Coupled Fixed Point Theorems for (<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.19961pt" id="M1" height="13.3426pt" version="1.1" viewBox="-0.0657574 -9.14299 30.0329 13.3426" width="30.0329pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g14-253" d="M613 273C613 384 553 457 461 457C403 457 313 408 282 288L233 62C201 72 161 113 161 190C161 311 225 390 306 425L290 467C135 418 22 343 22 197C22 96 84 7 221 -10C188 -172 178 -208 171 -229L185 -242C239 -237 300 -221 316 -191L346 -4C459 5 613 105 613 273ZM511 273C511 168 453 67 356 54L409 334C420 390 425 406 444 406C473 406 511 359 511 273Z"/></g><g transform="matrix(.017,0,0,-0.017,10.938,0)"><path id="g50-45" d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"/></g><g transform="matrix(.017,0,0,-0.017,17.915,0)"><path id="g14-245" d="M684 417L672 457C631 445 586 428 561 408C542 393 531 368 521 331L470 159C449 95 423 54 364 49L461 519L452 523L364 495L284 58C243 79 222 112 222 179C222 207 223 234 226 261C229 289 232 327 232 345C232 430 198 457 157 457C114 457 66 430 22 385L33 345C47 356 62 365 78 365C95 365 107 357 107 324C107 316 104 267 101 238C99 216 98 193 98 171C98 42 183 -9 274 -13L243 -187C238 -218 255 -238 267 -242L315 -195L351 -9C375 -3 406 8 428 20C473 45 512 79 547 158C575 221 605 289 619 329C632 364 646 395 684 417Z"/></g></svg>)-Contractive Mixed Monotone Mappings in Partially Ordered Metric Spaces and Applications

Jain, Manish; Gupta, Neetu; Kumar, Sanjay

doi:https://doi.org/10.1155/2014/586096

International Journal of Analysis

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Abstract Introduction and Preliminaries Conclusion References Copyright Related Articles

Research Article | Open Access

Volume 2014 | Article ID 586096 | https://doi.org/10.1155/2014/586096

Coupled Fixed Point Theorems for ()-Contractive Mixed Monotone Mappings in Partially Ordered Metric Spaces and Applications

Manish Jain,¹Neetu Gupta,²and Sanjay Kumar³

Academic Editor: Tohru Ozawa

Received14 Nov 2013

Accepted22 Jan 2014

Published18 Mar 2014

Abstract

The object of this paper is to establish the existence and uniqueness of coupled fixed points under a (, )-contractive condition for mixed monotone operators in the setup of partially ordered metric spaces. Presented work generalizes the recent results of Berinde (2011, 2012) and weakens the contractive conditions involved in the well-known results of Bhaskar and Lakshmikantham (2006), and Luong and Thuan (2011). The effectiveness of our work is validated with the help of a suitable example. As an application, we give a result of existence and uniqueness for the solutions of a class of nonlinear integral equations.

1. Introduction and Preliminaries

Fixed point theory is an important tool for studying the phenomenon of nonlinear analysis and is a bridge bond between pure and applied mathematics. The theory has its wide applications in engineering, computer science, physical and life sciences, economics, and other fields. Banach [1] introduced the well-known classical and valuable theorem in nonlinear analysis, which is named after him, known as the Banach contraction principle. This celebrated principle has been extended and improved by various authors in many ways over the years (see for instance [2–17]). Nowadays, fixed point theory has been receiving much attention in partially ordered metric spaces, that is, metric spaces endowed with a partial ordering. Ran and Reurings [17] were the first to establish the results in this direction. The results were then extended by Nieto and Rodríguez-López [10] for nondecreasing mappings. Works noted in [18–24] are some examples in this direction.

The work of Bhaskar and Lakshmikantham [25] is worth mentioning, as they introduced the new notion of fixed points for the mappings having domain the product space , which they called coupled fixed points, and thereby proved some coupled fixed point theorems for mappings satisfying the mixed monotone property in partially ordered metric spaces. As an application, they discussed the existence and uniqueness of a solution for a periodic boundary value problem.

Definition 1 (see [25]). Let be a partially ordered set. The mapping is said to have the mixed monotone property if is monotone nondecreasing in and monotone nonincreasing in ; that is, for any , ,

Definition 2 (see [25]). An element is called a coupled fixed point of the mapping if and .

Bhaskar and Lakshmikantham [25] gave the following result.

Theorem 3 (see [25]). 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 , , , with and .
If there exist two elements , with and , then there exist such that and .

It has also been shown in [25, Theorem 2.2] that the continuity assumption of in Theorem 3 can be replaced by an alternative condition imposed on the convergent nondecreasing and nonincreasing sequences in the space .

Assumption 4. has the property that(i)if a nondecreasing sequence converges to , then for all ,(ii)if a nonincreasing sequence converges to , then for all .
These results were then extended and generalized by several authors in the last six years (see [26–40]). Luong and Thuan [26, Theorem 2.1] extended the results of Bhaskar and Lakshmikantham [25] under the following contractive condition: with and , where , are the functions such that is continuous and nondecreasing; if and only if ;, for all , ,and the function satisfies
Berinde [27, Theorem 3] in an alternative way generalized the results of Bhaskar and Lakshmikantham [25] under a weaker contraction as follows: with and , where .
Berinde [28, Thoerem 2] further generalized and complemented the results noted in [25] and extended the work presented in [27] by considerably weakening the involved contractive condition as with and , where , : are the functions such that is continuous and (strictly) increasing; for all ;, for all ,and the function satisfies .

2. Coupled Fixed Points

In order to proceed with developing of our work and obtain our results, we need to consider the following.

Let denote all functions which satisfy the following: is lower semicontinuous and (strictly) increasing; for all ; for all , .We note that if and only if for .

Also, for , denote all functions which satisfy the following: if ; if for .

Now we are ready to prove our first main result as follows.

Theorem 5. Let be a partially ordered set and there is a metric on such that is a complete metric space. Suppose is a mapping having mixed monotone property on . Assume there exist and such that for all , , , with and (or and ).
Suppose either (a) is continuous or (b) satisfies Assumption 4.
If there exist two elements , with or then there exist , such that that is, has a coupled fixed point in .

Proof. Without loss of generality, assume that there exist two elements , such that (8) holds; that is, and . Let and . Then and . Similarly, let and . Since has the mixed monotone property, then we have and . Continuing in the same way, we can easily construct two sequences and in such that , and If for some , we have , then and ; that is, has a coupled fixed point. So from now onwards, we suppose , for all ; that is, we suppose that either or .
Now applying inequality (7) with and , for all , we get then for all , we have where .
Clearly, for all . Then for any , we obtain By monotonicity of , together with (14), we can easily see that is a nonnegative decreasing sequence in . So we have for some . If , then using the properties of , we get which is a contradiction. Hence and so we have We now prove that and are Cauchy sequences. Suppose, to the contrary, that at least one of the sequences , is not a Cauchy sequence. Then there exists for which 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 (17). Then, By (17), (18), and triangle inequality, we have Letting and using (16) in the last inequality, we have Again, by triangle inequality By monotonicity of and property , we have Since , , and .
Then by (7), we have By (22) and (23), we have Since is lower semicontinuous by taking limit as , we get a contradiction. Thus and are Cauchy sequences in complete metric space and hence and for some , .
Now suppose that assumption (a) holds. Then which shows that is a coupled fixed point of .
Suppose now assumption (b) holds. Since is a nondecreasing sequence that converges to , we have that for all . Similarly, for all .
Then So and hence which imply, by the monotonicity of and condition (7), Since is lower semicontinuous, by taking the limit as in the above inequality, we get Hence, we get and . Therefore, has a coupled fixed point .

Remark 6. In Theorem 5, substituting for implies the main result of Berinde [28, Theorem 2]. Note that the function satisfies all the conditions of our result. It is easy to verify this as in Theorem 2 [28] is a continuous function and for all such that . On the other hand, if for .

Remark 7. In Theorem 5, let be . Then it is easy to see that substituting for , where , yields the main result of Berinde [27, Theorem 3].
The following example shows that Theorem 5 is more general than Theorem 2.1 [25] (stated as Theorem 3 in the present paper) and Theorem 2.1 in [26], since the contractive condition (7) is more general than the contractive conditions (2) and (3).

Example 8. Let . Then is a partially ordered set with the natural ordering of real numbers. Let be defined by Then is a complete metric space.
Define by , .
Then is continuous, has mixed monotone property, and satisfies condition (7) but does not satisfy either condition (2) or condition (3). Indeed, assume there exists , such that (2) holds. Then, we must have by which, for , we get which for implies , a contradiction, since . Hence does not satisfy contractive condition (2).
Further, the contractive condition (3) is also not satisfied. Assume, to the contrary, that there exist , satisfying and , respectively, such that (3) holds. Then, we must have for all and . On taking , and letting , we obtain But by we have and hence, we deduce that, for all , , that is, , which contradicts . This shows that does not satisfy (3).

Now, we prove that (7) holds. Indeed, for and , we have

and by summing up the last two inequalities we get exactly (7) with , . Also, , are the two points in such that and .

So by Theorem 5 we obtain that has a coupled fixed point but none of Theorem 2.1 in [25] and Theorem 2.1 in [26] can be applied to in this example.

Next, we prove the uniqueness of coupled fixed point for our main result by defining a partial ordering on . If is a partially ordered set, then we endow the product with the following partial order: Then, we say , .

We say that and are comparable if or .

Theorem 9. Let all the conditions of Theorem 5 be fulfilled and for every , in , there exists a in that is comparable to and . Then has a unique coupled fixed point.

Proof. From Theorem 5, the set of coupled fixed points of is nonempty. Assume that and are two coupled fixed points of . We will show that and .
By assumption, there exists that is comparable to and . We define the sequences and as follows: Since is comparable to , we may assume . Following the proof of Theorem 5 we obtain inductively and therefore, by (7), we get Thus, we have , where . Inspired with the proof of Theorem 5 we can conclude that converges to for some . If , then we have which shows a contradiction. Thus ; that is, , which implies Similarly, we obtain that , and hence, by the uniqueness of limit and .

Theorem 10. Let all the conditions of Theorem 5 be fulfilled and suppose that , are comparable. Then has a unique fixed point; that is, there exist such that .

Proof. By Theorem 5, without loss of generality, we assume that Since and are comparable, we have or . We assume the second case. Then, by the mixed monotone property of , we have and, hence, by making use of induction, we can get Also, we have then, by the continuity of the distance function , we can obtain Further, on setting and in (7), we have Taking the limit as in the last inequality, we get If , then by we get a contradiction. So and hence . Similarly, uniqueness of can be easily established.

3. Application

As an application of the results proved in Section 2, we study the existence of unique solution of the following integral equation: Let denote the class of functions which satisfies the following conditions: for .

Clearly, is nonempty, as for given , is in , where , .

We assume that the functions , , , and fulfill the following conditions.

Assumption 11. (i) , , for all , .
(ii) There exist the positive numbers , , and , such that for all and , , with , the following conditions hold:
(iii) There exists some satisfying (57), such that where

Definition 12. An element with is called a coupled lower and upper solution of the integral equation (54) if and
for all .

Theorem 13. Consider the integral equation (54) with , for and . Suppose that there exists a coupled lower-upper solution for (54) and the Assumption 11 is satisfied. Then the integral equation (54) has a unique solution in .

Proof. Consider the natural order relation on ; that is, for , It is well known that is a complete metric space with respect to the sup metric It is easy to verify that condition (b) in Theorem 5 holds on the complete metric space . Also, is a partially ordered set under the following order relation in : For any , , and , for each , are in and are the upper and lower bounds of , respectively. Therefore, for every , , there exists a that is comparable to and .
Now we define the mapping by First, we show that has the mixed monotone property. To do this, let , and ; that is, for all . Then using Assumption 11, for any and all we obtain which implies that . Similarly, if , and , then for any . Let be as given in Assumption 11. Then for , , , such that and we get Since the function is nondecreasing and and , we have hence by (68), we obtain Similarly, we can obtain By summing up (70) and (71), multiplying by and dividing by 2, and then taking supremum with respect to we get, by using (60)-(61), Now, since is nondecreasing, we have and so by the definition of . Thus we get which is just the contractive condition (7) for , where . Now, let be a coupled upper-lower solution of (54). Then we have for all . Finally, Theorems 5 and 9 give that has a unique coupled fixed point . Since , then the hypotheses of Theorem 10 is satisfied and, therefore, there exists a unique such that for all ; that is, the integral equation (54) has a unique solution.

Remark 14. Trivial choices of the functions are , , , and , . Note that here we will assume that and .

4. Conclusion

We conclude that the obtained results improve, generalize, and enrich various recent coupled fixed point theorems in the framework of partially ordered metric spaces in a way that is essentially more natural, by considering the more general (symmetric) contractive condition (7). The theoretical results are accompanied by an applied example and an application to the nonlinear integral equation.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Copyright © 2014 Manish Jain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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