Nonlocal functionals and boundary value problems for functional differential equations
by
G. A. Kamenskii
Moscow State Aviation Institute, Department of Differential Equations, Moscow, 125 871, Volokolamskoe shosse 4, Russia

An integral with the integrand depending on deviating arguments is called the nonlocal functional. The Euler type equations that arise as necessary conditions of extrema of nonlocal functionals are the functional differential equations.

There are described different necessary and some sufficient conditions for extrema of nonlocal functionals. Theorems of existence and uniqueness of solutions to boundary value problems for functional differential equations are proved.

The spaces of solutions to these problems are, as a rule, the Sobolev spaces and it is not often possible to develop analytical methods of solution of these problems.

The different approximate methods of solution of boundary value problems for functional differential equations and direct methods of variational calculus for nonlocal functionals are described.

Date received: February 17, 2003