Existence Of Solutions Of Several Types Of Non-autonomous System And Solvability

The critical point theory mainly use variational and topological methods involving many fields in nonlinear analysis. It is the most vibrant frontier in the core of mathematics and one of the hotspots in nonlinear analysis. Critical point theory and the development of its applications have resolved a great number of important frontier issues in the areas of nonlinear analysis.As a branch of mathematics, the birth of the calculus of variations is the re-sult of continuously exploring many phenomenens in the real world. The calculus of variations in differential equation is considering a variational problem instead of the boundary value problem in order to prove the existence of solutions, the number of solutions and the method of obtaining the approximate solutions. Jo-hann Bernoulli(Johann Bernoulli,1667-1748) is often considered as the inventor of calculus of variations.In 1696 Johann Bernoulli challenged to the European mathematicians, and he made the famous "brachistochrone" problem. L 'Hospi-tal (Guillaume Francois Antonie de l'Hospital 1661-1704), Jacob Bibonuli (Jacob Bernoulli 1654-1705), Leibniz (Gottfried Wilhelm Leibniz,1646-1716) and New-ton (Isaac Newton1642-1727) have answered this question. Euler (Euler Lonhard, 1707-1783) and Lagrange (Lagrange, Joseph Louis,1736-1813) invented a general method of this kind of problem, thus establishing a new branch of mathematics-calculus of variations.The main idea of classical variational method is to ascertain the extreme value and extreme points. Under certain conditions, boundary value problem can be changed into a variational problem to be studied. Therefore, variational methods is a basic approach for studying boundary value problems.After the 1950s, due to the development of computer, the finite element method based on variational method have been widely used in physics, mechanics and engineering technology. It has become an important branch of computational mathematics.The past 20 years, modern variational methods (also called as large-scale variational method) has been significantly developed and obtained many impor-tant new results in the study of differential equations and partial differential equations. In this paper, we study the existance and the multiplicity of periodic solutions of two kinds of non-autonomous systems by using the least action principle, the Saddle theorem, the invariant sets of descending flow and the chain of rings theorem. The thesis is divided into two chapters.In Chapter 1, we discuss the p-Laplacian systems where p>1, F:R x RNâ†'R, satisfies the following assumption (A) F(t,x) is measurable in t for each x∈R, continuously differentiable in x for a.e. t∈[0, T], and there exist a∈C(R+, R+), b∈L1(0,T; R+) such that |F(t,x)|≤a(|x|)b(t),|â–½F(t,x)|≤a(|x|)b(t) for all x∈RN and a.e. t∈[0,T]. We can get the the existance of periodic solutions of (1.1) by using the least action principle and the Saddle theorem. In this chapter, we generalize the results of [1](p= 2) to the case of p> 1, and show the more general form. At the end of Chapter 1, we supplement the results of [2].In Chapter 2, we talk about the following second order non-autonomous systems Here V:R x RNâ†'R is a smooth function and is periodic in t with period 2Ï€,â–½uV is the gradient of V with respect to u. In [14], the authors get at least four periodic solutions of (2.1) by using the method of invariant sets of descending flow. In this chapter, we get at least seven periodic solutions of (2.1) by using chain of rings Theorem.