Hyers-Ulam-Rassias stability of the generalized fractional systems and the \(\rho\)-Laplace transform method (Q2029729)

In this article, are considered the following fractional linear differential systems \[\begin{cases} _0 D^{\alpha,\rho} y(t) = A y(t) + f(t), \ \ 0 < \alpha < 1, t>0, \\ ( _0 I^{1-\alpha, \rho} y)(0+)=b, \ \ b \in \mathbb{R}^n \end{cases} \tag{1}\] and \[\begin{cases} _0 D^{\alpha,\rho} y(t) = A \ _0 D^{\beta,\rho} y(t) + g(t), \ \ 0 < \beta < \alpha < 1, t>0, \\ ( _0 I^{1-\alpha, \rho} y)(0+)=b, \ \ b \in \mathbb{R}^n \end{cases} \tag{2}\] where \(_0 D^{\alpha,\rho} y(\cdot), \ _0 D^{\beta,\rho} y(\cdot)\) and \(_0 I^{1-\alpha, \rho} y(\cdot)\) are generalized left fractional derivative and integral operators, all in the sense of Katugampola, \(A\) is a \(n \times n\) constant matrix and \(f(t), g(t)\) are \(n\)-dimensional continuous vector-valued functions. For both systems sufficient conditions for Hyers-Ulam-Rassias stability are obtained by the use of a modified \(\rho\)-Laplace transform method, involving a parameter \(\rho\) and introduced from Abdeljawad for the so called conformable calculus. The obtained results are illustrated with two examples.