed element in X m , i.e., { x n } n N is a mixed -monotone sequence, from observation (d) of Remark 2.4, it follows that, for each n N , x n and x n + 1 are -mixed comparable. According to inequalities (10), we obtain
d ( f k ( x n + 1 ) , f k ( x n ) ) = d ( F k p ( x n ) , F k P ( x n 1 ) ) 1 m φ ( ρ ( f ( x n ) , f ( x n 1 ) ) ) ρ ( f ( x n ) , f ( x n 1 ) ) .
(11)
Since f has the sequentially mixed -monotone property, we see that { f ( x n ) } n N is a mixed -monotone sequence. From observation (d) of Remark 2.4, it follows that, for each n N , f ( x n ) and f ( x n + 1 ) are -mixed comparable. Then we have
ρ ( f ( x n + 1 ) , f ( x n ) ) k = 1 m d ( f k ( x n + 1 ) , f k ( x n ) ) ( by (9) ) φ ( ρ ( f ( x n ) , f ( x n 1 ) ) ) ρ ( f ( x n ) , f ( x n 1 ) ) ( by (11) ) .
(12)
Let
ξ n = ρ ( f ( x n ) , f ( x n 1 ) ) and κ ( t ) = 1 + φ ( t ) 2 .
Using (11) and Lemma 2.10, we obtain
d ( f k ( x n + 1 ) , f k ( x n ) ) 1 m φ ( ξ n ) ξ n < 1 m κ ( ξ n ) ξ n .
(13)
Using (12) and Lemma 2.10, we also obtain
ξ n + 1 φ ( ξ n ) ξ n < κ ( ξ n ) ξ n .
(14)
Since 0 < γ = sup n κ ( ξ n ) < 1 by Lemma 2.10 again, from (13) and (14), it follows that
d ( f k ( x n + 1 ) , f k ( x n ) ) < γ m ξ n and ξ n + 1 < γ ξ n ,
which implies
d ( f k ( x n + 1 ) , f k ( x n ) ) < γ n m ξ 1 .
(15)
For n 1 , n 2 N with n 1 > n 2 , since 0 < γ < 1 , from (15) we have
d ( f k ( x n 1 ) , f k ( x n 2 ) ) j = n 2 n 1 1 d ( f k ( x j + 1 ) , f k ( x j ) ) < ξ 1 m γ n 2 ( 1 γ n 1 n 2 ) 1 γ < ξ 1 m γ n 2 1 γ 0 as  n 2 ,
which also says that { f k ( x n ) } is a Cauchy sequence in X for any fixed k. Since f has the sequentially mixed -monotone property, i.e., { f k ( x n ) } n N is a mixed -monotone Cauchy sequence for k = 1 , , m , by the mixed -monotone completeness of X, there exists x ˆ ( k ) X such that f k ( x n ) x ˆ ( k ) as n for k = 1 , , m . Since the metrics d and d are compatible in the sense of preserving continuity, by Proposition 2.8, it follows that f ( x n ) x ˆ as n . Since each f k is continuous on X m , we also have
f k ( f ( x n ) ) f k ( x ˆ ) as  n .
Since F p is continuous with respect to f on X m , by Proposition 2.12, given any ϵ > 0 , there exists δ > 0 such that x X m with d ( x ˆ ( k ) , f k ( x ) ) < δ for all k = 1 , , m implies
d ( F k p ( x ˆ ) , F k p ( x ) ) < ϵ 2 for all  k = 1 , , m .
(16)
Since f k ( x n ) x ˆ ( k ) as n for all k = 1 , , m , given ζ = min { ϵ / 2 , δ } > 0 , there exists n 0 N such that
d ( f k ( x n ) , x ˆ ( k ) ) < ζ δ for all  n N  with  n n 0  and for all  k = 1 , , m .
(17)
For each n n 0 , by (16) and (17), it follows that
d ( F k p ( x ˆ ) , F k p ( x n ) ) < ϵ 2 for all  k = 1 , , m .
(18)
Therefore, we obtain
d ( F k p ( x ˆ ) , x ˆ ( k ) ) d ( F k p ( x ˆ ) , f k ( x n 0 + 1 ) ) + d ( f k ( x n 0 + 1 ) , x ˆ ( k ) ) = d ( F k p ( x ˆ ) , F k p ( x n 0 ) ) + d ( f k ( x n 0 + 1 ) , x ˆ ( k ) ) < ϵ 2 + ζ ( by (17) and (18) ) ϵ for all  k = 1 , , m .

Since ϵ is any positive number, we conclude that d ( F k p ( x ˆ ) , x ˆ ( k ) ) = 0 for all k = 1 , , m , which also says that F k p ( x ˆ ) = x ˆ ( k ) for all k = 1 , , m , i.e., F p ( x ˆ ) = x ˆ . This completes the proof. □

Remark 2.15 We have the following observations.

  • In Theorem 2.14, if we assume that the quasi-ordered metric space ( X , d , ) is complete (not mixed-monotonically complete), then the assumption for f having the sequentially mixed -monotone property can be dropped, since the proof is still valid in this case.

  • The assumption for inequalities (9) and (10) is weak since we just assume that it is satisfied for -mixed comparable elements. In other words, if x and y are not -mixed comparable, we do not need to check inequalities (9) and (10).

By considering the mixed I -monotone seed element instead of mixed -monotone seed element, the assumptions for inequalities (9) and (10) can be weakened, which is shown below.

Theorem 2.16 Suppose that the quasi-ordered metric space ( X , d , ) is mixed-monotonically complete, and that the metrics d and d are compatible in the sense of preserving continuity. Let I and J be any disjoint pair of { 1 , 2 , , m } . Consider the functions F : ( X m , d , I ) ( X m , d , I ) and f : ( X m , d , I ) ( X m , d , I ) satisfying F p ( X m ) f ( X m ) for some p N . Let x 0 be a mixed I -monotone seed element in X m . Assume that the functions F and f satisfy the following conditions:

  • F and f are commutative;

  • f has the sequentially mixed I -monotone property or the sequentially mixed -monotone property;

  • F p is continuous with respect to f on X m ;

  • each f k is continuous on X m for k = 1 , , m .

Suppose that there exist a function ρ : X m × X m R + and a function of contractive factor φ : [ 0 , ) [ 0 , 1 ) such that, for any x , y X m with y I x or x I y , the following inequalities are satisfied:
ρ ( x , y ) k = 1 m d ( x ( k ) , y ( k ) )
(19)
and, for each k = 1 , , m ,
d ( F k p ( x ) , F k p ( y ) ) 1 m φ ( ρ ( f ( x ) , f ( y ) ) ) ρ ( f ( x ) , f ( y ) ) .
(20)

Then F p has a fixed point x ˆ such that each component x ˆ ( k ) of x ˆ is the limit of the sequence { f k ( x n ) } n N constructed in (3) for all k = 1 , , m .

Proof We consider the sequence { x n } n N constructed from (3). Since x 0 is a mixed I -monotone seed element in X m , it follows that { x n } n N is a mixed I -monotone sequence, i.e., for each n N , x n 1 I x n or x n I x n 1 . According to inequalities (20), we obtain
d ( f k ( x n + 1 ) , f k ( x n ) ) = d ( F k p ( x n ) , F k P ( x n 1 ) ) 1 m φ ( ρ ( f ( x n ) , f ( x n 1 ) ) ) ρ ( f ( x n ) , f ( x n 1 ) ) .

Using the argument in the proof of Theorem 2.14, we can show that { f k ( x n ) } n N is a Cauchy sequence in X for any fixed k. Now, we consider the following cases.

  • Suppose that f has the sequentially mixed I -monotone property. We see that { f ( x n ) } n N is a mixed I -monotone sequence; that is, for each n N , f ( x n ) I f ( x n + 1 ) or f ( x n + 1 ) I f ( x n ) . Since { f k ( x n ) } n N is a Cauchy sequence in X for any fixed k, from observation (b) of Remark 2.4, we also see that { f k ( x n ) } n N is a mixed -monotone Cauchy sequence for k = 1 , , m .

  • Suppose that f has the sequentially mixed -monotone property. Since { x n } n N is a mixed I -monotone sequence, by part (b) of Remark 2.4, it follows that { x n ( k ) } n N in X is a mixed -monotone sequence for all k = 1 , , m . Therefore, we obtain that { f k ( x n ) } n N is a mixed -monotone Cauchy sequence for k = 1 , , m .

By the mixed -monotone completeness of X, there exists x ˆ ( k ) X such that f k ( x n ) x ˆ ( k ) as n for k = 1 , , m . The remaining proof follows from the same argument in the proof of Theorem 2.14. This completes the proof. □

Remark 2.17 We have the following observations.

  • In Theorem 2.16, if we assume that the quasi-ordered metric space ( X , d , ) is complete (not mixed-monotonically complete), then the assumption for f having the sequentially mixed I -monotone or -monotone property can be dropped, since the proof is still valid in this case.

  • From observation (a) of Remark 2.2, we see that the assumption for inequalities (20) and (19) are indeed weakened by comparing to inequalities (9) and (10).

Next, we shall study the coincidence point without considering the continuity of F p . However, we need to introduce the concept of mixed-monotone convergence given below.

Definition 2.18 Let ( X , d , ) be a metric space endowed with a quasi-order ‘’. We say that ( X , d , ) preserves the mixed-monotone convergence if and only if, for each mixed -monotone sequence { x n } n N that converges to x ˆ , we have x n x ˆ or x ˆ x n for each n N .

Remark 2.19 Let ( X , d , ) be a metric space endowed with a quasi-order ‘’ and preserve the mixed-monotone convergence. Suppose that { x n } n N is a sequence in the product space X m such that each sequence { x n ( k ) } n N is a mixed -monotone convergence sequence with limit point x ˆ ( k ) for k = 1 , , m . Then we have the following observations.
  1. (a)

    For each n N , x n and x ˆ are -mixed comparable.

     
  2. (b)

    For each n N , there exists a disjoint pair I n and J n (that depend on n) of { 1 , , m } such that x n I n x ˆ or x ˆ I n x n , where I n or J n is allowed to be an empty set.

     

Definition 2.20 Let I and J be a disjoint pair of { 1 , 2 , , m } . Given a quasi-ordered set ( X , ) , we also consider the quasi-ordered set ( X m , I ) defined in (1), and the function f : ( X m , I ) ( X m , I ) .

  • We say that the function f has the -comparable property if and only if, given any two -comparable elements x and y in X m , the function values f ( x ) and f ( y ) are -comparable.

  • We say that the function f has the I -comparable property if and only if, given any two I -comparable elements x and y in X m , the function values f ( x ) and f ( y ) are I -comparable.

Since we shall study the coincidence point without considering the continuity of F p , we can also consider the assumption that the metrics d and d are compatible in the sense of preserving convergence, which is weaker than that of preserving continuity considered in the previous theorems.

Theorem 2.21 Suppose that the quasi-ordered metric space ( X , d , ) is mixed-monotonically complete and preserves the mixed-monotone convergence. Assume that the metrics d and d are compatible in the sense of preserving convergence. Consider the functions F : ( X m , d ) ( X m , d ) and f : ( X m , d ) ( X m , d ) satisfying F p ( X m ) f ( X m ) for some p N . Let x 0 be a mixed -monotone seed element in X m . Assume that the functions F and f satisfy the following conditions:

  • F and f are commutative;

  • f has the -comparable property and the sequentially mixed -monotone property;

  • each f k is continuous on X m for k = 1 , , m .

Suppose that there exist a function ρ : X m × X m R + and a function of contractive factor φ : [ 0 , ) [ 0 , 1 ) such that, for any two -mixed comparable elements x and y in X m , the following inequalities are satisfied:
ρ ( x , y ) k = 1 m d ( x ( k ) , y ( k ) )
(21)
and, for each k = 1 , , m ,
d ( F k p ( x ) , F k p ( y ) ) 1 m φ ( ρ ( f ( x ) , f ( y ) ) ) ρ ( f ( x ) , f ( y ) ) .
(22)
Then the following statements hold true.
  1. (i)

    There exists x ˆ X m of F such that F p ( x ˆ ) = f ( x ˆ ) . If p = 1 , then x ˆ is a coincidence point of F and f.

     
  2. (ii)

    If there exists y ˆ X m such that x ˆ and y ˆ are -mixed comparable satisfying F p ( y ˆ ) = f ( y ˆ ) , then f ( x ˆ ) = f ( y ˆ ) .

     
  3. (iii)

    Suppose that x ˆ is obtained from part (i). If x ˆ and F ( x ˆ ) are -mixed comparable, then f q ( x ˆ ) is a fixed point of F for any q N .

     

Moreover, each component x ˆ ( k ) of x ˆ is the limit of the sequence { f k ( x n ) } n N constructed in (3) for all k = 1 , , m .

Proof From the proof of Theorem 2.14, we can construct a sequence { x n } n N in X m such that f k ( x n ) x ˆ ( k ) and f k ( f ( x n ) ) f k ( x ˆ ) as n , where { f k ( x n ) } n N is a mixed -monotone sequence for all k = 1 , , m . Since f k ( f ( x n ) ) f k ( x ˆ ) as n , given any ϵ > 0 , there exists n 0 N such that
d ( f k ( f ( x n ) ) , f k ( x ˆ ) ) < ϵ 2
(23)
for all n N with n n 0 and for all k = 1 , , m . Since { f k ( x n ) } n N is a mixed -monotone convergent sequence for all k = 1 , , m , from observation (a) of Remark 2.19, we see that, for each n N , f ( x n ) and x ˆ are -mixed comparable. Since f has the -comparable property, it follows that f ( f ( x n ) ) and f ( x ˆ ) are -mixed comparable. For each n n 0 , it follows that
d ( F k p ( x ˆ ) , F k p ( f ( x n ) ) ) 1 m φ ( ρ ( f ( x ˆ ) , f ( f ( x n ) ) ) ) ρ ( f ( x ˆ ) , f ( f ( x n ) ) ) ( by (22) ) < 1 m ρ ( f ( x ˆ ) , f ( f ( x n ) ) ) 1 m k = 1 m d ( f k ( x ˆ ) , f k ( f ( x n ) ) ) ( by (21) ) < ϵ 2 ( by (23) ) .
(24)
Since F and f are commutative, we have f ( F p ( x ) ) = F p ( f ( x ) ) for all x X m , which also implies
f k ( f ( x n ) ) = f k ( F p ( x n 1 ) ) = F k p ( f ( x n 1 ) ) .
Now, we obtain
d ( F k p ( x ˆ ) , f k ( x ˆ ) ) d ( F k p ( x ˆ ) , F k p ( f ( x n 0 ) ) ) + d ( F k p ( f ( x n 0 ) ) , f k ( x ˆ ) ) = d ( F k p ( x ˆ ) , F k p ( f ( x n 0 ) ) ) + d ( f k ( f ( x n 0 + 1 ) ) , f k ( x ˆ ) ) < ϵ 2 + ϵ 2 ( by (23) and (24) ) = ϵ .

Since ϵ is any positive number, we conclude that d ( F k p ( x ˆ ) , f k ( x ˆ ) ) = 0 , which says that F k p ( x ˆ ) = f k ( x ˆ ) for all k = 1 , , m , i.e., F p ( x ˆ ) = f ( x ˆ ) . This proves part (i).

To prove part (ii), since f has the -comparable property, it follows that f ( x ˆ ) and f ( y ˆ ) are -mixed comparable. If f k ( x ˆ ) f k ( y ˆ ) , i.e., d ( f k ( x ˆ ) , f k ( y ˆ ) ) 0 for some k, then we obtain
0 k = 1 m d ( f k ( x ˆ ) , f k ( y ˆ ) ) = k = 1 m d ( F k p ( x ˆ ) , F k p ( y ˆ ) ) 1 m k = 1 m φ ( ρ ( f ( x ˆ ) , f ( y ˆ ) ) ) ρ ( f ( x ˆ ) , f ( y ˆ ) ) ( by (22) ) < ρ ( f ( x ˆ ) , f ( y ˆ ) ) k = 1 m d ( f k ( x ˆ ) , f k ( y ˆ ) ) ( by (21) ) .

This contradiction says that f k ( x ˆ ) = f k ( y ˆ ) for all k = 1 , , m , i.e., f ( x ˆ ) = f ( y ˆ ) .

To prove part (iii), using the commutativity of F and f, we have
f ( F ( x ˆ ) ) = F ( f ( x ˆ ) ) = F ( F p ( x ˆ ) ) = F p ( F ( x ˆ ) ) .
(25)
By taking y ˆ = F ( x ˆ ) , equalities (25) say that f ( y ˆ ) = F p ( y ˆ ) . Since x ˆ and y ˆ = F ( x ˆ ) are -mixed comparable by the assumption, part (ii) says that
f ( x ˆ ) = f ( y ˆ ) = f ( F ( x ˆ ) ) = F ( f ( x ˆ ) ) ,
which says that f ( x ˆ ) is a fixed point of F. Given any q N , we have
F ( f q ( x ˆ ) ) = f q 1 ( F ( f ( x ˆ ) ) ) ( by the commutativity of  F  and  f ) = f q 1 ( f ( x ˆ ) ) = f q ( x ˆ ) ,

which says that f q ( x ˆ ) is a fixed point of F. This completes the proof. □

Theorem 2.22 Suppose that the quasi-ordered metric space ( X , d , ) is mixed-monotonically complete and preserves the mixed-monotone convergence. Assume that the metrics d and d are compatible in the sense of preserving convergence. Let I and J be any disjoint pair of { 1 , 2 , , m } . Consider the functions F : ( X m , d , I ) ( X m , d , I ) and f : ( X m , d , I ) ( X m , d , I ) satisfying F p ( X m ) f ( X m ) for some p N . Let x 0 be a mixed I -monotone seed element in X m . Assume that the functions F and f satisfy the following conditions:

  • F and f are commutative;

  • f has the sequentially mixed I -monotone property or the sequentially mixed -monotone property;

  • f has the I -comparable property for any disjoint pair I and J of { 1 , , m } ;

  • each f k is continuous on X m for k = 1 , , m .

Suppose that there exist a function ρ : X m × X m R + and a function of contractive factor φ : [ 0 , ) [ 0 , 1 ) such that, for any x , y X m and any disjoint pair I and J of { 1 , , m } with y I x or x I y , the following inequalities are satisfied:
ρ ( x , y ) k = 1 m d ( x ( k ) , y ( k ) )
(26)
and, for each k = 1 , , m ,
d ( F k p ( x ) , F k p ( y ) ) 1 m φ ( ρ ( f ( x ) , f ( y ) ) ) ρ ( f ( x ) , f ( y ) ) .
(27)
Then the following statements hold true.
  1. (i)

    There exists x ˆ X m of F such that F p ( x ˆ ) = f ( x ˆ ) . If p = 1 , then x ˆ is a coincidence point of F and f.

     
  2. (ii)

    If there exist a disjoint pair I and J of { 1 , , m } and y ˆ X m such that F p ( y ˆ ) = f ( y ˆ ) and that x ˆ and y ˆ are comparable with respect to the quasi-order I ’, then f ( x ˆ ) = f ( y ˆ ) .

     
  3. (iii)

    Suppose that x ˆ is obtained from part (i). If there exists a disjoint pair I and J of { 1 , , m } such that x ˆ and F ( x ˆ ) are comparable with respect to the quasi-order I ’, then f q ( x ˆ ) is a fixed point of F for any q N .

     

Moreover, each component x ˆ ( k ) of x ˆ is the limit of the sequence { f k ( x n ) } n N constructed in (3) for all k = 1 , , m .

Proof From the proof of Theorem 2.16, we can construct a sequence { x n } n N in X m such that f k ( x n ) x ˆ ( k ) and f k ( f ( x n ) ) f k ( x ˆ ) as n , where { f k ( x n ) } n N is a mixed -monotone sequence for all k = 1 , , m . Since f k ( f ( x n ) ) f k ( x ˆ ) as n , given any ϵ > 0 , there exists n 0 N such that
d ( f k ( f ( x n ) ) , f k ( x ˆ ) ) < ϵ 2
(28)
for all n N with n n 0 and for all k = 1 , , m . Since { f k ( x n ) } n N is a mixed -monotone convergent sequence for all k = 1 , , m , from observation (b) of Remark 2.19, we see that, for each n N , there exists a subset I n of { 1 , , m } such that
f ( x n ) I n x ˆ or x ˆ I n f ( x n ) .
(29)
Since f has the I -comparable property for any subset I of { 1 , , m } , it follows that
f ( f ( x n ) ) I n f ( x ˆ ) or f ( x ˆ ) I n f ( f ( x n ) ) .
(30)
For each n n 0 , we obtain
d ( F k p ( x ˆ ) , F k p ( f ( x n ) ) ) 1 m φ ( ρ ( f ( x ˆ ) , f ( f ( x n ) ) ) ) ρ ( f ( x ˆ ) , f ( f ( x n ) ) ) ( by (29) and (27) ) < 1 m ρ ( f ( x ˆ ) , f ( f ( x n ) ) ) 1 m k = 1 m d ( f k ( x ˆ ) , f k ( f ( x n ) ) ) ( by (30) and (26) ) < ϵ 2 ( by (28) ) .

The remaining proof follows from a similar argument in the proof of Theorem 2.21, and the proof is complete. □

Remark 2.23 Suppose that inequalities (21) and (22) in Theorem 2.21, and that inequalities (26) and (27) in Theorem 2.22 are satisfied for any x , y X m . Then, from the proofs of Theorems 2.21 and 2.22, we can see that parts (ii) and (iii) can be changed as follows.

(ii)′ If there exists y ˆ X m such that F p ( y ˆ ) = f ( y ˆ ) , then f ( x ˆ ) = f ( y ˆ ) .

(iii)′ Suppose that x ˆ is obtained from part (i). Then f q ( x ˆ ) is a fixed point of F for any q N .

The assumption that f has the I -comparable property for any disjoint pair I and J of { 1 , , m } in Theorem 2.22 can be dropped by strengthening inequalities (26) as shown below.

Theorem 2.24 Suppose that the quasi-ordered metric space ( X , d , ) is mixed-monotonically complete and preserves the mixed-monotone convergence. Assume that the metrics d and d are compatible in the sense of preserving convergence. Consider the functions F : ( X m , d , I ) ( X m , d , I ) and f : ( X m , d , I ) ( X m , d , I ) satisfying F p ( X m ) f ( X m ) for some p N . Let x 0 be a mixed I -monotone seed element in X m . Assume that the functions F and f satisfy the following conditions:

  • F and f are commutative;

  • f has the sequentially mixed I -monotone property or the sequentially mixed -monotone property;

  • each f k is continuous on X m for k = 1 , , m .

Suppose that there exists a function ρ : X m × X m R + such that, for any x , y X m , the following inequality is satisfied:
ρ ( x , y ) k = 1 m d ( x ( k ) , y ( k ) ) ,
(31)
and that there exists a function of contractive factor φ : [ 0 , ) [ 0 , 1 ) such that, for any x , y X m and any disjoint pair I and J of { 1 , , m } with y I x or x I y , the following inequality is satisfied:
d ( F k p ( x ) , F k p ( y ) ) 1 m φ ( ρ ( f ( x ) , f ( y ) ) ) ρ ( f ( x ) , f ( y ) ) for each  k = 1 , , m .
Then the following statements hold true.
  1. (i)

    There exists x ˆ X m of F such that F p ( x ˆ ) = f ( x ˆ ) . If p = 1 , then x ˆ is a coincidence point of F and f.

     
  2. (ii)

    If there exist a disjoint pair I and J of { 1 , , m } and y ˆ X m such that F p ( y ˆ ) = f ( y ˆ ) and that x ˆ and y ˆ are comparable with respect to the quasi-order I ’, then f ( x ˆ ) = f ( y ˆ ) .

     
  3. (iii)

    Suppose that x ˆ is obtained from part (i). If there exists a disjoint pair I and J of { 1 , , m } such that x ˆ and F ( x ˆ ) are comparable with respect to the quasi-order I ’, then f q ( x ˆ ) is a fixed point of F for any q N .

     

Moreover, each component x ˆ ( k ) of x ˆ is the limit of the sequence { f k ( x n ) } n N constructed in (3) for all k = 1 , , m .

Proof Since inequalities (31) are satisfied for any x and y, the arguments in the proof of Theorem 2.22 are still valid without considering (30). This completes the proof. □

Next, we shall consider the uniqueness for a common fixed point in the -mixed comparable sense.

Definition 2.25 Let ( X , ) be a quasi-ordered set. Consider the functions F : X m X m and f : X m X m defined on the product set X m into itself. The common fixed point x ˆ X m of F and f is unique in the -mixed comparable sense if and only if, for any other common fixed point x of F and f, if x and x ˆ are -mixed comparable, then x = x ˆ .

Theorem 2.26 Suppose that the quasi-ordered metric space ( X , d , ) is mixed-monotonically complete and preserves the mixed-monotone convergence. Assume that the metrics d and d are compatible in the sense of preserving continuity. Consider the functions F : ( X m , d ) ( X m , d ) and f : ( X m , d ) ( X m , d ) satisfying F p ( X m ) f ( X m ) for some p N . Let x 0 be a mixed -monotone seed element in X m . Assume that the functions F and f satisfy the following conditions:

  • F and f are commutative;

  • f has the -comparable property and the sequentially mixed -monotone property;

  • F p is continuous with respect to f on X m ;

  • each f k is continuous on X m for k = 1 , , m .

Suppose that there exist a function ρ : X m × X m R + and a function of contractive factor φ : [ 0 , ) [ 0 , 1 ) such that, for any two -mixed comparable elements x and y in X m , the following inequalities are satisfied:
ρ ( x , y ) k = 1 m d ( x ( k ) , y ( k ) )
(32)
and, for each k = 1 , , m ,
d ( F k p ( x ) , F k p ( y ) ) 1 m φ ( ρ ( f ( x ) , f ( y ) ) ) ρ ( f ( x ) , f ( y ) ) .
(33)
Then the following statements hold true.
  1. (i)

    F p and f have a unique common fixed point x ˆ in the -mixed comparable sense. Equivalently, if y ˆ is another common fixed point of F p and f, and is -mixed comparable with x ˆ , then y ˆ = x ˆ .

     
  2. (ii)

    For p 1 , suppose that F ( x ˆ ) and x ˆ obtained in (i) are -mixed comparable. Then F and f have a unique common fixed point x ˆ in the -mixed comparable sense.

     

Moreover, each component x ˆ ( k ) of x ˆ is the limit of the sequence { f k ( x n ) } n N constructed in (3) for all k = 1 , , m .

Proof To prove part (i), from Proposition 2.8 and part (i) of Theorem 2.21, we have f ( x ˆ ) = F p ( x ˆ ) . From Theorem 2.14, we also have F p ( x ˆ ) = x ˆ . Therefore, we obtain
x ˆ = f ( x ˆ ) = F p ( x ˆ ) .
This shows that x ˆ is a common fixed point of F p and f. For the uniqueness in the -mixed comparable sense, let y ˆ be another common fixed point of F p and f such that y ˆ and x ˆ are -mixed comparable, i.e., y ˆ = f ( y ˆ ) = F p ( y ˆ ) . By part (ii) of Theorem 2.21, we have f ( x ˆ ) = f ( y ˆ ) . Therefore, by the triangle inequality, we obtain
d ( x ˆ , y ˆ ) d ( x ˆ , f ( x ˆ ) ) + d ( f ( x ˆ ) , f ( y ˆ ) ) + d ( f ( y ˆ ) , y ˆ ) = 0 ,
(34)

which says that x ˆ = y ˆ . This proves part (i).

To prove part (ii), since F ( x ˆ ) and x ˆ are -mixed comparable, part (iii) of Theorem 2.21 says that f ( x ˆ ) is a fixed point of F, i.e., f ( x ˆ ) = F ( f ( x ˆ ) ) , which implies x ˆ = F ( x ˆ ) , since x ˆ = f ( x ˆ ) . This shows that x ˆ is a common fixed point of F and f. For the uniqueness in the -mixed comparable sense, let y ˆ be another common fixed point of F and f such that y ˆ and x ˆ are -mixed comparable, i.e., y ˆ = f ( y ˆ ) = F ( y ˆ ) . Then we have
y ˆ = f ( y ˆ ) = F ( y ˆ ) = F ( f ( y ˆ ) ) = F 2 ( y ˆ ) = = F p ( y ˆ ) .

By part (ii) of Theorem 2.21, we have f ( x ˆ ) = f ( y ˆ ) . From (34), we can similarly obtain x ˆ = y ˆ . This completes the proof. □

Since we consider a metric space ( X , d , ) endowed with a quasi-order ‘’, given any disjoint pair I and J of { 1 , , p } , we can define a quasi-order ‘ I ’ on X m as given in (1). Now, given any x X m , we define the chain C ( I , x ) containing x as follows:
C ( I , x ) = { y X m : y I x  or  x I y } = { y X m : x  and  y  are comparable with respect to ‘ I } .

Next, we shall introduce the concept of chain-uniqueness for a common fixed point.

Definition 2.27 Let ( X , ) be a quasi-ordered set. Consider the functions F : X m X m and f : X m X m defined on the product set X m into itself. The common fixed point x ˆ X m of F and f is called chain-unique if and only if, given any other common fixed point x of F and f, if x C ( I , x ˆ ) for some disjoint pair I and J of { 1 , , m } , then x = x ˆ .

Theorem 2.28 Suppose that the quasi-ordered metric space ( X , d , ) is mixed-monotonically complete and preserves the mixed-monotone convergence. Assume that the metrics d and d are compatible in the sense of preserving continuity. Consider the functions F : ( X m , d , I ) ( X m , d , I ) and f : ( X m , d , I ) ( X m , d , I ) satisfying F p ( X m ) f ( X m ) for some p N . Let x 0 be a mixed I -monotone seed element in X m . Assume that the functions F and f satisfy the following conditions:

  • F and f are commutative;

  • f has the sequentially mixed I -monotone property or the sequentially mixed -monotone property;

  • F p is continuous with respect to f on X m ;

  • each f k is continuous on X m for k = 1 , , m .

Suppose that there exist a function ρ : X m × X m R + and a function of contractive factor φ : [ 0 , ) [ 0 , 1 ) such that, for any x , y X m and any disjoint pair I and J of { 1 , , m } with y I x or x I y , the following inequalities are satisfied:
ρ ( x , y ) k = 1 m d ( x ( k ) , y ( k ) )
(35)
and, for each k = 1 , , m ,
d ( F k p ( x ) , F k p ( y ) ) 1 m φ ( ρ ( f ( x ) , f ( y ) ) ) ρ ( f ( x ) , f ( y ) ) .
(36)
Then the following statements hold true.
  1. (i)

    F p and f have a chain-unique common fixed point x ˆ . Equivalently, if y ˆ C ( I , x ˆ ) is another common fixed point of F p and f for some disjoint pair I and J of { 1 , , m } , then y ˆ = x ˆ .

     
  2. (ii)

    For p 1 , suppose that F ( x ˆ ) and x ˆ obtained in (i) are comparable with respect to the quasi-order I for some disjoint pair I and J of { 1 , , m } . Then F and f have a chain-unique common fixed point x ˆ .

     

Moreover, each component x ˆ ( k ) of x ˆ is the limit of the sequence { f k ( x n ) } n N constructed in (3) for all k = 1 , , m .

Proof To prove part (i), from Proposition 2.8 and part (i) of Theorem 2.22, we can show that x ˆ is a common fixed point of F p and f. For the chain-uniqueness, let y ˆ be another common fixed point of F p and f with y ˆ I x ˆ or x ˆ I y ˆ for some disjoint pair I and J of { 1 , , m } , i.e., y ˆ = f ( y ˆ ) = F p ( y ˆ ) . By part (ii) of Theorem 2.22, we have f ( x ˆ ) = f ( y ˆ ) . Therefore, according to (34), we can obtain x ˆ = y ˆ . This proves part (i). Part (ii) can be similarly obtained by applying Theorem 2.22 to the argument in the proof of part (ii) of Theorem 2.26. This completes the proof. □

Remark 2.29 We strongly assume that inequalities (32) and (33) in Theorem 2.26, and that inequalities (35) and (36) in Theorem 2.28 are satisfied for any x , y X m . Then, from Remark 2.23 and the proofs of Theorems 2.26 and 2.28, it follows that parts (i) and (ii) can be combined together to conclude that F and f have a unique common fixed point x ˆ .

3 Coincidence point theorems in the monotonically complete quasi-ordered metric space

Now, we are going to weaken the concept of mixed-monotone completeness for the quasi-ordered metric space. Let ( X , d , ) be a metric space endowed with a quasi-order ‘’. We say that the sequence { x n } n N in ( X , ) is -increasing if and only if x k x k + 1 for all k N . The concept of -decreasing sequence can be similarly defined. The sequence { x n } n N in ( X , ) is called -monotone if and only if { x n } n N is either -increasing or -decreasing.

Let I and J be a disjoint pair of { 1 , 2 , , m } . We say that the sequence { x n } n N in ( X m , I ) is I -increasing if and only if x n I x n + 1 for all n N . The concept of I -decreasing sequence can be similarly defined. The sequence { x n } n N in ( X m , I ) is called I -monotone if and only if { x n } n N is either I -increasing or I -decreasing.

Given a disjoint pair I and J of { 1 , 2 , , m } , let f : ( X m , I ) ( X m , I ) be a function defined on ( X m , I ) into itself. We say that f is I -increasing if and only if x I y implies f ( x ) I f ( y ) . The concept of I -decreasing function can be similarly defined. The function f is called I -monotone if and only if f is either I -increasing or I -decreasing.

In the previous section, we consider the mixed I -monotone seed element. Now, we shall consider another concept of seed element. Given a disjoint pair I and J of { 1 , 2 , , m } , we say that the initial element x 0 is a I -monotone seed element of X m if and only if the sequence { x n } n N constructed from (3) is a I -monotone sequence. It is obvious that if x 0 is a I -monotone seed element, then it is also a mixed I -monotone seed element.

Definition 3.1 Let ( X , d , ) be a metric space endowed with a quasi-order ‘’. We say that ( X , d , ) is monotonically complete if and only if each -monotone Cauchy sequence { x n } n N in X is convergent.

It is obvious that if ( X , d , ) is a mixed-monotonically complete quasi-ordered metric space, then it is also a monotonically complete quasi-ordered metric space. However, the converse is not true. In other words, the concept of monotone completeness is weaker than that of mixed-monotone completeness.

Theorem 3.2 Suppose that the quasi-ordered metric space ( X , d , ) is monotonically complete, and that the metrics d and d are compatible in the sense of preserving continuity. Consider the functions F : ( X m , d , I ) ( X m , d , I ) and f : ( X m , d , I ) ( X m , d , I ) satisfying F p ( X m ) f ( X m ) for some p N . Let x 0 be a I -monotone seed element in X m . Assume that the functions F and f satisfy the following conditions:

  • F and f are commutative;

  • f is I -monotone;

  • F p is continuous with respect to f on X m ;

  • each f k is continuous on X m for k = 1 , , m .

Suppose that there exist a function ρ : X m × X m R + and a function of contractive factor φ : [ 0 , ) [ 0 , 1 ) such that, for any x , y X m with y I x or x I y , the following inequalities
ρ ( x , y ) k = 1 m d ( x ( k ) , y ( k ) )
(37)
and
d ( F k p ( x ) , F k p ( y ) ) 1 m φ ( ρ ( f ( x ) , f ( y ) ) ) ρ ( f ( x ) , f ( y ) )
(38)

are satisfied for all k = 1 , , m . Then F p has a fixed point x ˆ such that each component x ˆ ( k ) of x ˆ is the limit of the sequence { f k ( x n ) } n N constructed in (3) for all k = 1 , , m .

Proof We consider the sequence { x n } n N constructed from (3). Since x 0 is a I -monotone seed element in X m , i.e., x n I x n + 1 for all n N or x n + 1 I x n for all n N , according to inequalities (38), we obtain
d ( f k ( x n + 1 ) , f k ( x n ) ) = d ( F k p ( x n ) , F k P ( x n 1 ) ) 1 m φ ( ρ ( f ( x n ) , f ( x n 1 ) ) ) ρ ( f ( x n ) , f ( x n 1 ) ) .
(39)
Since f is I -monotone, it follows that f ( x n ) I f ( x n + 1 ) for all n N or f ( x n + 1 ) I f ( x n ) for all n N . Then we have
ρ ( f ( x n + 1 ) , f ( x n ) ) k = 1 m d ( f k ( x n + 1 ) , f k ( x n ) ) ( by (37) ) φ ( ρ ( f ( x n ) , f ( x n 1 ) ) ) ρ ( f ( x n ) , f ( x n 1 ) ) ( by (39) ) .

According to the proof of Theorem 2.14, we can show that { f k ( x n ) } n N is a Cauchy sequence in X for any fixed k = 1 , , n . Since f is I -monotone and { x n } n N is a I -monotone sequence, it follows that { f ( x n ) } n N is a I -monotone sequence.

  • If { f ( x n ) } n N is a I -increasing sequence, then { f k ( x n ) } n N is a -increasing Cauchy sequence for k I , and is a -decreasing Cauchy sequence for k J .

  • If { f ( x n ) } n N is a I -decreasing sequence, then { f k ( x n ) } n N is a -decreasing Cauchy sequence for k I , and is a -increasing Cauchy sequence for k J .

By the monotone completeness of X, there exists x ˆ ( k ) X such that f k ( x n ) x ˆ ( k ) as n for k = 1 , , m . The remaining proof follows from the same argument in the proof of Theorem 2.14. This completes the proof. □

Next, we shall study the coincidence point without considering the continuity of F p . However, we need to introduce the concept of monotone convergence given below.

Definition 3.3 Let ( X , d , ) be a metric space endowed with a quasi-order ‘’. We say that ( X , d , ) preserves the monotone convergence if and only if, for each -monotone sequence { x n } n N that converges to x ˆ , either one of the following conditions is satisfied:

  • if { x n } n N is a -increasing sequence, then x n x ˆ for each n N ;

  • if { x n } n N is a -decreasing sequence, then x ˆ x n for each n N .

Remark 3.4 Let ( X , d , ) be a metric space endowed with a quasi-order ‘’ and preserve the monotone convergence. Given a disjoint pair I and J of { 1 , , m } , suppose that { x n } n N is a I -monotone sequence such that each sequence { x n ( k ) } n N converges to x ˆ ( k ) for k = 1 , , m . We consider the following situation.

  • If { x n } n N is a I -increasing sequence, then { x n ( k ) } n N is a -increasing sequence for k I , and is a -decreasing sequence for k J . By the monotone convergence, we see that, for each n N , x n ( k ) x ˆ ( k ) for k I and x n ( k ) x ˆ ( k ) for k J , which shows that x n I x ˆ for all n N .

  • If { x n } n N is a I -decreasing sequence, then { x n ( k ) } n N is a -decreasing sequence for k I , and is a -increasing sequence for k J . By the monotone convergence, we see that, for each n N , x n ( k ) x ˆ ( k ) for k I and x n ( k ) x ˆ ( k ) for k J , which shows that x n I x ˆ for all n N .

Therefore, we conclude that x n and x ˆ are comparable with respect to ‘ I ’ for all n N .

Theorem 3.5 Suppose that the quasi-ordered metric space ( X , d , ) is monotonically complete and preserves the monotone convergence. Assume that the metrics d and d are compatible in the sense of preserving convergence. Consider the functions F : ( X m , d , I ) ( X m , d , I ) and f : ( X m , d , I ) ( X m , d , I ) satisfying F p ( X m ) f ( X m ) for some p N . Let x 0 be a I -monotone seed element in X m . Assume that the functions F and f satisfy the following conditions:

  • F and f are commutative;

  • f is I -monotone;

  • each f k is continuous on X m for k = 1 , , m .

Suppose that there exist a function ρ : X m × X m R + and a function of contractive factor φ : [ 0 , ) [ 0 , 1 ) such that, for any x , y X m with y I x or x I y , the following inequalities
ρ ( x , y ) k = 1 m d ( x ( k ) , y ( k ) )
(40)
and
d ( F k p ( x ) , F k p ( y ) ) 1 m φ ( ρ ( f ( x ) , f ( y ) ) ) ρ ( f ( x ) , f ( y ) )
(41)
are satisfied for all k = 1 , , m . Then the following statements hold true.
  1. (i)

    There exists x ˆ X m of F such that F p ( x ˆ ) = f ( x ˆ ) . If p = 1 , then x ˆ is a coincidence point of F and f.

     
  2. (ii)

    If there exists y ˆ X m such that F p ( y ˆ ) = f ( y ˆ ) with x ˆ I y ˆ or y ˆ I x ˆ , then f ( x ˆ ) = f ( y ˆ ) .

     
  3. (iii)

    Suppose that x ˆ is obtained from part (i). If x ˆ and F ( x ˆ ) are comparable with respect to I ’, then f q ( x ˆ ) is a fixed point of F for any q N .

     

Moreover, each component x ˆ ( k ) of x ˆ is the limit of the sequence { f k ( x n ) } n N constructed in (3) for all k = 1 , , m .

Proof From the proof of Theorem 3.2, we can construct a sequence { x n } n N in X m such that f k ( x n ) x ˆ ( k ) and f k ( f ( x n ) ) f k ( x ˆ ) as n for all k = 1 , , m , where { f ( x n ) } n N is a I -monotone sequence. From Remark 3.4, it follows that, for each n N , f ( x n ) I x ˆ or f ( x n ) I x ˆ . Since f k ( f ( x n ) ) f k ( x ˆ ) as n , given any ϵ > 0 , there exists n 0 N such that
d ( f k ( f ( x n ) ) , f k ( x ˆ ) ) < ϵ 2
(42)
for all n N with n n 0 and for all k = 1 , , m . Since f is I -monotone, it follows that f ( f ( x n ) ) I f ( x ˆ ) or f ( f ( x n ) ) I f ( x ˆ ) . For each n n 0 , it follows that
d ( F k p ( x ˆ ) , F k p ( f ( x n ) ) ) 1 m φ ( ρ ( f ( x ˆ ) , f ( f ( x n ) ) ) ) ρ ( f ( x ˆ ) , f ( f ( x n ) ) ) ( by (41) ) < 1 m ρ ( f ( x ˆ ) , f ( f ( x n ) ) ) 1 m k = 1 m d ( f k ( x ˆ ) , f k ( f ( x n ) ) ) ( by (40) ) < ϵ 2 ( by (42) ) .

Using the same argument in the proof of part (i) of Theorem 2.21, part (i) of this theorem follows immediately.

To prove part (ii), since f is I -monotone, we immediately have f ( y ) I f ( x ) or f ( x ) I f ( y ) . If f k ( x ˆ ) f k ( y ˆ ) , i.e., d ( f k ( x ˆ ) , f k ( y ˆ ) ) 0 , then we obtain
0 k = 1 m d ( f k ( x ˆ ) , f k ( y ˆ ) ) = k = 1 m d ( F k p ( x ˆ ) , F k p ( y ˆ ) ) 1 m k = 1 m φ ( ρ ( f ( x ˆ ) , f ( y ˆ ) ) ) ρ ( f ( x ˆ ) , f ( y ˆ ) ) ( by (41) ) < ρ ( f ( x ˆ ) , f ( y ˆ ) ) k = 1 m d ( f k ( x ˆ ) , f k ( y ˆ ) ) ( by (40) ) .

This contradiction says that f k ( x ˆ ) = f k ( y ˆ ) for all k = 1 , , m , i.e., f ( x ˆ ) = f ( y ˆ ) . Finally, part (iii) follows from the same argument in the proof of part (iii) of Theorem 2.21 immediately. This completes the proof. □

Remark 3.6 Suppose that inequalities (40) and (41) in Theorem 3.5 are satisfied for any x , y X m . Then, from the proof of Theorem 3.5, we can see that parts (ii) and (iii) can be changed as follows.

(ii)′ If there exists y ˆ X m satisfying F p ( y ˆ ) = f ( y ˆ ) , then f ( x ˆ ) = f ( y ˆ ) .

(iii)′ Suppose that x ˆ is obtained from part (i). Then f q ( x ˆ ) is a fixed point of F for any q N .

Next, we shall study the I -chain-uniqueness for the common fixed point, which is different from the chain-uniqueness in Definition 2.27.

Definition 3.7 Let ( X , ) be a quasi-ordered set. Consider the functions F : X m X m and f : X m X m defined on the product set X m into itself. Given a disjoint pair I and J of { 1 , , m } , we recall that the chain C ( I , x ) containing x is given by
C ( I , x ) = { y X m : y I x  or  x I y } .

The common fixed point x ˆ X m of F and f is called I -chain-unique if and only if, for any other common fixed point x of F and f, if x C ( I , x ˆ ) , then x = x ˆ .

Theorem 3.8 Suppose that the quasi-ordered metric space ( X , d , ) is monotonically complete and preserves the monotone convergence. Assume that the metrics d and d are compatible in the sense of preserving continuity. Consider the functions F : ( X m , d , I ) ( X m , d , I ) and f : ( X m , d , I ) ( X m , d , I ) satisfying F p ( X m ) f ( X m ) for some p N . Let x 0 be a I -monotone seed element in X m . Assume that the functions F and f satisfy the following conditions:

  • F and f are commutative;

  • f is I -monotone;

  • F p is continuous with respect to f on X m ;

  • each f k is continuous on X m for k = 1 , , m .

Suppose that there exist a function ρ : X m × X m R + and a function of contractive factor φ : [ 0 , ) [ 0 , 1 ) such that, for any x , y X m with y I x or x I y , the following inequalities
ρ ( x , y ) k = 1 m d ( x ( k ) , y ( k ) )
(43)
and
d ( F k p ( x ) , F k p ( y ) ) 1 m φ ( ρ ( f ( x ) , f ( y ) ) ) ρ ( f ( x ) , f ( y ) )
(44)
are satisfied for all k = 1 , , m . Then the following statements hold true.
  1. (i)

    F p and f have a I -chain-unique common fixed point x ˆ .

     
  2. (ii)

    For p 1 , suppose that F ( x ˆ ) and x ˆ obtained in (i) are comparable with respect to I ’. Then F and f have a I -chain-unique common fixed point x ˆ .

     

Moreover, each component x ˆ ( k ) of x ˆ is the limit of the sequence { f k ( x n ) } n N constructed in (3) for all k = 1 , , m .

Proof To prove part (i), from Proposition 2.8 and part (i) of Theorem 3.5, we have f ( x ˆ ) = F p ( x ˆ ) . From Theorem 3.2, we also have F p ( x ˆ ) = x ˆ . Therefore, we obtain
x ˆ = f ( x ˆ ) = F p ( x ˆ ) .
This shows that x ˆ is a common fixed point of F p and f. For the I -chain-uniqueness, let y ˆ be another common fixed point of F p and f such that y ˆ and x ˆ are comparable with respect to ‘ I ’, i.e., y ˆ = f ( y ˆ ) = F p ( y ˆ ) . By part (ii) of Theorem 3.5, we have f ( x ˆ ) = f ( y ˆ ) . Therefore, by the triangle inequality, we have
d ( x ˆ , y ˆ ) d ( x ˆ , f ( x ˆ ) ) + d ( f ( x ˆ ) , f ( y ˆ ) ) + d ( f ( y ˆ ) , y ˆ ) = 0 ,
(45)

which says that x ˆ = y ˆ . This proves part (i). Part (ii) can be obtained by applying part (iii) of Theorem 3.5 to a similar argument in the proof of Theorem 2.26. This completes the proof. □

Theorem 3.9 Suppose that the quasi-ordered metric space ( X , d , ) is monotonically complete and preserves the monotone convergence. Assume that the metrics d and d are compatible in the sense of preserving continuity. Consider the functions F : ( X m , d , I ) ( X m , d , I ) and f : ( X m , d , I ) ( X m , d , I ) satisfying F p ( X m ) f ( X m ) for some p N . Let x 0 be a I -monotone seed element in X m . Assume that the functions F and f satisfy the following conditions:

  • F and f are commutative;

  • f is I -monotone for any disjoint pair I and J of { 1 , , m } ;

  • F p is continuous with respect to f on X m ;

  • each f k is continuous on X m for k = 1 , , m .

Suppose that there exist a function ρ : X m × X m R + and a function of contractive factor φ : [ 0 , ) [ 0 , 1 ) such that, for any x , y X m and any disjoint pair I and J of { 1 , , m } with y I x or x I y , the following inequalities
ρ ( x , y ) k = 1 m d ( x ( k ) , y ( k ) )
(46)
and
d ( F k p ( x ) , F k p ( y ) ) 1 m φ ( ρ ( f ( x ) , f ( y ) ) ) ρ ( f ( x ) , f ( y ) )
(47)
are satisfied for all k = 1 , , m . Then the following statements hold true.
  1. (i)

    F p and f have a chain-unique common fixed point x ˆ .

     
  2. (ii)

    For p 1 , suppose that F ( x ˆ ) and x ˆ obtained in (i) are comparable with respect to the quasi-order I for some disjoint pair I and J of { 1 , , m } . Then F and f have a chain-unique common fixed point x ˆ .

     

Moreover, each component x ˆ ( k ) of x ˆ is the limit of the sequence { f k ( x n ) } n N constructed in (3) for all k = 1 , , m .

Proof To prove part (i), from the proof of Theorem 3.8, we can show that x ˆ is a common fixed point of F p and f. For the chain-uniqueness, let y ˆ be another common fixed point of F p and f with y ˆ I x ˆ or x ˆ I y ˆ for some disjoint pair I and J of { 1 , , m } , i.e., y ˆ = f ( y ˆ ) = F p ( y ˆ ) . Since f is I -monotone for any disjoint pair I and J of { 1 , , m } , we also have f ( y ˆ ) I f ( x ˆ ) or f ( x ˆ ) I f ( y ˆ ) . If f k ( x ˆ ) f k ( y ˆ ) , i.e., d ( f k ( x ˆ ) , f k ( y ˆ ) ) 0 , then we obtain
0 k = 1 m d ( f k ( x ˆ ) , f k ( y ˆ ) ) <