On the Limiting Periodic Solutions of Integral Differential Volterra Equations and their Stability

Consideration was given to the question of asymptotic (exponential) stability of the maximum periodic solutions of the integrodifferential equations which have an asymptotically stable linear part and small periodic (exponential maximum periodic) perturbation. Under the unlimitedly increasing time, these solutions tend to the periodic modes. The sufficient conditions for asymptotic stability were indicated. In the resonance case where the linearized equation has a pair of purely imaginary roots with the corresponding oscillation frequency coinciding with the oscillation frequency of the periodic part of small perturbation (time function) and the coefficients of the power series expansion of the nonlinear terms, consideration was given to the problem of existence for the maximum periodic solutions of the integrodifferential equation. Conditions were established for existence of such solutions representable by the power series in the fractional degrees of the small parameter characterizing the value of small perturbation in the equation.