Positive Solutions For The Fourth-order Boundary Value Problems With Dependence On The First Order Derivative

Differential equation model can desribe the problems about the rate of change. So, differential equation plays a very important role in the actual application. As a part of the differential equation, boundary value problems apply more and more in the fileds, for example, chemical engineering, heat conduction, underground water flow, thenno-elasticity, plasma physics and so on, these can be solved by boundary value problems. The theory of boundary value problems with integral boundary conditions for ordinary differential equations arises in different areas of physics and applied mathematics. Although many scholars study the above problems by all sorts of fixed point theorem, and they have made great achievements, but many problems still haven't been solved.Aiming at these problems, positive solutions for fourth-order boundary value problems with dependence on the first derivative is mainly studied in this paper. It is divided into four chapters as follows:Firstly, the history, the current situation of the theory of boundary value problems for ordinary differential equations and the main content of this paper are introduced.Secondly, we obtain positive solutions for fourth-order boundary value problems with dependence on the first order derivative by the fixed point theorem which is proved by degree theory.Thirdly, by constructing an operator, we translate the problem of finding the solution for differential equation into the problem of finding out the fixed point of the operator by using a new fixed point theorem. We obtain positive solutions for nonlocal fourth-order boundary value problems with dependence on the first order derivative. the sufficient conditions of the existence of positive solutions for this kind of boundary value problems is obtained.Forthly, by using a new fixed point theorem which is obtained by degree theory, we obtain positive solutions for nonlocal fourth-order boundary value problems with dependence on all order derivatives. By constructing corresponding Green function for nonlocal fourth-order boundary value problems and using the properties of Green function and the fixed point theorem, we obtain sufficient conditions for the existence of positive solutions.