The process of searching for an approximate solution of an operator equation by the averaging method with a result
concretized in delay differential equations is studied using functional analysis in semiordered spaces. The majorant
method is used to determine sufficient conditions for the existence of a unique solution, process convergence, as
well as the approximation element and point-by-point error estimates. The use of functional analysis in semiordered
spaces makes it possible to weaken the requirements to the method applicability and to correct the error estimate.
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