Abstract
We introduce a generalized proximal weak contraction of rational type for the non-self-map and proved results to ensure the existence and uniqueness of best proximity point for such mappings in the setting of partially ordered metric spaces. Further, our results provides an extension of a result due to Luong and Thuan (2011) and also it provides an extension of Harjani (2010) to the case of self-mappings.
1. Introduction and Preliminaries
The fixed point theory of partially ordered metric space was introduced by Ran and Reurings [1], where they extended the Banach contraction principle in partially ordered sets with some applications to linear and nonlinear matrix equations. Subsequently, Nieto and Rodríguez-López [2] extended the result of Ran and Reurings and apply their results to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions. The following notion of an altering distance function was introduced by Khan et al. in [3].
Definition 1. A function is said to be an altering distance function if it satisfies the following conditions.(i)is continuous and nondecreasing.(ii) if and only if .
In [3], Khan et al. proved the fixed point theorems by using altering distance function together with contractive type condition.
Motivated by the interesting paper of Jaggi [4], in [5] Harjani et al. proved the following fixed point theorem in partially ordered metric spaces.
Theorem 2 (see [5]). Let be an ordered set and suppose that there exists a metric in such that is a complete metric space.
Let be a nondecreasing mapping such that Also, assume either is continuous or has the property thatIf there exists such that , then has a fixed point.
In [6] Luong and Thuan proved the following theorem.
Theorem 3 (see [6]). Let be an ordered set and suppose that there exists a metric in such that is a complete metric space. Let be a nondecreasing mapping such thatwhere is a lower semicontinuous function with if and only if , and .
Also, assume either is continuous or has the property (2).
If there exists such that , then has a fixed point.
In this article, we attempt to give a generalization of Theorem 3 by considering a non-self-map . Before getting into the details of our main theorem, let us give a brief discussion of best proximity point results.
1.1. Best Proximity Point
Let be a subset of a metric space . A mapping has a fixed point in if the fixed point equation has at least one solution. That is, is a fixed point of if . If the fixed point equation does not possess a solution, then for all . In such a situation, it is our aim to find an element such that is minimum in some sense. The best approximation theory and best proximity pair theorems are studied in this direction. Here we state the following well-known best approximation theorem due to Fan [7].
Theorem 4 (see [7]). Let be a nonempty compact convex subset of a normed linear space and be a continuous function. Then there exists such that .
Such an element in Theorem 4 is called a best approximant of in . Note that if is a best approximant, then need not be the optimum. Best proximity point theorems have been explored to find sufficient conditions so that the minimization problem has at least one solution. To have a concrete lower bound, let us consider two nonempty subsets of a metric space and a mapping . The natural question is whether one can find an element such that . Since , the optimal solution to the problem of minimizing the real valued function over the domain of the mapping will be the one for which the value is attained. point is called a best proximity point of if . Note that if , then the best proximity point is nothing but a fixed point of .
The existence and convergence of best proximity points is an interesting topic of optimization theory which recently attracted the attention of many authors [8–16]. Also one can find the existence of best proximity point in the setting of partially ordered metric space in [17–24].
The purpose of this article is to present best proximity point theorems for non-self-mappings in the setting of partially ordered metric spaces, thereby producing optimal approximate solutions for , where is a non-self-mapping. When the map is considered to be a self-map and is defined as identity function, then our result reduces to the fixed point theorem of Luong and Thuan [6].
Given nonempty subsets and of a metric space , the following notions are used subsequently: In [14], the authors discussed sufficient conditions which guarantee the nonemptiness of and . Moreover, in [12], the authors proved that is contained in the boundary of in the setting of normed linear spaces.
Definition 5 (see [17]). A mapping is said to be proximally increasing if it satisfies the condition that where .
One can see that, for a self-mapping, the notion of proximally increasing mapping reduces to that of increasing mapping.
Definition 6. A mapping is said to be generalized proximal weak contraction of rational type if it satisfies the condition that where , is an altering distance function, is a nondecreasing function with if and only if , and .
One can see that, for a self-mapping, the notion of generalized proximal weak contraction of rational type reduces to generalized weak contraction of rational type.
2. Main Results
Now, let us state our main result.
Theorem 7. Let be a nonempty set such that is a partially ordered set and is a complete metric space. Let and be nonempty closed subsets of the metric space such that . Let satisfy the following conditions.(i) is continuous, proximally increasing, and generalized proximal weak contraction of rational type such that .(ii)There exist and in such that Then, there exists an element in such that Further, the sequence , defined by converges to the element .
Proof. By hypothesis there exist elements and in such that Because of the fact that , there exists an element in such that Since is proximally increasing, we get .
Continuing this process, we can construct a sequence in such thatIf there exist such that , then . This means that is a best proximity point of and the proof is finished. Thus, we can suppose that for all .
Since , we getSuppose that there exists such that , and from (13), we have Hence, the sequence is monotone, nonincreasing and bounded. Thus, there exists such thatSince is a nonincreasing sequence, from (13), we getSuppose that . Then the inequality (16) implies thatBut, as and is nondecreasing function, and this gives us which contradicts (18). Hence,Now to prove that is a Cauchy sequence, suppose is not a Cauchy sequence. Then there exist for which we can find subsequences and of such that is smallest index for whichThis means thatLetting and using (20) we can conclude thatBy triangle inequality Letting in the above two inequalities, using (20) and (23), we getSince , , from (16), we have Using (25) and continuity of in the above inequality we can obtainBut, from we can find such that for any and consequently, Therefore, and this contradicts (27). Thus, is a Cauchy sequence in and hence converges to some element in . Since is continuous, we have .
Hence the continuity of the metric function implies that . But (12) shows that the sequence is a constant sequence with the value . Therefore, . This completes the proof.
Corollary 8. Let be a nonempty set such that is a partially ordered set and is a complete metric space. Let be a nonempty closed subset of the metric space . Let satisfy the following conditions.(i) is continuous, proximally increasing, and generalized proximal weak contraction of rational type.(ii)There exist elements and in such that with .Then, there exist an element in such that .
In what follows we prove that Theorem 7 is still valid for which is not necessarily continuous, assuming the following hypothesis in . has the property that
Theorem 9. Assume the conditions (30) and is closed in instead of continuity of in Theorem 7; then the conclution of Theorem 7 holds.
Proof. Following the proof of Theorem 7, there exists a sequence in satisfying the following condition:and converges to in . Note that the sequence in and is closed. Therefore, . Since , we get .
Since , there exist such thatSince is a nondecreasing sequence and , then . Particularly, for all . Since is proximally increasing and from (31) and (32), we obtain . But which implies . Therefore, we get that there exist elements and in such thatConsider the sequence that is constructed as follows:Arguing like above Theorem 7, we obtain that is a nondecreasing sequence and for certain . From (30), we have . Since for all , suppose that , is generalized proximal weak contraction of rational type; from (31) and (34) we have Taking limit as in the above inequality, we have which is a contradiction. Hence, . We have , and therefore , for all . From (34), we obtain is a best proximity point for . The proof is complete.
Corollary 10. Assume the condition (30) instead of continuity of in the Corollary 8; then the conclusion of Corollary 8 holds.
Now, we present an example where it can be appreciated that hypotheses in Theorems 7 and 9 do not guarantee uniqueness of the best proximity point.
Example 11. Let and consider the usual order and .
Thus, is a partially ordered set. Besides, is a complete metric space considering the euclidean metric. Let and be a closed subset of . Then, , and . Let be defined as . Then, it can be seen that is continuous, proximally increasing mappings such that . The only comparable pairs of elements in are for and there are no elements such that for . Hence, is generalized proximal weak contraction of rational type. It can be shown that the other hypotheses of the Theorems 7 and 9 are also satisfied.
However, has two best proximity points and .
Theorem 12. In addition to the hypotheses of Theorem 7 (resp., Theorem 9), suppose thatand then has a unique best proximity point.
Proof. From Theorem 7 (resp., Theorem 9), the set of best proximity points of is nonempty. Suppose that there exist elements in which are best proximity points. We distinguish two cases:
Case 1. If and are comparable.
Since and .
Since is a generalized proximal weak contraction of rational type, we get which implies , and by our assumption about , we get or .
Case 2. If is not comparable to .
By the condition (37) there exist comparable to and . We define a sequence as . Since is comparable with , we may assume that . Now using is proximally increasing, it is easy to show that for all .
Suppose that there exist such that , and again by using which is proximally increasing, we get . But, for all . Therefore, . Arguing like above, we obtain for all . Hence, as .
On the other hand, if for all , now using is a generalized proximal weak contraction of rational type, we have Since is nondecreasing, we get Hence, the sequence is monotone, nonincreasing and bounded. Thus, there exist such thatSuppose that . Then the inequality implies thatBut, as and is nondecreasing function, , and this gives which contradicts (42). Hence, . Analogously, it can be proved that . Finally, the uniqueness of the limit gives us .
Let us illustrate the above theorem with the following example.
Example 13. Let and consider the order and , where is usual order in .
Thus, is a partially ordered set. Besides, is a complete metric space where the metric is defined as . Let and be a closed subset of . Then, and . Let be defined as . Then, it can be seen that is continuous, proximally increasing mappings and proximally weak increasing such that . Now, we have to prove is a generalized proximal weak contraction of rational type. That is to prove where .
Note that and For and , we have and Now, we easily conclude that the mapping is a generalized proximal weak contraction of rational type. Hence all the hypotheses of the Theorem 12 are satisfied. Also, it can be observed that is the unique best proximity point of the mapping .
The following result, due to Luong and Thuan [6], is a corollary from the above Theorem 12, by taking .
Corollary 14. In addition to the hypothesis of Corollary 8 (resp., Corollary 10), suppose thatand then has a unique fixed point.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.