Abstract
In this paper, we study the periodic problem for the Liénard equation with an indefinite singularity of attractive type \t\t\tx″+f(x)x′+φ(t)x+r(t)xμ=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ x''+f(x)x'+\\varphi (t)x+\\frac{r(t)}{x^{\\mu }}=0, $$\\end{document} where f:(0,+infty )rightarrow R is continuous and may have singularities at zero, r, varphi : Rrightarrow R are T-periodic functions, and μ is a positive constant. Using the method of upper and lower functions, we obtain some new results on the existence of positive periodic solutions to the equation.
Highlights
As is well known, differential equations with singularities have a wide range of applications in physics, mechanics, and biology [1,2,3,4,5,6]
The paper [18] by Lazer and Solimini is a major milestone for the study of periodic problem to second-order differential equations with singularities
We study the periodic problem for Liénard equations with a singularity of attractive type in the case of r(t) ≥ 0 for a.e. t ∈ [0, T]
Summary
Differential equations with singularities have a wide range of applications in physics, mechanics, and biology [1,2,3,4,5,6]. The paper [18] by Lazer and Solimini is a major milestone for the study of periodic problem to second-order differential equations with singularities. The existence of periodic solutions was investigated for the singular equations x (t) + xα(t) = h(t) (1.1). (the singularity of attractive type) and x (t) – = h(t) (1.2) xα (t). (the singularity of repulsive type), where h : R → R is a continuous periodic function. Together with the method of lower and upper functions, they obtained that a necessary and sufficient condition for the existence of positive periodic.
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