Sufficient Conditions of Ergodicity of Solutions of Second Order Abstract Linear Differential Equations

Ярослав Володимирович Горбатенко


This paper is devoted to second order abstract linear differential equations in a Banach space. For such equations the Cauchy problem is stated, and the behavior of its solutions as \[t\rightarrow +\infty\] is examined. The aim of the paper is to study ergodicity and asymptotic behavior of the solutions of the strongly correct Cauchy problem. For this purpose the theory of complete second order linear differential equations in Banach spaces, developed by Fattorini, is used. As shown in the paper, for a wide class of equations the solutions are either ergodic or unbounded, depending on the initial values. For the solutions to be ergodic, conditions on the linear operators-coefficients of the differential equation and the initial values of the Cauchy problem are obtained. In case of ergodic solutions, exact values of ergodic limits are given. In case of unbounded solutions, asymptotic behavior of solutions is described. Results obtained in this paper are a generalization of the previously known results concerning ergodic properties of the solutions for the Cauchy problem for the incomplete second order equations.


Ergodicity; Asymptotic behavior; Banach space; Linear differential equations; Abstract Cauchy problem


E. Hille and R.S. Phillips, Functional Analysis and Semi-Groups. Providence: Amer. Math. Soc., 1957, 808 p.

Голдстейн Дж. Полугруппы линейных операторов и их приложения. – К.: Выща шк., 1989. – 348 с.

J.A. Goldstein et al., “Convergence rates of ergodic limits for semigroups and cosine functions,” Semigroup Forum, vol. 16, pp. 89–95, 1978.

S.-Y. Shaw, “Mean and pointwise ergodic theorems for cosine operator functions,” Math. J. Okayama Univ., vol. 27, is. 1, pp. 197–203, 1985.

R. Sato and S.-Y. Shaw, “Strong and uniform mean stability of cosine and sine operator functions,” J. Math. Anal. Appl., vol. 330, is. 2, pp. 1293–1306, 2007.

S.-Y. Shaw, “Growth order and stability of semigroups and cosine operator functions,” J. Math. Anal. Appl., vol. 357, is. 2, pp. 340–348, 2009.

Горбачук М.Л., Кочубей А.Н., Шкляр А.Я. О стабилизации решений дифференциальных уравнений в гильбертовом пространстве // Докл. акад. наук. – 1995. – 341, № 6. – С. 734–736.

Горбачук М.Л., Шкляр А.Я. О поведении на бесконечности решений дифференциальных уравнений в гильбертовом пространстве // Тр. семинара им. И.Г. Петровского. – 1996. – Вып. 19. – С. 174–201.

Горбатенко Я.В. Ергодичність розв’язків абстрактних лінійних диференціальних рівнянь другого порядку в банаховому просторі // Доп. НАНУкраїни – 2010. – № 9. – С. 10–19.

H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces. Amsterdam: Elsevier Science Publishers B.V., 1985, 314 p.

T.J. Xiao and J. Liang, “On complete second order linear differential equations in Banach spaces,” Pacific J. Math., vol. 142, is. 1, pp. 175–195, 1990.

Горбатенко Я.В. Розв’язність і сильна коректність задачі Коші для абстрактних лінійних диференціальних рівнянь у банахових просторах // Наук. вісті НТУУ “КПІ”. – 2010. – № 4. – С. 40–43.

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