Existence of solutions to a coupled system of implicit fractional differential equations with nonlocal boundary conditions

Abstract

Using Perov's fixed point theorem and a vector version of Krasnosel'skii's fixed point theorem in generalized metric spaces, the authors establish the existence and uniqueness of solutions to the coupled system of implicit fractional differential equations with nonlocal boundary conditions
\begin{equation*}
\begin{cases}
{}^cD^{\alpha}x(t) = g_{1}(t,x(t),y(t))+h_{1}(t, {}^cD^{\alpha}x(t)), & t\in J, \\
{}^cD^{\beta}y(t) = g_{2}(t,x(t),y(t))+h_{2}(t,{}^cD^{\beta}y(t)), & t\in J, \\
x(0) = L_1[x], \\
y(0) = L_2[y],
\end{cases}
\end{equation*}
where $\alpha$, $\beta \in [0,1)$, $J=[0,1]$, ${}^{c}D^{\alpha}$ and ${}^{c}D^{\beta}$ are Caputo fractional derivatives of order $\alpha$ and $\beta$, $g_{i}: [0,1]\times\mathbb{R}\times\mathbb{R} \to \mathbb{R}$ and $h_{i}: [0,1]\times\mathbb{R} \to \mathbb{R}$ are continuous functions for $i=1, 2$, and the functionals $L_{1}$ and $L_{2}$ are given by Stieltjes integrals. The topological structure of the solution set is also investigated where it is shown that the solution set is a nonempty, compact, and $R_{\delta}$ set.