The Existence Of Solutions For Two Class Of Higher Order Nonlocal Boundary Value Problems

A boundary value problem is an important definite solution problem in the theoreticalresearchs and applications for ordinary differential equations. It plays an very importanttheoretical and practical value in real life for solving some real problems. The research ofthe nonlinear m-point (the nonlocal) boundary value problems has been used in widely indifferent areas such as physics, applied mathematics and so on. We can get n or infinitesolutions by using the global bifurcation theory, and also can show that how many zeroand sign-changing times in a given range for these solutions. At present, it is a difficultand hot topic about sign-changing solutions for boundary value problems, many scholarsconsidered the sign-changing solutions for multi-point boundary value problems, and theyhave made great achievements, but there are many problems still haven’t been solved.Aiming at these problems, the existence of nodal solutions for a class of sixth-orderm-point boundary value problem and positive solutions of the nonlocal third-orderboundary value problems with dependence on the first order derivative are mainly studiedin this paper. It is divided into three chapters as follows:Firstly, the history of the theory of nonlocal boundary value problems for ordinarydifferential equations, the current situation of node solution of nonlocal boundary valueproblem and positive solutions for nonlocal boundary value problems and the maincontent of this paper are introduced.Secondly, we consider the existence of nodal solutions for a class of sixth-orderm-point boundary value problem. These spectral properties are then used to prove aRabinowitz-type global bifurcation theorem for a bifurcation problem related to the aboveproblem. Finally, we use the global bifurcation theorem to obtain nodal solutions of theabove problem, under various conditions on the asymptotic behaviour of f.Thirdly, giving the Green function for nonlocal third-order boundary value problems,and using the properties of Green function and a new fixed point theorem, we constructe aoperator to prove the positive solutions for nonlocal third-order boundary value problemswith dependence on the first order derivative. So, we obtain some new results.