The Mild Solutions Of Fractional Differential Equations With Nonlocal Conditions

The focus of this dissertation is to study the mild solutions for four types of fractional differential equations with nonlocal conditions. In this paper, we mainly devoted to the fixed point theory combined with solutions operators theorems and Krasnselskill theory to research the existence of mild solutions for the given systems, which is organized as follows:Firstly, we introduce the research background, the main work and innovations we did and analyze the development situations of fractional evolution equations system-atically.Chapter two is to research the existence of mild solutions for abstract fractional differential equations with nonlocal conditions. The definition of mild solutions is ob-tained by using Laplace Transform formula and piecewise functions. Then we give the relation between the solutions operators by using probability density function com-bined with semigroup operators theory. At last, A sufficient condition is derived for the existence and uniqueness of mild solution of the system.Chapter three is to talk the existence of mild solutions for impulsive fractional differential equations with nonlocal conditions. We make use of Laplace Transform formula and concerned operators theory to get the definition of the mild solutions. Then A sufficient condition is derived for the existence and uniqueness of mild solution of the system by applying contraction mapping principle.Chapter four is to study the existence of mild solutions for fractional differential equations with nonlocal conditions of order1<α<2. At first, we get the defi-nition of mild solutions with Laplace Transform formula and find a suitable path to estimate the norms of the solution operators. Then the existence and uniqueness of mild solution is obtained by applying the fixed point theorem combined with solution operators theorems and Krasnselskill theorem. Finally an example is given to verify our conclusions.In the fifth chapter, we also research the existence of mild solutions for abstract fractional differential equations with nonlocal conditions of order1<α<2. By the similar way used in chapter four, we obtain our main results.