Stability of perturbed \(n\)-dimensional Volterra differential equations (Q629280)

It is shown that the zero solution of the perturbed Volterra integrodifferential equation \[ \dot x(t) = A(t) x(t) + \int_0^t C(t,s) x(s)ds + f(t,x(t)) \] with \(f(t,0) = 0\) is asymptotically stable if there exists a non-negative function \(b(t)\) and a positive definite matrix \(B\) such that \(\| f(t,x(t))\|\leq b(t)\| x(t)\|\) and \[ {1 \over 2} \lambda_M^*(t) \lambda_m^{1/2}(B) + \lambda_M^{1/2}(B) \left( \int_0^\infty \left\| C(u,t) \right\| du + b(t) \right) \leq 0, \] where \(\lambda_M^*\) is the biggest eigenvalue of \(BA(t) + A(t)^TB\), and \(\lambda_m(B)\) and \(\lambda_M(B)\) are the smallest and largest eigenvalues of \(B\), respectively. Uniform stability is established under two different sets of additional assumptions.