Geodesic
Justification of the Averaging Method for Differential Equations with Large Rapidly Oscillating Summands and Boundary Conditions
Matematičeskie zametki, Tome 100 (2016) no. 1, pp. 94-108

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The averaging method is justified for normal systems of differential equations with rapidly oscillating summands proportional to the square root of the oscillation frequency in the case of the boundary-value problem on a finite interval and for the problem of bounded solutions on the positive semiaxis with boundary condition at its left endpoint.
Keywords: averaging method, boundary-value problem, problem on the semiaxis, rapidly oscillating summands. 
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     author = {V. B. Levenshtam and P. E. Shubin},
     title = {Justification of the {Averaging} {Method} for {Differential} {Equations} with {Large} {Rapidly} {Oscillating} {Summands} and {Boundary} {Conditions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {94--108},
     publisher = {mathdoc},
     volume = {100},
     number = {1},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2016_100_1_a6/}
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V. B. Levenshtam; P. E. Shubin. Justification of the Averaging Method for Differential Equations with Large Rapidly Oscillating Summands and Boundary Conditions. Matematičeskie zametki, Tome 100 (2016) no. 1, pp. 94-108. http://geodesic.mathdoc.fr/item/MZM_2016_100_1_a6/