Positive Solutions for a Second-Order p -Laplacian Boundary Value Problem with Impulsive Effects and Two Parameters

Abstract

The author considers an impulsive boundary value problem involving the one-dimensional p-Laplacian $-{({\phi }_p (u^{\prime }))}^{\prime }=\lambda \omega \left(t\right)f\left(t,u\right),$ $t\ne t_k,$ $\mathrm{\Delta }u{|}_{t=t_k}=\mu I_k\left(t_k,u\left(t_k\right)\right),$ $\mathrm{\Delta }{u}^{\mathrm{\prime }}{|}_{t=t_k}=0,$ $k=\mathrm{1,2},\dots ,n,$ $au(0)-b{u}^{\mathrm{\prime }}(0)={\int }_0^1\mathrm{‍}g(t)u(t)dt,u^{\mathrm{\prime }}(1)=0$, where $\lambda >0$ and $\mu >0$ are two parameters. Using fixed point theories, several new and more general existence and multiplicity results are derived in terms of different values of $\lambda >0$ and $\mu >0$. The exact upper and lower bounds for these positive solutions are also given. Moreover, the approach to deal with the impulsive term is different from earlier approaches. In this paper, our results cover equations without impulsive effects and are compared with some recent results by Ding and Wang.