A Nonlinear Variation of Constants Method for Integro-Differential and Integral Equations

Abstract

**Please note that the full text is embargoed** ABSTRACT: It is well known that a very important

technique in obtaining the asymptotic

behavior of solutions of linear and nonlinear

ordinary differential equations under

perturbations is through the use of the

variation of constants formula (see [1] and

[14]). Miller [16] used a well known

representation formula for perturbed linear

Volterra integral equations in terms of the

resolvent which has been successfully used in

recent years to analyze stability behavior of

solutions (see, for example, [12] and [17]

and references therein). In addition, Bownds

and Cushing [4] obtained another variation of

constants formula for linear integral

equations utilizing a fundamental matrix, and

thus extended in a very natural manner the

formula used in ordinary differential

equations. Using this formula in [5] and [6]

they have obtained significant results in the

stability of perturbed linear integral

equations and have successfully demonstrated

the contrasts and similarities between

stability theory of integral equations and

that of ordinary differential equations.