Multiplicity of positive periodic solutions to superlinear repulsive singular equations

Abstract

In this paper, we study positive periodic solutions to the repulsive singular perturbations of the Hill equations. It is proved that such a perturbation problem has at least two positive periodic solutions when the anti-maximum principle holds for the Hill operator and the perturbation is superlinear at infinity. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.

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…”

“…Formula (1.6) is crucial in [14][15][16][17] for applying some fixed point theorems on cones. Wang [25] studied the problem of periodic solutions for the singular delay Liénard equation of repulsive type…”

Positive solutions for nonlinear fractional semipositone differential equation with nonlocal boundary conditions

“…Motivated by the above papers and [7], the aim of this paper is to establish the existence and multiplicity of positive solutions of BVP (1.1). We obtain the existence of positive solutions by means of the LeraySchauder nonlinear alternative and a fixed-point theorem on cones.…”

On existence of positive periodic solutions of a kind of Rayleigh equation with a deviating argument

“…For example, in [1], [3]- [5], [7]- [8] the differential equations x ′′ (t) + a(t)x(t) = f (t, x(t)), x ′′ (t) + f (t, x(t), x ′ (t)) = 0 were studied. However, few results on the existence of positive periodic solutions for Rayleigh equations were found.…”