A New Approach to the Method of Nonlinear Variation of Parameters
Mitchell, A. Richard
Lord, M. E.
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As is well-known  the method of variation of parameters is a very useful tool in studying the properties of solutions of perturbed differential equations. Extending the classical variation of constants formula for linear systems, Alekseev  obtained a variation of constants formula for nonlinear systems. In , Lakshmikantham developed a nonlinear variation of constants formula for a scalar differential equation whose unperturbed term is of variable separable type, under rather mild assumptions. In this paper, we wish to study the general problem under weaker assumptions. Our approach is parallel to the classical one and yields two different formulas, one of which is shown to be equivalent to the Alekseev's formula under additional restrictions. Also, the formula given in  is shown to be a consequence of our results. To illustrate the flexibility of the results derived, we give as applications results on stability and asymptotic equilibrium.