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

AIMS Mathematics

article

Yang Wang¹  Yating Li²  Yansheng Liu² 

[1] School of Information Engineering, Shandong Management University;School of Mathematics and Statistics, Shandong Normal University

关键词: boundary value problems;    discontinuous differential equations;    fixed point theory;    fractional differential equation;    multiple solutions;   

DOI  :  10.3934/math.2023362

学科分类：地球科学(综合)

来源: AIMS Press

【 摘 要 】

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.

【 授权许可】

CC BY