Positive Solutions Of A Class Of Nonlinear∞-point Boundary Value Problems

The boundary value problem of ordinary differential equation is one basic theory of differential equations, many problems in engineering, mechanics, astronomy, cybernetics and biology can be summed up in the boundary value problem of ordinary differential equation. The boundary value problem of ordinary differential equation that can accurately describe many important physical phenomena has wide practical use, just because multipoint boundary value problem has its own inherent difficulty, the study of multipoint boundary value started relatively late. In reference [30] it studies the existence of the multiple positive solutions of a class of nonlinear second order diffenertial equations according to the fixed point theorem on cone. And then, there are related studies of infinite boundary value problem for ordinary differential equation, such as some studies in reference. According to the theory of compressing fixed point and cones expansion, this paper proves that a class of nonlinear second order ordinary differential equation u’’+α(t)u’+b(t)u+h(t)f(u)=0, t∈(0,1) They are the existence of positive solutions respectively in the following three kinds of boundary value conditions (1) u’(0)=0,u(1)=∑i=1∞αiu(ζi)(2) u’(0)=0,u(1)=∑i=1∞αiu(ζi)(3) u’(0)=∑i=1∞αiu(ζi),u(1)=∑i=1∞βiu(ζi)According to the research, this paper is mainly divided into the following three parts: The first part proves the existence of positive solutions in infinite-many point boundary value problemFirst according to the theory of compressing fixed point and cones expansion, this paper proves the existence of positive solution in n+2point boundary value problem First consider the case (2), the first step, case (2) is converted to be the integral equation. The second step, according to the theorem of fixed point in a cone, it proves that at least it has one positive solution when the nonlinear term in question (2) satisfies superlinear, and when n→∞,the limits of the situation (1) in case (2) comes out. In other words we will get the results we want. The third step, it proves that the positive solution s existence when the nonlinear term satisfies the sublinear.The second part proves the existence of positive solutions u"+a(t)u’+b(t)u+h(t)f(u)=0, te (0,1) In the boundary value conditions u(0)=0, u(1)=∑i=1∞αiu(ζi) under the existence of positive solutions the method of prove is similar to the first part, the difficulty is that in the process of proof the difficulty of converting the problem into the corresponding integral equation, the biggest difficulty is proof of the existence of positive solutions when n→∞.The third part proves the existence of positive solution in infinite point boundary value problem First it proves there is at least one positive solution in limited point boundary value problem and then it researches the existence of positive solution in infinite point boundary value problem when n→∞, because the corresponding Green function form in the problem (3) is more complex, so it is difficult to discuss the nature of it, at the same time it results in the complexity of integral equation corresponding to the problem (3),and brings certain difficulty to justify the corresponding result, but through the careful discussion we solve the above difficulties and get the desired results.