Boundary Value Problems Of Nonlinear Differential Equations

Along with science's and technology's development, various non-linear problemhas aroused people's widespread interest day by day, and so the nonlinear analysis has become one important research directions in modern mathematics. The nonlinear functional analysis is an important branch in nonlinear analysis, because it can explain well various the natural phenomenon. The boundary value problem of nonlinear differential equation stems from the applied mathematics, the physics, the biology and each kind of application discipline. It is one of most active domains of functional analysis studies at present. The impulsive nonlinear differential equation boundary value problem is also a hot spot which has been discussed in recent years. So it become a very important domain of differential equation research at present. Multiple positive solutions of impulsive differential equation, singular impulsive differential equation and multiple positive solutions in abstract space, received a great deal of recent attention.In this paper, we use the cone theory, the fixed point theory, the fixed point index, and so on, to study several kinds of boundary value problems for nonlinear differential equation. Examples and applications are given follow our main results, and the main results in essay [13,36,39,48,60,67] are spread and improved in this paper.The thesis is divided into three chapters according to contents.In chapter 1, we use the cone theory and Leggett-Williams fixed point theory to investigate the positive solutions of a class of boundary problems for third-order three-point nonlinear impulsive differential equations in Banach spaces as follows,where 0<η<1 and 1<α<(?),f∈C(R⁺,R⁺),h(t)∈C([0,1],R⁺) and is not identical zero on I_k∈C([0,1],R⁺),k=1,2…,m.â–³x"(t_k)=x"(t_k⁺)-x"(t_k^-).J=[0,1]ï¼¼{k₁,k₂…,k_m},R⁺=[0,+∞).First, we establish the existence of at least three positive solutions by using the well-known Leggett-Williams fixed point theorem. And then, we prove the existence of at least 2m-1 positive solutions for arbitrary integer m. This chapter generalize and improve the main results in [13,48](See the Corollary 1.3.1 in page 8, the Corollary 1.3.2 in page 9 and the Remark 1.3.2 in page 10).In chapter 2, we devote to the study of the positive solution of two-point boundary value problems for nonlinear second-order singular and impulsive dif-ferential systems as follows,whereα,β,γ,δ≥0,ρ=βγ+αγ+αγ> 0, J = (0,1), (?) = [0,1], 0 < t₁ < t₂ < ... < t_m < 1, J' = J\ {t₁,t₂,...,t_m}, f_i∈C(?)×R⁺×R⁺,R⁺), h_i∈C(J, (0,∞))(i = 1,2) may be singular at t = 0 or t = 1. where, I_i∈C(R⁺, R⁺), R⁺ = [0, +∞),â–³u(t_k)=u(t_k⁺)-u(t_k^-) =â–³v(t_k)=v(t_k⁺)-v(t_k^-),â–³v(t_k) = u'(t_k⁺) -u'(t_k^- ),â–³v(t_k)v(t_k⁺)-v(t_k^-), in which u'(t_k⁺),v'(t_k⁺) and u'(t_k^-),v'(t_k^-) denote the right and left limit of u'(t_k), v'(t_k), respectively. The fixed point theory and monotone assumption are used to investigate the nonnegative solution of the systems. In this chapter, a well-known decreasing fixed point theory is used to study the impulsive equation at first time, and we generalize and improve the main results in [36,67] (See the Remark 2.3.1 in page 19).In chapter 3. we use the fixed point index theory study the fourth order value problems in abstract space as followswhere J = [0,1],β<Ï€², g, f∈C(J, [0, +∞)). we studied whenλ,μchange, with f and g may be upperlinear or sublinear, the existence of none, one or two positive solutions for the fourth order value problems in abstract space of the BVP(3.1.1). we generalize and improve some results in [39,60](See the Remark 3.3.1 in page 33).