A generalization of the A. Vanderbauwhede variational principle

Let (X, Y,langle , ,rangle) be a dual pair of vector spaces and let B be a linear operator in X. We say that an operator Fcolon Xrightarrow Y is B-potential on an open set Msubset X with respect to a closed vector subspace X_0subset X if there exists a Gateaux-differentiable functional fcolon Mrightarrow bold R such that f'(x)h = langle F(x), Bh rangle for all xin M and hin X_0. The author presents conditions under which the operator F is B-potential and, consequently, solutions of the equation F(x)=0 correspond to critical points of f. Several special cases are discussed. The theorems presented generalize results of A. Vanderbauwhede and of I. Horová.