LAUSR

In the present paper, we deal with the existence and multiplicity of solutions for the following impulsive fractional boundary value problem $$\begin{aligned} {_{t}}D_{T}^{\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) + a(t)|u(t)|^{p-2}u(t)= & {} f(t,u(t)),\;\;t\ne t_j,\;\;\hbox {a.e.}\;\;t\in [0,T],\\ \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} I_j(u(t_j))\;\;j=1,2,\ldots ,n,\\ u(0)= & {} u(T) = 0. \end{aligned}$$tDTα0Dtαu(t)p-20Dtαu(t)+a(t)|u(t)|p-2u(t)=f(t,u(t)),t≠tj,a.e.t∈[0,T],ΔtIT1-α0Dtαu(tj)p-20Dtαu(tj)=Ij(u(tj))j=1,2,…,n,u(0)=u(T)=0.where $$\alpha \in (1/p, 1]$$α∈(1/p,1], $$1