Existence Of Solutions For Multi-point Boundary Value Problems At Resonance

The ordinary differential equation boundary value problem is one of the most important branches of ordinary differential equations. It is resulted from physics, chemistry, biology, medicine, economics, engineering, cybernetics and so on. Boundary value problem at resonance have been studied by several authors in recent ten years. But the existence of high-order with complex boundary value conditons and equation including p-Laplacian operator has rarely been studied. So in this paper, we consider the problem as following: In section I, we give an introduction of the origin of the problems, the status of the research on related issues, and the trend of the future development. In section 2, result for solvability of 4-point boundary value problems of fourth order differential problem at resonance is obtained by using coincidence degree theory due to Mawhin. An sufficient condition is given under non-linear growth restriction on f. In section 3, we study the existence of p-Laplace-like equtions subject to multi-point boundary value problems at resonance. By using degree theory, we obtained two sufficient conditions for solvability, some known results are improved. In section 4, we use Brouwer degree and Leray-Schauder degree to discuss solvability of multi-point boundary value problem with two critical conditions. the methods used in the paper are different from some known results.Later, we can consider boundary value problem at resonance with different tools to obtain the existence of postive solutions and multi-solutions.