Existence of periodic solutions of a Liénard equation with a singularity of repulsive type

Abstract

In this paper, the problem of positive periodic solutions is studied for the Liénard equation with a singularity of repulsive type,where f : (0, +∞) → R is continuous, α, h are continuous with T-periodic and α(t) ≥ 0 for all t ∈ R, μ ∈ (0, +∞) is a constant. By means of a Manásevich-Mawhin's continuation theorem, a sufficient and necessary condition is obtained for the existence of positive T-periodic solutions of the equation. The interesting point is that the weak singularity of restoring forcex μ at x = 0 is allowed and f may have a singularity at x = 0.

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.…”

“…In 2017, Lu [3] considered the existence of positive periodic solutions of the following Liénard equation with singularity:…”

A new result on the existence of periodic solutions for Rayleigh equation with a singularity

“…By using topological degree methods they obtained that a necessary and sufficient condition for the existence of positive periodic solutions for equation (1.1) is h > 0, and if we assume in addition that α ≥ 1, then a necessary and sufficient condition for the existence of positive periodic solutions for equation (1.2) is h < 0. After that, some methods associated with nonlinear functional analysis theory have been widely applied to the studied problem in many papers such as the variational methods used in [10][11][12][13], fixed point theorems used in [14][15][16][17][18][19], upper and lower solutions methods used in [20,21], and continuation theorems of coincidence degree used in [22][23][24][25][26][27][28][29][30][31]. For example, Torres [14] studied the periodic problem for the equation with singularity of repulsive type…”

Existence of positive periodic solutions for Liénard equations with an indefinite singularity of attractive type

“…In the past years, many mathematical researchers focused their attention on the equations with singularities [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. As is widely acknowledged, the paper [18] by Lazer and Solimini is a major milestone for the study of periodic problem to second-order differential equations with singularities.…”