Abstract:

In this paper, we provide new sufficient conditions for the existence of positive periodic solutions for a class of indefinite singular differential equation

\begin{equation*}

x''(t)+a(t)x(t)=\frac{h(t)}{x^\rho(t)}+g(t)x^\delta(t)+e(t),

\end{equation*}

where $\rho$ and $\delta$ are two positive constants and

$0<\delta\leq1$, $~h,~e\in

L^1(\mathbb{R}/T\mathbb{Z})$, $g\in L^1(\mathbb{R}/T\mathbb{Z})$ is

positive. Our proofs are based on the fixed point theorems (Schauder's fixed

point theorem and Krasnoselski$\breve{\mbox{i}}$-Guo's fixed point

theorem) and the positivity of the associated Green function.

Key words:
Schauder's fixed point theorem, Kasnoselskii-Guo's fixed point theorem, Indefinite singular, Sublinear and semilinear, Positive periodic solution