Approximation of bounded solutions and exponential dichotomy on the axis

Lyapunov transformations possessing certain properties are used to construct regular two-point boundary-value problems as approximations to the problem of determining a bounded solution in the general case. The concept of "limiting solutions as $t\to\infty$" is defined and the behaviour of solutions of linear ordinary differential equations as $t\to\infty$ is investigated. The necessary and sufficient conditions are derived under which a singular boundary-value problem with conditions assigned at infinity is uniquely solvable, and an appropriate approximating problem is constructed.