### 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.

#### Keywords

#### Full Text:

PDF#### References

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DOI: http://dx.doi.org/10.24193/subbmath.2021.4.09

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