Class Of Nonlinear Second Order Boundary Value Problems

Along with science's and technology's development, various non-linear problem has 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 cybernetics and each kind of application discipline. It is one of most active domains of functional analysis studiesin at present.In this paper, we use the cone theory, the fixed point theory, the maximum principle ,lower and upper solutions as well as the fixed point index theory and combined with a iterative technique and so on, to consider the solutions of several kinds of boundary value problems for nonlinear second order differential equation.The thesis is divided into three chapters according to contents.In chapter 1, we consider the existence of extreme solutions for a class of second-order impulsive integro-differential equation where 0 = t₀ < t₁ < t₂ < ...... < t_m = T, f : J×R³â†'R. We introducea mew definition of lower and upper solutions , which developedand extended the classic lower and upper solutions .We proved some new comparison theorem,which present that the method of lower and upper solutions coupled with monotone iterative techniqueis still valid for this kind of impulsive integro-differential equation, at last we applied new lower and upper solutions coupled with monotone iterative technique obtained the extreme solution of equation(1.1.1).This chapter improved-and extended the document[8,9],refer to note 1.2.1,note 1.2.2,note 1.3.1.In chapter 2,we by using the fixed point index theory , consider the existence of positive solutions for the semipositive integral boundary value problem of second-orderwhere f: [0,1]×[0, +∞)â†'R is continuous, and satisfies f(t, x)≥-Mx, where M > 0. In the case where f can be allowed to change sign,αandβ3 are right continuous on [0,1), left continuous at t=1, and nondecreasing on [0,1] withα(0) =β(0) = 0; (?)u(r)dα(r) and (?) u(r)dβ(r) denote the Riemann-Stieltjes integrals of u with respect toαandβ, respectively.There are requires of that the nonlinearfunction is nonnegative in the past thesis, we, however, can include the case that the nonlinear function is negative and get a positive solution in this chapter(note2.3.1).In chapter 3,we use the cone theory consider the existence of positive solution for singular second order three-point boundaryvalue problem with parameterwhereε>0,0 <η< 1,0 <α<(?),α∈C((0,1)â†'[0, +∞)), f∈C([0, +∞)â†'[0, +∞)). Document[6]will be the case of A = 0 in this chaper.It is not necessary that a is singular in document[6],however this chapter devoted to solve the case that a is singular with t = 0,t= 1.this chapter improved the document[6](note 3.2.1).