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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].
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