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Necessary and sufficient conditions for the solvability of a nonlinear two-point boundary value problem. (English) Zbl 0559.34014

The dual least action principle is used to prove a necessary and sufficient condition for the solvability of a Dirichlet problem of the form \(u''+u+f(x,u)=0\), \(u(0)=u(\pi)=0\) when f(x,\(\cdot)\) is nondecreasing and \(\int^{u}_{0}f(x,v)dv\) satisfies a suitable growth condition.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
49J35 Existence of solutions for minimax problems
58E30 Variational principles in infinite-dimensional spaces
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References:

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