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On asymptotic behaviour of certain integro-differential equations. (English) Zbl 0537.45006
The author studies the asymptotic behavior of solutions of the following integrodifferential equation $$ (1)\quad y'=f(t,y)+g(t,y,\int\sp{t}\sb{0}h(t,s,y)ds),\quad t\in R\sb+,\quad y\in R\sp n, $$ as a perturbation of the nonlinear differential equation $x'=f(t,x)$, $t\in R\sb+$, $x\in R\sp n$, where $f:R\sb+\times R\sp n\to R\sp n$ is a continuously differentiable function and $g:R\sb+\times R\sp n\times R\sp n\to R\sp n$ and $h:R\sb+\times R\sb+\times R\sp n\to R\sp n$ are continuous functions, $f(t,0)=g(t,0,0)=h(t,s,0)\equiv 0$. The results established in this paper give sufficient conditions which yield the boundedness and asymptotic behavior of solutions of (1). For similar results, see the reviewer’s paper [J. Math. Anal. Appl. 51, 550-556 (1975; Zbl 0313.34047)]. The main tools employed to establish the results are the two generalizations of the integral inequality established by the reviewer [ibid. 49, 794-802 (1975; Zbl 0305.26009)] and the nonlinear variation of constants formula due to {\it V. M. Alekseev} [Vestnik Mosk. Univ., Ser. 16, No.2, 28-36 (1961; Zbl 0105.293)].
Reviewer: B.G.Pachpatte

45J05Integro-ordinary differential equations
45M05Asymptotic theory of integral equations
26D15Inequalities for sums, series and integrals of real functions
34A34Nonlinear ODE and systems, general
34E10Perturbations, asymptotics (ODE)
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