On The Existence Of Positive Solutions Of Boundary Value Problems (for Systems) Of Fractional Differential Equations

Impulsive differential equation is an important branch of ordinary differential equa-tions.In recent years,fractional differential equation has been continue to in-depth study because of its theoretical system of continuous improvement and the close con-tact with many practical applications(such as:physics, mechanics,chemistry and engineering,etc.),by the international mathematical community and the importance of natural science.Fractional differential equations has become an important modern mathematics research direction.Impulsive differential equations is a hot topic in recent years,research in this area is a very important area.In this paper,using the cone theory,fixed point theorem for nonlinear functional methods such as the nonlinear impulsive differential equation exis-tence of positive solutions for serval difference boundary value problems,and obtained some new results.The thesis is divided into four sections according to contents.Chapter1Preference,we introduce the main contents,then give the related concepts and important lemma of this paper.Chapter2We consider the following second-order m-point boundary value prob-lem with impulse effect on an unbound domain Here J=(0,+∞),0<t1<t2<…<tn<∞,J1=J\{t1,t2,…,tn},0<ξ1<ξ2<…<ξm-2<+∞and ξi≠tk,i=1,2,…,m-2,k=1,2,…,n,αi>0,i=1,2,…,m-2.Δu’|t=tk=u’(tk+)-u’(tk-),Δu|t=tk=u(tk+)-u(tk-), where u’(tk+),u(tk+)and((u’(tk-),u(tk-))represent the right-hand limit and left-hand limit of u’(t)and u(t)at t=tk.Our results are based on a fixed point theorem of Schauder combined with the diagonalization method. Chapter3This chapter let us consider second order impulsve differential equa-tions involving Stieltjes where J=[0,1],0<t1<t2<…<t∞<1,Δu’|t=tk=u’(tk+)-u’(tk-),u’(tk+),u(tk+)((u’(tk-),u(tk-))represent the right-hand limit and left-hand limit of u’(t) and u(t)at t=tk,and we use the Avery-Peterson theorem,the existence of at least three positive solutions to second order differential equations is investigated.And we allow the with two difrerence Stieltjes integral.Chapter4This chapter we investigate the existence of solution for a four-point impulsive nonlocal boundary value problem of nonlinear differential equations of frac-tional order given by where J=[0,1],0<t1<t2<…<tp<1,J1=J\{t1,t2,…,tp),且f∈C[J×R×R,R],Ik,Ik∈C[R,R]Δu’|t=tk=u’(tk+)-u’(tk-),Δu|t=tk=u(tk+)-u(tk-),u’(tk+),u(tk+)((u’(tk-),u(tk-))represent the right-hand limit and left-hand limit of u’(t) and u(t) at t=tk,α>0,β>0且g1,g2:PC(J,R)â†'R are two continuous functions.By means of a fixed point theorem due to O’Regan,we establish sufficient conditions for the existence of at least one solution of the problem.