The Comparison Theorems Of Dynamic Equations On Time Scales And Its Applications

In this dissertation, firstly, we establish three comparison theorems of the solutions for dynamic equations with or without impulses on time scales, and we introduce a new definition of rd-piecewise continuous almost periodic functions; secondly, we propose two Lyapunov-type theorems of impulsive dynamic equation-s on time scales:then, as applications of these theorems, we study the dynamic properties of three types of population systems, respectively. By use of compari-son theorems of the solutions for dynamic equations on time scales, the hull theory of almost periodic functions and Lyapunov functional method, we obtained some sufficient conditions ensuring the permanence, the existence and global attractivi-ty of almost periodic solutions for multispecies Lotka-Volterra mutualism systems with time-varying delays on time scales. Besides, based on the impulsive theory on time scales, by means of the comparison theorems of the solutions for impulsive dynamic equations on time scales and Lyapunov functional theorems, we establish some sufficient conditions on the permanence, the existence and stability of almost periodic solutions for impulsive single-species systems and impulsive multispecies Lotka-Volterra competition systems with time delays on time scales. Our results are innovative even if the time scale T= R or Z. Finally, we present numerical examples to illustrate the feasibility of our theoretical results.