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Proceedings of the American Mathematical Society

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Necessary and sufficient conditions for the solvability of a nonlinear two-point boundary value problem


Authors: J. Mawhin, J. R. Ward and M. Willem
Journal: Proc. Amer. Math. Soc. 93 (1985), 667-674
MSC: Primary 34B15
DOI: https://doi.org/10.1090/S0002-9939-1985-0776200-X
MathSciNet review: 776200
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Abstract: The dual least action principle is used to prove a necessary and sufficient condition for the solvability of a Dirichlet problem of the form $ u'' + u + f\left( {x,u} \right) = 0$. $ u(0) = u(\pi ) = 0$ when $ f\left( {x, \cdot } \right)$ is nondecreasing and $ \int_0^u {f\left( {x,v} \right)dv} $ satisfies a suitable growth condition.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0776200-X
Keywords: Dirichlet problem, dual variational method, semilinear equations at resonance, jumping nonlinearities
Article copyright: © Copyright 1985 American Mathematical Society