The novel delay-dependent asymptotic stability of a differential and Riemann-Liouville fractional differential neutral system with constant delays and nonlinear perturbation is studied. We describe the new asymptotic stability criterion in the form of linear matrix inequalities (LMIs), using the application of zero equations, model transformation and other inequalities. Then we show the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with constant delays. Furthermore, we not only present the improved delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with single constant delay but also the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral equation with constant delays. Numerical examples are exploited to represent the improvement and capability of results over another research as compared with the least upper bounds of delay and nonlinear perturbation.

In this article, we initiate the study of new concepts of conformable \(q\)-fractional calculus. The conformable fractional \(q\)-derivative and \(q\)-integral are defined and their fundamental theorems are also proved. The uniform asymptotic stability of the \(q\)-deformed conformable fractional system with constant delay is investigated by using the Lyapunov-Razumikhin method. For application, a new asymptotic stability necessary condition for the conformable \(q\)-fractional linear system with constant delay is obtained in term of linear matrix inequality (LMI). A numerical example is demonstrated for the results given to illustrate the effectiveness.