Multiple solutions for a class of BVPs of fractional discontinuous differential equations with impulses

In this paper, we mainly study the following boundary value problems of fractional discontinuous differential equations with impulses:

$ \hskip 3mm \left\{ \begin{array}{lll} _{t}^{C} \mathcal {D}^{\mathfrak{R}}_{0^{+}}\Lambda(t) = \mathcal {E}(t)\digamma(t, \Lambda(t)), \ a.e.\ t\in Q, \\ \triangle \Lambda|_{t = t_{{\kappa}}} = \Phi_{{\kappa}}(\Lambda(t_{{\kappa}})), \ {\kappa} = 1, \ 2, \ \cdots, \ m, \\ \triangle \Lambda'|_{t = t_{{\kappa}}} = 0, \ {\kappa} = 1, \ 2, \ \cdots, \ m, \\ {\vartheta} \Lambda(0)-{\chi} \Lambda(1) = \int_{0}^{1}\varrho_{1}({\upsilon})\Lambda({\upsilon})d{\upsilon}, \\ {\zeta} \Lambda'(0)-\delta \Lambda'(1) = \int_{0}^{1}\varrho_{2}({\upsilon})\Lambda({\upsilon})d{\upsilon}, \end{array}\right. $

where $ {\vartheta} > {\chi} > 0, \ {\zeta} > \delta > 0 $, $ \Phi_{{\kappa}}\in C(\mbox{ $\mathbb{R}$ }^{+}, \mbox{ $\mathbb{R}$ }^{+}) $, $ \mathcal {E}, \ \varrho_{1}, \ \varrho_{2} \geq 0 $ a.e. on $ Q = [0, 1] $, $ \mathcal {E}, \ \varrho_{1}, \ \varrho_{2} \in L^{1}(0, 1) $ and $ \digamma:[0, 1]\times \mbox{ $\mathbb{R}$ }^{+}\rightarrow \mbox{ $\mathbb{R}$ }^{+} $, $ \mbox{ $\mathbb{R}$ }^{+} = [0, +\infty) $. By using Krasnosel skii's fixed point theorem for discontinuous operators on cones, some sufficient conditions for the existence of single or multiple positive solutions for the above discontinuous differential system are established. An example is given to confirm the main results in the end.