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Discontinuous solutions of semilinear differential-algebraic equations. II: \({\mathcal P}\)-consistency. (English) Zbl 0863.34001
Continuing their research in [Nonl. Anal., Theory Methods Appl. 27, 1241-1256 (1996)] the authors suggest a way of selecting certain \({\mathcal P}\) (perturbation)-consistent distribution solutions of implicit differential equations of the form: (1) \(A(t).x'= G(t,x)\) where \(A(t)\), \(t\in J\subset\mathbb{R}\) are singular \(n\times n\) matrices of constant rank \(r<n\). More precisely, assuming that equation (1) is a limit-case of the (perturbed) \({\mathcal P}_\mu\) equation (2) \(\overline{A}(\mu,t).x'= \overline{G}(\mu,t,x)\) in the sense that \(A(t)= \overline{A}(0,t)\), \(G(t,x)= \overline{G}(0,t,x)\), the authors introduce the concepts of \({\mathcal P}_\mu\)-consistent initial points and \({\mathcal P}_\mu\)-consistent solutions of (1) and discuss their relationships with the Tikhonov-Levinson theorem on singular perturbations of R. E. O’Malley [Appl. Math. Sci. 89, Springer, New York (1991; Zbl 0743.34059)]. The largest part of the paper is devoted to the discussion of some examples in nonlinear circuit theory.

MSC:
34A09 Implicit ordinary differential equations, differential-algebraic equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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