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

Mawhin, J.; Ward, J. R.; Willem, M.

doi:10.1090/S0002-9939-1985-0776200-X

Necessary and sufficient conditions for the solvability of a nonlinear two-point boundary value problem
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by J. Mawhin, J. R. Ward and M. Willem PDF

Proc. Amer. Math. Soc. 93 (1985), 667-674 Request permission

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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Lectures on critical point theory