Existence of Positive Solutions for Singular One-Dimensional $P$-Laplace BVP of the Second-Order Difference Systems
Received:December 15, 2011  Revised:October 12, 2012
Key Word: multiple solutions   singular   existence   discrete boundary value problem.
Fund ProjectL:Supported by the XJZDXK of China (Grant No.XJZDXK2011004) and the National Natural Science Foundation of China (Grant No.10971021).
 Author Name Affiliation Weimin HU School of Mathematics and Statistics, Ili Normal University, Xinjiang 835000, P. R. China Guli Bahaer School of Mathematics and Statistics, Ili Normal University, Xinjiang 835000, P. R. China Daqing JIANG School of Mathematics and Statistics, Northeast Normal University, Jilin 130024, P. R. China
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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 $(x,y)=(0,0)$.