Dynamic Analysis Of Essentially Stongly Order-Preserving Semiflows

In this thesis, we mainly study several properties on essentially strongly order-preserving semi?ows (essentially SOP semi?ows for short) and eventually essentiallystrongly monotone semi?ows of the ordered metric space from two aspects. On theone hand, we present and prove some results for the existence of a stable equilibriumand an asymptotically stable equilibrium, which extend some previous results on SOPsemi?ows to essentially SOP semi?ows. On the other hand, for the eventually essen-tially strongly monotone dynamical systems, global convergence is obtained under thecondition of the set of equilibrium points possessing minimal or stable property, whichextend some previous results on eventually strongly monotone semi?ows to eventuallyessentially strongly monotone semi?ows.This thesis is composed of three chapters.In the first chapter, the background, significance and current situation of essen-tially strongly order-preserving semi?ows we consider are brie?y reviewed. Further-more, we simply introduce the main research thingking in this thesis.In the second chapter, the related definition and basis knowledge of essentiallySOP semi?ows are firstly introduced. And then we present some stability properties ofessentially SOP semi?ows in normally ordered spaces. Then for the limitation of therequirement of normality in practical application, by introducing the notion of ordernorm and the new ordered topology space induced by the order norm, the stronglyordered Banach space become associated with the normal space. In this way, theresults we have got can generalized to strongly ordered Banach space. At the last ofthis chapter, su?cient conditions to ensure finiteness of the set of equilibrium pointsare established. Furthermore, some examples are given to illustrate the importance ofanalysis on solution semi?ows.In the third chapter, combining with the study idea given by current referenceswhich provide the concept of essentially strongly monotone semi?ows and eventuallyessentially strongly monotone semi?ows related to essentially SOP semi?ows, we brie?yintroduce the notion of the set of equilibrium points which possess a minimal property,obtain a series of consequences of essentially strongly monotone dynamical systemswhose set of equilibrium points possess a minimal property or stability. That in turnobtains global convergence of the systems, complements and completes the relatedresults of previous documents.