Ordinary Differential Equations with Delta Function Terms

This article is devoted to nonlinear ordinary differential equations with additive or multiplicative terms consisting of Dirac delta functions or derivatives thereof. Regularizing the delta function terms produces a family of smooth solutions. Conditions on the nonlinear terms, relating to the order of the derivatives of the delta function part, are established so that the regularized solutions converge to a limiting distribution. Introduction This paper is devoted to ordinary differential equations (and systems) of the form (0.1) y(t) = f(t, y(t)) + g(y(t))δ, y(t0) = y0, where δ(s) denotes the s-th derivative of the Dirac delta function. The case of constant g(y) ≡ α will be referred to as the additive case, the general case with s = 0 will be called the multiplicative case. We shall replace the delta function by a family of regularizations φε(t) = ε −1φ(t/ε) and ask under what conditions on f , g and s the family of regularized solutions admits a limit as ε → 0. In the case of partial differential equations, such weak limits have been termed delta waves and studied in various situations, see e.g. [15, 16, 21]. The interest in problem (0.1) comes also from the fact that such equations have been proven to admit solutions in the Colombeau algebra of generalized functions [5, 8, 10]. Equations of the type (0.1) with s = 0 arise in nonsmooth mechanics [1, 3, 7, 14] and are referred to as measure differential equations or impulsive differential equations, often considered under the form y(t) = f(t, y(t)) + N