On Lower Estimates of Solutions and Their Derivatives to a Fourth-Order Linear Integrodifferential Volterra Equation

Abstract

We examine solutions of the problem on sufficient conditions that guarantee a lower estimate and tending to infinity of solutions and their derivatives up to the third order to a fourth-order linear integrodifferential Volterra equation. For this purpose, we develop a method based on the nonstandard reduction method (S. Iskandarov), the Volterra transformation method, the method of shearing functions (S. Iskandarov), the method of integral inequalities (Yu. A. Ved’ and Z. Pakhyrov), the method of a priori estimates (N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina, and P. M. Simonov, 1991, 2001), the Lagrange method for integral representations of solutions to first-order linear inhomogeneous differential equations, and the method of lower estimate of solutions (Yu. A. Ved’ and L. N. Kitaeva). © 2018 Springer Science+Business Media, LLC, part of Springer Nature

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Journal of Mathematical Sciences, Vol. 230, No. 5, May, 2018
ON LOWER ESTIMATES OF SOLUTIONS AND THEIR DERIVATIVES
TO A FOURTH-ORDER LINEAR INTEGRODIFFERENTIAL
VOLTERRA EQUATION
S. Iskandarov and G. T. Khalilova UDC 517.968.74
Abstract. We examine solutions of the problem on sufficient conditions that guarantee a lower estimate
and tending to infinity of solutions and their derivatives up to the third order to a fourth-order linear
integrodifferential Volterra equation. For this purpose, we develop a method based on the nonstandard
reduction method (S. Iskandarov), the Volterra transformation method, the method of shearing func-
tions (S. Iskandarov), the method of integral inequalities (Yu. A. Ved’ and Z. Pakhyrov), the method
of a priori estimates (N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina, and P. M. Simonov, 1991,
2001), the Lagrange method for integral representations of solutions to first-order linear inhomogeneous
differential equations, and the method of lower estimate of solutions (Yu. A. Ved’ and L. N. Kitaeva).
Keywords and phrases:integrodifferential equation, a priori estimate, lower estimate, initial data,
instability.
AMS Subject Classification:53A40, 2015
Dedicated to the memory of Professor N. V. Azbelev and A. V. Chistyakov
We assume that all functions appearing below and their derivatives are continuous, all relations
holds for t≥t0,t≥τ≥t0,J=[t0,∞), and the abbreviation IDE means “integrodifferential
equation.”
As is known, the study of lower estimates of solutions and their derivatives to higher-order IDEs
of the Volterra type is one of the difficult branches of the asymptotic theory of such equations on the
semi-axis. In this paper, we consider the following problem.
Problem 1. Establish sufficient conditions that guarantee lower estimates and the tending to infinity
as t→∞of solutions and their derivatives up to the third order to the following fourth-order linear
IDE of the Volterra type:
x(4)(t)+a3(t)x (t)+a2(t)x (t)+a1(t)x(t)+a0(t)x(t)+
+
t

t0
[Q0(t, τ )x(τ)+Q1(t, τ )x(τ)+Q2(t, τ)x(τ)+Q3(t, τ )x(τ)]dτ =f(t),t≥t0.(1)
We consider solutions of IDE (1) of the class x(t)∈C4(J, R) with arbitrary initial data x(k)(t0),
k=0,1,2,3. Each solution of this type exists and is unique.
To our knowledge, this problem has not been studied earlier.
To solve this problem, we develop a method based on the nonstandard reduction method (S. Iskan-
darov), the Volterra transformation method, the method of shearing functions (S. Iskandarov), the
method of integral inequalities (Yu. A. Ved’ and Z. Pakhyrov), the method of a priori estimates
(N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina, P. M. Simonov, 1991, 2001), the Lagrange
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie
Obzory, Vol. 132,Proceedings of International Symposium “Differential Equations–2016,” Perm, 2016.
688 1072–3374/18/2305–0688 c
2018 Springer Science+Business Media, LLC
DOI 10.1007/s10958-018-3770-8
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