四阶常微分方程边值问题解的存在性和多重性
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摘要
本文为四阶常微分方程边值问题解的存在性的研究综述,以几类常见的边界条件的四阶常微分方程的研究为主线,简要回顾了最近十多年来四阶常微分方程边值问题的研究状况以及所取得的研究成果,集中对解的存在性,多重性和非存在性以及对所使用的研究方法进行阐述.
本文共分四章.第一章的上半部分回顾了常微分方程理论的产生和发展过程并引入了四阶常微分方程的各种边值问题,下半部分介绍了一些基本概念和预备引理.
第二章介绍了利用不动点指标理论、上下解方法和锥拉伸与锥压缩不动点定理来证明四阶常微分方程两点边值问题解的存在性和多重性的方法,所讨论的方程包括非线性项不依赖未知函数二阶导数的情形(奇异和非奇异的)和非线性项依赖未知函数二阶导数的一般情形.
首先利用不动点指标理论给出如下边值问题多重正解存在的充分条件.这里函数f满足
(H1)f：[0,1]×[0,∞)→[0,∞)连续；
(H3)令.存在p>0,使得当x∈[0,1],y∈[0,p]时,0 (H5)令.存在p>0,使得当x∈[1/4,3/4],y∈[1/24p,p]时,有f(z,y)>Bp.
定理1若f(x,y)满足假设(H1),(H2),(H3),则边值问题(1)至少存在两个正解u1,u2使得0 定理2若f(x,y)满足假设(H1),(H4),(H5),则边值问题(1)至少存在两个正解u1,u2满足0 对于奇异的情形,当f满足如下假设时,也可借助不动点指标理论证明问题(1)至少存在两个正解.
(H1')f∈C((0,1)×(0,∞),R+)且f(t,u)≤g(t)h(u),其中h∈C((0,+∞),R+),且对任意0 (H2')存在φ∈L1[0,1],使得关于t∈(0,1)一致成立,且满足
(H3')存在a(t),b(t)∈C([0,1],R+),使得其中a(t)满足在[0,1]的任意子区间上a(t)(?)0；
(H4')存在R>0使得
定理3假设条件(H1')-(H4')成立,且r(L)>1,则边值问题(1)至少有两个正解.对于非线性项依赖于未知函数二阶导数的情形借助于上下解方法可以得到如下的存在性结果.
定理4假设问题(2)存在上解a和下解β满足β≤α,β"≥α".函数f∶[0,1]×R×R→R是连续的,且满足f(t,u2,v)-f(t,u1,v)≥0,β(t)≤u1≤u2≤α(t),ν∈R,t∈[0,1], f(t,u,v2)-f(t,u,v1)≤0,α"(t)≤v1≤v2≤β"(t),u∈R,t∈[0,1].那么存在单调递减的序列{αn}n=0∞和单调递增的序列{βn}n=0∞,分别一致收敛于问题(2)于[β,α]上的最大解和最小解,这里α0=α,β0=β.
定理5假设问题(2)存在上解α和下解β满足β≤α,β"+r(α-β)≥α".函数f:[0,1]×R×R→R是连续的且满足当β(t)≤u1≤u2≤α(t),v∈R,t∈[0,1]时, f(t,u2,v)-f(t,u1,v)≥-b(u2-u1),当v2+r(α-β)≥v1,a"+r(α-β)≤v1,,v2≤β"+r(α-β),u∈R,t∈[0,1]时, f(t,u,v2)-f(t,u,v1)≤a(v2-v1),这里a,b≥0,a2-4b≥0,r1,2=(a±(?))/2.那么存在单调递减的序列{αn}n=0∞和单调递增的序列{βn}n=0∞,分别一致收敛于问题(2)于[β,α]上的最大解和最小解.
第三章介绍了利用上下解方法和锥拉伸与锥压缩不动点定理证明四阶常微分方程多点边值问题解的存在性的方法,所讨论的方程包括非线性项f不依赖弯矩项u"的情形,非线性项依赖于弯矩项u"的情形和依赖于未知函数三阶导数的情形.
对于问题其中a,b,c,d是非负常数,0≤ξ1<ξ2≤1,有如下存在性结果.
定理6若下列条件成立：(B1)a,b,c,d,ξ1,ξ2是非负常数,且满足0≤ξ1<ξ2≤1,b-aξ1≥0,d-c+cξ2≥0,δ=ad+bc+ac(ξ2-ξ1)≠0；(B2)f(t,u)∈C([0,1]×[0,∞),R+)关于u单调不减,当f∈(ξ1,ξ2)时,f(t,t(1-t))(?)0,并且存在常数0<μ<1使得对任意0≤k≤1,有kμf(t,u)≤f(t,ku).
则边值问题(3)至少存在一个正解.
当问题(3)中的非线性项f=f(t,u,u")时,记利用锥拉伸与锥压缩不动点定理可得
定理7若函数f满足下列条件：(C1)f∈C([0,1]×[0,∞)×(-∞,0],[0,∞))；(C2)f次线性,即min f0=+∞,max f∞=0.则边值问题(3)至少存在一个正解.
定理8若函数f满足条件(C1),且如下条件成立：(C3)f是超线性的,即max f0=0,min f∞=+∞.则问题(3)至少存在一个正解.通过改进上述正解存在性的条件,可以证明问题(3)存在多重正解.
定理9设函数f满足(C1),并且同时满足以下两个条件：(C4)min f0=min f∞=+∞；(C5)存在常数l1>0,使得对任意t∈[0,1],x∈[0,l1],-y∈[0,l1]有则问题(3)至少存在两个正解u1,u2满足0<‖u1‖2 定理10设函数f满足(C1),并且同时满足以下两个条件：(C6)max f0=max f∞=0；(C7)存在常数l2>0,使得对任意t∈[0,1],x∈[0,l2],-y∈[l2/4,l2]有则问题(3)至少存在两个正解u1,u2满足0<‖u1‖2 利用上下解方法和Schauder不动点定理,可以得到非线性项依赖于未知函数三阶导数的多点边值问题解的存在惟一性：这里a,b,c,d≥0,ρ=ad+bc+ac>0,f:[0,1]×R4→R是连续函数.
定理11设v,w是问题(4)的上、下解,且满足w"(t)≥v"(t),f满足关于w",v"的Nagumo条件,则问题(4)至少存在一个解满足w(t)≤u(t)≤v(t),w'(t)≤u'(t)≤v'(t), v"(t)≤u"(t)≤w"(t),t∈[0,1].
定理12设w,v是问题(4)的下解与上解,f满足关于w",v"的Nagumo条件.若f(t,x1,x2,x3,x4)关于x1,x2单调递减,关于x4严格单调递增,则问题(4)存在惟一解u(t)满足w(t)≤u(t)≤v(t),w'(t)≤u'(t)≤v'(t),v"(t)≤u"(t)≤w"(t),t∈[0,1].
第四章介绍了利用上下解方法和不动点指标定理来证明四阶常微分方程周期边值问题解的存在性,多重性和不存在性的方法.
对如下形式的四阶方程周期边值问题：假设非线性项f满足下面两个条件：
(D1)对任意给定的β,α∈C[0,2π],β(t)≤α(t),t∈[0,2π],存在0 (D2)对几乎所有的t∈[0,2π]和一切v∈R,当β(t)≤u1≤u2≤α(t)时,有
关于问题(5)有下述存在性结果：
定理13设问题(5)存在下解β(f)与上解α(f)满足β(t)≤α(t),t∈[0,2π],函数f(t,u,v)满足Caratheodory条件.若条件(D1),(D2)成立,则存在一个单调非减序列{βj}j=0∞和一个单调非增序列{αj}j=0∞。分别单调一致收敛于问题(5)在序区间[β,α]上的最小解和最大解,这里β0=β,α0=α,[β,α]={u∈C[0,2π]:β(t)≤u(t)≤α(t),t∈[0,2π]}.
当非线性项f仅满足单边Lipschitz条件时,也可证明周期边值问题(5)存在解.
定理14设问题(5)存在下解β(t)与上解α(f)满足β(t)≤α(t),t∈[0,2π],函数f(t,u,v)是Caratheodory函数且满足条件
(D3)对任意给定的β,α∈C[0,2π],β(t)≤α(t),t∈[0,2π],存在a,b>0,b2≥4a,使得当β(t)≤u1≤u2≤α(t),v1,v2∈R,v1≤v2时,对几乎所有的t∈[0,2π],都有f(t,u2,v2)-f(t,u1,v1)≥-a(u2-u1)+b(v2-v1).则问题(5)存在解u∈W4,1[0,2π]满足β(t)≤u(t)≤α(t).
定理15设问题(5)存在下解β(t)与上解α(t)满足β(t)≤α(t),t∈[0,2π],函数f(t,u,v)满足条件
(D4)存在常数C,D>0,D<4C+1/4,D2>4C,使得当β(t)≤u1≤u2≤α(t), v1,v2∈R,v1≤v2时,对几乎所有的t∈[0,2π],都有f(t,u2,v1)-f(t,u1,v2)≥-C(u2-u1)-D(v1-v2).则问题(5)存在解u∈W4,1[0,2π],且满足β(t)≤u(t)≤α(t).
最后介绍应用锥中的不动点指标定理证明周期边值问题正解的存在性和多重性的方法和相应的结果,其中f：[0,1]×R+→R+是连续的,a,b∈R满足00,a/π4+b/π2+1>0.记
定理16若函数f满足下列条件之一(E1)f0a；(E2)f0>a,f∞ 定理17若存在两个正常数c,d使得φ(c)a,则问题(6)至少存在一个正解u∈K满足min{c,d}<||u|| 定理18假设存在n+1个正数a1 推论如果下列条件之一成立：(E23)φ(l)a,l∈(0,+∞).则问题(6)在K中没有解.
This thesis is a survey of the recent results in investigating the boundary value problems of fourth-order ordinary differential equations. We briefly overview the recent situation for studying this kind of problems and also survey the results with concentration on some im-portant boundary value problems, such as two-point boundary value problems, multi-point boundary value problems and periodic boundary value problems.
We introduce the existence of (positive) solutions, the existence of multiple (positive) solutions to some fourth-order two-point boundary value problems; introduce the existence of solutions and the existence of multiple solutions to some fourth-order multiple-point boundary value problems; we also introduce some results to the existence, multiplicity and nonexistence of solutions to the fourth-order periodic boundary value problems.
The thesis consists of 4 Chapters. In the first Chapter, by reviewing the theory of or-dinary differential equations and its development, we introduce the fourth-order boundary value problems with different boundary conditions, introduce the background of the prob-lems and outline the basic concepts and related theorems. We also introduce some basic definitions in nonlinear analysis and several important theorems including cone-stretching and cone-compressing fixed-point theorems, fixed-point index theorems of completely con-tinuous operators, Schauder fixed-point theorem and Arzela-Ascoli'lemma.
In Chapter 2, we introduce the method of proving the existence and multiplicity of so-lutions to the fourth-order two-point boundary value problems with the use of fixed-point in-dex theories, upper and lower solutions methods and cone-stretching and cone-compressing fixed-point theorems. What we discuss includes both the case when the nonlinearity f de-pends on u" and the case when f does not depend on u".
We first list some sufficient conditions to the multiplicity of solutions to the following boundary value problem with the use of fixed-point index theories. Under the following hypotheses:
(H1)f:[0,1]×[0,∞)→[0,∞) is continuous;
(H3) Set A= (∫01G(τ(s), s)ds)-1. There exists a p> 0 such that 0< f(x, y)< Ap for all x∈[0,1], y∈[0,p];
(H5) Denote such that f(x, y)> Bp for all x∈[1/4,3/4],y∈[1/24 p,p].
Theorem 1 Assume that f(x,y) satisfies (H1), (H2), (H3), then Problem (1) has at least two positive solutions u1, u2 satisfying 0< u1< p< u2.
Theorem 2 Assume that f(x,y) satisfies (H1), (H4), (H5), then Problem (1) admits at least two positive solutions-u1, u2 satisfying 0 For the singular case, we can also prove that Problem (1) has at least two positive solu-tions by using the theories of fixed-point index when f satisfies the following hypotheses:
(H1')f∈C((0,1) x (0,∞), R+) and f(t, u)≤g(t)h(u), where h∈C((0,+∞), R+), and for any 0< r< R, it holds
Here hr,R(t)(?) max{h(u):u∈[q(t)r,R]}, q(t) is given in (2.2);
(H2') There exists aφ∈L1[0,1] satisfying 0<∫01 G(τ(s), s)φ(s)ds<+∞such that (?)
(H3') There exist a(t), b(t)∈C([0,1], R+) such that f(t, u)≥a(t)u-b(t),t∈(0,1),u∈(0,+∞), where a(t) (?) 0 in any subinterval of [0,1];
(H4') There exists an R> 0 such that
Theorem 3 Assume that (H1')-(H4') hold and that r(L)> 1, then Problem (1) has at least two positive solutions.
When f depends on u", one can get the existence result of the following problem with the use of the upper and lower solutions method. The main results are the following,
Theorem 4 If there exist an upper solution a and a lower solutionβof Problem (2) such thatβ≤α,β"≥α", and i f:[0,1]×R×R→R is continuous and satisfies f(t, u2, v)-f(t, u1,v)≥0,β(t)≤u1≤u2≤a(t), v∈R, t∈[0,1], f(t, u, v2)-f(t, u,v1)≤0,α"(t)≤v1≤v2≤β"(t), u∈R,t∈[0,1].
Then there exist two monotone sequences{αn}n=0∞and {β0}n=0∞, nonincreasing and nondecreas-ing, respectively, withα0=α,β0=β, which converge uniformly to the extremal solution to Problem(2) in[β,α].
Theorem 5 If there exist an upper solution a and a lower solutionβof Problem (2) which satisfyβ≤α,β"+r(α-β)≥α", and if f:[0,1]×R×R→R is continuous and satisfies f(t, u2, v)-f(t, u1,v)≥-b(u2-u1), for allβ(t)≤u1≤u2≤α(t), v∈R, t∈[0,1]; f(t, u, v2)-f(t, u, v1)≤a(v2-v1), for v2+r(α-β)≥v1, a"+r(α-β)≤v1,v2≤β"+r(α-β), u∈R, t∈[0,1], where a,b≥0, a2-4b≥0, r1,2= (a±(?))/2. Then there exist two monotone sequences {an}a=0∞and {βn}n=0∞, nonincreasing and nondecreasing, respectively, with a0=α,β0=β, which converge uniformly to the extremal solution to Problem (2) in [β,α].
In Chapter 3, we use the upper and lower solutions method and cone-stretching and cone-compressing fixed-point theorems to introduce the existence of solutions to the multi-point boundary value problem of fourth-order equations. The problems under consideration include the case f= f(t, u), the case f= f(t, u, u") and the case f= f(t, u, u'u", u'"). We consider the following four-point boundary value problem: where a, b, c, d≥0 and 0≤ξ1≤ξ2≤1.
The main conclusion is the following theorem.
Theorem 6 If the following conditions hold:
(B1) a,b, c, d≥0,0≤ξ1≤ξ2≤1 and b-aξ1≥0, d-c+cξ2≥0,δ= ad+be+ ac(ξ2-ξ1)≠0;
(B2) f(t,u)∈C([0,1] x [0,∞),R+) is nondecreasing relative to u, f(t,t(1-t))(?) 0 for t∈(ξ1,ξ2) and there exists a constant 0<μ< 1 such that kμf(t, u)≤f(t, ku) for any 0≤k≤1. Then Problem (3) has at least one positive solution.
When the nonlinearity f= f(t, u, u"), denote
With the use of cone-stretching and cone-compressing fixed-point theorems one can prove the following theorem.
Theorem 7 Assume that f satisfies:
(C1)f∈C([0,1] x [0,∞) x (-∞,0], [0,∞));
(C2)f is sublinear, namely min f0=+∞, max f∞= 0.
Then Problem (3) has at least one positive solution.
Theorem 8 Assume (C1) and
(C3) f is superliner, namely max f0= 0, min f∞=+∞.
Then Problem (3) has at least one positive solution.
One can prove the existence of multiple solutions to Problem (3) by modifying the above conditions.
Assume that
(C4) min f0= min f∞,=+∞;
(C5) There exists an l1> 0 such that for all t∈[0,1], x∈[0,l1],-y∈[0,l1];
(C6) max f0= max f∞,= 0;
(C7) There exists an l2> 0 such that for any t∈[0,1], x∈[0,l2],-y∈[l2/4,l2].
Theorem 9 Assume (C1), (C4) and (C5). Then Problem (3) admits at least two posi-tive solutions u1, u2 satisfying 0<||u1||2< l1< ||u2||2.
Theorem 10 Assume (C1), (C6) and (C7). Then Problem (3) admits at least two positive solutions u1, u2 satisfying 0<||u1||2< l2< ||u2||2.
As for the following problem where a, b, c, d≥0,ρ= ad+be+ac> 0,f:[0,1]×R4→R is continuous. One can get the existence and uniqueness of solutions with the use of the upper and lower solutions method and Schauder fixed point theorem.
Theorem 11 Suppose that v,w are upper and lower solutions to Problem (4) such that w"(t)≥v"(t) and f satisfies the Nagumo condition with respect to w" and v". Then Problem (4) has at least one solution u such that w(t)≤u(t)≤v(t), w'(t)≤u'(t)≤v'(t), v"(t)≤u"(t)≤w"(t), t∈[0,1].
Theorem 12 Suppose that v, w are upper and lower solutions to Problem (4) and f satisfies the Nagumo condition with respect to w" and v". If f(t,x1,x2, x3, x4) is decreasing in x1,x2 and strictly increasing in x4, then Problem (4) has a unique solution u(t) such that w(t)≤u(t)≤v(t), w'(t)≤u'(t)≤v'(t), v"(t)≤u"(t)≤w"(t), t∈[0,1].
In Chapter 4, we study the existence, multiplicity and nonexistence of solutions to the following periodic boundary value problem
Assume that
(D1) For any givenβ,α∈C[0,2π] withβ(t)<α(t), t∈[0,2π], there exist 0< A< B such that A(v2-v1)< f(t, u, v2)-(t, u,v1)< B(v2-v1) for a.e. t∈[0,2π] wheneverβ(t)< u< a(t), v1,v2∈R, v1 (D2) Inequality holds for a. e.t∈[0,2π] wheneverβ(t)< u1 The main theorem is
Theorem 13 Suppose that there exist a lower solutionβ(t) and an upper solutinα(t) such thatβ(t)<α(t), t∈[0,2π], and f(t, u, v) is a Caratheodory function satisfying the hypotheses (D1) and (D2). Then there exist two sequences {βj}j=0∞and{αj}j=0∞nondecreas-ing and nonincreasing, respectively, withβ0=β,α0=α, which converge uniformly and monotonically to the extremal solutions to Problem (5) in [β,α].
When f satisfies only one-side Lipschitz condition, one can also prove the existence of solutions to Problem (5).
Theorem 14 Suppose that there exist a lower solutionβ(t) and an upper solutionα(t) such thatβ(t)≤a(t), t∈[0,2π], and f(t, u, v) is a Caratheodory function satisfying the hypothesis (D3):
(D3) For any givenβ,α∈C[0,2π],β(t)< a(t), t∈[0,2π], there exist a, b> 0 such that b2≥4a and f(t,u2,v2)-f(t,u1,v1)≥-a(u2-u1)+b(v2-v1); for a. e. t∈[0,2π] whenever B(t)≤u1≤u2≤α(t), v1, v2∈R, v1< v2.
Then Problem (5) has a solution u∈W4,1 [0,2π] satisfyingβ(t)≤u(t)≤α(t).
Theorem 15 Suppose that there exist a lower solutionβ(t)and an upper solutionα(t) such thatβ(t)≤α(t),t∈[0,2π],and f(t,u,v)satisfies the hypothesis
(D4)There exist C,D>0 such that D<4C+1/4,D2>4C and f(t,u2,V1)-f(t,u1,v2)≥-C(u2-u1)-D(v1-v2) for a.e.t∈[0,2π]wheneverβ(t)≤u1≤u2≤α(t),v1,v2∈R,v1≤v2. Then Problem(5)has a solution u∈W4,1[0,2π]satisfyingβ(t)≤u(t)≤α(t).
At the end of the thesis,we outline the results in studying the existence,multiplicity and nonexistence of solutions to the following problem under suitable conditions to f where f:[0,1]×R+→R+is continuous,a,b∈R such that 00, a/π4+b/π2+1>0.
Denote (?)=(?) (?)=(?)
Theorem 16 Assume that one of the following two conditions holds: (E1)f0a；(E2)f0>a,f∞ Theorem 17 Suppose that there exist two positive constants c,d such thatφ(c)< a,φ(d)>a,then Problem(6)has at least one positive solution satisfying min{c,d}<||u||< max{c,d}.Hereφ(l)=max{f(t,c)/c:t∈[0,1],c∈[σ-l,l]},φ(l)=min{f(t,c)/c:t∈[0,1],c∈[σ-l,l]),
Theorem 18 Suppose that there exist n+1 positive constants a1< a2<… Theorem 19 If one of the following conditions holds: (E23)φ(l)a, l∈(0,+∞).
Then Problem (6) has no positive solution u∈K.
本文共分四章.第一章的上半部分回顾了常微分方程理论的产生和发展过程并引入了四阶常微分方程的各种边值问题,下半部分介绍了一些基本概念和预备引理.
第二章介绍了利用不动点指标理论、上下解方法和锥拉伸与锥压缩不动点定理来证明四阶常微分方程两点边值问题解的存在性和多重性的方法,所讨论的方程包括非线性项不依赖未知函数二阶导数的情形(奇异和非奇异的)和非线性项依赖未知函数二阶导数的一般情形.
首先利用不动点指标理论给出如下边值问题多重正解存在的充分条件.这里函数f满足
(H1)f：[0,1]×[0,∞)→[0,∞)连续；
(H3)令.存在p>0,使得当x∈[0,1],y∈[0,p]时,0
定理1若f(x,y)满足假设(H1),(H2),(H3),则边值问题(1)至少存在两个正解u1,u2使得0
(H1')f∈C((0,1)×(0,∞),R+)且f(t,u)≤g(t)h(u),其中h∈C((0,+∞),R+),且对任意0
(H3')存在a(t),b(t)∈C([0,1],R+),使得其中a(t)满足在[0,1]的任意子区间上a(t)(?)0；
(H4')存在R>0使得
定理3假设条件(H1')-(H4')成立,且r(L)>1,则边值问题(1)至少有两个正解.对于非线性项依赖于未知函数二阶导数的情形借助于上下解方法可以得到如下的存在性结果.
定理4假设问题(2)存在上解a和下解β满足β≤α,β"≥α".函数f∶[0,1]×R×R→R是连续的,且满足f(t,u2,v)-f(t,u1,v)≥0,β(t)≤u1≤u2≤α(t),ν∈R,t∈[0,1], f(t,u,v2)-f(t,u,v1)≤0,α"(t)≤v1≤v2≤β"(t),u∈R,t∈[0,1].那么存在单调递减的序列{αn}n=0∞和单调递增的序列{βn}n=0∞,分别一致收敛于问题(2)于[β,α]上的最大解和最小解,这里α0=α,β0=β.
定理5假设问题(2)存在上解α和下解β满足β≤α,β"+r(α-β)≥α".函数f:[0,1]×R×R→R是连续的且满足当β(t)≤u1≤u2≤α(t),v∈R,t∈[0,1]时, f(t,u2,v)-f(t,u1,v)≥-b(u2-u1),当v2+r(α-β)≥v1,a"+r(α-β)≤v1,,v2≤β"+r(α-β),u∈R,t∈[0,1]时, f(t,u,v2)-f(t,u,v1)≤a(v2-v1),这里a,b≥0,a2-4b≥0,r1,2=(a±(?))/2.那么存在单调递减的序列{αn}n=0∞和单调递增的序列{βn}n=0∞,分别一致收敛于问题(2)于[β,α]上的最大解和最小解.
第三章介绍了利用上下解方法和锥拉伸与锥压缩不动点定理证明四阶常微分方程多点边值问题解的存在性的方法,所讨论的方程包括非线性项f不依赖弯矩项u"的情形,非线性项依赖于弯矩项u"的情形和依赖于未知函数三阶导数的情形.
对于问题其中a,b,c,d是非负常数,0≤ξ1<ξ2≤1,有如下存在性结果.
定理6若下列条件成立：(B1)a,b,c,d,ξ1,ξ2是非负常数,且满足0≤ξ1<ξ2≤1,b-aξ1≥0,d-c+cξ2≥0,δ=ad+bc+ac(ξ2-ξ1)≠0；(B2)f(t,u)∈C([0,1]×[0,∞),R+)关于u单调不减,当f∈(ξ1,ξ2)时,f(t,t(1-t))(?)0,并且存在常数0<μ<1使得对任意0≤k≤1,有kμf(t,u)≤f(t,ku).
则边值问题(3)至少存在一个正解.
当问题(3)中的非线性项f=f(t,u,u")时,记利用锥拉伸与锥压缩不动点定理可得
定理7若函数f满足下列条件：(C1)f∈C([0,1]×[0,∞)×(-∞,0],[0,∞))；(C2)f次线性,即min f0=+∞,max f∞=0.则边值问题(3)至少存在一个正解.
定理8若函数f满足条件(C1),且如下条件成立：(C3)f是超线性的,即max f0=0,min f∞=+∞.则问题(3)至少存在一个正解.通过改进上述正解存在性的条件,可以证明问题(3)存在多重正解.
定理9设函数f满足(C1),并且同时满足以下两个条件：(C4)min f0=min f∞=+∞；(C5)存在常数l1>0,使得对任意t∈[0,1],x∈[0,l1],-y∈[0,l1]有则问题(3)至少存在两个正解u1,u2满足0<‖u1‖2
定理11设v,w是问题(4)的上、下解,且满足w"(t)≥v"(t),f满足关于w",v"的Nagumo条件,则问题(4)至少存在一个解满足w(t)≤u(t)≤v(t),w'(t)≤u'(t)≤v'(t), v"(t)≤u"(t)≤w"(t),t∈[0,1].
定理12设w,v是问题(4)的下解与上解,f满足关于w",v"的Nagumo条件.若f(t,x1,x2,x3,x4)关于x1,x2单调递减,关于x4严格单调递增,则问题(4)存在惟一解u(t)满足w(t)≤u(t)≤v(t),w'(t)≤u'(t)≤v'(t),v"(t)≤u"(t)≤w"(t),t∈[0,1].
第四章介绍了利用上下解方法和不动点指标定理来证明四阶常微分方程周期边值问题解的存在性,多重性和不存在性的方法.
对如下形式的四阶方程周期边值问题：假设非线性项f满足下面两个条件：
(D1)对任意给定的β,α∈C[0,2π],β(t)≤α(t),t∈[0,2π],存在0
关于问题(5)有下述存在性结果：
定理13设问题(5)存在下解β(f)与上解α(f)满足β(t)≤α(t),t∈[0,2π],函数f(t,u,v)满足Caratheodory条件.若条件(D1),(D2)成立,则存在一个单调非减序列{βj}j=0∞和一个单调非增序列{αj}j=0∞。分别单调一致收敛于问题(5)在序区间[β,α]上的最小解和最大解,这里β0=β,α0=α,[β,α]={u∈C[0,2π]:β(t)≤u(t)≤α(t),t∈[0,2π]}.
当非线性项f仅满足单边Lipschitz条件时,也可证明周期边值问题(5)存在解.
定理14设问题(5)存在下解β(t)与上解α(f)满足β(t)≤α(t),t∈[0,2π],函数f(t,u,v)是Caratheodory函数且满足条件
(D3)对任意给定的β,α∈C[0,2π],β(t)≤α(t),t∈[0,2π],存在a,b>0,b2≥4a,使得当β(t)≤u1≤u2≤α(t),v1,v2∈R,v1≤v2时,对几乎所有的t∈[0,2π],都有f(t,u2,v2)-f(t,u1,v1)≥-a(u2-u1)+b(v2-v1).则问题(5)存在解u∈W4,1[0,2π]满足β(t)≤u(t)≤α(t).
定理15设问题(5)存在下解β(t)与上解α(t)满足β(t)≤α(t),t∈[0,2π],函数f(t,u,v)满足条件
(D4)存在常数C,D>0,D<4C+1/4,D2>4C,使得当β(t)≤u1≤u2≤α(t), v1,v2∈R,v1≤v2时,对几乎所有的t∈[0,2π],都有f(t,u2,v1)-f(t,u1,v2)≥-C(u2-u1)-D(v1-v2).则问题(5)存在解u∈W4,1[0,2π],且满足β(t)≤u(t)≤α(t).
最后介绍应用锥中的不动点指标定理证明周期边值问题正解的存在性和多重性的方法和相应的结果,其中f：[0,1]×R+→R+是连续的,a,b∈R满足0
定理16若函数f满足下列条件之一(E1)f0
This thesis is a survey of the recent results in investigating the boundary value problems of fourth-order ordinary differential equations. We briefly overview the recent situation for studying this kind of problems and also survey the results with concentration on some im-portant boundary value problems, such as two-point boundary value problems, multi-point boundary value problems and periodic boundary value problems.
We introduce the existence of (positive) solutions, the existence of multiple (positive) solutions to some fourth-order two-point boundary value problems; introduce the existence of solutions and the existence of multiple solutions to some fourth-order multiple-point boundary value problems; we also introduce some results to the existence, multiplicity and nonexistence of solutions to the fourth-order periodic boundary value problems.
The thesis consists of 4 Chapters. In the first Chapter, by reviewing the theory of or-dinary differential equations and its development, we introduce the fourth-order boundary value problems with different boundary conditions, introduce the background of the prob-lems and outline the basic concepts and related theorems. We also introduce some basic definitions in nonlinear analysis and several important theorems including cone-stretching and cone-compressing fixed-point theorems, fixed-point index theorems of completely con-tinuous operators, Schauder fixed-point theorem and Arzela-Ascoli'lemma.
In Chapter 2, we introduce the method of proving the existence and multiplicity of so-lutions to the fourth-order two-point boundary value problems with the use of fixed-point in-dex theories, upper and lower solutions methods and cone-stretching and cone-compressing fixed-point theorems. What we discuss includes both the case when the nonlinearity f de-pends on u" and the case when f does not depend on u".
We first list some sufficient conditions to the multiplicity of solutions to the following boundary value problem with the use of fixed-point index theories. Under the following hypotheses:
(H1)f:[0,1]×[0,∞)→[0,∞) is continuous;
(H3) Set A= (∫01G(τ(s), s)ds)-1. There exists a p> 0 such that 0< f(x, y)< Ap for all x∈[0,1], y∈[0,p];
(H5) Denote such that f(x, y)> Bp for all x∈[1/4,3/4],y∈[1/24 p,p].
Theorem 1 Assume that f(x,y) satisfies (H1), (H2), (H3), then Problem (1) has at least two positive solutions u1, u2 satisfying 0< u1< p< u2.
Theorem 2 Assume that f(x,y) satisfies (H1), (H4), (H5), then Problem (1) admits at least two positive solutions-u1, u2 satisfying 0
(H1')f∈C((0,1) x (0,∞), R+) and f(t, u)≤g(t)h(u), where h∈C((0,+∞), R+), and for any 0< r< R, it holds
Here hr,R(t)(?) max{h(u):u∈[q(t)r,R]}, q(t) is given in (2.2);
(H2') There exists aφ∈L1[0,1] satisfying 0<∫01 G(τ(s), s)φ(s)ds<+∞such that (?)
(H3') There exist a(t), b(t)∈C([0,1], R+) such that f(t, u)≥a(t)u-b(t),t∈(0,1),u∈(0,+∞), where a(t) (?) 0 in any subinterval of [0,1];
(H4') There exists an R> 0 such that
Theorem 3 Assume that (H1')-(H4') hold and that r(L)> 1, then Problem (1) has at least two positive solutions.
When f depends on u", one can get the existence result of the following problem with the use of the upper and lower solutions method. The main results are the following,
Theorem 4 If there exist an upper solution a and a lower solutionβof Problem (2) such thatβ≤α,β"≥α", and i f:[0,1]×R×R→R is continuous and satisfies f(t, u2, v)-f(t, u1,v)≥0,β(t)≤u1≤u2≤a(t), v∈R, t∈[0,1], f(t, u, v2)-f(t, u,v1)≤0,α"(t)≤v1≤v2≤β"(t), u∈R,t∈[0,1].
Then there exist two monotone sequences{αn}n=0∞and {β0}n=0∞, nonincreasing and nondecreas-ing, respectively, withα0=α,β0=β, which converge uniformly to the extremal solution to Problem(2) in[β,α].
Theorem 5 If there exist an upper solution a and a lower solutionβof Problem (2) which satisfyβ≤α,β"+r(α-β)≥α", and if f:[0,1]×R×R→R is continuous and satisfies f(t, u2, v)-f(t, u1,v)≥-b(u2-u1), for allβ(t)≤u1≤u2≤α(t), v∈R, t∈[0,1]; f(t, u, v2)-f(t, u, v1)≤a(v2-v1), for v2+r(α-β)≥v1, a"+r(α-β)≤v1,v2≤β"+r(α-β), u∈R, t∈[0,1], where a,b≥0, a2-4b≥0, r1,2= (a±(?))/2. Then there exist two monotone sequences {an}a=0∞and {βn}n=0∞, nonincreasing and nondecreasing, respectively, with a0=α,β0=β, which converge uniformly to the extremal solution to Problem (2) in [β,α].
In Chapter 3, we use the upper and lower solutions method and cone-stretching and cone-compressing fixed-point theorems to introduce the existence of solutions to the multi-point boundary value problem of fourth-order equations. The problems under consideration include the case f= f(t, u), the case f= f(t, u, u") and the case f= f(t, u, u'u", u'"). We consider the following four-point boundary value problem: where a, b, c, d≥0 and 0≤ξ1≤ξ2≤1.
The main conclusion is the following theorem.
Theorem 6 If the following conditions hold:
(B1) a,b, c, d≥0,0≤ξ1≤ξ2≤1 and b-aξ1≥0, d-c+cξ2≥0,δ= ad+be+ ac(ξ2-ξ1)≠0;
(B2) f(t,u)∈C([0,1] x [0,∞),R+) is nondecreasing relative to u, f(t,t(1-t))(?) 0 for t∈(ξ1,ξ2) and there exists a constant 0<μ< 1 such that kμf(t, u)≤f(t, ku) for any 0≤k≤1. Then Problem (3) has at least one positive solution.
When the nonlinearity f= f(t, u, u"), denote
With the use of cone-stretching and cone-compressing fixed-point theorems one can prove the following theorem.
Theorem 7 Assume that f satisfies:
(C1)f∈C([0,1] x [0,∞) x (-∞,0], [0,∞));
(C2)f is sublinear, namely min f0=+∞, max f∞= 0.
Then Problem (3) has at least one positive solution.
Theorem 8 Assume (C1) and
(C3) f is superliner, namely max f0= 0, min f∞=+∞.
Then Problem (3) has at least one positive solution.
One can prove the existence of multiple solutions to Problem (3) by modifying the above conditions.
Assume that
(C4) min f0= min f∞,=+∞;
(C5) There exists an l1> 0 such that for all t∈[0,1], x∈[0,l1],-y∈[0,l1];
(C6) max f0= max f∞,= 0;
(C7) There exists an l2> 0 such that for any t∈[0,1], x∈[0,l2],-y∈[l2/4,l2].
Theorem 9 Assume (C1), (C4) and (C5). Then Problem (3) admits at least two posi-tive solutions u1, u2 satisfying 0<||u1||2< l1< ||u2||2.
Theorem 10 Assume (C1), (C6) and (C7). Then Problem (3) admits at least two positive solutions u1, u2 satisfying 0<||u1||2< l2< ||u2||2.
As for the following problem where a, b, c, d≥0,ρ= ad+be+ac> 0,f:[0,1]×R4→R is continuous. One can get the existence and uniqueness of solutions with the use of the upper and lower solutions method and Schauder fixed point theorem.
Theorem 11 Suppose that v,w are upper and lower solutions to Problem (4) such that w"(t)≥v"(t) and f satisfies the Nagumo condition with respect to w" and v". Then Problem (4) has at least one solution u such that w(t)≤u(t)≤v(t), w'(t)≤u'(t)≤v'(t), v"(t)≤u"(t)≤w"(t), t∈[0,1].
Theorem 12 Suppose that v, w are upper and lower solutions to Problem (4) and f satisfies the Nagumo condition with respect to w" and v". If f(t,x1,x2, x3, x4) is decreasing in x1,x2 and strictly increasing in x4, then Problem (4) has a unique solution u(t) such that w(t)≤u(t)≤v(t), w'(t)≤u'(t)≤v'(t), v"(t)≤u"(t)≤w"(t), t∈[0,1].
In Chapter 4, we study the existence, multiplicity and nonexistence of solutions to the following periodic boundary value problem
Assume that
(D1) For any givenβ,α∈C[0,2π] withβ(t)<α(t), t∈[0,2π], there exist 0< A< B such that A(v2-v1)< f(t, u, v2)-(t, u,v1)< B(v2-v1) for a.e. t∈[0,2π] wheneverβ(t)< u< a(t), v1,v2∈R, v1
Theorem 13 Suppose that there exist a lower solutionβ(t) and an upper solutinα(t) such thatβ(t)<α(t), t∈[0,2π], and f(t, u, v) is a Caratheodory function satisfying the hypotheses (D1) and (D2). Then there exist two sequences {βj}j=0∞and{αj}j=0∞nondecreas-ing and nonincreasing, respectively, withβ0=β,α0=α, which converge uniformly and monotonically to the extremal solutions to Problem (5) in [β,α].
When f satisfies only one-side Lipschitz condition, one can also prove the existence of solutions to Problem (5).
Theorem 14 Suppose that there exist a lower solutionβ(t) and an upper solutionα(t) such thatβ(t)≤a(t), t∈[0,2π], and f(t, u, v) is a Caratheodory function satisfying the hypothesis (D3):
(D3) For any givenβ,α∈C[0,2π],β(t)< a(t), t∈[0,2π], there exist a, b> 0 such that b2≥4a and f(t,u2,v2)-f(t,u1,v1)≥-a(u2-u1)+b(v2-v1); for a. e. t∈[0,2π] whenever B(t)≤u1≤u2≤α(t), v1, v2∈R, v1< v2.
Then Problem (5) has a solution u∈W4,1 [0,2π] satisfyingβ(t)≤u(t)≤α(t).
Theorem 15 Suppose that there exist a lower solutionβ(t)and an upper solutionα(t) such thatβ(t)≤α(t),t∈[0,2π],and f(t,u,v)satisfies the hypothesis
(D4)There exist C,D>0 such that D<4C+1/4,D2>4C and f(t,u2,V1)-f(t,u1,v2)≥-C(u2-u1)-D(v1-v2) for a.e.t∈[0,2π]wheneverβ(t)≤u1≤u2≤α(t),v1,v2∈R,v1≤v2. Then Problem(5)has a solution u∈W4,1[0,2π]satisfyingβ(t)≤u(t)≤α(t).
At the end of the thesis,we outline the results in studying the existence,multiplicity and nonexistence of solutions to the following problem under suitable conditions to f where f:[0,1]×R+→R+is continuous,a,b∈R such that 0
Denote (?)=(?) (?)=(?)
Theorem 16 Assume that one of the following two conditions holds: (E1)f0
Theorem 18 Suppose that there exist n+1 positive constants a1< a2<…
Then Problem (6) has no positive solution u∈K.
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