矩阵方程的正定解和分数阶微分方程的谱问题
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摘要
微分方程的研究是伴随着微积分的出现而发展起来的,具有300多年的悠久历史。微分方程领域研究内容丰富,是研究自然现象强有力的数学工具之一,同时也与其它学科有着紧密的联系,在自然科学、环境生态、工程技术、社会经济等方面有着广泛的应用。本文主要研究与微分方程理论和相关应用具有紧密联系的两大方面的内容,即矩阵方程的正定解及其扰动分析和分数阶微分方程的谱问题,其研究动机和内容详述如下。
一方面对于许多来源于实际问题的用常微分方程,积分方程,积分微分方程刻画的数学物理问题,常用的一种处理方法是通过差分方法将其离散化,从而可以把原问题转化为某种矩阵方程来进行研究.矩阵方程是矩阵理论和数值代数领域的重要内容之一。近些年来,形式如X-∑i=1m Ai*Xpi Ai=Q的非线性矩阵方程由于来源广泛,包括微分方程,物理计算中的大型线性方程组,插值理论的极值问题,控制理论,梯形网络,动态规划,随机滤波等而引起了众多学者的关注,对这类方程正定解的相关理论和数值方法的研究已经取得了一系列的成果.由于正定解在实际问题中应用较多,所以我们只考虑矩阵方程的正定解。对此类非线性矩阵方程的研究主要考虑三方面内容：
(i)存在正定解的充分和必要条件；
(ii)求得正定解的有效的数值方法；
(iii)关于方程正定解的扰动分析,包括扰动界,条件数,后向误差,剩余界.
我们知道非线性矩阵方程的可解性是进行数值求解的理论基础,而有效的数值求解方法为定量求出解提供了可行的计算过程。由于非线性矩阵方程来源于工程和物理中的大数据计算问题,在求解过程中通常存在两类误差影响计算结果的精度,即数值计算方法引起的截断误差和计算环境引起的舍入误差,为了分析这些误差对原问题的解的影响,我们需要研究原始数据的扰动对解的影响,从而需要对矩阵方程的正定解进行扰动分析。用扰动界和条件数来揭示矩阵方程自身的稳定性,用后向误差和剩余界检验算法的数值稳定性和估计近似解的精确程度。
本文在已有成果的基础上,研究下面形式的非线性矩阵方程：
1.非线性矩阵方程X-∑i=1m Ai*Xpi Ai=Q(pi>0)来源于控制系统最优问题和插值理论最优问题,对此矩阵方程我们分m=1和m>1两种情况进行讨论.
(i)当m=1时,原方程退化为非线性矩阵方程X-A*X-p A=Q(p>0),此时我们对p≥1和00)的积分表示,推导出关于方程唯一正定解的扰动边界和由Rice定义的条件数的显式表达式。当0 (ii)当m>1时,对非线性矩阵方程X-∑i=1m Ai*X-pi Ai=Q,当pi>0时,我们讨论了此方程的可解性,得到了方程存在正定解的充分和必要条件,并且推导出方程存在唯一正定解的条件,构造了得到此唯一正定解的迭代方法.利用不动点定理,Kronecker积和矩阵范数的性质,我们给出了方程正定解的扰动界和方程近似解的后向误差估计并且利用矩阵函数X-p (p>0)的积分表示和算子理论,推导出方程正定解的条件数的显式表达式。特别地,当0 2.非线性矩阵方程X-∑i=1m Ai*Xpi Ai=Q来源于数学物理问题,对此矩阵方程当pi>0时,我们先利用矩阵分解定理给出方程存在正定解的充分必要条件,然后分01两种情况进行讨论.当01时,我们推导出方程存在唯一正定解的充分条件,而且得到了方程正定解的一些性质,利用矩阵函数在Frobenius范数下的不等式,推导出方程正定解的一个扰动界.
另一方面,对微分方程的一个重要的研究方法是通过研究微分方程谱的性质来对微分方程的解进行研究,也就是微分方程谱理论.整数阶微分方程谱问题也称为Sturm-Liouville问题,其相关理论于170多年前被提出来,自此它的相关理论在诸如科学,工程和数学领域占据重要地位。Sturm-Liouville问题源于常微分方程边值问题,而常微分方程的边值问题一部分直接来源于现实问题本身,另外的一大部分是源于偏微分方程,如热传导(或扩散)问题、弦(膜)振动问题、电磁学中的Maxwell方程问题等。进入19世纪,Fourier系统地提出了分离变量法,在将这一方法应用于更为复杂的物理现象产生的那些偏微分方程问题时,就会产生两个或多个常微分方程的边值问题。从算子的角度来看,Sturm-Liouville算子是一类十分重要的微分算子,在经典微分算子和近代量子物理学中均有重要的应用背景。另外,自二十世纪末开始,分数阶微积分理论的迅速发展和应用的日趋广泛,促进了分数阶微分方程的出现和发展。人们发现将分数阶微积分的观点引入微分方程更能准确地描述事物的变化规律和本质属性,于是分数阶微分方程在实际中有了广泛应用,如：分形动力学、连续力学、自动控制、流体力学、生物力学、粘弹性力学、量子力学、统计学、工程学、布朗运动、地震分析、神经的分数模型和描述种群繁殖的数学模型等。因此,分数阶微分方程越来越多的引起数学家的关注。很多时候,为了实际问题的需求也要考虑分数阶微分方程的谱问题。研究分数阶微分方程的谱问题既是解决实际问题的需要,同时又能丰富和完善分数阶微分方程的相关理论。分数阶微分方程的谱问题被相关学者提出后,一直没有得到深入的研究,目前,有学者用数值计算的方法来研究分数阶微分方程的谱问题,但就连普通的特征值和特征函数的性质也没有从理论上加以说明。到目前为止,从理论上讨论分数阶微分方程特征值和特征函数性质的文章非常少,基于此,本文主要讨论下面的分数阶微分方程谱问题：的谱问题,其中q∈L2(0.1)是实值函数,D0+α和D1-α分别是α阶的左,右分数阶Riemann-Liouville导数,1<α<3/2,μ是实数,λ是谱参数。基于Hilbert空间中的自伴紧算子的谱理论,我们证明了此谱问题的谱仅有可数个有限重的实特征值,相应的特征函数在Hilbert空间中构成完备正交系,并且估计出特征值的下界。的谱问题.其中q∈L2(0，1)是实值函数D0+α和D1-α分别是α阶的左,右分数阶Riemann-Liouville导数0<α<1/2,μ是实数,λ是谱参数。我们利用Hilbert空间中的自伴紧算子的谱理论证明了此类谱问题的谱仅有可数个有限重的实特征值,相应的特征函数在Hilbert空间中构成完备正交系,并且估计出特征值的下界。
关于仅含左或右分数阶导数的微分方程的初值问题的理论已经相当完善,解的存在性,解关于参数的连续依赖性,可微性及解的延拓定理已经建立,但对同时含有左、右分数阶导数的微分方程,其“初值问题”的提法不甚清楚,现有的关于分数阶微分方程的专著及相关文献中也很少陈述。但在整数阶微分方程边值问题的研究中,其相应的初值问题的理论是一种十分有效的方法和工具,例如Prufer变换,解关于参数的可微性等。基于此,本章我们首先提出了同时含左、右分数阶导数的微分方程的两类“初值问题”,在适当的条件下我们可以证明此类“初值问题”解的存在性和唯一性,尔后,利用上述结果我们专门研究了特征值问题中特征值的几何重数,建立了特征值为单的一系列问题：
1.在区间(0,1)上首先建立分数阶微分方程初值问题的相关结论,其中q∈L(0,1)是实值函数,D0+α和D1-α分别是α阶的左,右分数阶Riemann-Liouville导数,0<α<1,μ,λ是实数.而后利用这些结论证明了分数阶微分方程的特征值的几何重数是单的.
2.在区间(0,1)上首先建立分数阶微分方程初值问题的相关结论,其中q∈L(0,1)是实值函数,D0+α和CD1-α分别是a阶的左分数阶Riemann-Liouville导数和a阶的右分数阶Caputo导数,0<α<1/2,μ,λ是实数。而后利用这些结论证明了分数阶微分方程的特征值的几何重数是单的.
The research on differential equations has been developed along with the appear-ance of Calculus with a history of more than300years. There are plentiful research fields about differential equations which are considered one of the most powerful math-ematic tools in studying natural phenomena. The research in this field is closely related with other subjects and widely applied in the fields of natural science, ecological en-vironment, engineering technology and social economy. This paper mainly focuses on two aspects closely related to the theory and application of differential equations, which refers to positive definite solutions and perturbation analysis of matrix equations as well as the spectral problems of fractional differential equations. The motive and content of the research are detailed as follows.
On the one hand, as to many mathematical physics problems originated from prac-tical issues, which are depicted by ordinary differential equations, integral equations and integral differential equations, the usually applied method is to discretize them through the difference method and therefore the original problem can be transformed into a certain kind of matrix equations to be researched. Matrix equation is one of the important substance of the matrix theory and numerical algebra. In recent years there has been a constantly increasing interest in developing the theory and numerical approaches of the positive definite solutions for the nonlinear matrix equation of the form X-∑i=1m Ai*X-pi Ai=Q. This kind of matrix equations arises in solving a large-scale system of linear equations in many physical calculations, extremal interpolation problem, differential equations, control theory, ladder network, dynamic programming, stochastic filtering and so on. From the application point of view, the positive definite solutions are more important. In the sequel, a solution always means a positive definite one. The research results of this kind of matrix equations mainly concentrate on three basic problems:
(i) Necessary and sufficient conditions for the existence of positive definite solutions;
(ⅱ) Effective numerical ways for obtaining the positive definite solutions;
(ⅲ) Perturbation analysis of the positive definite solutions, including perturbation bound, condition number, backward error, residual bound,
where the solvability is the theoretical basis of the numerical solution for the nonlinear matrix equation, and numerical solution methods provide a feasible calculation process. Since nonlinear matrix equations stem from calculation problems with mass data in engineering and physics, there usually exist two kinds of errors during solution proce-dure influencing the accuracy of the results which refer to truncation errors caused by the numerical calculation method and rounding errors caused by calculating environ-ment. In order to analyze the impact of errors on the solutions of original problems, one needs to study the impact on the solution from the turbulence of original data, therefore the perturbation analysis of positive definite solutions for matrix equations is required. The perturbation bound and condition number will be applied to illustrate the stability of matrix equations, while the backward error and residual bound will be applied to check the numeral stability of the algorithm and the accuracy of the ap-proximately estimated solutions. Motivated by the work and applications of this kind of matrix equations, we study the following nonlinear matrix equations.
1. For the nonlinear matrix equation X-∑Ai*X-pi Ai=Q(pi>0), we consider two cases:the case m=1and the case m>1.
(i) When m=1, the above equation can be reduced to X-A*X-p A=Q(p>0). Two cases are considered here:the case p≥1and the case00), we derive a perturbation es-timate for the positive definite solution. Moreover, explicit expressions of the condition number defined by Rice for the positive definite solution are giv-en. In the case0 (ii) When m>1, some necessary and sufficient conditions for the existence of pos-itive definite solutions of the matrix equation X-∑Ai*X-pi Ai=Q(pi>0) are obtained. Simultaneously, the condition that the equation has a unique positive definite solution is also given. An effective iterative method to obtain the unique solution is established. By using Brouwer's fixed point theorem, the properties of the Kronecker product and matrix norm, we evaluate a perturba-tion bound and a backward error of an approximate solution to this equation. Moreover, based on the integral representation of matrix function X-p (p>0) and operator theory, we obtain the explicit expressions of the condition number for the positive definite solution. When0 2. For the nonlinear matrix equation X-∑Ai*X-pi Ai=Q(pi>0), we obtain necessary and sufficient conditions for the existence of positive definite solutions of this matrix equation. we consider two cases:the case00.
(i) In the case0 (ii) In the case pi>1, sufficient conditions for the existence of a unique positive definite solutions for the matrix equation are obtained. Some properties of positive definite solutions are derived. By using the elegant properties of the Frobenius norm, we derive a perturbation bound for this matrix equation.
On the other hand, one important method to deal with differential equations is to study their solutions through the research on the nature of their spectrums, i.e. the spectral theory of differential equations. The Sturm-Liouville problem is the basic problem in the spectral theory of differential equations and its related theories were raised170years ago, and since then its related theories have played important roles in scientific, engineering and mathematical fields.
Sturm-Liouville problem is originated from the boundary value problem of ordi-nary differential equations which are partly from practical issues directly and partly from the problems of partial differential equations, such as the heat conduction (or dif-fusion) problem, the string (film) vibration problem and the Maxwell equation problem in electromagnetics. At the beginning of the19th century, Fourier systematically put forward the method of separation of variables and applied this method to problems of partial differential equations equations with boundary value conditions caused by more complicated physical phenomena. From operator point of view, Sturm-Liouville operator is an extremely important kind which has significant application background in both classical differential operators and modern quantum physics. Besides, since the end of the20th century, the rapid development and widespread application of the fractional calculus theory have promoted the appearance and growth of fractional dif-ferential equations. It has been discovered that to introduce the concept of fractional calculus can describe the change rule and essential attribute of objects, which brought to the wide application of fractional differential equations in reality, such as in the fields of fractal dynamics, continuum mechanics, auto control, hydrodynamics, biomechanics, viscoelastic mechanics, quantum mechanics, statistics, engineering science, Brownian motion, earthquake analysis, fractional models of nerve and mathematical models de-scribing the population reproduction. Therefore, fractional differential equations have more and more drawn the interest of mathematicians. The research on the theory of fractional differential equations can both enrich existing mathematical theories and provide better mathematical models for studies in physical, biological and economical process and phenomena. In many cases, the consideration in the spectrum problem of fractional differential equations is needed to deal with practical issues. The study on spectrum problem of fractional equations can satisfy practical requirement as well as enrich and perfect related theories of fractional differential equations. Since the spec-trum problem of fractional differential equations was raised in early research papers, it has never been researched intensively. Recently, many scholars have applied the method of numerical calculation to study this problem, whereas even the natures of eigenvalue and eigenfunction have not been theoretically illustrated. Until now, there have been very few articles theoretically referring to the natures of eigenvalue and eigenfunction in fractional differential equations. Based on these findings, this paper will mainly focus on following spectrum problems of fractional differential equations:
1. The spectral problem is considered, where q∈L2(0,1) is a real-valued function, D0+α and D1-α are the left and right Riemann-Liouville fractional derivatives of order α, respectively,1<α<3/2, μ is a real constant and λ is the spectral parameter. This equa-tion aries from the non-local continuum mechanics. It is a governing equilibrium equation of an elastic bar of finite length L with long-range interactions among non-adjacent particles. In this study, by using the spectral theory of self-adjoint compact operators in Hilbert spaces, we prove that the spectrum of the spectral problem associated to this equation with1<α<3/2consists of only countable and real eigenvalues with finite multiplicity and the orthogonal completeness of the corresponding eigenfunction system in the Hilbert spaces. Furthermore, we obtain a lower bound of the eigenvalues.
2. The spectral problem is considered, where q∈L2(0,1) is a real-valued function, D0+α, D1-α are respectively the left and right Riemann-Liouville fractional derivatives of order α,0<α<1/2, μ is a real constant and A is the spectral parameter. We prove that the spectrum of this problem consists of only real eigenvalues and its eigenfunctions corresponding to different eigenvalues are orthogonal by the spectral theory of self-adjoint compact operators in Hilbert spaces. Furthermore, we demonstrate that the real eigenvalues of this equation are countable and each eigenvalue has finite multiplicity. We also prove that the set of all corresponding orthogonal eigenfunctions forms a complete system. A lower bound of the eigenvalues is obtained.
The theory about initial value problems of differential equations involving only left or right fractional derivatives has been considerably perfect. The existence of solutions, the consistent dependence on parameters of solutions, the differentiability of solutions and the continuation theorem have been established. However, the initial value prob-lems of differential equations with both left and right fractional derivatives have not been put forward clearly enough, and seldom mentioned in the existing publications and papers on fractional differential equations. Nevertheless, on the research of bound-ary problems of differential equations with integer orders, the corresponding theory of initial value problems proves to be a very effective method and tool, such as the Prufer transformation and the differentiability of solutions on parameters. Therefore, in this paper, we first put forward the initial value problems of differential equations with both left and right fractional derivatives. Under proper conditions, we can prove the existence and uniqueness of solutions for this kind of initial value problems. Then, ap-plying results above, we specifically research the geometric multiplicity of eigenvalues for eigenvalue problems and form a series of problems with single eigenvalue.
1. First, the initial value problem is considered, where q∈L(0,1) is a real-valued function, D0+α and D1-α are the left and right Riemann-Liouville fractional derivatives of order α, respectively,0<α <1, μ and λ are real constants. The uniqueness of the solution for this kind of initial value problem is proved. And then, by means of the above results, we prove the eigenvalue of eigenvalue problem is simple.
2. The initial value problem is considered, where q∈L(0,1) is a real-valued function, D0+α is the left Riemann-Liouville fractional derivatives of order a and D1-α is the right Caputo fractional derivatives of order a, respectively,0<α<1/2,μ and λ are real constants. The uniqueness of the solution for this kind of initial value problem is proved. And then, by means of the above results, we prove the eigenvalue of eigenvalue problem is simple.
一方面对于许多来源于实际问题的用常微分方程,积分方程,积分微分方程刻画的数学物理问题,常用的一种处理方法是通过差分方法将其离散化,从而可以把原问题转化为某种矩阵方程来进行研究.矩阵方程是矩阵理论和数值代数领域的重要内容之一。近些年来,形式如X-∑i=1m Ai*Xpi Ai=Q的非线性矩阵方程由于来源广泛,包括微分方程,物理计算中的大型线性方程组,插值理论的极值问题,控制理论,梯形网络,动态规划,随机滤波等而引起了众多学者的关注,对这类方程正定解的相关理论和数值方法的研究已经取得了一系列的成果.由于正定解在实际问题中应用较多,所以我们只考虑矩阵方程的正定解。对此类非线性矩阵方程的研究主要考虑三方面内容：
(i)存在正定解的充分和必要条件；
(ii)求得正定解的有效的数值方法；
(iii)关于方程正定解的扰动分析,包括扰动界,条件数,后向误差,剩余界.
我们知道非线性矩阵方程的可解性是进行数值求解的理论基础,而有效的数值求解方法为定量求出解提供了可行的计算过程。由于非线性矩阵方程来源于工程和物理中的大数据计算问题,在求解过程中通常存在两类误差影响计算结果的精度,即数值计算方法引起的截断误差和计算环境引起的舍入误差,为了分析这些误差对原问题的解的影响,我们需要研究原始数据的扰动对解的影响,从而需要对矩阵方程的正定解进行扰动分析。用扰动界和条件数来揭示矩阵方程自身的稳定性,用后向误差和剩余界检验算法的数值稳定性和估计近似解的精确程度。
本文在已有成果的基础上,研究下面形式的非线性矩阵方程：
1.非线性矩阵方程X-∑i=1m Ai*Xpi Ai=Q(pi>0)来源于控制系统最优问题和插值理论最优问题,对此矩阵方程我们分m=1和m>1两种情况进行讨论.
(i)当m=1时,原方程退化为非线性矩阵方程X-A*X-p A=Q(p>0),此时我们对p≥1和0
另一方面,对微分方程的一个重要的研究方法是通过研究微分方程谱的性质来对微分方程的解进行研究,也就是微分方程谱理论.整数阶微分方程谱问题也称为Sturm-Liouville问题,其相关理论于170多年前被提出来,自此它的相关理论在诸如科学,工程和数学领域占据重要地位。Sturm-Liouville问题源于常微分方程边值问题,而常微分方程的边值问题一部分直接来源于现实问题本身,另外的一大部分是源于偏微分方程,如热传导(或扩散)问题、弦(膜)振动问题、电磁学中的Maxwell方程问题等。进入19世纪,Fourier系统地提出了分离变量法,在将这一方法应用于更为复杂的物理现象产生的那些偏微分方程问题时,就会产生两个或多个常微分方程的边值问题。从算子的角度来看,Sturm-Liouville算子是一类十分重要的微分算子,在经典微分算子和近代量子物理学中均有重要的应用背景。另外,自二十世纪末开始,分数阶微积分理论的迅速发展和应用的日趋广泛,促进了分数阶微分方程的出现和发展。人们发现将分数阶微积分的观点引入微分方程更能准确地描述事物的变化规律和本质属性,于是分数阶微分方程在实际中有了广泛应用,如：分形动力学、连续力学、自动控制、流体力学、生物力学、粘弹性力学、量子力学、统计学、工程学、布朗运动、地震分析、神经的分数模型和描述种群繁殖的数学模型等。因此,分数阶微分方程越来越多的引起数学家的关注。很多时候,为了实际问题的需求也要考虑分数阶微分方程的谱问题。研究分数阶微分方程的谱问题既是解决实际问题的需要,同时又能丰富和完善分数阶微分方程的相关理论。分数阶微分方程的谱问题被相关学者提出后,一直没有得到深入的研究,目前,有学者用数值计算的方法来研究分数阶微分方程的谱问题,但就连普通的特征值和特征函数的性质也没有从理论上加以说明。到目前为止,从理论上讨论分数阶微分方程特征值和特征函数性质的文章非常少,基于此,本文主要讨论下面的分数阶微分方程谱问题：的谱问题,其中q∈L2(0.1)是实值函数,D0+α和D1-α分别是α阶的左,右分数阶Riemann-Liouville导数,1<α<3/2,μ是实数,λ是谱参数。基于Hilbert空间中的自伴紧算子的谱理论,我们证明了此谱问题的谱仅有可数个有限重的实特征值,相应的特征函数在Hilbert空间中构成完备正交系,并且估计出特征值的下界。的谱问题.其中q∈L2(0，1)是实值函数D0+α和D1-α分别是α阶的左,右分数阶Riemann-Liouville导数0<α<1/2,μ是实数,λ是谱参数。我们利用Hilbert空间中的自伴紧算子的谱理论证明了此类谱问题的谱仅有可数个有限重的实特征值,相应的特征函数在Hilbert空间中构成完备正交系,并且估计出特征值的下界。
关于仅含左或右分数阶导数的微分方程的初值问题的理论已经相当完善,解的存在性,解关于参数的连续依赖性,可微性及解的延拓定理已经建立,但对同时含有左、右分数阶导数的微分方程,其“初值问题”的提法不甚清楚,现有的关于分数阶微分方程的专著及相关文献中也很少陈述。但在整数阶微分方程边值问题的研究中,其相应的初值问题的理论是一种十分有效的方法和工具,例如Prufer变换,解关于参数的可微性等。基于此,本章我们首先提出了同时含左、右分数阶导数的微分方程的两类“初值问题”,在适当的条件下我们可以证明此类“初值问题”解的存在性和唯一性,尔后,利用上述结果我们专门研究了特征值问题中特征值的几何重数,建立了特征值为单的一系列问题：
1.在区间(0,1)上首先建立分数阶微分方程初值问题的相关结论,其中q∈L(0,1)是实值函数,D0+α和D1-α分别是α阶的左,右分数阶Riemann-Liouville导数,0<α<1,μ,λ是实数.而后利用这些结论证明了分数阶微分方程的特征值的几何重数是单的.
2.在区间(0,1)上首先建立分数阶微分方程初值问题的相关结论,其中q∈L(0,1)是实值函数,D0+α和CD1-α分别是a阶的左分数阶Riemann-Liouville导数和a阶的右分数阶Caputo导数,0<α<1/2,μ,λ是实数。而后利用这些结论证明了分数阶微分方程的特征值的几何重数是单的.
The research on differential equations has been developed along with the appear-ance of Calculus with a history of more than300years. There are plentiful research fields about differential equations which are considered one of the most powerful math-ematic tools in studying natural phenomena. The research in this field is closely related with other subjects and widely applied in the fields of natural science, ecological en-vironment, engineering technology and social economy. This paper mainly focuses on two aspects closely related to the theory and application of differential equations, which refers to positive definite solutions and perturbation analysis of matrix equations as well as the spectral problems of fractional differential equations. The motive and content of the research are detailed as follows.
On the one hand, as to many mathematical physics problems originated from prac-tical issues, which are depicted by ordinary differential equations, integral equations and integral differential equations, the usually applied method is to discretize them through the difference method and therefore the original problem can be transformed into a certain kind of matrix equations to be researched. Matrix equation is one of the important substance of the matrix theory and numerical algebra. In recent years there has been a constantly increasing interest in developing the theory and numerical approaches of the positive definite solutions for the nonlinear matrix equation of the form X-∑i=1m Ai*X-pi Ai=Q. This kind of matrix equations arises in solving a large-scale system of linear equations in many physical calculations, extremal interpolation problem, differential equations, control theory, ladder network, dynamic programming, stochastic filtering and so on. From the application point of view, the positive definite solutions are more important. In the sequel, a solution always means a positive definite one. The research results of this kind of matrix equations mainly concentrate on three basic problems:
(i) Necessary and sufficient conditions for the existence of positive definite solutions;
(ⅱ) Effective numerical ways for obtaining the positive definite solutions;
(ⅲ) Perturbation analysis of the positive definite solutions, including perturbation bound, condition number, backward error, residual bound,
where the solvability is the theoretical basis of the numerical solution for the nonlinear matrix equation, and numerical solution methods provide a feasible calculation process. Since nonlinear matrix equations stem from calculation problems with mass data in engineering and physics, there usually exist two kinds of errors during solution proce-dure influencing the accuracy of the results which refer to truncation errors caused by the numerical calculation method and rounding errors caused by calculating environ-ment. In order to analyze the impact of errors on the solutions of original problems, one needs to study the impact on the solution from the turbulence of original data, therefore the perturbation analysis of positive definite solutions for matrix equations is required. The perturbation bound and condition number will be applied to illustrate the stability of matrix equations, while the backward error and residual bound will be applied to check the numeral stability of the algorithm and the accuracy of the ap-proximately estimated solutions. Motivated by the work and applications of this kind of matrix equations, we study the following nonlinear matrix equations.
1. For the nonlinear matrix equation X-∑Ai*X-pi Ai=Q(pi>0), we consider two cases:the case m=1and the case m>1.
(i) When m=1, the above equation can be reduced to X-A*X-p A=Q(p>0). Two cases are considered here:the case p≥1and the case0
(i) In the case0
On the other hand, one important method to deal with differential equations is to study their solutions through the research on the nature of their spectrums, i.e. the spectral theory of differential equations. The Sturm-Liouville problem is the basic problem in the spectral theory of differential equations and its related theories were raised170years ago, and since then its related theories have played important roles in scientific, engineering and mathematical fields.
Sturm-Liouville problem is originated from the boundary value problem of ordi-nary differential equations which are partly from practical issues directly and partly from the problems of partial differential equations, such as the heat conduction (or dif-fusion) problem, the string (film) vibration problem and the Maxwell equation problem in electromagnetics. At the beginning of the19th century, Fourier systematically put forward the method of separation of variables and applied this method to problems of partial differential equations equations with boundary value conditions caused by more complicated physical phenomena. From operator point of view, Sturm-Liouville operator is an extremely important kind which has significant application background in both classical differential operators and modern quantum physics. Besides, since the end of the20th century, the rapid development and widespread application of the fractional calculus theory have promoted the appearance and growth of fractional dif-ferential equations. It has been discovered that to introduce the concept of fractional calculus can describe the change rule and essential attribute of objects, which brought to the wide application of fractional differential equations in reality, such as in the fields of fractal dynamics, continuum mechanics, auto control, hydrodynamics, biomechanics, viscoelastic mechanics, quantum mechanics, statistics, engineering science, Brownian motion, earthquake analysis, fractional models of nerve and mathematical models de-scribing the population reproduction. Therefore, fractional differential equations have more and more drawn the interest of mathematicians. The research on the theory of fractional differential equations can both enrich existing mathematical theories and provide better mathematical models for studies in physical, biological and economical process and phenomena. In many cases, the consideration in the spectrum problem of fractional differential equations is needed to deal with practical issues. The study on spectrum problem of fractional equations can satisfy practical requirement as well as enrich and perfect related theories of fractional differential equations. Since the spec-trum problem of fractional differential equations was raised in early research papers, it has never been researched intensively. Recently, many scholars have applied the method of numerical calculation to study this problem, whereas even the natures of eigenvalue and eigenfunction have not been theoretically illustrated. Until now, there have been very few articles theoretically referring to the natures of eigenvalue and eigenfunction in fractional differential equations. Based on these findings, this paper will mainly focus on following spectrum problems of fractional differential equations:
1. The spectral problem is considered, where q∈L2(0,1) is a real-valued function, D0+α and D1-α are the left and right Riemann-Liouville fractional derivatives of order α, respectively,1<α<3/2, μ is a real constant and λ is the spectral parameter. This equa-tion aries from the non-local continuum mechanics. It is a governing equilibrium equation of an elastic bar of finite length L with long-range interactions among non-adjacent particles. In this study, by using the spectral theory of self-adjoint compact operators in Hilbert spaces, we prove that the spectrum of the spectral problem associated to this equation with1<α<3/2consists of only countable and real eigenvalues with finite multiplicity and the orthogonal completeness of the corresponding eigenfunction system in the Hilbert spaces. Furthermore, we obtain a lower bound of the eigenvalues.
2. The spectral problem is considered, where q∈L2(0,1) is a real-valued function, D0+α, D1-α are respectively the left and right Riemann-Liouville fractional derivatives of order α,0<α<1/2, μ is a real constant and A is the spectral parameter. We prove that the spectrum of this problem consists of only real eigenvalues and its eigenfunctions corresponding to different eigenvalues are orthogonal by the spectral theory of self-adjoint compact operators in Hilbert spaces. Furthermore, we demonstrate that the real eigenvalues of this equation are countable and each eigenvalue has finite multiplicity. We also prove that the set of all corresponding orthogonal eigenfunctions forms a complete system. A lower bound of the eigenvalues is obtained.
The theory about initial value problems of differential equations involving only left or right fractional derivatives has been considerably perfect. The existence of solutions, the consistent dependence on parameters of solutions, the differentiability of solutions and the continuation theorem have been established. However, the initial value prob-lems of differential equations with both left and right fractional derivatives have not been put forward clearly enough, and seldom mentioned in the existing publications and papers on fractional differential equations. Nevertheless, on the research of bound-ary problems of differential equations with integer orders, the corresponding theory of initial value problems proves to be a very effective method and tool, such as the Prufer transformation and the differentiability of solutions on parameters. Therefore, in this paper, we first put forward the initial value problems of differential equations with both left and right fractional derivatives. Under proper conditions, we can prove the existence and uniqueness of solutions for this kind of initial value problems. Then, ap-plying results above, we specifically research the geometric multiplicity of eigenvalues for eigenvalue problems and form a series of problems with single eigenvalue.
1. First, the initial value problem is considered, where q∈L(0,1) is a real-valued function, D0+α and D1-α are the left and right Riemann-Liouville fractional derivatives of order α, respectively,0<α <1, μ and λ are real constants. The uniqueness of the solution for this kind of initial value problem is proved. And then, by means of the above results, we prove the eigenvalue of eigenvalue problem is simple.
2. The initial value problem is considered, where q∈L(0,1) is a real-valued function, D0+α is the left Riemann-Liouville fractional derivatives of order a and D1-α is the right Caputo fractional derivatives of order a, respectively,0<α<1/2,μ and λ are real constants. The uniqueness of the solution for this kind of initial value problem is proved. And then, by means of the above results, we prove the eigenvalue of eigenvalue problem is simple.
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