Existence Results to Tripled System of Fractional Langevin Equations via Measure of Noncompactness Technique
Main Article Content
Abstract
In this study, boundary-constrained tripled systems of fractional Langevin equations with some specified operators and generalized boundary conditions are taken into account. The uniqueness of the solution for the system under consideration are discovered using fractional calculus and the Banach fixed point theorem. We examine the assumptions and conditions that require solutions utilizing the tripled fixed point and the continuity coefficient by introducing noncompactness measures. To demonstrate the outcomes, we create a control problem examples to verify this correctness.
Article Details
References
- A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, 2006.
- K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer Berlin Heidelberg, 2010. https://doi.org/10.1007/978-3-642-14574-2.
- V. Lakshmikantham, A. Vatsala, Basic Theory of Fractional Differential Equations, Nonlinear Anal. Theory Methods Appl. 69 (2008), 2677–2682. https://doi.org/10.1016/j.na.2007.08.042.
- A. Salem, H. Malaikah, N. Alsobhi, Fractional Mathieu Equation with Two Fractional Derivatives and Some Applications, Fractal Fract. 9 (2025), 80. https://doi.org/10.3390/fractalfract9020080.
- L. Almaghamsi, A. Salem, Fractional Langevin Equations with Infinite-Point Boundary Condition: Application to Fractional Harmonic Oscillator, J. Appl. Anal. Comput. 13 (2023), 3504–3523. https://doi.org/10.11948/20230124.
- K. Shah, J. Wang, H. Khalil, R.A. Khan, Existence and Numerical Solutions of a Coupled System of Integral BVP for Fractional Differential Equations, Adv. Differ. Equ. 2018 (2018), 149. https://doi.org/10.1186/s13662-018-1603-1.
- A. Khan, Y. Li, K. Shah, T.S. Khan, On Coupled p-Laplacian Fractional Differential Equations with Nonlinear Boundary Conditions, Complexity 2017 (2017), 8197610. https://doi.org/10.1155/2017/8197610.
- D. Luo, A. Zada, S. Shaleena, M. Ahmad, Analysis of a Coupled System of Fractional Differential Equations with Non-Separated Boundary Conditions, Adv. Differ. Equ. 2020 (2020), 590. https://doi.org/10.1186/s13662-020-03045-6.
- A. Salem, F. Alzahrani, M. Alnegga, Coupled System of Nonlinear Fractional Langevin Equations with Multipoint and Nonlocal Integral Boundary Conditions, Math. Probl. Eng. 2020 (2020), 7345658. https://doi.org/10.1155/2020/7345658.
- A. Salem, L. Almaghamsi, Existence Solution for Coupled System of Langevin Fractional Differential Equations of Caputo Type with Riemann–Stieltjes Integral Boundary Conditions, Symmetry 13 (2021), 2123. https://doi.org/10.3390/sym13112123.
- I. Petras, R.L. Magin, Simulation of Drug Uptake in a Two Compartmental Fractional Model for a Biological System, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 4588–4595. https://doi.org/10.1016/j.cnsns.2011.02.012.
- D.T. Gillespie, The Chemical Langevin Equation, J. Chem. Phys. 113 (2000), 297–306. https://doi.org/10.1063/1.481811.
- M.M. Matar, I.A. Amra, J. Alzabut, Existence of Solutions for Tripled System of Fractional Differential Equations Involving Cyclic Permutation Boundary Conditions, Bound. Value Probl. 2020 (2020), 140. https://doi.org/10.1186/s13661-020-01437-x.
- H. Afshari, H. Shojaat, M. Siahkali Moradi, Existence of the Positive Solutions for a Tripled System of Fractional Differential Equations via Integral Boundary Conditions, Results Nonlinear Anal. 4 (2021), 186–199. https://doi.org/10.53006/rna.938851.
- S. Etemad, M.M. Matar, M.A. Ragusa, S. Rezapour, Tripled Fixed Points and Existence Study to a Tripled Impulsive Fractional Differential System via Measures of Noncompactness, Mathematics 10 (2022), 25. https://doi.org/10.3390/math10010025.
- A. Salem, F. Alzahrani, L. Almaghamsi, Fractional Langevin Equations with Nonlocal Integral Boundary Conditions, Mathematics 7 (2019), 402. https://doi.org/10.3390/math7050402.
- A. Salem, B. Alghamdi, Multi-Point and Anti-Periodic Conditions for Generalized Langevin Equation with Two Fractional Orders, Fractal Fract. 3 (2019), 51. https://doi.org/10.3390/fractalfract3040051.
- B. Ahmad, M. Alghanmi, A. Alsaedi, H.M. Srivastava, S.K. Ntouyas, The Langevin Equation in Terms of Generalized Liouville–Caputo Derivatives with Nonlocal Boundary Conditions Involving a Generalized Fractional Integral, Mathematics 7 (2019), 533. https://doi.org/10.3390/math7060533.
- H. Fazli, J.J. Nieto, Fractional Langevin Equation with Anti-Periodic Boundary Conditions, Chaos Solit. Fractals 114 (2018), 332–337. https://doi.org/10.1016/j.chaos.2018.07.009.
- H. Fazli, H. Sun, S. Aghchi, Existence of Extremal Solutions of Fractional Langevin Equation Involving Nonlinear Boundary Conditions, Int. J. Comput. Math. 98 (2021), 1–10. https://doi.org/10.1080/00207160.2020.1720662.
- A. Salem, Existence Results of Solutions for Anti-Periodic Fractional Langevin Equation, J. Appl. Anal. Comput. 10 (2020), 2557–2574. https://doi.org/10.11948/20190419.
- H. Baghani, A. Salem, Solvability and Stability of a Class of Fractional Langevin Differential Equations with the Mittag–Leffler Function, Bol. Soc. Mat. Mex. 30 (2024), 46. https://doi.org/10.1007/s40590-024-00618-3.
- A. Salem, A. Al-Dosari, Positive Solvability for Conjugate Fractional Differential Inclusion of (k,n-k) Type without Continuity and Compactness, Axioms 10 (2021), 170. https://doi.org/10.3390/axioms10030170.
- A. Salem, M. Alnegga, Measure of Noncompactness for Hybrid Langevin Fractional Differential Equations, Axioms 9 (2020), 59. https://doi.org/10.3390/axioms9020059.
- A. Seemab, M. Rehman, Existence of Solution of an Infinite System of Generalized Fractional Differential Equations by Darbo’s Fixed Point Theorem, J. Comput. Appl. Math. 364 (2020), 112355. https://doi.org/10.1016/j.cam.2019.112355.
- M. Mursaleen, B. Bilalov, S. Rizvi, Applications of Measures of Noncompactness to Infinite System of Fractional Differential Equations, Filomat 31 (2017), 3421–3432. https://doi.org/10.2298/fil1711421m.
- J. Banaśs, M. Krajewska, Existence of Solutions for Infinite Systems of Differential Equations in Spaces of Tempered Sequences, Electron. J. Differ. Equ. 2017 (2017), no. 60, 1–28. https://ejde.math.txstate.edu/Volumes/2017/60/abstr.html.
- A. Salem, H. Malaikah, E.S. Kamel, An Infinite System of Fractional Sturm–Liouville Operator with Measure of Noncompactness Technique in Banach Space, Mathematics 11 (2023), 1444. https://doi.org/10.3390/math11061444.
- A. Salem, H.M. Alshehri, L. Almaghamsi, Measure of Noncompactness for an Infinite System of Fractional Langevin Equation in a Sequence Space, Adv. Differ. Equ. 2021 (2021), 132. https://doi.org/10.1186/s13662-021-03302-2.
- A. Salem, L. Almaghamsi, F. Alzahrani, An Infinite System of Fractional Order with p-Laplacian Operator in a Tempered Sequence Space via Measure of Noncompactness Technique, Fractal Fract. 5 (2021), 182. https://doi.org/10.3390/fractalfract5040182.
- K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., 1993.
- I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
- J. Banas, K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1980.
- J. Banaś, M. Jleli, M. Mursaleen, B. Samet, C. Vetro, Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, Springer Singapore, 2017. https://doi.org/10.1007/978-981-10-3722-1.
- J. Banaś, L. Olszowy, On a Class of Measures of Noncompactness in Banach Algebras and Their Application to Nonlinear Integral Equations, Z. Anal. Anwend. 28 (2009), 475–498. https://doi.org/10.4171/zaa/1394.
- Z. Baitiche, K. Guerbati, M. Benchohra, Y. Zhou, Boundary Value Problems for Hybrid Caputo Fractional Differential Equations, Mathematics 7 (2019), 282. https://doi.org/10.3390/math7030282.
- M. Jleli, E. Karapinar, D. O’Regan, B. Samet, Some Generalizations of Darbo’s Theorem and Applications to Fractional Integral Equations, Fixed Point Theory Appl. 2016 (2016), 11. https://doi.org/10.1186/s13663-016-0497-4.
- S. Szufla, On the Application of Measure of Noncompactness to Existence Theorems, Rend. Semin. Mat. Univ. Padova 75 (1986), 1–14. https://www.numdam.org/item/RSMUP_1986__75__1_0/.