| The qualitative analysis of asymptotic periodic solutions of differential equations of fractional order is an important class of research problems in differential equations.Some real-life phenomena exhibit periodic variations.The presence of disturbing factors can lead to the fact that the periodic phenomena are not accurately fixed.Therefore,the asymptotic periodic solutions of neutral fractional order differential equations are of great theoretical importance and research value.In this paper,we investigate the asymptotic periodic solutions of neutral fractional order differential equations under different conditions by using fractional order calculus,operator semigroup theory,Arzela-Ascoli theorem,cone immobility theorem,Hausdorff non-compact measure,M（？）nch immobility theorem,stochastic analysis,Banach immobility theorem and Gronwall inequality.The existence and qualitative analysis of asymptotic and asymptotically positive periodic solutions to fractional differential equations of neutral order under different conditions.The study is divided into the following three parts.Firstly,we study the existence of asymptotic periodic solutions of fractional order differential equations of neutral type，which is a further study of the asymptotic periodic solutions of fractional order differential equations in general perfect spaces,bridging the gap between the existing studies of asymptotic periodic solutions of differential equations,and the mild solutions of the equations by using the Laplace transformation.Ascoli’s theorem to obtain the existence of at least one asymptotic positive solution of the equation,and then the cone immobility theorem to obtain the existence of a unique asymptotic positive solution.The existence and stability of asymptotically periodic positive solutions of neutral fractional order differential equations with impulses is then studied as a further study of asymptotically periodic positive solutions of neutral fractional order differential equations.The existence uniqueness of asymptotically periodic positive solutions on the positive cone and the Ulam-Hyers stability are obtained.Finally,the existence of asymptotically periodic positive solutions of the fractional order stochastic differential equation with neutral-type impulses with standard Brownian motion and fractional Brownian motion is investigated.This is a further study of fractional order differential equations with random terms,adding both Brownian and fractional Brownian motion.After obtaining the mild solution of the equation,the existence and asymptotic stability of asymptotic periodic solutions of such stochastic fractional order differential equations are obtained by combining stochastic analysis,Banach’s immobility point theorem and Gronwall’s inequality. |
