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With the development of modern physics and applied mathematics, it de-mands the mathematical ability of analyzing and controling the objective phe-nomena toward to the overall high and precision level, which made the results of the nonlinear analysis was accumulated, and gradually formed an important branch subject of the present analysis mathematics-Nonlinear functional analy-sis. Along with science’s and technology’s development, various non-linear prob-lem has come up from the fields of physics, chemisty, mathematics, biology, medicine, economics, engineering, cybernetics, and these problems has aroused people’s widespread attention day by day. However, the nonlinear functional analysis offers effective theoretic tools for these problems, and it is a subject of profound theories and broad applications. Nonlinear impulsive equation is a more important direction of non-linear functional analysis, because the impulsive equa-tions have the impulsive phenomenon, the continuous of the solutions is affected by the qualities of impulsive, using the method of nonlinear analysis to study nonlinear impulsive equations is valuable questions. In this paper using upper and lower solutions, and monotone iterative technique,we discuss boundary value problems of nonlinear impulsive equations and give some sufficient conditions of the existences of solutions.The thesis is divided into three chapters according to the contents:In Chapter 1, we use fixed point theorem to study the following first-order integro-differential equations where J= [0,T], J0= J/{t1,t2,…,tp}> 0<t1< t2<…<tp<T, f∈C{J×R×R×R,R), Ik∈C{R,R),Δx{tk)= x(tk+),x(tk-) denote the right and left limits of x(t) at tk{k= 1,2,…, p), k∈C(D,R+),D={(t,s)∈J×J:t≥s},h∈C(J×J, R+).Under new comparison principle,we get some existence solutions for the equations.Finally, we give some applications to demostrare the main results of this paper.In Chapter 2,we consider the following third-order impulsive differential equations where,Ik,Ik1,Ik2∈C(R,R),0<t1<…<tp<T,Δx(tk)=x(tk+)-x(tk),Δx′(tk)=x′(tk+)-x(tk),Δx″(tk)=x″(tk+)-x″(tk)(k=1,2,…,p),f:[0,T]×R→R is a Caratheodory function. By using upper and lower solutions and monotone iterative technique,we obtain the existence and uniqueness of coupled solutions.In Chapter 3,we discuss the following impulsive functional integro-differential equations: where,f∈C(J×R4,R),,Ik∈C(R,R),0≤θ(t)≤t,t∈J,0<t1<t2<…<tp<T,Δx(tk)=x(tk+)-x(tk-),0<σk-1≤(tk-tk-1)/2,0≤τk-1≤(tk-tk-1)／2(k=1,2,…p),λ∈R is constant, K∈C(D,R+),D={(t,s)∈J×J:t≥s},H∈C(J×J,R+).By using upper and lower solutions and monotone iterative technique,we obtain the existence theorems of solutions for impulsive functional integro-differential equations.
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