Study On The Existence Of Mild Solutions For Several Types Of Fractional Integro-differential Equations

Fractional calculus is an old and novel branch of mat,hematics.With the successful application of fractional calculus in differe.nt fields ranging from physics,biology,finance to optimal control system,the theory and application of fractional differential equations have been developed rapidly.Compared with integer order differential equations,fractional differential equations have memory in time and genetic properties,which are more suitable for describing many problems in anomalous diffusion,viscous fluid mechanics,porous media mechanics,electrical engineering and bioengineering.However,because of the memory in time and nonlocality of fractional differential operators.it also increases the difficulty of studying fractional differential equations.In addition,random factors are everywhere in nature and real life,so the deterministic mathematical model can not fully reflect the random influence.For instance,the deterministic models arising in finance environment,weather derivative,often fluctuate in view of the presence of some kind of noise.Hence,it is of great theoretical and practical significance to study the existence and uniqueness of mild solutions for fractional(stochastic)differential equations.Nonlinear functional analysis is an important part of modern mathematics,in which semigroup of operator,fixed point theorem,partial order method and other nonlinear analysis methods are powerful tools to study many nonlinear mathematical models.In the past decades,a lot of scholars at home and abroad have accomplished series of work on the theory and application of nonlinear functional analysis.In this dissertation,we focus on several types of fractional integro-differential equations,including a class of fractional semilinear integro-differential equations of mixed type with delay,fractional semilinear impulsive integro-differential equations with nonlocal initial conditions,periodic boundary value problems for fractional semilinear nonautonomous differential equations with noninstantaneous impulses,nonlinear fractional partial differential equations of order 1