On the Cauchy problem for nonlinear fractional systems with Lipschitzian matrices under the generalized metric spaces (Q6608691)

The paper under review deals with the existence and uniqueness results and the Ulam-Hyers stability of solutions for the following system of differential equations under generalized Caputo derivative:\N\begin{align*}\N^{c}D_{a^{+}}^{\tau,\Theta} \varpi_1(\varsigma)=f_1(\varsigma,\varpi_1(\varsigma), \varpi_2(\varsigma),..., \varpi_n(\varsigma)),\\\N^{c}D_{a^{+}}^{\tau,\Theta} \varpi_2(\varsigma)=f_2(\varsigma,\varpi_1(\varsigma), \varpi_2(\varsigma),..., \varpi_n(\varsigma)),\\\N\vdots\\\N^{c}D_{a^{+}}^{\tau,\Theta} \varpi_n(\varsigma)=f_n(\varsigma,\varpi_1(\varsigma), \varpi_2(\varsigma),..., \varpi_n(\varsigma))\N\end{align*}\Nsubject to the coupled nonlocal conditions\N\begin{align*}\N\varpi_1(0)=\mu_{11}[\varpi_1]+\mu_{12}[\varpi_2]+...+\mu_{1n}[\varpi_n],\\\N\varpi_2(0)=\mu_{21}[\varpi_1]+\mu_{22}[\varpi_2]+...+\mu_{2n}[\varpi_n],\\\N\vdots\\\N\varpi_n(0)=\mu_{n1}[\varpi_1]+\mu_{n2}[\varpi_2]+...+\mu_{nn}[\varpi_n],\N\end{align*}\Nwhere \(^{c}D_{a^{+}}^{\tau,\Theta}\) is the \(\Theta\)-Caputo fractional derivative of order \(\tau\in(0,1]\); \(\Theta\in C^1([a,b],\mathbf{R}^n)\) is an increasing differentiable function such that \(\Theta^{\prime} (\varsigma)\neq 0\) for each \(\varsigma\in[a,b]\); \(f_i:[a,b]\times\mathbf{R}^n\rightarrow\mathbf{R}\), \(i=1,2,...,n\) are given continuous functions and \(\mu_{ij}:C([a,b],\mathbf{R})\longrightarrow\mathbf{R}\) (\(i,j=1,2,...,n\)) are continuous and linear functions.