C-1 solutions for fully nonlinear systems of differential equations of first order
(2009) Journal of Differential Equations — Vol. 247, n° 3, p. 770-778 (2009)
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- This note is concerned with the existence of continuously differentiable solutions for the nonlinear system of differential equations f(x'(t)) = g(t, x(t)), x(0) = x(0), where Omega is an open set containing (0, x(0)), g : Omega subset of R x R-n -> R-n is continuous and f : R-n -> R-n satisfies Im(g) subset of Im(f). The set of points x such that f is not locally Lipschitz in an open neighborhood of x is denoted by Lambda(f). We prove the existence of at least one C-1 solution x : [0, T] -> R-n to the system if f is continuous, coercive and if each y in the set f(Lambda(f)boolean OR {x is not an element of Lambda(f) : partial derivative f(x) is not of maximal rank}) has exactly one preimage in R-n. (C) 2009 Elsevier Inc. All rights reserved.
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Goblet, J. (2009). C-1 solutions for fully nonlinear systems of differential equations of first order. Journal of Differential Equations, 247(3), 770-778. https://doi.org/10.1016/j.jde.2009.02.013 (Original work published 2009)