Study On Mixed Monotone Operators With Concavity And Convexity And Impulsive Equations

The purpose of this paper is to discuss some class of fixed point theoremsof mixed monotone operators and some impulsive differential equations. Bymeans of partial ordering method, sub-solution sup-solution method and fixedpoint index theorem, some new conclusions are studied. The paper includesthree chapters.In chapter 1, we provide a research summary of several class of mixedmonotone operators. At the same time, we introduce briefly the main works inour papers.In chapter 2, we get the fixed point theorem of mixed monotone operatoron nonlinear noncompact condition, get some new conclusions. All conclusionswe get developed and generated the relevant conclusions.Theorem 2.1 discuss the t-α(t,u,v) mixed monotone operator A withoutinclude sub-solution and sup-solution condition, then we use sub-solutionsup-solution method in Banach space, get the unique fixed point in P_u0, and thenthe iterative converge sequence are given. At the same time, we get a newmethod of fixed point on weaker condition.Theorem 2.2 discussη-sequential contraction operators which deducefrom sequential contraction operators, using sub-solution sup-solution method,combine the condition ofη-sequential contraction operator, get the uniqueness of the fixed point, then the iterative converge sequence are given. At last, we usetwo methods to show the correctness of this theorem.Theorem 2.3 and theorem 2.4 discuss the mixed monotone operators withe-concavity and convexity, using sub-solution and sup-solution method inordered Banach space, combining the theorems of positive operator semigroupand their main characteristic, discusses the sufficient and necessary condition ofthe existence of nonlinear evolution equation has coupled periodic solution, andgive the iterative converge sequence of coupled periodic solution, provides amethod to judge if this class of evolution equation have coupled periodicsolution, recovers the limitation of only necessary condition.As application, we give their apply in differential equation and sufficientand necessary condition for the existence of coupled periodic solutions on n-thorder dynamical system.Chapter 3. We consider the second order impulsive boundary valueproblemThis equation not only have impulsive on function, but also have impulsive onderivative, and f have relationship on t,x,x'. Finally obtained the existenceinterval of two solution of this equation. And in the same way, we show theapplication of the theorem in real equation.