Impulsive integral boundary value problems of the higher-order fractional differential equation with eigenvalue arguments

Abstract

We consider a class of higher-order nonlinear Riemann-Liouville fractional differential equation with Riemann-Stieltjes integral boundary value conditions and impulses as follows: $$ \left \{\textstyle\begin{array}{@{}l} -D_{0^{+}}^{\alpha}u(t)=\lambda a(t)f(t,u(t)),\quad t\in(0,1)\setminus\{ t_{k}\}_{k=1}^{m}, \Delta u(t_{k})=I_{k}(u(t_{k})), \quad t=t_{k}, u(0)=u'(0)=\cdots=u^{(n-2)}(0),\qquad u'(1)=\int_{0}^{1}u(s)\,dH(s). \end{array}\displaystyle \right . $$ By converting the boundary value problem into an equivalent integral equation and applying the Schauder fixed-point theorem, fixed-point index theorem, we have established sufficient conditions for the existence and multiplicity of positive solutions. The eigenvalue intervals are also given. Some examples are presented to illustrate the validity of our main results.