Existence, uniqueness and Ulam's stabilities for a class of implicit impulsive Langevin equation with Hilfer fractional derivatives

Xiaoming Wang; Rizwan Rizwan; Jung Rey Lee; Akbar Zada; Syed Omar Shah; Xiaoming Wang; Rizwan Rizwan; Jung Rey Lee; Akbar Zada; Syed Omar Shah

doi:10.3934/math.2021288

AIMS Mathematics

2021, Volume 6, Issue 5: 4915-4929. doi: 10.3934/math.2021288

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Research article

Existence, uniqueness and Ulam's stabilities for a class of implicit impulsive Langevin equation with Hilfer fractional derivatives

1.
School of Mathematics and Computer Science, Shangrao Normal University, Shangrao, China
2.
Department of Mathematics, University of Buner, Buner, Pakistan
3.
Department of Mathematics, Daejin University, Kyunggi 11159, Korea
4.
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
5.
Department of Physical and Numerical Sciences Qurtaba University of Science and Technology Peshawar, DI Khan, Pakistan

Received: 15 October 2020 Accepted: 04 January 2021 Published: 02 March 2021
MSC : 26A33, 34A08, 34B27

In this manuscript, a class of implicit impulsive Langevin equation with Hilfer fractional derivatives is considered. Using the techniques of nonlinear functional analysis, we establish appropriate conditions and results to discuss existence, uniqueness, Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stability results of our proposed model, with the help of Banach's fixed point theorem. An example is provided at the end to illustrate our results.
- Langevin equation,
- Hilfer fractional derivative,
- instantaneous impulses,
- Ulam-Hyers-Rassias stability
Citation: Xiaoming Wang, Rizwan Rizwan, Jung Rey Lee, Akbar Zada, Syed Omar Shah. Existence, uniqueness and Ulam's stabilities for a class of implicit impulsive Langevin equation with Hilfer fractional derivatives[J]. AIMS Mathematics, 2021, 6(5): 4915-4929. doi: 10.3934/math.2021288

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Abstract

In this manuscript, a class of implicit impulsive Langevin equation with Hilfer fractional derivatives is considered. Using the techniques of nonlinear functional analysis, we establish appropriate conditions and results to discuss existence, uniqueness, Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stability results of our proposed model, with the help of Banach's fixed point theorem. An example is provided at the end to illustrate our results.

References

[1]	R. P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973–1033. doi: 10.1007/s10440-008-9356-6
[2]	B. Ahmad, J. J. Nieto, A. Alsaedi, M. El-Shahed, A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal. Real, 13 (2012), 599–606. doi: 10.1016/j.nonrwa.2011.07.052
[3]	B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 58 (2009), 1838–1843. doi: 10.1016/j.camwa.2009.07.091
[4]	Z. Ali, F. Rabiei, K. Shah, On Ulam's type stability for a class of impulsive fractional differential equations with nonlinear integral boundary conditions, J. Nonlinear Sci. Appl., 10 (2017), 4760–4775. doi: 10.22436/jnsa.010.09.19
[5]	Z. Ali, A. Zada, K. Shah, Ulam satbility to a toppled systems of nonlinear implicit fractional order boundary value problem, Bound. Value Probl., 2018 (2018), 1–16. doi: 10.1186/s13661-017-0918-2
[6]	Z. Ali, A. Zada, K. Shah, On Ulam's stability for a coupled systems of nonlinear implicit fractional differential equations, B. Malays. Math. Sci. So., 42 (2019), 2681–2699. doi: 10.1007/s40840-018-0625-x
[7]	Z. Bai, On positive solutions of a non-local fractional boundary value problem, Nonlinear Anal. Theor., 72 (2010), 916–924. doi: 10.1016/j.na.2009.07.033
[8]	D. Baleanu, H. Khan, H. Jafari, R. A. Khan, M. Alipure, On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions, Adv. Differ. Equ., 2015 (2015), 1–14. doi: 10.1186/s13662-014-0331-4
[9]	M. Benchohra, J. R. Graef, S. Hamani, Existence results for boundary value problems with nonlinear fractional differential equations, Appl. Anal., 87 (2008), 851–863. doi: 10.1080/00036810802307579
[10]	M. Benchohra, D. Seba, Impulsive fractional differential equations in Banach spaces, Electron. J. Qual. Theo., 8 (2009), 1–14.
[11]	J. B. Diaz, B. Margolis, A fixesd point theorem of the alternative, for contractions on a generalized complete matric space, Bull. Amer. Math. Soc., 74 (1968), 305–309. doi: 10.1090/S0002-9904-1968-11933-0
[12]	K. S. Fa, Generalized Langevin equation with fractional derivative and long-time correlation function, Phys. Rev. E, 73 (2006), 061104. doi: 10.1103/PhysRevE.73.061104
[13]	M. Feckan, Y. Zhou, J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci., 17 (2012), 3050–3060. doi: 10.1016/j.cnsns.2011.11.017
[14]	D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), 222–224. doi: 10.1073/pnas.27.4.222
[15]	A. Khan, J. F. Gomez–Aguilar, T. S. Khan, H. Khan, Stability analysis and numerical solutions of fractional order HIV/AIDS model, Chaos, Soliton. Fract., 122 (2019), 119–128. doi: 10.1016/j.chaos.2019.03.022
[16]	Z. G. Hu, W. B. Liu, T. V. Chen, Existence of solutions for a coupled system of fractional differential equations at resonance, Bound. Value Probl., 98 (2012), 1–13.
[17]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equation, Elsevier Science Inc., 2006.
[18]	N. Kosmatov, Initial value problems of fractional order with fractional impulsive conditions, Results Math., 63 (2013), 1289–1310. doi: 10.1007/s00025-012-0269-3
[19]	V. Lakshmikantham, S. Leela, J. V. Devi, Theory of fractional dynamic systems, Cambridge Scientific Publishers, 2009.
[20]	S. C. Lim, M. Li, L. P. Teo, Langevin equation with two fractional orders, Phys. Lett. A, 372 (2008), 6309–6320. doi: 10.1016/j.physleta.2008.08.045
[21]	F. Mainardi, P. Pironi, The fractional Langevin equation: Brownian motion revisited, Extracta Math., 11 (1996), 140–154.
[22]	P. A. Naik, Global dynamics of a fractional-order SIR epidemic model with memory, Int. J. Biomath., 13 (2020), 2050071. doi: 10.1142/S1793524520500710
[23]	P. A. Naik, K. M. Owolabi, M. Yavuz, J. Zu, Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells, Chaos, Soliton. Fract., 140 (2020), 110272. doi: 10.1016/j.chaos.2020.110272
[24]	P. A. Naik, M. Yavuz, J. Zu, The role of prostitution on HIV transmission with memory: A modeling approach, Alex. Eng. J., 59 (2020), 2513–2531. doi: 10.1016/j.aej.2020.04.016
[25]	P. A. Naik, J. Zu, K. M. Owolabi, Modeling the mechanics of viral kinetics under immune control during primary infection of HIV-1 with treatment in fractional order, Physica A, 545 (2020), 123816. doi: 10.1016/j.physa.2019.123816
[26]	I. Podlubny, Fractional differential equations, Academic Press, 1999.
[27]	T. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math., 72 (1978), 297–300. doi: 10.1090/S0002-9939-1978-0507327-1
[28]	R. Rizwan, Existence theory and stability snalysis of fractional Langevin equation, Int. J. Nonlin. Sci. Num., 20 (2019).
[29]	R. Rizwan, A. Zada, X. Wang, Stability analysis of non linear implicit fractional Langevin equation with non-instantaneous impulses, Adv. Differ. Equ., 2019 (2019), 1–31. doi: 10.1186/s13662-018-1939-6
[30]	R. Rizwan, A. Zada, Nonlinear impulsive Langevin equation with mixed derivatives, Math. Method. Appl. Sci., 43 (2020), 427–442. doi: 10.1002/mma.5902
[31]	I. A. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babes Bolyai Math., 54 (2009), 125–133.
[32]	R. Shah, A. Zada, A fixed point approach to the stability of a nonlinear volterra integro diferential equation with delay, Hacet. J. Math. Stat., 47 (2018), 615–623.
[33]	S. O. Shah, A. Zada, A. E. Hamza, Stability analysis of the first order non–linear impulsive time varying delay dynamic system on time scales, Qual. Theor. Dyn. Syst., 18 (2019), 825–840. doi: 10.1007/s12346-019-00315-x
[34]	S. Tang, A. Zada, S. Faisal, M. M. A. El-Sheikh, T. Li, Stability of higher–order nonlinear impulsive differential equations, J. Nonlinear Sci. Appl., 9 (2016), 4713–4721. doi: 10.22436/jnsa.009.06.110
[35]	V. E. Tarasov, Fractional dynamics: Application of fractional calculus to dynamics of particles, fields and media, Springer-Verlag Berlin Heidelberg, 2010.
[36]	S. M. Ulam, A collection of mathematical problems, Interscience Publishers Inc., 1968.
[37]	J. Wang, M. Feckan, Y. Zhou, Ulams type stability of impulsive ordinary differential equation, J. Math. Anal. Appl., 35 (2012), 258–264.
[38]	J. Wang, Y. Zhou, M. Feckan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl., 64 (2012), 3389–3405. doi: 10.1016/j.camwa.2012.02.021
[39]	J. Wang, Y. Zhou, Z. Lin, On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242 (2014), 649–657.
[40]	A. Zada, S. Ali, Stability Analysis of Multi-point Boundary Value Problem for Sequential Fractional Differential Equations with Non-instantaneous Impulses, Int. J. Nonlin. Sci. Num., 19 (2018), 763–774 doi: 10.1515/ijnsns-2018-0040
[41]	A. Zada, S. Ali, Y. Li, Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition, Adv. Differ. Equ., 2017 (2017), 1–26. doi: 10.1186/s13662-016-1057-2
[42]	A. Zada, W. Ali, S. Farina, Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses, Math. Method. Appl. Sci., 40 (2017), 5502–5514. doi: 10.1002/mma.4405
[43]	A. Zada, W. Ali, C. Park, Ulam's type stability of higher order nonlinear delay differential equations via integral inequality of Gr$\ddot{o}$nwall-Bellman-Bihari's type, Appl. Math. Comput., 350 (2019), 60–65.
[44]	A. Zada, R. Rizwan, J. Xu, Z. Fu, On implicit impulsive Langevin equation involving mixed order derivatives, Adv. Differ. Equ., 2019 (2019), 1–26. doi: 10.1186/s13662-018-1939-6
[45]	A. Zada, S. O. Shah, Hyers-Ulam stability of first-order non-linear delay dierential equations with fractional integrable impulses, Hacet. J. Math. Stat., 47 (2018), 1196–1205.
[46]	A. Zada, O. Shah, R. Shah, Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems, Appl. Math. Comput., 271 (2015), 512–518.

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