Infinitely many solutions for a class of fourth-order impulsive differential equations

References

[1] G. A. Afrouzi and A. Hadjian, Infinitely many solutions for a class of Dirichlet quasilinear elliptic systems, J. Math. Anal. Appl. 393 (2012), no. 1, 265–272. 10.1016/j.jmaa.2012.04.013Search in Google Scholar

[2] G. A. Afrouzi, A. Hadjian and S. Shokooh, Infinitely many solutions for a Dirichlet boundary value problem with impulsive condition, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 77 (2015), no. 4, 9–22. Search in Google Scholar

[3] G. A. Afrouzi, S. Heidarkhani and S. Shokooh, Infinitely many solutions for Steklov problems associated to non-homogeneous differential operators through Orlicz–Sobolev spaces, Complex Var. Elliptic Equ. 60 (2015), no. 11, 1505–1521. 10.1080/17476933.2015.1031122Search in Google Scholar

[4] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Contemp. Math. Appl. 2, Hindawi Publishing, New York, 2006. 10.1155/9789775945501Search in Google Scholar

[5] G. Bonanno and G. M. Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl. 2009 (2009), Article ID 670675. 10.1155/2009/670675Search in Google Scholar

[6] G. Bonanno, B. Di Bella and J. Henderson, Existence of solutions to second-order boundary-value problems with small perturbations of impulses, Electron. J. Differential Equations 2013 (2013), Paper No. 126. Search in Google Scholar

[7] P. Candito and G. Molica Bisci, Existence of solutions for a nonlinear algebraic system with a parameter, Appl. Math. Comput. 218 (2012), no. 23, 11700–11707. 10.1016/j.amc.2012.05.058Search in Google Scholar

[8] J. Chen, C. C. Tisdell and R. Yuan, On the solvability of periodic boundary value problems with impulse, J. Math. Anal. Appl. 331 (2007), no. 2, 902–912. 10.1016/j.jmaa.2006.09.021Search in Google Scholar

[9] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, 2013. Search in Google Scholar

[10] G. D’Aguì and A. Sciammetta, Infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous Neumann conditions, Nonlinear Anal. 75 (2012), no. 14, 5612–5619. 10.1016/j.na.2012.05.009Search in Google Scholar

[11] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations and Inclusions, World Scientific, Singapore, 1989. 10.1142/0906Search in Google Scholar

[12] J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, Topological Methods for Ordinary Differential Equations (Montecatini Terme 1991), Lecture Notes in Math. 1537, Springer, Berlin (1993), 74–142. 10.1007/BFb0085076Search in Google Scholar

[13] J. J. Nieto and D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl. 10 (2009), no. 2, 680–690. 10.1016/j.nonrwa.2007.10.022Search in Google Scholar

[14] L. A. Peletier, W. C. Troy and R. C. A. M. Van der Vorst, Stationary solutions of a fourth order nonlinear diffusion equation (in Russian), Differentsialnye Uravneniya 31 (1995), 327–337; translation in Differential Equations 31 (1995), 301–314. Search in Google Scholar

[15] D. Qian and X. Li, Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. Math. Anal. Appl. 303 (2005), no. 1, 288–303. 10.1016/j.jmaa.2004.08.034Search in Google Scholar

[16] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), no. 1–2, 401–410. 10.1016/S0377-0427(99)00269-1Search in Google Scholar

[17] A. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises 14, World Scientific Publishing, River Edge, 1995. 10.1142/2892Search in Google Scholar

[18] J. Sun, H. Chen and L. Yang, Variational methods to fourth-order impulsive differential equations, J. Appl. Math. Comput. 35 (2011), no. 1–2, 323–340. 10.1007/s12190-009-0359-xSearch in Google Scholar

[19] Y. Tian and W. Ge, Applications of variational methods to boundary-value problem for impulsive differential equations, Proc. Edinb. Math. Soc. (2) 51 (2008), no. 2, 509–527. 10.1017/S0013091506001532Search in Google Scholar

[20] W. Wang and J. Shen, Eigenvalue problems of second order impulsive differential equations, Comput. Math. Appl. 62 (2011), no. 1, 142–150. 10.1016/j.camwa.2011.04.061Search in Google Scholar

[21] W. Wang and X. Yang, Multiple solutions of boundary-value problems for impulsive differential equations, Math. Methods Appl. Sci. 34 (2011), no. 13, 1649–1657. 10.1002/mma.1472Search in Google Scholar

[22] J. Xiao and J. J. Nieto, Variational approach to some damped Dirichlet nonlinear impulsive differential equations, J. Franklin Inst. 348 (2011), no. 2, 369–377. 10.1016/j.jfranklin.2010.12.003Search in Google Scholar

[23] J. Xie and Z. Luo, Solutions to a boundary value problem of a fourth-order impulsive differential equation, Bound. Value Probl. 2013 (2013), Paper No. 154. 10.1186/1687-2770-2013-154Search in Google Scholar