Home Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem
Article
Licensed
Unlicensed Requires Authentication

Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem

  • Dandan Min and Fangqi Chen EMAIL logo
Published/Copyright: August 23, 2021

Abstract

In this paper, we consider a class of nonlinear fractional impulsive differential equation involving Sturm-Liouville boundary-value conditions and p-Laplacian operator. By making use of critical point theorem and variational methods, some new criteria are given to guarantee that the considered problem has infinitely many solutions. Our results extend some recent results and the conditions of assumptions are easily verified. Finally, an example is given as an application of our fundamental results.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No: 11872201, 11572148). There are no conflicts of interest to this work.

References

[1] G.A. Afrouzi, A. Hadjian, A variational approach for boundary value problems for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 21, No 6 (2018), 1565–1584; DOI: 10.1515/fca-2018-0082; https://www.degruyter.com/journal/key/FCA/21/6/html.Search in Google Scholar

[2] L. Bai, B. Dai, Three solutions for a p-Laplacian boundary value problem with impulsive effects. Appl. Math. Comput. 217, No 24 (2011), 9895–9904.Search in Google Scholar

[3] G. Bognar, Similarity solution of boundary layer flows for nonlinear Newtonian fluids. Int. J. Nonlinear Sci. Numer. 10 (2009), 1555–1566.Search in Google Scholar

[4] G. Bonanno, R. Rodríguez-López, S. Tersian, Existence of solutions to boundary value problems for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17, No 3 (2014), 717–744; DOI: 10.2478/s13540-014-0196-y; https://www.degruyter.com/journal/key/FCA/17/3/html.Search in Google Scholar

[5] G. Chai, J. Chen, Existence of solutions for impulsive fractional boundary value problems via variational method. Bound. Value Probl. 2017, No 23 (2017), 1–20; doi:10.1186/s13661-017-0755-3.Search in Google Scholar

[6] A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal. 33 (1998), 181–186.10.1016/S0362-546X(97)00525-7Search in Google Scholar

[7] D. Gao, J. Li, Infinitely many solutions for impulsive fractional differential equations through variational methods. Quaest. Math. 43, No 9 (2020), 1285–1301.10.2989/16073606.2019.1609619Search in Google Scholar

[8] D. Guo, Nonlinear Functional Analysis. Science and Technology Press of Shang Dong, China (2004).Search in Google Scholar

[9] S. Heidarkhani, A. Cabada, G.A. Afrouzi, S. Moradi, G. Caristi, A variational approach to perturbed impulsive fractional differential equations. J. Comput. Appl. Math. 341 (2018), 42–60.10.1016/j.cam.2018.02.033Search in Google Scholar

[10] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000).10.1142/3779Search in Google Scholar

[11] 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.Search in Google Scholar

[12] F. Jiao, Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl. 62 (2011), 1181–1199.10.1016/j.camwa.2011.03.086Search in Google Scholar

[13] A. Khalip, M. Rehman, On variational methods to non-instantaneous impulsive fractional differential equation. Appl. Math. Lett. 83 (2018), 95–102.10.1016/j.aml.2018.03.014Search in Google Scholar

[14] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006).Search in Google Scholar

[15] A.A. Kilbas, J.J. Trujillo, Differential equations of fractional orders: Methods, results and problems, II. Appl. Anal. 81 (2002), 435–493.10.1080/0003681021000022032Search in Google Scholar

[16] D. Li, F. Chen, Y. An, The existence of solutions for an impulsive fractional coupled system of (p, q)-Laplacian type without the Ambrosetti-Rabinowitz condition. Math. Meth. Appl. Sci. 42, No 5 (2019), 1449–1464.10.1002/mma.5435Search in Google Scholar

[17] D. Li, F. Chen, Y. An, The multiplicity of solutions for a class of nonlinear fractional Dirichlet boundary value problems with p-Laplacian type via variational approach. Int. J. Nonlinear Sci. Numer. Simul. (2019), 1–11; doi:10.1515/ijnsns-2018-0102.Search in Google Scholar

[18] Z. Liu, H. Chen, T. Zhou, Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem. Comput. Math. Appl. 61, No 6 (2011), 1687–1699.10.1016/j.camwa.2011.01.042Search in Google Scholar

[19] D. Ma, L. Liu, Y. Wu, Existence of nontrivial solutions for a system of fractional advection-dispersion equations. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113 (2019), 1041–1057; doi:10.1007/s13398-018-0527-7.Search in Google Scholar

[20] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems. Springer-Verlag, Berlin (1989).10.1007/978-1-4757-2061-7Search in Google Scholar

[21] Z. Mehraban, S. Heidarkhani, S.A. Tersian, A variational approach to impulsive Sturm-Liouville differential equations with nonlinear derivative dependence. Int. J. Appl. Math. 32, No 5 (2019), 805–828.Search in Google Scholar

[22] D. Min, L. Liu, Y. Wu, Uniqueness of positive solutions for the singular fractional differential equations involving integral boundary value conditions. Bound. Value Probl. 2018, No 23 (2018), 1–18; doi:10.1186/s13661-018-0941-y.Search in Google Scholar

[23] J.J. Nieto, D. O'Regan, Variational approach to impulsive differential equations. Nonlinear Anal. Real World Appl. 10, No 2 (2009), 680–690.10.1016/j.nonrwa.2007.10.022Search in Google Scholar

[24] N. Nyamoradi, R. Rodríguez-López, Multiplicity of solutions to fractional Hamiltonian systems with impulsive effects. Chaos Solitons Fractals 102 (2017), 254–263.10.1016/j.chaos.2017.05.020Search in Google Scholar

[25] N. Nyamoradi, S. Tersian, Existence of solutions for nonlinear fractional order p-Laplacian differential equations via critical point theory. Fract. Calc. Appl. Anal. 22, No 4 (2019), 945–967; DOI: 10.1515/fca-2019-0051; https://www.degruyter.com/journal/key/FCA/22/4/html.Search in Google Scholar

[26] I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).Search in Google Scholar

[27] Y.A. Rossikhin, M.V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50 (1997), 15–67.10.1115/1.3101682Search in Google Scholar

[28] G. Tcvetkova, S. Tersian, Multiplicity of solutions of Dirichlet's problem for second-order p-Laplacian differential equations with variable coefficients. Int. J. Appl. Math. 33, No 5 (2020), 929–936; DOI:10.12732/ijam.v33i5.13.Search in Google Scholar

[29] Y. Tian, J.J. Nieto, The applications of critical-point theory to discontinuous fractional-order differential equations. Proc. Edinb. Math. Soc. 60, No 4 (2017), 1021–1051.10.1017/S001309151600050XSearch in Google Scholar

[30] C.E. Torres, N. Nyamoradi, Impulsive fractional boundary value problem with p-Laplacian operator. J. Appl. Math. Comput. 55 (2017), 257–278.10.1007/s12190-016-1035-6Search in Google Scholar

[31] Y. Wang, Y. Liu, Y. Cui, Infinitely many solutions for impulsive fractional boundary value problem with p-Laplacian. Bound. Value Probl. 2018, No 94 (2018), 1–16; doi:10.1186/s13661-018-1012-0.Search in Google Scholar

[32] Y. Zhao, H. Chen, B. Qin, Multiple solutions for a coupled system of nonlinear fractional differential equations via variational methods. Appl. Math. Comput. 257 (2015), 417–427.Search in Google Scholar

[33] Y. Zhao, H. Chen, Q. Zhang, Infinitely many solutions for fractional differential system via variational method. J. Appl. Math. Comput. 50 (2016), 589–609.10.1007/s12190-015-0886-6Search in Google Scholar

[34] Y. Zhou, Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014).10.1142/9069Search in Google Scholar

Received: 2020-01-10
Revised: 2021-05-22
Published Online: 2021-08-23
Published in Print: 2021-08-26

© 2021 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 24–4–2021)
  4. Research Paper
  5. Three representations of the fractional p-Laplacian: Semigroup, extension and Balakrishnan formulas
  6. Tutorial paper
  7. The bouncing ball and the Grünwald-Letnikov definition of fractional derivative
  8. Research Paper
  9. Fractional diffusion-wave equations: Hidden regularity for weak solutions
  10. Censored stable subordinators and fractional derivatives
  11. Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem
  12. Multivariable fractional-order PID tuning by iterative non-smooth static-dynamic H synthesis
  13. Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity
  14. The rate of convergence on fractional power dissipative operator on compact manifolds
  15. Fractional Langevin type equations for white noise distributions
  16. Local existence and non-existence for a fractional reaction–diffusion equation in Lebesgue spaces
  17. Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function
  18. The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation
  19. Robust fractional-order perfect control for non-full rank plants described in the Grünwald-Letnikov IMC framework
  20. Properties of the set of admissible “state control” pair for a class of fractional semilinear evolution control systems
Downloaded on 16.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2021-0046/pdf
Scroll to top button