The Existence Of Solutions For Fractional Impulsive Integro-Differential Equations

Nonlinear functional analysis is a significant research field in modern mathematics with with profound theoretical relevance and broad practical applications.It encompasses various essential theories and methods,such as topology degree theory,critical point theory,fixed point theory,and partial order methods.The theories and and methodologies have been found wide-ranging applications across diverse domains,including differential equations,partial equations,integral-differential equations,and so on.Fractional differential equations are commonly used to describe many phenomena in various scientific and engineering fields such as aerodynamics,viscoelasticity,electromagnetics,control theory,chemistry,biology,economics.In recent years,fractional differential equations have rapidly developed in engineering,physics,electronics,chemistry,and other fields due to their profound physical background and applications.Fractional differential equations possess singular and nonlocal characteristics,which endow them unique advantages in describing complex systems with memory effects,historical dependencies,and global interactions.Due to their extensive application value,fractional differential equations have attracted numerous experts and scholars to conduct systematic and in-depth research,resulting in numerous meaningful research achievements in both theory and application.Based on the existing research findings,we intend to further investigate the existence of solutions for initial value problems of nonlinear fractional impulsive integro-differential equations.This thesis investigates the existence of solutions for a some class of Caputo fractional impulse integral-differential equations through the application of non-compactness measure theory,resolvent operator theory and fixed point theory.Firstly,we provide an overview of the research background,the evolution of fractional order differential equations and fundamental theoretical principles.On this basis,the existence of solutions for fractional integro-differential equations with nonlocal conditions and instantaneous impulses,non-instantaneous impulses,and integral impulses are discussed on finite intervals;Then,the existence of solutions for fractional order integral differential equations with nonlocal conditions and instantaneous and non-instantaneous impulses are discussed on infinite intervals.The main research work of this thesis includes:1.For fractional impulse integral-differential development equation of 1<β≤2 with nonlocal conditions,the existence theorem of mild solutions of the equation is obtained by means of non-compact measure theory,Monch fixed point theorem and stepwise estimation method under the condition that the impulse terms are continuously bounded.The methods and constraints of the existing literature are improved.2.For fractional integral-differential equations with non-instantaneous impulses and nonlocal conditions.We remove the Lipschitz condition and the additional condition to ensure that the compression coefficient is less than 1,and obtain the existence theorem of the solution by using non-compact measure theory,resolvent operator theory,Monch fixed point theorem and stepwise estimation method.This result generalizes and improves some existing results.3.For fractional order integro-differential equations with integral impulse conditions.In the case that Lipschitz coefficients are Lebesgue integrable functions,the existence results are obtained by using the principle of compression mapping.Then,using the noncompactness measure,the resolvernt operator theory and the generalized Darbo fixed point theorem,the Lipschitz condition of nonlinear term is removed and the existence result is obtained.Furthermore,in our assumptions,the coefficients of the non-compactness measure are Lebesgue integrable functions,and the operator A depends on t.4.For fractional integral-differential equations featuring instantaneous impulses and non-instantaneous impulses on an infinite interval.In the case of equations with impulse terms,mixed integral terms and nonlocal conditions.We give the basic space of this thesis,and then the existence theorem of the solutions of fractional instantaneous integro-differential equations is obtained by using resolvent operator theory and Schauder fixed point theorem.By using the same method,we obtain the existence results of fractional integral-differential equations with non-instantaneous impulses.The results improve and generalize the results of some literatures.