Asymptotic integration of impulsive differential equations

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2017
Doğru Akgöl, Sibel
The main objective of this thesis is to investigate asymptotic properties of the solutions of differential equations under impulse effect, and in this way to fulfill the gap in the literature about asymptotic integration of impulsive differential equations. In this process our main instruments are fixed point theorems; lemmas on compactness; principal and nonprincipal solutions of impulsive differential equations and Cauchy function for impulsive differential equations. The thesis consists of five chapters. In Chapter ref{intro}, the statement of the problem and a review of related literature are given. Chapter ref{pre} contains preliminary concepts about impulsive differential equations and necessary theorems from functional analysis. Moreover, it provides characterizations of principal and nonprincipal solutions of impulsive differential equations with continuous solutions, and new results about existence of principal and nonprincipal solutions for impulsive differential equations with discontinuous solutions. In Chapter ref{viaprinc}, new results, stating asymptotic properties of the solutions, are expressed via principal and nonprincipal solutions. In the first section, impulsive differential equations with discontinuous solutions are considered. By dividing these equations into two groups according to the type of impulse effects, various asymptotic representations for the solutions of each group are given. Moreover, sufficient conditions for existence of positive and monotone solutions are obtained. A new lemma consisting of compactness criteria for sets of piecewise continuous functions is also presented. The second section is devoted to impulsive differential equations with continuous solutions, and for these equations, both analogous results to previous theorems and a general asymptotic formula depending on a parameter is obtained. Several subsidiary examples are placed at the end of the chapter. In Chapter ref{viacauchy}, asymptotic representation of solutions for impulsive differential equations with discontinuous solutions is produced with the help of Cauchy functions, and an example is presented. The last chapter contains a summary of the thesis and suggests some open problems for further studies.  

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Citation Formats
S. Doğru Akgöl, “Asymptotic integration of impulsive differential equations,” Ph.D. - Doctoral Program, Middle East Technical University, 2017.