The Study Of Solutions For Some Classes Of Initial And Boundary Value Problems Of Nonlinear Fractional Differential Systems

The theory of fractional calculus has a history of nearly four centuries. As a result of the contradiction with classical calculus in many aspects and the lack of practical background, it has been at the initial stage of slow progress. Fractional theory begins to enter a new accelerated period as the mathematician Mandelbrot pointing out that there are a large number of fractional phenomenon in nature. With the advent of fractional academic monographs, conferences and magazines, the theory of fractional calculus has entered the stage of rapid development.Fractional differential model with nonlocal and memory properties can describe the mathematical model with practical background with less parameters, and can overcome the weakness of errors between integer order differential model and experimental results. However, the divergence has been going on throughout the decades between the current fractional calculus theory and the classical integer order calculus, and international aca-demic debates also exist constantly. Therefore, the research on fractional calculus has very important theoretical and practical significance. In this paper, we discuss the solu-tions of initial and boundary value problems for nonlinear fractional differential systemsChapter 1 provides the historical background, research significance, current research situation, and the main work of this paper.Chapter 2 is devoted to discussing the exact controllability of a class of semilinear fractional evolution systems. We introduce a new and weaker concept of exact control-lability and the semigroup{T(t):t≥ 0} is not supposed to be compact. By using re-solvent operators theory, Sadovskii’s fixed point theorem, and the Kuratowski’s measure of noncompactness, we investigate the exact controllability of the considered fractional evolution system without the Lipschitz continuity and other growth conditions imposed on the nonlinear function f. Actually, the nonlinear function f is only supposed to be continuous. Our results can be regarded as a development of previous conclusions.Chapter 3 investigates the existence of positive solutions for two classes of high-order nonlinear fractional differential coupled systems. Firstly, we consider a high-order Caputo fractional differential system which is coupled between the derivative term and nonlinear term. Through the study of Green function, an appropriate cone is established. Using the differentiable property of integral operator at θ and ∞ along cone, and the theory of spectrum of bounded positive linear operator together with some corresponding fixed point index theorems on cone, we give some existence theorems of positive solutions. Secondly, we consider a high-order Riemann-Liouville fractional differential system which is coupled at integral boundary conditions. By utilizing the nonlinear alternative of Leray-Schauder type, Krasnoselskii’s fixed point theorems, and the theory of fixed point index on cone, we derive an interval of parameter λ such that for any λ lying in this interval, the considered system has at least two positive solutions.Chapter 4 is assigned to study the multiple solutions for two classes of nonlinear fractional differential systems. Firstly, a class of fractional differential equation with nonlinear term dependent on the derivative term is considered. By studying the Green function and its properties, a suitable cone is established. Applying Guo-Krasnoselskii’s fixed point theorem, Leggett-Williams fixed point theorem, and a new extension of Kras-noselskii’s fixed point theorem, we give some new existence criteria of multiple positive solutions. Secondly, by using of the fixed point index theorems on cone for differentiable operators, we investigate the existence of multiple positive solutions for nonlinear frac-tional differential equation with integral boundary value conditions and parameters. The other methods involved are the theory of spectrum of bounded positive linear operator, bounded linear inverse operator theory and fixed point index theorems on cone.In Chapter 5, we investigate the eigenvalue problems of a class of nonlinear impulsive singular boundary value problem in Banach spaces. By virtue of a special transformation, we first convert the considered system into another solvable form such that the associated operator can be used to overcome the influence of impulse and parameter. Then a special cone is constructed to deal with the singularity of the system. Finally, by using the fixed point index theorems on cone and Kuratowski’s measure of noncompactness, some existence theorems of eigenvalue problems are given under different hypotheses. In order to overcome the abstraction of the infinite-dimensional Banach space, a specific example is also given to illustrate the main results. The problems studied and the methods used in this chapter extend and improve the existing research results.