Existence and multiplicity results for nonlinear periodic boundary value problems
Consider the following periodic boundary value problems \[ \begin{aligned} -u^{\prime \prime }+a\left( t\right) u &=\lambda f\left( t,u\right) ,\text{ \;\;}0\leq t\leq 2\pi , \\ u\left( 0\right) &=u\left( 2\pi \right) ,\text{ \;\;}u^{\prime }\left( 0\right) =u^{\prime }\left( 2\pi \right) ,\end{aligned} \] and \[ \begin{aligned} u^{\prime \prime }+a\left( t\right) u &=\lambda f\left( t,u\right) ,\text{ \;\;}0\leq t\leq 2\pi , \\ u\left( 0\right) &=u\left( 2\pi \right) ,\text{ \;\;}u^{\prime }\left( 0\right) =u^{\prime }\left( 2\pi \right) ,\end{aligned} \] where \(a\in L_{1}\left( 0,2\pi \right) \), \(f:\left[ 0,2\pi \right] \times [ 0,+\infty )\rightarrow [ 0,+\infty )\)\ is continuous, \(\lambda \) is a positive parameter. The criteria for the existence, nonexistence and multiplicity of positive solutions are established by using the Global continuation theorem, fixed point index theory and approximate method. The results obtained herein generalize and complement some previous findings of [\textit{J. R. Graef, L. Kong} and \textit{H. Wang}, J. Differ. Equations 245, No.~5, 1185--1197 (2008; Zbl 1203.34028)] and some other known results.
scientific article; zbMATH DE number 5687526
scientific article