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.
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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 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 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 an empty set.
Remark 2.2 For any , we have the following observations.
-
(a)
If for some disjoint pair I and J of , then x and y are ⪯-mixed comparable.
-
(b)
If x and y are ⪯-mixed comparable, then there exists a disjoint pair I and J of such that .
Definition 2.3 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 2.4 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 2.5 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 element as follows.
-
(A)
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; 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 2.4, it follows that if is a mixed -monotone seed element, then it is also a mixed ⪯-monotone seed element.
Example 2.6 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 2.7 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 can consider a product metric space in which the metric is induced by the original metric d. For example, the following distance functions
and
make to be the product metric spaces. For the general product metric , we consider the following concepts.
-
We say that the metrics and d are compatible in the sense of preserving convergence if and only if, given a sequence in , the following statement holds:
-
We say that the metrics and d are compatible in the sense of preserving continuity if and only if, given any , there exists a positive constant (which depends on ϵ) such that the following statement holds:
We can check that the product metric defined in (5) or (6) is compatible with d in the sense of preserving convergence and continuity.
Proposition 2.8 If and d are compatible in the sense of preserving continuity, then and d are compatible in the sense of preserving convergence.
Proof Suppose that . By definition, given any , there exists such that for all , i.e., for all and . For the converse, given any , there exist such that for all , where . Let
It follows that for all and all , i.e., for all . This completes the proof. □
Mizoguchi and Takahashi [16, 17] considered the mapping that satisfies the following condition:
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 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 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
Then, for any strictly decreasing sequence in , we have
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 .
Suppose that and d are compatible in the sense of preserving continuity. Then F is continuous at if and only if each is continuous at for . Indeed, it is obvious that if F is continuous at , then each is continuous at for . For the converse, given any , there exists such that implies , where . Let
It follows that implies for all , i.e., . Next, we propose another concept of continuity.
Definition 2.11 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 2.12 Let be a metric space, and let and be functions defined on into itself. Suppose that and d are compatible in the sense of preserving continuity. Then F is continuous with respect to f at if and only if, given any , there exists such that with for all implies for all .
Proof Suppose that F is continuous with respect to f at . By definition, given any , there exists such that with implies . Let . It follows that for all if and only if , which implies , i.e., for all . For the converse, given any , there exists such that for all implies for all . Let . It follows that if and only if for all , which implies for all , i.e., . This completes the proof. □
Lemma 2.13 Let be a metric space. If as with respect to the metric d, then, given any fixed , as .
Theorem 2.14 Suppose that the quasi-ordered metric space is mixed-monotonically complete, and that the metrics and d are compatible in the sense of preserving continuity. 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;
-
eachis continuous onfor.
Suppose that there exist a function and a function of contractive factor such that, for any two ⪯-mixed comparable elements x and y in , the following inequalities are satisfied:
and, for each ,
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 2.4, it follows that, for each , and are ⪯-mixed comparable. According to 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 2.4, it follows that, for each , and are ⪯-mixed comparable. Then we have
Let
Using (11) and Lemma 2.10, we obtain
Using (12) and Lemma 2.10, we also obtain
Since by Lemma 2.10 again, from (13) and (14), it follows that
which implies
For with , since , from (15) we have
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 . Since the metrics and d are compatible in the sense of preserving continuity, by Proposition 2.8, it follows that as . Since each is continuous on , we also have
Since is continuous with respect to f on , by Proposition 2.12, given any , there exists such that with for all implies
Since as for all , given , there exists such that
For each , by (16) and (17), 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 2.15 We have the following observations.
-
In Theorem 2.14, 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 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 -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 is mixed-monotonically complete, and that the metrics and d are compatible in the sense of preserving continuity. Let I and J be any disjoint pair of . 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 or the sequentially mixed ⪯-monotone property;
-
is continuous with respect to f on;
-
eachis continuous onfor.
Suppose that there exist a function and a function of contractive factor such that, for any with or , the following inequalities are satisfied:
and, for each ,
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 inequalities (20), we obtain
Using the argument in the proof of Theorem 2.14, 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 2.4, 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 2.4, it follows that in X is a mixed ⪯-monotone sequence for all . Therefore, we obtain 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 2.14. 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 is complete (not mixed-monotonically complete), then the assumption for f having the sequentially mixed -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 . However, we need to introduce the concept of mixed-monotone convergence given below.
Definition 2.18 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 2.19 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 (that depend on n) of such that or , where or is allowed to be an empty set.
Definition 2.20 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 .
-
We say that the function f has the ⪯-comparable property if and only if, given any two ⪯-comparable elements x and y in , the function values and are ⪯-comparable.
-
We say that the function f has the -comparable property if and only if, given any two -comparable elements x and y in , the function values and are -comparable.
Since we shall study the coincidence point without considering the continuity of , we can also consider the assumption that the metrics 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 is mixed-monotonically complete and preserves the mixed-monotone convergence. Assume that the metrics and d are compatible in the sense of preserving 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;
-
eachis continuous onfor.
Suppose that there exist a function and a function of contractive factor such that, for any two ⪯-mixed comparable elements x and y in , the following inequalities are satisfied:
and, for each ,
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 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 2.14, 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 (a) of Remark 2.19, we see that, for each , and are ⪯-mixed comparable. Since f has the ⪯-comparable property, it follows that and are ⪯-mixed comparable. For each , it follows that
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., for some k, 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 , equalities (25) say 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. □
Theorem 2.22 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence. Assume that the metrics and d are compatible in the sense of preserving convergence. Let I and J be any disjoint pair of . 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 or the sequentially mixed ⪯-monotone property;
-
f has the-comparable property for any disjoint pairandof;
-
eachis continuous onfor.
Suppose that there exist a function and a function of contractive factor such that, for any and any disjoint pair and of with or , the following inequalities are satisfied:
and, for each ,
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 such that and that and are comparable with respect to the quasi-order ‘’, 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 2.16, 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 2.19, 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
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 . 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 such that , then .
(iii)′ Suppose that is obtained from part (i). Then is a fixed point of F for any .
The assumption that f has the -comparable property for any disjoint pair and of in Theorem 2.22 can be dropped by strengthening inequalities (26) as shown below.
Theorem 2.24 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence. Assume that the metrics and d are compatible in the sense of preserving 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 sequentially mixed-monotone property or the sequentially mixed ⪯-monotone property;
-
eachis continuous onfor.
Suppose that there exists a function such that, for any , the following inequality is satisfied:
and that there exists a function of contractive factor such that, for any and any disjoint pair and of with or , the following inequality is satisfied:
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 such that and that and are comparable with respect to the quasi-order ‘’, 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 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 be a quasi-ordered 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 2.26 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence. Assume that the metrics and d are compatible in the sense of preserving continuity. 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;
-
eachis continuous onfor.
Suppose that there exist a function and a function of contractive factor such that, for any two ⪯-mixed comparable elements x and y in , the following inequalities are satisfied:
and, for each ,
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 Proposition 2.8 and part (i) of Theorem 2.21, we have . From Theorem 2.14, 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 2.21, we have . Therefore, by the triangle inequality, we obtain
which says that . This proves part (i).
To prove part (ii), since and are ⪯-mixed comparable, part (iii) of Theorem 2.21 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 2.21, we have . From (34), we can similarly obtain . This completes the proof. □
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 2.27 Let be a quasi-ordered 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 2.28 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence. Assume that the metrics and d are compatible in the sense of preserving continuity. 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 or the sequentially mixed ⪯-monotone property;
-
is continuous with respect to f on;
-
eachis continuous onfor.
Suppose that there exist a function and a function of contractive factor such that, for any and any disjoint pair and of with or , the following inequalities are satisfied:
and, for each ,
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 Proposition 2.8 and part (i) of Theorem 2.22, 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 2.22, we have . Therefore, according to (34), we can obtain . 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 . 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 .
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 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 3.1 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 3.2 Suppose that the quasi-ordered metric space is monotonically complete, and that the metrics and d are compatible in the sense of preserving continuity. Consider the functions and satisfying for some . Let be a -monotone seed element in . 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;
-
eachis continuous onfor.
Suppose that there exist a function and a function of contractive factor such that, for any with or , the following 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 inequalities (38), we obtain
Since f is -monotone, it follows that for all or for all . Then we have
According to the proof of Theorem 2.14, 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 2.14. This completes the proof. □
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 3.3 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 3.4 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 3.5 Suppose that the quasi-ordered metric space is monotonically complete and preserves the monotone convergence. Assume that the metrics and d are compatible in the sense of preserving convergence. Consider the functions and satisfying for some . Let be a -monotone seed element in . Assume that the functions F and f satisfy the following conditions:
-
F and f are commutative;
-
f is-monotone;
-
eachis continuous onfor.
Suppose that there exist a function and a function of contractive factor such that, for any with or , the following 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 3.2, we can construct a sequence in such that and as for all , where is a -monotone sequence. From Remark 3.4, 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 2.21, 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 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 . 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 satisfying , then .
(iii)′ Suppose that is obtained from part (i). Then is a fixed point of F for any .
Next, we shall study the -chain-uniqueness for the common fixed point, which is different from the chain-uniqueness in Definition 2.27.
Definition 3.7 Let be a quasi-ordered 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 3.8 Suppose that the quasi-ordered metric space is monotonically complete and preserves the monotone convergence. Assume that the metrics and d are compatible in the sense of preserving continuity. Consider the functions and satisfying for some . Let be a -monotone seed element in . 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;
-
eachis continuous onfor.
Suppose that there exist a function and a function of contractive factor such that, for any with or , the following 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 2.8 and part (i) of Theorem 3.5, we have . From Theorem 3.2, 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 3.5, 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 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 is monotonically complete and preserves the monotone convergence. Assume that the metrics and d are compatible in the sense of preserving continuity. Consider the functions and satisfying for some . Let be a -monotone seed element in . Assume that the functions F and f satisfy the following conditions:
-
F and f are commutative;
-
f is-monotone for any disjoint pairandof;
-
is continuous with respect to f on;
-
eachis continuous onfor.
Suppose that there exist a function and a function of contractive factor such that, for any and any disjoint pair and of with or , the following 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 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 the proof of Theorem 3.8, 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., . Since f is -monotone for any disjoint pair and of , we also have or . If , i.e., , then we obtain
This contradiction says that for all , i.e., . Therefore, according to (45), we also have . 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 3.10 We strongly assume that inequalities (43) and (44) in Theorem 3.8 and inequalities (46) and (47) in Theorem 3.9 are satisfied for any . Then, from Remark 3.6 and the proofs of Theorems 3.8 and 3.9, we can see that parts (i) and (ii) can be combined as F and f have a unique common fixed point .
4 Fixed points of functions having mixed monotone property in the product spaces
We shall study the fixed points of functions having mixed monotone property in the product space. The concept of mixed monotone property for functions was adopted for presenting the coupled fixed point theorems.
Let be a quasi-ordered set, and let I and J be the disjoint pair of . Consider the function .
-
We say that F is -increasing if and only if implies .
-
We say that F is -decreasing if and only if implies .
From (2), we see that F is -decreasing if and only if it is -increasing, and F is -increasing if and only if it is -decreasing.
Example 4.1 For , we take the disjoint pair and of . We say that the function has the I-mixed monotone property if and only if the following conditions are satisfied:
-
is increasing in the corresponding variables , , ;
-
is decreasing in the corresponding variables , .
According to (1), it follows that implies , i.e., F is I-increasing. Now, we assume that the function has the J-mixed monotone property, i.e., the following conditions are satisfied:
-
is increasing in the corresponding variables , .
-
is decreasing in the corresponding variables , , .
It follows that implies , i.e., F is J-decreasing on .
Based on Example 4.1, the general definition is given below.
Definition 4.2 Let be a quasi-ordered set, and let and be the disjoint pair of . We say that the function has the I-mixed monotone property if and only if the following conditions are satisfied:
-
F is increasing in the positions, respectively; that is, if for , then
for all fixed with and ;
-
F is decreasing in the positions, respectively; that is, if for , then
for all fixed with and .
It can be realized that the function has the J-mixed monotone property if and only if the following conditions are satisfied:
-
F is increasing in the positions, respectively; that is, if , then
for each ;
-
F is decreasing in the positions, respectively; that is, if , then
for each .
Remark 4.3 According to (1), if the function has the I-mixed monotone property, then F is -increasing. Also, if the function has the J-mixed monotone property, then F is -increasing (i.e., -decreasing) on .
Now, considering the function , we are going to define many monotonic concepts of F on as follows.
Definition 4.4 Let be a quasi-ordered set, and let I and J be the disjoint pair of . Consider the quasi-ordered 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 4.5 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 4.4 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 4.4 can be reduced to only consider the -increasing and -decreasing cases.
Definition 4.6 Let be a quasi-ordered set, and let and be the disjoint pair of . We say that the function has the I-mixed monotone property on if and only if its k th component function satisfies the following conditions:
-
has the I-mixed monotone property for ; in other words, is increasing in the positions and is decreasing in the positions, respectively;
-
has the J-mixed monotone property for ; in other words, is decreasing in the positions and is increasing in the positions, respectively.
The meaning of the J-mixed monotone property for F can be similarly realized.
Remark 4.7 According to (1), if the function has the I-mixed monotone property, then F is -increasing. Also, if the function has the J-mixed monotone property, then F is -increasing, i.e., -decreasing.
Example 4.8 Continued from Example 4.1, the function has the I-mixed monotone property if and only if the component functions of F for satisfy the following conditions:
-
the component functions have the I-mixed monotone property for ;
-
the component functions have the J-mixed monotone property for .
Example 4.9 Continued from Example 4.8, given the initial element , we consider the iteration . Then, according to (4), we can define two quasi-ordered sets and as follows.
-
(i)
Assume that , i.e., for and for .
-
♦ For , we are going to claim . This can be realized by simply checking the case of . By definition, we see that is increasing in the corresponding variables , , , and is decreasing in the corresponding variables , . Therefore, we obtain
-
♦ For , we are going to claim . This can be realized by simply checking the case of . By definition, we see that is increasing in the corresponding variables , , and is decreasing in the corresponding variables , , . Therefore, we obtain
By induction, we can show that the sequences are ⪯-increasing for , and are ⪯-decreasing for . Therefore, according to (4), we can induce a quasi-ordered set .
-
-
(ii)
Assume that , i.e., for and for . Equivalently, we have by (2). We can similarly show that the sequences are ⪯-decreasing for , and are ⪯-increasing for . Therefore, according to (4), we can induce a quasi-ordered set .
By referring to Example 4.9, we have the following general result.
Lemma 4.10 Let be a quasi-ordered set, and let I and J be the disjoint pair of . Assume that the function has the I-mixed monotone property on . Given the initial element , we define the sequence by . Then the following statements hold true.
-
(i)
Suppose that . Then the sequences are ⪯-increasing for each and are ⪯-decreasing for each . In other words, is a -increasing sequence.
-
(ii)
Suppose that . Then the sequences are ⪯-decreasing for each and are ⪯-increasing for each . In other words, is a -decreasing sequence.
Moreover, the initial element is a -monotone seed element in .
Proof To prove part (i), we have the following cases:
-
If , then has the I-mixed monotone property. Therefore, we have
by checking one component at each time.
-
If , then has the J-mixed monotone property. Therefore, we have
by checking one component at each time.
Therefore, by induction, part (i) follows immediately. A similar argument can also apply to part (ii), and the proof is complete. □
Theorem 4.11 Suppose that the quasi-ordered metric space is monotonically complete, and that the metrics and d are compatible in the sense of preserving continuity. Given a disjoint pair I and J of , assume that the function is continuous on and has the I-mixed monotone property, and that there exist a function and a function of contractive factor such that, for any with or , the following inequalities are satisfied:
and
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 below
for all .
Proof According to (48), we have or . From Lemma 4.10, the initial element is a -monotone seed element in . Therefore, the results follow immediately from Theorem 3.2 by taking f as the identity function. This completes the proof. □
Next, we can consider the chain-uniqueness and drop the assumption of continuity of F by assuming that preserves the monotone convergence.
Theorem 4.12 Suppose that the quasi-ordered metric space is monotonically complete and preserves the monotone convergence, and that the metrics and d are compatible in the sense of preserving convergence. Given a disjoint pair I and J of , assume that the function has the I-mixed monotone property, and that there exist a function and a function of contractive factor such that, for any with or , the following inequalities are satisfied:
and
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 (48) for all .
Proof According to (48), we have or . From Lemma 4.10, the initial element is a -monotone seed element in . Therefore, the results follow immediately from Theorem 3.5 by taking f as the identity function. This completes the proof. □
The following fixed point theorem considers the different monotone conditions for F.
Theorem 4.13 Suppose that the quasi-ordered metric space is monotonically complete, and that the metrics and d are compatible in the sense of preserving continuity. Given a disjoint pair I and J 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 contractive factor such that, for any with or , the following 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 (48) 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 (48), 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 3.2 by taking f as the identity function. This completes the proof. □
Next, we can consider the chain-uniqueness and drop the assumption of continuity of F by assuming that preserves the monotone convergence.
Theorem 4.14 Suppose that the quasi-ordered metric space is monotonically complete and preserves the monotone convergence, and that the metrics and d are compatible in the sense of preserving convergence. Given a disjoint pair I and J 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 contractive factor such that, for any with or , the following 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 (48) for all .
Proof From the proof of Theorem 4.13, we see that the initial element is a -monotone seed element in . Therefore, the results follow immediately from Theorem 3.5 by taking f as the identity function. This completes the proof. □
The following fixed point theorem considers the odd integer and the different monotone conditions for F.
Theorem 4.15 Suppose that the quasi-ordered metric space is mixed-monotonically complete, and that the metrics and d are compatible in the sense of preserving continuity. Given a disjoint pair I and J of , assume that the function is continuous on and -decreasing, and that there exist a function and a function of contractive factor such that, for any with or , the following inequalities are satisfied:
and
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 from (48) for all .
Proof Since F is -decreasing and p is an odd integer, we see that is -decreasing. It follows that implies , and 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 2.16 by taking f as the identity function. This completes the proof. □
Next, we can consider the chain-uniqueness and drop the assumption of continuity of F by assuming that preserves the mixed-monotone convergence.
Theorem 4.16 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence, and that the metrics and d are compatible in the sense of preserving convergence. Given a disjoint pair I and J of , suppose that there exists such that the following conditions are satisfied:
-
the functionis-decreasing;
-
or.
Assume that there exist a function and a function of contractive factor such that, for any and any disjoint pair and of with or , the following 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 (48) for all .
Proof From the proof of Theorem 4.15, we see that the initial element is a mixed -monotone seed element in . Therefore, the results follow immediately from Theorem 2.22 by taking f as the identity function. This completes the proof. □
5 Fixed points of functions having mixed comparable property in the product spaces
We shall study the fixed points of functions having mixed comparable property in the product space. Here, we shall consider the mixed-monotonically complete quasi-ordered metric space.
Definition 5.1 Let I and J be a disjoint pair of . Given a quasi-ordered set , we consider the corresponding quasi-ordered set .
-
We say that the function has 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.
-
We say that the function has 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 5.2 Suppose that the quasi-ordered metric space is mixed-monotonically complete, and that the metrics and d are compatible in the sense of preserving continuity. Assume that the function is continuous on and has the ⪯-mixed comparable property, and that there exist a function and a function of contractive factor such that, for any two ⪯-mixed comparable elements x and y in , the following 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 (48) for all .
Proof According to (48), we see 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 2.4, 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 2.14 immediately by taking f as the identity function. This completes the proof. □
Next, we can drop the assumption of continuity of F by assuming that preserves the mixed-monotone convergence.
Theorem 5.3 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence, and that the metrics and d are compatible in the sense of preserving convergence. Assume that the function has the ⪯-mixed comparable property, and that there exist a function and a function of contractive factor such that, for any two ⪯-mixed comparable elements x and y in , the following 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 metrics and d are compatible in the sense of preserving continuity, 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 (48) for all for all .
Proof According to argument in the proof of Theorem 5.2, we see that the initial element is a mixed ⪯-monotone seed element in . Therefore, part (i) follows from Theorem 2.21 immediately by taking f as the identity function. Also, part (ii) follows from Theorem 2.26 immediately by taking f as the identity function. This completes the proof. □
Theorem 5.4 Suppose that the quasi-ordered metric space is mixed-monotonically complete, and that the metrics and d are compatible in the sense of preserving continuity. Given a disjoint pair I and J of , assume that the following conditions are satisfied:
-
the functionis continuous onand has the-comparable property;
-
there existssuch thatandare comparable with respect to the quasi-order ‘’ for some.
Assume that there exist a function and a function of contractive factor such that, for any with or , the following inequalities
and
are satisfied for all . Then has a fixed point such that each component of is the limit of the sequence constructed from (48) for all .
Proof According to (48), 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 2.16 immediately by taking f as the identity function. This completes the proof. □
Next, we can drop the assumption of continuity of F by assuming that preserves the mixed-monotone convergence.
Theorem 5.5 Suppose that the quasi-ordered metric space is mixed-monotonically complete and preserves the mixed-monotone convergence, and that the metrics and d are compatible in the sense of preserving convergence. Given a disjoint pair I and J of , assume that the following conditions are satisfied:
-
the functionhas the-comparable property;
-
there existssuch thatandare comparable with respect to the quasi-order ‘’ for some.
Suppose that there exist a function and a function of contractive factor such that, for any and any disjoint pair and of with or , the following 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 metrics and d are compatible in the sense of preserving continuity, 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 (48) for all .
Proof According to the argument in the proof of Theorem 5.4, we see that the initial element is a mixed -monotone seed element in . Therefore, part (i) follows from Theorem 2.22 immediately by taking f as the identity function. Also, part (ii) follows from Theorem 2.28 immediately by taking f as the identity function. This completes the proof. □
6 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 system of integral equations (49) 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 function by . Then the system of integral equations as shown in (49) can be written as the following vectorial form of integral equation:
where . Equivalently, we shall find such that (50) is satisfied, which also says that is a solution of (50).
Definition 6.1 Consider the quasi-ordered metric space .
-
(a)
We say that is a unique solution of the system of integral equations (50) in the ⪯-mixed comparable sense if and only if the following conditions are satisfied:
-
♦ is a solution of (50);
-
♦ if is another solution of (50) 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 (50) if and only if the following conditions are satisfied:
-
♦ is a solution of (50);
-
♦ if is another solution of (50) satisfying or (i.e., and are comparable with respect to ≼), then .
-
Theorem 6.2 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
where is defined in (5) or (6). Suppose that the following conditions are satisfied:
-
♦ F is-increasing;
-
♦ there exist a functionand a function of contractive factorsuch that, for anywithor, the following inequalities
(51)
and
are satisfied for all ;
-
♦ there existssuch thator.
Then there exists a -chain-unique solution of the system of integral equations (50).
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 4.14, 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 integral equation (50). This completes the proof. □
Corollary 6.3 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
where is defined in (5) or (6). Suppose that the following conditions are satisfied:
-
♦ F is-increasing;
-
♦ there exists a function of contractive factorsuch that, for anywithor, the following inequality
is satisfied for all ;
-
♦ there existssuch thator.
Then there exists a -chain-unique solution of the system of integral equations (50).
Proof For , we define the function by
Then the desired result follows from Theorem 6.2 immediately. □
Remark 6.4 We have the following observations.
-
♦ The assumptions for inequalities (51) and (52) 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 inequalities (51) and (52).
-
♦ In Theorem 6.2, according to Remark 4.7, if the function F is assumed to have the I-mixed monotone property on instead of assuming it to be -increasing, then the results are still valid.
Lemma 6.5 For any , we define
Then the quasi-ordered metric space preserves the monotone convergence.
Proof Let be a ⪯-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 6.6 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 6.7 Let be a quasi-ordered metric space with the metric and the quasi-order ⪯∗ defined in (54) and (53), respectively. Let I and J be a disjoint pair of . Define the function by
where is defined in (1) according to ⪯∗, and is defined in (5) or (6) according to . Suppose that the following conditions are satisfied:
-
♦ F is-increasing;
-
♦ there exists a functionsuch that, for anywithor, the following inequality
is satisfied;
-
♦ there exists a function of contractive factorsuch that, for anywithor, the following inequality
(55)
is satisfied for , where , and the functions and satisfy the following inequalities: for ,
and
-
♦ there existssuch thator.
Then there exists a -chain-unique solution of the system of integral equations (50).
Proof Lemmas 6.5 and 6.6 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 6.2, we complete the proof. □
Corollary 6.8 Let be a quasi-ordered metric space with the metric and the quasi-order ⪯∗ defined in (54) and (53), respectively. Let I and J be a disjoint pair of . Define the function by
where is defined in (1) according to ⪯∗, and is defined in (5) or (6) according to . Suppose that the following conditions are satisfied:
-
F is-increasing;
-
there exists a function of contractive factorsuch that, for anywithor, the following inequality
is satisfied for , where , and the function satisfies the following inequality
-
there existssuch thator.
Then there exists a -chain-unique solution of the system of integral equations (50).
Proof For , we define the function by
and the function by
Since
we have
for . The desired result follows from Theorem 6.7 immediately, and the proof is complete. □
Compared to Corollary 6.8, we consider a different type of inequalities below.
Theorem 6.9 Let be a quasi-ordered metric space with the metric and the quasi-order ⪯∗ defined in (54) and (53), respectively. Let I and J be a disjoint pair of . Define the function by
where is defined in (1) according to ⪯∗, and is defined in (5) or (6) according to . Suppose that the following conditions are satisfied:
-
♦ F is-increasing;
-
♦ there exists a function of contractive factorsuch that, for anywithor, the following inequality
(61)
is satisfied for , where , and the function satisfies the following inequality: for ,
for ;
-
♦ there existssuch thator.
Then there exists a -chain-unique solution of the system of integral equations (50).
Proof For , we define a function by (59). Now, we have
Using Theorem 6.2, we complete the proof. □
7 Applications to the system of ordinary differential equations
We consider the quasi-ordered metric space and the following system of ordinary differential equations:
where , and are continuous functions for .
Definition 7.1 Consider the quasi-ordered metric space and the product space .
-
(a)
We say that is a unique solution of the system of boundary value problem (63) in the ⪯-mixed comparable sense
-
is a solution of (63);
-
if is another solution of (63) such that and are ⪯-mixed comparable, then .
-
is a solution of (63);
-
if is another solution of (63) satisfying or (i.e., and are comparable with respect to ≼), then .
if and only if the following conditions are satisfied:
-
Let I and J be the disjoint pair of .
-
(b)
We say that is a -chain-unique solution of the system of boundary value problem (63) if and only if the following conditions are satisfied:
Definition 7.2 Let be a quasi-ordered set.
-
We say that the quasi-order ‘⪯’ is compatible with the addition if and only if, for any with , we have for any .
-
We say that the quasi-order ‘⪯’ is compatible with the nonnegative multiplication if and only if, for any with , we have for any nonnegative function .
Remark 7.3 Suppose that the quasi-order ‘⪯’ is compatible with the addition. If and , we want to claim . By definition, we immediately have
The transitivity proves the desired claim.
Remark 7.4 Suppose that the quasi-order ‘⪯’ in is compatible with the addition and nonnegative multiplication. Then we have the following observations.
-
For . If , and , then, from Remark 7.3, we have .
-
If with , then for any nonnegative function by the compatibility for nonnegative function.
Consider a disjoint pair I and J of . We say that the function defined by
is -increasing if and only if with implies . By referring to Definition 4.4, we can similarly define the other kinds of monotonic concepts.
Definition 7.5 Let be a quasi-ordered set. Consider the function such that for any fixed and for any fixed . We say that the integral of function is compatible with the quasi-order ‘⪯’ if and only if, for any fixed , implies
Example 7.6 For any , we define
If the function satisfies for any fixed and for any fixed , then the integral of function is compatible with the quasi-order ‘⪯’. Indeed, for any fixed , means that for all by (65), which also says that, for any ,
This shows (64) by (65) again.
Theorem 7.7 Suppose that the quasi-ordered metric space is monotonically complete and preserves the monotone convergence. Let I and J be a disjoint pair of , and let . Define the function by
and the function by
where is defined in (5) or (6). Suppose that the following conditions are satisfied:
-
the quasi-order ‘⪯’ inis compatible with the addition and nonnegative multiplication;
-
g is-increasing;
-
for any, the integrals of the following functions
(67)
are compatible with the quasi-order ‘⪯’ for .
-
there exist a functionand a function of contractive factorsuch that, for anywithor, the following inequalities
and
are satisfied for all ;
-
there existssuch thator.
Then the system of boundary value problem (63) has a -chain-unique solution.
Proof First of all, problem (63) can be written as follows:
We shall rewrite problem (68) as a system of integral equations. Multiplying on both sides, we have
which is equivalent to
By taking integration on both sides, we have
Since , we obtain
which implies
From (69) and (70), for , we have
Therefore, we obtain the following integral equations:
where is given in (66). We see that if is a solution of integral equation (71), then is a solution of problem (68), which also says that is a solution of the original boundary value problem (63).
Since , the quasi-order ‘⪯’ is compatible with the addition and nonnegative multiplication, and g is -increasing, it follows that, for each fixed , the function is -increasing by Remark 7.4. Since the integrals of the functions defined in (67) are compatible with the quasi-order ‘⪯’ for , it shows that the function F is -increasing. Theorem 6.2 says that the system of integral equations (71) has a -chain-unique solution , which also says that is a solution of the original boundary value problem (63). However, this -chain-uniqueness is about the system of integral equations (71), which is not the -chain-uniqueness about the original boundary value problem (63). Now, let be another solution of the system of boundary value problem (63) such that and are comparable with respect to ‘’. By referring to the derivation of (71), we can show that is also a solution of the system of integral equations (71). By the -chain-uniqueness given in Theorem 6.2, it follows that ; that is, is a -chain-unique solution of the system of boundary value problem (63). This completes the proof. □
Corollary 7.8 Suppose that the quasi-ordered metric space is monotonically complete and preserves the monotone convergence. Let I and J be a disjoint pair of , and let . Define the function by
and the function by
where is defined in (5) or (6). Suppose that the following conditions are satisfied:
-
the quasi-order ‘⪯’ inis compatible with the addition and nonnegative multiplication;
-
g is-increasing;
-
for any, the integrals of the following functions
(72)
are compatible with the quasi-order ‘⪯’ for .
-
there exists a function of contractive factorsuch that, for anywithor, the following inequality
is satisfied for all ;
-
there existssuch thator.
Then the system of boundary value problem (63) has a -chain-unique solution.
Proof For , we define the function by
Then the desired result follows from Theorem 7.7 immediately. □
Theorem 7.9 Let be a quasi-ordered metric space with the metric and the quasi-order ⪯∗ defined in (54) and (53), respectively. Let I and J be a disjoint pair of , and let be defined in (1) according to ⪯∗. Let . Define the function by
Suppose that the following conditions are satisfied:
-
there exists a functionsuch that, for anywithor, the following inequality
is satisfied;
-
there existswithsuch that, for anywithor, the following inequality
(74)
is satisfied for , where the function satisfies the following inequality: for ,
-
there exists such that
or
Then the system of boundary value problem (63) has a -chain-unique solution.
Proof It is easy to see that the quasi-order ‘⪯∗’ in is compatible with the addition and nonnegative multiplication. It follows that g is -increasing. According to Example 7.6, for any , we see that the integrals of the following functions
are compatible with the quasi-order ‘⪯∗’ for . Now, we define
In order to obtain
from the proof of Theorem 6.7, we need to check inequalities (55) and (57). Now, we take as a constant function with value , and φ as a constant function with value . Then φ is a function of contractive factor. According to (74), we see that inequality (55) is satisfied. We also have
which shows that inequality (57) is satisfied. According to the proof of Theorem 6.7, we see that (75) is satisfied. Therefore, the desired result follows from Theorem 7.7, and the proof is complete. □
Definition 7.10 For , we define some solution concepts as follows.
-
We say that w is a sub-solution of problem (63) if and only if
-
We say that w is a sup-solution of problem (63) if and only if
-
Let I and J be a disjoint pair of . We say that w is an -mixed solution of problem (63) if and only if
where I is allowed to be a nonempty set.
We have to emphasize that the -mixed solution and -mixed solution are essentially the same. It is obvious that if , then the -mixed solution is also a sup-solution, and if , then the -mixed solution is also a sub-solution.
Theorem 7.11 Let be a quasi-ordered metric space with the metric and the quasi-order ⪯∗ defined in (54) and (53), respectively. Let I and J be a disjoint pair of , and let be defined in (1) according to ⪯∗. Suppose that the following conditions are satisfied:
-
there exists a functionsuch that, for anywithor, the following inequality
is satisfied;
-
there existswithsuch that, for anywithor, the following inequality
is satisfied for , where the function satisfies the following inequality: for ,
-
there exists an-mixed solution of problem (63).
Then the system of boundary value problem (63) has a -chain-unique solution.
Proof Let be an -mixed solution of problem (63). Then, for , we have
which implies
Multiplying on both sides, we have
which is equivalent to
By taking integration on both sides, we have
Since , we obtain
which implies
From (76) and (77), for , we have
Therefore, we obtain
where is given in (66). We can similarly show that
which says that
Therefore, the desired result follows from Theorem 7.9 immediately. □
The assumption for the existence of -mixed solution in Theorem 7.11 can be replaced by the assumption for the existence of sub-solution or sup-solution in which J or I is taken to be an empty set, respectively.
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Wu, HC. 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. J Inequal Appl 2014, 518 (2014). https://doi.org/10.1186/1029-242X-2014-518
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DOI: https://doi.org/10.1186/1029-242X-2014-518