非线性奇异问题的正解和非平凡解
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摘要
随着社会的发展和科学技术的进步,人们广泛研究了越来越多的非线性问题.作为研究各种非线性问题的学科,非线性泛函分析是现代数学中既有深刻理论意义,又有广泛应用价值的研究方向.它以数学及自然科学领域中出现的非线性问题为背景,提供了处理许多非线性问题的诸多方法,包括拓扑度理论、锥理论、临界点理论和单调算子理论.很自然地,获得的研究成果可以广泛地应用于各种非线性微分方程、积分方程和其他类型的方程以及计算数学、控制理论、最优化理论、动力系统、经济数学中.
边值问题是非线性常微分方程和非线性偏微分方程理论研究中极为活跃且成果丰硕的领域.它源于应用数学、物理学和控制论等应用学科,因此,边值问题的研究具有重要的理论意义和应用价值.常微分方程边值问题通常可以转化成积分方程问题求解,故积分方程的研究也有重要的价值.
近几年,研究奇异微分方程方程的文章逐渐增多.以常微分方程奇异边值问题为例,这些文章仅能做到在区间[0,1]两端点处奇异,在整个区间[0,1]上是不能做到奇异的.本文利用锥理论、不动点指数理论和拓扑度理论,致力于研究奇异Sturm-Liouville边值问题、高阶奇异常微分方程积分边值问题、奇异Hammerstein积分方程问题正解或非平凡解的存在性,本文的创新之处在于:将奇异条件推广到整个区间[0,1],即不仅在[0,1]两端点处可奇异,在整个区间[0,1]内都可奇异,且非线性项的超线性和次线性增长条件都是用线性积分算子的第一特征值刻画的.这说明,所得结果在某种程度上是最优结果.
本文共分为四章:
在第一章中,我们用Krasnoselskii不动点定理研究下列奇异Sturm-Liouville问题的正解的存在性:(?)其中f∈C([0,1]×R+,R+), g∈L(0,1), g在[0,1]上可奇异且几乎处处非负,满足01 g(t)dt > 0, a∈C1([0,1],R+) ,b∈C([0,1],R+),αi≥0,βi≥0,αi2 +βi2 = 0(i=1,2);α,β在[0,1]上单增,在[0,1)上右连续,在t = 1处左连续,α(0) =β(0) = 0,γ0∈[0,π/2],γ1∈[0,π/2] .本章的研究价值在于:一方面,边值条件由两点边值条件或多点边值条件推广为一般的积分边值条件;另一方面,非线性项g允许在[0,1]上奇异.
在第二章中,我们利用拓扑度理论建立下列高阶非线性奇异常微分方程的非平凡解的存在性: (?)其中f∈C([0,1]×R,R), a∈L(0,1), a在[0,1]上可奇异且几乎处处非负,满足01 a(t)dt >0.在这一章还利用锥上的郭大钧-Krasoselskii不动点定理证明了高阶边值问题:(?)正解的存在性.其中引理2改正了葛渭高专著中的一个错误结果.
在第三章,首先用Krasnoselskii不动点定理研究非线性奇异Hammerstein积分方程:(?), a在[0,1]上可奇异且非负,满足01 a(t)dt > 0, k∈C([0,1]×[0,1],R+).对超线性和次线性条件下都做到了第一特征值,本质推广了和改进了现有文献的结果.作为应用,还讨论了一个二阶奇异Sturm-Liouville问题的正解及多重正解的存在性问题.本章还研究了扰动后的非线性奇异Hammerstein积分方程的正解的存在性:(?)其中f∈C([0,1]×R+,R+), g∈L(0,1), g在[0,1]上可奇异且几乎处处非负,满足01 g(t)dt > 0, k∈C([0,1]×[0,1],R+), ?i∈C[0,1],对t∈(0,1),满足?i(t) > 0,αi在[0,1]上单调不减,在[0,1)上右连续,在t = 1处左连续,且αi(0) = 0,i = 1,2,...,n.
在第四章中,用不动点指数理论研究了非线性二阶常微分方程组边值问题(?)正解的存在性,其中f1,f2∈C([0,1]×R+×R+,R+);αi为定义在[0,1]上的单增非常值函数,且αi(0) = 0; Hi∈C(R+,R+)(i = 1,2,3,4).所得结果改进和推广了现有文献的结果,本质不同于意大利数学家Gennaro Infante, Paolamaria Pietramal 2009年在文献[40]中的相应结果.
With the development of society and the progress of science and technology, moreand more nonlinear problems have been extensively investigated. As a result of this, aim-ing at dealing with various nonlinear problems, nonlinear functional analysis is not onlyof profound theoretical significance but also is widely applied. Based on various nonlinearproblems arising in mathematical sciences themselves and natural sciences, it provides uswith many methods for tackling all kinds of nonlinear problems, such as topological de-gree theory, cone theory, critical point theory and monotone operator theory. Naturally,the results established therein can be widely applied to various nonlinear equations, in-cluding di?erential and integral, as well as to computational mathematics, control theory,optimization theory, dynamical systems, economic mathematics, etc.
The theory of boundary value problems for nonlinear ordinary and partial di?erentialequations is among the most active and fruitful fields. Such problems can find roots inapplied mathematics, physics, control theory, and other applied sciences. Therefore, theresearch of boundary value problems is not only of great theoretical significance and butalso of wide applicability. A boundary value problem for a nonlinear ordinary di?erentialequation, usually, can be transformed into an equivalent integral equation. This meansthat the study of integral equations is also important.
In recent years, many papers have been published on boundary value problems forsingular ordinary di?erential equations, but with nonlinearities being merely singular atone or two endpoints of the interval [0,1] instead of being singular on the whole interval[0,1]. By using cone theory, topological degree theory and fixed point index theory, in thisthesis, we study existence of positive or nontrivial solutions for singular Sturm-Liouvilleboundary value problems, higher order singular problems with integral boundary condi-tions, and singular Hammerstein integral equations. The novelty of this thesis include twoaspects. First, our nonlinearities may be singular on the whole interval [0,1], in contrastto ones in the existing literature, which are only allowed to be singular at one or two end-points of [0,1]. Second, growth conditions of our nonlinearities, superlinear and sublinear,are described in terms of first eigenvalues of associated linear integral operators. Thismeans that our results presented here are optimal in some sense.This thesis contains four chapters.
In Chapter One, by using the Krosnoselskii fixed point theorem, we study the ex- istence of positive solutions for the following singular Sturm-Liouville boundary valueproblem (?)ingular and is almost everywhere positiveon [0,1] with 01 g(t)dt > 0; a∈C1([0,1],R+),b∈C([0,1],R+),αi≥0,βi≥0,αi2 +βi2 =0(i = 1,2);α,βare increasing on [0,1] and right continuous on∈[0,1), left continuousat t = 1 withα(0) =β(0) = 0;γ0∈[0,π/2],γ1∈[0,π/2]. The novelty of this is twofold.
First, our boundary conditions are expressed in terms of Riemann-Stieltjes integrals, incontrast to two-point or multipoint boundary conditions in the literature. Second, ourweight function g may be singular on the whole interval [0,1].
In Chapter Two, by using topological degree theory, we first investigate the existenceof nontrivial solutions for the following higher order nonlinear singular boundary valueproblem (?)where f∈C([0,1]×R,R), and a∈L(0,1) may be singular on [0,1], with a > 0 a.e. [0,1]and 01 a(t)dt > 0.
Next, by using the Guo-Krasoselskii fixed point theorem, we study the existence ofpositive solutions for the following higher order boundary value problem(?)Our Lemma 2 in this chapter corrects an error in monograph [13, section6.5.1] due to GeWeigao.
In Chapter Three, by using the Krosnoselskii fixed point theorem, we first discussthe existence and multiplicity of positive solutions for the following nonlinear singular Hammerstein integral equation(?)where f∈C([0,1]×R+,R+), a∈L(0,1) may be singular and almost nonnegative every-where on [0,1] with 01 a(t)dt > 0, and k∈C([0,1]×[0,1],R+). Our main results for boththe superlinear case and the sublinear case are described in terms of first eigenvalues ofassociated linear integral operators, extending and improving the results in the existingliteratures essentially. As applications, we apply our main results to discuss the existenceand mutiplicity of positive solutions for a second-order singular Sturm-Liouville problem.Next, we study the existence of positive solutions for the following perturbed Hammersteinintegral equation(?)where f∈C([0,1]×R+,R+), and g∈L(0,1) is nonnegative and is allowed to be singularon [0,1] with 01 g(t)dt > 0; k∈C([0,1]×[0,1],R+); ?i∈C[0,1] satisfies ?i(t) > 0 forall t∈(0,1);αi is increasing on [0,1] and right continuous on∈[0,1), left continuous att = 1 withαi(0) = 0,i = 1,2,...,n.
In Chapter Four, by using fixed point index theory, we study the existence of positivesolutions for the following systems of nonlinear second-order ordinary di?erential equations(?)where f1,f2∈C([0,1]×R+×R+,R+);αi is an increasing and nonconstant function on[0,1] withαi(0) = 0(i = 1,2,3,4); Hi∈C(R+,R+)(i = 1,2,3,4). Our main resultsobtained extend and improve ones in the existing literatures, and are di?erent from thecorresponding ones due to G. Infante and P. Pietramala in 2009.
边值问题是非线性常微分方程和非线性偏微分方程理论研究中极为活跃且成果丰硕的领域.它源于应用数学、物理学和控制论等应用学科,因此,边值问题的研究具有重要的理论意义和应用价值.常微分方程边值问题通常可以转化成积分方程问题求解,故积分方程的研究也有重要的价值.
近几年,研究奇异微分方程方程的文章逐渐增多.以常微分方程奇异边值问题为例,这些文章仅能做到在区间[0,1]两端点处奇异,在整个区间[0,1]上是不能做到奇异的.本文利用锥理论、不动点指数理论和拓扑度理论,致力于研究奇异Sturm-Liouville边值问题、高阶奇异常微分方程积分边值问题、奇异Hammerstein积分方程问题正解或非平凡解的存在性,本文的创新之处在于:将奇异条件推广到整个区间[0,1],即不仅在[0,1]两端点处可奇异,在整个区间[0,1]内都可奇异,且非线性项的超线性和次线性增长条件都是用线性积分算子的第一特征值刻画的.这说明,所得结果在某种程度上是最优结果.
本文共分为四章:
在第一章中,我们用Krasnoselskii不动点定理研究下列奇异Sturm-Liouville问题的正解的存在性:(?)其中f∈C([0,1]×R+,R+), g∈L(0,1), g在[0,1]上可奇异且几乎处处非负,满足01 g(t)dt > 0, a∈C1([0,1],R+) ,b∈C([0,1],R+),αi≥0,βi≥0,αi2 +βi2 = 0(i=1,2);α,β在[0,1]上单增,在[0,1)上右连续,在t = 1处左连续,α(0) =β(0) = 0,γ0∈[0,π/2],γ1∈[0,π/2] .本章的研究价值在于:一方面,边值条件由两点边值条件或多点边值条件推广为一般的积分边值条件;另一方面,非线性项g允许在[0,1]上奇异.
在第二章中,我们利用拓扑度理论建立下列高阶非线性奇异常微分方程的非平凡解的存在性: (?)其中f∈C([0,1]×R,R), a∈L(0,1), a在[0,1]上可奇异且几乎处处非负,满足01 a(t)dt >0.在这一章还利用锥上的郭大钧-Krasoselskii不动点定理证明了高阶边值问题:(?)正解的存在性.其中引理2改正了葛渭高专著中的一个错误结果.
在第三章,首先用Krasnoselskii不动点定理研究非线性奇异Hammerstein积分方程:(?), a在[0,1]上可奇异且非负,满足01 a(t)dt > 0, k∈C([0,1]×[0,1],R+).对超线性和次线性条件下都做到了第一特征值,本质推广了和改进了现有文献的结果.作为应用,还讨论了一个二阶奇异Sturm-Liouville问题的正解及多重正解的存在性问题.本章还研究了扰动后的非线性奇异Hammerstein积分方程的正解的存在性:(?)其中f∈C([0,1]×R+,R+), g∈L(0,1), g在[0,1]上可奇异且几乎处处非负,满足01 g(t)dt > 0, k∈C([0,1]×[0,1],R+), ?i∈C[0,1],对t∈(0,1),满足?i(t) > 0,αi在[0,1]上单调不减,在[0,1)上右连续,在t = 1处左连续,且αi(0) = 0,i = 1,2,...,n.
在第四章中,用不动点指数理论研究了非线性二阶常微分方程组边值问题(?)正解的存在性,其中f1,f2∈C([0,1]×R+×R+,R+);αi为定义在[0,1]上的单增非常值函数,且αi(0) = 0; Hi∈C(R+,R+)(i = 1,2,3,4).所得结果改进和推广了现有文献的结果,本质不同于意大利数学家Gennaro Infante, Paolamaria Pietramal 2009年在文献[40]中的相应结果.
With the development of society and the progress of science and technology, moreand more nonlinear problems have been extensively investigated. As a result of this, aim-ing at dealing with various nonlinear problems, nonlinear functional analysis is not onlyof profound theoretical significance but also is widely applied. Based on various nonlinearproblems arising in mathematical sciences themselves and natural sciences, it provides uswith many methods for tackling all kinds of nonlinear problems, such as topological de-gree theory, cone theory, critical point theory and monotone operator theory. Naturally,the results established therein can be widely applied to various nonlinear equations, in-cluding di?erential and integral, as well as to computational mathematics, control theory,optimization theory, dynamical systems, economic mathematics, etc.
The theory of boundary value problems for nonlinear ordinary and partial di?erentialequations is among the most active and fruitful fields. Such problems can find roots inapplied mathematics, physics, control theory, and other applied sciences. Therefore, theresearch of boundary value problems is not only of great theoretical significance and butalso of wide applicability. A boundary value problem for a nonlinear ordinary di?erentialequation, usually, can be transformed into an equivalent integral equation. This meansthat the study of integral equations is also important.
In recent years, many papers have been published on boundary value problems forsingular ordinary di?erential equations, but with nonlinearities being merely singular atone or two endpoints of the interval [0,1] instead of being singular on the whole interval[0,1]. By using cone theory, topological degree theory and fixed point index theory, in thisthesis, we study existence of positive or nontrivial solutions for singular Sturm-Liouvilleboundary value problems, higher order singular problems with integral boundary condi-tions, and singular Hammerstein integral equations. The novelty of this thesis include twoaspects. First, our nonlinearities may be singular on the whole interval [0,1], in contrastto ones in the existing literature, which are only allowed to be singular at one or two end-points of [0,1]. Second, growth conditions of our nonlinearities, superlinear and sublinear,are described in terms of first eigenvalues of associated linear integral operators. Thismeans that our results presented here are optimal in some sense.This thesis contains four chapters.
In Chapter One, by using the Krosnoselskii fixed point theorem, we study the ex- istence of positive solutions for the following singular Sturm-Liouville boundary valueproblem (?)ingular and is almost everywhere positiveon [0,1] with 01 g(t)dt > 0; a∈C1([0,1],R+),b∈C([0,1],R+),αi≥0,βi≥0,αi2 +βi2 =0(i = 1,2);α,βare increasing on [0,1] and right continuous on∈[0,1), left continuousat t = 1 withα(0) =β(0) = 0;γ0∈[0,π/2],γ1∈[0,π/2]. The novelty of this is twofold.
First, our boundary conditions are expressed in terms of Riemann-Stieltjes integrals, incontrast to two-point or multipoint boundary conditions in the literature. Second, ourweight function g may be singular on the whole interval [0,1].
In Chapter Two, by using topological degree theory, we first investigate the existenceof nontrivial solutions for the following higher order nonlinear singular boundary valueproblem (?)where f∈C([0,1]×R,R), and a∈L(0,1) may be singular on [0,1], with a > 0 a.e. [0,1]and 01 a(t)dt > 0.
Next, by using the Guo-Krasoselskii fixed point theorem, we study the existence ofpositive solutions for the following higher order boundary value problem(?)Our Lemma 2 in this chapter corrects an error in monograph [13, section6.5.1] due to GeWeigao.
In Chapter Three, by using the Krosnoselskii fixed point theorem, we first discussthe existence and multiplicity of positive solutions for the following nonlinear singular Hammerstein integral equation(?)where f∈C([0,1]×R+,R+), a∈L(0,1) may be singular and almost nonnegative every-where on [0,1] with 01 a(t)dt > 0, and k∈C([0,1]×[0,1],R+). Our main results for boththe superlinear case and the sublinear case are described in terms of first eigenvalues ofassociated linear integral operators, extending and improving the results in the existingliteratures essentially. As applications, we apply our main results to discuss the existenceand mutiplicity of positive solutions for a second-order singular Sturm-Liouville problem.Next, we study the existence of positive solutions for the following perturbed Hammersteinintegral equation(?)where f∈C([0,1]×R+,R+), and g∈L(0,1) is nonnegative and is allowed to be singularon [0,1] with 01 g(t)dt > 0; k∈C([0,1]×[0,1],R+); ?i∈C[0,1] satisfies ?i(t) > 0 forall t∈(0,1);αi is increasing on [0,1] and right continuous on∈[0,1), left continuous att = 1 withαi(0) = 0,i = 1,2,...,n.
In Chapter Four, by using fixed point index theory, we study the existence of positivesolutions for the following systems of nonlinear second-order ordinary di?erential equations(?)where f1,f2∈C([0,1]×R+×R+,R+);αi is an increasing and nonconstant function on[0,1] withαi(0) = 0(i = 1,2,3,4); Hi∈C(R+,R+)(i = 1,2,3,4). Our main resultsobtained extend and improve ones in the existing literatures, and are di?erent from thecorresponding ones due to G. Infante and P. Pietramala in 2009.
引文
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