Startseite Switched coupled system of nonlinear impulsive Langevin equations involving Hilfer fractional-order derivatives
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Switched coupled system of nonlinear impulsive Langevin equations involving Hilfer fractional-order derivatives

  • Rizwan Rizwan EMAIL logo , Akbar Zada ORCID logo , Hira Waheed und Usman Riaz
Veröffentlicht/Copyright: 25. August 2021
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International Journal of Nonlinear Sciences and Numerical Simulation
Aus der Zeitschrift International Journal of Nonlinear Sciences and Numerical Simulation Band 24 Heft 6

Abstract

In this manuscript, switched coupled system of nonlinear impulsive Langevin equations involving four Hilfer fractional-order derivatives is considered. Using the techniques of nonlinear functional analysis, we establish appropriate conditions and results to discuss the existence, uniqueness, and Ulam’s type stability results of our proposed model, with the help of Schaefer’s fixed point theorem. An example is provided at the end to illustrate our results.

Keywords: Hilfer fractional derivative; Langevin equation; instantaneous impulses; Ulam–Hyers–Rassias stability
MSC 2010: 26A33; 34A08; 34B27
Corresponding author: Rizwan Rizwan, Department of Mathematics, University of Buner, Buner, Pakistan, E-mail:

  1. Author contribution: All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare that they have no competing interests regarding this research work.

References

[1] 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 Anal.: Real World Appl., vol. 13, no. 2, pp. 599–602, 2012. https://doi.org/10.1016/j.nonrwa.2011.07.052.Suche in Google Scholar

[2] K. S. Fa, “Generalized Langevin equation with fractional derivative and long-time correlation function,” Phys. Rev. E, vol. 73, no. 6, p. 061104, 2006. https://doi.org/10.1103/PhysRevE.73.061104.Suche in Google Scholar PubMed

[3] S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,” Phys. Lett., vol. 372, no. 42, pp. 6309–6320, 2008. https://doi.org/10.1016/j.physleta.2008.08.045.Suche in Google Scholar

[4] F. Mainardi and P. Pironi, “The fractional Langevin equation: Brownian motion revisited,” Extracta Math., vol. 11, no. 1, pp. 140–154, 1996.Suche in Google Scholar

[5] R. Rizwan, “Existence theory and stability analysis of fractional Langevin equation,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 20, nos 7–8, pp. 833–848, 2019. https://doi.org/10.1515/ijnsns-2019-0053.Suche in Google Scholar

[6] 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 Appl. Math., vol. 109, pp. 973–1033, 2010. https://doi.org/10.1007/s10440-008-9356-6.Suche in Google Scholar

[7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equation, North Holland, Elsevier Science B.V., 2006.Suche in Google Scholar

[8] V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge, Cambridge Scientific Publishers, 2009.Suche in Google Scholar

[9] I. Podlubny, Fractional Differential Equations, San Diego, Academic Press, 1999.Suche in Google Scholar

[10] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Moscow, Springer, HEP, 2011.10.1007/978-3-642-14003-7Suche in Google Scholar

[11] S. Tang, A. Zada, S. Faisal, M. M. A. El-Sheikh, and T. Li, “Stability of higher-order nonlinear impulsive differential equations,” J. Nonlinear Sci. Appl., vol. 9, pp. 4713–4721, 2016. https://doi.org/10.22436/jnsa.009.06.110.Suche in Google Scholar

[12] R. Rizwan, A. Zada, and X. Wang, “Stability analysis of non linear implicit fractional Langevin equation with non-instantaneous impulses,” Adv. Differ. Equ., vol. 2019, p. 85, 2019. https://doi.org/10.1186/s13662-019-1955-1.Suche in Google Scholar

[13] Y. Ding, J. Jiang, D. ORegan, and J. Xu, “Positive solutions for a system of Hadamard-type fractional differential equations with semipositone nonlinearities,” Hind. Compl., p. 9742418, 2020. https://doi.org/10.1155/2020/9742418.Suche in Google Scholar

[14] J. Xu, J. Jiang, and D. ORegan, “Positive solutions for a class of p-Laplacian Hadamard fractional-order three-point boundary value problems,” Mathematics, vol. 8, p. 308, 2020. https://doi.org/10.3390/math8030308.Suche in Google Scholar

[15] Z. Ali, A. Zada, and K. Shah, “Ulam stability to a toppled systems of nonlinear implicit fractional order boundary value problem,” Bound. Value Probl., vol. 2018, p. 175, 2018. https://doi.org/10.1186/s13661-018-1096-6.Suche in Google Scholar

[16] M. Benchohra and D. Seba, “Impulsive fractional differential equations in Banach spaces,” Electron. J. Qual. Theor. Differ. Equ., vol. 8, pp. 1–14, 2009. https://doi.org/10.14232/ejqtde.2009.4.8.Suche in Google Scholar

[17] M. Feckan, Y. Zhou, and J. Wang, “On the concept and existence of solution for impulsive fractional differential equations,” Commun. Nonlinear Sci. Numer. Simulat., vol. 17, pp. 3050–3060, 2012. https://doi.org/10.1016/j.cnsns.2011.11.017.Suche in Google Scholar

[18] J. Wang, Y. Zhou, and M. Feckan, “Nonlinear impulsive problems for fractional differential equations and Ulam stability,” Comput. Math. Appl., vol. 64, pp. 3389–3405, 2012. https://doi.org/10.1016/j.camwa.2012.02.021.Suche in Google Scholar

[19] N. Kosmatov, “Initial value problems of fractional order with fractional impulsive conditions,” Results Math., vol. 63, pp. 1289–1310, 2013. https://doi.org/10.1007/s00025-012-0269-3.Suche in Google Scholar

[20] A. Zada, W. Ali, and S. Farina, “Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses,” Math. Methods Appl. Sci., vol. 40, no. 15, pp. 5502–5514, 2017. https://doi.org/10.1002/mma.4405.Suche in Google Scholar

[21] A. Zada and S. O. Shah, “Hyers–Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses,” Hacettepe J. Math. Stat., vol. 47, no. 5, pp. 1196–1205, 2018. https://doi.org/10.15672/HJMS.2017.496.Suche in Google Scholar

[22] A. Zada and S. Ali, “Stability analysis of multi-point boundary value problem for sequential fractional differential equations with non-instantaneous impulses,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 19, no. 7, pp. 763–774, 2018. https://doi.org/10.1515/ijnsns-2018-0040.Suche in Google Scholar

[23] R. Rizwan and A. Zada, “Nonlinear impulsive Langevin equation with mixed derivatives,” Math. Methods Appl. Sci., vol. 43, no. 1, pp. 427–442, 2020. https://doi.org/10.1002/mma.5902.Suche in Google Scholar

[24] D. Baleanu, H. Khan, H. Jafari, R. A. Khan, and M. Alipure, “On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions,” Adv. Differ. Equ., vol. 2015, 2015, Art No. 318. https://doi.org/10.1186/s13662-015-0651-z.Suche in Google Scholar

[25] J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete matric space,” Bull. Am. Math. Soc., vol. 74, pp. 305–309, 1968. https://doi.org/10.1090/s0002-9904-1968-11933-0.Suche in Google Scholar

[26] S. M. Ulam, A Collection of Mathematical Problems, New York, Interscience Publishers, 1968.Suche in Google Scholar

[27] D. H. Hyers, “On the stability of the linear functional equation,” Proc. Natl. Acad., vol. 27, pp. 222–224, 1941. https://doi.org/10.1073/pnas.27.4.222.Suche in Google Scholar PubMed PubMed Central

[28] T. M. Rassias, “On the stability of linear mappings in Banach spaces,” Proc. Am. Math., vol. 72, pp. 297–300, 1978. https://doi.org/10.1090/s0002-9939-1978-0507327-1.Suche in Google Scholar

[29] I. A. Rus, “Ulam stability of ordinary differential equations,” Stud. Univ. Babes Bolyai Math., vol. 54, pp. 125–133, 2009.Suche in Google Scholar

[30] S. O. Shah, A. Zada, and A. E. Hamza, “Stability analysis of the first order non-linear impulsive time varying delay dynamic system on time scales,” Qual. Theory Dyn. Syst., vol. 18, pp. 825–840, 2019. https://doi.org/10.1007/s12346-019-00315-x.Suche in Google Scholar

[31] A. Zada, W. Ali, and C. Park, “Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Gröwall–Bellman–Bihari’s type,” Appl. Math. Comput., vol. 350, pp. 60–65, 2019. https://doi.org/10.1016/j.amc.2019.01.014.Suche in Google Scholar

[32] A. Zada, O. Shah, and R. Shah, “Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems,” Appl. Math. Comput., vol. 271, pp. 512–518, 2015. https://doi.org/10.1016/j.amc.2015.09.040.Suche in Google Scholar

[33] A. Zada, R. Rizwan, J. Xu, and Z. Fu, “On implicit impulsive Langevin equation involving mixed order derivatives,” Adv. Differ. Equ., vol. 2019, p. 489, 2019. https://doi.org/10.1186/s13662-019-2408-6.Suche in Google Scholar

[34] H. Waheed, A. Zada, and J. Xu, “Well-posedness and Hyers–Ulam results for a class of impulsive fractional evolution equations,” Math. Methods Appl. Sci., vol. 44, pp. 749–771, 2020. https://doi.org/10.1002/mma.6784.Suche in Google Scholar

[35] A. Khan, J. F. Gomez-Aguilar, T. S. Khan, and H. Khan, “Stability analysis and numerical solutions of fractional order HIV/AIDS model,” Chaos, Solit. Fractals, vol. 122, pp. 119–128, 2019. https://doi.org/10.1016/j.chaos.2019.03.022.Suche in Google Scholar

[36] R. Rizwan, A. Zada, M. Ahmad, S. O. Shah, and H. Waheed, “Existence theory and stability analysis of switched coupled system of nonlinear implicit impulsive Langevin equations with mixed derivatives,” Math. Methods Appl. Sci., vol. 44, pp. 8963–8985, 2021. https://doi.org/10.1002/mma.7324.Suche in Google Scholar

[37] Z. Bai, “On positive solutions of a non-local fractional boundary value problem,” Nonlinear Anal. Theor. Methods Appl., vol. 72, no. 2, pp. 916–924, 2010. https://doi.org/10.1016/j.na.2009.07.033.Suche in Google Scholar

[38] M. Benchohra, J. R. Graef, and S. Hamani, “Existence results for boundary value problems with nonlinear fractional differential equations,” Hist. Anthropol., vol. 87, no. 7, pp. 851–863, 2008. https://doi.org/10.1080/00036810802307579.Suche in Google Scholar

[39] R. Shah and A. Zada, “A fixed point approach to the stability of a nonlinear Volterra integro differential equation with delay,” Hacettepe J. Math. Stat., vol. 47, no. 3, pp. 615–623, 2018.Suche in Google Scholar

[40] K. M. Furati, M. D. Kassim, and N. E. Tatar, “Existence and uniqueness for a problem involving Hilfer fractional derivative,” Comput. Math. Appl., vol. 64, no. 6, pp. 1616–1626, 2012. https://doi.org/10.1016/j.camwa.2012.01.009.Suche in Google Scholar

[41] J. Wang, Y. Zhou, and Z. Lin, “On a new class of impulsive fractional differential equations,” Appl. Math. Comput., vol. 242, pp. 649–657, 2014. https://doi.org/10.1016/j.amc.2014.06.002.Suche in Google Scholar

[42] A. Zada, S. Ali, and Y. Li, “Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition,” Adv. Differ. Equ., vol. 2017, p. 317, 2017. https://doi.org/10.1186/s13662-017-1376-y.Suche in Google Scholar

[43] B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Comput. Math. Appl., vol. 58, pp. 1838–1843, 2009. https://doi.org/10.1016/j.camwa.2009.07.091.Suche in Google Scholar

[44] Z. G. Hu, W. B. Liu, and T. Y. Chen, “Existence of solutions for a coupled system of fractional differential equations at resonance,” Bound. Value Probl., vol. 2012, p. 98, 2012. https://doi.org/10.1186/1687-2770-2012-98.Suche in Google Scholar

[45] Z. Ali, A. Zada, and K. Shah, “On Ulam’s stability for a coupled systems of nonlinear implicit fractional differential equations,” Bull. Malays. Math. Sci. Soc., vol. 42, pp. 2681–2699, 2019. https://doi.org/10.1007/s40840-018-0625-x.Suche in Google Scholar

[46] J. Wang and Y. Zhang, “Nonlocal initial value problems for differential equations with Hilfer fractional derivative,” Appl. Math. Comput., vol. 266, pp. 850–859, 2015. https://doi.org/10.1016/j.amc.2015.05.144.Suche in Google Scholar

[47] Z. Ali, F. Rabiei, and K. Shah, “On Ulam’s type stability for a class of impulsive fractional differential equations with nonlinear integral boundary conditions,” J. Nonlinear Sci. Appl., vol. 10, pp. 4760–4775, 2017. https://doi.org/10.22436/jnsa.010.09.19.Suche in Google Scholar
Received: 2020-10-22
Revised: 2021-03-15
Accepted: 2021-07-20
Published Online: 2021-08-25

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Original Research Articles
  3. Frequency responses for induced neural transmembrane potential by electromagnetic waves (1 kHz to 1 GHz)
  4. Investigating existence results for fractional evolution inclusions with order r ∈ (1, 2) in Banach space
  5. Optimal control of non-instantaneous impulsive second-order stochastic McKean–Vlasov evolution system with Clarke subdifferential
  6. Generalized forms of fractional Euler and Runge–Kutta methods using non-uniform grid
  7. Controllability of coupled fractional integrodifferential equations
  8. A new generalized approach to study the existence of solutions of nonlinear fractional boundary value problems
  9. Rational soliton solutions in the nonlocal coupled complex modified Korteweg–de Vries equations
  10. Buoyancy driven flow characteristics inside a cavity equiped with diamond elliptic array
  11. Analysis and numerical effects of time-delayed rabies epidemic model with diffusion
  12. Two occurrences of fractional actions in nonlinear dynamics
  13. Multiwave interaction solutions for a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics
  14. Effects of mixed time delays and D operators on fixed-time synchronization of discontinuous neutral-type neural networks
  15. Solvability and stability of nonlinear hybrid ∆-difference equations of fractional-order
  16. Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator
  17. Simulation and modeling of different cell shapes for closed-cell LM-13 alloy foam for compressive behavior
  18. Pandemic management by a spatio–temporal mathematical model
  19. Adaptive ADI difference solution of quenching problems based on the 3D convection–reaction–diffusion equation
  20. Null controllability of Hilfer fractional stochastic integrodifferential equations with noninstantaneous impulsive and Poisson jump
  21. Application of modified Mickens iteration procedure to a pendulum and the motion of a mass attached to a stretched elastic wire
  22. Modeling the spatiotemporal intracellular calcium dynamics in nerve cell with strong memory effects
  23. Switched coupled system of nonlinear impulsive Langevin equations involving Hilfer fractional-order derivatives
  24. Improving the dynamic behavior of the plate under supersonic air flow by using nonlinear energy sink
https://doi.org/10.1515/ijnsns-2020-0240
Schlagwörter für diesen Artikel
Hilfer fractional derivative; Langevin equation; instantaneous impulses; Ulam–Hyers–Rassias stability

Artikel in diesem Heft

  1. Frontmatter
  2. Original Research Articles
  3. Frequency responses for induced neural transmembrane potential by electromagnetic waves (1 kHz to 1 GHz)
  4. Investigating existence results for fractional evolution inclusions with order r ∈ (1, 2) in Banach space
  5. Optimal control of non-instantaneous impulsive second-order stochastic McKean–Vlasov evolution system with Clarke subdifferential
  6. Generalized forms of fractional Euler and Runge–Kutta methods using non-uniform grid
  7. Controllability of coupled fractional integrodifferential equations
  8. A new generalized approach to study the existence of solutions of nonlinear fractional boundary value problems
  9. Rational soliton solutions in the nonlocal coupled complex modified Korteweg–de Vries equations
  10. Buoyancy driven flow characteristics inside a cavity equiped with diamond elliptic array
  11. Analysis and numerical effects of time-delayed rabies epidemic model with diffusion
  12. Two occurrences of fractional actions in nonlinear dynamics
  13. Multiwave interaction solutions for a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics
  14. Effects of mixed time delays and D operators on fixed-time synchronization of discontinuous neutral-type neural networks
  15. Solvability and stability of nonlinear hybrid ∆-difference equations of fractional-order
  16. Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator
  17. Simulation and modeling of different cell shapes for closed-cell LM-13 alloy foam for compressive behavior
  18. Pandemic management by a spatio–temporal mathematical model
  19. Adaptive ADI difference solution of quenching problems based on the 3D convection–reaction–diffusion equation
  20. Null controllability of Hilfer fractional stochastic integrodifferential equations with noninstantaneous impulsive and Poisson jump
  21. Application of modified Mickens iteration procedure to a pendulum and the motion of a mass attached to a stretched elastic wire
  22. Modeling the spatiotemporal intracellular calcium dynamics in nerve cell with strong memory effects
  23. Switched coupled system of nonlinear impulsive Langevin equations involving Hilfer fractional-order derivatives
  24. Improving the dynamic behavior of the plate under supersonic air flow by using nonlinear energy sink
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