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A generalization of Rolle's Theorem and an application to a nonlinear equation

Part of: Boundary value problems

Published online by Cambridge University Press:  09 April 2009

Antonio Tineo
Antonio Tineo
Affiliation:
Departamento de Matematicas, Universidad de Los Andes, Facultad de Ciencias Merida, Edo Merida, Venezuela
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Abstract

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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′).

MSC classification

Secondary: 34B15: Nonlinear boundary value problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 3 , June 1989 , pp. 395 - 401
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Graines, R. T. and Mawhin, J. L., Coincidence degree and nonlinear differential equations (Lectures Notes in Math., 568, Springer-Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
[2]Granas, A., Guenther, R. B. and Lee, J. W., ‘Nonlinear boundary value problems for some class of ordinary differential equations’, Rocky Mountain J. Math. 10 (1980), 35–58.CrossRefGoogle Scholar