Two-point Boundary Value Problems For Impulsive Differential Equations

The mutation of subject in it's changing process is called impulse. Impulse is a com-mon phenomenon in the technological field and the mathematical modeling in the area it is named impulsive differential system. Impulsive differential equations have been studied extensively in recent years. Such equations arise in many applications such as spacecraft control,impact mechanics, the transfer device for satellite trajectory, the robot, neurotic network, muddleheaded controlling, as well as the study of confidential communication. So reseach work about these is of great value both in theory and application.In 1980s, there was some fundamental theory about impulsive differential equa-tions[4]. Subsequently,many reseach workers contiuned to study and develop the the-ory of these equations. Also there were some existence results under some conditions. e.g. Jiang Daqing, Wei zhongli,[6], [25],[13] in China and Donal O'Regan, L.H.Erbel, Y.H.Lee,Xinzhi Liu[8-10],[28-30] abroad have done a lot work. In these reseach work, the nonlinearities may be singular. Most methods they applied are fixed point theorem, fixed point index theory on cone, the method of upper and lower solutions and so on. The thesis contains two chapters.Using the method of upper and lower solutions and the fixed point theory on cone and approximation technique, we overcome the difficulties which lie in sign changing and singularities. Thus we can get the existence of positive solutions for second order impulsive differential equations with singuar boundary value problems.In chapter one, we consider singular boundary value problems for second order sublinear impulsive differential equations where f∈C[(0,1)×(0,∞),(0,∞)]f/(t,1)(?)0, t∈(0,1), I∈C(R+,R+),I∈ (R+,(-∞,0]),R+= [0,+∞), f(t,x) may be singular with t=0 or t=1,I is con-tinuous and increasing on[0,∞).In [10], Yong-Hoon Lee,Xinzhi Liu, by using the method of upper and lower solutions and the monotone interative technique, discussed singular boundary value problems for second order sublinear impulsive differential equations and gave criteria of the existence of extremal solutions, where a, b∈R,△u+|t=t1=u(t1+)-u(t1),△u'|t=t1=u'(t1+)-u'(t1) and f:D (?) (0,1)×R→R,I:R×R,N:R×R→R is continuous,f may be singular at t=0 or t=1.This paper gives new conditions,'first we establish the upper and lower solutions technique for impulsive two-piont boundary problems (1). Second,using above technique, we prove (1)the necessary and sufficient conditions for the existence of positive solutions. Finally, an example is given to show the effectiveness of the theorems.''And when the two operations of I and I are different, the necessary and sufficient conditions for the existence of positive solution of research are rare.In chapter two, we consider semipositone singular boundary value problems for second order impulsive differential equations where f∈C[(0,1)×(0,∞), (0,∞)], q∈C(0,1), I∈C(R+,R+),I∈(R+,(-∞,0]),R+= [0,+∞), and t1∈(0,1) be given, and assume f may be singular at t=0,1 or x=0,I is continuous and nondecreasing on [0,00).In [24],Gaoyun Feng, by the fixed point theory on cone, gave sufficient conditions for the existence of positive solutions to the semipositone singular boundary value problems for second order impulsive differential equations where f∈C[(0,1)×(0,∞),(0,∞)],q∈C(0,1),I∈C(R+,R+),R+=[0,+∞),and t1∈(0,1)be given. Assume f may be singular at t=0,1 or x=0,I is continuous and nondecreasing on[0,∞).This paper increased△x|t=t1=I(x(t1)),we also use the fixed point theory on cone gaving sufficient conditions for the existence of positive solutions of(2). Finally,an example is given to show the effectiveness of the theorems.