# Lieva Beklaryan

## The Boundary Value Problem for the Functional-Differential Equation

Many applied problems reduce to studing of solutions of traveling
waves type for infinite-dimensional dynamic systems. In
particular, there is a studing of the infinite-dimensional
dynamic system in the theory of a plastic deformation

m \ddot{y}_i = \phi (y_i) + y_ {i+1} - 2y_i + y_ {i-1}, \qquad i
\in \Bbb Z, \quad t\in \Bbb R (1)

where the potential $\phi(\cdot)$ is setted by smooth periodic
function. The equation (1) is a system with a potential of
Fraenkel - Kontorova [1]. This system models a behaviour of a
countable number of balls with masses equal to $m$, located in
integer points of the real exis, where any two neighboring balls
are connected by a flexible spring. The studing of such systems
with different potentials is one of the intensively developing
directions in the theory of dynamic systems. The studing of
solutions of traveling waves type, as one of observed classes of
waves, is the central problem for these system. A connection
between solutions of traveling wave tape of the equation (1) and
solutions of the induced differential equation with deviating
argument is established.