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, \\ \tr...
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AIMS Press
2023-01-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023362?viewType=HTML |
| _version_ | 1797936835578560512 |
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| author | Yang Wang Yating Li Yansheng Liu |
| author_facet | Yang Wang Yating Li Yansheng Liu |
| author_sort | Yang Wang |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-10T18:36:27Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-10T18:36:27Z |
| publishDate | 2023-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-894647d6415147b5bf6d6a18027633122023-02-02T01:19:47ZengAIMS PressAIMS Mathematics2473-69882023-01-01837196722410.3934/math.2023362Multiple solutions for a class of BVPs of fractional discontinuous differential equations with impulsesYang Wang0Yating Li1Yansheng Liu21. School of Information Engineering, Shandong Management University, Jinan 250357, Shandong, China2. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, Shandong, China2. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, Shandong, ChinaIn 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.https://www.aimspress.com/article/doi/10.3934/math.2023362?viewType=HTMLboundary value problemsdiscontinuous differential equationsfixed point theoryfractional differential equationmultiple solutions |
| spellingShingle | Yang Wang Yating Li Yansheng Liu Multiple solutions for a class of BVPs of fractional discontinuous differential equations with impulses AIMS Mathematics boundary value problems discontinuous differential equations fixed point theory fractional differential equation multiple solutions |
| title | Multiple solutions for a class of BVPs of fractional discontinuous differential equations with impulses |
| title_full | Multiple solutions for a class of BVPs of fractional discontinuous differential equations with impulses |
| title_fullStr | Multiple solutions for a class of BVPs of fractional discontinuous differential equations with impulses |
| title_full_unstemmed | Multiple solutions for a class of BVPs of fractional discontinuous differential equations with impulses |
| title_short | Multiple solutions for a class of BVPs of fractional discontinuous differential equations with impulses |
| title_sort | multiple solutions for a class of bvps of fractional discontinuous differential equations with impulses |
| topic | boundary value problems discontinuous differential equations fixed point theory fractional differential equation multiple solutions |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023362?viewType=HTML |
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