Necessary and Sufficient Conditions for the Solvability of a Nonlinear Two-Point Boundary Value Problem

The dual least action principle is used to prove a necessary and sufficient condition for the solvability of a Dirichlet problem of the form <!-- MATH $u'' + u + f\left( {x,u} \right) = 0$ --> . <!-- MATH $u(0) = u(\pi ) = 0$ --> when <!-- MATH $f\left( {x, \cdot } \right)$ --> is nondecreasing and <!-- MATH $\int_0^u {f\left( {x,v} \right)dv}$ --> satisfies a suitable growth condition.