Solutions To Initial And Boundary Value Problems Of Nonlinear Differential Equations

This thesis mainly investigates the existence of solutions to initial and boundaryvalue problems for several nonlinear differential equations of integer order and frac-tional order by utilizing the fixed point theorems, fixed point index theory of nonlinearfunctional analysis, iterative technique and cone theory. There are six chapters in thisthesis.In Chapter 1, the research background and results of this thesis is brie?y pre-sented.In Chapter 2, the contents are about preliminaries, mainly including the defini-tions, some properties of fractional integral and derivative and the basic definitions andtheorems of cone.In Chapter 3, the following third-order three-point boundary value problems(?)is studied. By utilizing the connection between u and u , the fixed point indextheory and the fixed point theorem, we get the existence of multiple positive solutions.In Chapter 4, two classes of boundary value problems of impulsive differentialequations in Banach spaces are discussed.In Section 4.1, the existence of solutions for the following second order singulardifferential equations in Banach spaces(?)is considered. By constructing a convex closed set to overcome the singularity, andusing the fixed point theorem for continuous operators, we obtain the existence ofsolution. In Section 4.2, the following second order differential equations with integralboundary condition in Banach spaces(?)is investigated. Combining the properties of Green function with cone theory and thefixed point index theory, we show the existence of multiple positive solutions.In Chapter 5, we deal with m-point boundary value problems for nonlinear frac-tional differential equations(?)Using the properties of the Green function, analysis technique and iterative technique,we get the existence of minimal and maximal positive solutions, and the method usedis different from the method of upper and lower solutions.In Chapter 6, we consider two classes of initial value problems of fractional dif-ferential equations.In Section 6.1, the following fractional differential equations with integral bound-ary condition (?)is studied. By an estimation and the Mo¨nch fixed point theorem, we obtain the exis-tence of solution.In Section 6.2, the existence of solutions for fractional differential equations(?)is investigated. By using a comparability theorem, iterative technique and cone theory,we get the existence of minimal and maximal solutions and minimal positive solution.