THE lower estimations of solutions of linear VOLTERRA integral-differential equation of fourth order

Abstract

The article is devoted to solving the problem of establishing the sufficient conditions for the lower estimations and tends to infinity of solutions of linear fourth order Volterra integraldifferential equation. To solve the problem we develop a method based on the ideas of the method non-standard reduce to the system, the method Volterra for conversion of equations, the author’s method of cutting functions, the Y.A. Ved’s method of integral inequalities, the Lagrange method for the integral representation of solutions of linear non homogeneous first order differential equations and the Y.A. Ved’s method for lower estimations of solutions. Note that the problem is solved for the solutions of the equation with the initial Cauchy data of a certain initial manifold. Note that the study of lower estimations for solutions of high order Volterra integro-differential equations is one of the difficult problems of the asymptotic theory of solutions of these equations on the half-axis. We have also demonstrated the fundamental possibility of studying this question.