Perturbation theory for Volterra integrodifferential systems

Principal Matrix Solutions and Variation of Parameters for Volterra Integro-Dynamic Equations on Time Scales

“…Changing the order of integration (see [1, Lemma 2.1]) we get (13). Conversely, one may show by using Theorem 4 that a differentiable function x(t) solving (13) satisfies (11) and x(s) = x 0 .…”

Transient behaviour and stability points of the Poiseuille flow of a KBKZ-fluid

“…The behaviour of the perturbed solution for t > to determines the stability of the unperturbed solution: If the perturbed solution remains bounded (or tends to zero) then the unperturbed solution is (asymptotically) stable. The definitions of stability of solutions of integrodifferential equations used here are similar to the usual definitions for ordinary differential equations (see [5], [6] and [7]). …”

“…Under these conditions Grossman and Miller [5] proved that the stability of (4.8) is determined by the stability of its linear form (with g = 0)…”

“…The resolvent R associated with this linear equation, is defined as the unique solution of as A and a are locally integrable (see [5] for details). Here, the function C is defined by …”

A fixed point theorem, a perturbed differential equation, and a multivariable Volterra integral equation

“…and J. S. W. Wong [13], [14], Miller [11], [12], M. Z. Nashed and Wong [16], and S, I. Grossman and Miller [4]). …”