C 1 Solutions for Semi-Implicit Systems of Differential Equations
(2012) Journal of Dynamics and Differential Equations — Vol. 24, n° 3, p. 483-494 (2012)
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- Abstract
- The present note is a continuation of the author's effort to study the existence of continuously differentiable solutions to the semi-implicit system of differential equations f(x′ (t)) = g(t,x(t)) (1) x(0) = x <inf>0</inf> (2) where - Ω <inf>g</inf>⊆ ℝ × ℝ <sup>n</sup>is an open set containing (0, x <inf>0</inf>) and Ω <inf>g</inf> → ℝ <sup>n</sup> is a continuous function,Ω <inf>g</inf>⊆ ℝ <sup>n</sup> is an open set and f: Ω <inf>g</inf> → ℝ <sup>n</sup> is a continuous function. The transformation of (1)-(2) into a solvable explicit system of differential equations is trivial if f is locally injective around an element γ ε Ω <inf>f</inf> ∩ f <sup>-1</sup>(g(0,x_0)). In this paper, we study (1)-(2) when such a translation is not possible because of the inherent multivalued nature of f <sup>-1</sup>. © 2012 Springer Science+Business Media, LLC.
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Goblet, J. (2012). C 1 Solutions for Semi-Implicit Systems of Differential Equations. Journal of Dynamics and Differential Equations, 24(3), 483-494. https://doi.org/10.1007/s10884-012-9257-2 (Original work published 2012)