On the boundary value problems of piecewise differential equations with left-right fractional derivatives and delay

In this paper, we study the multi-point boundary value problems for a new kind of piecewise differential equations with left and right fractional derivatives and delay. In this system, the state variables satisfy the different equations in different time intervals, and they interact with each other through positive and negative delay. Some new results on the existence, no-existence and multiplicity for the positive solutions of the boundary value problems are obtained by using Guo–Krasnoselskii’s fixed point theorem and Leggett–Williams fixed point theorem. The results for existence highlight the influence of perturbation parameters. Finally, an example is given out to illustrate our main results.