VARIATIONAL METHODS TO THE FOURTH-ORDER LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS WITH NON-INSTANTANEOUS IMPULSES

Shihan Wang; Yu Tian

doi:10.11948/20190413

2020 Volume 10 Issue 6

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Shihan Wang, Yu Tian. VARIATIONAL METHODS TO THE FOURTH-ORDER LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS WITH NON-INSTANTANEOUS IMPULSES[J]. Journal of Applied Analysis & Computation, 2020, 10(6): 2521-2536. doi: 10.11948/20190413

Citation:

Shihan Wang, Yu Tian. VARIATIONAL METHODS TO THE FOURTH-ORDER LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS WITH NON-INSTANTANEOUS IMPULSES[J]. Journal of Applied Analysis & Computation, 2020, 10(6): 2521-2536. doi: 10.11948/20190413

VARIATIONAL METHODS TO THE FOURTH-ORDER LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS WITH NON-INSTANTANEOUS IMPULSES

Shihan Wang¹,
Yu Tian^1, ,

1.
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

Corresponding author: Email address:tianyu2992@163.com(Y. Tian)

Abstract

Abstract

In this paper, the existence of solutions for the fourth-order linear and nonlinear differential equations with non-instantaneous impulses is studied by applying variational methods. The interesting point lies in that the variational structures corresponding to the fourth-order linear and nonlinear differential equations with non-instantaneous impulses are established for the first time.
- Variational methods /
- non-instantaneous impulses /
- fourth-order /
- linear and nonlinear differential equations
MSC: 58E30, 35A15, 34A37

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References

[1]	R. Agarwal, S. Hristova and D. O'Regan, Non-instantaneous Impulses in Differential Equations, Springer, New York, 2017. Google Scholar
[2]	R. Agarwal, D. O'Regan and S. Hristova, Stability by Lyapunov functions of nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput., 2017, 53, 147–168. doi: 10.1007/s12190-015-0961-z CrossRef Google Scholar
[3]	Z. Bai, X. Dong and C. Yin, Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions, Bound. Value Probl., 2016, 63, 1–11. Google Scholar
[4]	G. Bonanno and B.D. Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 2008, 343, 1166–1176. doi: 10.1016/j.jmaa.2008.01.049 CrossRef Google Scholar
[5]	M. Chipot, Elements of Nonlinear Analysis, Birkh$\ddot{a}$user, 2000. Google Scholar
[6]	K. Chang, Methods in Nonlinear Analysis, in: Springer Monogr. Math., Springer-Verlag, Berlin, 2005. Google Scholar
[7]	V. Colao, L. Muglia and X. Hong, Existence of solutions for a second-order differential equation with non-instantaneous impulses and delay, Ann. Mat. Pura Appl., 2016, 195, 697–716. doi: 10.1007/s10231-015-0484-0 CrossRef Google Scholar
[8]	E. Hern$\acute{a}$ndez and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 2013, 141, 1641–1649. Google Scholar
[9]	J. Henderson and R. Luca, Systems of Riemann-Liouville fractional equations with multi-point boundary conditions, Applied Mathematics and Computation., 2017, 309, 303–323. doi: 10.1016/j.amc.2017.03.044 CrossRef Google Scholar
[10]	B. Liang and J.J. Nieto, Variational approach to differential equations with not-instantaneous impulses, Appl. Mat. Lett., 2017, 73, 44–48. doi: 10.1016/j.aml.2017.02.019 CrossRef Google Scholar
[11]	M. Muslim, A. Kumar and M. Fe$\breve{c}$kan, Existence, uniqueness and stability of solutions to second order nonlinear differential equations with non-instantaneous impulses, Journal of King Saud University-Science., 2018, 30, 204–213. doi: 10.1016/j.jksus.2016.11.005 CrossRef Google Scholar
[12]	R. Ma and J. Xu, Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem, Nonlinear Anal. Theory, 2010, 72, 113–122. doi: 10.1016/j.na.2009.06.061 CrossRef Google Scholar
[13]	J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, Berlin, 1989. Google Scholar
[14]	J. J. Nieto and R. Rodriguez-Lopez, Periodic boundary value problem for non-lipschitzian impulsive functional differential equations, J. Math. Anal. Appl., 2006, 318, 593–610. doi: 10.1016/j.jmaa.2005.06.014 CrossRef Google Scholar
[15]	J. J. Nieto and D. O'Regan, Variational approach to impulsive differential equations. Nonlinear Analysis: Real Word Application, 2009, 10, 680–690. Google Scholar
[16]	M. Pierri, D. O'Regan and V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput., 2013, 219(12), 6743–6749. Google Scholar
[17]	P. H. Rabinowitz, Minimax methods in critical point theory with application to differential equations, Amer. Math. Soc., 1986, 65. Google Scholar
[18]	J. Sun, H. Chen and L. Yang, Variational methods to fourth-order impulsive differential equations, J. Appl. Math. Comput., 2011, 35, 323–340. doi: 10.1007/s12190-009-0359-x CrossRef Google Scholar
[19]	Y. Tian and J. Henderson, Three anti-periodic solutions for second-order impulsive differential inclusions via non-smooth critical point theory, Nonlinear Anal., 2012, 75, 6496–6505. doi: 10.1016/j.na.2012.07.025 CrossRef Google Scholar
[20]	Y. Tian and W. Ge, Applications of variational methods to boundary-value problem for impulsive differential equations, Proc. Edinb. Math. Soc., 2008, 51, 509–527. doi: 10.1017/S0013091506001532 CrossRef Google Scholar
[21]	Y. Tian and M. Zhang, Variational method to differential equations with instantaneous and not-instantaneous impulses, Appl. Mat. Lett., 2019, 94, 160–165. doi: 10.1016/j.aml.2019.02.034 CrossRef Google Scholar
[22]	J. Wang and X. Li, Periodic BVP for integer/fractional order nonlinear differential equations with non instantaneous impulses, J. Appl. Math. Comput., 2014, 46, 321–334. doi: 10.1007/s12190-013-0751-4 CrossRef Google Scholar
[23]	Y. Wei, Q. Song and Z. Bai, Existence and iterative method for some fourth order nonlinear boundary value problems, Appl. Math. Lett., 2019, 87, 101–107. doi: 10.1016/j.aml.2018.07.032 CrossRef Google Scholar
[24]	P. Yang, J. Wang, D. O'Regan and M. Fe$\breve{c}$kan, Inertial manifold for semi-linear non-instantaneous impulsive parabolic equations in an admissible space, Commun. Nonlinear Sci. Numer. Simul., 2019, 75, 174–195. doi: 10.1016/j.cnsns.2019.03.029 CrossRef Google Scholar