Levin- and Lyapunov-Type Inequalities Tầm quan trọng của sự bất bình đẳng so sánh cơ bản đã được công nhận từ lâu trong việc nghiên cứu hành vi của chất lượng các giải pháp của phương trình vi phân thông thường thứ hai để. Lịch sử của những sự bất bình đẳng cho các hệ thống khác biệt giữa các liên tục đi xa trở lại bắt đầu với các bài báo nổi tiếng của Sturm [414] trong đó cung cấp cho sự bất bình đẳng cho các số không của các giải pháp của phương trình vi phân. | Chapter 5 Levin- and Lyapunov-Type Inequalities Introduction The importance of basic comparison inequalities has been long recognized in the study of qualitative behavior of solutions of ordinary second-order differential equations. The history of these inequalities for continuous differential systems goes far back starting with the famous paper of Sturm 414 which gives inequalities for the zeros of solutions of linear second-order differential equations. In his fundamental work 201 Lyapunov has given one of the most basic and inspiring inequalities which provides a lower bound for the distance between consecutive zeros of the solutions of the linear second-order differential equation. Lyapunov s inequality has become a versatile tool in the study of qualitative nature of solutions of ordinary second-order differential equations. Over the years there have appeared a number of generalizations extensions variants and applications related to the basic Sturmain comparison theorem and the original Lyapunov inequality. This chapter considers basic inequalities developed in the literature related to the Sturmain comparison theorem and to the Lyapunov inequality which occupies a fundamental place in the theory of ordinary differential equations. Inequalities of Levin and Others In this section we give results involving comparison of the solutions of linear and nonlinear second-order differential equations investigated by Levin 187 Kreith 171 and Ladas 177 . Here we consider only solutions which are defined on the whole interval of definition of the independent variable and their existence and uniqueness will be assumed without further mention. An oscillatory solution is by definition one which has arbitrary large zeros. 485 486 Chapter 5. Levin- and Lyapunov-Type Inequalities The basic Sturmain comparison theorem deals with functions u x and v x satisfying u c x u 0 v Y x v 0. If Y x f c x then solutions of oscillate more rapidly than solutions of
