Infinitely many solutions for a class of fractional boundary value problem with $p$-Laplacian with impulsive effects
DOI:
https://doi.org/10.5269/bspm.47913Abstract
The existence of infinitely many solutions for a class of impulsive fractional boundary value problems with $p$-Laplacian with Neumann conditions is established. Our approach is based on recent variational methods for smooth functionals defined on reflexive Banach spaces. One example is presented to demonstrate the application of our main results.
References
1. Bai, C., Infinitely many solutions for a perturbed nonlinear fractional boundary-value problem, Electron. J. Differ. Equ. 2013, 1-12, (2013).
2. Bai, Z., Qiu, Z., Existence of positive solution for singular fractional differential equation, Appl. Math. Comput. 215, 2761-2767, (2009). https://doi.org/10.1016/j.amc.2009.09.017
3. Benson, D., Wheatcraft, S., Meerschaert, M., Application of a fractional advection dispersion equation, Water Resour. Res. 36, 1403-1412, (2000). https://doi.org/10.1029/2000WR900031
4. Benson, D., Wheatcraft, S., Meerschaert, M., The fractional-order governing equation of L'evy motion, Water Resour. Res. 36, 1413-1423, (2000). https://doi.org/10.1029/2000WR900032
5. Bonanno, G., Candito, P., Nonlinear difference equations investigated via critical point methods, Nonlinear Anal. TMA 70, 3180-3186, (2009). https://doi.org/10.1016/j.na.2008.04.021
6. Bonanno, G., D'Agu'ı, G., Multiplicity results for a perturbed elliptic Neumann problem, Abstr. Appl. Anal. 2010, doi:10.1155/2010/564363, 10 pages, (2010). https://doi.org/10.1155/2010/564363
7. Bonanno, G., Molica Bisci, G., A remark on perturbed elliptic Neumann problems, Studia Univ. "Babe¸s-Bolyai", Mathematica, Volume LV, Number 4, December (2010).
8. Bonanno, G., Molica Bisci, G., Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl. 2009, 1-20, (2009). https://doi.org/10.1155/2009/670675
9. G. D'Agu'ı, G., Heidarkhani, S., Sciammetta, A., Infinitely many solutions for a class of quasilinear two-point boundary value problems, Electron. J. Qualitative Theory of Differential Equations 2015, No. 8, 1-15. https://doi.org/10.14232/ejqtde.2015.1.8
10. P. Candito, P., D'Agu'ı, G., Three solutions for a discrete nonlinear Neumann problem involving the p-Laplacian, Adv. Differ. Equ. 2010, 1-11, (2010). https://doi.org/10.1155/2010/862016
11. K. Diethelm, K., The Analysis of Fractional Differential Equation, Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14574-2_8
12. J. Eggleston, J. Rojstaczer, S., Identification of large-scale hydraulic conductivity trends and the influence of trends on contaminant transport, Water Resources Researces 34, 2155-2168, (1998). https://doi.org/10.1029/98WR01475
13. F. Faraci, F., Multiple solutions for two nonlinear problems involving the p-Laplacian, Nonlinear Anal. TMA 63, e1017-e1029, (2005). https://doi.org/10.1016/j.na.2005.02.066
14. M. Galewski, M., Molica Bisci, G., Existence results for one-dimensional fractional equations, Math. Meth. Appl. Sci., 39, 1480-1492, (2016). https://doi.org/10.1002/mma.3582
15. Graef, J.R., Heidarkhani, S., Kong, L., Infinitely many solutions for problems of multi-point boundary value equations, Topol. Meth. Nonl. Anal. 42, 105-118.
16. Graef, J.R., Kong, L., Yang, B., Positive solutions for a semipositone fractional boundary value problem with a forcing term, Fract. Calc. Appl. Anal. 15, 8-24, (2012). https://doi.org/10.2478/s13540-012-0002-7
17. Heidarkhani, S.,Multiple solutions for a nonlinear perturbed fractional boundary value problem, Dynamic. Sys. Appl. 23., 317-331, (2014).
18. Heidarkhani, S., Henderson, J., Infinitely many solutions for nonlocal elliptic problems of (p1, . . . , pn)-Kirchhoff type, Electron. J. Differential Equations 2012, 1-15, 69.
19. Heidarkhani, S., Moradi, S., Existence results for impulsive fractional differential equations with p-Laplacian via variational methods, preprint.
20. Graef, J.R., Heidarkhani, Kong, L., S., Moradi, S., Existence results for impulsive fractional differential equations with p-Laplacian via variational methods, Mathematica Bohemica, to appear. .
21. Graef, J.R., Heidarkhani, Kong, L., S., Moradi, S., Three solutions for impulsive fractional boundary value problem with p-Laplacian, Bull. Iran. Math. Soc. (2021). https://doi.org/10.1007/s41980-021-00589-5
22. Heidarkhani, S., Salari, S., Nontrivial solutions for impulsive fractional differential problems through variational methods, Mathematical Methods in the Applied Sciences 43, 6529-6541, (2020). https://doi.org/10.1002/mma.6396
23. S. Heidarkhani, S., Zhao, Y., Caristi, G., Afrouzi, G.A., Moradi, S., Infinitely many solutions for perturbed impulsive fractional differential problems, Appl. Anal. 96, 1401-1424, (2017). https://doi.org/10.1080/00036811.2016.1192147
24. Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000). https://doi.org/10.1142/3779
25. Jiao, F., Zhou, Y., Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl. 62, 1181-1199, (2011). https://doi.org/10.1016/j.camwa.2011.03.086
26. Jing, W.X., Huang, X., Guo, W., Zhang, Q., The existence of positive solutions for the singular fractional differential equation, Appl. Math. Comput. 41, 171-182, (2013). https://doi.org/10.1007/s12190-012-0603-7
27. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam (2006).
28. Kong, L., Existence of solutions to boundary value problems arising from the fractional advection dispersion equation, Electron. J. Diff. Equ., Vol. 2013, 1-15, (2013).
29. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S., Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientifc, Teaneck, NJ, USA, (1989). https://doi.org/10.1142/0906
30. Li, W., He, Z.,The applications of sums of ranges of accretive operators to nonlinear equations involving the p-Laplacian operator, Nonlinear Anal. TMA 24, 185-193, (1995). https://doi.org/10.1016/0362-546X(94)E0051-H
31. Ricceri, B., A general variational principle and some of its applications, J. Comput. Appl. Math. 113, 401-410, (2000). https://doi.org/10.1016/S0377-0427(99)00269-1
32. Risken, H., The Fokker-Planck Equation, Springer, Berlin, (1988). https://doi.org/10.1007/978-3-642-61544-3
33. Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach, Longhorne, PA, (1993).
34. Samoilenko, A.M., Perestyuk, N.A., Impulsive Differerential Equations, vol. 14 of World Scientifc Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientifc, River Edge, NJ, USA, (1995). https://doi.org/10.1142/2892
35. Wang, J., Li, X., Wei, W., On the natural solution of an impulsive fractional differential equation of order q ∈ (1, 2), Commun. Nonlinear Sci. Numer. Simul. 17, 4384-4394, (2012). https://doi.org/10.1016/j.cnsns.2012.03.011
36. Wang, Y., Liu, Y., Cui, Y., Infinitely many solutions for impulsive fractional boundary value problem with p-Laplacian, Bound. Value Prob. 2018, 1-16, (2018). https://doi.org/10.1186/s13661-018-1012-0
37. Zhang, Y., Liu, L., Wu, Y., Variational structure and multiple solutions for a fractional advection-dispersion equation, Comput. Math. Appl. 61, 1032-1047, (2011).
38. Zhao, Y., Tang, L., Multiplicity results for impulsive fractional differential equations with p-Laplacian via variational methods, Bound. Value Probl. 2017, 1-15, (2017). https://doi.org/10.1186/s13661-017-0855-0
39. Y. Zhou, Y., Basic Theory of Fractional Differential Equations, World Scientific, Singapore (2014). https://doi.org/10.1142/9069
2. Bai, Z., Qiu, Z., Existence of positive solution for singular fractional differential equation, Appl. Math. Comput. 215, 2761-2767, (2009). https://doi.org/10.1016/j.amc.2009.09.017
3. Benson, D., Wheatcraft, S., Meerschaert, M., Application of a fractional advection dispersion equation, Water Resour. Res. 36, 1403-1412, (2000). https://doi.org/10.1029/2000WR900031
4. Benson, D., Wheatcraft, S., Meerschaert, M., The fractional-order governing equation of L'evy motion, Water Resour. Res. 36, 1413-1423, (2000). https://doi.org/10.1029/2000WR900032
5. Bonanno, G., Candito, P., Nonlinear difference equations investigated via critical point methods, Nonlinear Anal. TMA 70, 3180-3186, (2009). https://doi.org/10.1016/j.na.2008.04.021
6. Bonanno, G., D'Agu'ı, G., Multiplicity results for a perturbed elliptic Neumann problem, Abstr. Appl. Anal. 2010, doi:10.1155/2010/564363, 10 pages, (2010). https://doi.org/10.1155/2010/564363
7. Bonanno, G., Molica Bisci, G., A remark on perturbed elliptic Neumann problems, Studia Univ. "Babe¸s-Bolyai", Mathematica, Volume LV, Number 4, December (2010).
8. Bonanno, G., Molica Bisci, G., Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl. 2009, 1-20, (2009). https://doi.org/10.1155/2009/670675
9. G. D'Agu'ı, G., Heidarkhani, S., Sciammetta, A., Infinitely many solutions for a class of quasilinear two-point boundary value problems, Electron. J. Qualitative Theory of Differential Equations 2015, No. 8, 1-15. https://doi.org/10.14232/ejqtde.2015.1.8
10. P. Candito, P., D'Agu'ı, G., Three solutions for a discrete nonlinear Neumann problem involving the p-Laplacian, Adv. Differ. Equ. 2010, 1-11, (2010). https://doi.org/10.1155/2010/862016
11. K. Diethelm, K., The Analysis of Fractional Differential Equation, Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14574-2_8
12. J. Eggleston, J. Rojstaczer, S., Identification of large-scale hydraulic conductivity trends and the influence of trends on contaminant transport, Water Resources Researces 34, 2155-2168, (1998). https://doi.org/10.1029/98WR01475
13. F. Faraci, F., Multiple solutions for two nonlinear problems involving the p-Laplacian, Nonlinear Anal. TMA 63, e1017-e1029, (2005). https://doi.org/10.1016/j.na.2005.02.066
14. M. Galewski, M., Molica Bisci, G., Existence results for one-dimensional fractional equations, Math. Meth. Appl. Sci., 39, 1480-1492, (2016). https://doi.org/10.1002/mma.3582
15. Graef, J.R., Heidarkhani, S., Kong, L., Infinitely many solutions for problems of multi-point boundary value equations, Topol. Meth. Nonl. Anal. 42, 105-118.
16. Graef, J.R., Kong, L., Yang, B., Positive solutions for a semipositone fractional boundary value problem with a forcing term, Fract. Calc. Appl. Anal. 15, 8-24, (2012). https://doi.org/10.2478/s13540-012-0002-7
17. Heidarkhani, S.,Multiple solutions for a nonlinear perturbed fractional boundary value problem, Dynamic. Sys. Appl. 23., 317-331, (2014).
18. Heidarkhani, S., Henderson, J., Infinitely many solutions for nonlocal elliptic problems of (p1, . . . , pn)-Kirchhoff type, Electron. J. Differential Equations 2012, 1-15, 69.
19. Heidarkhani, S., Moradi, S., Existence results for impulsive fractional differential equations with p-Laplacian via variational methods, preprint.
20. Graef, J.R., Heidarkhani, Kong, L., S., Moradi, S., Existence results for impulsive fractional differential equations with p-Laplacian via variational methods, Mathematica Bohemica, to appear. .
21. Graef, J.R., Heidarkhani, Kong, L., S., Moradi, S., Three solutions for impulsive fractional boundary value problem with p-Laplacian, Bull. Iran. Math. Soc. (2021). https://doi.org/10.1007/s41980-021-00589-5
22. Heidarkhani, S., Salari, S., Nontrivial solutions for impulsive fractional differential problems through variational methods, Mathematical Methods in the Applied Sciences 43, 6529-6541, (2020). https://doi.org/10.1002/mma.6396
23. S. Heidarkhani, S., Zhao, Y., Caristi, G., Afrouzi, G.A., Moradi, S., Infinitely many solutions for perturbed impulsive fractional differential problems, Appl. Anal. 96, 1401-1424, (2017). https://doi.org/10.1080/00036811.2016.1192147
24. Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000). https://doi.org/10.1142/3779
25. Jiao, F., Zhou, Y., Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl. 62, 1181-1199, (2011). https://doi.org/10.1016/j.camwa.2011.03.086
26. Jing, W.X., Huang, X., Guo, W., Zhang, Q., The existence of positive solutions for the singular fractional differential equation, Appl. Math. Comput. 41, 171-182, (2013). https://doi.org/10.1007/s12190-012-0603-7
27. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam (2006).
28. Kong, L., Existence of solutions to boundary value problems arising from the fractional advection dispersion equation, Electron. J. Diff. Equ., Vol. 2013, 1-15, (2013).
29. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S., Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientifc, Teaneck, NJ, USA, (1989). https://doi.org/10.1142/0906
30. Li, W., He, Z.,The applications of sums of ranges of accretive operators to nonlinear equations involving the p-Laplacian operator, Nonlinear Anal. TMA 24, 185-193, (1995). https://doi.org/10.1016/0362-546X(94)E0051-H
31. Ricceri, B., A general variational principle and some of its applications, J. Comput. Appl. Math. 113, 401-410, (2000). https://doi.org/10.1016/S0377-0427(99)00269-1
32. Risken, H., The Fokker-Planck Equation, Springer, Berlin, (1988). https://doi.org/10.1007/978-3-642-61544-3
33. Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach, Longhorne, PA, (1993).
34. Samoilenko, A.M., Perestyuk, N.A., Impulsive Differerential Equations, vol. 14 of World Scientifc Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientifc, River Edge, NJ, USA, (1995). https://doi.org/10.1142/2892
35. Wang, J., Li, X., Wei, W., On the natural solution of an impulsive fractional differential equation of order q ∈ (1, 2), Commun. Nonlinear Sci. Numer. Simul. 17, 4384-4394, (2012). https://doi.org/10.1016/j.cnsns.2012.03.011
36. Wang, Y., Liu, Y., Cui, Y., Infinitely many solutions for impulsive fractional boundary value problem with p-Laplacian, Bound. Value Prob. 2018, 1-16, (2018). https://doi.org/10.1186/s13661-018-1012-0
37. Zhang, Y., Liu, L., Wu, Y., Variational structure and multiple solutions for a fractional advection-dispersion equation, Comput. Math. Appl. 61, 1032-1047, (2011).
38. Zhao, Y., Tang, L., Multiplicity results for impulsive fractional differential equations with p-Laplacian via variational methods, Bound. Value Probl. 2017, 1-15, (2017). https://doi.org/10.1186/s13661-017-0855-0
39. Y. Zhou, Y., Basic Theory of Fractional Differential Equations, World Scientific, Singapore (2014). https://doi.org/10.1142/9069
Downloads
Published
2022-12-26
Issue
Section
Research Articles
License
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
