Several Classes Of Second-order Cycle And Anti-periodic Boundary Value Problem Solution

Along with the development of modern physics and applied mathematics, many kinds of non-linear problems have emerged, which greatly urged the nonlinearfunctional analysis to be better improved. Non-linear ordinary differential equation is a more important direction of non-linear functional analysis, and the differential equation with periodical boundary value conditions is a hot spot recently.In particular, with deep research and the requirement of some physical phenomenon, the anti-periodic boundary value problems have also been deeply studied. In this paper using cone theorem, fixed point theorem as well as lower and upper solutions coupled with montone iterative technique, we discusse several classes of periodic and anti-periodic problems and give some sufficient conditions of the existences of solutions. Meanwhile we apply the main results to the existenceof solutions for boundary value problems.The thesis is divided into three chapters according the contents.In Chapter 1, we use cone expansion and compression fixed point theorem of norm type to study the following class of singular second order periodic boundary value problemwhere f : [0,2Ï€]×[0,∞)â†'[0,∞) is continuous and g : (0, 2Ï€)â†'[0,∞) is continuous. However g may be singular at both end points and a(t) doesn't need to ensure the Green function is positive. In particular, when a(t)=(?) is true, the Green function of this equation is zero on a line. In this paper we obtain the existence theorems with one and two solutions. The main results obtained improve and generalize the the existing resluts. Finally, two examples are given to demostrare the main results of this paper.In Chapter 2, the following periodic boundary value problem for second order integro-differential equationis well discussed, whereBy developing a new maximum principle, this paper introduces a new definition of lower and upper solutions and presents that the method of lower and upper solutions coupled with monotone iterative technique is still valid. Meanwhile, we extend previous results.In Chapter 3, we consider a class of second anti-periodic boundary value problemwhereλ> 0, f∈C(J×R, R). we establish several existence results by using the Leray-Schauder fixed point theorem and the lower and upper solutions methods. In this paper we needn't require f to have increasing condition when using lower and upper solution method, which is different from the the classical lower and upper solution method.