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The subject of this paper is solutions of an autoresonance equation. We look for a connection between the parameters of the solution bounded as t -> -infinity, and the parameters of two two-parameter families of solutions as t -> infinity. One family consists of the solutions which are not captured into resonance, and another of those increasing solutions which are captured into resonance. In this way we describe the transition through the separatrix for equations with slowly varying parameters and get an estimate for parameters before the resonance of those solutions which may be captured into autoresonance. (C) 2014 AIP Publishing LLC.
We investigate nonlinear problems which appear as Euler-Lagrange equations for a variational problem. They include in particular variational boundary value problems for nonlinear elliptic equations studied by F. Browder in the 1960s. We establish a solvability criterion of such problems and elaborate an efficient orthogonal projection method for constructing approximate solutions.