On the generalization of a direct method of the calculus of variations

Abstract

1. Introduction. Consider a topological linear space V, with adjoint space V*> and a f unction defined on a domain in F and taking on values in V*; one wishes to prove the existence of a solution of the equation (x) = 0. A "standard" method is to show that is the Frchet differential of a convex, real-valued functional <f> and use the (virtually automatic) lower-semicontinuity of <f> to show that </> has a minimum on some compact set in V (which is usually taken with a "weak" topology). With appropriate asymptotic conditions on </>> the compact set can be taken very large so that the minimum occurs at an interior point x f which then satisfies the equation. In this problem and a class of related problems, it has been found that the essential property of is that it is monotone in the sense: for all xi 9 x^ we have (xi-Xz, /(^j)-/(x 2 ))O, and the existence of the scalar <j> can often be dispensed with. (See