| Summary: | 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.
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