| The theory of impulsive differential system describes a process of development and change expressed by differential equations. The most prominent feature of impulsive dif-ferential system is able to fully take the instantaneous impact on the state into account, to overcome the shortcomings that many continuous system is unable to accurately ex-press the practical mathematical model. After decades of development, it has had a preliminary theoretical framework. The research achievements in many aspects appear constantly [1-28].Fractional impulsive differential system has become a research topic of the gen-eral mathematicians and been widely applied in many fields such as physics, chemistry, aerodynamics, electro dynamics of complex medium and so on. In addition, the re-lated theory of development ([1-3]) has led to extensive research of fractional differential equations with initial or boundary value problems, see [4-22] and the references therein.Many changes depend not only on the state at that time, also depend on the past state. Therefore the differential equation cannot be accurately described the objective things and replaced by differential difference equations especially delay differential equa-tions model. Impulsive functional differential system has been widely applied in neural network, optical control, population dynamics, biotechnology, economics etc [31-37]. This paper is focused on investigating the existence and multiplicity of positive solutions of the boundary value problem for nonlinear fractional impulsive differential equations and fractional impulsive functional differential equations. This paper is divided into two chapters.In chapter one, we study the following boundary value problem for nonlinear frac-tional impulsive differential equations where 1<α< 2 is a real number. Dα is the standard Riemann-Liouville fractional derivative,f € Car((0,1)× (0,+∞)) and f is positive, Q∈C(R+, R+),I∈ C(R+, (-∞,0 J=[0,1],J’= J\{t1},△u(t1)= u(t+1)-u(t-1), where u(t-1) and u(t-1) denote the right and the left limit of u(t) at t=t1 respectively. △u’(t1) has a similar meaning for u’(t).Ravi P. Agarwal etc. studied the existence of positive solutions for the singular fractional boundary value problem. Xu Xian investigated multiplicity results for positive solutions of some semi-positone three-point boundary value problems. In this paper, we discuss the existence and multiplicity of positive solutions of system (1) according to the ideas of two papers respectively. Unlike the previous papers, we consider the effect of pulse and draw a conclusion.In chapter two, we consider the impulsive functional differential system where 1<α<2 is a real number. Here f(t,x)∈C((0,1) x R+, R) is singular at t 0, 1,x= 0 and Da is the standard Riemann-Liouville fractional derivative.0<T< 1, η(t)∈C([-T,0]),η}(t)> 0 for t∈[-T,0) and η(0)= 0.Xinwei Su considered the existence of positive solutions by means of the Krasnosel-skii fixed point theorem. In this paper, we study the effect of pulse on the existence of solutions with the same method. As we all know, there are fewer papers considered the problem.The thesis not only prove each theorem, but also give examples to support the main theorems. |
