Positive solutions for first-order nonlinear Caputo-Hadamard fractional differential equations
Authors :
Abdelouaheb Ardjouni 1 * and Ahcene Djoudi 2
Author Address :
1 Department of Mathematics and Informatics, University of Souk Ahras, P.O. Box 1553, Souk Ahras, 41000, Algeria.
2 Faculty of Sciences, Department of Mathematics, University of Annaba, P.O. Box 12, Annaba 23000, Algeria.
*Corresponding author.
Abstract :
In this paper, we study the existence and uniqueness of positive solutions of the first-order nonlinear Caputo-Hadamard fractional differential equation%
\begin{equation*}
\left\{
\begin{array}{l}
\mathfrak{D}_{1}^{\alpha }\left( x\left( t\right) -g\left( t,x\left(
t\right) \right) \right) =f\left( t,x\left( t\right) \right) ,\text{\ }%
1<t\leq e, \\
x\left( 1\right) =x_{0}>g\left( 1,x_{0}\right) >0,%
\end{array}%
\right.
\end{equation*}%
where $0<\alpha \leq 1$. In the process we convert the given fractional
differential equation into an equivalent integral equation. Then we
construct appropriate mappings and employ the Krasnoselskii and Banach fixed
point theorems and the method of upper and lower solutions to show the
existence and uniqueness of a positive solution of this equation. Finally,
an example is given to illustrate our results.
Keywords :
Fixed points, fractional differential equations, positive solutions, existence, uniqueness.
DOI :
Article Info :
Received : December 24, 2019; Accepted : April 12, 2020.
