Asymptotic integration of impulsive differential equations

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.  


Sturm comparison theory for impulsive differential equations
Özbekler, Abdullah; Ağacık, Zafer; Department of Mathematics (2005)
In this thesis, we investigate Sturmian comparison theory and oscillation for second order impulsive differential equations with fixed moments of impulse actions. It is shown that impulse actions may greatly alter the oscillation behavior of solutions. In chapter two, besides Sturmian type comparison results, we give Leightonian type comparison theorems and obtain Wirtinger type inequalities for linear, half-linear and non-selfadjoint equations. We present analogous results for forced super linear and super...
Asymptotic integration of second-order impulsive differential equations
Akgol, S. D.; Zafer, Ağacık (2018-02-01)
We initiate a study of the asymptotic integration problem for second-order nonlinear impulsive differential equations. It is shown that there exist solutions asymptotic to solutions of an associated linear homogeneous impulsive differential equation as in the case for equations without impulse effects. We introduce a new constructive method that can easily be applied to similar problems. An illustrative example is also given.
Energy bounds for some nonstandard problems in partial differential equations
Özer, Özge; Çelebi, Okay; Department of Mathematics (2005)
This thesis is a survey of the studies of Ames,Payne and Schaefer about the partial differential equations with nonstandard auxiliary conditions; this is where the values of the solution are prescribed as a combination of initial time t=0 and at a later time t=T. The first chaper is introductory and contains some historical background of the problem,basic definitions and theorems.In Chapter 2 energy bounds and pointwise bounds for the solutions of the nonstandard hyperbolic problems have been investigated a...
Discontinuous dynamics with grazing points
Kıvılcım, Ayşegül; Akhmet, Marat; Department of Mathematics (2016)
The scope of this thesis is to investigate the periodic solutions of impulsive systems with grazing and modeling through differential equations with impulses. By means of differential equations with impacts, the system which is modeled through two distinct differential equations is taken into account and such models are named as models with impact deformations. The surfaces as well as the coefficient of restitution are determined to be dependent on the impact velocity. The simulations are obtained for the r...
Inverse problems for a semilinear heat equation with memory
Kaya, Müjdat; Çelebi, Okay; Department of Mathematics (2005)
In this thesis, we study the existence and uniqueness of the solutions of the inverse problems to identify the memory kernel k and the source term h, derived from First, we obtain the structural stability for k, when p=1 and the coefficient p, when g( )= . To identify the memory kernel, we find an operator equation after employing the half Fourier transformation. For the source term identification, we make use of the direct application of the final overdetermination conditions.
Citation Formats
S. Doğru Akgöl, “Asymptotic integration of impulsive differential equations,” Ph.D. - Doctoral Program, Middle East Technical University, 2017.