On Variational Formulations for Non-Linear, Non-Potential Operators

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Abstract
This paper consists of two parts. In the first part the method for the choice of a bilinear form rendering a non-linear operator potential has been developed. The method constitutes an extension of Magri's approach, the last being valid for linear operators only. The second part of the paper is concerned with the method of adding the adjoint operator. It has been proved that a system of two coupled operator equations consisting of a given non-linear operator equation and the equation determined by the adjoint of its Frechet derivative is described by a potential operator. The variational functional has been derived. Some particular cases have been discussed.

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