Twin positive solutions of functional differential equations for the one-dimensional $p$-laplacian

Abstract

For the boundary value problem (BVP) of second order functional differential equations for the one-dimensional $p$-Laplac-ian: \begin{eqnarray*} &\mbox{}& (\Phi_p (y^{\prime}))^{\prime}(t) + m(t) f(t, y^t) = 0 \mbox{} \hspace{0.5cm} {\rm for} \ t \in [0, 1],\\ &\mbox{}& \hspace{12mm} y(t) = \eta (t) \mbox{} \hspace{ 0.5cm} {\rm for} \ t \in [- \sigma, 0],\\ &\mbox{}& \hspace{12mm} y^{\prime}(t) = \xi (t) \mbox{} \hspace{ 0.5cm} {\rm for} \ t \in [1, d], \end{eqnarray*} suitable conditions are imposed on $f(t, y^t)$ which yield the existence of at least two positive solutions. Our result generalizes the main result of Avery, Chyan and Henderson.