Subharmonic Solutions for Some Second-Order Differential Equations with Singularities

Abstract

The existence of infinitely many subharmonic solutions is proved for the periodically forced nonlinear scalar equation u" + g(u) e(t), where g is a continuous function that is defined on a open proper interval (A, B) C ]. The nonlinear restoring field g is supposed to have some singular behaviour at the boundary of its domain. The following two main possibilities are analyzed:(a) The domain is unbounded and g is sublinear at infinity. In this case, via critical point theory, it is possible to prove the existence of a sequence of subharmonics whose amplitudes and minimal periods tend to infinity.(b) The domain is bounded and the periodic forcing term e(t) has minimal period T 0. In this case, using the generalized Poincar-Birkhoff fixed point theorem, it is possible to show that for any m E 1, there are infinitely many periodic solutions having mT as minimal period.Applications are given to the dynamics of a charged particle moving on a line over which one has placed some electric charges of the same sign.

A multiplicity result for periodic solutions of second order differential equations with a singularity

“…This fact could be related with a result in [9], where the case of superlinear growth at +∞ was considered, as well as the case of a second repulsive singularity at a pointb > 0. In both cases, the existence of infinitely many solutions was proved.…”

Periodic solution for p-Laplacian Rayleigh equation with attractive singularity and time-dependent deviating argument

“…The authors found a new method for estimating a lower a priori bounds of the periodic solutions to the given equation. Besides, many articles have been published about Liénard equation with repulsive singularity (see [4][5][6][7][8][9][10][11][12][13]). Recently, some good deal of works have been performed on the existence of periodic solutions of Rayleigh equations with singularity (see [14][15][16]).…”

“…In the past years, researchers paid much attention to investigating the problem of periodic solutions for second-order equations with singularities (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]). Among those studies, the study of properties of repulsive singularities can be traced back to 1996.…”

On oscillatory behavior of two-dimensional time scale systems

“…For example, Bartolini et al [4][5][6] considered a secondorder system in order to control uncertain nonlinear systems by some control techniques, for example, sliding mode and approximate linearization. In addition to nonoscillatory solutions of second-order systems, periodic and subharmonic solutions were also considered in [7][8][9], and important results were obtained. If T = R, then system (1) turns out to be the system of first-order differential equations, and this system was considered by Li [10].…”