摘要：This paper investigates the existence of infinitely many positive solutions for the second-order n-dimensional impulsive singular Neumann system − x ″ ( t ) + M x ( t ) = λ g ( t ) f ( t , x ( t ) ) , t ∈ J , t ≠ t k , − Δ x ′ t = t k = μ I k ( t k , x ( t k ) ) , 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 = [ x 1 , x 2 , … , x n ] ⊤ , g ( t ) = diag [ g 1 ( t ) , … , g i ( t ) , … , g n ( t ) ] , $$\begin{aligned}& \mathbf{x}=[x_),x_,,\dots ,x_{n}]^{\top }, \qquad \mathbf{g}(t)=\operatorname{diag} \bigl[g_)(t), \ldots ,g_{i}(t), \ldots , g_{n}(t) \bigr], \end{aligned}$$ where g i ∈ L p [ 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 , 1 2 ) $[0,\frac),)$ . Our methods employ the fixed point index theory and the inequality technique.
关键词：Multi-parameter ; n -dimensional impulsive Neumann system ; Infinitely many singularities ; Matrix theory ; Fixed point index theory and inequality technique
