Some Studies On Initial Boundary Value Problems Of Nonlinear Fractional Differential Equation

The differential equations are often used to establish mathematical models to describe various complex systems in many fields,such as mechanics,physics,information science and biomedicine,and have attracted the attention of many scholars for many years.Because of the fractional differential equations can accurately describe various nonlinear states or phenomena with memory and genetic characteristics,it’s applied in many fields such as electrical conduction in biological systems,Brownian motion,turbulence problems,chaos and fractal dynamics,etc.Therefore,the existence and related properties of solutions for nonlinear fractional differential equations have very important practical significance and potential application value.In this dissertation,we study several initial-boundary value problems of fractional differential equations involving p-Laplacian and Riemann-Stieltjes(R-S)integral boundary conditions,and discuss a kind of Caputo fractional differential equation models and applications in the field of biomedicine.In theory,the existence,uniqueness and multiplicity of solutions to several fractional differential equations are investigated by employing nonlinear operator theory,fixed point theory and monotone iterative technique,some related conclusions are obtained.Firstly,we investigate the initial value problems and boundary value problems of several fractional differential equations.(1)For Riemann-Liouville(R-L)fractional differential equations integral boundary value problems involving p-Laplacian,the existence and uniqueness of solutions are obtained by employing mixed monotone operator fixed point theorems.(2)For high-order fractional differential equations with R-S integral boundary conditions,we obtain the existence and uniqueness of solutions,and we can construct the continuous monotone iterative sequences to approximate the unique solutions by using sum-type operator and increasing φ-(h,σ)-concave operator fixed point theorems.(3)For high-order R-L fractional differential equations boundary value problems involving p-Laplacian,the existence of multiple solutions is obtained by employing multiple solution theory of nonlinear operator equations and Leggett-Williams fixed point theorem.(4)For high-order conformal fractional differential equations boundary value problems involving p-Laplacian,the existence of solutions is obtained by employing GuoKrasnoselskii fixed point theorem.(5)For two point boundary value problems of conformal fractional differential equations with sum-type nonlinear terms,the existence and uniqueness of positive solutions are obtained by using sum-type operators fixed point theorem.(6)For conformal fractional differential equations initial value problems with impulsive terms,the existence-uniqueness of positive solutions is obtained by employing mixed monotone operators fixed point theory,and the monotone iterative sequences are constructed to approximate unique solutions.For all kinds of fractional differential equations initial value and boundary value problems mentioned above,we also give some examples to verify the main conclusions.Secondly,in order to obtain the existence conclusion of solutions of fractional differential equations involving p-Laplacian,we study the certain composite type operators T=A(?)B,and obtain a new fixed point theorem based on set Ph.Furthermore,we discuss a class of two points boundary value problems of fractional differential equations involving p-Laplacians by employing the operator T=A(?)B fixed point theorem.We generalize and improve the relevant conclusions in the literature.Finally,we study a class of fractional impulsive differential equations initial value problems with control terms,the existence and uniqueness of the solutions are obtained,and investigate the optimal control problem of unique solution for the differential systems.The research in this thesis enriches the theories and conclusions of nonlinear functional analysis and differential equations,and brings profound theoretical significance.