##
Existence theory for single and multiple
solutions to singular positone discrete Dirichlet
boundary value problems to the
one-dimension $p$-Laplacian

##
*Daqing Jiang, Lili Zhang, Donal O'Regan
and Ravi P. Agarwal*

**Address.**

Department of Mathematics,
Northeast Normal University,
Changchun 130024, P. R. China

Department of Mathematics,
National University of Ireland, Galway, Ireland

Department of Mathematical Science,
Florida Institute of Technology,
Melbourne, Florida 32901-6975, USA

**E-mail. **agarwal@fit.edu

**Abstract.**

In this paper we establish the existence of
single and multiple solutions to the positone discrete Dirichlet
boundary value problem
$$
\left\{
\begin{array}{l}
\Delta\big[\phi (\Delta u(t-1))\big]+ q(t) f(t,u(t))=0\,,\quad
t\in \{1,2,\dots,T\}\\[3pt]
u(0)=u(T+1)=0\,,
\end{array}
\right.
$$
where $\phi(s) = |s|^{p-2}s$, $p>1$ and our nonlinear
term $f(t,u)$ may be
singular at $u=0$.

**AMSclassification. **34B15.

**Keywords. ** Multiple solutions, singular, existence, discrete
boundary value problem.