ANALYTICAL SOLUTION OF INITIAL VALUE PROBLEM FORORDINARY DIFFERENTIAL EQUATION WITH SINGULARPERTURBATION AND PIECEWISE CONSTANT ARGUMENT

Abstract

The study of differential equations with piecewise constant arguments has been treated widely inthe literature. This type of equation, in which techniques of differential and difference equationsare combined, models, among others, some biological phenomena , the stabilization of hybridcontrol systems with feedback discrete controller or damped oscillators.The first studies in thisfield have been given in 1984, after this, some papers related with stability, oscillation propertiesand existence of periodic outcomes have been treated by several authors.The manuscript is craftedas follows: Section 2 outlines the primary methodologies adopted throughout the inquiry.Section3 is dedicated to obtaining the exclusive outcome to the issue. We formulate a series of differenceequations overseeing the vector(y(θi)y′(θi)), i=1,pwhich portray the constituents of theoutcome. This generalized approach allows for a broader understanding of how to tackle suchdifferential equations across various scenarios. These equations now form a recognized branch ofthe field of differential equations, and they are frequently used in biological and economic models.Undoubtedly, their applications will continue to increase in the future.