The author studies the asymptotic behavior of solutions of the following integrodifferential equation
as a perturbation of the nonlinear differential equation , , , where is a continuously differentiable function and and are continuous functions, . 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 V. M. Alekseev [Vestnik Mosk. Univ., Ser. 16, No.2, 28-36 (1961; Zbl 0105.293)].