APPROXIMATION OF BOUNDED SOLUTIONS AND EXPONENTIAL DICHOTOMY ON AN 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 --> infinity" is defined and the behaviour of solutions of linear ordinary differential equations as t --> infinity 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.