Vol. 19, No. 2, 1966

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Some existence theorems in the calculus of variations

David Alan Sánchez

Vol. 19 (1966), No. 2, 357–363

In this paper are discussed theorems of existence of a minimum for nonparametric integrals of the calculus of variations defined on an infinite interval, depending on an unknown function, its derivative, and on a convolution integral. The approach of the direct methods of the calculus of variations will be employed.

The author has shown previously that under the usual conditions of convexity on the integrand the class of functionals to be considered are lower semicontinuous with respect to uniform convergence on −∞ < x < but not with respect to uniform convergence on every compact set. Therefore additional hypotheses on the admissible class of functions and the integrand must be imposed to assure that a minimizing sequence of elements converges in the stronger sense.

The reason for the study of such functionals arose from a certain class of optimization problems in communication theory (see W. M. Brown and C. Palermo [1] for example). The author discussed existence of a minimum and lower semicontinuity of these functionals in [4], and the theorems given here represent an improvement over those previously given and consider a special linear case as well. Necessary conditions for an extremum are not discussed, but have been considered in [6].

Mathematical Subject Classification
Primary: 49.00
Received: 1 July 1965
Published: 1 November 1966
David Alan Sánchez