The Research On Quasilinearization Method And Existence Of Solutions For Some Nonlinear Fractional Differential Equations

Recently, fractional calculus have been widely used in physical, biological, mechanical,financial fields and so on. The description of some phenomena by using fractional derivatives is more accurate than integer derivatives, so more and more scholars begin to research this area. By using the fixed point method, monotone iterative technique, mixed monotone iterative technique, operator semigroup, etc., we investigate the quasilinearization method and existence of solutions of some nonlinear fractional dierential equations. Our results improve and develop corresponding ones in existing literatures, some new results are obtained.The structure of this thesis is as follows:In Chapter 1, we briefly introduce the research background, overseas and domestic research status and present the main work of this thesis.In Chapter 2, we list some relevant preparatory knowledges used in this thesis, including fractional calculus, cone and mixed monotone operator, measures of noncompactness,operators semigroup.In chapter 3, we mainly study the quasilinearization method for fractional dierential equations of Riemann-Liouville type. In view of previous studies, most of them focused on Caputo fractional dierential equations. This dissertation uses a new comparison principle,get a new result of fractional dierential equations of Riemann-Liouville type.In chapter 4, we mainly study the quasilinearization method of higher order impulsive fractional dierential equations. we develp the quasilinearization method of fractional dierential equations from 0 < q ≤1 to n- 1 < q ≤ n, get a new result.In chapter 5, we mainly study the quasilinearization method of fractional dierential equations with delayed arguments. The form of equations are more complex than the previous,so it brings us some diculties. By constructing a new comparison principle, using monotone iterative technique, we get a monotone sequence which converge quadratically to the unique generalized solution of problem.In chapter 6, we mainly study the existence and uniqueness of solutions of fractional dierential equations of Volterra type with nonlocal boundary condition. By constructing a comparison principle, using lower and upper solutions method and monotone iterative technique, we get the existence and uniqueness of solutions.In chapter 7, we mainly study the existence of the mild solutions to the fractional impulsive evolution equations. By definitions of the lower and upper quasi-solutions, using mixed monotone iterative technique and operator semigroup, we get the existence of the mild solutions.In chapter 8, we give the conclusion of our works and present the prospect of further work.