Several Types Of Nonlinear Differential Equations Boundary Value Problems

With the development of science and technology, various non-linear problem has aroused people's widespread interest day by day, The nonlinear functional analysis is an important branch in nonlinear analysis, because it can explain well various the natural phenomenon.The boundary value problems of nonlinear differential equations stem from the applied mathematics, the engineering, the biology, the physics, the cybernetics and each kind of application discipline. It is one of most active domains of functional analysis studies at present. In this paper, we use the cone theory, the fixed point theory as well as the fixed point index theory, to study the existence of solutions for several kinds of boundary value problems for nonlinear differential equation.The thesis is divided into three chapters according to contents.In Chapter 1, we study the existence of positive solutions for three-order integral boundary value problems with p-Laplacian whereφp(s)=|s|p-2s, p> 1, (φp)-1=φq,1/p+1/q= 1,α,β,γ,δ≥0,ρ=γβ+αγ+αδ> 0, g, h∈C([0,1], [0,+∞)). 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.In Chapter 2, by using the generalized zero point index of L- N operators in multi-point boundary value problem with p-Laplacian at resonance. whereΦp(u)=|u|p-2u,p>1,Φp=Φq-1,1/p+1/q=1,f:[0,1]×Rn�'R is continuous,R+=[0,∞),α∈C((0,1),[0,∞)),a(t)may be singular at t=0 and t=1,e(t)∈C[0,1],αi∈R+(i=1,2,…,m-2),∑i=1m-2αi=1,0<ξ1<ξ2<…<ξm=2<1.If∑i=1m-2αi=1,then BVP(2.1.1)is the so-called resonance case.In Chapter 3,by using a fixed point theorem in a cone,we discuss the exis-tence of at least three positive solutions to the following second-order boundary value problem whereφ:R�'R is an increasing homeomorphism and positive homeomorphism andφ(0)=0.αi∈R+(i=1,2,…,m-2),0<ξ1<ξ2<…<ξm-2<1.The problem we investigated is more general than that is considered,and our results generalize and extend previous results in the field.