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

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2014
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.

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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.