| Nonlinear functional analysis is a research discipline in analysis mathematics both to have the profound theory and to have the widespread application. It takes the nonlinear problems appearing in mathematics and the natural sciences as background to establish some general theories and methods to handle nonlinear problem. Because it can commendably explain all kinds of natural phenomenal, in recent years it has received highly attention of the domestic and foreign mathematics and natural science field, and gradually formed an important subject. The boundary value problem of nonlinear differential equation stems from the applied mathematics, the physics, the cybernetics and each kind of application discipline. It is one of most active domains of functional analysis studiesin at present. The singular nonlinear differential equation boundary value problem is also the hot spot which has been discussed in recent years. So it become a very important domain of differential equation research at present. In this paper, we use the cone theory, the fixed point theory, the topological degree theory as well as the fixed point index theory and combined with upper-lower solutions and so on, to study several kinds of boundary value problems for nonlinear singular differential equation and equation systems .The thesis is divided into three chapters according to contents.In chapter 1, by establishing a new comparison result and constructing upper and lower solutions, using the Schauder fixed point theorems, a sufficient condition of the existence of positive solutions for the four-order singular eigenvalue value problem with integral boundary conditions is investigated.(?) (1.1.1)whereλ> 0 is a parameter, f : (0,1)×(0, +∞)'R+ is continuous and f(t,u)may be singular at t = 0.1and u = 0; gi,hi∈L1([0,1],R+) (i = 1,2), R+ = [0, +∞). The problem we investigated is more general than that is considered in [8], and our results generalize and extend previous results in the field. In chapter 2, we discuss three-point boundary-value problems for second order impulsive differential equations with advanced arguments. We establish sufficient conditions under which such problems have positive solutions. To obtain the results we use the fixed point index.(?) (2.1.1)where J = [0,1].0 = t0 1 <…< tm < tm+1 = 1, J' = (0, 1)\{t1,t2,…,tm},R+= [0,+∞), Jk = (tk,tk=1], k = 0,1,…,m-1, Jm = (tm,tm=1);△x(tk) = x(tk+)-x(tk-).△x'(tk) = x'(tk+)-x'(tk-); f,j∈C(R+.R+).α∈C(J, (0,1]), t≤α(t)≤1, t∈J.In chapter 3, by employing a well-known fixed point index theorem and combining with a varication substitution, we study the existence of positive solutions for a singular semipositone impulsive differential system with integral boundary conditions. A new existence result is established, which is in essence different from the known results.Whereα,β,δ,γ≥0 are constants such thatρ=βγ+αδ+αγ> 0, f, g : (0,1)×[0, +∞)'[0, +∞) arc continuous and may be singular atl= 0 or l= 1; q : (0,1)'(-∞.+∞) is Lcbcsguc intcgrable; h∈L1[0,1] is non-negative, Ik∈C([0. +∞), [0. +∞)).△y'|t=tk = y'(tk+) - y'(tk-), where y'(tk-), y'(tk+) denote the left and right limits of y'(t) at t = tk, respectively. |
PDF Full Text Request