Previous |  Up |  Next

Article

Title: Existence results for impulsive fractional differential equations with $p$-Laplacian via variational methods (English)
Author: Graef, John R.
Author: Heidarkhani, Shapour
Author: Kong, Lingju
Author: Moradi, Shahin
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 147
Issue: 1
Year: 2022
Pages: 95-112
Summary lang: English
.
Category: math
.
Summary: This paper presents several sufficient conditions for the existence of at least one classical solution to impulsive fractional differential equations with a $p$-Laplacian and Dirichlet boundary conditions. Our technical approach is based on variational methods. Some recent results are extended and improved. Moreover, a concrete example of an application is presented. (English)
Keyword: fractional $p$-Laplacian
Keyword: impulsive effect
Keyword: classical solution
Keyword: variational method
MSC: 26A33
MSC: 34A08
MSC: 34B15
MSC: 34B37
MSC: 34K45
idZBL: Zbl 07547243
idMR: MR4387470
DOI: 10.21136/MB.2021.0104-19
.
Date available: 2022-04-17T18:10:12Z
Last updated: 2022-09-06
Stable URL: http://hdl.handle.net/10338.dmlcz/149591
.
Reference: [1] Afrouzi, G. A., Hadjian, A., Bisci, G. Molica: Some remarks for one-dimensional mean curvature problems through a local minimization principle.Adv. Nonlinear Anal. 2 (2013), 427-441. Zbl 1283.34014, MR 3199740, 10.1515/anona-2013-0021
Reference: [2] Agarwal, R. P., Franco, D., O'Regan, D.: Singular boundary value problems for first and second order impulsive differential equations.Aequationes Math. 69 (2005), 83-96. Zbl 1073.34025, MR 2126188, 10.1007/s00010-004-2735-9
Reference: [3] Agarwal, R. P., Hristova, S., O'Regan, D.: Non-Instantaneous Impulses in Differential Equations.Springer, Cham (2017). Zbl 1426.34001, MR 3726876, 10.1007/978-3-319-66384-5
Reference: [4] Baek, H.: Extinction and permanence of a three-species Lotka-Volterra system with impulsive control strategies.Discrete Dyn. Nat. Soc. 2008 (2008), Article ID 752403, 18 pages. Zbl 1167.34350, MR 2452458, 10.1155/2008/752403
Reference: [5] Bai, C.: Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative.J. Math. Anal. Appl. 384 (2011), 211-231. Zbl 1234.34005, MR 2825176, 10.1016/j.jmaa.2011.05.082
Reference: [6] Bai, C.: Solvability of multi-point boundary value problem of nonlinear impulsive fractional differential equation at resonance.Electron. J. Qual. Theory Differ. Equ. 2011 (2011), Article ID 89, 19 pages. Zbl 1340.34007, MR 2854028, 10.14232/ejqtde.2011.1.89
Reference: [7] Bhairat, S. P., Dhaigude, D.-B.: Existence of solutions of generalized fractional differential equation with nonlocal initial condition.Math. Bohem. 144 (2019), 203-220. Zbl 07088846, MR 3974188, 10.21136/MB.2018.0135-17
Reference: [8] Bonanno, G., Candito, P.: Three solutions to a Neumann problem for elliptic equations involving the $p$-Laplacian.Arch. Math. 80 (2003), 424-429. Zbl 1161.35382, MR 1982841, 10.1007/s00013-003-0479-8
Reference: [9] Bonanno, G., D'Aguì, G.: Multiplicity results for a perturbed elliptic Neumann problem.Abstr. Appl. Anal. 2010 (2010), Article ID 564363, 10 pages. Zbl 1207.35118, MR 2674389, 10.1155/2010/564363
Reference: [10] Bonanno, G., Bisci, G. Molica: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities.Bound. Value Probl. 2009 (2009), Article ID 670675, 20 pages. Zbl 1177.34038, MR 2487254, 10.1155/2009/670675
Reference: [11] Bonanno, G., Rodríguez-López, R., Tersian, S.: Existence of solutions to boundary value problem for impulsive fractional differential equations.Fract. Calc. Appl. Anal. 17 (2014), 717-744. Zbl 1308.34010, MR 3260304, 10.2478/s13540-014-0196-y
Reference: [12] Chen, L., Sun, J.: Nonlinear boundary value problem of first order impulsive functional differential equations.J. Math. Anal. Appl. 318 (2006), 726-741. Zbl 1102.34052, MR 2215181, 10.1016/j.jmaa.2005.08.012
Reference: [13] Chen, J., Tang, X. H.: Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory.Abstr. Appl. Anal. 2012 (2012), Article ID 648635, 21 pages. Zbl 1235.34011, MR 2872321, 10.1155/2012/648635
Reference: [14] Chu, J., Nieto, J. J.: Impulsive periodic solutions of first-order singular differential equations.Bull. Lond. Math. Soc. 40 (2008), 143-150. Zbl 1144.34016, MR 2409187, 10.1112/blms/bdm110
Reference: [15] Diethelm, K.: The Analysis of Fractional Differential Equation: An Application-Oriented Exposition Using Differential Operators of Caputo Type.Lecture Notes in Mathematics 2004. Springer, Berlin (2010). Zbl 1215.34001, MR 2680847, 10.1007/978-3-642-14574-2
Reference: [16] Faraci, F.: Multiple solutions for two nonlinear problems involving the $p$-Laplacian.Nonlinear Anal., Theory Methods Appl., Ser. A 63 (2005), e1017--e1029. Zbl 1224.35152, 10.1016/j.na.2005.02.066
Reference: [17] Galewski, M., Bisci, G. Molica: Existence results for one-dimensional fractional equations.Math. Methods Appl. Sci. 39 (2016), 1480-1492. Zbl 1381.34012, MR 3483509, 10.1002/mma.3582
Reference: [18] Gao, D., Li, J.: Infinitely many solutions for impulsive fractional differential equations through variational methods.Quaest. Math. 43 (2020), 1285-1301. Zbl 07307597, MR 4168249, 10.2989/16073606.2019.1609619
Reference: [19] Gao, Z., Yang, L., Liu, G.: Existence and uniqueness of solutions to impulsive fractional integro-differential equations with nonlocal conditions.Appl. Math., Irvine 4 (2013), 859-863. 10.4236/am.2013.46118
Reference: [20] Graef, J. R., Kong, L., Kong, Q.: Multiple solutions of systems of fractional boundary value problems.Appl. Anal. 94 (2015), 1288-1304. Zbl 1323.34007, MR 3325346, 10.1080/00036811.2014.930822
Reference: [21] Guo, L., Zhang, X.: Existence of positive solutions for the singular fractional differential equations.J. Appl. Math. Comput. 44 (2014), 215-228. Zbl 1300.34017, MR 3147738, 10.1007/s12190-013-0689-6
Reference: [22] Heidarkhani, S.: Multiple solutions for a nonlinear perturbed fractional boundary value problem.Dyn. Syst. Appl. 23 (2014), 317-331. Zbl 1321.34012, MR 3241882
Reference: [23] Heidarkhani, S., Afrouzi, G. A., Ferrara, M., Caristi, G., Moradi, S.: Existence results for impulsive damped vibration systems.Bull. Malays. Math. Sci. Soc. (2) 41 (2018), 1409-1428. Zbl 1401.34031, MR 3818425, 10.1007/s40840-016-0400-9
Reference: [24] Heidarkhani, S., Afrouzi, G. A., Moradi, S., Caristi, G., Ge, B.: Existence of one weak solution for $p(x)$-biharmonic equations with Navier boundary conditions.Z. Angew. Math. Phys. 67 (2016), Article ID 73, 13 pages. Zbl 1353.35153, MR 3508996, 10.1007/s00033-016-0668-5
Reference: [25] Heidarkhani, S., Zhao, Y., Caristi, G., Afrouzi, G. A., Moradi, S.: Infinitely many solutions for perturbed impulsive fractional differential systems.Appl. Anal. 96 (2017), 1401-1424. Zbl 1367.34007, MR 3633869, 10.1080/00036811.2016.1192147
Reference: [26] (ed.), R. Hilfer: Applications of Fractional Calculus in Physics.World Scientific, Singapore (2000). Zbl 0998.26002, MR 1890104, 10.1142/3779
Reference: [27] Jiao, F., Zhou, Y.: Existence of solutions for a class of fractional boundary value problems via critical point theory.Comput. Math. Appl. 62 (2011), 1181-1199. Zbl 1235.34017, MR 2824707, 10.1016/j.camwa.2011.03.086
Reference: [28] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations.North-Holland Mathematics Studies 204. Elsevier, Amsterdam (2006). Zbl 1092.45003, MR 2218073, 10.1016/s0304-0208(06)x8001-5
Reference: [29] Kong, L.: Existence of solutions to boundary value problems arising from the fractional advection dispersion equation.Electron. J. Differ. Equ. 2013 (2013), Article ID 106, 15 pages. Zbl 1291.34016, MR 3065059
Reference: [30] Li, W., Zhen, H.: The applications of sums of ranges of accretive operators to nonlinear equations involving the $p$-Laplacian operator.Nonlinear Anal., Theory Methods Appl. 24 (1995), 185-193. Zbl 0828.35041, MR 1312589, 10.1016/0362-546X(94)E0051-H
Reference: [31] Nieto, J. J., O'Regan, D.: Variational approach to impulsive differential equations.Nonlinear Anal., Real World Appl. 10 (2009), 680-690. Zbl 1167.34318, MR 2474254, 10.1016/j.nonrwa.2007.10.022
Reference: [32] Nieto, J. J., Uzal, J. M.: Nonlinear second-order impulsive differential problems with dependence on the derivative via variational structure.J. Fixed Point Theory Appl. 22 (2020), Article ID 19, 13 pages. Zbl 1442.34057, MR 4050175, 10.1007/s11784-019-0754-3
Reference: [33] Oldham, K. B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order.Mathematics in Science and Engineering 111. Academic Press, New York (1974). Zbl 0292.26011, MR 0361633, 10.1016/S0076-5392(09)60219-8
Reference: [34] Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations.Regional Conference Series in Mathematics 65. American Mathematical Society, Providence (1986). Zbl 0609.58002, MR 0845785, 10.1090/cbms/065
Reference: [35] Ricceri, B.: A general variational principle and some of its applications.J. Comput. Appl. Math. 113 (2000), 401-410. Zbl 0946.49001, MR 1735837, 10.1016/S0377-0427(99)00269-1
Reference: [36] Tian, Y., Ge, W.: Applications of variational methods to boundary-value problem for impulsive differential equations.Proc. Edinb. Math. Soc., II. Ser. 51 (2008), 509-527. Zbl 1163.34015, MR 2465922, 10.1017/S0013091506001532
Reference: [37] Wang, G., Ahmad, B., Zhang, L., Nieto, J. J.: Comments on the concept of existence of solution for impulsive fractional differential equations.Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 401-403. Zbl 07172418, MR 3111618, 10.1016/j.cnsns.2013.04.003
Reference: [38] Wang, Y., Liu, Y., Cui, Y.: Infinitely many solutions for impulsive fractional boundary value problem with $p$-Laplacian.Bound. Value Probl. 2018 (2018), Article ID 94, 16 pages. MR 3814794, 10.1186/s13661-018-1012-0
Reference: [39] Wei, L., Agarwal, R. P.: Existence of solutions to nonlinear Neumann boundary value problems with generalized $p$-Laplacian operator.Comput. Math. Appl. 56 (2008), 530-541. Zbl 1155.35360, MR 2442671, 10.1016/j.camwa.2008.01.013
Reference: [40] Zhao, Y., Tang, L.: Multiplicity results for impulsive fractional differential equations with $p$-Laplacian via variational methods.Bound. Value Probl. 2017 (2017), Article ID 123, 15 pages. Zbl 06767364, MR 3690093, 10.1186/s13661-017-0855-0
Reference: [41] Zhou, Y.: Basic Theory of Fractional Differential Equations.World Scientific, Hackensack (2014). Zbl 1336.34001, MR 3287248, 10.1142/9069
.

Files

Files Size Format View
MathBohem_147-2022-1_7.pdf 302.9Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo