Positive solution of Hilfer fractional differential equations with integral boundary conditions

Abstract

In this article, we have interested the study of the existence and
uniqueness of positive solutions of the first-order nonlinear Hilfer
fractional differential equation
$$D_{0^{+}}^{\alpha ,\beta }y(t)=f(t,y(t)),\text{ }0<t\leq 1,$$
with the integral boundary condition
$$I_{0^{+}}^{1-\gamma }y(0)=\lambda \int_{0}^{1}y(s)ds+d,$$
where $0<\alpha \leq 1,$ $0\leq \beta \leq 1,$ $\lambda \geq 0,$
$d\in \mathbb{R}^{+},$ and $D_{0^{+}}^{\alpha ,\beta }$, $I_{0^{+}}^{1-\gamma }$ are
fractional ope\-rators in the Hilfer, Riemann-Liouville concepts,
respectively. In this approach, we transform the given fractional
differential equation into an equivalent integral equation. Then we
establish sufficient conditions and employ the Schauder fixed point theorem
and the method of upper and lower solutions to obtain the existence of a
positive solution of a given problem. We also use the Banach contraction
principle theorem to show the existence of a unique positive solution.
The result of existence obtained by structure the upper and lower control
functions of the nonlinear term is without any monotonous conditions.
Finally, an example is presented to show the effectiveness of our main results.