Existence of Positive Solutions for Singular One-Dimensional $P$-Laplace BVP of the Second-Order Difference Systems

Abstract:

      In this paper we establish the existence of single and multiple positive solutions to the following singular discrete boundary value problem $$\left\{\begin{array}{l}\Delta[\phi (\Delta x(i-1))]+ q_{1}(i)f_{1}(i,x(i),y(i))=0, ~~i\in \{1,2,\ldots,T\}\\\Delta[\phi (\Delta y(i-1))]+ q_{2}(i)f_{2}(i,x(i),y(i))=0,\\x(0)=x(T+1)=y(0)=y(T+1)=0,\end{array}\right.\tag 1.1$$ where $\phi(s)=|s|^{p-2}s$, $p>1$ and the nonlinear terms $f_{k}(i,x,y)~(k=1,2)$ may be singular at $(x,y)=(0,0)$.