On preserving stability of Volterra integral equations under a general class of perturbations

for all T > a > to. The matr ix A(t, s) is assumed to be locally in L 1 in (t, s) for t > s > to. Here u, v and ~o are cont inuous (but not necessarily differentiable) n-vector-valued functions. The per turbat ion te rm p(t, s, ~(s)) is, for each t > s > to, a functional defined for all f E S(b) = {~ ~ C O [to, +oo ) : I~1o = maxt>__to I~(t)l -< b} for some b > 0 which is sufficiently smooth so tha t solutions of (P) exist locally and are extendable. (For example, p might be cont inuous for t >_ s >_ to and all ~ ~ S(b) [2].) Also p(t, s, 0) = 0 for all t > s > to. H y p o thesis (H1) is satisfied for example if A(t, s) is continuous. I f ( H I ) holds, then for each ~o(t), cont inuous for t > t o, the linear system (L) has a unique solution existing for all t _> a (see [2]). Also the existence and

[1]  Richard K Miller,et al.  Nonlinear Volterra Integral Equations , 1970 .

[2]  J. Cushing,et al.  Some stability theorems for systems of volterra integral equations , 1975 .

[3]  R. K. Miller.  On the linearization of Volterra integral equations. , 1968 .

[4]  H. Antosiewicz,et al.  Differential Equations: Stability, Oscillations, Time Lags , 1967 .

[5]  F. V. Vleck,et al.  Stability and Asymptotic Behavior of Differential Equations , 1965 .

[6]  ASYMPTOTIC STABILITY AND PERTURBATIONS FOR LINEAR VOLTERRA INTEGRODIFFERENTIAL SYSTEMS , 1972 .

[7]  J. Cushing,et al.  Some stability criteria for linear systems of Volterra integral equations , 1972 .

[8]  J. Cushing,et al.  A representation formula for linear Volterra integral equations , 1973 .

[9]  S. I. Grossman.  Existence and stability of a class of nonlinear Volterra integral equations , 1970 .

[10]  A. R. Forsyth.  Theory of Differential Equations , 1961 .

[11]  R. K. Miller,et al.  Perturbation theory for Volterra integrodifferential systems , 1970 .

[12]  Admissibility and nonlinear Volterra integral equations , 1970 .

[13]  J. M. Cushing,et al.  On strong stability for linear integral equations , 2005, Mathematical systems theory.

[14]  J. Yorke,et al.  Perturbation theorems for ordinary differential equations , 1967 .

[15]  C. Corduneanu.  Problèmes globaux dans le théorie des équations intégrales de Volterra , 1965 .

[16]  J. Schwartz,et al.  Linear Operators. Part I: General Theory. , 1960 .

[17]  Wolfgang Hahn,et al.  Stability of Motion , 1967 .

[18]  R. K. Miller,et al.  Asymptotic stability properties of linear Volterra integrodifferential equations. , 1971 .

[19]  R. Bellman.  Stability theory of differential equations , 1953 .

[20]  W. Mydlarczyk.  Remarks on a nonlinear Volterra equation , 1991 .

[21]  R. K. Miller,et al.  On Volterra integral equations with nonnegative integrable resolvents , 1968 .

[22]  On stability in ordinary differential equations. , 1969 .

[23]  A. Fuller,et al.  Stability of Motion , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[24]  J. Nohel.  Perturbations of volterra equations and admissibility , 1971 .

[25]  J. Nohel,et al.  Perturbations of Volterra integral equations , 1969 .

[26]  Aaron Strauss.  On a perturbed Volterra integral equation , 1970 .

[27]  J. M. Cushing.  Strong stability of perturbed systems of Volterra integral equations , 2005, Mathematical systems theory.