Positive Solutions for a \(n\)th-Order Impulsive Differential Equation with Integral Boundary Conditions
In this paper we study the existence of positive solutions for the following \(n\)th-order impulsive boundary value problem $$\begin{aligned} \left\{ \begin{array}{l} u^{(n)}(t)+f(t,u(t))=0,\quad t\in [0,1],t\not =t_k,\\ -\Delta u^{(n-1)}|_{t=t_k}=I_k(u(t_k)),\quad k=1,2,\ldots ,m,\\ u(0)=\int _0^1 u(t)\mathrm d \alpha (t),\qquad u(1)=\int _0^1 u(t)\mathrm d \beta (t),\\ u^{\prime }(0)=\cdots =u^{(n-3)}(0)=u^{(n-2)}(0)=0. \end{array}\right. \end{aligned}$$Here \(f\in C([0,1]\times \mathbb{R }^+,\mathbb{R }^+), I_k\in C(\mathbb{R }^+,\mathbb{R }^+) ( {\mathbb{R }^+:=[0,\infty )})\) and \(\int _0^1 u(t)\mathrm d \alpha (t), \int _0^1 u(t)\mathrm d \beta (t)\) are RiemannStieltjes integrals (i.e., \(\alpha (t)\) and \(\beta (t)\) have bounded variation). We use the KrasnoselskiiZabreiko fixed point theorem to establish our main results. Furthermore, our nonlinear term \(f\) is allowed to grow superlinearly and sublinearly.
positive solution boundary value problem riemannstieltjes integral krasnoselskiizabreiko fixed point theorem impulsive effect