The Research Of Some Problems For Nonlinear Fractional Diferential Equations

This thesis mainly investigates the existence of solutions to initial and boundary valueproblems for several nonlinear fractional diferential equations and fractional impulsive dif-ferential equations by utilizing the half ordering method, the fixed point method, method ofupper-lower solution and monotone iterative technique of nonlinear analysis, we obtainedsome new results. We completed fifteen papers during the master’s degree, seven of themhave been published, the main publications of the papers publishing are Computers and Math-ematics with Applications, Applied Mathematics and Computation, Journal of ComputationalAnalysis and Applications, Dynamic Systems and Applications, Bulletin of the MalaysianMathematical Sciences Society and so on. Because of the length limited, this thesis onlyselects six papers to come the key introduction.The whole thesis is divided into eight chapters.In Chapter1, we briefly introduce the development, research background and researchstatus of fractional calculus and fractional diferential equations, and introduce the main re-sults of our thesis.In Chapter2and3, we mainly considered existence of solutions for nonlocal problems offractional diferential equations and higher order fractional integro-diferential equations withintegral boundary value conditions. We first established two new maximum principles, then inthe weaker conditions we discussed existence of extremal solutions and quasi-solutions of twokinds of nonlinear fractional diferential equations in Banach space by the method of upperand lower solutions combining with monotone iterative technique.In Chapter4, we mainly investigated existence and uniqueness of solutions of higher-order nonlinear fractional impulsive diferential equations of Caputo type. In this chapter,we first proved a new lemma (Lemma4.2.3) and generalized the results in the literature[3](Lemma2.22), secondly, we proved the existence and uniqueness of solutions for the equa-tions by Schauder fixed point theorem and Banach fixed point theorem, lastly, we present someexamples to verify our results.In Chapter5, we mainly discussed existence of solutions for nonlocal problems of non-linear fractional impulsive diferential equations. In this chapter, we first constructed a com-parison principle for nonlocal problems of fractional impulsive diferential equations, which was a new result, then we considered the existence of extremal solutions and quasi-solutionsfor the equations by the method of upper and lower solutions combining with monotone iter-ative technique.In Chapter6, we mainly considered existence and uniqueness of solutions for nonlin-ear fractional impulsive diferential equations of Riemann-Liouville type with generalizedanti-periodic boundary value conditions. Similar to Chapter4, we first proved a new lemma(Lemma6.2.2) and generalized the results in the literature [3](Lemma2.5), secondly, weproved the existence and uniqueness of solutions for the equations by Schauder fixed pointtheorem and Banach fixed point theorem, lastly, we present some examples to verify our re-sults.In Chapter7, we mainly researched existence of solutions for nonlinear fractional im-pulsive diferential systems with nonlinear boundary conditions. In this chapter, we first con-structed a maximum principle for the nonlinear fractional impulsive diferential systems tookadvantage of the inner relationship between the equations and systems, which was a new re-sult. Then we discussed the existence of extremal solutions for the systems by the method ofupper and lower solutions combining with monotone iterative technique.In Chapter8, based on the research at present, we proposed some ideas for our futurework.