Lyapunov type inequalities and their applications for linear and nonlinear systems under impulse effect

Kayar, Zeynep
In this thesis, Lyapunov type inequalities and their applications for impulsive systems of linear/nonlinear differential equations are studied. Since systems under impulse effect are one of the fundamental problems in most branches of applied mathematics, science and technology, investigation of their theory has developed rapidly in the last three decades. In addition, Lyapunov type inequalities have become a popular research area in recent years due to the fact that they provide not only better understanding of the qualitative nature of the solutions of ordinary and impulsive systems, for instance oscillation, disconjugacy, stability and asymptotic behavior of solutions, but also deeper analysis for boundary and eigenvalue problems. This thesis consists of 7 chapters. Chapter 1 is introductory and contains detailed literature review, and brief information about the linear systems of impulsive differential equations and Hamiltonian systems. The main contributions of the thesis, which are presented in the second and third chapters, are to derive Lyapunov type inequalities for the linear 2nx 2n Hamiltonian system with impulsive perturbations and to prove the existence and uniqueness criteria for the solutions of inhomogenous boundary value problems to such systems, respectively. Since changing the impulsive perturbation or assuming different conditions on the impulses leads to different inequalities, presence of the impulse effect provides various Lyapunov type inequalities. This shows that the systems of impulsive equations is richer and more fruitful in the applications than the systems of ordinary differential equations and that is why we are interested in these systems. Besides, the obtained inequalities are new even in the nonimpulsive case and therefore they improve and generalize the previous ones existing in the literature. In Chapter 3, the connection, which has not been noticed even for the nonimpulsive case, between Lyapunov type inequalities and boundary value problems has been revealed for the first time and two existence and uniqueness criteria for the solutions of inhomogenous BVPs are proved by using the Lyapunov type inequalities obtained in the previous chapter. Furthermore, the unique solution of inhomogenous BVPs is expressed in terms of Green’s function (pair) and the properties of Green’s function (pair) are listed. Chapter 4 is devoted to the stability theory, which is the application of Lyapunov type inequalities, for the linear planar Hamiltonian systems with impulsive perturbations. Two pairs of stability criteria are obtained, one of which is the generalization of the results obtained for systems of ordinary differential equations to the impulsive case and the latter is new and alternative to the former. In Chapter 5 and 6, we establish several Lyapunov type inequalites, some of which are generalizations of the nonimpulsive case while the others are new for nonlinear and quasilinear impulsive systems, respectively. As an application of Lyapunov type inequalities, we investigate disconjugacy intervals and study the asymptotic behaviour of oscillatory solutions for the systems under considerations and find a lower bound for the eigenvalues of the associated eigenvalue problems. The last chapter serves as a conclusion and is a summary of our findings.


Asymptotic integration of impulsive differential equations
Doğru Akgöl, Sibel; Ağacık, Zafer; Özbekler, Abdullah; Department of Mathematics (2017)
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....
Boundary value problems for higher order linear impulsive differential equations
Uğur, Ömür; Akhmet, Marat (2006-07-01)
In this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied. Green's formula is defined for piecewise differentiable functions. Properties of Green's functions for higher order impulsive boundary value problems are introduced. An appropriate example of the Green's function for a boundary value problem is provided. Furthermore, eigenvalue problems and basic properties of eigensolutions are considered. (c) 2006 Elsevier In...
Lyapunov-type inequalities for nonlinear impulsive systems with applications
Kayar, Zeynep; Zafer, Agacik (University of Szeged, 2016)
We obtain new Lyapunov-type inequalities for systems of nonlinear impulsive differential equations, special cases of which include the impulsive Emden-Fowler equations and half-linear equations. By applying these inequalities, sufficient conditions are derived for the disconjugacy of solutions and the boundedness of weakly oscillatory solutions.
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...
Two studies on backward stochastic differential equations
Tunç, Vildan; Sezer, Ali Devin; Department of Financial Mathematics (2012)
Backward stochastic differential equations appear in many areas of research including mathematical finance, nonlinear partial differential equations, financial economics and stochastic control. The first existence and uniqueness result for nonlinear backward stochastic differential equations was given by Pardoux and Peng (Adapted solution of a backward stochastic differential equation. System and Control Letters, 1990). They looked for an adapted pair of processes {x(t); y(t)}; t is in [0; 1]} with values i...
Citation Formats
Z. Kayar, “Lyapunov type inequalities and their applications for linear and nonlinear systems under impulse effect,” Ph.D. - Doctoral Program, Middle East Technical University, 2014.