Existence And Uniqueness Of Solutions For Nonlinear Fractional Two-Point Boundary Value Problems

As an important part of the study of calculus theory,fractional calculus has a wide and profound application in physics and mechanics.It is especially suitable for describing physical and mechanical processes with memory and genetic phenomena.This dissertation mainly studies the existence and uniqueness of solutions of nonlinear fractional two-point boundary value problems from two aspects.On the one hand,the monotone iterative techniques and existence of solutions for nonlinear fractional differential equations of Riemann-Liouville type are studied.Here,a new comparison principle is constructed.By using monotone iterative technique and upper and lower solutions method,the extremum solution of the nonlinear fractional two-point boundary value problem in Banach space under weak conditions is obtained,and numerical examples are given to demonstrate it.On the other hand,The quasi-linearization methods and uniqueness of solutions for nonlinear fractional differential equations of Riemann-Liouville-type are studied.Firstly,the newly established comparison principle and the upper and lower solutions method are used to consider the solutions of the corresponding linear problems,and a set of monotone sequences are obtained.Then,it is proved that the sequence converges uniformly and quadratically to the unique solution of the original problem.Finally,numerical examples are given to verify the effectiveness of the theoretical results.At the same time,the method is extended to a class of fractional two-point boundary value problems of Riemann-Liouville type with advanced arguments.A new results of quasi-linearization method for fractional differential equations with advanced arguments are obtained,and two numerical examples are given to demonstrate the method.