Existence of homoclinic solutions for difference equations on integers via variational method
DOI:
https://doi.org/10.5269/bspm.47232Abstract
In this paper, we are concerned with the existence of at least three distinct solutions for discrete boundary value problems. The proof of the main result is based on variational methods. We also provide two examples in order to illustrate the main results.
References
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24. H. Liang, P. Weng, Existence and multiple solutions for a second-order difference boundary value problem via critical point, J. Math. Anal. Appl. 326 (2007), 511-520. https://doi.org/10.1016/j.jmaa.2006.03.017
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26. B. Ricceri, On a three critical points theorem, Arch. Math. 75 (2000), 220-226. https://doi.org/10.1007/s000130050496
27. R. Steglinski, On homoclinic solutions for a second order difference equation with p-Laplacian, Discrete Continuous Dyn. Syst.-Ser. B 23 (2018), 487-492. https://doi.org/10.3934/dcdsb.2018033
28. R. Steglinski, On sequences of large homoclinic solutions for a difference equations on integers, Adv. Differ. Equ. 2016 (2016), 1-11. https://doi.org/10.1186/s13662-016-0771-0
29. R. Steglinski, On sequences of large solutions for discrete anisotropic equations, Electronic J. Qual. Theo. Differ. Equ. 2015, No. 25, 1-10. https://doi.org/10.14232/ejqtde.2015.1.25
30. R. Steglinski, Sequences of small homoclinic solutions for difference equations on integers, Electron. J. Differential Equations, Vol. 2017 (2017), No. 228, pp. 1-12. https://doi.org/10.1186/s13662-016-0771-0
31. L. Zhilong, Existence of positive solutions of superlinear second-order Neumann boundary value problem, Nonlinear Anal. 72 (2010), 3216-3221. https://doi.org/10.1016/j.na.2009.12.021
32. D. B. Wang, W. Guan, Three positive solutions of boundary value problems for p-Laplacian difference equations, Comput. Math. Appl. 55 (2008), 1943-1949. https://doi.org/10.1016/j.camwa.2007.08.033
33. P. J. Y. Wong, L. Xie, Three symmetric solutions of Lidstone boundary value problems for difference and partial difference equations, Comput. Math. Appl. 45 (2003), 1445-1460. https://doi.org/10.1016/S0898-1221(03)00102-0
2. C. Bereanu, P. Jebelean, C. Serban, Periodic and Neumann problems for discrete p-Laplacian, J. Math. Anal. Appl. 399 (2013), 75-87. https://doi.org/10.1016/j.jmaa.2012.09.047
3. L. H. Bian, H. R. Sun, Q. G. Zhang, Solutions for discrete p-Laplacian periodic boundary value problems via critical point theory, J. Differ. Equ. Appl. 18 (2012), 345-355. https://doi.org/10.1080/10236198.2010.491825
4. M. Bohner, G. Caristi, S. Heidarkhani, S. Moradi, Existence of at least one homoclinic solution for a nonlinear secondorder difference equation, Inter. J. Nonlinear Sci. Numerical Simul. 20 (2019), 433-439. https://doi.org/10.1515/ijnsns-2018-0223
5. G. Bonanno, A critical points theorem and nonlinear differential problems, J. Global Optim. 28 (2004), 249-258. https://doi.org/10.1023/B:JOGO.0000026447.51988.f6
6. G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75 (2012), 2992-3007. https://doi.org/10.1016/j.na.2011.12.003
7. G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal. 54 (2003), 651-665. https://doi.org/10.1016/S0362-546X(03)00092-0
8. G. Bonanno, P. Candito, Infinitely many solutions for a class of discrete nonlinear boundary value problems, Appl. Anal. 884 (2009), 605-616. https://doi.org/10.1080/00036810902942242
9. G. Bonanno, P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differ. Equ. 244 (2008), 3031-3059. https://doi.org/10.1016/j.jde.2008.02.025
10. G. Bonanno, P. Candito, Nonlinear difference equations investigated via critical point methods, Nonlinear Anal. 70 (2009), 3180-3186. https://doi.org/10.1016/j.na.2008.04.021
11. G. Bonanno, B. Di Bella, A boundary value problem for fourth-ordere lastic beam equations, J. Math. Anal. Appl. 343 (2008), 1166-1176. https://doi.org/10.1016/j.jmaa.2008.01.049
12. A. Cabada, C. Li, S. Tersian, On homoclinic solutions of a semilinear p-Laplacian difference equation with periodic coefficients, Adv. Differ. Equ. 2010 (2010), 1-17. https://doi.org/10.1155/2010/195376
13. P. Candito, G. D'Aguı, Three solutions to a perturbed nonlinear discrete Dirichlet problem, J. Math. Anal. Appl. 375 (2011), 594-601. https://doi.org/10.1016/j.jmaa.2010.09.050
14. S. Heidarkhani, G. A. Afrouzi, G. Caristi, J. Henderson, S. Moradi, A variational approach to difference equations, J. Differ. Equ. Appl. 22 (2017), 1761-1776. https://doi.org/10.1080/10236198.2016.1243671
15. S. Heidarkhani, G. A. Afrouzi, S. Moradi, G. Caristi, Existence of three solutions for multi-point boundary value problems, J. Nonlinear Funct. Anal. 2017 (2017), Article ID 47. https://doi.org/10.23952/jnfa.2017.47
16. 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. https://doi.org/10.1016/j.cam.2018.02.033
17. S. Heidarkhani, A.L.A. De Araujo, G. A. Afrouzi, S. Moradi, Multiple solutions for Kirchhoff- type problems with variable exponent and nonhomogeneous Neumann conditions, Math. Nachr. 291 (2018), 326-342. https://doi.org/10.1002/mana.201600425
18. S. Heidarkhani, S. Moradi, S. A. Tersian, Three solutions for second-order boundary-value problems with variable exponents, Electron. J. Qual. Theory Differ. Equ. 33 (2018), 1-19. https://doi.org/10.14232/ejqtde.2018.1.33
19. J. Henderson, H. B. Thompson, Existence of multiple solutions for second order discrete boundary value problems, Comput. Math. Appl. 43 (2002), 1239-1248. https://doi.org/10.1016/S0898-1221(02)00095-0
20. A. Iannizzotto, S. Tersian, Multiple homoclinic solutions for the discrete p-Laplacian via critical point theory, J. Math. Anal. Appl. 403 (2013), 173-182. https://doi.org/10.1016/j.jmaa.2013.02.011
21. L. Kong, Existence of solutions to boundary value problems arising from the fractional advection dispersion equation, Electron. J. Diff. Equ. 2013 (2013), pp. 1-15.
22. L. Kong, Homoclinic solutions for a higher order difference equation with p-Laplacian, Indag. Math. 27 (2016), no.1, 124-146. https://doi.org/10.1016/j.indag.2015.08.007
23. L. Kong, Homoclinic solutions for a second order difference equation with p-Laplacian, Appl. Math. Comput. 247 (2014), 1103-1121. https://doi.org/10.1016/j.amc.2014.09.069
24. H. Liang, P. Weng, Existence and multiple solutions for a second-order difference boundary value problem via critical point, J. Math. Anal. Appl. 326 (2007), 511-520. https://doi.org/10.1016/j.jmaa.2006.03.017
25. B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401-410. https://doi.org/10.1016/S0377-0427(99)00269-1
26. B. Ricceri, On a three critical points theorem, Arch. Math. 75 (2000), 220-226. https://doi.org/10.1007/s000130050496
27. R. Steglinski, On homoclinic solutions for a second order difference equation with p-Laplacian, Discrete Continuous Dyn. Syst.-Ser. B 23 (2018), 487-492. https://doi.org/10.3934/dcdsb.2018033
28. R. Steglinski, On sequences of large homoclinic solutions for a difference equations on integers, Adv. Differ. Equ. 2016 (2016), 1-11. https://doi.org/10.1186/s13662-016-0771-0
29. R. Steglinski, On sequences of large solutions for discrete anisotropic equations, Electronic J. Qual. Theo. Differ. Equ. 2015, No. 25, 1-10. https://doi.org/10.14232/ejqtde.2015.1.25
30. R. Steglinski, Sequences of small homoclinic solutions for difference equations on integers, Electron. J. Differential Equations, Vol. 2017 (2017), No. 228, pp. 1-12. https://doi.org/10.1186/s13662-016-0771-0
31. L. Zhilong, Existence of positive solutions of superlinear second-order Neumann boundary value problem, Nonlinear Anal. 72 (2010), 3216-3221. https://doi.org/10.1016/j.na.2009.12.021
32. D. B. Wang, W. Guan, Three positive solutions of boundary value problems for p-Laplacian difference equations, Comput. Math. Appl. 55 (2008), 1943-1949. https://doi.org/10.1016/j.camwa.2007.08.033
33. P. J. Y. Wong, L. Xie, Three symmetric solutions of Lidstone boundary value problems for difference and partial difference equations, Comput. Math. Appl. 45 (2003), 1445-1460. https://doi.org/10.1016/S0898-1221(03)00102-0
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2022-01-30
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