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Journal of Function Spaces
Volume 2016, Article ID 6275367, 11 pages
http://dx.doi.org/10.1155/2016/6275367
Research Article

Discussion on Some Recent Order-Theoretic Metrical Coincidence Theorems Involving Nonlinear Contractions

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received 22 December 2015; Accepted 16 February 2016

Academic Editor: Tomonari Suzuki

Copyright © 2016 Aftab Alam 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.

Abstract

We prove some coincidence theorems involving a pair of self-mappings and defined on an ordered metric space wherein is -increasing -contractive mapping. In our results, neither the whole space nor the range subspaces ( or ) are required to be complete. Instead, we use the completeness of a subspace of satisfying suitable conditions.

1. Introduction

The appearance of two noted papers by Goebel [1] and Jungck [2] containing two metrical coincidence theorems has attracted the attention of several researchers of this domain. The main difference between these two results lies in the fact that Goebel [1] employed the completeness of one of the range subspaces of the involved mappings while Jungck [2] assumed the completeness of entire metric space together with commutativity of the involved mappings. Specifically, Jungck’s theorem ensures the uniqueness of common fixed point while Goebel’s theorem is confined to mere coincidence points.

The origin of the order-theoretic metric fixed point theory can be traced back to Turinici [3, 4] which was later undertaken by Ran and Reurings [5] and Nieto and Rodríguez-López [6]. In recent years, the results of Nieto and Rodríguez-López [6] are generalized and extended by many authors (e.g., [720]). In order-theoretic results, the contractivity condition is weaker than the usual contractivity condition as it is merely required to be compatible with the involved partial ordering. The techniques involved in the proofs of such results are the combination of ideas used in the proof of contraction principle together with the one involved in monotone iterative technique.

Recently, Alam et al. [18] obtained order-theoretic versions of Goebel as well as Jungck coincidence theorems under Boyd-Wong type nonlinear contraction (cf. [21]). Most recently, Alam et al. [19] weakened the relevant metrical notions, namely, completeness, continuity, -continuity, and compatibility, by defining their order-theoretic analogues and utilized the same to enrich the results of Alam et al. [18]. In these two papers [18, 19], the authors assumed that either the whole space is complete or the range of any of the underlying mappings is complete.

The aim of this paper is to improve the coincidence theorems of Alam et al. [18, 19] without the completeness requirements of whole space (or range subspaces). Instead, we assume the completeness of a subspace satisfying suitable conditions. In this continuation, we also introduce the order-theoretic analogue of the notion of closed sets and utilize the same to derive consequences of our main results under completeness of whole space. Finally, as an application of one of our newly proved results, we indicate a result ensuring existence and uniqueness of solution of a first-order periodic boundary value problem.

2. Preliminaries

In this section, to make our exposition self-contained, we recall some basic definitions, relevant notions, and auxiliary results. Throughout this paper, stands for the set of natural numbers and for the set of whole numbers (i.e., ).

Definition 1 (see [22]). A set together with a partial order (often denoted by ) is called an ordered set. As expected, denotes the dual order of (i.e., means ).

Definition 2 (see [22]). Two elements and of an ordered set are called comparable if either or . For brevity, one denotes it by .
Clearly, the relation is reflexive and symmetric, but not transitive in general (cf. [23]).

Definition 3 (see [22]). A subset of an ordered set is called totally or linearly ordered if every pair of elements of is comparable; that is,

Definition 4 (see [3]). A sequence in an ordered set is said to be(i)increasing or ascending if, for any ,(ii)decreasing or descending if, for any ,(iii)monotone if it is either increasing or decreasing,(iv)bounded above if there is an element such thatso that is an upper bound of ,(v)bounded below if there is an element such thatso that is a lower bound of .

Definition 5 (see [9]). Let and be two self-mappings on an ordered set . One says that is -increasing if, for any ,Notice that, under the restriction , the identity mapping on , the notion of -increasing mapping transforms to the notion of increasing mapping.

Definition 6 (see [1, 2, 24]). Let and   be two self-mappings on a nonempty set . Then(i)an element is called a coincidence point of and if(ii)if is a coincidence point of and and such that , then is called a point of coincidence of and ,(iii)if is a coincidence point of and such that , then is called a common fixed point of and ,(iv) and are called commuting if, for all ,(v) and are called weakly compatible (or partially commuting or coincidentally commuting) if and commute at their coincidence points; that is, for any ,

Definition 7 (see [25, 26]). Let and be two self-mappings on a metric space . One says that and are(i)weakly commuting if, for all ,(ii)compatible if, for any sequence and for any ,

Definition 8 (see [27]). Let and   be two self-mappings on a metric space and . One says that is -continuous at if, for any sequence ,Moreover, is called -continuous if it is -continuous at each point of .
Notice that, particularly with , the identity mapping on , Definition 8 reduces to the definition of continuity.

Definition 9 (see [8]). A triplet is called an ordered metric space if is a metric space and is an ordered set. Moreover, if the metric space is complete, then is called an ordered complete metric space.

Definition 10. Let be an ordered metric space and a nonempty subset of . Then and , respectively, induce a metric and a partial order on so thatThus is an ordered metric space, which is called a subsapce of .
Conventionally, we opt to refer to as a subspace of rather than saying is a subspace and continue to write and instead of and , respectively.
Let be an ordered metric space and a sequence in . We adopt the following notations:(i)If is increasing and , then we denote it symbolically by .(ii)If is decreasing and , then we denote it symbolically by .(iii)If is monotone and , then we denote it symbolically by .

Definition 11 (see [19]). An ordered metric space is called(i)-complete if every increasing Cauchy sequence in converges,(ii)-complete if every decreasing Cauchy sequence in converges,(iii)-complete if every monotone Cauchy sequence in converges.

Remark 12 (see [19]). In an ordered metric space, completeness -completeness -completeness as well as -completeness.

Definition 13. Let be an ordered metric space. A subset of is called(i)-closed if, for any sequence ,(ii)-closed if, for any sequence ,(iii)-closed if, for any sequence ,

Remark 14. In an ordered metric space, closedness -closedness -closedness as well as -closedness.

Proposition 15. (i) An -complete subspace of an ordered metric space is -closed. (ii) An -complete subspace of an ordered metric space is -closed. (iii) An -complete subspace of an ordered metric space is -closed.

Proof. We prove here only part (i). Proceeding on similar lines, one can prove part (ii) as well as part (iii). Let be an ordered metric space. Suppose that is an -complete subspace of . Take a sequence such that . As each convergent sequence is Cauchy, is an increasing Cauchy sequence in . Hence, -completeness of implies that the limit of must lie in ; that is, . Therefore, is -closed.

Proposition 16. (i) An -closed subspace of an -complete ordered metric space is -complete. (ii) An -closed subspace of an -complete ordered metric space is -complete. (iii) An -closed subspace of an -complete ordered metric space is -complete.

Proof. We prove here only part (i). Proceeding on similar lines, one can prove part (ii) as well as part (iii). Let be an -complete ordered metric space. Suppose that is an -closed subspace of . Take an increasing Cauchy sequence in . As is -complete, such that and so . Hence, -closedness of implies that . Therefore, is -complete.

Definition 17 (see [19]). Let be an ordered metric space, a mapping, and . Then, is called(i)-continuous at if, for any sequence ,(ii)-continuous at if, for any sequence ,(iii)-continuous at if, for any sequence ,Moreover, is called -continuous (resp., -continuous, -continuous) if it is -continuous (resp., -continuous, -continuous) at each point of .

Remark 18 (see [19]). In an ordered metric space, continuity -continuity -continuity as well as -continuity.

Definition 19 (see [19]). Let be an ordered metric space, and two self-mappings on , and . Then, is called(i)-continuous at if, for any sequence ,(ii)-continuous at if, for any sequence ,(iii)-continuous at if, for any sequence ,Moreover, is called -continuous (resp., -continuous, -continuous) if it is -continuous (resp., -continuous, -continuous) at each point of .
Notice that, on setting (the identity mapping on ), Definition 19 reduces to Definition 17.

Remark 20 (see [19]). In an ordered metric space, -continuity -continuity -continuity as well as -continuity.

Definition 21 (see [19]). Let be an ordered metric space and and two self-mappings on . One says that and are(i)-compatible if, for any sequence and for any ,(ii)-compatible if, for any sequence and for any ,(iii)-compatible if, for any sequence and for any ,Here, it can be pointed out that the notion of -compatibility is slightly weaker than the notion of -compatibility defined by Luong and Thuan [28]. Luong and Thuan [28] assumed that only the sequence is monotone but we assume that both and are monotone.

Remark 22 (see [19]). In an ordered metric space, commutativity weak commutativity compatibility -compatibility -compatibility as well as -compatibility weak compatibility.
In the following lines, we formulate some definitions using certain properties utilized by earlier authors especially from [6, 9, 29, 30] besides some other ones.

Definition 23 (see [18]). Let be an ordered metric space. One says that(i) has ICU (increasing-convergence-upper bound) property if every increasing convergent sequence in is bounded above by its limit (as an upper bound); that is,(ii) has DCL (decreasing-convergence-lower bound) property if every decreasing convergent sequence in is bounded below by its limit (as a lower bound); that is,(iii) has MCB (monotone-convergence-boundedness) property if it has both ICU and DCL property.

Definition 24 (see [18]). Let be an ordered metric space and a self-mapping on One says that(i) has -ICU property if -image of every increasing convergent sequence in is bounded above by -image of its limit (as an upper bound); that is,(ii) has -DCL property if -image of every decreasing convergent sequence in is bounded below by -image of its limit (as a lower bound); that is,(iii) has -MCB property if it has both -ICU and -DCL property.Notice that, under the restriction , the identity mapping on , Definition 24 reduces to Definition 23.
Now, we recall the following notions which are relatively weaker than their corresponding earlier definitions.

Definition 25 (see [19]). Let be an ordered metric space. One says that(i) has ICC (increasing-convergence-comparable) property if every increasing convergent sequence in has a subsequence such that every term of is comparable with the limit of ; that is,(ii) has DCC (decreasing-convergence-comparable) property if every decreasing convergent sequence in has a subsequence such that every term of is comparable with the limit of ; that is,(iii) has MCC (monotone-convergence-comparable) property if every monotone convergent sequence in has a subsequence such that every term of is comparable with the limit of ; that is,

Remark 26 (see [19]). In an ordered metric space, we have the following:ICU property ICC property.DCL property DCC property.MCB property MCC property ICC property as well as DCC property.

Definition 27 (see [19]). Let be an ordered metric space and a self-mapping on . One says that(i) has -ICC property if every increasing convergent sequence in has a subsequence such that every term of is comparable with -image of the limit of ; that is,(ii) has -DCC property if each decreasing convergent sequence in has a subsequence such that every term of is comparable with -image of the limit of ; that is,(iii) has -MCC property if each monotone convergent sequence in has a subsequence such that every term of is comparable with -image of the limit of ; that is,Observe that under the restriction , the identity mapping on , Definition 27 reduces to Definition 25.

Remark 28 (see [19]). In an ordered metric space, we have the following:-ICU property -ICC property.-DCL property -DCC property.-MCB property -MCC property -ICC property as well as -DCC property.

Definition 29 (see [21, 31]). One denotes by the family of functions satisfying(a) for each ,(b) for each .Finally, we record the following results in the form of lemmas needed in our results.

Lemma 30 (see [18]). Let . If is a sequence such that , then .

Lemma 31 (see [14, 32, 33]). Let be a metric space and a sequence in . If is not Cauchy, then there exist and two subsequences and of such that(i),(ii),(iii). Moreover, suppose that ; then,(iv),(v).

Lemma 32 (see [18]). Let and be two self-mappings defined on an ordered set . If is -increasing and , then .

Lemma 33 (see [34]). Let be a nonempty set and a self-mapping on . Then, there exists a subset such that and is one-one.

3. Main Results

Now, we prove one of our main results on coincidence points which runs as follows.

Theorem 34. Let be an ordered metric space and an -complete subspace of . Let and be two self-mappings on . Suppose that the following conditions hold:(a).(b) is -increasing.(c)There exists such that .(d)There exists such that(e)   and are -compatible.   is -continuous. Either is -continuous or has -ICC property. Or alternately  . Either is -continuous or and are continuous or has ICC property.Then, and have a coincidence point.

Proof. Firstly, we notice that assumption (a) is equivalent to saying that and . Now, in view of assumption (c), if , then is a coincidence point of and and hence we are through. Otherwise, using assumption , we construct a sequence of joint iteration of and based at point ; that is,Now, we assert that is an increasing sequence; that is, We prove this fact by mathematical induction. On using (37) with and assumption (c), we haveso that (38) holds for Suppose that (38) holds for ; that is, which on using (37) and assumption (b) gives rise tothat is, (38) holds for . Hence, by induction, (38) holds for all .
In view of (37) and (38), the sequence is also an increasing sequence; that is,If for some , then, using (37), we have ; that is, is a coincidence point of and so that we are through. On the other hand, if for each , we can define a sequence , whereOn using (37), (38), (43), and assumption (d), we obtainso that Hence, by Lemma 30, we obtainNext, we show that is a Cauchy sequence. On the contrary, suppose that is not Cauchy; then, owing to Lemma 31, there exist and two subsequences and of such that , , and , where . Further, in view of (46), Lemma 31 assures us thatDenote . As , due to (38), we have . On using (37) and assumption (c), we obtainso that On taking limit superior as in (49) and using (47) and the definition of , we havewhich is a contradiction so that the sequence is Cauchy.
Owing to (37), so that is an increasing Cauchy sequence in . As is -complete, there exists such that , which together with (38) gives rise toOn using (37), (42), and (51), we obtainNow, using assumptions (e) and , we accomplish the rest of the proof. Assume that (e) holds. Using assumption (e2) (i.e., -continuity of ) in (51) and (52), we haveOn using (51), (52), and assumption (e1) (i.e., -compatibility of and ), we obtainNow, we show that is a coincidence point of and . To accomplish this, we use assumption (e3). Suppose that is -continuous. On using (51) and -continuity of , we obtainOn using (54), (55), (56), and continuity of , we obtainso that Thus, is a coincidence point of and and hence we are through. Alternately, suppose that has -ICC property; then, due to availability of (51), there exists a subsequence of such thatOn using (59) and assumption (d), we obtainNow, we assert thatOn account of two different possibilities arising here, we consider a partition of ; that is, and verifying that(i),(ii).In case (i), on using Lemma 32, we get and hence (61) holds for all . In case (ii), owing to the definition of , we have and hence (61) holds for all . Thus, (61) holds for all .
On using triangular inequality, (53), (54), (55), and (61), we getso that Thus, is a coincidence point of and .
Now, assume that holds. Owing to assumption (i.e., ), we can find some such that Hence, (51) and (52), respectively, reduce toNow, we show that is a coincidence point of and . To accomplish this, we use assumption . Firstly, suppose that is -continuous; then using (64), we getOn using (65) and (66), we getHence, we are done. Secondly, suppose that and are continuous. Owing to Lemma 33, there exists a subset such that and is one-one. Without loss of generality, we are able to choose such that . Now, define byAs is one-one and , is well defined. As and are continuous, it follows that is continuous. Since and , there exists such that . By using Lemma 33, we get . On using (64) and (65), we getMaking use of (68), (69), and continuity of , we getThus, is a coincidence point of and and hence we are through. Finally, suppose that has the ICC property. Then, due to availability of (64), there exists a subsequence of such thatOn using (37), (71), and assumption (d), we obtainWe assert thatOn account of two different possibilities arising here, we consider a partition of ; that is, and verifying that(i),(ii).In case (i), on using Lemma 33, we get , which, in view of (37), gives rise to and hence (73) holds for all . In case (ii), by the definition of , we have and hence (73) holds for all . Thus, (73) holds for all .
On using (64), (73), and continuity of , we getso that Hence, is a coincidence point of and . This completes the proof.

Now, as a consequence, we particularize Theorem 34 by assuming the -completeness of whole space .

Corollary 35. Let be an -complete ordered metric space. Let and be two self-mappings on . Suppose that the following conditions hold:(a).(b) is -increasing.(c)There exists such that .(d)There exists such that   and are -compatible.   is -continuous. Either is -continuous or has -ICC property. Or alternately there exists an -closed subspace of such that . Either is -continuous or and are continuous or has ICC property.Then, and have a coincidence point.

Proof. The result corresponding to part (e) follows easily on setting in Theorem 34 while the same (result) in the presence of part follows using Proposition 16.

Remark 36. If is onto in Corollary 35, then we can drop assumption (a) as in this case it trivially holds. Also, we can remove assumption as it trivially holds for using Proposition 15. Whenever is onto, owing to assumption (a), must be onto and hence earlier conclusion is immediate.
Next, we present a dual result corresponding to Theorem 37.

Theorem 37. Theorem 34 (similarly Corollary 35) remains true if certain involved terms, namely, -complete, -closed, -compatible, -continuous, -continuous, ICC property, and -ICC property, are, respectively, replaced by -complete, -closed, -compatible, -continuous, -continuous, DCC property, and -DCC property provided assumption (c) is replaced by the following condition:there exists such that .

Proof. The scheme of the proof is similar to the one adopted in the proof of Theorem 34 (or Corollary 35). In this case, instead of (38) and (42), we, respectively, infer that the sequences and are decreasing so that we have to use -analogues of completeness, closedness, compatibility, continuity, and -continuity besides DCC property and -DCC property instead of ICC property and -ICC property. Therefore, proceeding on the lines of the proof of Theorem 34 (or Corollary 35), the rest of the proof of this theorem can be completed.

Now, combining Theorems 34 and 37 and making use of Remarks 12, 14, 18, 20, 22, 26, and 28, we obtain the following result.

Theorem 38. Theorem 34 (similarly Corollary 35) remains true if certain involved terms, namely, -complete, -closed, -compatible, -continuous, -continuous, ICC property, and -ICC property, are, respectively, replaced by -complete, -closed, -compatible, -continuous, -continuous, MCC property, and -MCC property provided assumption (c) is replaced by the following condition:there exists such that .

On setting with , in the previous results, we get the corresponding results for linear contractions in the form of the following consequence.

Corollary 39. Theorem 34 (similarly, Theorems 37 and 38) remains true if one replaces condition (d) by the following condition:there exists such that

The following consequences of our main results are apparent in view of Remarks 26 and 28.

Corollary 40. Theorem 34 remains true if the term ICC property (resp., -ICC property) is replaced by ICU property (resp., -ICU property).

Corollary 41. Theorem 37 remains true if the term DCC property (resp., -DCC property) is replaced by DCL property (resp., -DCL property).

Corollary 42. Theorem 38 remains true if the term MCC property (resp., -MCC property) is replaced by MCB property (resp., -MCB property).

On using Remarks 12, 14, 18, 20, and 22, we obtain the more natural versions of our main results in the form of the following consequence.

Corollary 43. Theorem 34 (resp., Theorems 37 and 38) remains true if the usual metrical notions, namely, completeness, closedness, compatibility (or commutativity/weak commutativity), continuity, and -continuity, are used instead of their respective -analogues (resp., -analogues and -analogues).

4. Uniqueness Results

In this section, we prove certain results ensuring the uniqueness of a point of coincidence, coincidence point, and common fixed point corresponding to earlier established results.

Definition 44 (see [19]). Let be an ordered set and and two self-mappings on . One says that is -directed if for each pair , , such that and . In case (where denotes identity mapping on ), is called -directed.
In particular, for , Definition 44 coincides with the usual notion of directed set suggested by Jleli et al. [14]. In other words, we use the term directed set instead of -directed set.
The following notion is inspired by Turinici [23].

Definition 45 (see [19]). Let be an ordered set, , and . A subset of is called -chain between and in if(i),(ii) and ,(iii).We denote by the class of all -chains between and in . In particular, for , we write instead of .

Theorem 46. In addition to the hypotheses of Theorem 34 (similarly Theorems 37 and 38), suppose that the following condition holds: is nonempty, for each .Then, and have a unique point of coincidence.

Corollary 47. Theorem 46 remains true if one replaces condition by one of the following conditions: is totally ordered. is -directed.

Theorem 48. In addition to the hypotheses of Theorem 46, suppose that the following condition holds:one of and is one-one.Then, and have a unique coincidence point.

Theorem 49. In addition to the hypotheses embodied in condition of Theorem 46, suppose that the following condition holds: and are weakly compatible.Then, and have a unique common fixed point.

We skip the proofs of the above results as the same can be completed on the lines of the proofs of the relevant results of Alam et al. [19].

5. Fixed Point Theorems

On setting , the identity mapping on , in earlier results, we deduce the following corresponding fixed point theorems.

Theorem 50. Let be an ordered metric space and a self-mapping on . Let be an -complete subspace of such that . Suppose that the following conditions hold:(i) is increasing.(ii)Either is -continuous or has ICC property.(iii)There exists such that .(iv)There exists such thatThen, has a fixed point.

Theorem 51. Theorem 50 remains true if certain involved terms, namely, -complete, -continuous, and ICC property, are, respectively, replaced by -complete, -continuous, and DCC property provided assumption (iii) is replaced by the following condition:there exists such that .

Theorem 52. Theorem 50 remains true if certain involved terms, namely, -complete, -continuous, and ICC property, are, respectively, replaced by -complete, -continuous, and MCC property provided assumption (iii) is replaced by the following condition:there exists such that .

Corollary 53. Theorem 50 (similarly Theorems 51 and 52) remains true if condition (iv) is replaced by the following condition:there exists such that

Corollary 54. Theorem 50 remains true if the term ICC property is replaced by ICU property.

Corollary 55. Theorem 51 remains true if the term DCC property is replaced by DCL property.

Corollary 56. Theorem 52 remains true if the term MCC property is replaced by MCB property.

Corollary 57. Theorem 50 (resp., Theorems 51 and 52) remains true if the usual metrical notions, namely, completeness and continuity, are used instead of their respective -analogues (resp., -analogues and -analogues).

Theorem 58. In addition to the hypotheses of Theorem 50 (similarly Theorems 51 and 52), suppose that the following condition holds: is nonempty for each .Then, has a unique fixed point.

Corollary 59. Theorem 58 remains true if one replaces condition by one of the following conditions: is totally ordered. is -directed.

Notice that all the above-mentioned results in this section enrich and sharpen the fixed point theorems of Kutbi et al. [20].

Finally, inspired by [6], we present an example, where Theorem 58 can be applied. We study the existence and uniqueness of solution for the following first-order periodic boundary value problem:where and is a continuous function.

Let denote the space of all continuous functions defined on . Now, we recall the following definitions.

Definition 60 (see [6]). A function is called a lower solution of (80), if

Definition 61 (see [6]). A function is called an upper solution of (80), ifLet denote the family of functions satisfying the following conditions:(i) is continuous and increasing.(ii) for each .Typical examples of are , , , and . Also, clearly, .
Now, we record the following result regarding the existence and uniqueness of a solution of problem (80) in the presence of a lower solution or an upper solution.

Theorem 62. In addition to problem (80), suppose that there exists and such that for all with

Then, the existence of a lower solution or an upper solution of problem (80) provides the existence and uniqueness of a solution of problem (80).

The proof of Theorem 62 can be completed on the lines of the proof of Theorem due to Kutbi et al. [20], wherein the authors used Theorem 58 (with ).

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All the authors contributed equally. Thus, formally, all the authors read and approved the final paper.

Acknowledgments

The second named author (i.e., Qamrul Haq Khan) is grateful to University Grant Commission, Government of India, New Delhi, India.

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