高阶微分方程的周期积分边值问题
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摘要
具有积分边值的非线性微分方程具有广泛的应用性,例如热传导,等离子物理等许多实际问题都可以归结为带有积分边值条件的问题.因此,积分边值问题是国内外的研究热点.本文的主要工作是：利用拓扑度理论,同伦连续法等非线性分析理论和方法,研究高阶非线性微分方程的周期积分边值问题解的存在性和唯一性.
第一部分我们主要研究高阶微分方程的周期积分边值问题.在推广的Laz-er型限制条件下得到偶数阶微分方程周期积分边值问题解存在性的充分条件.并将结论推广到x∈Rn的情形.
这部分首先考虑下面的2n阶微分方程周期积分边值问题其中t∈[0,2π],x∈R,ki,(i=0,1,…,n-1)是一些常数,f：[0,2π]×R→R.
主要结果为下面定理：
定理2.1.1假设函数f(t,x)是连续可微的,并且存在常数k,M1,M2,使得其中是Z+上的单调增函数,则周期积分边值问题(0.0.1)有唯一解.
其次,我们进一步考虑n维向量方程的周期积分边值问题其中,f∈[0,2π],x(t)=(x1(t),x2(f),…,xn(t)),αj,(j=0,1,…,k-1),是一些常数.
在下面的假设条件下进行讨论：
(H1).f∈C1([0,2π]×Rn),并且f关于x的Jacobi矩阵fx为n×n实对称矩阵.
(H2)存在n×n实对称矩阵A和B以及正整数Ni,(i=1,2,…,n)满足其中λ1≤λ2≤,…,λ。和μ1≤μ2≤,…,μn分别是矩阵A和B的特征值,
定理2.4.1在满足假设(H1)和(H2)条件下,边值问题(0.0.2)有唯一解.
第二部分我们进一步讨论较为一般的二阶非线性微分方程的周期积分边值问题.通过引进变换得到方程(0.0.3)的周期积分边值问题的研究等价于研究问题
假设如下：
(A1)存在常数M1>0和M2>0,函数p(t)满足M1≤p(t)≤M2；
(A2)存在常数a和b,使得对于所有的(t,x)∈[0,T]×R满足
(A3)存在N∈Z+,使得
定理3.3.1在满足假设(A1),(A2)和(A3)的条件下,周期积分边值问题(0.0.4)有唯一解.
第三部分我们考虑线性增长条件的另外一个情况,即函数比值的情况,获得了前面所讨论方程的周期积分边值问题解的存在性.
具体地,考虑下面周期积分边值问题
其中t∈[0,2π],x∈R,f：[0,2π]×R→R.
假设
(A1)函数f连续,即f∈C([0,2π]×R,R)；
(A2)存在m,N和ε,使得对于所有的(t,x)∈[0,2π]×((-∞,-m]∪[m,∞)),都有其中,m>0,N是非负整数,ε是一个小正数.主要结果为以下定理：
定理4.2.7在满足假设(A1)和(A2)的条件下,周期积分边值问题(0.0.5)至少有一解.
The theory of boundary value problems with integral conditions for or-dinary diferential equations has widely application in areas of applied math-ematics and physics such as heat conduction, plasma physics. The existenceof solutions for boundary value problems with integral conditions has beenwidely studied. The main content of this paper is to study the existence anduniqueness of solutions for several types of periodic-integral boundary valueproblems for high diferential equations by using a combination of the nonlin-ear functional analysis theory and method such as degree theory, homotopycontinuation method.
In the first part, we research the existence of solutions of periodic-integralboundary value problems for high diferential equations. we extend the re-sults of periodic integral boundary value problem of the second order Dufngequation. We obtain some sufcient conditions for the existence and unique-ness of solutions of periodic-integral boundary value problems for even-orderdiferential equations. Furthermore we consider the case of x∈Rn.
In this part,we first consider the2nth-order differential equation: where t∈[0,2π],x∈R,ki,i=0,1,…,n-1are some constants,and f:[0,2π]×R→R.
Theorem2.1.1The function f is continnous and differentiable,and there exist constants k,M1,M2,such that where is monotone increasing on Z+Then periodic integral boundary value problem(0.0.6)has a unique solution.
Furthermore,we also consider the vector equations where,x(t)=(x1(t),x2(t),…,xn(t)),t∈[0,2π],αj,j=0,1,…,k-1are some constants.
We need the following hypotheses:(H1) f∈C1([0,2π]×Rn),and Jacobian matrix fx is a symmetric n×n matrix.(H2)There exist two constant symmetric n×n matrices A and B such that and, if λ1≤λ2≤,…,λn and μ1≤μ2≤,…,μn are the eigenvalues of A and B, respectively, then there exist integers Ni, i=1,2,…, n, satisfying the condition where is monotone increasing on Z+.
Theorem2.4.1Assumed (H1) and (H2) are satisfied, the problem (0.0.7) has a unique solution.
In the second part, we study the periodic-integral boundary value problem of nonlinear differential equation By the transform of the periodic-integral boundary value problem of differential equation (0.0.8) is equivalent to where p(t)∈C1([0,2π], R) and f∈C1([0,2π]×R,R).
By applying the homotopy method we obtain the existence and uniqueness of solutions to this kind boundary value problems.
We need the following hypotheses:(Al) there exist two constants M1>0and M2>0such that p(t) satisfies for al t∈[0,2π];
(A2) there exist constants a and b, such that for all (t,x)∈[0,T]×R;
(A3) there exists∈E Z+, such that
Theorem3.3.1Assume that (Al),(A2) and (A3) are satisfied. Then problem (0.0.9) has a unique solution.
In the third part, we consider periodic integral boundary value problem under the linear increasing conditions. We obtain the existence of solutions for periodic integral boundary value problem.
In this part, we main consider the following second order Duffing equation: where t E [0,2π],x∈R.
Assume that
(A1)f∈C([0,2π]×R,R);
(A2) there exist N E Z+and ε>0, such that for all (t,x)∈[0,2π]×((-∞,-m]∪[m,∞)). we have the following theorem:
Theorem4.2.7
Assume that (A1) and (A2) are satisfied. Then periodic integral boundaryvalue problem (0.0.10) has at least one solution.
第一部分我们主要研究高阶微分方程的周期积分边值问题.在推广的Laz-er型限制条件下得到偶数阶微分方程周期积分边值问题解存在性的充分条件.并将结论推广到x∈Rn的情形.
这部分首先考虑下面的2n阶微分方程周期积分边值问题其中t∈[0,2π],x∈R,ki,(i=0,1,…,n-1)是一些常数,f：[0,2π]×R→R.
主要结果为下面定理：
定理2.1.1假设函数f(t,x)是连续可微的,并且存在常数k,M1,M2,使得其中是Z+上的单调增函数,则周期积分边值问题(0.0.1)有唯一解.
其次,我们进一步考虑n维向量方程的周期积分边值问题其中,f∈[0,2π],x(t)=(x1(t),x2(f),…,xn(t)),αj,(j=0,1,…,k-1),是一些常数.
在下面的假设条件下进行讨论：
(H1).f∈C1([0,2π]×Rn),并且f关于x的Jacobi矩阵fx为n×n实对称矩阵.
(H2)存在n×n实对称矩阵A和B以及正整数Ni,(i=1,2,…,n)满足其中λ1≤λ2≤,…,λ。和μ1≤μ2≤,…,μn分别是矩阵A和B的特征值,
定理2.4.1在满足假设(H1)和(H2)条件下,边值问题(0.0.2)有唯一解.
第二部分我们进一步讨论较为一般的二阶非线性微分方程的周期积分边值问题.通过引进变换得到方程(0.0.3)的周期积分边值问题的研究等价于研究问题
假设如下：
(A1)存在常数M1>0和M2>0,函数p(t)满足M1≤p(t)≤M2；
(A2)存在常数a和b,使得对于所有的(t,x)∈[0,T]×R满足
(A3)存在N∈Z+,使得
定理3.3.1在满足假设(A1),(A2)和(A3)的条件下,周期积分边值问题(0.0.4)有唯一解.
第三部分我们考虑线性增长条件的另外一个情况,即函数比值的情况,获得了前面所讨论方程的周期积分边值问题解的存在性.
具体地,考虑下面周期积分边值问题
其中t∈[0,2π],x∈R,f：[0,2π]×R→R.
假设
(A1)函数f连续,即f∈C([0,2π]×R,R)；
(A2)存在m,N和ε,使得对于所有的(t,x)∈[0,2π]×((-∞,-m]∪[m,∞)),都有其中,m>0,N是非负整数,ε是一个小正数.主要结果为以下定理：
定理4.2.7在满足假设(A1)和(A2)的条件下,周期积分边值问题(0.0.5)至少有一解.
The theory of boundary value problems with integral conditions for or-dinary diferential equations has widely application in areas of applied math-ematics and physics such as heat conduction, plasma physics. The existenceof solutions for boundary value problems with integral conditions has beenwidely studied. The main content of this paper is to study the existence anduniqueness of solutions for several types of periodic-integral boundary valueproblems for high diferential equations by using a combination of the nonlin-ear functional analysis theory and method such as degree theory, homotopycontinuation method.
In the first part, we research the existence of solutions of periodic-integralboundary value problems for high diferential equations. we extend the re-sults of periodic integral boundary value problem of the second order Dufngequation. We obtain some sufcient conditions for the existence and unique-ness of solutions of periodic-integral boundary value problems for even-orderdiferential equations. Furthermore we consider the case of x∈Rn.
In this part,we first consider the2nth-order differential equation: where t∈[0,2π],x∈R,ki,i=0,1,…,n-1are some constants,and f:[0,2π]×R→R.
Theorem2.1.1The function f is continnous and differentiable,and there exist constants k,M1,M2,such that where is monotone increasing on Z+Then periodic integral boundary value problem(0.0.6)has a unique solution.
Furthermore,we also consider the vector equations where,x(t)=(x1(t),x2(t),…,xn(t)),t∈[0,2π],αj,j=0,1,…,k-1are some constants.
We need the following hypotheses:(H1) f∈C1([0,2π]×Rn),and Jacobian matrix fx is a symmetric n×n matrix.(H2)There exist two constant symmetric n×n matrices A and B such that and, if λ1≤λ2≤,…,λn and μ1≤μ2≤,…,μn are the eigenvalues of A and B, respectively, then there exist integers Ni, i=1,2,…, n, satisfying the condition where is monotone increasing on Z+.
Theorem2.4.1Assumed (H1) and (H2) are satisfied, the problem (0.0.7) has a unique solution.
In the second part, we study the periodic-integral boundary value problem of nonlinear differential equation By the transform of the periodic-integral boundary value problem of differential equation (0.0.8) is equivalent to where p(t)∈C1([0,2π], R) and f∈C1([0,2π]×R,R).
By applying the homotopy method we obtain the existence and uniqueness of solutions to this kind boundary value problems.
We need the following hypotheses:(Al) there exist two constants M1>0and M2>0such that p(t) satisfies for al t∈[0,2π];
(A2) there exist constants a and b, such that for all (t,x)∈[0,T]×R;
(A3) there exists∈E Z+, such that
Theorem3.3.1Assume that (Al),(A2) and (A3) are satisfied. Then problem (0.0.9) has a unique solution.
In the third part, we consider periodic integral boundary value problem under the linear increasing conditions. We obtain the existence of solutions for periodic integral boundary value problem.
In this part, we main consider the following second order Duffing equation: where t E [0,2π],x∈R.
Assume that
(A1)f∈C([0,2π]×R,R);
(A2) there exist N E Z+and ε>0, such that for all (t,x)∈[0,2π]×((-∞,-m]∪[m,∞)). we have the following theorem:
Theorem4.2.7
Assume that (A1) and (A2) are satisfied. Then periodic integral boundaryvalue problem (0.0.10) has at least one solution.
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