A Nonlinear Variation of Constants Method for Integro-Differential and Integral Equations
Lord, M. E.
Bernfeld, Stephen R.
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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  and ). Miller  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,  and  and references therein). In addition, Bownds and Cushing  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  and  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.