Existence Of Solutions Of Fractional Functional Differential Equations With Causal Operators

During the past decades, with the development of fractional calculus, frac-tional di?erential equations haved developed rapidly due to its application invarious sciences. Although a lot of results have been obtained for the existenceof solutions for functional di?erential equations with causal operators, the resultsfor the initial value problems of fractional functional di?erential equations withcausal operators is scarce. This dissertation focuses on the existence of solutionsof fractional functional di?erential equations with causal operators. This thesisconsists of four chapters. Main contents are organized as follows:In chapter 1, we first introduce the basic concepts for the fractional calculusand fractional di?erential equations, the definition and properties of the causaloperator. Next, we also introduce some application background for fractionaldi?erential equations and some important fixed theorems. Finally, we introducethe main work of this thesis.In chapter 2, we discuss initial value problem for fractional functional di?er-ential equations which contains causal operator in Banach space. We first estab-lish a fractional integral equations which is equivalent to the fractional functionaldi?erential equations. By using the theory of fractional calculus and measure ofnoncompactness, we obtain the existence results of solutions for fractional dif-ferential systems. We also discuss the continuation of solutions for fractionalfunctional di?erential systems.In chapter 3, we study initial value problem for neutral fractional functionaldi?erential equation with causal operator. Based on the theory of fractionalcalculus, we establish a fractional integral equation which is equivalent to thefractional functional di?erential equations. By using Krasnoselskii's fixed pointtheorem, we obtain the existence and uniqueness results of solutions for the sys-tem.In chapter 4, we discuss the initial value problems for two kinds of fractionalneutral functional di?erential equation with causal operator. First, we establishtwo fractional integral equations which are equivalent to the two systems, respec-tively. Next, by using the hybrid fixed point theorem, we obtain the existenceresults of extremal solutions.Keywords: Fractional di?erential equation, Fractional functional di?erentialequations, Riemann-Liouville derivative, Caputo derivative, initial value problem,the measure of noncompactness, fixed point theorem.