tailieunhanh - Báo cáo hóa học: "BOUNDEDNESS IN FUNCTIONAL DYNAMIC EQUATIONS ON TIME SCALES"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: BOUNDEDNESS IN FUNCTIONAL DYNAMIC EQUATIONS ON TIME SCALES | BOUNDEDNESS IN FUNCTIONAL DYNAMIC EQUATIONS ON TIME SCALES ELVAN AKIN-BOHNER AND YOUSSEF N. RAFFOUL Received 1 February 2006 Revised 25 March 2006 Accepted 27 March 2006 Using nonnegative definite Lyapunov functionals we prove general theorems for the boundedness of all solutions of a functional dynamic equation on time scales. We apply our obtained results to linear and nonlinear Volterra integro-dynamic equations on time scales by displaying suitable Lyapunov functionals. Copyright 2006 E. Akin-Bohner and Y. N. Raffoul. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction In this paper we consider the boundedness of solutions of equations of the form xA t G t x s 0 s t G t x - on a time scale T a nonempty closed subset of real numbers where x e R and G 0 to X R R is a given nonlinear continuous function in t and x. For a vector x e R we take xh to be the Euclidean norm of x. We refer the reader to 8 for the continuous case that is T R. In 6 the boundedness of solutions of xA t G t x t x t0 x0 t0 0 x0 e R is considered by using a type I Lyapunov function. Then in 5 the authors considered nonnegative definite Lyapunov functions and obtained sufficient conditions for the exponential stability of the zero solution. However the results in either 5 or 6 do not apply to the equations similar to c t x a t x j B t s f x s As Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 79689 Pages 1-18 DOI ADE 2006 79689 2 Boundedness in functional dynamic equations on time scales which is the Volterra integro-dynamic equation. In particular we are interested in applying our results to with f x xn where n is positive and rational. The authors are confident that there is nothing in the literature that deals with the qualitative analysis of Volterra .