Asymptotic integration of certain Volterra integro-differential equations with oscillatory decreasing kernels (Q2116180)

The paper studies the dynamics of solutions of some Volterra integro-differential equations. The author constructs their asymptotic representations for sufficiently large values of the independent variable \(t\). While obtaining the formulas, he uses the method proposed for the asymptotic integration of the linear dynamical systems containing oscillatory decreasing coefficients. Note that the integrations \( \int t^{- \rho} dt\) in the asymptotic solution of the equations are replaced by \[ \dfrac{t^{1 -\rho}}{1-\rho} (1+o(1)).\] The study shows that the asymptotics for the solutions of the equations as \(t \rightarrow \infty \) are not in the usual form and the dynamics of the solutions of the equations are not always defined by the fundamental solutions of the harmonic oscillator. The author analyses the formulas of the equations and discusses the qualitative and quantitative differences of the asymptotic solutions.