Completely Continuous Scalars and Variational Methods

Abstract

The notion of a completely continuous scalar in a real Hilbert-space E has been introduced in a previous paper, to which we refer for the exact definition. Roughly speaking it is a real valued function I(t) of the point X C E such that its gradient2 a(X) is a mapping which is completely continuous in the ordinary sense ([8], Theorem 5.1). The quadratic integral forms occurring in the theory of non-singular linear integral equations are special cases.3 As is well known (see e.g. [2], p. 109) these integral forms play a basic role in the treatment of linear integral equations with a symmetric kernel. It is the object of the present paper to show that the completely continuous scalars form likewise an adequate tool for the treatment by variational methods of linear or non-linear functional equations of the form