A class of impulsive vibration equation with fractional derivatives

Xue Wang; Xiping Liu; Mei Jia; Xue Wang; Xiping Liu; Mei Jia

doi:10.3934/math.2021120

AIMS Mathematics

2021, Volume 6, Issue 2: 1965-1990. doi: 10.3934/math.2021120

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Research article

A class of impulsive vibration equation with fractional derivatives

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received: 25 October 2020 Accepted: 30 November 2020 Published: 07 December 2020
MSC : 26A33, 34A08, 34A37, 34B15

In this paper, we study a class of second order impulsive vibration equation containing fractional derivatives. By using monotone iterative technique, some new results on the multiplicity for solutions of the equations under nonlinear boundary conditions are obtained, and the properties of the solutions are discussed. Finally, the practicability of our results is discussed through a concrete example.
- fractional derivative,
- impulsive vibration equation,
- Caputo derivative,
- nonlinear boundary conditions
Citation: Xue Wang, Xiping Liu, Mei Jia. A class of impulsive vibration equation with fractional derivatives[J]. AIMS Mathematics, 2021, 6(2): 1965-1990. doi: 10.3934/math.2021120

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Abstract

In this paper, we study a class of second order impulsive vibration equation containing fractional derivatives. By using monotone iterative technique, some new results on the multiplicity for solutions of the equations under nonlinear boundary conditions are obtained, and the properties of the solutions are discussed. Finally, the practicability of our results is discussed through a concrete example.

References

[1]	A. Arshad, S. Kamal, J. Fahd, Existence and stability analysis to a coupled system of implicit type impulsive boundary value problems of fractional-order differential equations, Adv. Differ. Equ., 2019 (2019), 1–21. doi: 10.1186/s13662-018-1939-6
[2]	H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620–709. doi: 10.1137/1018114
[3]	Z. Bai, Z. Du, S. Zhang, Iterative method for a class of fourth-order p-Laplacian beam equation, J. Appl Anal. Comput., 9 (2019), 1443–1453.
[4]	C. Promsakon, E. Suntonsinsoungvon, S. K. Ntouyas, J. Tariboon, Impulsive boundary value problems containing Caputo fractional derivative of a function with respect to another function, Adv. Differ. Equ., 2019 (2019), 1–17. doi: 10.1186/s13662-018-1939-6
[5]	J. Cao, H. Chen, Impulsive fractional differential equations with nonlinear boundary conditions, Math. Comp. Model., 55 (2012), 303–311. doi: 10.1016/j.mcm.2011.07.037
[6]	I. Dassios, D. Baleanu, Caputo and related fractional derivatives in singular systems, Appl. Math. Comput., 337 (2018), 591–606.
[7]	S. Dhar, Q. Kong, M. McCabe, Fractional boundary value problems and Lyapunov-type inequalities with fractional integral boundary conditions, Electron. J. Qual. Theo., 2016 (2016),
[8]	K. Diethelm, The analysis of fractional differential equations, Spring-Verlag Berlin, Heidelberg, 2010.
[9]	M. Feckan, Y. Zhou, J. R. 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
[10]	D. Guo, J. Sun, Z. Liu, Nonlinear ordinary differential equation functional method, Shandong Science and Technology Press, Jinan (in Chinese), 2005.
[11]	Z. Gao, T. Hu, H. Pang, Existence and uniqueness theorems for a fractional differential equation with impulsive effect under Band-Like integral boundary conditions, Adv. Math. Phys., 2020 (2020), 1–8.
[12]	M. Jia, X. Liu, Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions, Appl. Math. Comput., 232 (2014), 313–323.
[13]	M. Jia, L. Li, X. Liu, J. Song, Z. Bai, A class of nonlocal problems of fractional differential equations with composition of derivative and parameters, Adv. Differ. Equ., 2019 (2019), 1–26. doi: 10.1186/s13662-018-1939-6
[14]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science, 2006.
[15]	M. Lázaro, J. L. Pérez-Aparicio, Dynamic analysis of frame structures with free viscoelastic layers: New closed-form solutions of eigenvalues and a viscous approach, Eng. Struct., 54 (2013), 69–81.
[16]	R. Lewandowski, Z. Pawlak, Dynamic analysis of frames with viscoelastic dampers modelled by rheological models with fractional derivatives, J. Sound Vib., 330 (2011), 923–936. doi: 10.1016/j.jsv.2010.09.017
[17]	X. Liu, M. Jia, Solvability and numerical simulations for BVPs of fractional coupled systems involving left and right fractional derivatives, Appl. Math. Comput., 353 (2019), 230–242.
[18]	X. Liu, M. Jia, W. Ge, The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator, Appl. Math. Lett., 65 (2017), 56–62. doi: 10.1016/j.aml.2016.10.001
[19]	X. Liu, M. Jia, The method of lower and upper solutions for the general boundary value problems of fractional differential equations with p-Laplacian, Adv. Differ. Equ., 2018 (2018), 1–15. doi: 10.1186/s13662-017-1452-3
[20]	I. Podluny, Fractional differential equtions, Academic press, San Diego, 1999.
[21]	T. Sandev, R. Metzler, Z. Tomovski, Correlation functions for the fractional generalized Langevin equation in the presence of internal and external noise, J. Math. Phys., 55 (2014), 1–23.
[22]	P. J. Torvik, R. L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294–298. doi: 10.1115/1.3167615
[23]	A. Ullah, K. Shah, T. Abdeljawad, R. A. Khan, I. Mahariq, Study of impulsive fractional differential equation under Robin boundary conditions by topological degree method, Bound. Value Probl., 2020 (2020), 1–17. doi: 10.1186/s13661-019-01311-5
[24]	B. S. Vadivoo, R. Ramachandran, J. Cao, H. Zhang, X. Li, Controllability analysis of nonlinear neutral-type fractional-order differential systems with state delay and impulsive effects, Int. J. Control. Autom., 16 (2018), 659–669. doi: 10.1007/s12555-017-0281-1
[25]	B. Zhu, L. Liu, Y. Wu, Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Comput. Math. Appl., 78 (2019), 1811–1818. doi: 10.1016/j.camwa.2016.01.028
[26]	T. Zhang, L. Xiong, Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative, Appl. Math. Lett., 101 (2020), 1–7.

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