Study Of The Existence And Uniqueness Of Positive Solutions About Fractional Differential Equations

In recent decades, the development of fractional calculus has aroused more andmore attention, the most simple linear fractional di?erential equations have been devel-oped to slightly complicated nonlinear ones. The research of existence and uniquenessof solutions or solutions has been an important subject that has a strong practicalsignificance.This paper of master degree chie?y studies the existence and uniqueness of positivesolutions for some category of nonlinear fractional di?erential equations, the di?erentialoperator we discussed is defined in Riemann-Liouville sense, here what di?erent fromother literatures is that each nonlinear term is related to both unknown functions, andthe research of existence and uniqueness is separated independently.In the first chapter, we give the historical background, current situation of theproblem we studied. In the second chapter, we transform the di?erential equations ofinitial conditions under Caputo definition into Fredholm integral equations by usingrelevant properties. Then we define the integral operator and prove its completelycontinuous. In the third chapter, we mainly use the upper and lower solution method,Schauder fixed point theorem and Banach contractive mapping theory in nonlinearfunctional analysis to get some existence results and uniqueness results. At first, thenonlinear terms are positive and monotone increasing, and then they are weakenedto some special growth conditions. When we give the uniqueness results, this articleextends the Lipschitz condition L < 1 to arbitrary positive numbers which is an im-portant work of this paper. We also give some simple relative examples to illustratewhat we said above, thus we make previous research results more plentiful. In the last,we give a summary and some prospect about this thesis.