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

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] $.

Keywords

• singularity
• multiplicity
• tempered fractional equation
• reducing order technique