The Study Of Anti-Periodic Solutions And Anti-Periodic Boundary Value Problems For First-order Evolution Equations Associated With Maximal Monotone Mappings

With the development of science and technology, the nonlinear problems, existing in the field of natural science and social science extensively, have attracted the attention of people,and the research of many nonlinear problems eventually boil down to the nonlinear evolution equation.The nonlinear evolution equations have a very important role in many fields,and it is a hot research subject to prove the existence of anti-periodic solutions, there are many current international and domestic scholars engaging in the research of this aspect. However,because the nonlinear theory is extremely complex and the principle of superpostion is not established, it is difficult to prove the existence of anti-periodic solutions for nonlinear evolution equations.Especially, when a first-order nonlinear evolution equations contain the maximal monotone mappings, because not all the conditions of the maximal monotone mappings exist anti-periodic solutions, it is quite difficult to study the existence of anti-periodic solutions and anti-periodic boundary value problems.Moreover, the current research in this area is not a lot,so it is quite necessary to study the existence of anti-periodic solutions for nonlinear evolution equations associated with maximal monotone mappings.As is known to us, periodicity is a common phenomenon in nature,so are the nonlinear evolution equations.Many experts and scholars have studied the periodic solutions about nonlinear evolution equations for a long time,and got many useful conclusions.The research is also quite perfect and tend to be mature day after day.The research of anti-periodic solutions for nonlinear evolution equations is closely related to the study of periodic solutions. The scholar H Okochi in Japan gave an example of the parabolic equation that didn't exist a periodic solution, but had an anti-periodic solution. This was the origin of the existence of anti-periodic solutions. Then further researchs were carried out. By means of the methods of studying periodic solutions,other scholars studied the anti-periodic solutions,and made a lot of meaningful results.Now the anti-periodic problems have been widely used in physics, biology, and medicine and so on. In this paper, we mainly study the existence probem of anti-periodic solutions and anti-periodic boundary value problems for first-order evolution equations associated with maximal monotone mappings in a real Hilbert space. This paper contents three chapters.In the first chapter is an introduction part.We introduce in detail the concept of nonlinear evolution equations, classification, general form and research background, as well as the development of anti-periodic solutions for nonlinear evolution equations and the main contents of this paper.In the second chapter, we mainly introduce some important conclusions and theorems about maximal monotone mappings and anti-periodic solution problems.In the third chapter,we mainly study the existence of anti-periodic solutions for first-order nonlinear evolution equations associated with maximal monotone mappings in a real Hilbert and their application. Anti-periodic solutions for first-order nonlinear evolution equations associated with maximal monotone mappings in Hilbert,are worked out and further research about Okochi's result is carried out.In the fourth chapter, we mainly study the anti-periodic boundary value problems for first-order semilinear evolution equations associated with maximal monotone mappings in a real Hilbert. Using by Leray-Schauder's topology degree theory in nonlinear analysis, the existence of anti-periodic boundary value problems for first-order semilinear evolution equations associated with maximal monotone mappings in Hilbert space are considered, and further research about the results that have been obtained are carried out.