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

  • Dandan Min und Fangqi Chen EMAIL logo
Veröffentlicht/Copyright: 23. August 2021
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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.Suche 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.Suche 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.Suche 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.Suche 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.Suche 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)00525-7Suche 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), 1285–1301.10.2989/16073606.2019.1609619Suche in Google Scholar

[8] D. Guo, Nonlinear Functional Analysis. Science and Technology Press of Shang Dong, China (2004).Suche 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.2018.02.033Suche in Google Scholar

[10] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000).10.1142/3779Suche 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.Suche 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.2011.03.086Suche 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.2018.03.014Suche 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).Suche in Google Scholar

[15] A.A. Kilbas, J.J. Trujillo, Differential equations of fractional orders: Methods, results and problems, II. Appl. Anal. 81 (2002), 435–493.10.1080/0003681021000022032Suche 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.5435Suche 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.Suche 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.2011.01.042Suche 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.Suche in Google Scholar

[20] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems. Springer-Verlag, Berlin (1989).10.1007/978-1-4757-2061-7Suche 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.Suche 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.Suche 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.2007.10.022Suche 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.2017.05.020Suche 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.Suche in Google Scholar

[26] I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).Suche 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), 15–67.10.1115/1.3101682Suche 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.Suche 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/S001309151600050XSuche 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/s12190-016-1035-6Suche 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.Suche 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.Suche 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/s12190-015-0886-6Suche in Google Scholar

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

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