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

Abstract

Abstract In 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 11 $p>1$, 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.

Keywords

• Conformable fractional derivative
• Time scales
• Fixed-point theorems on cone
• p-Laplacian operator