Abstract
We prove a coupled coincidence point theorem for mappings F : and g : , where F has the mixed g-monotone property, in partially ordered metric spaces via implicit relations. Our result extends and improves several results in the literature. Examples are also given to illustrate our work.
1. Introduction and Preliminaries
The notion of coupled fixed point was introduced by Guo and Lakshmikantham [1] in 1987. Later, Bhaskar and Lakshmikantham [2] defined the notions of mixed monotone mapping and proved some coupled fixed point theorems for the mixed monotone mappings. In this pioneer paper [2], they also discussed the existence and uniqueness of solution for a periodic boundary value problem. We start with recalling these basic concepts.
Definition 1.1 (see [2]). Let be a partially ordered set and . The mapping is said to have the mixed monotone property if is monotone nondecreasing in and is monotone nonincreasing in , that is, for any ,
Definition 1.2 (see [2]). An element is called a coupled fixed point of the mapping if
The main results of Bhaskar and Lakshmikantham in [2] are the following theorems.
Theorem 1.3 (see [2]). Let be a partially ordered set and suppose there exists a metric on such that is a complete metric space. Let be a continuous mapping having the mixed monotone property on . Assume that there exists a with for all and . If there exist two elements with then there exist such that
Theorem 1.4 (see [2]). Let be a partially ordered set and suppose there exists a metric on such that is a complete metric space. Assume that has the following property: (i)if a nondecreasing sequence , then for all ,(ii) if a nonincreasing sequence , then for all .
Let be a mapping having the mixed monotone property on . Assume that there exists a with
for all and . If there exist two elements with
then there exist such that
Afterwards, a number of coupled coincidence/fixed point theorems and their application to integral equations, matrix equations, and periodic boundary value problem have been established (e.g., see [3β28] and references therein). In particular, Lakshmikantham and ΔiriΔ [7] established coupled coincidence and coupled fixed point theorems for two mappings and , where has the mixed -monotone property and the functions and commute, as an extension of the fixed point results in [2]. Choudhury and Kundu in [15] introduced the concept of compatibility and proved the result established in [7] under a different set of conditions. Precisely, they established their result by assuming that and are compatible mappings. For the sake of completeness, we remind these characterizations.
Definition 1.5 (see [7]). Let be a partially ordered set and let and are two mappings. We say has the mixed -monotone property if is -nondecreasing in its first argument and is -nonincreasing in its second argument, that is, for any ,
Definition 1.6 (see [7]). An element is called a coupled coincident point of the mappings and if
Definition 1.7 (see [15]). The mappings and where , are said to be compatible if where and are sequences in such that and for all are satisfied.
Luong and Thuan [11] slightly extended the concept of compatible mappings into the context of partially ordered metric spaces, namely, -compatible mappings and proved some coupled coincidence point theorems for such mappings in partially ordered generalized metric spaces.
The concept of -compatible mappings is stated as follows.
Definition 1.8 (cf. [11]). Let be a partially ordered metric space. The mappings and are said to be -compatible if where and are sequences in such that , are monotone and for all are satisfied.
Let be a partially metric space. If and are compatible then they are -compatible. However, the converse is not true. The following example shows that there exist mappings that are -compatible but not compatible.
Example 1.9 (see [11]). Let with the usual metric , for all . We consider the following order relation on : Let be given by and be defined by Then and are -compatible but not compatible.
Indeed, let in such that are monotone and for some . Since for all , . The case is impossible. In fact, if . Then since are monotone, for all , for some . That is for all . This implies , for all , which is a contradiction. Thus . That implies for all , for some . That is for all . Thus, for all , Hence hold. Therefore and are -compatible.
Now let in be defined by We have but Thus, and are not compatible.
Implicit relation on metric spaces has been used in many articles (see, e.g., [29β31] and references therein). In this paper, we use the following implicit relation to prove a coupled coincidence point theorem for mappings and , where has the mixed -monotone property and are -compatible.
Let denote all functions which satisfy(i) is continuous,(ii) for each .
Obviously, if then .
Let denote all continuous functions which satisfy(H1) is nonincreasing in and ,(H2) there exists a function such that
It is easy to check that the following functions are in :(i), where are nonnegative real numbers satisfying ;(ii), where ;(iii), where .
In this paper, we prove a coupled coincidence point theorem for mappings satisfying such implicit relations.
2. Coupled Coincidence Point Theorem
Now we are going to prove our main result.
Theorem 2.1. Let be a partially ordered complete metric space. Suppose and are mappings such that has the mixed -monotone property. Assume that there exists such that
for all with and . Suppose , is continuous and is -compatible with . Suppose either(a) is continuous or;(b) has the following property:(i)if a nondecreasing sequence , then for all ,(ii)if a nonincreasing sequence , then for all .
If there exist two elements with
then and have a coupled coincidence point in .
Proof. Let be such that and . Since , we construct the sequences and in as follows:
By using the mathematical induction and the mixed -monotone property of , we can show that
If there is some such that and then
that means is a coupled coincidence point of and . Thus we may assume that for all .
Since and , from (2.1), we have
or
By the properties of , we have
which implies that
Similarly, one can show that
From (2.9) and (2.10), we have
which implies
This means that is a decreasing sequence of positive real numbers. So there is a such that
We will show that . Assume, to the contrary, that . Taking in (2.11), we have
which is a contradiction. Thus .
In what follows, we will show that and are Cauchy sequences. Suppose, to the contrary that at least one of or is not a Cauchy sequence. This means that there exists an for wich we can find subsequences of and of with such that
Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfies (2.15). Then
Using the triangle inequality and (2.16), we have
From (2.15) and (2.17), we have
Letting in the inequalities above and using (2.13) we get
By the triangle inequality
From the last two inequalities and (2.15), we have
Again, by the triangle inequality,
Therefore,
From (2.21) and (2.23), we have
Taking in the inequalities above and using (2.13), we get
From (2.19) and (2.25), the sequences , , , and have subsequences converging to , , and , respectively, and . We may assume that
We first assume that . Since , and . From (2.1), we have
or
or
Letting , we have
Thus,
which implies . That is a contradiction.
Using the same argument as above for the case , we also get a contradiction. Thus and are Cauchy sequences. Since is complete, there exist such that
Thus
Since and are -compatible, from (2.33), we have
Now, suppose that assumption (a) holds. We have
Taking the limit as in (2.36) and by (2.32), (2.34) and the continuity of and we get .
Similarly, we can show that . Therefore, and .
Finally, suppose that assumption (b) holds. Since is nondecreasing sequence and and is nonincreasing sequence and , by the assumption, we have and for all .
Since is continuous, from (2.32), (2.34), and (2.35) we have
We have
Letting and using (2.37), we have
which implies that . Hence . Similarly, one can show that .
Thus proved that and have a coupled coincidence point in .
Example 2.2 (see, e.g., [11]). Let , and be defined as in Example 1.9. Then(i) is complete and has the property(a)if a nondecreasing sequence , then for all ,(b) if a nonincreasing sequence , then for all ;(ii);(iii) is continuous and and are -compatible;(iv) there exist such that and ;(v) has the mixed -monotone property. Indeed, for every , let such that (a) if then or or or . Thus,
otherwise ,(b) if , then and , that is, and . Thus
therefore, is the -nondecreasing in its first argument. Similarly, is the -nonincreasing in its second argument;(vi) for , if and then . Indeed,(a) if and then and . Thus ,(b) if and then and . Thus if or then , otherwise . Similarly, if and then ,(c)if and then both are in one of the sets , , or and both are also in one of the sets , , or . Thus if or and or , otherwise, .
Therefore, all the conditions of Theorem 2.1 are satisfied with . Applying Theorem 2.1, we conclude that and have a coupled coincidence point.
Note that, we cannot apply the result of Choudhury and Kundu [15], the result of Choudhury et al. [32] as well as the result of Lakshmikantham and ΔiriΔ [7] to this example.