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A generalization of Rolle's Theorem and an application to a nonlinear equation
Published online by Cambridge University Press: 09 April 2009
Abstract
Given two C1 -functions g: R → R, u: [0,1] → R such that u(0) = u(1) = 0, g(0) = 0, we prove that there exists c, with 0 < c < 1, such that u′(c) = g(u(c)). This result implies the classical Rolle's Theorem when g ≡ 0. Next we apply our result to prove the existence of solutions of the Dirichlet problem for the equation x′ = f(t, x, x′).
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- Copyright © Australian Mathematical Society 1989
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