Lower estimates of solutions and their derivatives for the linear fourth order Volterra integro-differential equation

The article is devoted to solving the problem of establishing the sufficient conditions for the lower estimates and tendency to infinity of solutions and their derivatives up to the third order of the linear fourth order Volterra integro-differential equation. To solve the problem, we develop a method based on the ideas of the method for non-standard reduction to a system, the method of converting Volterra equations, the author's method of cutting functions, Yu.A. Ved's method of integral inequalities, the Lagrange method for the integral representation of solutions of linear inhomogeneous first order differential equations and Yu.A. Ved's method for lower estimations of solutions. The study has the following design: first, a priori estimates are established on a half-axis for solutions and their derivatives, then lower estimates are produced using the integral representations for solutions and their first, second, third derivatives, with the formation of the manifold to the initial data. Thus, the problem is solved for solutions and their derivatives of the equation with the initial Cauchy data of a certain initial manifold, respectively. It should be noted that the study of lower estimates for solutions of high order Volterra integro-differential equations is one of complex problems within the asymptotic theory of solving such equations on the half-axis. The connection of this problem with instability and non-oscillation of solutions is discussed, as well as with the absence of singular points in the sense of Ya.V. Bykov for the considered Volterra integro-differential equation of the fourth order. It is emphasized that the task under study is new for the corresponding linear fourth-order differential equation. The fundamental possibility of studying this question have been demonstrated.