Doundary Value Problems For Second Order Nonlinear Impulsive Singular Integro-Differential Equations

Impulsive differential equations have been investigated extensively because they have a lot of backgrounds in Biology,Medicine,Economics and Space technique,see[6]-[24]. In these results,the books "Theory of Impulsive Differential Equations"[24]and "Introduction to the Impulsive Differential Systems"[10] have summarized all the results on this area in recent 20 years.In the literature[38],the system is as follows: where J=[0,+∞),J+=(0,+∞),00,Iik∈C[P0λ×P1λ×…×Pn-1λ,P],(?)λ>0(i=0,1,…,n-1,k=1,2,3,…),u0i≥ui*(i= 0,1,…,n-1),β>1,u(n-1)(∞)=t→∞lim u(n-1)(t),and(Tu)(t)=(?)0tK(t,s)u(s)ds, (Su)(t)=(?)0∞H(t,s)u(s)ds,where K∈C[D,J],D={(t,s)∈J×J:t≥s),H∈C[J×J,J].△u(i)|t=tk=u(i)(tk+)-u(i)(tk-).In this paper,Professor Guo Dajun discussed the existence of positive solutions for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces by using the fixed point theorem.There are some open questions for impulsive differential equations. On the one hand,there are fewer results on this area when the impulsive functions are singular and the nonlinearity is singular at the same time；on the other hand,there are fewer results on this area when the nonlinearity has integral items.This paper discusses the special case in the literature above,when n=2,β=0. In chapter 1,we obtain the existence of positive solutions of two point boundary value problems for second-order nonlinear impulsive singular differential equations: where△u|t=tk=u(tk+)-u(tk-),△u'|t=tk=u'(tk+)-u'(tk-),u(t) is left continuous at tk In this paper,f(t,x,y)∈C((0,+∞)3,(-∞,+∞)),I0,k,I1,k∈C((0,+∞)2,(O,+∞)), and not only f(t,x,y) is singular at t=0,x=0,y=0,but also I0,k(x,y),I1,k(x,y) are singular at x=0,y=0.In chapter 2,we discuss the existence of positive solutions of two point boundary value problems for second-order nonlinear impulsive singular integro-differential equa-tions: where△u|t=tk=u(tk+)-u(tk-),△u'|t=tk=u'(tk+)-u'(tk-),u(t) is left continuous at tk,(Au)(t)=f0tK(t,s)u(s)ds,(Su)(t)=f0∞(t,s)u(s)ds,f∈C((0,+∞)5,(-∞,+∞)), I0,k,I1,k∈C((0,+∞)2,(0,+∞)),similarly, the nonlinearity is singulaWe use Schauder fixed point theorem to present the existence of positive solutions for two point boundary value problem in finite intervals,by using Arzela-Ascoli theo-rem,we obtain the convergent subsequences of the corresponding approximate solutions, so the limits are just the solutions,and then extend it to the infinite intervals. In this paper,theorem 1.3.1 presents the results on the existence of positive solutions in(1.1.1), when the impulsive functions are singular and f(t,x,y)is singular at t=0,x=0 and y=0 at the same time. Theorem 2.3.1 presents the existence of positive solutions in (2.1.1),when the nonlinearity has integral items.