Solution of Fractional Differential Equations Utilizing Symmetric Contraction

Hussain, Aftab

doi:https://doi.org/10.1155/2021/5510971

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Recent Advances in Fixed Point Theory in Abstract Spaces

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Volume 2021 | Article ID 5510971 | https://doi.org/10.1155/2021/5510971

Solution of Fractional Differential Equations Utilizing Symmetric Contraction

Aftab Hussain¹

Academic Editor: Huseyin Isik

Received11 Feb 2021

Revised08 Mar 2021

Accepted04 Apr 2021

Published22 May 2021

Abstract

The aim of this paper is to present another family of fractional symmetric --contractions and build up some new results for such contraction in the context of -metric space. The author derives some results for Suzuki-type contractions and orbitally -complete and orbitally continuous mappings in -metric spaces. The inspiration of this paper is to observe the solution of fractional-order differential equation with one of the boundary conditions using fixed-point technique in -metric space.

1. Preliminaries and Scope

Fixed-point theory has been promoted by a few particular works in the most recent decades [1–3]. One of the intriguing methodologies was presented in Karapinar et al.’s work [4] which starts a thought of interpolative kind of contractions and sest up shiny new fixed-point results in partial metric space. Recently, Jleli and Samet [5] introduced a new generalization of metric space and named it as -metric space.

Definition 1. (see [5]). Let be the set function that meets the following conditions: is nondecreasing; that is, for all , it implies . For each iteration , we haveThe generalized notion of metric space is as follows.

Definition 2. (see [5]). Let with be a given mapping. Suppose that there exists such that , . for all . Each , , , and for each with , we haveThen, it is said that is an -metric on .
Here, the pair is called an -metric space and it is abbreviated as -MS. A sequence in is -Cauchy, if . Furthermore, is -complete, if every -Cauchy sequence is -convergent in .
The following example is stated in [5].

Example 1. (see [5]). The set of natural numbers is an -MS if we define byfor all , , and . Moreover, does not form a metric but it is an -MS.
Jleli and Samet proposed a simple Banach fixed-point theorem as follows.

Theorem 1. (see [5]). Let be an -MS. Let be a self-mapping. Suppose that the following conditions are met:(i) is -complete.(ii) a constant such thatThen, attains a unique fixed-point .
In 2012, Samet et al. introduced a class of -admissible mappings as follows.

Definition 3. (see [6]). Let and . is said to be -admissible if , and implies that .
Next, Salimi et al. [7] modified the concept of -admissible mapping as follows.

Definition 4. (see [7]). Let and be two functions. is called an -admissible mapping with respect to , if , and implies that .
If , then the above definition reduces to Definition 3. If , then is called an -subadmissible mapping.

Definition 5. (see [8]). Consider a metric space and assume that and are two functions. A mapping is considered as --continuous mapping in whenever is given, and the sequence is as follows:For more details, see, for example, [9, 10].
A mapping is called orbitally continuous in if implies that . mapping is orbitally continuous on if is orbitally continuous .

2. Fractional Symmetric --Contraction of Type-I

In this segment, first we present a new fractional symmetric --contraction of type-I.

Definition 6. Let be an -metric space and two functions. We consider that is a fractional symmetric --contraction of type-I along with constants and such that, whenever , we havewherewhere , for all .

Example 2. Let with grace of -metric defined byand consider and . Define byand byIf , clearly such thatBy taking any value of constants and , clearly, (6) holds for all , . Point out that has two fixed points, which are 0 and 1.
Now, we initiate brand new fixed-point theorems for fractional symmetric --contraction of type-I in the configuration of -complete -MS.

Theorem 2. Let be a complete -metric space and is a fractional symmetric --contraction of type-I satisfying the following statements:(i) is an -admissible mapping concerning (ii)There exists to such an extent that (iii) is --continuousAt that point, possesses a fixed point at .

Proof. Consider in with the goal that . For , we build a chain in such a way that and . Proceeding with this exercise, , for every . Presently, as long as mapping is -admissible with respect to , at that time . Carrying on in this way, we getProvided that for some , then is a fixed point of . So, we assume that accompanied byAs is fractional symmetric --contraction of type-I, a part of , we havewhich implies thatand we deduce thatWe conclude that is a nonincreasing sequence with nonnegative terms. Thus, there is a nonnegative constant such that . Note that . From (16), we haveThis providesConsidering for as much asthere subsist some corresponding toLet be fixed and let be cognate and is satisfied. By , there exists which connotes thatHence, by (21) and , we getwhere with the goal that together with . Therefore, by using and (22), we havewhich implies that by , we haveConsequently, is an -Cauchy sequence. Meanwhile is an -complete metric space and there exists such that is -convergent to ; that is,and is --continuous as well as , each one of at that point as ; in other words, . Now we are going to prove that is a fixed point of . We argue by contradiction by supposing that . By , we haveBy using () and the contractive condition, we havefor all . In other words, by using () and (25), we getwhich gives a contradiction. Therefore ; hence possesses a fixed point of .

Theorem 3. Let be an -complete -metric space and let be a fractional symmetric --contraction of type-I fulfilling the accompanying affirmations:(i) is an -admissible mapping concerning (ii)There exists a to such extent that (iii)An iteration in is such that escorted by at the same time ; after that holds for each Afterwards, possesses a fixed point in .

Proof. On closing lines of the proof of Theorem 2, we acquire for each . Using (), we haveFrom (6) connecting , we haveEmploy (25) with certitude thatand we obtainand making use of , we havewhich is a contradiction. Therefore, ; in other words, possesses a fixed point of .

Example 3. Let with an -metric byaccompanied by together with . Define along withand by(i)Case I. If , clearly . Hence, every condition of Theorem 2 is satisfied.(ii)Case II. If , clearly is an -admissible mapping with respect to , whenever , such that(iii)By taking constant , and , for all .(iv)Case III. If any , then we haveTherefore, whole constraints of Theorem 2 are satisfied. Hence, is fractional symmetric --contraction of type-I.

Definition 7. Consider an -metric space and two functions. Then an -metric space on is said to be --complete if and only if every -Cauchy sequence , along with-converges in .

Remark 1. Theorems 2 and 3 also hold for --complete -metric space instead of -complete -metric space (for details, see [10]).

3. Fractional Symmetric --Contraction of Type-II

In this section, a fractional symmetric --contraction of type-II is introduced and in the structure of -complete -metric space. Using this notion, we shall provide a fixed-point theorem.

Definition 8. Consider a self-map on an -metric space and two functions . We presume that is a fractional symmetric --contraction of type-II provided that there are constants and such that, whenever , we ownwherewhere , for all .
Now we show and demonstrate our next theorem.

Theorem 4. Let be an -complete -metric space and let be a fractional symmetric --contraction of type-II fulfilling the accompanying affirmations:(i) is an -admissible mapping concerning (ii)There exists to such an extent that (iii) is --continuousAfter that, possesses a fixed point in .

Proof. Consider in correspondent to . For , we build an iteration in such a way that and . Proceeding with this exercise, , for all . Now, as long as mapping is -admissible with respect to , at that time . Carrying on in this way, we ownProvided that for some , then is a fixed point of . So, we assume that accompanied byAs is fractional symmetric --contraction of type-II, a part of , we ownwhich implies thatand we deduce thatWe conclude that is a nonincreasing sequence with nonnegative terms. As a result, there is a nonnegative constant such that . We shall indicate that . Indeed, from (46), we derive thatThe rest of the test follows the same lines of Theorem 2.

Theorem 5. Consider an -complete -metric space and let be a fractional symmetric --contraction of type-II meeting the following assertions:(i) is an -admissible mapping with respect to (ii)There exists such that (iii)An iteration in is such that escorted by at the same time ; after that holds for each Afterwards, possesses a fixed point in .

Proof. Similar to the lines of Theorem 3, since, by (iii), holds for every . Using (), we meetFrom (40) and , we haveMaking use of (25), we getand we procureUsing , we havewhich is a logical inconsistency. Along these lines ; that is, possesses a fixed point of .

4. Fractional Symmetric --Contraction of Type-III

In this section, fractional symmetric --contraction of type-III is considered in the environment of -complete -metric space. After stating a fixed-point theorem for such maps, we set up fractional symmetric --contraction of type-III as follows.

Definition 9. Consider an -metric space with a self-map and two functions . We say that is fractional symmetric --contraction of type-III along with constants and such that, whenever , we havewherewhere , for all .
Now we declare and demonstrate our next theorem.

Theorem 6. Let an -complete be an -metric space along with being a fractional symmetric --contraction of type-III which meets the following assertions:(i) is an -admissible mapping concerning (ii)There exists such that (iii) is --continuousAfter that, possesses a fixed point in .

Proof. Consider in with the aim that . Take any ; we erect an recapitulate in such a way that and . Continuing with this practice, , every . As long as mapping is -admissible with respect to , at that time . Carrying on in this way, we findProvided that for some , then is a fixed point of . So, we assume that accompanied byAs is fractional symmetric --contraction of type-III, a part of , we ownProvided that , at that time,which is a contradiction. We deduce thatWe conclude that is a nonincreasing sequence with nonnegative terms. As a result, there is a nonnegative constant such that . We shall indicate that . Indeed, from (59), we derive thatPause the proof and go behind the closing lines of Theorem 2.

Theorem 7. Consider an -complete -metric space and let be a fractional symmetric --contraction of type-III meeting the following assertions:(i) is an -admissible mapping with respect to (ii)There exists such that (iii)An iteration in is such that escorted by at the same time ; after that holds for each Afterwards, possesses a fixed point in .

Proof. Similar to the same lines of Theorem 3, considering (iii), for all . By (), we haveUsing (40) along with , we haveUsing (25) the factuality isand we obtainUtilizing , we havewhich is a logical inconsistency. Along these lines, ; that is, possesses a fixed point of .

5. Fractional Symmetric --Contraction of Type-IV

In this part, we propose a new notion, fractional symmetric --contraction of type-IV, in the framework of -complete -metric space.(i) is an -admissible mapping concerning (ii)There exists which connotes that (iii) is --continuous

Definition 10. Consider an -metric space with a self-map and two functions . We say that is a fractional symmetric --contraction type-IV along with constants and with such that, whenever , we havewherewhere , for all .
Now we declare and demonstrate our next theorem.

Theorem 8. Let an -complete be an -metric space along with being a fractional symmetric --contraction of type-IV that meets the following assertions:

After that, possesses a fixed point in .

Proof. Consider in with the aim that . Take any ; we build a chain in such a way that and . Proceeding with this exercise, , for every . As long as mapping is -admissible with respect to , at that time . Carrying on in this way, we getProvided that for some , then is a fixed point of . So, we assume that accompanied byAs is fractional symmetric --contraction of type-IV, a part of , we haveOn condition that , at that time,which is a contradiction. We deduce thatLet up the closing lines of Theorem 2.

Theorem 9. Consider an -complete -metric space and suppose that is a fractional symmetric --contraction of type-IV fulfilling the accompanying affirmations:(i) is an -admissible mapping concerning (ii)There exists to such an extent that (iii)An iteration in is analogous to escorted by at the same time ; after that holds for each Afterwards, possesses a fixed point in .
Whether , in Theorems 2, 3, 4, and 5, we introduce the following corollaries.

Corollary 1. Consider an -complete -metric space and suppose that is a fractional symmetric --contraction of type-I fulfilling the accompanying affirmations:(i) is an -admissible mapping(ii)There subsist parallel to (iii) is --continuousAfterwards, possesses a fixed point in .

Corollary 2. Consider an -complete -metric space and let be fractional symmetric --contraction of type-I fulfilling the accompanying affirmations:(i) is an -admissible mapping(ii)There subsist parallel to (iii)An iteration in is analogous to escorted by at the same time ; after that holds for each Afterwards, possesses a fixed point in .

Corollary 3. Let an -complete be -metric space and let be a fractional symmetric --contraction of type-II meeting the accompanying affirmations:(i) is an -admissible mapping(ii)There subsist parallel to (iii) is --continuousThen gets a fixed point in .

Corollary 4. Let an -complete be -metric space and let be a fractional symmetric --contraction of type-II meeting the accompanying affirmations:(i) is an -admissible mapping(ii)There subsist parallel to (iii)An iteration in is analogous to escorted by at the same time ; after that holds for each Afterwards, possesses a fixed point in .
In similar fashion, we can deduce Corollaries 1, 2, 3, and 4 for fractional symmetric --contraction of type-III and that of type-IV, respectively.

6. Consequences

As a consequence of our results, we derive some effect for Suzuki-type fractional symmetric contractions and orbitally -complete and orbitally continuous mappings in -metric spaces.

Theorem 10. Consider an -metric space and let be a continuous self-mapping on . Assume that there exists in addition to such thatwherewhere , for all .
At that time, possesses a fixed point in .

Proof. Describe byand , and . It is clear thatwhich means that conditions (i)-(iii) of our Theorem 2 hold true. Letwhich implies contractive condition:Finally, every constraint of Theorem 2 holds true. Hence, possesses a fixed point in .

Theorem 11. Consider an -metric space and let be a self-mapping of . Suppose that the following assertions hold:(i) is an orbitally -complete -metric space.(ii)There exists in addition to such that(i)where(ii), for all for some , where is an orbit of ,(iii)if is a sequence such that with as , then .Then, possesses a fixed point.

Proof. Describe , by on and ; otherwise, for all (see Remark 6 [11]). Then is an --complete -metric and is an -admissible mapping with respect to . Let ; then , and then, from (ii), we havewhereThat is, is a fractional symmetric --contraction of type-I. Let be a sequence commensurate with together with for . So, . From (iii), ; that is, . Hence, every norm of Theorem 3 holds true. Thus, possesses a fixed point.

Theorem 12. Consider an -metric space and let be a self-mapping of . Suppose that the following assertions hold:(i)For all , there exists along with , such that(ii)where(iii), for some ,(iv)the operator is orbitally continuous.Afterwards, possesses a fixed point.

Proof. Describe , by on and ; otherwise, (see Remark 1.1 [12]), and we know that is an --continuous mapping. Let ; then . So ; that is, . Therefore, is an -admissible mapping with respect to . From (i), we havewhereand aforesaid is a fractional symmetric --contraction of type-I. Hence, each constraint of Theorem 2 holds true. Thus, gets a fixed point.

Theorem 13. Consider an -metric space and let be a self-mapping of . Suppose that the following assertions hold:(i) is an orbitally -complete -metric space;(ii)there subsist parallel to such that(iii)where(iv), for all for some , where is an orbit of ;(v)if is a sequence such that with as , then .Then, possesses a fixed point.

Theorem 14. Consider an -metric space and let be a self-mapping of . Suppose that the following assertions hold:(i)for all , there subsist along with , such that(ii)where(iii), for some ;(iv)the operator is orbitally continuous.Afterwards, possesses a fixed point.
Theorems 10, 11, and 12 can be derived easily for fractional symmetric contraction of type-III and that of type-IV, respectively.

7. Application to Fractional-Order Differential Equations

The local and nonlocal fractional differential equations have been recently proved to be significant tools in the modeling of many phenomena in numerous fields of science and building. The fractional-order differential equations have numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetism, and so forth. For more details, see [2, 13–19]. Our aim is to give the existence and uniqueness of bounded solution of local fractional-order differential equation given in (93). Consider a function . The conformable derivative of order of at is defined by [20]

The conformable fractional integral associated with (91) is defined by [20, 21]

We consider the following boundary value problem of a conformable fractional-order differential equation:

The integral representation of the solution to the boundary value problem (93) iswhere is a Green’s function defined byand denotes the Riemann integrable of with respect to and is a continuous function.

So,

Let be the linear space of all continuous functions defined on , and let for all . Then, is an -complete metric space.

We consider the following conditions:(a)There exists , and : is a function for each with , such that where .(b)There exists such that for all ;(c)For each , there exists such that for all ;(d)For any cluster point of a sequence of points in with

Theorem 15. Suppose that conditions (a)-(d) are satisfied. Then, (93) has at least one solution .

Proof. We know that is a solution of (93) if and only if is a solution of the fractional-order integral equationWe define a map byThen, problem (93) is equivalent to finding , that is, a fixed point of . Let , such that , for all . For using (a), we getThus,for all such that for all . We define byThen, for all , , we haveObviously, for all . If for each , then . From (c), we have and so . Thus, is an -admissible map concerning . From (b), there subsist parallel to . By (d), we have that, for any cluster point of a sequence of points in with , . By applying Theorem 2, if has a fixed point in , there exists such that , and is a solution of (93).

7.1. Applications

The fractional-order differential equations emerge in various areas of engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, control theory, biology, economics, blood flow phenomena, signal and image processing, biophysics, aerodynamics, and fitting of experimental data.

7.2. Open Problem

What are the conditions for making a power of the contraction a nonnegative real number for fixed point and coincidence fixed point for two or more maps in various spaces?

8. Conclusion

The aim of this paper is to produce four new classes of type contractions. This research focuses on new idea of fractional symmetric --contractions of type-I, type-II, type-III, and type-IV in the structure of -metric space, which is different from and more general than ordinary metric. This paper will open a new conspiracy of fractional fixed-point theory. We develop here Suzuki-type fixed point results in orbitally complete -metric space. These new investigations and applications would enhance the impact of new setup.

Data Availability

The data used to support the findings of this study are available upon request.

Conflicts of Interest

The author declares that there are no conflicts of interest.

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Copyright

Copyright © 2021 Aftab Hussain. 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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