Common Fixed-Point Results in Ordered Left (Right) Quasi-<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.20973 12.533" width="8.20973pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g></svg>-</span>Metric Spaces and Applications

Nashine, Hemant Kumar; Shil, Sourav; Garai, Hiranmoy; Dey, Lakshmi Kanta; Parvaneh, Vahid

doi:https://doi.org/10.1155/2020/8889453

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Abstract Introduction and Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 8889453 | https://doi.org/10.1155/2020/8889453

Common Fixed-Point Results in Ordered Left (Right) Quasi--Metric Spaces and Applications

Hemant Kumar Nashine,¹Sourav Shil,¹Hiranmoy Garai,²Lakshmi Kanta Dey,²and Vahid Parvaneh³

Academic Editor: Xiaolong Qin

Received03 Sept 2020

Revised06 Oct 2020

Accepted19 Oct 2020

Published02 Dec 2020

Abstract

We use the notions of left- and right-complete quasi--metric spaces and partial ordered sets to obtain a couple of common fixed-point results for strictly weakly isotone increasing mappings and relatively weakly increasing mappings, which satisfy a pair of almost generalized contractive conditions. To illustrate our results, throughout the paper, we give several relevant examples. Further, we use our results to establish sufficient conditions for existence and uniqueness of solution of a system of nonlinear matrix equations and a pair of fractional differential equations. Finally, we provide a nontrivial example to validate the sufficient conditions for nonlinear matrix equations with numerical approximations.

1. Introduction and Preliminaries

We denote by the set of real numbers; ; we denote by the set of natural numbers and . Also, for the mappings , we denote by CFP and CFP the set of all common fixed points of and , respectively.

The metric fixed-point theory has been extended in many directions by many renowned mathematicians. One important direction of such ones is to revise the underlying metric space to some other spaces by making suitable changes obtained by Czerwik. He introduced the notion of -metric spaces (see [1]), which is further extended as quasi--metric spaces by Shah and Hussain [2].

Definition 1 (see [2]). Let be a set and let be a given real number. A function is a quasi--metric on if, for all , (M1) (M2) (M3) The pair is then termed as a quasi--metric space with constant .
It is to be noted that every metric space is quasi-metric space, and quasi-metric space is a quasi--metric space but the converses need not be true. The above space is further extended with the introduction of right and left quasi--metric spaces (in the line of [3]).

Definition 2 (see [4]). Let be a quasi--metric space and let be a sequence in . Then is said to be(i)left-Cauchy if, for every , we get such that for all (ii)right-Cauchy if, for every , we get such that for all

Definition 3 (see [4]). Let be a quasi--metric space. Then is called(i)left-complete if every left-Cauchy sequence in is convergent(ii)right-complete if every right-Cauchy sequence in is convergentOn the other hand, an extension of fixed-point results for various types of contractions in metric spaces is secured by adding an (partial) ordering structure on the underlying structure . Some early results in this direction were established by Turinici in [5, 6]; one may note that their starting points were “amorphous” contributions in the area due to Matkowski [7, 8]. These types of results have been reinvestigated by Ran and Reurings [9] and also by Nieto and Ródríguez-López [10, 11]. In 2019, Gu and Shatanawi [12] obtained some common coupled fixed-point results in partial metric spaces and some recent results of Latif et al. [13] and Malhotra et at. [14] are also important. In [15], Nashine et al. used the concept of -weakly isotone increasing mappings to extend Ćirić’s [16] result in ordered metric spaces. The main importance of their results is that they obtained their results without considering any kind of commutativity condition. After all such generalizations and extensions, Nashine and Altun [17] introduced a new notion of increasing mapping, which they designated as -strictly weakly isotone increasing mapping, and then they obtained some results by considering this new type of increasing mappings. After this, in [18], Nashine and Samet introduced relatively weakly increasing mappings and proved some fixed-point results in ordered metric spaces and applied their results to integral equations. In this sequel, we like to recall some useful definitions in the context of a partially ordered set .

Definition 4 (see [17–20]). Let be a partially ordered set and let be three mappings. Then,(1) is called dominating if for each .(2)the pair is called weakly increasing if and for each .(3) is called -weakly isotone increasing if, for each , we have .(4)the mapping is said to be -strictly weakly isotone increasing if, for satisfying , we have .(5) and are said to be weakly increasing with respect to if and and, for each , we haveLet be a -metric space. Then two mappings are said to be compatible if , for each sequence in with and for some . If is a quasi--metric space and is a partially ordered set, then the triplet is called an ordered quasi--metric space. The space is called regular if whenever is a nondecreasing sequence in with respect to and as , then holds.
In the literature of fixed point, one may note that, to find common fixed point of two or more mappings in the setting of different abstract spaces, more specifically in left- and right-complete quasi--metric spaces, commutativity condition of the mappings plays crucial roles. So it is a challenging work to obtain common fixed point of two or more mappings in such spaces without considering the commutativity condition. One of the main motivations of the paper is to resolve this issue. To proceed with this, we utilize the approaches of Nashine and Altun [17] and Nashine and Samet [18] to obtain some common fixed-point results in the setting of ordered left-complete and right-complete quasi--metric spaces. Firstly, we establish some common fixed-point theorems for a pair of mappings using the -strictly weakly isotone increasing condition and without using any kind of commutativity condition in ordered left-complete quasi--metric spaces. Secondly, we obtain a common fixed-point result for a triplet of mappings satisfying relatively weakly increasing condition and almost generalized contractive conditions in ordered right-complete quasi--metric spaces.
Another important motivation of this paper is to show how we can apply our obtained results in at least two different applicable areas. These are connected to get solutions of a pair of nonlinear matrix equations and also a pair of fractional differential equations. Further, we provide some nontrivial examples to illustrate our obtained results. Finally, our attempts give extensions of the works discussed in [2, 3, 9–11, 15, 18, 21] and other related results in the sense of generalized contractive conditions and generalized weakly increasing mappings in the crucial setting with new applications to the functional equations.

2. Results for Pair of Mappings

In this section, at first, we prove a common fixed-point result of a pair of mappings involving -strictly weakly isotone increasing condition. Before this, we state the following important lemma regarding the left- (right-) Cauchyness of a sequence in quasi--metric context.

Lemma 1. Let be a quasi--metric space and let be a sequence in . Then, we have the following:(1)If there exists satisfying then is a left-Cauchy sequence.(2)If there exists satisfyingthen is a right-Cauchy sequence.

Proof. The proof of this lemma can be done in the line of ([22], p. 3, Lemma [23]).

Theorem 1. Let be an ordered left-complete quasi--metric space with constant . Suppose that are two mappings such thatfor all comparable , where , , andIn addition, let be -strictly weakly isotone increasing, there exists an such that , and one of and is continuous. Then .

Proof. Let be such that and construct a sequence in satisfyingAs is -weakly isotone increasing,and, proceeding with this argument, we getLet . Then, for all , we show thatFrom (10), we have that for all . Then, from (4), with and , we getBy (8), we have(i)If , by (12), we have a contradiction.(ii)If , we have(iii)If , we have(iv)If , we haveConsequently, for all . Similarly, using (5) with (7), we can show that . Therefore, (11) holds for all and so, from Lemma 1, we can conclude that is a left-Cauchy sequence.
From the left-completeness of , there exists such that as . Clearly, if or is continuous, then or . Thus, .By the next result, we show that the continuity of or in the previous theorem can be replaced by some other conditions.

Theorem 2. If one replaces the continuity in Theorem 1 by regularity of , then conclusion of Theorem 1 is valid provided .

Proof. Following the lines of proof of Theorem 1, we have that there exists such thatUsing (5) for and , we haveLetting in the above equation, we getHence and so . Analogously, using (4) for and , we get . It follows that ; that is, .
Next, we characterize the common fixed-points set in the following theorem.

Theorem 3. Let all the conditions of Theorems 1 and 2 hold. Then is totally ordered if and only if CFP contains exactly one element.

Proof. First, we assume that CFP is totally ordered. Let with . Consider (4) for and , and we getthat is,Again, using (5) for and and by calculation we getAdding (22) and (23), we getwhich gives a contradiction. Thus, is singleton. The converse is trivial.
Putting in Theorem 3, we obtain the following result.

Corollary 1. Let be an ordered left-complete quasi--metric space with constant . Suppose that is a mapping such thatfor all comparable , where , , andAlso suppose that for all with . If there exists an element such that and either is continuous at or is regular, then has a fixed point. Moreover, is totally ordered if and only if it is a singleton.

Next, we come up with the following example, which illustrates Theorem 1.

Example 1. Let and define byWe define a relation “” on by if and only if either and or and . Then is an ordered left-complete quasi--metric space with constant . Next, we define two mappings byFor with , we have , , and . So and hence . Again, for and , we have , , and . So and hence . Thus, is -strictly weakly isotone increasing. It is clear that is continuous. We choose .
Now assume that is arbitrary such that and are comparable. Then the following cases arise: Case I: let and ; that is, . First, we assume that . Then and and . Therefore, Next, we assume that . Then and so Thus, On the similar arguments, we can get Case II: let and ; that is, . Then, we can show in a similar manner that Case III: let and . Then and . Also, we haveTherefore,where .
Hence, from all three cases, we haveThus, all the conditions of Theorem 1 are fulfilled and the pair has a unique common fixed point (which is ).

3. Results for Three Mappings

In this section, we obtain a result for three mappings involving weakly increasing condition.

Theorem 4. Let be an ordered right-complete quasi--metric space with constant . Let be three mappings such that and , andfor all comparable , where , , and

We assume the following hypotheses:(i) and are weakly increasing with respect to (ii) and are dominating maps

Assume either of the following:(a) and are compatible; is continuous at (b) and are compatible; is continuous at

Then . In addition, is totally ordered if and only if contains exactly one element.

Proof. Start with defining a sequence in asWe claim thatUsing hypothesis (i) and (41),Since , , and we getAgain,Since , we getHence, by induction, (42) holds.
Let . Now we claim that, for all , we haveUsing (38) and (42) with and ,where(i)If , by (48), we have that is, .(ii)If , we get(iii)If , we getThus, in all cases, we have for all . Similarly, using (39), we can show for all . Therefore, we conclude that (47) holds for all ; that is,Therefore, from Lemma 1, it follows that is a right-Cauchy sequence. From the right-completeness of , there exists such thatWe will prove that is a common fixed point of the three mappings , and .
We haveFirst, suppose that (a) holds. ThenFrom (54) and the continuity of , we haveAlso, we have . Then, from (38), we getwhereLimiting in (59) and using (55)–(58), we obtainNow, and as , so by the assumption we have . Then, using (38), we getwhereLetting in (62) and using (55) and (56), we getAgain, and as , so by the assumption we have and hence using (39) giveswhereLetting in (65) and using (55) and (56), we haveTherefore, . Hence is a common fixed point of , and . The proof is similar when (b) holds.
Now, suppose that is totally ordered. We claim that there is a unique . Assume to the contrary that and but . Using (38) with and , we havewhereTherefore,a contradiction. Hence, . The converse is trivial.
Taking the identity mapping on in Theorem 4, we have the following consequence.

Corollary 2. Let be an ordered right-complete quasi--metric space with constant . Let the pair of mappings satisfyfor all comparable , , , and

Further, assume that(i) and are weakly increasing mapping(ii) and are dominating maps

Then . In addition, is totally ordered if and only if contains exactly one element.

Now, we provide the following example to authenticate Theorem 4.

Example 2. We consider and define byAlso, we define a relation on by if and only if . Then, we can easily check that is an ordered right-complete quasi--metric space with constant . Next, we define three mappings byThen, clearly and . We choose . Now we assume that are arbitrary such that and are comparable. Then the following cases arise: Case I: let and . First, we assume that . Then and . Therefore, we have Next, we assume that . Then and , and so we have In a similar way, we can show that Case II: let and . Then, proceeding in the same way as that in Case I, we can show that Case III: let and . Then , , and . Therefore, which implies that Case IV: let and . This case is similar to the previous case and so is omitted. Case V: let . In this case, it is trivial to check that (38) and (39) hold.So, combining all the five cases, we see that (38) and (39) hold for all comparable .
Now let . Then, we have if and if . Therefore, if , then and , so . If , then and so . Similarly, for , we can show that . Hence and are weakly increasing with respect to . Further, we can easily show that and are dominating maps; and are compatible; and is continuous. Thus, we can apply Theorem 4 to deduce that , and have a unique common fixed point .

4. Application to Nonlinear Matrix Equations

In this section, we will apply the common fixed-point results in quasi--metric spaces of the previous section to obtain variants of the results of Garai and Dey [23] on existence of common solution to systems of NMEs. For other variants on solution to systems of NMEs, one is referred to [24, 25]. For a matrix , any singular value of will be denoted by , and the sum of these values, that is, the trace norm of , will be denoted by . We will use the standard partial order on given by if and only if is a positive semidefinite matrix. We define a function by

Then is an ordered right-complete quasi--metric space with .

Utilizing this quasi--metric space, we now prove the following theorem regarding the solution (s) of a pair of nonlinear matrix equations.

Theorem 5. Consider the systemwhere , , , and the operators are continuous in the trace norm. Let, for some , and, for any with , hold for all singular values of and , respectively. Assume the following:(1), where (2)For any with , ( stands for null matrix of order ) or holds(3)For any with , hold(4)For any with , if then hold(5)There exists such that, for any ,(a) holds for all and with , or , where(b)holds for all and with , or , where

Then system (82) has a solution, and if is a solution of the system, then with . Further, the iterative sequence , where, for ,and is an arbitrary element of satisfying , converges to a unique solution of the system, if or .

Proof. Let us consider the set . Then, is right-complete with respect to the metric . For any , we haveSince , we have for all singular values of so, by summing the singular values of , we get . Using this in (86), we getSimilarly, for any , we can show thatTherefore, the mappings defined on byfor are self-maps on . Next, for any , using assumption (3), we have and . So the mappings are dominating mappings. Again, for , we have and . So, for any , using assumption (4), we have and . Therefore, are weakly increasing mappings. We choose . Now let be arbitrary such that either or . Then, we haveIf , then we haveIf , then we can similarly show thatAgain iforthen we can show thatorThus, combining equations (91)–(96) and using the definition of the quasi--metric , we getIn a similar way, we can show thatHence, by Corollary 2, it is implied that the pair of mappings has a common fixed point in , say . So is a solution to system (82). By assumption (2), we see that any solution of system (82) must be positive definite, and since , we have . Again, by the order preserving (or order reversing) of the successive elements of the sequence converging to , we see that system (82) cannot have more than one solution. Thus, system (82) has a unique solution with and .
Now we present an example in order to validate the above theorem.

Example 3. Consider system (82) for and , with and ; that is,whereAfter calculations, we get , , , and .
Let . The conditions of Theorem (99) can be checked numerically, taking various special values for matrices involved. For example, they can be tested (and verified to be true) forwhere,To see the convergence of the sequence defined in (85), we start with initial values,with ,with ,with and, after 20 iterations, we have the following approximation of the unique positive definite solution of system (99):with .with .with . Also, the elements of each sequence are order preserving. The graphical representation of convergence is shown below.
Convergence behaviour

5. Application to Nonlinear Fractional Differential Equations

Consider a pair of nonlinear fractional differential equations (FDEs, in short):where , , with the two-point boundary conditionswhere is a continuous function.

The Caputo derivative of fractional-order is defined aswhere is a continuous function, denotes the integer part of the positive real number , and is the gamma function.

The Riemann-Liouville fractional derivative of order for a continuous function is defined byprovided the right-hand side is point-wise defined on .

The FDEs (109) are equivalent to the integral equationwhere the Green function is

Following [4], consider that endowed with the metricis a right-complete quasi--metric space with , whereis the usual supremum norm. It is noticed that is a right-complete ordered quasi--metric space, where denotes if for all .

Theorem 6. Let be two operators defined bywhere . Assume the following:(i) is a continuous function, which is nondecreasing in the second variable(ii)For all ,(iii)For all ,(iv)For all , ,Then, the integral equations (113) have a solution .

Proof. Using (II), for all , we haveSimilarly,Then, we have and for all . This implies that and are weakly increasing. Using (III), for all , and are dominating operators.
To check contraction conditions, we start withthat is, for ,NowAlsoTherefore, we getHencefor all comparable such that . Also, it is an obvious fact that the above inequality holds true if . Therefore, for and , Corollary 2 implies that there is a unique common fixed point of the operators ; that is, is also a solution to the integral equation (117) and the FDEs (109). To see the uniqueness of solution, let be another solution of integral equation (117); then, using condition (II), we havethat is, , a contradiction, as .

6. Concluding Remark

We obtain some common fixed-point results for a pair of mappings and also for a triplet of mappings in quasi--metric spaces using the left- and right-completeness in the absence of commutativity condition. We utilize these results to obtain solutions of a pair of nonlinear matrix equations and a pair of fractional differential equations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first and second authors are thankful to SERB, India, for providing fund under the project CRG/2018/000615. The third author is thankful to CSIR, New Delhi, India, for their financial support under CSIR-SRF fellowship scheme (Award no. 09/973 (0018)/2017-EMR-I).

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Copyright © 2020 Hemant Kumar Nashine 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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