Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Damped Vibration Problems with Impulsive Effects

Abstract

This paper is concerned with the existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems with impulsive effects. Some new results are obtained under more relaxed conditions by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory. The results obtained in this paper generalize and improve some existing works in the literature.

Existence of solutions for a class of second-order impulsive Hamiltonian system with indefinite linear part

“…Since impulsive problems are important in real world, many researchers have extensively studied the theory and applications of impulsive differential equations, see [2,4,5,8,11] for more details. In recent years, variational methods have been widely used to study Hamiltonian system and impulsive problems, see [1,3,6,7,9,[12][13][14][15][16][17] and references therein. In [7], Kyritsi and Papageorgion investigated problem (1.2) and obtained an existence result by using Morse critical groups.…”

Existence of fast homoclinic solutions for a class of second-order damped vibration systems

“…For example, see [8][9][10][11][12][13][14][15][16][17][18][19], and the references cited therein. For the existence of homoclinic solutions for damped vibration problem (1.4), please see the literature [20][21][22][23][24][25], and the references cited therein. For other kinds of damped vibration problem, please see the literature [26] and [27].…”

“…Then E with its standard norm · is a reflexive Banach space. Similar to [20][21][22][23][24][25], the definition of fast homoclinic solutions is given in the following. The main results are given in the following.…”

Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Nonlinear Second-Order Nonautonomous Systems in a Weighted Sobolev Space

“…Zhang et al [18] considered a special nonautonomous problem and obtained some results of fast homoclinic orbits. Zhang [19] investigated a class of damped vibration problems with impulsive effects and established some existence and multiplicity results of fast homoclinic solutions. For the applications of weighted Sobolev space in elliptic equations, please see [20,21] and references cited therein.…”

“…Similar to [16][17][18][19], we have the following definition of fast homoclinic solutions. Then problem (1) has at least one nontrivial fast homoclinic solution.…”