Existence Of Positive Solutions For Boundary Value Problems Of Second-order Time Lag Differential Equations

Nonlinear functional analysis is an important branch of modern analysis mathematics,it is developed progressively by the process of studying biology, modern physics, economics and other disciplines .Because it not only provided a fruitful theoretical tools in the field of science and technology, and played an irreplaceable role in dealing with the practical problems of corresponding nonlinear integral equation and differential equation , but also was a good explanation for a wide range of natural phenomena in the natural world, more and more mathematics are devoting their time to it. Among them, the nonlinear boundary value problem comes from a lot of branches of applied mathematics and physics, and it summarizes many issues in the field of applied mathematics, and it has high practical value in nonlinear proliferation, gas heat ignited, biochemistry field, it becomes a strong interesting problem for people, also it is one of the most active fields that is studied in analysis mathematics at present.The present paper employs the cone theory, and Krasnoselskii fixed point theorem and so on, to investigate the existence of positive solutions of several classes of second-order nonlinear three point boundary value problems, the obtained results are either new or intrinsically generalize and improve the previous relevant ones. This paper considers these problems from the following aspects: In Section 1, we give the origin of the problem, the status of research on related issues, and the trend of the future development. In Section 2, we study the existence of positive solutions of a simple eigenvalues problem with singularity and the superlinear semipositone problem of second-order delay differential equations, the results of this section has been accepted by the "differential equation annual". In Section 3, we changed the boundary conditions and tried to use the method of Section 1 for inspecting the nature of this kind of boundary value problem, we have achieved through controlling the parameter and adjusting the interval of eigenvalues. moreover, we give some examples to illustrate our results.Later, we can also combine it with some practical mathematical models in the field of nonlinear proliferation,control theory,biochemistry, to make them stand together, achieve better results.