Positive solutions for first-order nonlinear Caputo-Hadamard fractional differential equations

Abstract

In this paper, we study the existence and uniqueness of positive solutions of the first-order nonlinear CaputoHadamard fractional differential equation
\(
\left\{\begin{array}{l}
\mathfrak{D}_1^\alpha(x(t)-g(t, x(t)))=f(t, x(t)), 1<t \leq e, \\
x(1)=x_0>g\left(1, x_0\right)>0,
\end{array}\right.
\)
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.