The study focuses on analyzing a multi-frequency system

\begin{equation*}
\frac{da}{d\tau} = X_0(\tau,a) + \varepsilon X_1(\tau,a,\varphi), \\\
\end{equation*}
\begin{equation*}
\frac{d\varphi}{d\tau} = \frac{\omega(\tau,a)}{\varepsilon} + Y_1(\tau,a,\varphi),
\end{equation*}

subjected to multi-point conditions within the subinterval [0, L].

The complexity of the study of multi-frequency systems is the resonance phenomena inhe \linebreak rent in them, which consist in the rational or almost rational commensurability of frequencies. As a result, the solution of the system of equations averaged over fast variables in the general case may deviate from the solution of the exact problem by the amount O(1).

An asymptotic solution of a system of linear differential equations with a small parameter at some derivatives and coefficients dependent on this parameter is constructed by reducing this system to a simpler form by averaging over fast variables $\phi$. The question of existence and construction of the global solution of systems of differential equations with a deviating argument and a small parameter are investigated. Employing the method of oscillatory integrals estimation, it was demonstrated that solutions for both the averaged and exact problems exhibit unity within small neighborhoods. An unenhanced approximation of the averaging error, the order of which i distinctly reliant on a minor parameter, has been derived. The obtained outcome is further elucidated through the presentation of a model example.

The acquired findings and methodologies have the potential to further the understanding of multifrequency systems with delayed arguments, aiding in the formulation of theories. Additionally, they can contribute to the creation of asymptotic and comprehensive solutions for singularly perturbed sets of differential equations. These outcomes hold significance in advancing mathematical models for oscillatory processes under conditions of frequency resonance, potentially impacting various domains.

[1] Samoilenko A.M., Petryshyn R.I. Mathematical aspects of the theory of nonlinear oscillations. - K.: Scientific opinion, 2004. – 474 p.
[2] Bihun Ya.Y. Existence of solution and averaging of multipoint boundary problems for multifrequency systems with a linearly transformed argument. Nonlinear Oscillations 2008, 11(4), 462-471
[3] Bihun Ya.Y., Skutar I.D. Averaging in multi-frequency systems with delay and locally integral conditions. Bukovinsk Mathematical Journal. 2020. Vol. 8, № 2. P. 14–23.
[4] Bihun Ya.Y., Petryshyn R.I., Krasnokutska I.V. Averaging in multi-frequency systems with linearly transformed arguments and multi-point and integral conditions. Bukovinsky Mathematical Journal. 2016. Т.4, No 3. С. 30-35.
[5] Harry Bateman; Arthur Erdelyi; Higher transcendental functions. New York : McGraw-Hill, 1953. Vol. II. 316 p.