Existence Of Solutions To Singular Boundary Value Problems For Several Kinds Of Differential Equations

In last few years, singular boundary value problems (SBVP,for short) have resulted from mathematics,physics,chemistry, biology,medicine,economics,engineering,sybernetics and so on. Prom 1980s, such problems have received a great deal of attention by many researchers. In solving these problems, many important methods and theories such as partial ordering method, topological degree method,the theory of cone and the variational method have been developed gradually. They become very effective theoretical tool to solve many nonlinear problems in the fields of the science and technology.This paper deeply discusses the existence of multiple solutions to SBVP for second order,third order and fourth order nonlinear differential equations mainly by making use of fixed point theorem and the theorem of cone. Because of the importance of singular and parameters, we also study the effect of singular and parameters on the solutions of nonlinear differential equations, and we obtain some useful results.There are four chapters in the dissertation.In the first chapter, by using Avery-Peterson fixed point theorem, we deal with p-Laplacian singular boundary problems where f∈C([0,+∞)×(-∞,0], [0,+∞)), and w(t) maybe singular at t=0 and t=1.In this chapter, we obtain three positive solutions under certain conditins.In the second chapter, by using the Leggett-Williams fixed point theorem, we deal with the existence of three positive solutions and infinitely many positive solutions for the following third-order three-point SBVP where w(t),f(t,u) may be singular at t=0,t=1 and u=0,0<η<1.1<α<1/ηIn the third chapter, we investigate the existence of positive solutions of the following SBVP where the nonlinearity f(t,u) and p(t) may be singular at t=0,t=1,λ∈(0.+∞) is a parameter,α,β,γ,δ≥0,βγ+αγ+αδ>0. In this chapter, by using the fixed point index theory in cones., we derive aλ* which is a constant,such that forλ=λ*,0<λ<λ*.λ>λ*, the boundary value problem has at least one positive solution,two positive solutions or nothing solution if / is superlinear. An example is worked out to indicate that our conditions are reasonable.In the fourth chapter, we investigate the existence of positive solutions,of the fol-lowing singular boundary value problems where the nonlinearity f(t,v) and g(t.u) may be singular at t =0,t=1,α≥0,β≥0,γ≥0,δ≥0 and△=αγ+αδ+βγ>0. In this chapter, by constructing a special cone and using cone compression and expansion fixed point theorem, the existence of double positive solutions to the above SBVP is guaranteed under certain conditions.