Abstract |
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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. |