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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
##### MSC:
 47J25 Iterative procedures involving nonlinear operators 49K27 Optimality conditions for problems in abstract spaces