I.The Basic Information of the Course
Course Number:202012420017
The EnglishName of theCourse：Nonlinear Analysis and Application
The ChineseName of theCourse：非线性分析及其应用
In-classHours andAllocationTotal class hours:32, classroom teaching:20classhours, classroom discussion:6classhours,6 classhours on computer, etc.
Credit(s):2
Semester:1
AppliedDiscipline(ProfessionalDegreeCategory)：Mathematics
CourseObjectOriented：Academic Master and AcademicDoctor
EvaluationMode:Big Project
TeachingMethod:Seminar-style Teaching
CourseOpening Department:School of Mathematical Science
II.Prerequisite Course
Function of Real Variable, Functional Analysis and Topology,etc.
III. The Objectives and Requirements of the Course
The aim of the course is to cultivate students'
The course isaprofessional elective coursesfor mathematics and applied mathematics, the aim of the course is to cultivatetheoretical thinking ability.of the students; grasp the basic thoughts, theories, methods and skills of nonlinear functional analysis, and cultivate students' understanding of the theory of nonlinear analysis. Possess strong abstract thinking ability, logical reasoning ability and the ability to solve some nonlinear problems encountered in scientific research, thereby forming a strict and accurate mathematical literacy, which will lay a certain foundation for the further study of mathematics related disciplines and the teaching of mathematics disciplines theoretical basis.
By studying this course, students should meet the following basic requirements:
(1)Thoroughly understand the basic knowledge in the book, pay attention to flexible application.
(2)Pay attention to abstract thinking ability and logical reasoning ability, and require some theoretical proofs.
(3)Ability to consult relevant literature at home and abroad to understand relevant theories.
IV. The Content of the Course
Graspthe current main research methods of nonlinear analysis: critical point theory, topological degree, Morse index and bifurcation methods; students need to master the differences and connections of various research methods as much as possible, and use appropriate research techniques to solve different practical application problemsrelatedproblems. The details are as follows:
Chapter 1Differentiation of Banach Spaces(8classhours)
Thissectionmainly introduces the Frechet differential and Gateaux differential of nonlinear operators, gives the implicit function theorem and inverse function theorem and some applications, and briefly introduces thebifurcationtheory. Specifically, understand the concept of boundedness and continuity of nonlinear mapping,graspthe differential theory of nonlinear mapping, understand the conditions and application scope of compact continuous mapping and implicit function theorem.
Ideological and political courses: The continuity of mapping and fable:“Pulling Up the Seedlings to Help Them Grow”. In life, many changes are continuous, such as the growth of plants, the change of temperature, the accumulation of knowledge, etc., cannot be eager to achieve success, and must follow its original law. Such as learning, the accumulation of knowledge requires time and enduring unremitting efforts. The idea of vainly seeking shortcuts is unscientific and can only be counterproductive. The ancients used story metaphors to promote the violation of the objective laws of the development of things, eager for success, but bad things. The continuity of the mapping also confirms this principle.
Chapter2Topological degree theory（10 classhours）
The Brouver degree of continuous mapping in finite dimensional space and Leray-Schauder degree of fully continuous field in Banch space are established, and some important fixed point theorems are given. Specifically, understand the definition of Brouwer degree, grasp the nature of Brouwer degree, understand the definition of Leray-Schauder degree, and grasp the nature of Leray-Schauder degree, and be able to use fixed point theorem to solve some classic analysis problems.
Ideological and political courses: introduced by two examples:
(1)Open a map of the world in one place. There is a point on it. Its position on the map overlaps its actual position.
(2)Imagine a spherical hedgehog with fur on its surface. Now, if you comb all the hairs, that is, map the points on the sphere to another point, then there is one hair that cannot be turned down.
The above example shows that there is always a permanent part in the phenomenon of dynamic change. That is, no point. When explaining the principle of Banach compression image, students will not only continuously experience the perfection of the construction proof process, but also think about the eternal truth. For example,
(1)If life is a functional relation (y = f (x)), then most people will play the role of x and y. they are fickle and have countless changeable characters. How many people will do the eternal x, that is, x= f (x);
(2)When we talk about fixed points, we can use the verse of Zhi-Zhang He, a poet of Tang Dynasty: "I left home young and not till old do i come back, my accent is unchanged,my hair no
longer black.", Shen Yang of Ming Dynasty "Green hills and wide rivers remain the same, how many times has the setting sun spread its flame!", Hu Cui of Tang Dynasty "Now her face is nowhere to be seen at all.Only the peach blossoms still smile in the spring breeze" .
Chapter3Variational principle（6 classhours）
It mainly introduces the classical variational method, functional extremum and gradient, minimum sequence method, Ekeland variational principle, steepest descent method and so on. Specifically, understand some classic extreme-value problems, master two deformation lemmas, maximum-minimum principles, and index theory, and use these methods to solve some nonlinear eigenvalue problems.
Ideological and Political Courses: In learning the variational method, teachers can draw out the dialectical relationship between "process and result". Variational method is an important knowledge point in nonlinear analysis, and its derivation process is very complicated. Students often only pay attention to the energy functional results, but ignore the derivation process, but the derivation process is the core of the knowledge point. Teachers should let students understand that the process of struggle is far more important than the final result, let students learn how to face success and failure, and learn the scientific spirit of seeking truth from facts.
Chapter4Minimax principle（8 classhours）
It mainly introduces the deformation lemma, the minimax principle, the mountain lemma and the orbit. Specifically, master the proofs of mountain theorem, fountain theorem, dual mountain theorem and orbital theorem, analyze the conditions for the establishment of these theorems, and proficiently apply some mathematical physics partial differential equations to solve problems.
Ideological and political courses: Northern Song Dynasty, Su Shi’s verse: "It's a rangeviewed in face and peaks viewed from the side, assuming different shapes viewed from far and wide. Of mountain Lu we cannot make out the true face, for we are lost in the heart of the very place.”It depicts the different appearance of Lu Mountain with different angles of observers. In the minimax principle, the knowledge point of functional extreme value, which is drawn after the combination of number and shape, is like the mountain of Lushan Mountain. The maximum value is obtained at the top of the mountain, and the minimum value appears in the valley. And also giveverse"Written on the Wall at West Forest Temple" to show theconcept of extreme value, this technique will bring a touch of poetry and painting to the abstract mathematics class.When explaining the knowledge of extreme value, we should not only teach students to seek the extreme value and extreme value of function, but also let them realize that life is like a continuous curved surface. Ups and downs are the only way to go. They are the needs of growth. They are not discouraged when they fall into the trough, willing to be plain and not indulged, and they are not open when they stand at the peak. This is called broad-minded. To learn to look at problems from the perspective of sports, low and peak is just a turning point in our life. To know the truth and the whole picture of things, we must go beyond the narrow scope and get rid of the subjective prejudice.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1.X.P.Xue,S.T.Qin, Y.H.Wu. NonlinearAnalysis (2nd Edition). Science Press，2018.
2.D.J.Guo.NonlinearFunctionalAnalysis(2nd Edition).Higher Education Press,2015.
3.K.C.Chang. CriticalPointTheory and itsApplication. Shanghai Science and Technology Press,1986.
4. K.C.Chang.Methods in Nonlinear Analysis, Springer-Berlin, 2005.
B. Learning Resources
1.https://www.dektw.com/course/377/
Outline Writer (Signature):Ge Bin
Leader in charge of teaching at the College (Signature)：
Date：2020/6/22
中文
