The existence of one solution for impulsive differential equations via variational methods
DOI:
https://doi.org/10.5269/bspm.48439Resumo
We prove the existence of at least one non-trivial weak solution for a nonlinear Dirichlet boundary value problem subject to perturbations of impulsive terms via employing a critical point theorem for differentiable functionals.
Referências
1. G.A. Afrouzi, A. Hadjian, G. Molica Bisci, Some remarks for one-dimensional mean curvature problems through a local minimization principle, Adv. Nonlinear Anal. 2, 427-441, (2013). https://doi.org/10.1515/anona-2013-0021
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6. H. Chen, J. Li, Variational approach to impulsive differential equations with Dirichlet boundary conditions, Bound. Value Probl. 2010, 1-16, (2010). https://doi.org/10.1155/2010/325415
7. P. Chen, X. H. Tang, New existence and multiplicity of solutions for some Dirichlet problems with impulsive effects, Math. Comput. Model. 55, 723-739, (2012). https://doi.org/10.1016/j.mcm.2011.08.046
8. J. Chu, J. J. Nieto, Impulsive periodic solutions of first-order singular differential equations, Bull. Lond. Math. Soc. 40, 143-150 (2008). https://doi.org/10.1112/blms/bdm110
9. M. Galewski, G. Molica Bisci, Existence results for one-dimensional fractional equations, Math. Meth. Appl. Sci. 39, 1480-1492, (2016). https://doi.org/10.1002/mma.3582
10. J. R. Graef, S. Heidarkhani, L. Kong, Existence of solutions to an impulsive Dirichlet boundary value problem, Fixed Point Theory, 19, 225-234, (2018). https://doi.org/10.24193/fpt-ro.2018.1.18
11. J. R. Graef, S. Heidarkhani, L. Kong, Infinitely many periodic solutions to a class of perturbed second-order impulsive Hamiltonian systems, Differ. Equ. Appl. 9, 195-212, (2017). https://doi.org/10.7153/dea-09-16
12. J. R. Graef, S. Heidarkhani, L. Kong, Nontrivial solutions of a Dirichlet boundary value problem with impulsive effects, Dynamic Syst. Appl. 25, 335-350, (2016).
13. S. Heidarkhani, G.A. Afrouzi, M. Ferrara, G. Caristi, S. Moradi, Existence results for impulsive damped vibration systems, Bull. Malays. Math. Sci. Soc. 41, 1409-1428, (2018). https://doi.org/10.1007/s40840-016-0400-9
14. S. Heidarkhani, G.A. Afrouzi, S. Moradi, G. Caristi, B. Ge, Existence of one weak solution for p(x)-biharmonic equations with Navier boundary conditions, Z. Angew. Math. Phys. 67, 1-13, (2016). https://doi.org/10.1007/s00033-016-0668-5
15. S. Heidarkhani, M. Ferrara, G.A. Afrouzi, G. Caristi, S. Moradi, Existence of solutions for Dirichlet quasilinear systems including a nonlinear function of the derivative, Electron. J. Diff. Equ. 2016, 1-12, (2016).
16. S. Heidarkhani, M. Ferrara, A. Salari, Infinitely many periodic solutions for a class of perturbed second-order differential equations with impulses, Acta. Appl. Math. 139, 81-94, (2014). https://doi.org/10.1007/s10440-014-9970-4
17. S. Heidarkhani, Y. Zhou, G. Caristi, G.A. Afrouzi, S. Moradi, Existence results for fractional differential systems through a local minimization principle, Comput. Math. Appl. (2016), https://doi.org/10.1016/j.camwa.2016.04.012
18. E. L. Lee, Y. H. Lee, Multiple positive solutions of two point boundary value problems for second order impulsive differential equations, Appl. Math. Comput. 158, 745-759, (2004). https://doi.org/10.1016/j.amc.2003.10.013
19. Z. Liu, H. Chen, H. Zhou, Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem, Comput. Math. Appl. 61, 1687-1699, (2011). https://doi.org/10.1016/j.camwa.2011.01.042
20. J. Sun, H. Chen, Multiplicity of solutions for class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems, Nonlinear Anal. RWA 11, 4062-4071, (2010). https://doi.org/10.1016/j.nonrwa.2010.03.012
21. J. Sun, H. Chen, Variational method to the impulsive equation with Neumann boundary conditions, Bound. Value Probl. 2009, 1-17, (2009). https://doi.org/10.1155/2009/316812
22. J. Sun, H. Chen, L. Yang, The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method, Nonlinear Anal. TMA 73, 440-449 (2010). https://doi.org/10.1016/j.na.2010.03.035
23. H.-R. Sun, Y.-N. Li, J.J. Nieto, Q. Tang, Existence of solutions for Sturm-Liouville boundary value problem of impulsive differential equations, Abstract and Applied Analysis, 2012, 1-19, (2012). https://doi.org/10.1155/2012/707163
24. B. Ricceri, A general variational principle and some of its application, J. Comput. Appl. Math. 113, 401-410, (2000). https://doi.org/10.1016/S0377-0427(99)00269-1
25. W. Wang, X. Yang, Multiple solutions of boundary-value problems for impulsive differential equations, Math. Meth. Appl. Sci. 34, 1649-1657, (2011). https://doi.org/10.1002/mma.1472
26. J. Xiao, J. J. Nieto, Z. Luo, Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods, Commun. Nonlinear Sci. Numer. Simul. 17, 426-432, (2012). https://doi.org/10.1016/j.cnsns.2011.05.015
27. D. Zhang, B. Dai, Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions, Comput. Math. Appl. 61, 3153-3160, (2011). https://doi.org/10.1016/j.camwa.2011.04.003
28. Z. Zhang, R. Yuan, An application of variational methods to Dirichlet boundary value problem with impulses, Nonlinear Anal. RWA, 11, 155-162, (2010). https://doi.org/10.1016/j.nonrwa.2008.10.044
29. J. Zhou, Y. Li, Existence and multiplicity of solutions for some Dirichlet problems with impulse effects, Nonlinear Anal. TMA 71, 2856-2865, (2009). https://doi.org/10.1016/j.na.2009.01.140
2. L. Bai, B. Dai, An application of variational methods to a class of Dirichlet boundary value problems with impulsive effects, J. Franklin Inst. 348, 2607-2624, (2011). https://doi.org/10.1016/j.jfranklin.2011.08.003
3. M. Bohner, G. Caristi, S. Heidarkhani, S. Moradi, A critical point approach to boundary value problems on the real line, Appl. Math. Lett. 76, 215-220, (2018). https://doi.org/10.1016/j.aml.2017.08.017
4. G. Bonanno, B. Di Bella, J. Henderson, Existence of solutions to second-order boundary-value problems with small perturbations of impulses, Electron. J. Diff. Equ. 2013, 1-14, (2013). https://doi.org/10.1186/1687-2770-2013-278
5. G. Bonanno, B. Di Bella, J. Henderson, Infinitely many solutions for a boundary value problem with impulsive effects, Bound. Value Probl. 2013, 1-14, (2013). https://doi.org/10.1186/1687-2770-2013-278
6. H. Chen, J. Li, Variational approach to impulsive differential equations with Dirichlet boundary conditions, Bound. Value Probl. 2010, 1-16, (2010). https://doi.org/10.1155/2010/325415
7. P. Chen, X. H. Tang, New existence and multiplicity of solutions for some Dirichlet problems with impulsive effects, Math. Comput. Model. 55, 723-739, (2012). https://doi.org/10.1016/j.mcm.2011.08.046
8. J. Chu, J. J. Nieto, Impulsive periodic solutions of first-order singular differential equations, Bull. Lond. Math. Soc. 40, 143-150 (2008). https://doi.org/10.1112/blms/bdm110
9. M. Galewski, G. Molica Bisci, Existence results for one-dimensional fractional equations, Math. Meth. Appl. Sci. 39, 1480-1492, (2016). https://doi.org/10.1002/mma.3582
10. J. R. Graef, S. Heidarkhani, L. Kong, Existence of solutions to an impulsive Dirichlet boundary value problem, Fixed Point Theory, 19, 225-234, (2018). https://doi.org/10.24193/fpt-ro.2018.1.18
11. J. R. Graef, S. Heidarkhani, L. Kong, Infinitely many periodic solutions to a class of perturbed second-order impulsive Hamiltonian systems, Differ. Equ. Appl. 9, 195-212, (2017). https://doi.org/10.7153/dea-09-16
12. J. R. Graef, S. Heidarkhani, L. Kong, Nontrivial solutions of a Dirichlet boundary value problem with impulsive effects, Dynamic Syst. Appl. 25, 335-350, (2016).
13. S. Heidarkhani, G.A. Afrouzi, M. Ferrara, G. Caristi, S. Moradi, Existence results for impulsive damped vibration systems, Bull. Malays. Math. Sci. Soc. 41, 1409-1428, (2018). https://doi.org/10.1007/s40840-016-0400-9
14. S. Heidarkhani, G.A. Afrouzi, S. Moradi, G. Caristi, B. Ge, Existence of one weak solution for p(x)-biharmonic equations with Navier boundary conditions, Z. Angew. Math. Phys. 67, 1-13, (2016). https://doi.org/10.1007/s00033-016-0668-5
15. S. Heidarkhani, M. Ferrara, G.A. Afrouzi, G. Caristi, S. Moradi, Existence of solutions for Dirichlet quasilinear systems including a nonlinear function of the derivative, Electron. J. Diff. Equ. 2016, 1-12, (2016).
16. S. Heidarkhani, M. Ferrara, A. Salari, Infinitely many periodic solutions for a class of perturbed second-order differential equations with impulses, Acta. Appl. Math. 139, 81-94, (2014). https://doi.org/10.1007/s10440-014-9970-4
17. S. Heidarkhani, Y. Zhou, G. Caristi, G.A. Afrouzi, S. Moradi, Existence results for fractional differential systems through a local minimization principle, Comput. Math. Appl. (2016), https://doi.org/10.1016/j.camwa.2016.04.012
18. E. L. Lee, Y. H. Lee, Multiple positive solutions of two point boundary value problems for second order impulsive differential equations, Appl. Math. Comput. 158, 745-759, (2004). https://doi.org/10.1016/j.amc.2003.10.013
19. Z. Liu, H. Chen, H. Zhou, Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem, Comput. Math. Appl. 61, 1687-1699, (2011). https://doi.org/10.1016/j.camwa.2011.01.042
20. J. Sun, H. Chen, Multiplicity of solutions for class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems, Nonlinear Anal. RWA 11, 4062-4071, (2010). https://doi.org/10.1016/j.nonrwa.2010.03.012
21. J. Sun, H. Chen, Variational method to the impulsive equation with Neumann boundary conditions, Bound. Value Probl. 2009, 1-17, (2009). https://doi.org/10.1155/2009/316812
22. J. Sun, H. Chen, L. Yang, The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method, Nonlinear Anal. TMA 73, 440-449 (2010). https://doi.org/10.1016/j.na.2010.03.035
23. H.-R. Sun, Y.-N. Li, J.J. Nieto, Q. Tang, Existence of solutions for Sturm-Liouville boundary value problem of impulsive differential equations, Abstract and Applied Analysis, 2012, 1-19, (2012). https://doi.org/10.1155/2012/707163
24. B. Ricceri, A general variational principle and some of its application, J. Comput. Appl. Math. 113, 401-410, (2000). https://doi.org/10.1016/S0377-0427(99)00269-1
25. W. Wang, X. Yang, Multiple solutions of boundary-value problems for impulsive differential equations, Math. Meth. Appl. Sci. 34, 1649-1657, (2011). https://doi.org/10.1002/mma.1472
26. J. Xiao, J. J. Nieto, Z. Luo, Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods, Commun. Nonlinear Sci. Numer. Simul. 17, 426-432, (2012). https://doi.org/10.1016/j.cnsns.2011.05.015
27. D. Zhang, B. Dai, Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions, Comput. Math. Appl. 61, 3153-3160, (2011). https://doi.org/10.1016/j.camwa.2011.04.003
28. Z. Zhang, R. Yuan, An application of variational methods to Dirichlet boundary value problem with impulses, Nonlinear Anal. RWA, 11, 155-162, (2010). https://doi.org/10.1016/j.nonrwa.2008.10.044
29. J. Zhou, Y. Li, Existence and multiplicity of solutions for some Dirichlet problems with impulse effects, Nonlinear Anal. TMA 71, 2856-2865, (2009). https://doi.org/10.1016/j.na.2009.01.140
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2022-12-23
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