Some Existence Results for High Order Fractional Impulsive Differential Equation on Infinite Interval

Abstract

In this paper, we consider the high order impulsive differential equation on infinite interval $\left\{\begin{array}{c}{D}_{0+}^{\alpha }u\left(t\right)+f\left(t,u\left(t\right),J_{0+}^{\beta }u\left(t\right),D_{0+}^{\alpha -1}u\left(t\right)\right)=0,t\in \left[0,\infty \right)\mathrm{\setminus }{\left\{t_k\right\}}_{k=1}^m\hfill \\ △u\left(t_k\right)=I_k\left(u\left(t_k\right)\right),t=t_k,k=1,\dots ,m\hfill \\ u\left(0\right)=u^{\prime }\left(0\right)=\cdots =u^{\left(n-2\right)}\left(0\right)=0,D_{0+}^{\alpha -1}u\left(\infty \right)=u_0\hfill \end{array}\right.$ By applying Schauder fixed points and Altman fixed points, we obtain some new results on the existence of solutions. The nonlinear term of the equation contains fractional integral operator $J^{\beta }u\left(t\right)$ and lower order derivative operator $D_{0+}^{\alpha -1}u\left(t\right)$ . An example is presented to illustrate our results.