Abstract
AbstractIn this paper, the problem of periodic solutions is studied for second order differential equations with indefinite singularities$$\begin{array}{} \displaystyle x''(t)+ f(x(t))x'(t)+\varphi(t)x^m(t)-\frac{\alpha(t)}{x^\mu(t)}+\frac{\beta(t)}{x^y (t)}=0, \end{array}$$wheref∈C((0, +∞), ℝ) may have a singularity at the origin, the signs ofφandαare allowed to change,mis a non-negative constant,μandyare positive constants. The approach is based on a continuation theorem of Manásevich and Mawhin with techniques of a priori estimates.
Highlights
In the past years, the problem of existence of periodic solutions to second order di erential equations with de nite singularities, either attractive type or repulsive type, was extensively studied by many researchers [1]-[14]
In [15], Hakl and Torres investigated the problem of periodic solutions to the equation g(t) h(t)
In [18], the authors considered the existence of positive periodic solutions to the equation like α(t)
Summary
The problem of existence of periodic solutions to second order di erential equations with de nite singularities, either attractive type or repulsive type, was extensively studied by many researchers [1]-[14]. In [18], the authors considered the existence of positive periodic solutions to the equation like α(t) Compared with the case where the signs of functions φ(t), α(t) and β(t) are in de nite, the work of obtaining the estimates of periodic solutions to (1.3) is more di cult.
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