Positive solutions to fractional boundary-value problems with p-Laplacian on time scales

Abstract

In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator: $$\begin{aligned}& D^{\alpha }\bigl(\phi_{p}\bigl(D^{\alpha }u(t)\bigr)\bigr)= f \bigl(t, u(t)\bigr),\quad t\in [0,1]_{T}, \\& u(0)= u\bigl(\sigma (1)\bigr)= D^{\alpha }u(0)= D^{\alpha }u\bigl(\sigma (1)\bigr)=0, \end{aligned}$$ where $1<\alpha \leq 2$ is a real number, the time scale T is a nonempty closed subset of $\mathbb{R}$ . $D^{\alpha }$ is the conformable fractional derivative on time scales, $\phi_{p}(s)=\vert s \vert ^{p-2}s$ , $p>1$ , $\phi_{p}^{-1}=\phi_{q}$ , $1/p+1/q=1$ , and $f:[0, \sigma (1)]\times [0,+ \infty )\to [0,+\infty )$ is continuous. By the use of the approach method and fixed-point theorems on cone, some existence and multiplicity results of positive solutions are acquired. Some examples are presented to illustrate the main results.