| Nonlinear functional analysis is an important branch of modern analysis mathematics, many problems arise from models of chemical reators, population biology, contagious disease, economics and other systems. Because it can explain many kinds of natural phenomena, more and more mathematicians are devoting their time to it. The study of multi-point boundary-value problems for linear second order ordinary differential equations was initiated by Il'in and Moiseev. Then Gupta studied three-piont boundary-value problems for nonlinear ordinary differential equations. Since then, the more general nonlinear multi-point boundary-value problems have been studied by several authors by using the Leray-Schauder Continuation Theorem, Nonlinear Alternatives of Leray-Schauder and coincidence degree theory. Also, they have had many useful results[3-6]. Otherwise few paper studied global behavior of positive solutions. Recently by using the fixed point theorem [7] studied one kind of the global behavior of positive solutions of nonlinear three-point boundary-value problems.Motivated by this, we consider the global behavior of positive solutions of three-point boundary-value problems for second-order differential equations, under two different kinds of boundary-value problems, using fixed point theorem, fixed point index theorem and cone theory. When the nonlinear function is superlinear case or sublinear case, we have the result that the positive solutions of the two boundary-value problems have a continuum, which means an nonempty, closed and connected subset.In the first section of this paper, we study the history of the problems above and the development nowadays. The second and third sections are the main parts of the paper. We consider the global behavior of positive solutions of the same differential equation under different boundary-value problems. At last, we have some other ideas about this problem, hoping more and better results.
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