Multipoint BVP for the Langevin Equation under <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="12.3952pt" version="1.1" viewBox="-0.0657574 -7.8684 10.1561 12.3952" width="10.1561pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-253" d="M559 271C559 395 493 448 433 448C370 448 304 394 275 297C259 245 235 121 219 37C157 54 113 110 113 201C113 310 180 386 276 429L265 457C131 410 23 326 23 186C23 83 86 5 211 -10C180 -159 167 -220 158 -252L165 -261C198 -255 236 -246 251 -218L287 -5C421 9 559 119 559 271ZM484 273C484 144 400 40 293 33L352 331C364 389 382 406 405 406C431 406 484 370 484 273Z"/></g></svg>-</span>Hilfer Fractional Operator

Almalahi, Mohammed A.; Panchal, Satish K.; Jarad, Fahd

doi:https://doi.org/10.1155/2022/2798514

Journal of Function Spaces

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Research Article | Open Access

Volume 2022 | Article ID 2798514 | https://doi.org/10.1155/2022/2798514

Multipoint BVP for the Langevin Equation under -Hilfer Fractional Operator

Mohammed A. Almalahi,¹Satish K. Panchal,²and Fahd Jarad^3,4

Academic Editor: Calogero Vetro

Received04 Feb 2022

Revised11 Apr 2022

Accepted13 Apr 2022

Published14 May 2022

Abstract

In this research paper, we consider a class of boundary value problems for a nonlinear Langevin equation involving two generalized Hilfer fractional derivatives supplemented with nonlocal integral and infinite-point boundary conditions. At first, we derive the equivalent solution to the proposed problem at hand by relying on the results and properties of the generalized fractional calculus. Next, we investigate and develop sufficient conditions for the existence and uniqueness of solutions by means of semigroups of operator approach and the Krasnoselskii fixed point theorems as well as Banach contraction principle. Moreover, by means of Gronwall’s inequality lemma and mathematical techniques, we analyze Ulam-Hyers and Ulam-Hyers-Rassias stability results. Eventually, we construct an illustrative example in order to show the applicability of key results.

1. Introduction

In recent decades, the subject of fractional calculus (FC) becomes a very significant tool to characterize memory phenomena in many branches of engineering and sciences. Some properties of solutions like the existence and uniqueness of solutions for fractional boundary value problems (FBVPs) have been widely investigated [1–5], and a broad rundown of references is given in that regard. The importance of FBVPs comes from their applicability in several fields like science and engineering. Langevin [6] devised an equation to describe the progression of physical processes in fluctuating settings in 1908, which is known as the Langevin equation. Mainardi and Pironi in 1990 [7] presented the fractional Langevin equation (FLE). Yukunthorn et al. [8] via Banach’s, Krasnoselskii’s, and Leray-Schauder’s nonlinear alternative and Leray-Schauder studied the existence and uniqueness of solution for a sequential nonlinear fractional Caputo-Langevin equation. Baghani [9] discussed the solvability of initial value problems for nonlinear Langevin equation involving two fractional orders. Fazli and Nieto [10] by means of coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces investigated the existence and uniqueness of solutions for the nonlinear Langevin equation involving two fractional orders with antiperiodic boundary conditions. Salem et al. [11] studied the existence and uniqueness of solution for fractional integrodifferential Langevin equation involving two fractional orders with three-point multiterm fractional integral boundary conditions. For further works on characteristics of solutions, such as existence, uniqueness, and stability results for fractional Langevin equations (FLE), see [12–15]. We also include some recent works on qualitative analysis of similar situations with the generalized fractional operators (see [16–20]). Samet et al. [21, 22] introduced a new concept of fixed point theorems for mappings in complete metric spaces. System stability is one of the most essential qualitative characteristics of solutions to FDEs. However, there are few results of Ulam-Hyers (UH) and generalized Ulam-Hyers (GUH) stability of solutions of FDEs in the literature. On the other hand, Guo et al. [23] studied the existence and Hyers-Ulam stability of the virtually periodic solution to the fractional differential equation with impulse and fractional Brownian motion under nonlocal conditions via the semigroups of operator approach and the Mönch fixed point method. Li et al. [24] by using Krasnoselskii’s fixed point method investigated the existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delay. Shu and Shi [25] studied the mild solution of impulsive fractional evolution equations under Caputo fractional derivative.

Recently, Almalahi and Panchal [26] by means of some fixed point theorems studied the qualitative properties of solution for the following problem: where is the -Hilfer FD of order with type , , and is the -RL fractional integral of order , and is an increasing function with for all .

The recent works regarding of the FLE can be found in [27–29]. Li et al. [27] via some fixed point techniques and degree theory obtained some existence results of Caputo-type Langevin FBVPs with infinite-point boundary conditions where are the Caputo FDs of order .

Seemab et al. in [29] discussed the existence and uniqueness of solution for the following problem: where denotes the -Caputo FD of order to

Nuchpong et al. [28] by using Banach, Leray–Schauder, and Krasnoselskii fixed point theorems discussed the existence and uniqueness results of -Hilfer-type Langevin FBVP nonlocal integral boundary conditions

Motivated by the aforementioned discussions, in this research paper, we study the existence and uniqueness as well as Ulam-Hyers stability results for Langevin-type generalized FDEs with infinite-point boundary conditions of the form where are the -Hilfer FDs of order with type and is a -RL fractional integral of order such that , and is an increasing function with for all , and is a given function.

Observe that the results that will be obtained according to the problem (5) cover results of [27–29] as follows: (i)Li et al. in [27] (for , and )(ii)Nuchpong et al. [28] results (for , and )(iii)Seemab et al., in [29] (for , and )

We anticipate that the results presented in this paper will be groundbreaking and contribute to the existence of knowledge on the Langevin equation. The results acquired in this study are applicable to a wide range by choosing suitable values of function and may be used to a variety of other challenges.

The arrangement of this paper is as per the following. In Section 2, we will give some definitions and lemmas to demonstrate our primary outcomes. In Section 3, Krasnoselskii and Banach fixed point techniques are used to acquire the existence and uniqueness of solutions of the suggested problem (5). In Section 4, we analyze the stability results in Ulam-Hyers sense. In the last section, we introduce a numerical model to represent the fundamental results.

2. Auxiliary Results

In this section, to analyze our main results, we present here some important definitions and auxiliary lemmas. Assume that and denote the Banach space of all continuous functions defined from into with the norm . Let be an increasing function with for all .

Definition 1 (see [4]). Let and Then, the following representation is called -RL fractional integral of of order .

Definition 2 (see [31]). Let , and the functions . Then, the -Hilfer FD of with order and type is defined by where

Remark 3. In Definition 2, type and function allow to interpolate continuously between the Riemann-Liouville FD and the Caputo FD. More precisely, we have the following: (i)-Hilfer FD corresponds to the Riemann-Liouville FD for ( ), i.e.,(ii)-Hilfer FD corresponds to the Caputo FD for ( ), i.e.,(iii)-Hilfer FD corresponds to the -Riemann-Liouville FD for , i.e.,(iv)-Hilfer FD corresponds to the -Caputo FD for , i.e.,

Theorem 4 (see [4]). Let be a function. Then, .

Lemma 5 (see [4, 30]). Let and . Then,

Lemma 6 (see [30]). If , and , then

Theorem 7 (see [31]) (Banach theorem). Let be a closed subset of Banach space , and the operator be a strict contraction that means for some , Then, has a fixed point in .

Theorem 8 (see [32]) (Krasnoselskii’s theorem). Let be a nonempty, closed, convex, and bounded subset from Banach space . If there exist two operators such that (i) for all , imply , (ii) is compact and continuous, and (iii) is a contraction mapping, then there exists a function such that

3. Equivalent Integral Equations for Problem (5)

In order to convert the problem (5) into a fixed point problem, we will present the following lemma with linear function .

Lemma 9. For let , . Then, the function is a solution of the linear problem if and only if where

Proof. Let be a solution of the problem (15). Take taking into consideration that on both sides of the following equation: Using Lemma 6, we have where is an arbitrary constant. By conditions and we obtain and hence, (19) reduces to Taking again taking into consideration that on both sides of (20) and using Lemmas 6 and 5, we have According to the condition we obtain and hence, (21) reduces to By replacing with in equation (22), we get Replacing again with in equation (22) with multiplication by with the use of semigroup property defined in Lemma 5, we get Now, by equations (23) and (24) and second condition , we get Hence, Putting into (22), we get (16). Conversely, we assume that the solution satisfies (16). Then, one can get and . Furthermore, applying on both sides of (16) replacing by and multiplying by , we get Thus, all conditions are satisfied. Next, applying on both sides of (16), we have Using Lemmas 5 and 6, we get Applying on both sides of equation (29), we have Using Lemma 5, we get The proof is completed.

Lemma 10. For let , , and be a continuous function Then, the solution of the problem (5) is given by In order to simplify our analysis, we will use the following notation: where and

4. Existence and Uniqueness Solution

In this section, we will discuss the existence and uniqueness of solutions for -Hilfer FDE (5) by applying Theorems 8 and 7. To demonstrate our main results, the following assumptions must be fulfilled.

() is a continuous and there exists a constant number such that

() is continuous and there exists such that with .

Theorem 11. Assume that () and () hold. Then, the problem (5) has at least one solution, provided that , where

Proof. Consider the continuous operator , which is defined by Clearly, the fixed points of the operator defined by (37) is a solution of the problem (5). Now, we will prove that the operator has a fixed point by using Theorem 8. Let be a closed ball defined by with Let be two operators such that , where In order to achieve conditions of Theorem 8, the proof is divided into the following steps.
Step 1. for all For any , we have Due to the fact that is an increasing function, we have and hence, Thus, Thus,
Step 2. is compact and continuous.
Let be a sequence such that in Then, That means is continuous. Moreover, the operator is bounded on due to Step 1. Thus, is uniformly bounded on . Next, we will prove that is equicontinuous. Let such that . Then, For and continuity of , we obtain Hence, is equicontinuous. As a result of the Arzelá-Ascoli theorem, we deduce that the operator is compact in . Thus, is completely continuous.
Step 3. is contraction mapping. For and , we obtain Thus, we conclude that is a contraction.
According to the above steps and Theorem 8, we infer that problem (5) has at least one solution on .

Theorem 12. Assume that () holds. If then the problem (5) has a unique solution on .

Proof. We noted that the fixed points of the operator defined in (37) are a solution of problem (5). Define a closed ball set as with , where and

First, we show that By (33) and condition (), we have

Thus, for , we obtain

Thus,

Next, we prove that is contraction map. Indeed, for and , we obtain

From (48), is a contraction map. Hence, in view of Theorem 7, we conclude that the problem (5) has a unique solution on .

5. Stability Analysis

In 1940 [33], the problem of stability of functional equations was created by Ulam’s question regarding the stability of group homomorphisms. In the following year, Hyers [34] gave a positive interpretation of the Ulam question within the Banach spaces, and this was the first major advance and a step towards more solutions in this field. Since then, many papers have been published regarding various generalizations of the Ulam problem and Hyers theory. In 1978, Rassias [35] succeeded in extending Hyers’ theory of mappings between Banach spaces. Rassias’s result attracted many mathematicians around the world who began their investigations of the problems of stability of functional equations.

In this regard, we discuss the stability results in the frame of Ulam-Hyers- (UH-) Rassias (UHR). Let and a continuous function such that it satisfies the following inequalities:

Lemma 13 (see [36]). Let and be two nonnegative functions locally integrable on Assume that is nonnegative and nondecreasing, and let be an increasing function such that for all . If then If is a nondecreasing function on . then, we have

Definition 14. We say that the problem (5) is UH stable if for everythat satisfies an inequality (54) and is a solution of the problem (5), there exists a constant number such that

Definition 15. We say that the problem (5) is UHR stable with respect to nondecreasing function if for every that satisfies an inequality (55) and is a solution of the problem (5), there exists such that

Remark 16. A function satisfies an inequality (54) if and only if there exist a functions such that

Lemma 17. Let , , and If a function satisfies (54), then satisfies where

Proof. By Remark 16, we have Then we get It follows that

Theorem 18. Let us assume that holds Then, is UH stable

Proof. Let and be a function that satisfies an inequality (54) and let be a solution of Then, by using Theorem 12, we have where

Since we can easily prove that . Hence, according to () and Lemma 17, then for each

Thus, by Lemma 13, we get

It follows that where . Hence, the problem (5) is UH stable.

Corollary 19. Under the assumptions of Theorem 18, if there exists a function , then the problem (5) is generalized UH stable.
In the forthcoming theorem, we discuss Ulam-Hyers-Rassias stability. For that, the following hypothesis must be satisfied.
() There exists an increasing function , and there exists such that for any , where

Remark 20. A function satisfies an inequality (55) if and only if there exist a functions such that

Lemma 21. Let , , and . If a function satisfies (55), then satisfies where

Proof. Indeed, by Remark 20 and Theorem 12, one can easily prove that

Theorem 22. Assume that () and () hold. Then, is UHR and generalized UHR stable.

Proof. By the same technique in Theorem 18, one can prove that Thus, by () and Lemma 13, we have It follows that where . Hence, the problem (5) is UHR stable as well as generalized UHR stable.

6. An Example

Consider the Langevin-type fractional integrodifferential equation such that , , , and

Clearly, for each , we have with . By the given data, we can get and hence, Thus, all assumptions in Theorem 12 hold. Then, problem (84) has a unique solution. Moreover, we have

Thus, all the assumptions in Theorem 11 hold. Then, problem (84) has at least one solution. Furthermore, for , we find that is satisfied. Then, equation (67) is Ulam-Hyers stable with where and Finally, we consider for ]. Then, is nondecreasing continuous function. Hence, by Lemma 5, we get where for and Therefore, Theorem 22 applicable. Furthermore, for and a continuous function , we find that is satisfied. Then, equation (80) is UHR stable with where

7. Conclusion

Because the idea of fractional operators in the context of -Hilfer is innovative and important, several academics have explored and established various qualitative features of FDE solutions incorporating such operators. To extend these qualitative features, we developed and investigated sufficient conditions for the existence and uniqueness of solutions, as well as Ulam-Hyers stability results for a nonlinear fractional integrodifferential Langevin equation involving -Hilfer FD with respect to an increasing function, for a nonlinear fractional integrodifferential Langevin equation involving -Hilfer FD.

Our technique was based on reducing the problem (5) to a fractional integral equation and using certain conventional Banach-type and Krasnoselskii-type fixed point theorems. Furthermore, we investigated the stability data in the Ulam-Hyers sense using mathematical analytic tools. To support the main results, an example was given.

In fact, our outcomes generalize those in [26–29] and cover many results not study yet. Due to the wide recent investigations and applications of the Langevin equation, we believe that acquired results here will be important for future investigations on the theory of fractional calculus.

Data Availability

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

Disclosure

This work is conducted during our work at Hajjah University (Yemen).

Conflicts of Interest

All authors declare that they have no conflict of interest.

References

B. Ahmad and R. Luca, “Existence of solutions for sequential fractional integro-differential equations and inclusions with nonlocal boundary conditions,” Applied Mathematics and Computation, vol. 339, pp. 516–534, 2018.
View at: Publisher Site | Google Scholar
R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.
View at: Publisher Site | Google Scholar
B. Ahmad, J. J. Nieto, A. Alsaedi, and M. El-Shahed, “A study of nonlinear Langevin equation involving two fractional orders in different intervals,” Nonlinear Analysis: Real World Applications, vol. 13, no. 2, pp. 599–606, 2012.
View at: Publisher Site | Google Scholar
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
I. Podlubny, “Fractional differential equations,” Mathematics in Science and Engineering; Academic Press, New York, NY, USA, 1999.
View at: Google Scholar
W. T. Coffey, Y. P. Kalmykov, and J. T. Waldron, The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering, World Scientific, Singapore, 2004.
View at: Publisher Site
F. Mainradi and P. Pironi, “The fractional Langevin equation: Brownian motion revisted,” Extracta Mathematicae, vol. 10, pp. 140–154, 1996.
View at: Google Scholar
W. Yukunthorn, S. K. Ntouyas, and J. Tariboon, “Nonlinear fractional Caputo Langevin equation with nonlocal Riemann Liouville fractional integral conditions,” Advances in Difference Equations, vol. 2014, Article ID 315, 2014.
View at: Publisher Site | Google Scholar
H. Baghani, “Existence and uniqueness of solutions to fractional Langevin equations involving two fractional orders,” Journal of Fixed Point Theory and Applications, vol. 20, no. 2, 2018.
View at: Publisher Site | Google Scholar
H. Fazli and J. J. Nieto, “Fractional Langevin equation with anti-periodic boundary conditions,” Chaos, Solitons and Fractals, vol. 114, pp. 332–337, 2018.
View at: Publisher Site | Google Scholar
W. Sudsutad and J. Tariboon, “Nonlinear fractional integro-differential Langevin equation involving two fractional orders with three-point multi-term fractional integral boundary conditions,” Journal of Applied Mathematics and Computing, vol. 43, no. 1-2, pp. 507–522, 2013.
View at: Publisher Site | Google Scholar
Z. Baitiche, C. Derbazi, and M. M. Matar, “Ulam stability for nonlinear-Langevin fractional differential equations involving two fractional orders in the -Caputo sense,” Applicable Analysis, vol. 100, 2021.
View at: Publisher Site | Google Scholar
A. Boutiara, M. S. Abdo, M. A. Alqudah, and T. Abdeljawad, “On a class of Langevin equations in the frame of Caputo function-dependent-kernel fractional derivatives with antiperiodic boundary conditions,” AIMS Mathematics, vol. 6, no. 6, pp. 5518–5534, 2021.
View at: Publisher Site | Google Scholar
A. Salem, F. Alzahrani, and L. Almaghamsi, “Fractional Langevin equations with nonlocal integral boundary conditions,” Mathematics, vol. 7, no. 5, p. 402, 2019.
View at: Publisher Site | Google Scholar
M. A. Almalahi, F. Ghanim, T. Botmart, O. Bazighifan, and S. Askar, “Qualitative analysis of Langevin integro-fractional differential equation under Mittag–Leffler functions power law,” Fractal and Fractional, vol. 5, no. 4, p. 266, 2021.
View at: Publisher Site | Google Scholar
M. A. Almalahi, S. K. Panchal, and F. Jarad, “Stability results of positive solutions for a system of ψ -Hilfer fractional differential equations,” Chaos, Solitons & Fractals, vol. 147, article 110931, 2021.
View at: Publisher Site | Google Scholar
M. A. Almalahi, M. S. Abdo, and S. K. Panchal, “Existence and Ulam-Hyers stability results of a coupled system of ψ-Hilfer sequential fractional differential equations,” Results in Applied Mathematics, vol. 10, article 100142, 2021.
View at: Publisher Site | Google Scholar
M. S. Abdo, S. T. Thabet, and B. Ahmad, “The existence and Ulam–Hyers stability results for -Hilfer fractional integrodifferential equations,” ournal of Pseudo-Differential Operators and Applications, vol. 11, no. 4, pp. 1757–1780, 2020.
View at: Publisher Site | Google Scholar
S. S. Redhwan, S. L. Shaikh, M. S. Abdo et al., “Investigating a generalized Hilfer-type fractional differential equation with two-point and integral boundary conditions,” AIMS Mathematics, vol. 7, no. 2, pp. 1856–1872, 2022.
View at: Publisher Site | Google Scholar
A. M. Saeed, M. A. Almalahi, and M. S. Abdo, “Explicit iteration and unique solution for -Hilfer type fractional Langevin equations,” AIMS Mathematics, vol. 7, no. 3, pp. 3456–3476, 2021.
View at: Google Scholar
B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for --contractive type mappings,” Nonlinear Analysis: theory, methods & applications, vol. 75, no. 4, pp. 2154–2165, 2012.
View at: Publisher Site | Google Scholar
P. Salimi, C. Vetro, and P. Vetro, “Some new fixed point results in non-Archimedean fuzzy metric spaces,” Nonlinear Analysis: Modelling and Control, vol. 18, no. 3, pp. 344–358, 2013.
View at: Publisher Site | Google Scholar
Y. Guo, M. Chen, X. B. Shu, and F. Xu, “The existence and Hyers-Ulam stability of solution for almost periodical fractional stochastic differential equation with fBm,” Stochastic Analysis and Applications, vol. 39, no. 4, pp. 643–666, 2021.
View at: Publisher Site | Google Scholar
S. Li, L. Shu, X. B. Shu, and F. Xu, “Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays,” Stochastics, vol. 91, no. 6, pp. 857–872, 2019.
View at: Publisher Site | Google Scholar
X. B. Shu and Y. Shi, “A study on the mild solution of impulsive fractional evolution equations,” Applied Mathematics and Computation, vol. 273, pp. 465–476, 2016.
View at: Publisher Site | Google Scholar
M. A. Almalahi and S. K. Panchal, “On the theory of -Hilfer nonlocal Cauchy problem,” Journal of Siberian Federal University. Mathematics & Physics, vol. 14, no. 2, pp. 159–175, 2021.
View at: Publisher Site | Google Scholar
B. Li, S. Sun, and Y. Sun, “Existence of solutions for fractional Langevin equation with infinite-point boundary conditions,” Journal of Applied Mathematics and Computing, vol. 53, no. 1-2, pp. 683–692, 2017.
View at: Publisher Site | Google Scholar
C. Nuchpong, S. K. Ntouyas, D. Vivek, and J. Tariboon, “Nonlocal boundary value problems for ψ-Hilfer fractional-order Langevin equations,” Boundary Value Problems, vol. 2021, 12 pages, 2021.
View at: Publisher Site | Google Scholar
A. Seemab, M. u. Rehman, J. Alzabut, Y. Adjabi, and M. S. Abdo, “Langevin equation with nonlocal boundary conditions involving a -Caputo fractional operators of different orders,” AIMS Mathematics, vol. 6, no. 7, pp. 6749–6780, 2021.
View at: Publisher Site | Google Scholar
J. V. C. Sousa and C. E. de Oliveira, “On the -Hilfer fractional derivative,” Communications in Nonlinear Science and Numerical Simulation, vol. 60, pp. 72–91, 2018.
View at: Publisher Site | Google Scholar
K. Deimling, Nonlinear Functional Analysis, Springer, New York, 1985.
View at: Publisher Site
M. A. Krasnoselskii, “Two remarks on the method of successive approximations,” Uspekhi Matematicheskikh Nauk, vol. 10, pp. 123–127, 1995.
View at: Google Scholar
S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, 8, Inter-science, New York-London, 1960.
D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222–224, 1941.
View at: Publisher Site | Google Scholar
T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
View at: Publisher Site | Google Scholar
J. V. C. Sousa and C. E. de Oliveira, “A Gronwall inequality and the Cauchy-type problem by means of -Hilfer operator,” Differential Equations & Applications, vol. 11, no. 1, pp. 87–106, 2009.
View at: Google Scholar

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Copyright © 2022 Mohammed A. Almalahi 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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