Fractional-order Boundary Value Problems With Nonlocal Conditions And Positive Solutions

Fractional order calculus is the theory of arbitrary order differential and integral. It is unified with integer order differential and integral calculus, and generalizes integer order calculus. Because of the memory properties of fractional order derivative, we can apply it more efficiently in physical memory and genetic properties, and it can better simulate natural physical process and power system process. Thus fractional calculus is used wider and wider in control theory, biological engineering, electrochemical processes, mechanical engineering, condensed matter physics. So the investigation of fractional order differential equations not only has theoretical significance, but also has important practical value. Based on the theory of fractional differential equations, in this work, we investigate the existence of solutions and positive solutions for several kinds of fractional differential equations with boundary value conditions.Firstly, we introduce the research background, analyze the development situations of fractional differential equations systematically. Then we point out the main work and innovations we did.Chapter two is to study the existence of positive solutions for impulsive fractional differential equations with boundary value conditions. The solution of the equation is obtained by using the properties of Caputo's fractional order calculus and piecewise function. Applying Kuratowski measure of non-compactness and fixed point theorem on cone, we get sufficient conditions for the existence of positive mild solutions.Chapter three is to talk the existence of positive mild solutions for fractional evolution equations with boundary value conditions. We discuss the properties of positive operator first. Then sufficient conditions are derived for the existence of positive mild solutions of the system by using fixed point theorem on cone.Chapter four is to investigate the existence of mild solution for impulsive fractional evolution equations with nonlocal conditions and infinite delay. We get the definition of mild solution with classical Mittag-Leffler function and Laplace transform. Then the existence of mild solution is obtained by applying Krasnselskill's fixed point theorem.