Solvability And Stability Of Fractional-order Equations

The theory of fractional calculus has been widely used in modern mathematics for more than 300 years.The study of solutions of fractional differential(difference)equations is a topic of general interest in natural sciences and engineering and has important application value in the research of medical image processing,quantitative finance,population mobility,neural networks and large-scale climate.Therefore,the qualitative research and application of the solutions of fractional equations is a very meaningful research work.This paper investigates the existence and stability of solutions of several typical fractional order equations(systems)by using the fixed point theorem,the fractional comparison principle,the lower-upper solution method,Lyapunov stability theory,differential inclusion and setvalued mapping theory,Mittag-Leffler function estimation,inequality techniques,etc.As an application,this paper also discusses the existence and stability of solutions of fractional memristive neural networks and the conclusions are verified by simulation.The results of this paper enrich the study of solutions of fractional order equations.The whole paper is divided into five chapters.Chapter 1,we introduce the origin of the studied topic,its historical background,the current state of research at home and abroad,and some basic concepts and properties related to fractional calculus.In Chapter 2,the existence of unique solutions and multi-solvability of fractional q-difference equations with integral boundary value problems is investigated.In Section 1,the existence-uniqueness conditions of solutions of a class of fractional q-difference equations with Stieltjes integral boundary conditions,where the Lipschitz constants are related to the first eigenvalues of the corresponding operators,are obtained by using the properties of u0-positive linear operators.And the existence of multiple positive solutions is obtained by Guo-Krasnoselskii and Leggett-Williams fixed point theorems.In Section 2,the existence of extremal solutions of a class of fractional difference equations with integral boundary value conditions is proved by the fractional comparison principle and the lower-upper solution method.The Stieltjes integral condition is introduced into the fractional q-difference equation,which has not been seen in the literature.The results obtained enrich the study of boundary value problems for fractional q-difference equations.In Chapter 3,the existence and uniqueness of solutions of fractional differential equation coupled systems are studied.In Section 1,the multi-point boundary value problem for p-Laplacian generalized fractional systems with Riesz-Caputo derivatives is discussed.First,the definition of the mixed upper and lower solutions is given based on the previous chapter,combined with the monotone iterative method,the existence of solutions for the studied system are obtained.Secondly,in order to prove the uniqueness of the solution of the equation when p=2,we establish a fixed point theorem for ?-(h,e)-convex operators,and discuss the sum-type operator equation A(x,x)+Bx+e=x in Banach space,several conclusions for this operator equation are obtained without requiring the existence of the upper-lower solutions or compactness conditions,which provides a new approach for the study of boundary value problems.In Section 2,the criterion of compact operator on infinite interval is given.By using the fixed point theorem,the existence and uniqueness of the solution of the fractional differential systems on infinite interval are discussed.The nonlinear term depends on the lower-order derivative and the boundary condition contains the perturbation parameter,compared with the existing literature,the system studied in this section is more general.In Chapter 4,we study the uniqueness and stability of solutions for two classes of generalized fractional differential systems.In Section 1,we establish a comparison theorem by using new fractional differential inequalities,and obtain the global Mittag-Leffler stability criteria for generalized differential systems by using Lyapunov direct method.When the system contains time-delay,we give the stability conditions for Lyapunov functions with time delays,Gronwall inequality is used to deal with the case of time delay.Compared with Razumikhin's tool,it is less conservative.Furthermore,the theoretical results are applied to the generalized fractional memristive neural networks.Due to the effect of time-varying delay and the parameter p,the system we study is more complex,and the Mittag-Leffler stability criterion for the solution is obtained under weaker conditions.In Section 2,we studied the uniqueness and finite-time stability of solutions of generalized neutral-type fractional systems with time-delay.On the one hand,a new estimate for the Mittag-Leffler function is given,and a Gronwall integral inequality(without time-delay)based on the multi-parameter Mittag-Leffler function is constructed.Combined with the p-Laplace transform,a finite-time stability criterion is indirectly obtained.On the other hand,for neutral systems,a generalized time-delay fractional Gronwall integral inequality is given,which directly yields a criterion for the finite-time stability of the system.As an application,the finite-time stability of the neutral generalized fractional-order memristive neural network is discussed and numerical simulations are given to verify the validity of the theoretical results.The stability of neutral generalized fractional-order systems has not yet been studied in the literature,and this chapter extends and improves the results of related literature.Chapter 5 summarizes the research contents and provides an outlook for future research.