Positive Periodic Solution to a Lienard Equation with Indefinite Weights
Fukun Li *
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China.
Shujing Yuan
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China.
*Author to whom correspondence should be addressed.
Abstract
This study establishes sufficient conditions for the existence of positive T-periodic solutions to a Lienard equation of the form \(x^{\prime \prime}+f(t, x) x^{\prime}+a(t) x=\frac{h(t)}{x^\lambda}+e(t),\) where a, h, and e are continuous T-periodic functions, h may change sign, λ > 0, and f(t, x) may depend explicitly on time and may be singular as x → 0+.
The analysis addresses the difficulty caused by the time-dependent resistance term, for which the integral cancellation available in the classical f(x)x′ setting is not generally valid, and also permits both a and h to change sign. By constructing appropriate cones in C and using positive Green functions, the periodic problem is reformulated as a fixedpoint problem. The Krasnoselski˘ı–Guo fixed-point theorem is then applied under explicit growth and sign conditions. Two regimes are considered according to the sign of the forcing term e. For positive forcing, the argument uses the Green function associated with x′′ +Nx; for negative forcing, it uses the Green function associated with −x′′ +a−(t)x. The resulting conditions cover both ρ > 1 and ρ = 1 in the estimate \(|f(t, x)| \leq \frac{m}{x^\rho} .\)
A concrete example illustrates the optimal choice of the splitting parameter in the first theorem. These results provide sufficient criteria for positive periodic solutions under indefinite weights and singular resistance. The findings remain confined to the stated sufficient hypotheses.
Keywords: Positive periodic solution, Li´enard equation, Indefinite singularity, Krasnoselski˘ı–Guo fixed point theorem