Existence of positive solutions of second-order periodic boundary value problems with sign-changing Green’s function

Abstract

In this paper, we consider the existence of positive solutions of second-order periodic boundary value problem

$$u'' + {\left( {\frac{1}{2} + \varepsilon } \right)^2}u = \lambda g\left( t \right)f\left( u \right),t \in \left[ {0,2\pi } \right],u\left( 0 \right) = u\left( {2\pi } \right),u'\left( 0 \right) = u'\left( {2\pi } \right)$$

, where 0 < ε < 1/2, g: [0, 2π] → ℝ is continuous, f: [0,∞) → ℝ is continuous and λ > 0 is a parameter.