A class of singular n-dimensional impulsive Neumann systems

• Main Authors: Ping Li, Meiqiang Feng, Minmin Wang
• Format: Article
• Language: English
• Published: SpringerOpen 2018-03-01
• Series: Advances in Difference Equations
• Subjects: Multi-parameter; n-dimensional impulsive Neumann system; Infinitely many singularities; Matrix theory; Fixed point index theory and inequality technique
• Online Access: http://link.springer.com/article/10.1186/s13662-018-1558-2

Abstract This paper investigates the existence of infinitely many positive solutions for the second-order n-dimensional impulsive singular Neumann system −x″(t)+Mx(t)=λg(t)f(t,x(t)),t∈J,t≠tk,−Δx′|t=tk=μIk(tk,x(tk)),k=1,2,…,m,x′(0)=x′(1)=0. $$\begin{aligned}& -\mathbf{x}^{\prime\prime}(t)+ M\mathbf{x}(t)=\lambda {\mathbf{g}}(t)\mathbf{f} \bigl(t,\mathbf{x}(t) \bigr),\quad t\in J, t\neq t_{k}, \\& -\Delta {\mathbf{x}}^{\prime}|_{t=t_{k}}=\mu {\mathbf{I}}_{k} \bigl(t_{k},\mathbf{x}(t_{k}) \bigr),\quad k=1,2,\ldots ,m, \\& \mathbf{x}^{\prime}(0)=\mathbf{x}^{\prime}(1)=0. \end{aligned}$$ The vector-valued function x is defined by x=[x1,x2,…,xn]⊤,g(t)=diag[g1(t),…,gi(t),…,gn(t)], $$\begin{aligned}& \mathbf{x}=[x_{1},x_{2},\dots ,x_{n}]^{\top }, \qquad \mathbf{g}(t)=\operatorname{diag} \bigl[g_{1}(t), \ldots ,g_{i}(t), \ldots , g_{n}(t) \bigr], \end{aligned}$$ where gi∈Lp[0,1] $g_{i}\in L^{p}[0,1]$ for some p≥1 $p\geq 1$, i=1,2,…,n $i=1,2,\ldots , n$, and it has infinitely many singularities in [0,12) $[0,\frac{1}{2})$. Our methods employ the fixed point index theory and the inequality technique.