On a Method of Study of Specific Asymptotic Stability of Solutions to a Sixth-Order Linear Integrodifferential Volterra Equation

Abstract

We state sufficient conditions of the asymptotic stability on the semi-axis of solutions to a linear, homogeneous, sixth-order integrodifferential Volterra-type equation in the case where solutions of the corresponding linear, homogeneous, sixth-order differential equation are asymptotically unstable. We also present a new method and an illustrating example.

Full text

Journal of Mathematical Sciences, Vol. 230, No. 5, May, 2018
ON A METHOD OF STUDY OF SPECIFIC ASYMPTOTIC
STABILITY OF SOLUTIONS TO A SIXTH-ORDER LINEAR
INTEGRODIFFERENTIAL VOLTERRA EQUATION
S. Iskandarov UDC 517.968.74
Abstract. We state sufficient conditions of the asymptotic stability on the semi-axis of solutions to a
linear, homogeneous, sixth-order integrodifferential Volterra-type equation in the case where solutions
of the corresponding linear, homogeneous, sixth-order differential equation are asymptotically unstable.
We also present a new method and an illustrating example.
Keywords and phrases:integrodifferential equation, specific asymptotic stability, method of squar-
ing, method of shearing functions, Lyusternik–Sobolev lemma.
AMS Subject Classification:53A40, 20M15
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.” The asymptotic stability of a solution to a sixth-order linear IDE means that the solution
and their derivatives up to the fifth order tend to zero as t→∞.
Problem. Establish sufficient conditions of asymptotic stability of any solution to a sixth-order linear
homogeneous Volterra-type IDE of the form
x(6)(t)−a5(t)x(5) (t)+a4(t)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)+Q4(t, τ )x(4)(τ)
+Q5(t, τ )x(5)(τ)]dτ =0,t≥t0,(1)
under the condition a5(t)≥0 i.e., in the case where any nonzero solution of the differential equation
x(6)(t)−a5(t)x(5) (t)+a4(t)x(4)(t)+a3(t)x(t)+a2(t)x (t)+a1(t)x(t)+a0(t)x(t)=0,
t≥t0, is not asymptotically stable, which follows from the Ostrogradsky–Liouville formula.
We call this problem the specific problem; it has not been examined earlier.
Asolutionx(t)∈C6(J, R) of the IDE (1) exists for arbitrary initial data xk(t0), k=0,1,2,3,4,5,
and is unique.
A method of solution of the problem stated above is as follows. We perform the following nonstan-
dard substitution in the IDE (1) (see [5]):
x(4)(t)+p3x (t)+p2x (t)+p1x(t)+p0x(t)=W(t)y(t),(2)
where pkare certain auxiliary parameters, pk>0, k=0,1,2,3, 0 <W(t) is a weight function, and
y(t) is the new unknown function.
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.
1072–3374/18/2305–0683 c
2018 Springer Science+Business Media, LLC 683
DOI 10.1007/s10958-018-3769-1
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