| Along with the development of scientific technology,modern physics and applied mathmatics, many kinds of non-linear problems have emerged. Those problems have increasingly aroused people's widespread attentions, which greatly urge the non-linear functional analysis to be better improved. The impulsive differential equations and the singular nonlinear differential equation are the topic points in recent years. Many authors have studied in every asperts.In this paper using cone theorem, fixed point index theorem, upper and lower solutions as well as fixed point theorem to study several kinds of boundary value problems for nonlinear singular differential equation and we discuss the existence of postive solutions to the boundary value problem for the differential equation.The thesis is divided into three chapters according to contents.In chapter 1, It is concerned impulsive boundary value problems with p-Laplacian satisfying Sturm-Liouville boundary conditions The existence of positive solution are obtained by the method of the cone theory and fixed point index theorem, whereφp(s)=|s|p-2s,p>1, (φp)-1=φq,1/p+1/q= 1, a≥0, b≥0,≥0, d≥0, The problem we investigated is more general than that is considered, and our results generalize and extend previous results in the field.In Chapter 2, It is considered the following n-order differential equations based upon a specially constructed cone and the fixed point theory in cone for a strict set contraction operator satisfying we can obtain the existence of positive solutions in abstract spaces, Where a,f are singular, which imposed some known results.In Chapter 3, we investigate the postive solutions of fourth-order singular boundary value problem whereα≥0,β≥0,γ≥0,δ≥0,Δ=αγ+αδ+βγ>0 and a(t)is singular at t=0,1, By using the fixed-point index theorem in a cone, the existence of at least two positive solutions is established. Meanwhile, the existence of many solutions is also obtained, which is the complement of previously known results. |
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