In this paper, we consider the continuous Hopfield model with a weak
interaction of network neurons. This model is described by a system of
differential equations with linear boundary conditions. Also, we consider
the questions of finding necessary and sufficient conditions of solvability
and constructive construction of solutions of the given problem, which turn
into solutions of the linear generating problem, as the parameter
$\varepsilon$ tends to zero. An iterative algorithm for finding solutions
has been constructed. The problem of finding the extremum of the target
functions on the given problem solution is considered. To minimize a
functional, an accelerated method of conjugate gradients is used. Results
are illustrated with examples for the case of three neurons.