Nonexistence of Finite-Time Stable Equilibria in a Class of Nonlinear Integral Equations

This brief report studies conditions to ensure the nonexistence of finite-time stable equilibria in a class of systems that are described by means of nonlinear integral equations, whose kernels are part of some Sonine kernel pairs. It is firstly demonstrated that, under certain criteria, a real-valued function that converges in finite-time to a constant value, different from the initial condition, and remains there afterwards, cannot have a Sonine derivative that also remains at zero after some finite time. Then, the concept of equilibrium is generalized to the case of equivalent equilibrium, and it is demonstrated that a nonlinear integral equation, whose kernel is part of some Sonine kernel pair, cannot possess equivalent finite-time stable equilibria. Finally, illustrative examples are presented.