Existence Of Solutions For Some Fractional Order Differential Equations

In recent decades, fractional differential equations have become an important subject due to their application in various sciences and have been widely concerned, the research of fractional differential equations has a strong practical significance.This paper mainly studies the existence of solutions for two kinds of fractional order differential equations. In the first section, the paper gives the historical background and significance, and current situation of the problem we studied. In the second section, the relevant definitions and theorems used in this article are given, they are prepared to the following main conclusions to be proved. In the third section, we study the existence of solutions for fractional order functional differential equations with finite delay, we transform the fractional order functional differential equations into integral equations by the related lemma, we define the appropriate integral operator in product space, then based on certain conditions the existence and uniqueness of solutions are obtained by means of the Banach contraction mapping principle,Leary-Schauder fixed point theorem and Schauder fixed point theorem, the results that are obtained on basis of the other literatures are some new results, relevant examples are given to illustrate the conclusions. In the fourth section, we also discuss the existence of quasi-solutions of fractional order differential equation, we give the definition of quasi-solutions of fractional order differential equation, the related lemma is obtained by the definition of Mittag-Leffler function in fractional differential equations, on this basis, by mixed monotone operator fixed point theorem we get existence results of the maximum and minimum quasi-solutions, iterative sequences of the maximum and minimum quasi-solutions and the convergence, similarly, the related examples are given to verify the conclusions of the theorems. In the last, we give a summary and point out some advantages and weaknesses about the thesis.