Periodic Solutions And Boundary Value Problems Of Ordinary Differential Equations

In this paper, we are mainly concerned with periodic solutions and boundary value problems of ordinary differential equations. We aim at findingperiodic solutions of the higher order Duffing equations using the Newton continuation method, giving a constructive proof for the existence of periodic solutions of the higher order Duffing equations via the Homotopy method, introducing a continuation theorem on the existence of solutions for some nonlinear problems with p-Laplacian like operators and solving the upper and lower solutions to nonlinear boundary value problems whith the continuationtheorem.The periodic solutions of differential equations can express not only some periodic motions but also some aperiodic motions. The most typical problems is the resonance problems. Being the most simple equations of mathematical physics and mathematical models, the Duffing equations play a central role among the studies of nonlinear resonance problems.There are basically three cases on the studies of the second order Duffing equationi.e., (a) superlinear case; (b) semilinear case; (c) sublinear increase case. Cesari[4], Morris [68], Micheletti[66] studied the existence of the 2Ï€-peri-odic solutions by various methods, respectively. Jacobowitz[35] and Ding Weiyue[88] studied the periodic problems of the semilinear Duffing equationsvia the Poincare-Birkhoff theorem. Especially, under the conditions of superlinearity and existence and uniqueness of solutions of the initial values problems, Ding Weiyue proved that there exist limitless periodic solutions. In 1995, Li Yong and Lin Zhenhua gave a constructive proof on Poincare-Birkhoff theorem [50] via the Homotopy method which can find periodic solutions numerically.Over the past thirty years, Duffing equations have become an important research object in many branches of science [20, 21, 24, 32, 47, 49, 70, 74, 78, 79]. The problem of the existence of periodic solutions for the higher order Duffing equations was investigated in [14, 46, 54, 56, 58] and numerous other papers which are impossible to list here. It has become one of the most intriguing topics in the study of differential equations. In this paper, we consider the simple high order Duffing equation of the formwhere g : R→R is a C¹ function, e : R→R is a continuous function, and e(t) = e(t + 2Ï€), for t∈R.The main aim of this part is to give out proof of the existence and uniqueness of periodic solutions and to find periodic solutions numerically using the proposed Newton formula.Let x(t, P) be the solution of equation (2) with the initial valueswhere P = (P₀,…,P_2n-1)^?.Define a homotopy map H : R²ⁿ×R²ⁿ×[0, 1]→R²ⁿ by whereλ∈[0,1], P, P∈R²ⁿ, P = P whenλ= 0, a = (a₀,…, a_2n-1)^?, andTheorem 1 Let g(x) in equation (2) be satisfiedwhere N is a nonnegative integer. Then for all P∈R²ⁿ, there exists a C¹ curve (P(λ,λ) of the homotopy equationstarting at P(0) = P such that following this curve (P(λ),λ) toλ= 1, one can obtain a point P(1). Then x(t, P(1)) is the unique 2Ï€-periodic solution of equation (1).Numerically tracing the path equation, we reduce the problems of findingperiodic solutions of (2) to the problem of finding the zero point of the following mapwhere X = (x(t),x'(t),…,x^2n-1)(?) is the solution of (2) with the initial value P = (P₀, P₀,…, P_2n-1)(?). Then, homotopy map (4) can be written aswhere a = (a₀, a₁,…, a_2n-1)(?) is defined by (5). According to the discussions above, we know that there exists a C¹ solution curve of (8) starting at (P, 0), and one can find a piont (P^*, 1) such that H(P^*, 1) = 0. Then P^* is the optimal initial value for the periodic solution of (2).It is easy to see that the solution of equation (8) can be expanded as a series inλ where Y₀ is the initial value of P, while Y₁, Y₂,…need to be determined later. Whenλ→1, (9) is the optimal initial value for the periodic solution of (2), i.e.,Expanding F(P) into Taylor series and substituting it into (8), we can obtainwhere B_i =(?)²F_i/(?)P_j(?)P_k (0≤j, k≤2n - 1) is the Hessian matrix of F_i(P),Comparing the coefficients of the vavious powers of A and replacing the derivatives by the differences of P, we obtain the iterative formulaThe formula (10) is called the Newton like formula with second-order approximation. The approximate solution obtained by (10) converges to its optimal solution faster than the solution obtained by the Newton formula.In 1974, Scarf, Kellogg, Li and Yorke [43] proved the Brouwer fixed point theroem. In 1978, Chow, Mallet-Paret and Yorke [12] stated the Ho-motopy continuous method. Some related results can be found in [2, 3, 26, 31, 50, 52, 53, 55, 60, 80, 81, 82]. In 1990, Li Yong proved the boundedness theorem on periodic solutions and boundary value problems, and stated a calculus of convergence in the large extent for the numerical evalutions of those problems.In the second part, we give a constructive proof for the existence of periodic solutions of the higher order Duffing equationwithvia the homotopy method, where g : R→R is a C¹ function, e : R→R is a continuous function, and e(t) = e(t + 2Ï€), t∈R.Consider the following auxiliary boundary value problemwhereλe∈[0,1],λ₀ = 1/2[N²ⁿ + (N + l)²ⁿ], P = (P₀,…, P_2n-1)(?), x(t, P,λ) is the solution of (12) with the initial valuesand for every P∈R²ⁿ, there existsDefine a homotopy map whereλ∈[0,1], a = (a₀,…, a_2n-1)(?).Theorem 2 If there is a A > 0, such thatwhere N is a nonnegative integer,α> 0,β> 0, then (11) has at least one 2Ï€-periodic solution.The problems of finding periodic solutions of (11) can be reduced to the problem of finding the zero point of the following homotopy mapwhere X = (x(t),x'(t),…,x^(2n-1)(?), x(t) is the solution of the following initial value problemIt is obviously that H(P, P,0) = 0, while the solution P_* of H(P, P, 1) = 0 is the optimal initial value of periodic solution of (11).For the global convergence property of the proposed method, we have the following theorem.Theorem 3 Consider the homotopy map (15) and assume that the conditionsof Theorem 2 are satisfied. If for all P∈R²ⁿ, there exists a C¹ curve (P(s),λ(s)) of (15) starting at P(0) = P such that following the curve (P(s), X(s)) toλ= 1, then one can obtain a zero point (P^*,1) of H(P, P, X) = 0.In the natural world, the essence of many phenomenons is the study of solutions of nonlinear differential equations. The Brouwer degree theory, the fixed points theorems and the Leray-Schauder degree theory are all important tools on the studies of nonlinear differential equations.Among the studies on the nonlinear boundary value problems [8, 9, 22, 23, 27, 30, 34, 36, 42, 44, 45, 51, 61], recently, problems of periodic solutions for the p-Laplacian have become popular. Some works can be found in [1, 6, 7, 10, 63] and references therein. In [6] and [7], the Manasevich-Mawhin continuation theorem had been stated and some existence results for various boundary value problems were proved, which contained Dirichlet, periodic, Neuman problems and so on.In the third part, we introduce a main result on a continuation theorem. At last we give upper and lower solutions results for (16), there we use the continuation Theorem 4.Consider the nonlinear boundary value problem of the formwhereφ_p : R¹â†’R¹ is an increasing homeomorphism which includes p-Laplacian like operatorφ_p(s) = |s|^p-2s,φ_p(0) = 0, f : [0,T]×R¹Ã—R¹â†’R¹, A(t)(?)0 are continuous functions, v =φ_p(u'), a, b, c, d≥0, and a + b > 0, c + d > 0. Letdenote the Banach space, equipped with the normWe denote by (?)_p the inverseφ_p^-1 such thatφ_po(?)_p = id. It is clear that (?)_p is also an increasing homeomorphism. Thus problem (16) can be written in the equivalent formwith the corresponding boundary conditions. The following theorem is the main result of the third part.Theorem 4 Assume thatΩis an open bounded set in X such that for eachλ∈[0,1] the problemhas no solution on (?)Ω, where v =φ_p(u'), then problem (16) has a solution inΩ.As an application of Theorem 4, we consider the upper and lower solutionsto the nonlinear boundary value problemTheorem 5 Suppose:(a) There exist a lower solution and an upper solutionα,β∈C²([0, T], R¹)such thatα(t)≤β(t), (φ_P(α'))'≥f(t,α,α'), (φ_p(β'))'≤f(t,β,β') for t∈[0,T].(b) The boundary conditions satisfy:(c) There exists a continuous function h : [0, +∞)→(0, +∞) such thatandfor t∈[0,T],α(t)≤u(t)≤β(t).Then problem (18) has a solution u(t) such thatα(t)≤u(t)≤β(t) for all t∈[0,T].