Class Of Nonlinear Differential Equations, Positive Solutions

Recently along with the development of scientific technology, physics and ap-plied mathmatics,many kind of non-linear problems have emerged. Those prob-lems have increasingly aroused people's widespread attentions, which greatly urge the non-linear functional analysis to be better improved. In this paper using topological degree theory, fixed point index theorem of con map, the fixed point theorems of mixed monotone operators. we discuss several classes of Sturm-Liouville boundary value problems and talk about their existences of postive solutions.In Chapter 1, we use topological degree theory to study the existence of postive solutions for third-order Sturm-Liouville 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.In Chapter 2, we use fixed point index theorem of con map to study the mul-tiple positive solutions of second-order Sturm-Liouville boundary value problems on the half line where f:[0,+00)×[0,+∞)×(-∞,0]�'[0,+∞) is a continuous, non-negative function and may be singular at t=0; u:[0,+∞)�'[0,+∞) is a Lebesgue in-tergrable function; p∈C([0,+∞), [0,+∞))∩C1[0,+∞)) wifh p>0 on (0,+∞), whereIn Chapter 3. we use the fixed point theorems of mixed monotone operators to investigate the existence and uniqueness of positive solutions of third-order boundary value problem whereλ>0,α,β,γ,δ≥0,andρ=βγ+αγ+αδ>0,φ∈C((0,1),(0,+∞)),φmay be singular at t=0 and t=1.