Existence Of Anti-periodic Solutions For Ordinary Differential Equations

Nonlinear boundary value problem is one of the important research topic in the field of nonlinear analysis. It was applied widely in physics, chemistry, biology and economic and other fields, and also received people's attentions and concerns all the time. the ordinary differential equation boundary value problem is one of the most active orientations in nonlinear boundary value problems. Because there are only a few type of ordinary differential equations can get the analytic solutions, so it is very difficult to get the analytic solution for boundary value problem. In order to adapt to the needs of practical problems, similar method must be used, therefore, the first question we should answer is: whether the solutions of boundary value problem exist or not? This is the basic subject of boundary value problems.Non-linear vibration is a very important branch in ordinary differential equations theory, which has a profound physical background and theoretical significance. In recent years, this theory gets widely attention and develops rapidly. As the simplest mathematical equation, Duffing equation was widely applied in the mechanical and elec- method, afterwards, Morris[66], Micheletti[1], Respectively, used different methods to deal with solvability for the periodic boundary value problem of Duffing equation under the super-linear growth.Thereafter, many scholars from different perspectives to promote the conditions of him(See[72, 35, 63, 43]).For the semi-linear case, in 1967, Loud [60] considered the existence of solutions for equation(1) under the periodic boundary value condition. He proved that if f satisfyDuffing equation(1) has 2Ï€-periodic Solutions. In 1975, Reissing[75] further studied the existence of solutions for equation(1)under the periodic boundary value condition and obtained the following solvability conditionsSince {n²} is the eigenvalue of operator - y" under the periodic boundary value condition, the solvability condition in the above-mentioned periodic boundary value problem shows that the ratio (?) stays strictly between two consecutive eigenvalues at infinity. Usually, this condition is called non-resonance conditions in the non-linear vibration theory, while the ratio (?) is usually called frequency function. In 1982, in the [80], Ding, Tongren made advatage of the following conditionto get the existing of 2Ï€-periodic Solutions for the equation(1). In 1987, Omari and Zanolin[71]proved, if the following conditions being satisfied,then the equation (1) has 2Ï€-periodic solutions. What deserve our attention is here, (?) still stays between the two continuous eigenvalues at infinity. However,(?) could take eigenvalues n² and (n + 1)². In physics, this conditions correspond to the double resonance, and we call it the double resonance conditions in the essay.On the other hand, in 1993, Fonda[36] used the ratio (?) to replace the (?). Assuming that (?) stays between the first characteristic value and the first Fucik spectrum at infinity, he got the existence of the solution of equation (1).Under other boundary value conditions, by using resonance and non-resonance conditions, people have achieved a series of results([49, 40, 83, 32, 33, 53, 68, 76, 84]) about the existence of solutions for the equation(1). Such as, Mawhin[64] has studied the solvability of the equation((1) under Neumann boundary value condition. In recent years, the anti-periodic boundary value problems for ordinary differential equations frequently appeared in physics, medicine, construction and other fields. In 1988, Okochi[69] firstly studied the existence problem of anti-periodic solutions for the non-linear evolution equations, under the anti-periodic boundary value condition. For the abstract type evolution equation:whereψan even function on a real Hilbert space H. By applying a fixed point theorem for a non-expansive mapping, that the problem has a solution. In 1990, Okochi[70] investigated the anti-periodic problem for the parabolic type evolution equation. Afterwards, from 1990 to 1992, Aftabizadeh[3, 4, 6] studied the anti-periodic solutions for higher order differential equation by means of the Leray-Schauder degree theory. Chen Yuqing[17] and his parters respectively researched the existence of anti-periodic solutions of the following problemswhereψ: H→R is even function,f : R→R and f(t + T) = f(t), t∈R. In 2003, Djiakov and Mityagin[73, 74] studied the antiperiodic boundary value problems of the Schroinger operator and the Hill operator.This thesis mainly used the following methods. 1.Variational Methods.The methods of variations, or critical point theory,is synchronously-produced with differential and integral calculus, which is a mathematics branch of researching functional extremum. It is not only closely related to numerous mathematic branches, but also closely linked to mechanics, physics and engineering etc. Variational Methods offer important axiom for those subjects and also be wildly appropriate.The basic idea of Variational Methods is: turning the problems to solve functional extremum into the problems to solve operator equation, or the reverse conclude, turning the problems to solve non-linear operator equation to the problems to solve critical point of some functional, especially functional extremum.2. Homotopy continuous method.In 1976, Kellogg , Li Yorke[46] applied the homotopy continuous method to prove the Brouwer fixed point theorem. Afterwards, a lot of scholars used this method to deal with many kinds of boundary value problems.In 1990, using homotopy continuous method, Li Yong proved the bounded-ness theorem on periodic solutions and boundary value problems, and stated a calculus of convergence in the large extent for the numerical evacuations of those problems. In 1995, Li Yong and Lin Zhenghua[56] gave a constructive proof on Poincare-Birkhoff theorem via the homotopy continuous method which make can be directly used in finding periodic solutions numerically.3.Leray-Schauder degree theory.The concept of topological degree originated from the research of fixed point using some methods of Algebraic Topology. In the theory of classical analysis, some scholars found that, under certain conditions, some characteristic of solutions of equations don't change along of the continuous deformation of equations, which leads to the emerge of the theory of topological degree. The degree theory turns the solvability of equations into the calculations of some topological invariants.In 1912, Holland mathematician Brouwer firstly established the concept of Brouwer degree for continuous functions in the space of finity dimensional using Algebraic Topology. The purpose is to define a integerdeg(f,Ω),a) such that it relates the number of solutions of equation f(x) = a, where (?),Ω(?) Rⁿ is a bounded open set. In 1934, France mathematician Leray and Poland mathematician Schauder established Leray-Schauder degree theory in Banach space, which extended the Brouwer degree. Subsequently, they pulled the degree theory into the methods of analysis in 1951.On the basis of Leray-Schauder degree, Mawhin, who is the academician of Belgium royal science institute, introduced the concept of coincidence degree in 1976, which contributed to the convenience of the application of the topological degree theory in solving differential equations. Solvability of topological degree is the ground of exporting fixedpoint theorem from the topological degree. Some problem of solving equations can be transformed into problem of solving fixed-point of operator equations. Then one could receive the existence and multiplicity of solutions for all kinds of non-linear system.In Chapter two, we consider the anti-periodic boundary problem for the following second-order Duffing equation:where f∈C²([0,Ï€]×R).Theorem 2.1.1 Assume that there exists M > 0 andδ> 0 suchthat if |s|≥M, thenholds for a.e. t∈[0,Ï€], where N∈N⁺, and F(t,s) =∫₀^sf(t,Ï„)dÏ„. Then problem (5) admits a solution.Furthermore, we study the following more general anti-periodic boundary value problemwhereρ> 0 is a constant, f∈C²([0,2Ï€],R).Using similar methods, we can getTheorem 2.2.2 Suppose thatρ∈R, and f∈C²([0,Ï€],R). If the following conditionsholds for a.e. t∈[0,Ï€], where p, q > 0, then problem(6) admits a solution.In the third part, under the frame of the Fucik spectrum, we study the existence of anti-periodic solutions for the Duffing equationLetΣdenote the Fucik spectrum of operator -y" under the antiperiodic boundary value condition, which is the set of real number pair (λ₊,λ_-)∈R² such that the anti-periodic boundary value problemhas nontrivial solutions. By simple calculations we getCombining prior estimate with Leray-Schauder degree theory, we canget the following result.Theorem 3.2.1 Assume that f∈C([0,T]×R,R), f(t + (?), -s) =-f(t, s). If the following conditions hold(i) There exist positive constantsρ, C₁, M such that (ii) There exist connect subsetΓ(?) R²\Σ, constants p₁, q₁,p₂,q₂ > 0 and a point of the type (λ,λ)∈R² such thatanduniformly for all a.e. t∈[0, T], then the equation(7) admits a (?)-anti-periodic solution.