Limit periodic solutions of the Volterra integro-differential equations (Q2895478)

In the critical case of a pair of pure imaginary roots for an integro-differential equation of Volterra type, the question of the existence of limit-periodic solutions is investigated under the condition that the frequency of the periodic part of a limit-periodic small perturbation in the equation coincides with the eigenfrequency of the linearized homogeneous equation. It is shown that the integro-differential equation admits a family of limit-periodic solutions that are representable by power series in terms of the small parameter that characterizes the perturbation quantity and in terms of small arbitrary initial values of the noncritical variables of the problem. Existence conditions are proven for those solutions that are expressible by quantities that determine the Lyapunov constant \(g_3\) computable from terms of third-order equations.NEWLINENEWLINEPeriodic solutions of integro-differential equations of Volterra type with an infinite sequence (with lower limit \(-\infty\) in the integral terms) were considered in the monographs [1,2]. The construction of periodic solutions for equations of special type close to systems of equations of second order that emerge in mechanical problems was carried out in the works [3--5] both in the non-resonance and in the resonance case. The convergence of successive approximations in the constructions [3--5] is based on the method of majorant functions.NEWLINENEWLINEFor the entire collection see [Zbl 1236.70003].