Two-point Boundary Value Problems For Impulsive Differential Equations

The mutation of subject in it’s changing process is called impulse. Impulse isa common phenomenon in the technological field and the mathematical modelingin the area is named impulsive diferential system. In recent years, impulsive difer-ential systems have important applications in the field of chaos control, confiden-tial communications, aerospace technology, risk management, information science,medicine,economics. In1980s, there was some fundamental theory about impulsivediferential equations[1]. Subsequently, many people continued to study and developthe theory of these equations. Also there were some existence results under someconditions[222]. For instance, Jiang Daqing[56], Wei Zhongli[16], Xu Xian[15][22]inChina and L.H.Erbel[7], Y.H.Lee[810], Donal O’Regan[1113], Xinzhi Liu[14]abroadhave done a lot work. In these research work, the nonlinearities may be singular ornonsingular. Most methods they applied are fixed point theorem, fixed point indextheory on a cone, the method of upper and lower solutions and so on.The thesis contains two chapters. We use the fixed point index theory on a coneto discuss the diferential impact of nonlinear singular as well as impulse conditions.We can get the existence of positive solutions for second order impulsive diferentialequations with singular boundary value problems.In chapter one, we consider two-point singular boundary value problems forsecond order impulsive diferential equationswhere f∈C[（0,1）×（0,+∞）×R,（0,+∞）], I∈C（R+, R+）, R+=[0,+∞), t₁∈（0,1）, f may be singular with y=0or y=0, I is continuous and increasingon[0,+∞), y|t=t₁=y（t₁^-） y（t₁）.In [15], by using the fixed point theory, Xu Xian discussed boundary value problems for second order impulsive diferential equationsand gave criteria of the existence of extremal solutions, where ai∈C[0,1],1≤i≤m.1<αi<0,1≤i≤k;0≤αi<1, k+1≤i≤m; k≥0. I∈C（R+, R+）,R+=[0,+∝). y|t=t₁=y（t+1） y（t₁）, J=[0,1],J₁=[0, t₁],J₂=[t₁,1], J=J/{0,1, t₁}.In [23], by using the fixed point index theory, Zhang Lili discussed initial sin-gular problems for second order diferential equationswheref（t, y, y） at y=0or y=0singular. This paper gives new conditions topromote the above literature. Nonlinear term is extended to the more general formand increased y|t=t₁=I（y（t₁））. We use the the fixed point index theory to provethe sufcient conditions for the existence of positive solutions. Finally, examples aregiven to show the applications of theorems.In chapter two, we consider singular boundary value problems for second orderimpulsive diferential equationswhere f∈C[（0,1）×（0,∞）×R,（0,∞）], q∈C（0,1）, I∈C（R+, R+）,Iˉ∈（R⁺,（∞,0]）,R+=[0,+∞）, and t₁∈（0,1） be given, and assume f may be singular at y=0ory=0, I is continuous and nondecreasing on [0,∞). In [27], by the fixed point index theory on a cone, Gong Qin gave sufcientconditions for the existence of positive solutions to the singular boundary valueproblems for second order impulsive diferential equationswhere f∈C[（0,1）×（0,∞）,（0,∞）], q∈C（0,1）, I∈C（R⁺, R⁺）, R⁺=[0,+∞), t₁in （0,1）, f may be singular at t=0,1or x=0, I is continuous and nondecreasingon [0,∞). In this paper, nonlinear term extended to a more general form. We alsouse the fixed point theory on a cone giving sufcient conditions for the existence ofpositive solutions of （2）. Finally, an example is given to show the applications oftheorems.