*(English)*Zbl 0537.45006

The author studies the asymptotic behavior of solutions of the following integrodifferential equation

as a perturbation of the nonlinear differential equation ${x}^{\text{'}}=f(t,x)$, $t\in {R}_{+}$, $x\in {R}^{n}$, where $f:{R}_{+}\times {R}^{n}\to {R}^{n}$ is a continuously differentiable function and $g:{R}_{+}\times {R}^{n}\times {R}^{n}\to {R}^{n}$ and $h:{R}_{+}\times {R}_{+}\times {R}^{n}\to {R}^{n}$ 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 reviewerâ€™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 *V. M. Alekseev* [Vestnik Mosk. Univ., Ser. 16, No.2, 28-36 (1961; Zbl 0105.293)].

##### MSC:

45J05 | Integro-ordinary differential equations |

45M05 | Asymptotic theory of integral equations |

26D15 | Inequalities for sums, series and integrals of real functions |

34A34 | Nonlinear ODE and systems, general |

34E10 | Perturbations, asymptotics (ODE) |