We study cosmological solutions in R + βR N -gravity for an isotropic Universe filled with ordinary matter with the equation of state parameter γ. Using the BogolyubovKrylov-Mitropol'skii averaging method we find asymptotic oscillatory solutions in terms of new functions, which have been specially introduced by us for this problem and appeared as a natural generalization of the usual sine and cosine. It is shown that the late-time behaviour of the Universe in the model under investigation is determined by the sign of the difference γ − γ crit where γ crit = 2N/(3N − 2). If γ < γ crit , the Universe reaches the regime of small oscillations near values of Hubble parameter and matter density, corresponding to General Relativity solution. Otherwise higher-curvature corrections become important at late times. We also study numerically basins of attraction for the oscillatory and phantom solutions, which are present in the theory for N > 2. Some important differences between N = 2 and N > 2 cases are discussed. *

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We construct a sequence that converges to a solution of the Cauchy problem for a singularly perturbed linear inhomogeneous differential equation of an arbitrary order. This sequence is also an asymptotic sequence in the following sense: the deviation (in the norm of the space of continuous functions) of its nth element from the solution of the problem is proportional to the (n + 1)th power of the parameter of perturbation. This sequence can be used for justification of asymptotics obtained by the method of boundary functions.Date: May 12, 2018. Key words and phrases. Singular perturbations, Banach fixed-point theorem, method of asymptotic iterations, method of boundary functions, Routh-Hurwitz stability criterion.Recall that for the fulfillment of the conditions (3) it is necessary (and for m ∈ {1, 2} is also sufficiently) that all a i (x) be negative.

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