existence of solutions to nonlinear p-laplacian fractional differential equations with higher-order derivative terms

In this article, we discuss the existence of positive solution to a nonlinear p-Laplacian fractional differential equation whose nonlinearity contains a higher-order derivative $$\displaylines{ D_{0^+}^{\beta}\phi_p\big(D_{0^+}^{\alpha}u(t)\big) +f\big(t,u(t),u'(t),\dots,u^{(n-2)}(t)\big)=0,\quad t\in ( 0,1 ),\cr u(0)=u'(0)=\dots=u^{(n-2)}(0)=0,\cr u^{(n-2)}(1)=au^{(n-2)}(\xi)=0,\quad D_{0^+}^{\alpha}u(0)=D_{0^+}^{\alpha}u(1)=0, }$$ where ${n-1}<\alpha \leq n$, $n\geq 2$, $1<\beta \leq 2$, $0<\xi <1$, $0\leq a\leq 1$ and $0\leq a\xi ^{\alpha-n}\leq 1$, $\phi_{p}(s)=|s|^{p-2}s$, $p>1$, $\phi_{p}^{-1}=\phi_q$, $\frac{1}{p}+\frac{1}{q}=1$. $D_{0^+}^{\alpha}$, $D_{0^+}^{\beta}$ are the standard Riemann-Liouville fractional derivatives, and $f\in C((0,1)\times[0,+\infty)^{n-1},[0,+\infty))$. The Green's function of the fractional differential equation mentioned above and its relevant properties are presented, and some novel results on the existence of positive solution are established by using the mixed monotone fixed point theorem and the upper and lower solution method. The interesting of this paper is that the nonlinearity involves the higher-order derivative, and also, two examples are given in this paper to illustrate our main results from the perspective of application.