Consider the functions , belonging to the space , the set , the function , such that exists the partial derivative
Let , where for .
The authors prove sufficient conditions for -uniform stability of trivial solutions of the nonlinear system
and the nonlinear Volterra integro-differential system
which can be seen as perturbed system of
or the variational system
associated with system (3), where is the solution of (3) with for .
The fundamental matrix solution of (4) is given by
The authors prove that the trivial solutions of (1), (2), (3), (4) are -uniformly stable and -stable on .