Abstract
The main aim of this paper is to study and establish some new coincidence point and common fixed point theorems in the product space of mixed-monotonically complete quasi-ordered metric space. Especially, we shall study the fixed points of functions having the monotone property or the comparable property in the product space of quasi-ordered metric space. An interesting application is to investigate the existence and uniqueness of a solution for the system of integral equations.
MSC:47H10, 54H25.
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1 Introduction
The existence of coincidence point has been studied in [1–4] and the references therein. Also, the existence of common fixed point has been studied in [5–15] and the references therein. In this paper, we shall introduce the concept of mixed-monotonically complete quasi-ordered metric space, and establish some new coincidence point and common fixed point theorems in the product space of those quasi-ordered metric spaces. We shall also present the interesting applications to the existence and uniqueness of solution for system of integral equations.
In Section 2, we shall derive the coincidence point theorems in the product space of mixed-monotonically complete quasi-ordered metric space. In Section 3, 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 4, the fixed point theorems for the functions having the comparable property in the product space of mixed-monotonically complete quasi-ordered metric space will be derived. Finally, in Section 5, we shall present the interesting application to investigate the existence and uniqueness of solutions for the system of integral equations.
2 Coincidence point theorems in product spaces
Let X be a nonempty set. We consider the product set
The element of is represented by the vectorial notation , where for . We also consider the function defined by
where for all . The vectorial element is a fixed point of F if and only if ; that is,
for all .
Definition 2.1 Let X be a nonempty set. Consider the functions and by and , where and for .
-
The element is a coincidence point of F and f if and only if , i.e., for all .
-
The element is a common fixed point of F and f if and only if , i.e., for all .
-
The functions F and f are said to be commutative if and only if for all .
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, is called a quasi-ordered set.
For any , we say that x and y are ⪯-mixed comparable if and only if, for each , one has either or . Let I be a subset of and . In this case, we say that I and J are the disjoint pair of . We can define a binary relation on as follows:
It is obvious that is a quasi-ordered set that depends on I. We also have
We need to mention that I or J is allowed to be empty set.
Remark 2.1 For any , we have the following observations.
-
If for some disjoint pair I and J of , then x and y are ⪯-mixed comparable.
-
If x and y are ⪯-mixed comparable, then there exists a disjoint pair I and J of such that .
Definition 2.2 Let I and J be a disjoint pair of . Given a quasi-ordered set , we consider the quasi-ordered set defined in (1).
-
The sequence in X is said to be a mixed ⪯-monotone sequence if and only if or (i.e., and are comparable with respect to ‘⪯’) for all .
-
The sequence in is said to be a mixed ⪯-monotone sequence if and only if each sequence in X is a mixed ⪯-monotone sequence for all .
-
The sequence in is said to be a mixed -monotone sequence if and only if or (i.e., and are comparable with respect to ‘’) for all .
Remark 2.2 Let I and J be a disjoint pair of . We have the following observations.
-
(a)
in is a mixed -monotone sequence if and only if it is a mixed -monotone sequence.
-
(b)
If in is a mixed -monotone sequence, then it is also a mixed ⪯-monotone sequence; that is, each sequence in X is a mixed ⪯-monotone sequence for all .
-
(c)
If in is a mixed ⪯-monotone sequence, then, given any , there exists a disjoint pair of and (which depends on n) of such that or .
-
(d)
in is a mixed ⪯-monotone sequence if and only if, for each , and are ⪯-mixed comparable.
Definition 2.3 Let I and J be a disjoint pair of . Given a quasi-ordered set , we also consider the quasi-ordered set defined in (1), and the function .
-
The function f is said to have the sequentially mixed ⪯-monotone property if and only if, given any mixed ⪯-monotone sequence in , is also a mixed ⪯-monotone sequence.
-
The function f is said to have the sequentially mixed -monotone property if and only if, given any mixed -monotone sequence in , is also a mixed -monotone sequence.
It is obvious that the identity function on has the sequentially mixed -monotone and ⪯-monotone property.
Let X be a nonempty set. We consider the functions and satisfying for some , where for any . Therefore, we have for . Given an initial element , where for , since , there exists such that . Similarly, there also exists such that . Continuing this process, we can construct a sequence such that
for all ; that is,
for all . We introduce the concepts of mixed-monotone seed elements as follows.
-
(A)
The initial element is said to be a mixed ⪯-monotone seed element of if and only if the sequence constructed from (3) is a mixed ⪯-monotone sequence; that is, each sequence in X is a mixed ⪯-monotone sequence for .
-
(B)
Given a disjoint pair I and J of , we say that the initial element is a mixed -monotone seed element of if and only if the sequence constructed from (3) is a mixed -monotone sequence.
From observation (b) of Remark 2.2, it follows that if is a mixed -monotone seed element, then it is also a mixed ⪯-monotone seed element.
Example 2.1 Suppose that the initial element can generate a sequence such that, for each , the generated sequence is either ⪯-increasing or ⪯-decreasing. In this case, we define the disjoint pair I and J of as follows:
It means that if , then the sequence is ⪯-decreasing. Therefore, the sequence satisfies for any . In this case, the initial element is a mixed -monotone seed element with the disjoint pair I and J defined in (4).
Definition 2.4 Let be a metric space endowed with a quasi-order ‘⪯’. We say that is mixed-monotonically complete if and only if each mixed ⪯-monotone Cauchy sequence in X is convergent.
It is obvious that if the quasi-ordered metric space is complete, then it is also mixed-monotonically complete. However, the converse is not necessarily true.
For the metric space , we consider the product metric space in which the metric is defined by
or
Remark 2.3 We can check that the product metric defined in (5) or (6) satisfies the following concepts.
-
Given a sequence in , the following statement holds:
-
Given any , there exists a positive constant (which depends on ϵ) such that the following statement holds:
Mizoguchi and Takahashi [16, 17] considered the mapping that satisfies the following condition:
in the contractive inequality, and generalized the Nadler 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 2.5 We say that is a function of contractive factor if and only if, for any strictly decreasing sequence in , we have
Using the routine arguments, we can show that the function satisfies (7) if and only if φ is a function of the contractive factor. Throughout this paper, we shall assume that the mapping φ satisfies (8) in order to prove the various types of coincidence and common fixed point theorems in the product space.
Let be a metric space, and let be a function defined on into itself. If F is continuous at , then, given , there exists such that with implies . From Remark 2.3, we see that F is continuous at if and only if each is continuous at for . Next, we propose another concept of continuity.
Definition 2.6 Let be a metric space, and let be the corresponding product metric space. Let and be functions defined on into itself. We say that F is continuous with respect to f at if and only if, given any , there exists such that with implies . We say that F is continuous with respect to f on if and only if it is continuous with respect to f at each .
It is obvious that if the function F is continuous at with respect to the identity function, then it is also continuous at .
Proposition 2.1 The function F is continuous with respect to f at if and only if, given any , there exists such that with for all imply for all .
Theorem 2.1 Suppose that the quasi-ordered metric space is mixed-monotonically complete. Consider the functions and satisfying for some . Let be a mixed ⪯-monotone seed element in . Assume that the functions F and f satisfy the following conditions:
-
F and f are commutative;
-
f has the sequentially mixed ⪯-monotone property;
-
is continuous with respect to f on ;
-
each is continuous on for .
Suppose that there exist a function and a function of the contractive factor such that, for any two ⪯-mixed comparable elements x and y in , the inequalities
and
are satisfied for all . Then has a fixed point such that each component of is the limit of the sequence constructed in (3) for all .
Proof We consider the sequence constructed from (3). Since is a mixed ⪯-monotone seed element in , i.e., is a mixed ⪯-monotone sequence, from observation (d) of Remark 2.2, it follows that, for each , and are ⪯-mixed comparable. According to the inequalities (10), we obtain
Since f has the sequentially mixed ⪯-monotone property, we see that is a mixed ⪯-monotone sequence. From observation (d) of Remark 2.2, it follows that, for each , and are ⪯-mixed comparable. Let
Then, using (9) and (11), we obtain
which also says that the sequence is strictly decreasing. Let . From (12), it follows that
which implies
For with , since , from (13), it follows that
which also says that is a Cauchy sequence in X for any fixed k. Since f has the sequentially mixed ⪯-monotone property, i.e., is a mixed ⪯-monotone Cauchy sequence for , by the mixed ⪯-monotone completeness of X, there exists such that as for . By Remark 2.3, it follows that as . Since each is continuous on , we also have
Since is continuous with respect to f on , by Proposition 2.1, given any , there exists such that with for all imply
Since as for all , given , there exists such that
For each , by (14) and (15), it follows that
Therefore, we obtain
Since ϵ is any positive number, we conclude that for all , which also says that for all , i.e., . This completes the proof. □
Remark 2.4 We have the following observations.
-
In Theorem 2.1, if we assume that the quasi-ordered metric space 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 assumptions for the inequalities (9) and (10) are 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 the inequalities (9) and (10).
In Theorem 2.1, we can consider a different function ρ that is defined on instead of . Then we can have the following result.
Theorem 2.2 Suppose that the quasi-ordered metric space is mixed-monotonically complete. Consider the functions and satisfying for some . Let be a mixed ⪯-monotone seed element in . Assume that the functions F and f satisfy the following conditions:
-
F and f are commutative;
-
f has the sequentially mixed ⪯-monotone property;
-
is continuous with respect to f on ;
-
each is continuous on for .
Suppose that there exist a function and a function of the contractive factor such that, for any two ⪯-mixed comparable elements x and y in , the inequalities
and
are satisfied for all . Then has a fixed point such that each component of is the limit of the sequence constructed in (3) for all .
Proof Using a similar argument to the proof of Theorem 2.1, we can obtain the desired results. □
By considering the mixed -monotone seed element instead of mixed ⪯-monotone seed element, the assumptions for the inequalities (9) and (10) can be weaken, which is shown below.
Theorem 2.3 Suppose that the quasi-ordered metric space is mixed-monotonically complete. Consider the functions and satisfying for some . Let be a mixed -monotone seed element in , and let be a quasi-ordered set induced by . Assume that the functions F and f satisfy the following conditions:
-
F and f are commutative;
-
f has the sequentially mixed -monotone property or the sequentially mixed ⪯-monotone property;
-
is continuous with respect to f on ;
-
each is continuous on for .
Suppose that there exist a function and a function of the contractive factor such that, for any with or , the inequalities
and
are satisfied for all . Then has a fixed point such that each component of is the limit of the sequence constructed in (3) for all .
Proof We consider the sequence constructed from (3). Since is a mixed -monotone seed element in , it follows that is a mixed -monotone sequence, i.e., for each , or . According to the inequalities (20), we obtain
Using the argument in the proof of Theorem 2.1, we can show that is a Cauchy sequence in X for any fixed k. Now, we consider the following cases.
-
Suppose that f has the sequentially mixed -monotone property. We see that is a mixed -monotone sequence; that is, for each , or . Since is a Cauchy sequence in X for any fixed k, from observation (b) of Remark 2.2, we also see that is a mixed ⪯-monotone Cauchy sequence for .
-
Suppose that f has the sequentially mixed ⪯-monotone property. Since is a mixed -monotone sequence, by part (b) of Remark 2.2, it follows that in X is a mixed ⪯-monotone sequence for all . Therefore, we see that is a mixed ⪯-monotone Cauchy sequence for .
By the mixed ⪯-monotone completeness of X, there exists such that as for . The remaining proof follows from the same argument in the proof of Theorem 2.1. This completes the proof. □
Remark 2.5 We have the following observations.
-
In Theorem 2.3, if we assume that the quasi-ordered metric space is complete (not mixed-monotonically complete), then the assumption for f having the sequentially mixed -monotone can be dropped, since the proof is still valid in this case.
-
From the observation (a) of Remark 2.1, we see that the assumptions for the inequalities (19) and (20) are indeed weaken by comparing to the inequalities (9) and (10).
-
We can also obtain a similar result when the inequalities (19) and (20) in Theorem 2.3 are replaced by the inequalities (17) and (18), respectively.
Next, we shall study the coincidence point without considering the continuity of . However, we need to introduce the concept of mixed-monotone convergence given below.
Definition 2.7 Let be a metric space endowed with a quasi-order ‘⪯’. We say that preserves the mixed-monotone convergence if and only if, for each mixed ⪯-monotone sequence that converges to , we have or for each .
Remark 2.6 Let be a metric space endowed with a quasi-order ‘⪯’ and preserve the mixed-monotone convergence. Suppose that is a sequence in the product space such that each sequence is a mixed ⪯-monotone convergence sequence with limit point for . Then we have the following observations.
-
(a)
For each , and are ⪯-mixed comparable.
-
(b)
For each , there exists a disjoint pair and (which depend on n) of such that or , where or is allowed to be empty set.
Definition 2.8 Let I and J be a disjoint pair of . Given a quasi-ordered set , we consider the quasi-ordered set defined in (1), and the function .
-
The function f is said to have the ⪯-comparable property if and only if, given any two ⪯-comparable elements x and y in , the function values and are ⪯-comparable.
-
The function f is said to have the -comparable property if and only if, given any two -comparable elements x and y in , the function values and are -comparable.
Theorem 2.4 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence. Consider the functions and satisfying for some . Let be a mixed ⪯-monotone seed element in . 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 is continuous on for .
Suppose that there exist a function and a function of the contractive factor such that, for any two ⪯-mixed comparable elements x and y in , the inequalities
and
are satisfied for all . Then the following statements hold true.
-
(i)
There exists of F such that . If , then is a coincidence point of F and f.
-
(ii)
If there exists another such that and are ⪯-mixed comparable satisfying , then .
-
(iii)
Suppose that is obtained from part (i). If and are ⪯-mixed comparable, then is a fixed point of F for any .
Moreover, each component of is the limit of the sequence constructed in (3) for all .
Proof From the proof of Theorem 2.1, we can construct a sequence in such that and as , where is a mixed ⪯-monotone sequence for all . Since as , given any , there exists such that
for all with and for all . Since is a mixed ⪯-monotone convergence sequence for all , from observation (a) of Remark 2.6, we see that and are ⪯-mixed comparable for each . Since f has the ⪯-comparable property, it follows that and are ⪯-mixed comparable. For each , we have
Since F and f are commutative, we have for all , which also implies
Now, we obtain
Since ϵ is any positive number, we conclude that , which says that for all , i.e., . This proves part (i).
To prove part (ii), since f has the ⪯-comparable property, it follows that and are ⪯-mixed comparable. If , i.e., , then we obtain
This contradiction says that for all , i.e., .
To prove part (iii), using the commutativity of F and f, we have
By taking , the equalities (25) says that . Since and are ⪯-mixed comparable by the assumption, part (ii) says that
which says that is a fixed point of F. Given any , we have
which says that is a fixed point of F. This completes the proof. □
Remark 2.7 We have the following observations.
-
In Theorem 2.4, if we assume that the quasi-ordered metric space 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.
-
We can also obtain a similar result when the inequalities (21) and (22) in Theorem 2.4 are replaced by the inequalities (17) and (18), respectively.
Theorem 2.5 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence. Consider the functions and satisfying for some . Let be a mixed -monotone seed element in , and let be a quasi-ordered set induced by . Assume that the functions F and f satisfy the following conditions:
-
F and f are commutative;
-
f has the sequentially mixed -monotone property or the sequentially mixed ⪯-monotone property;
-
each is continuous on for ;
-
f has the -comparable property for any disjoint pair and of .
Suppose that there exist a function and a function of the contractive factor such that, for any and any disjoint pair and of with or , the inequalities
and
are satisfied for all . Then the following statements hold true.
-
(i)
There exists of F such that . If , then is a coincidence point of F and f.
-
(ii)
If there exist a disjoint pair and of and another such that and are comparable with respect to the quasi-order ‘ ’ satisfying , then .
-
(iii)
Suppose that is obtained from part (i). If there exists a disjoint pair and of such that and are comparable with respect to the quasi-order ‘ ’, then is a fixed point of F for any .
Moreover, each component of is the limit of the sequence constructed in (3) for all .
Proof From the proof of Theorem 2.3, we can construct a sequence in such that and as , where is a mixed ⪯-monotone sequence for all . Since as , given any , there exists such that
for all with and for all . Since is a mixed ⪯-monotone convergent sequence for all , from observation (b) of Remark 2.6, we see that, for each , there exists a subset of such that
Since f has the -comparable property for any subset of , it follows that
For each , we obtain
Using the same argument in the proof of Theorem 2.4 immediately, we complete the proof. □
Remark 2.8 We have the following observations.
-
Suppose that the inequalities (21) and (22) in Theorem 2.4, and that the inequalities (26) and (27) in Theorem 2.5 are satisfied for any . Then, from the proofs of Theorems 2.4 and 2.5, we can see that parts (ii) and (iii) can be changed as follows.
-
(ii)′
If there exists another satisfying , then .
-
(iii)′
Suppose that is obtained from part (i). Then is a fixed point of F for any .
-
(ii)′
-
We can also obtain a similar result when the inequalities (26) and (27) in Theorem 2.5 are replaced by the inequalities (17) and (18), respectively.
Next, we shall consider the uniqueness for a common fixed point in the ⪯-mixed comparable sense.
Definition 2.9 Let be a quasi-order set. Consider the functions and defined on the product set into itself. The common fixed point 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 are ⪯-mixed comparable, then .
Theorem 2.6 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence. Consider the functions and satisfying for some . Let be a mixed ⪯-monotone seed element in . 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;
-
is continuous with respect to f on ;
-
each is continuous on for .
Suppose that there exist a function and a function of the contractive factor such that, for any two ⪯-mixed comparable elements x and y in , the inequalities
and
are satisfied for all . Then the following statements hold true.
-
(i)
and f have a unique common fixed point in the ⪯-mixed comparable sense. Equivalently, if is another common fixed point of and f, and is ⪯-mixed comparable with , then .
-
(ii)
For , suppose that and obtained in (i) are ⪯-mixed comparable. Then F and f have a unique common fixed point in the ⪯-mixed comparable sense.
Moreover, each component of is the limit of the sequence constructed in (3) for all .
Proof To prove part (i), from Remark 2.3 and part (i) of Theorem 2.4, we have . From Theorem 2.1, we also have . Therefore, we obtain
This shows that is a common fixed point of and f. For the uniqueness in the ⪯-mixed comparable sense, let be another common fixed point of and f such that and are ⪯-mixed comparable, i.e., . By part (ii) of Theorem 2.4, we have . Therefore, by the triangle inequality, we have
which says that . This proves part (i).
To prove part (ii), since and are ⪯-mixed comparable, part (iii) of Theorem 2.4 says that is a fixed point of F, i.e., , which implies , since . This shows that is a common fixed point of F and f. For the uniqueness in the ⪯-mixed comparable sense, let be another common fixed point of F and f such that and are ⪯-mixed comparable, i.e., . Then we have
By part (ii) of Theorem 2.4, we have . From (33), we can similarly obtain . This completes the proof. □
Remark 2.9 We can also obtain a similar result when the inequalities (31) and (32) in Theorem 2.6 are replaced by the inequalities (17) and (18), respectively.
Since we consider a metric space endowed with a quasi-order ‘⪯’, given any disjoint pair I and J of , we can define a quasi-order ‘’ on as given in (1). Now, given any , we define the chain containing x as follows:
Next, we shall introduce the concept of chain-uniqueness for a common fixed point.
Definition 2.10 Let be a quasi-order set. Consider the functions and defined on the product set into itself. The common fixed point of F and f is called chain-unique if and only if, given any other common fixed point x of F and f, if for some disjoint pair and of , then .
Theorem 2.7 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence. Consider the functions and satisfying for some . Let be a mixed -monotone seed element in , and let be a quasi-ordered set induced by . Assume that the functions F and f satisfy the following conditions:
-
F and f are commutative;
-
f has the sequentially mixed -monotone property or the ⪯-monotone property;
-
is continuous with respect to f on ;
-
each is continuous on for .
Suppose that there exist a function and a function of the contractive factor such that, for any and any disjoint pair and of with or , the inequalities
and
are satisfied for all . Then the following statements hold true.
-
(i)
and f have a chain-unique common fixed point . Equivalently, if is another common fixed point of and f for some disjoint pair and of , then .
-
(ii)
For , suppose that and obtained in (i) are comparable with respect to the quasi-order ‘ ’ for some disjoint pair and of . Then F and f have a chain-unique common fixed point .
Moreover, each component of is the limit of the sequence constructed in (3) for all .
Proof To prove part (i), from Remark 2.3 and part (i) of Theorem 2.5, we can show that is a common fixed point of and f. For the chain-uniqueness, let be another common fixed point of and f with or for some disjoint pair and of , i.e., . By part (ii) of Theorem 2.5, we have . Therefore, by the triangle inequality, we have
which says that . This proves part (i). Part (ii) can be similarly obtained by applying Theorem 2.5 to the argument in the proof of part (ii) of Theorem 2.6. This completes the proof. □
Remark 2.10 We have the following observations.
-
Suppose that the inequalities (31) and (32) in Theorem 2.6, and that the inequalities (34) and (35) in Theorem 2.7 are satisfied for any . Then, from Remark 2.8 and the proofs of Theorems 2.6 and 2.7, we can see that parts (i) and (ii) can be combined together to conclude that F and f have a unique common fixed point .
-
We can also obtain a similar result when the inequalities (34) and (35) in Theorem 2.7 are replaced by the inequalities (17) and (18), respectively.
Now, we are going to weaken the concept of mixed-monotone completeness for the quasi-ordered metric space. Let be a metric space endowed with a quasi-order ‘⪯’. We say that the sequence in is ⪯-increasing if and only if for all . The concept of ⪯-decreasing sequence can be similarly defined. The sequence in is called ⪯-monotone if and only if is either ⪯-increasing or ⪯-decreasing.
Let I and J be a disjoint pair of . We say that the sequence in is -increasing if and only if for all . The concept of -decreasing sequence can be similarly defined. The sequence in is called -monotone if and only if is either -increasing or -decreasing.
Given a disjoint pair I and J of , let be a function defined on into itself. We say that f is -increasing if and only if implies . The concept of -decreasing function can be similarly defined. The function f is called -monotone if and only if f is either -increasing or -decreasing.
In the previous section, we consider the mixed -monotone seed element. Now, we shall consider another concept of seed element. Given a disjoint pair I and J of , we say that the initial element is a -monotone seed element of if and only if the sequence constructed from (3) is a -monotone sequence. It is obvious that if is a -monotone seed element, then it is also a mixed -monotone seed element.
Definition 2.11 Let be a metric space endowed with a quasi-order ‘⪯’. We say that is monotonically complete if and only if each ⪯-monotone Cauchy sequence in X is convergent.
It is obvious that if 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 2.8 Suppose that the quasi-ordered metric space is monotonically complete. Consider the functions and satisfying for some . Let be a -monotone seed element in , and let be a quasi-ordered set induced by . Assume that the functions F and f satisfy the following conditions:
-
F and f are commutative;
-
f is -monotone;
-
is continuous with respect to f on ;
-
each is continuous on for .
Suppose that there exist a function and a function of the contractive factor such that, for any with or , the inequalities
and
are satisfied for all . Then has a fixed point such that each component of is the limit of the sequence constructed in (3) for all .
Proof We consider the sequence constructed from (3). Since is a -monotone seed element in , i.e., for all or for all , according to the inequalities (37), we obtain
Since f is -monotone, it follows that for all or for all . Let
Then, using (36) and (38), we obtain
Let . From (39), it follows that
which implies
Using the argument in the proof of Theorem 2.1, we can show that is a Cauchy sequence in X for any fixed . Since f is -monotone and is a -monotone sequence, it follows that is a -monotone sequence.
-
If is a -increasing sequence, then is a ⪯-increasing Cauchy sequence for , and is a ⪯-decreasing Cauchy sequence for .
-
If is a -decreasing sequence, then is a ⪯-decreasing Cauchy sequence for , and is a ⪯-increasing Cauchy sequence for .
By the monotone completeness of X, there exists such that as for . The remaining proof follows from the same argument in the proof of Theorem 2.1. This completes the proof. □
Remark 2.11 We can also obtain a similar result when the inequalities (36) and (37) in Theorem 2.8 are replaced by the inequalities (17) and (18), respectively.
Next, we shall study the coincidence point without considering the continuity of . However, we need to introduce the concept of monotone convergence given below.
Definition 2.12 Let be a metric space endowed with a quasi-order ‘⪯’. We say that preserves the monotone convergence if and only if, for each ⪯-monotone sequence that converges to , either one of the following conditions is satisfied:
-
if is a ⪯-increasing sequence, then for each ;
-
if is a ⪯-decreasing sequence, then for each .
Remark 2.12 Let be a metric space endowed with a quasi-order ‘⪯’ and preserve the monotone convergence. Given a disjoint pair I and J of , suppose that is a -monotone sequence such that each sequence converges to for . We consider the following situation.
-
If is a -increasing sequence, then is a ⪯-increasing sequence for , and is a ⪯-decreasing sequence for . By the monotone convergence, we see that, for each , for and for , which shows that for all .
-
If is a -decreasing sequence, then is a ⪯-decreasing sequence for , and is a ⪯-increasing sequence for . By the monotone convergence, we see that, for each , for and for , which shows that for all .
Therefore, we conclude that and are comparable with respect to ‘’ for all .
Theorem 2.9 Suppose that the quasi-ordered metric space is monotonically complete and preserves the monotone convergence. Consider the functions and satisfying for some . Let be a -monotone seed element in , and let be a quasi-ordered set induced by . Assume that the functions F and f satisfy the following conditions:
-
F and f are commutative;
-
f is -monotone;
-
each is continuous on for .
Suppose that there exist a function and a function of the contractive factor such that, for any with or , the inequalities
and
are satisfied for all . Then the following statements hold true.
-
(i)
There exists of F such that . If , then is a coincidence point of F and f.
-
(ii)
If there exists such that with or , then .
-
(iii)
Suppose that is obtained from part (i). If and are comparable with respect to ‘ ’, then is a fixed point of F for any .
Moreover, each component of is the limit of the sequence constructed in (3) for all .
Proof From the proof of Theorem 2.8, we can construct a sequence in such that and as for all , where is a -monotone sequence. From Remark 2.12, it follows that, for each , or . Since as , given any , there exists such that
for all with and for all . Since f is -monotone, it follows that or . For each , it follows that
Using the same argument in the proof of part (i) of Theorem 2.4, part (i) of this theorem follows immediately.
To prove part (ii), since f is -monotone, we immediately have or . If , i.e., , then we obtain
This contradiction says that for all , i.e., . Finally, part (iii) follows from the same argument as in the proof of part (iii) of Theorem 2.4 immediately. This completes the proof. □
Remark 2.13 We have the following observations.
-
Suppose that the inequalities (40) and (41) in Theorem 2.9 are assumed to be satisfied for any . Then, from the proof of Theorem 2.9, we can see that parts (ii) and (iii) can be changed as follows.
-
(ii)′
If there exists satisfying , then .
-
(iii)′
Suppose that is obtained from part (i). Then is a fixed point of F for any .
-
(ii)′
-
We can also obtain a similar result when the inequalities (40) and (41) in Theorem 2.8 are replaced by the inequalities (17) and (18), respectively.
Next, we shall study the -chain-uniqueness for the common fixed point, which is the different concept from Definition 2.10.
Definition 2.13 Let be a quasi-order set. Consider the functions and defined on the product set into itself. Given a disjoint pair I and J of , we recall that the chain containing x is given by
The common fixed point of F and f is called -chain-unique if and only if, for any other common fixed point x of F and f, if , then .
Theorem 2.10 Suppose that the quasi-ordered metric space is monotonically complete and preserves the monotone convergence. Consider the functions and satisfying for some . Let be a -monotone seed element in , and let be a quasi-ordered set induced by . Assume that the functions F and f satisfy the following conditions:
-
F and f are commutative;
-
f is -monotone;
-
is continuous with respect to f on ;
-
each is continuous on for .
Suppose that there exist a function and a function of the contractive factor such that, for any with or , the inequalities
and
are satisfied for all . Then the following statements hold true.
-
(i)
and f have a -chain-unique common fixed point .
-
(ii)
For , suppose that and obtained in (i) are comparable with respect to ‘ ’. Then F and f have a -chain-unique common fixed point .
Moreover, each component of is the limit of the sequence constructed in (3) for all .
Proof To prove part (i), from Proposition 2.3 and part (i) of Theorem 2.9, we have . From Theorem 2.8, we also have . Therefore, we obtain
This shows that is a common fixed point of and f. For the -chain-uniqueness, let be another common fixed point of and f such that and are comparable with respect to ‘’, i.e., . By part (ii) of Theorem 2.9, we have . Therefore, by the triangle inequality, we have
which says that . This proves part (i). Part (ii) can be obtained by applying part (iii) of Theorem 2.9 to the similar argument in the proof of Theorem 2.6. This completes the proof. □
Remark 2.14 We can also obtain a similar result when the inequalities (43) and (44) in Theorem 2.10 are replaced by the inequalities (17) and (18), respectively.
3 Fixed points of functions having monotone property
We shall study the fixed points of functions having monotone property in the product space. Considering the function , we shall define many monotonic concepts of F as follows.
Definition 3.1 Let be a quasi-order set, and let I and J be the disjoint pair of . Consider the quasi-order set and the function .
-
We say that F is -increasing if and only if implies .
-
We say that F is -increasing if and only if implies .
-
We say that F is -increasing if and only if implies .
-
We say that F is -increasing if and only if implies .
-
We say that F is -decreasing if and only if implies .
-
We say that F is -decreasing if and only if implies .
-
We say that F is -decreasing if and only if implies .
-
We say that F is -decreasing if and only if implies .
Remark 3.1 From (2), we see that it suffices to consider the increasing cases. On the other hand, we also see that F is -increasing if and only if it is -increasing, and F is -increasing if and only if it is -increasing. Therefore, the cases in Definition 3.1 can be reduced to only consider the -increasing and -increasing cases. Since the -increasing case is equivalent to the -decreasing case, it follows that the cases in Definition 3.1 can be reduced to only consider the -increasing and -decreasing cases.
Theorem 3.1 Suppose that the quasi-ordered metric space is monotonically complete. Let I and J be a disjoint pair of . Assume that the function is continuous on and satisfies any one of the following conditions:
-
(a)
F is -increasing;
-
(b)
p is an even integer and F is -decreasing.
Assume that there exist a function and a function of the contractive factor such that, for any with or , the inequalities
and
are satisfied for all and for some . If there exists such that or , then the function has a fixed point , where each component of is the limit of the sequence constructed from (51) for all .
Proof We consider the following cases.
-
If F is -increasing, then it follows that is -increasing.
-
If F is -decreasing and p is an even integer, then is also -increasing.
According to (51), we have or . Since and , it follows that implies , and implies . Therefore, if , then we can generate a -increasing sequence , and if , then we can generate a -decreasing sequence , which also says that the initial element is a -monotone seed element in . Therefore, the results follow immediately from Theorem 2.8 by taking f as the identity function. This completes the proof. □
Remark 3.2 We can also obtain a similar result when the inequalities (45) and (46) in Theorem 3.1 are replaced by the inequalities (17) and (18), respectively.
Next, we can consider the chain-uniqueness and drop the assumption of continuity of F by assuming that preserves the monotone convergence.
Theorem 3.2 Suppose that the quasi-ordered metric space is monotonically complete and preserves the monotone convergence. Let I and J be a disjoint pair of . Assume that the function satisfies any one of the following conditions:
-
(a)
F is -increasing;
-
(b)
p is an even integer and F is -decreasing.
Assume that there exist a function and a function of the contractive factor such that, for any with or , the inequalities
and
are satisfied for all and for some . If there exists such that or , then the function has a -chain-unique fixed point , where each component of is the limit of the sequence constructed from (51) for all .
Proof From the proof of Theorem 3.1, we see that the initial element is a -monotone seed element in . Therefore, the results follow immediately from Theorem 2.9 by taking f as the identity function. This completes the proof. □
Remark 3.3 We can also obtain a similar result when the inequalities (47) and (48) in Theorem 3.2 are replaced by the inequalities (17) and (18), respectively.
Theorem 3.3 Suppose that the quasi-ordered metric space is mixed-monotonically complete. Let I and J be a disjoint pair of . Assume that the function is continuous on and -decreasing, and that there exist a function and a function of the contractive factor such that, for any with or , the inequalities
and
are satisfied for all and for some odd integer . If there exists such that or , then the function has a fixed point , where each component of is the limit of the sequence constructed below
for all .
Proof Since F is -decreasing and p is an odd integer, it follow that is -decreasing. We see that implies , and that implies . Therefore, we can generate a -mixed-monotone sequence , which also says that the initial element is a mixed -monotone seed element in . Therefore, the results follow immediately from Theorem 2.3 by taking f as the identity function. This completes the proof. □
Remark 3.4 We can also obtain a similar result when the inequalities (49) and (50) in Theorem 3.3 are replaced by the inequalities
and
respectively.
Next, we can consider the chain-uniqueness and drop the assumption of continuity of F by assuming that preserves the mixed-monotone convergence.
Theorem 3.4 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence. Suppose that there exists a disjoint pair I and J of and such that the following conditions are satisfied:
-
the function is -decreasing.
-
or .
Assume that there exists a function and a function of the contractive factor such that, for any and any disjoint pair and of with or , the inequalities
and
are satisfied for all and for some odd integer . Then the function has a chain-unique fixed point , where each component of is the limit of the sequence constructed from (51) for all .
Proof From the proof of Theorem 3.3, we see that the initial element is a mixed -monotone seed element in . Therefore, the results follow immediately from Theorem 2.5 by taking f as the identity function. This completes the proof. □
Remark 3.5 We can also obtain a similar result when the inequalities (54) and (55) in Theorem 3.4 are replaced by the inequalities (52) and (53), respectively.
4 Fixed points of functions having comparable property
We shall study the fixed points of functions having the comparable property in the product space.
Definition 4.1 Let I and J be a disjoint pair of . Given a quasi-ordered set , we consider the corresponding quasi-ordered set .
-
The function is said to have the ⪯-mixed comparable property if and only if, for any two ⪯-mixed comparable elements x and y in , the function values and in are ⪯-mixed comparable.
-
The function is said to have the -comparable property if and only if, for any two elements with or (i.e., x and y are comparable with respect to ‘’), one has either or (i.e., the function values and in are comparable with respect to ‘’).
It is obvious that if F is -increasing or -decreasing, then it also has the -comparable property.
Theorem 4.1 Suppose that the quasi-ordered metric space is mixed-monotonically complete. Assume that the function is continuous on and has the ⪯-mixed comparable property, and that there exist a function and a function of the contractive factor such that, for any two ⪯-mixed comparable elements x and y in , the inequalities
and
are satisfied for all and for some . If there exists such that and are ⪯-mixed comparable, then has a fixed point such that each component of is the limit of the sequence constructed from (51) for all .
Proof According to (51), it follows that and are ⪯-mixed comparable. Since F has the ⪯-mixed comparable property, we see that has also the ⪯-mixed comparable property. It follows that and are also ⪯-mixed comparable. Therefore, we can generate a mixed ⪯-monotone sequence by observation (d) of Remark 2.2, which also says that the initial element is a mixed ⪯-monotone seed element in . Since F is continuous on , it follows that is also continuous on . Therefore, the result follows from Theorem 2.1 immediately by taking f as the identity function. This completes the proof. □
Remark 4.1 We can also obtain a similar result when the inequalities (56) and (57) in Theorem 4.1 are replaced by the inequalities (52) and (53), respectively.
Next, we can drop the assumption of continuity of F by assuming that preserves the mixed-monotone convergence.
Theorem 4.2 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence. Assume that the function has the ⪯-mixed comparable property, and that there exist a function and a function of the contractive factor such that, for any two ⪯-mixed comparable elements x and y in , the inequalities
and
are satisfied for all and for some . Suppose that there exists such that and are ⪯-mixed comparable. Then the following statements hold true.
-
(i)
There exists a unique fixed point of in the ⪯-mixed comparable sense.
-
(ii)
For , we further assume that the function is continuous on , and that and obtained in (i) are ⪯-mixed comparable. Then is a unique fixed point of F in the ⪯-mixed-monotone sense.
Moreover, each component of is the limit of the sequence constructed from (51) for all for all .
Proof According to the argument in the proof of Theorem 4.1, we see that the initial element is a mixed ⪯-monotone seed element in . Therefore, part (i) follows from Theorem 2.4 immediately by taking f as the identity function. Also, part (ii) follows from Theorem 2.6 immediately by taking f as the identity function. This completes the proof. □
Remark 4.2 We can also obtain a similar result when the inequalities (58) and (59) in Theorem 4.2 are replaced by the inequalities (52) and (53), respectively.
Theorem 4.3 Suppose that the quasi-ordered metric space is mixed-monotonically complete. Given a disjoint pair I and J of , assume that the following conditions are satisfied:
-
the function is continuous on and has the -comparable property;
-
there exists such that and are comparable with respect to the quasi-order ‘ ’ for some .
Assume that there exist a function and a function of the contractive factor such that, for any two ⪯-mixed comparable elements x and y in , the inequalities
and
are satisfied for all . Then has a fixed point such that each component of is the limit of the sequence constructed from (51) for all .
Proof According to (51), we see that and are comparable with respect to ‘’. Since F has the -comparable property, we see that has also the -comparable property. It follows that and are also comparable with respect to ‘’. Therefore, we can generate a mixed -monotone sequence , which also says that the initial element is a mixed -monotone seed element in . Since F is continuous on , it follows that is also continuous on . Therefore, the result follows from Theorem 2.3 immediately by taking f as the identity function. This completes the proof. □
Remark 4.3 We can also obtain a similar result when the inequalities (60) and (61) in Theorem 4.3 are replaced by the inequalities (52) and (53), respectively.
Next, we can drop the assumption of continuity of F by assuming that preserves the mixed-monotone convergence.
Theorem 4.4 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence. Given a disjoint pair I and J of , assume that the following conditions are satisfied:
-
the function has the -comparable property;
-
there exists such that and are comparable with respect to the quasi-order ‘ ’ for some .
Suppose that there exists a function and a function of the contractive factor such that, for any and any disjoint pair and of with or , the inequalities
and
are satisfied for all . Then the following statements hold true.
-
(i)
There exists a chain-unique fixed point of .
-
(ii)
For , we further assume that the function is continuous on , and that and obtained in (i) are comparable with respect to ‘ ’ for some disjoint pair and of . Then is a chain-unique fixed point of F.
Moreover, each component of is the limit of the sequence constructed from (51) for all .
Proof According to the argument in the proof of Theorem 4.3, we see that the initial element is a mixed -monotone seed element in . Therefore, part (i) follows from Theorem 2.5 immediately by taking f as the identity function. Also, part (ii) follows from Theorem 2.7 immediately by taking f as the identity function. This completes the proof. □
Remark 4.4 We can also obtain a similar result when the inequalities (62) and (63) in Theorem 4.4 are replaced by the inequalities (52) and (53), respectively.
5 Applications to the system of integral equations
Let be the space of all continuous functions from into ℝ. We also denote by the product space of for m times. In the sequel, we shall consider a metric d and a quasi-order ‘⪯’ on such that is monotonically complete or mixed-monotonically complete and preserves the monotone convergence.
Given continuous functions and for , we consider the following system of integral equations:
for , where . We shall find such that the systems of integral equations (64) are all satisfied, where is the k th component of for . The solution will be in the sense of chain-uniqueness.
For the vector-valued function defined on , the k th component function of h is denoted by for . The integral of h on is defined as the following vector in :
Now, we define a vector-valued functions by . Then the system of integral equations as shown in (64) can be written as the following vectorial form of integral equation:
where . Equivalently, we shall find such that (65) is satisfied, which also says that is a solution of (65).
Definition 5.1 Consider the quasi-ordered metric space .
-
(a)
We say that is a unique solution of the system of integral equations (65) in the ⪯-mixed comparable sense if and only if the following conditions are satisfied:
-
is a solution of (65);
-
if is another solution of (65) such that and are ⪯-mixed comparable, then .
Given a disjoint pair I and J of , consider the product space .
(b) We say that is a -chain-unique solution of the system of integral equations (65) if and only if the following conditions are satisfied:
-
is a solution of (65);
-
if is another solution of (65) satisfying or (i.e., and are comparable with respect to ≼), then .
Theorem 5.1 Suppose that the quasi-ordered metric space is monotonically complete and preserves the monotone convergence. Let I and J be a disjoint pair of . Define the function by
Suppose that the following conditions are satisfied:
-
F is -increasing;
-
there exist a function and a function of the contractive factor such that, for any with or , the inequalities
(66)
and
are satisfied for all ;
-
there exists such that or .
Then there exists a -chain-unique solution of the system of integral equations (65).
Proof Since is defined in (5) or (6), we immediately have that the metrics and d are compatible in the sense of preserving convergence. Using condition (a) and considering in Theorem 3.2, we see that F has a -chain-unique fixed point in . In other words, we have
which says that is a -chain-unique solution of the vectorial form of the integral equation (65). This completes the proof. □
Remark 5.1 The assumption for the inequalities (66) and (67) are really weak, since we just assume that they are satisfied for -comparable elements. In other words, if x and y are not -comparable, we do not need to check the inequalities (66) and (67).
Theorem 5.2 Suppose that the quasi-ordered metric space is monotonically complete and preserves the monotone convergence. Let I and J be a disjoint pair of . Define the function by
Suppose that the following conditions are satisfied:
-
F is -increasing;
-
there exist a function and a function of the contractive factor such that, for any with or , the inequalities
(68)
and
are satisfied for all ;
-
there exists such that or .
Then there exists a -chain-unique solution of the system of integral equations (65).
Proof By applying Remark 3.3 to the argument in the proof of Theorem 5.1, we can obtain the desired result. □
Corollary 5.1 Suppose that the quasi-ordered metric space is monotonically complete and preserves the monotone convergence. Let I and J be a disjoint pair of . Define the function by
Suppose that the following conditions are satisfied:
-
F is -increasing;
-
there exists a function of the contractive factor such that, for any with or , the inequalities
are satisfied for all ;
-
there exists such that or .
Then there exists a -chain-unique solution of the system of integral equations (65).
Proof By taking
the desired result follows from Theorem 5.2 immediately. □
Lemma 5.1 For any , we define
Then the quasi-ordered metric space preserves the monotone convergence.
Proof Let be an ⪯-increasing sequence in , and let be the d-limit of . Suppose that there exists such that ; that is, there exists such that . Since for all and , it follows that for all , which contradicts the convergence . Therefore, we must have for all , i.e., . If is a ⪯-decreasing sequence in and converges to , then we can similarly show that for all . This completes the proof. □
The following result is well known.
Lemma 5.2 For any , we define
Then the quasi-ordered metric space is complete.
Given a disjoint pair I and J of , we can consider a quasi-ordered set that depends on I, where, for any ,
Then we have the following interesting existence.
Theorem 5.3 Let be a quasi-ordered metric space with the metric and the quasi-order ⪯∗ defined in (71) and (70), respectively. Let I and J be a disjoint pair of . Define the function by
where is defined in (1) according to ⪯∗. Suppose that the following conditions are satisfied:
-
F is -increasing;
-
there exists a function such that, for any with or , the inequalities are satisfied for all ;
-
there exists a function of the contractive factor such that, for any with or , the inequalities
(72)
are satisfied for , where , and the functions and satisfy the following inequalities: for
and
for ;
-
there exists such that or .
Then there exists a -chain-unique solution of the system of integral equations (65).
Proof Lemmas 5.1 and 5.2 say that the quasi-ordered metric space is complete and preserves the monotone convergence. For or , it means, for each ,
or
which also says that
Then we have
Using Theorem 5.1, we complete the proof. □
Remark 5.2 The assumption for the inequalities (72) is really weak, since we just assume that it is satisfied for -comparable elements. In other words, if x and y are not -comparable, we do not need to check the inequalities (72).
Corollary 5.2 Let be a quasi-ordered metric space with the metric and the quasi-order ⪯∗ defined in (71) and (70), respectively. Let I and J be a disjoint pair of . Define the function by
where is defined in (1) according to ⪯∗. Suppose that the following conditions are satisfied:
-
F is -increasing;
-
there exists a function of the contractive factor such that, for any with or , the inequalities
are satisfied for , where , and the function satisfies the inequalities: for ,
for ;
-
there exists such that or .
Then there exists a -chain-unique solution of the system of integral equations (65).
Proof We take and define the function by
Since
the desired result follows from Theorem 5.3 immediately, and the proof is complete. □
Theorem 5.4 Let be a quasi-ordered metric space with the metric and the quasi-order ⪯∗ defined in (71) and (70), respectively. Let I and J be a disjoint pair of . Define the function by
where is defined in (1) according to ⪯∗. Suppose that the following conditions are satisfied:
-
F is -increasing;
-
there exists a function such that, for any with or , the inequalities are satisfied for all ;
-
there exists a function of the contractive factor such that, for any with or , the inequalities
(76)
are satisfied for , where , and the functions and satisfy the following inequalities: for ,
and
-
there exists such that or .
Then there exists a -chain-unique solution of the system of integral equations (65).
Proof We first have
By applying Theorem 5.2 to the argument in the proof of Theorem 5.3, the desired result can be obtained immediately. □
Corollary 5.3 Let be a quasi-ordered metric space with the metric and the quasi-order ⪯∗ defined in (71) and (70), respectively. Let I and J be a disjoint pair of . Define the function by
where is defined in (1) according to ⪯∗. Suppose that the following conditions are satisfied:
-
F is -increasing;
-
there exists a function of the contractive factor such that, for any with or , the inequalities
are satisfied for , where , and the function satisfies the following inequality: for ,
-
there exists such that or .
Then there exists a -chain-unique solution of the system of integral equations (65).
Proof For , we define the function by
and the function by
Since
we have
for all . The desired result follows from Theorem 5.4 immediately, and the proof is complete. □
Compared to Corollary 5.3, we consider the different type of inequalities below.
Theorem 5.5 Let be a quasi-ordered metric space with the metric and the quasi-order ⪯∗ defined in (71) and (70), respectively. Let I and J be a disjoint pair of . Define the function by
where is defined in (1) according to ⪯∗. Suppose that the following conditions are satisfied:
-
F is -increasing;
-
there exists a function of the contractive factor such that, for any with or , the inequalities
(81)
are satisfied for , where , and the function satisfies the inequalities: for ,
for ;
-
there exists such that or .
Then there exists a -chain-unique solution of the system of integral equations (65).
Proof For , we define a function by (79). Now, we have
Using Theorem 5.2, we complete the proof. □
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Wu, HC. Fixed point theorems for the functions having monotone property or comparable property in the product spaces. Fixed Point Theory Appl 2014, 230 (2014). https://doi.org/10.1186/1687-1812-2014-230
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DOI: https://doi.org/10.1186/1687-1812-2014-230