Positive Solutions for a $n$th-Order Impulsive Differential Equation with Integral Boundary Conditions

Abstract

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 Riemann–Stieltjes integrals (i.e., $\alpha (t)$ and $\beta (t)$ have bounded variation). We use the Krasnoselskii–Zabreiko fixed point theorem to establish our main results. Furthermore, our nonlinear term $f$ is allowed to grow superlinearly and sublinearly.