ON THE HORIZONTAL ASYMPTOTE OF THE SOLUTIONS OF VOLTERRA TYPE SECOND ORDER WEAKLY NONLINEAR INTEGRO- DIFFERENTIAL EQUATION ON THE SEMI-AXIS
Sufficient conditions for the tendency to finite limits of solutions and to zero of their first derivatives of a weakly nonlinear integro- differential equation of the second order of Volterra type are established. To this end, a method of transforming the equations of V. Volterra is being developed, weight function method, partial cutting method and integral inequality method, as well as the Lyusternik-Sobolev lemma. It is also shown that the sufficient conditions obtained in this article differ from those sufficient conditions that were previously established using the method of weight and cutting functions, in other words, it is shown that the above listed asymptotic properties are true for solutions of new classes of Volterra-type integro-differential equations on the half shaft. It is noted that the conditions for the presence of a horizontal asymptote with an unlimited increase in the argument of the solutions of the considered weakly nonlinear integro-differential equation of the second order can be used to predict some stabilization processes from the theory of automatic control. An illiustrative example is given