@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 = {boundary value problems, discontinuous differential equations, fixed point theory, fractional differential equation, multiple solutions},
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.}
}