# Triple positive solution to the one-dimensional $p$-Laplacian equation

### Xuemei Zhang

North China Electric Power University, Beijing, China### Meiqiang Feng

Beijing Institute of Technology, China### Weigao Ge

Beijing Institute of Technology, China

## Abstract

We obtain sufficient conditions for the existence of at least three positive solutions to the second-order nonlinear delay differential equation with one-dimensional $p$-Laplacian

$⎩⎨⎧ (Φ_{p}(x_{′}(t)))_{′}+w(t) f(t, x(t), x(t − τ), x_{′}(t))=0,t∈(0,1),x(t)=0,x(1)=0, τ>0,−τ≤t≤0, $

where $Φp(s)$ is the $p$-Laplacian operator, i.e., $Φ_{p}(s)=∣s∣_{p−2}s$, $p>1$, $(Φ_{p})_{−1}=Φ_{q}$, $p1 +q1 =1$. The arguments are based upon a new fixed point theorem in a cone introduced by Avery and Peterson.

## Cite this article

Xuemei Zhang, Meiqiang Feng, Weigao Ge, Triple positive solution to the one-dimensional $p$-Laplacian equation. Port. Math. 65 (2008), no. 1, pp. 143–155

DOI 10.4171/PM/1803