Infinite Delay Fractional Differential Equations In Banach Spaces

Fractional calculus is the generalization of classical calculus, which deals with the derivative and integral of arbitrary order. It has been applied in many fields of science and technology such as Optical and thermal system, Capacitor, Signal processing and system identification, Control theory, Biological problem and so on. It has attracted much attention due to its wide application, and there has been a significant development in fractional differential equations in the past decades. Now the theory of fractional differential equations has become research focus of mathematics.This paper is devoted to the study of weighted fractional differential equations with infinite delay in Banach spaces modeled as where 0<a<1,Dα is the Riemann-Liouville fractional derivative, y(t)= t1-a y(t), f:(0,b]×Bâ†'B is a given function satisfying some assumptions, and 93 is the phase space. We give the definition of solutions, and investigate the existence and continuous dependence of solution to such equations in the space C1-α ((a,b];X).The full text is divided into five parts. In the first chapter, a brisf history of fractional calculus is introduced. Chapter 2 is preliminaries, including the basic definitions and theorems of fractional integral and derivative, phase spaces, the theory of the measure of noncompactness and related fixed point theorems. Chapter 3 is about the auxiliary results, where a Gronwall-type inequality is prove specific for fractional differential equations, and the comparison property of fractional integral is discussed. Chapter 4 is devoted to the main results. Existence and uniqueness results are obtained for the weighted fractional differential equations with infinite delay in Banach spaces, based on the theory of measure of non-compactness, Schauder’s and Banach’s fixed point theorems. In Chapter 5, the continuous dependence on the initial date, especially on the fractional order are studied.