LAUSR

AbstractIn this article, we consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator: Dα(ϕp(Dαu(t)))=f(t,u(t)),t∈[0,1]T,u(0)=u(σ(1))=Dαu(0)=Dαu(σ(1))=0, $$\begin{aligned}& D^{\alpha }\bigl(\phi_{p}\bigl(D^{\alpha }u(t)\bigr)\bigr)= f \bigl(t, u(t)\bigr),\quad t\in [0,1]_{T}, \\& u(0)= u\bigl(\sigma (1)\bigr)= D^{\alpha }u(0)= D^{\alpha }u\bigl(\sigma (1)\bigr)=0, \end{aligned}$$ where 1<α≤2$1<\alpha \leq 2$ is a real number, the time scale T is a nonempty closed subset of R$\mathbb{R}$. Dα$D^{\alpha }$ is the conformable fractional derivative on time scales, ϕp(s)=|s|p−2s$\phi_{p}(s)=\vert s \vert ^{p-2}s$, p>1$p>1$, ϕp−1=ϕq$\phi_{p}^{-1}=\phi_{q}$, 1/p+1/q=1$1/p+1/q=1$, and f:[0,σ(1)]×[0,+∞)→[0,+∞)$f:[0, \sigma (1)]\times [0,+ \infty )\to [0,+\infty )$ 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.