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On quasiclassical solutions of an inverse problem in the calculus of variations. (English. Russian original) Zbl 0605.47062
Sov. Math., Dokl. 32, 646-649 (1985); translation from Dokl. Akad. Nauk SSSR 285, 53-56 (1985).
The usual basis for a variational method for solving a nonlinear equation \(N(u)=f\) is a functional \(\phi\) such that \((\nabla \phi)(u)=N(u)-f\) for all \(u\in D(N)\). The existence of such a \(\phi\) is guaranteed only under restrictive conditions on N. However, all that is really needed is that a functional F exists such that \((\delta F)(\bar u)=0\) if and only if \(\bar u\in D(N)\) and \(N(\bar u)=f\), where \(\delta\) is the Gateaux variation. Several examples of such functionals are given. Based on one of them, the variational theory for finding generalized solutions of \(N(u)=f\) is developed.
Reviewer: H.W.Engl
47J25 Iterative procedures involving nonlinear operators
49K27 Optimality conditions for problems in abstract spaces