@article{Wang2023, 
author = {Yang Wang and Yating Li and Yansheng Liu},
title = {Multiple solutions for a class of BVPs of fractional discontinuous differential equations with impulses},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {3},
pages = {7196-7224},
keywords = {fractional differential equation, fixed point theory, boundary value problems, multiple solutions, discontinuous differential equations},
url = {https://www.sciopen.com/article/10.3934/math.2023362},
doi = {10.3934/math.2023362},
abstract = {In this paper, we mainly study the following boundary value problems of fractional discontinuous differential equations with impulses:           {                                                                              t                                      C                                                          D                                                      0                                  +                                                                                    R                                              Λ          (          t          )          =                      E                    (          t          )          ϝ          (          t          ,          Λ          (          t          )          )          ,                    a          .          e          .                    t          ∈          Q          ,                                      △          Λ                                    |                                      t              =                              t                                                      κ                                                                                =                      Φ                                          κ                                              (          Λ          (                      t                                          κ                                              )          )          ,                                κ                    =          1          ,                    2          ,                    ⋯          ,                    m          ,                                      △                      Λ            ′                                              |                                      t              =                              t                                                      κ                                                                                =          0          ,                                κ                    =          1          ,                    2          ,                    ⋯          ,                    m          ,                                                  ϑ                    Λ          (          0          )          −                      χ                    Λ          (          1          )          =                      ∫                          0                                      1                                            ϱ                          1                                (                      υ                    )          Λ          (                      υ                    )          d                      υ                    ,                                                  ζ                                Λ            ′                    (          0          )          −          δ                      Λ            ′                    (          1          )          =                      ∫                          0                                      1                                            ϱ                          2                                (                      υ                    )          Λ          (                      υ                    )          d                      υ                    ,                        where        ϑ    >      χ    >  0  ,        ζ    >  δ  >  0,         \Phi_{{\kappa}}\in C(\text{  \mathbb{R}       }^{+}, \text{  \mathbb{R}       }^{+})  ,          E    ,        ϱ          1        ,        ϱ          2        ≥  0 a.e. on    Q  =  [  0  ,  1  ],         E    ,        ϱ          1        ,        ϱ          2        ∈      L          1        (  0  ,  1  ) and        \digamma:[0, 1]\times \text{  \mathbb{R}       }^{+}\rightarrow \text{  \mathbb{R}       }^{+}  ,         \text{  \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.}
}