The existence of solutions of infinite boundary value problems for first-order impulsive differential systems in Banach spaces
Consider in a Banach space the boundary value problem \[ \begin{cases} x'(t) & =f(t,x(t),y(t)),\\ y'(t) & =g(t,x(t),y(t)),\;t\in J',\\ \Delta x|_{t=t_k} & =I_k(x(t_k),y(t_k)),\\ \Delta y|_{t=t_k} & =J_k(x(t_k), y(t_k)),\;k=1,2,\dots,\\ x(\infty) &=\beta x(0),\quad y(\infty)= \delta y(0),\end{cases} \tag{*} \] where \(J=[0,\infty)\), \(0<t_1<\cdots<t_k <\cdots\), \(\lim_{k\to\infty}t_k= \infty\), \(J'=J\setminus\{t_1,\dots,t_k, \dots\}\), \(x(\infty)=\lim_{t\to\infty} x(t),y(\infty)=\lim_{t\to\infty} y(t)\), \(\beta>1\), \(\delta>1\), \(\Delta x|_{t=t_k}\) and \(\Delta y|_{t=t_k}\) denote the jumps of \(x(t)\) and \(y(t)\) at \(t=t_k\), respectively, i.e., \[ \Delta x|_{t=t_k}=x(t_k^+)-x(t_k^-),\quad \Delta y|_{t=t_k}=y(t_k^+)-y(t_k^-), \] here \(x(t_k^+)\), \(x(t_k^-)\) and \(y(t_k^+)\), \(y(t_k^-)\) represent the right and left limits of \(x(t)\) and \(y(t)\) at \(t=t_k\), respectively. The authors derive conditions such that (*) has at least one nonnegative solution. The proof is based on Schauder's fixed point theorem.
scientific article; zbMATH DE number 5368387