The Global Stability Of The N-dimensional System

In the paper, we study the global stability for n-dimensional cooperative system and we introduce the theory of monotone flow into the systems and obtain thefollowing theorem: Theorem A: let F :X Rn be a C1 cooperative vector field,suppose the following conditions hold : (a) X = Rnor intRn+ or [[p,q]]; (b) every forward semi-orbit has compact closure in X ; (c) there is not more than one equilibrium p ; then there is a unique equilibrium p and it is globally asymptoticallystable. We extend this result which is similar to ones of Hirsch[1l] and Jiang[2] for 3-dimensional cooperative system.Furthermore, in the paper we discusse the cone of Km, and obtain the similar result: Theorem B: let F be a continuously differentiable cooperative vector field on D, suppose the following conditions hold : (a)D is pm -convex; (b) everyforward semi-orbit has compact closure in D ; (c) there is not more than one equilibrium p, then there is a unique equilibrium p and it is globally asymptoticallystable .The paper divide into three chapters. The first chapter gives the origin of the problem and notation and some definitions. The second chapter discusses and provesthe Theorem A. The third chapter introduces the cone of Km and proves the Theorem B.