Periodic solutions of singular nonautonomous second order differential equations

Abstract

Abstract. We consider a class of singular nonautonomous second order differential equations. We prove the existence of at least one periodic solution. We use a variational approach based on the mountain pass theorem.2000 Mathematics Subject Classification: 34B15 -34C15 -34C25 -58E05.

Existence for Singular Periodic Problems: A Survey of Recent Results

“…During the last two decades, singular differential equations have attracted many researchers [1][2][3][4][5][6][7][8][9][10][11] because such equations describe many problems in the applied sciences, such as the Brillouin focusing system [12][13][14], nonlinear elasticity [15], and gravitational forces [3]. Besides these important applications, it has been found that a particular case of singular equations, the Ermakov-Pinney equation, plays an important role in studying the Lyapunov stability of periodic solutions of Lagrangian equations [16][17][18].…”

“…In the literature, two different approaches have been used to establish the existence results for singular equations. The first one is the variational approach [3,4,6,19,20] and the second one is topological methods [1,10,[21][22][23][24][25][26][27][28]. In our opinion, the first important result was proved in the pioneering paper of Lazer and Solimini [29].…”

Impulsive Periodic Solutions for Singular Problems via Variational Methods

“…Singular problems without impulse effects have also been investigated extensively in the literature [2,3,5,9,10,[12][13][14] by using topological methods and variational methods. For example, Boucherif and Daoudi-Merzagui [5] considered a class of singular differential equations and obtained the existence of periodic solutions when the nonlinearity is bounded from above on u by using the mountain pass theorem.…”

“…Inspired by [5,22], in this paper we shall study the existence of periodic solutions for impulsive singular problems. The study of impulsive singular problems is more recent and the number of references is small; see [1,11,25].…”

Positive periodic solutions to impulsive delay differentialequations

“…In fact, it seems that the work [20] is the first paper along this line. We must emphasize that singular boundary value problems without impulses have attracted the attention of many researchers [1,2,6,14].…”