J Austral Math Soc Ser A 46 pp395--401, 1989.

A Generalization of Rolle's Theorem and an Application to a Nonlinear Equation

Antonio Tineo

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 º 0. Next we apply our result to prove the existnece of solutions of the Dirichlet problem for the equation x¢¢ = f(t, x, x¢).

1980 AMS Subject Classification (1985 Revision): 34B15

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Authors

Antonio Tineo

Departamento de Matematicas, Universidad de Los Andes, Facultad de Ciencias, Merida, Edo Merida, Venezuela.