一维$P$-Laplace二阶差分系统奇异边值问题正解的存在性

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

中文关键词:  分数阶微分方程  格林函数  正解  不动点定理  边值问题.

英文关键词:multiple solutions  singular  existence  discrete boundary value problem.

中文摘要:

      应用锥压缩锥拉伸不动点定理和Leray-Schauder 抉择定理研究了一类具有P-Laplace算子的奇异离散边值问题$$\left\{\begin{array}{l}\Delta[\phi (\Delta x(i-1))]+ q_{1}(i)f_{1}(i,x(i),y(i))=0, ~~~i\in \{1,2,...,T\}\\\Delta[\phi (\Delta y(i-1))]+ q_{2}(i)f_{2}(i,x(i),y(i))=0,\\x(0)=x(T+1)=y(0)=y(T+1)=0,\end{array}\right.$$的单一和多重正解的存在性，其中$\phi(s) = |s|^{p-2}s, ~p>1$，非线性项$f_{k}(i,x,y)(k=1,2)$在$(x,y)=(0,0)$具有奇性.

英文摘要:

      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 $(x,y)=(0,0)$.