Multiple positive solutions for a singular tempered fractional equation with lower order tempered fractional derivative

Xinguang Zhang; Yongsheng Jiang; Lishuang Li; Yonghong Wu; Benchawan Wiwatanapataphee; Xinguang Zhang; Yongsheng Jiang; Lishuang Li; Yonghong Wu; Benchawan Wiwatanapataphee

doi:10.3934/era.2024091

Electronic Research Archive

2024, Volume 32, Issue 3: 1998-2015. doi: 10.3934/era.2024091

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Multiple positive solutions for a singular tempered fractional equation with lower order tempered fractional derivative

1.
School of Mathematical and Informational Sciences, Yantai University, Yantai 264005, China
2.
Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia
3.
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China

Received: 30 December 2023 Revised: 21 February 2024 Accepted: 28 February 2024 Published: 06 March 2024

Let $ \alpha\in (1, 2], \beta\in (0, 1) $ with $ \alpha-\beta > 1 $. This paper focused on the multiplicity of positive solutions for a singular tempered fractional boundary value problem

$ \begin{equation*} \left\{\begin{aligned}\ & -{^R _0}{{\mathbb{D}_t}^{\alpha,\lambda}} u(t) = p(t)h\left(e^{\lambda t} u(t), {^R _0}{{\mathbb{D}_t}^ {\beta,\lambda}}u(t)\right), t\in(0,1),\\& {^R _0}{{\mathbb{D}_t}^ {\beta,\lambda}}u(0) = 0, \ \ {^R _0}{{\mathbb{D}_t}^ {\beta,\lambda}}u(1) = 0, \end{aligned}\right. \end{equation*} $

where $ h\in C([0, +\infty)\times[0, +\infty), [0, +\infty)) $ and $ p \in L^1([0, 1], (0, +\infty)) $. By applying reducing order technique and fixed point theorem, some new results of existence of the multiple positive solutions for the above equation were established. The interesting points were that the nonlinearity contained the lower order tempered fractional derivative and that the weight function can have infinite many singular points in $ [0, 1] $.
- singularity,
- multiplicity,
- tempered fractional equation,
- reducing order technique
Citation: Xinguang Zhang, Yongsheng Jiang, Lishuang Li, Yonghong Wu, Benchawan Wiwatanapataphee. Multiple positive solutions for a singular tempered fractional equation with lower order tempered fractional derivative[J]. Electronic Research Archive, 2024, 32(3): 1998-2015. doi: 10.3934/era.2024091

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Abstract

Let $ \alpha\in (1, 2], \beta\in (0, 1) $ with $ \alpha-\beta > 1 $. This paper focused on the multiplicity of positive solutions for a singular tempered fractional boundary value problem

$ \begin{equation*} \left\{\begin{aligned}\ & -{^R _0}{{\mathbb{D}_t}^{\alpha,\lambda}} u(t) = p(t)h\left(e^{\lambda t} u(t), {^R _0}{{\mathbb{D}_t}^ {\beta,\lambda}}u(t)\right), t\in(0,1),\\& {^R _0}{{\mathbb{D}_t}^ {\beta,\lambda}}u(0) = 0, \ \ {^R _0}{{\mathbb{D}_t}^ {\beta,\lambda}}u(1) = 0, \end{aligned}\right. \end{equation*} $

where $ h\in C([0, +\infty)\times[0, +\infty), [0, +\infty)) $ and $ p \in L^1([0, 1], (0, +\infty)) $. By applying reducing order technique and fixed point theorem, some new results of existence of the multiple positive solutions for the above equation were established. The interesting points were that the nonlinearity contained the lower order tempered fractional derivative and that the weight function can have infinite many singular points in $ [0, 1] $.

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