The Qualitative Theory Of Several Differential Systems And Its Application

During 1881-1886, the French mathematician J.H.Poincare (1854-1912), published four papers on integral curve determined by differential equations, in which the qualita-tive ideas were firstly introduced during solving the differential equation, this opened up a new world to discuss the nature and application of differential equations. At the same period, the Russian mathematician A.M.Liapunov (1857-1918) proposed the stability theory of ordinary differential equations, known as motion stability theory, which improve and develop the qualitative theory in the study of concrete questions further. Then two hundred years from then on, with the tireless efforts of mathemati-cians from different countries, the qualitative theory of differential equations develops rapidly, many disciplines and branches appear. The differential equations researched also undergo a course from linear to nonlinear, from low-order to high-order. These differential equation models constructed in different areas can describe the macroscopic and microscopic worlds more accurate, which have attracted wide attention in applica-tions. Therefore, qualitative theory of differential equations and applications become a hot topic of research in recent years.In the study of the qualitative theory of differential equations, along with the rapid development of fractal geometry, biology, automatic control, physics, control systems and fractional controllers, rheological, electrical analytical chemistry and other disci-plines, the spectral theory of differential operators and fractional differential equations have drawn more attention of many scholars, and have obtained some breakthrough developments. As one of important kind of differential operators, the Dirac operator presents important issues in the description of the energy quantum mechanics and atomic internal forces. If the mechanical systems are affected by external forces, or the atomic systems are disturbed by outside electromagnetic field effects, the origi-nal systems are likely no longer continuous, which result the characteristics function with discontinuities in the equations describing the system. These physical phenom-ena are abstracted to Dirac system with transmission conditions. Nowadays, a few researcher study the nature of the eigenvalues of Dirac system. The papers [110] s-tudies the inverse spectral problem of a simple form of Dirac system (diagonal form) with a transmission condition. Spectral problems of Dirac system in the generally form with the transmission conditions, such as the self-adjoint of Dirac operators, the completeness of eigenfunctions, etc, have not been found to our best knowledge.Since the 19th century, along with the grown up and development of the general-ized operator theory, the theory of fractional differential equations have gained rapid development, both in theory research and in applications. There are many monographs and papers majoring in this theory, numerous research directions appear. Each branch has certain development, see references [1], [25], [26], [79] for details. However, since the fractional integral operator has the property of non-locality, and the kernel of integral operator is weak singular, it is very difficult to study the fractional differential system using the theory and methods of integer order differential equations, so it is necessary to establish an independent fractional calculus theory. Due to its computational com-plexity, and some of their explanations of physical significance has not yet been widely recognized, the theory of fractional differential equations is still in its infancy with respect to the theory of integer order differential equations in general. The qualita-tive theory and application of fractional differential equations is one of the hot research fields in recent years, and it is one of the most direct applications of fractional calculus, which has been widely used in many areas of natural sciences. Although the research achievements of fractional differential equations are rich, the qualitative properties of fractional differential, integral equation theory are not widely researched relatively, some of aspects of qualitative properties are in a state of expectation research.The dissertation is divided into six chapters. The main research are the properties of basic solution and eigenvalues of Dirac system with transmission conditions, some integral inequalities with weakly singular kernel and their applications to fractional differential systems, oscillation theorems for matrix Hamiltonian systems, oscillation criteria for nonlinear fractional differential equations, and generalized variational oscil-lation principles for second order differential equations with mixed nonlinearities. Some new interesting results under weaker conditions have been obtained, most of which have been published in Appl. Math. Comput.(SCI), Abstr. Appl. Anal.(SCI), Discrete Dyn. Nat. Soc.(SCI), Adv. Difference Equ.(SCI), and J. Appl. Math.(SCI) etc.The chapter Ⅰ briefly describes the historical background of the qualitative theory of differential equations, focuses on the histories of spectral theory of differential op-erators, and qualitative theory of fractional order differential equation. It is divided into five sections to introduce the research questions, methods and conclusions for each chapter.The chapter Ⅱ studies with the spectrum of Dirac operator with an internal sin-gular point, we add a transmission condition at the singular point, then we discuss the spectral problem of the Dirac system’s with transmission conditions in details. § 2.2 gives a transformation of Dirac system; § 2.3 gives the description of the question to be studied; § 2.4 investigates the operator defined in suitable Hilbert space by the boundary conditions and transmission conditions, the eigenvalues of the operator are involved; § 2.5 discusses the fundamental solutions and properties of the Dirac sys-tem, and the correspondence between the zeros of the Wronski determinant and the eigenvalues of the Dirac operator, and the multiplicity of eigenvalues; § 2.6 gives the resolvent operator and Green functions.The chapter Ⅲ discusses several integral inequalities containing weakly singular kernels and their applications in fractional differential systems. In § 3.2, integral in-equality with weakly singular kernel for discontinuous function and its application to pulse fractional differential systems is studied. Due to the presence of a weakly singular kernel, the research methods of inequalities for discontinuous function are different to classic Gronwall-Bellman-Bihari type inequalities. Using the Mittag-Leffler function Eβ(·), and successive iteration method, we prove a new type of integral inequality for non-continuous function. In § 3.4, two categories of Gronwall-Bellman type inequali-ties containing weakly singular kernels are gotten, where the second category contains delay term, and they are used in the research of boundedness and other properties of the solution of fractional differential equations.In Chapter IV, oscillation theorems for matrix Hamiltonian systems are obtained. First in § 4.2, the interval oscillation criteria of Hamiltonian system (4.1) are given by using monotonous function in matrix space and negative-preserve functional, combined with the use of Riccati transformation and integral means method. These criteria de-pend on the nature of the coefficient matrices only in a sequence of sub-interval on the real line, thereby improved many existing results. Secondly, using a linear transforma-tion keeping oscillation, and a generalized Riccati transformation, integration means method in § 4.3, oscillation criteria for Hamiltonian system (4.1) are established, which improve many of the existing results, and simplify the proof of the known theorems. Finally, examples are given to illustrate the accuracy of our results.In Chapter V, oscillation criteria for two kinds of nonlinear fractional differential equation are given. In §5.2, new oscillation criteria are obtained for the more general fractional differential equation, different hypotheses on the nonlinear term f(u) imply different oscillation criteria. These criteria are different from most known ones in the sense that they are based on the information only on a sequence of interval of half-line, rather than on the whole half-line. In §5.3, using the Riemann-Liouville fractional derivative and Caputo fractional derivative separately, oscillation criteria including mixed nonlinear term for fractional differential equations are gotten, and several examples are given to verify the validity of the theorems.In Chapter VI, oscillation criteria of second order forced differential equation with mixed nonlinearities are obtained based on Leighton’s variational principles. Our em- phasis will be directed towards oscillation criteria that are closely related to the gen-eralized energy functional (the generalization of (α+1)-degree energy functional) for half-linear equations, which improve the results mention above. Examples are also given to illustrate the effectiveness of our main results.