  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 \begin{equation*} D_{0^{+}}^{\alpha ,\beta
}y(t)=f(t,y(t)),\text{ }0<t\leq 1, \end{equation*}% with the integral boundary
condition$\ $% \begin{equation*} I_{0^{+}}^{1-\gamma }y(0)=\lambda
\int_{0}^{1}y(s)ds+d, \end{equation*}% where $0<\alpha \leq 1,$ $0\leq \beta
\leq 1,$ $\lambda \geq 0,$ $d\in %TCIMACRO{\U{211d} }% %BeginExpansion
\mathbb{R} %EndExpansion ^{+},$ and $D_{0^{+}}^{\alpha ,\beta }$,
$I_{0^{+}}^{1-\gamma }$ are fractional operators 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.

General Mathematics