Multiple Solutions For Fourth Order M-Point Boundary Value Problem

In later years, all sorts of nonlinear problems have resulted from mathematics, physics, chemistry, biology, medicine, economics, engineering, cybernetics and so on. During the development of solving such problems, nonlinear functional analysis has been one of the most important research fields in modern mathematics. It mainly includespartial ordering method, topological degree method and the variational method. Also it provides a much effect theoretical tool for solving many nonlinear problems in the fields of the science and technology. And what is more, it is an important approach for studying nonlinear differential equations arising from many applied mathematics. L. E. J. Brouwerhad established the conception of topological degree for finite dimensionalspace in 1912. J. Lerry and J. Schauder had extend the conception to completely continuous field of Banach space in 1934. afterward E. Rother, M. A. Kranoselskii, P. H. Rabinnowitz, H. Amann, K. Deimling had carried on embedded research on topologicaldegree and cone theory. Many well known mathematicians in china, such as Zhang Gongqing, Guo Dajun, Chen Wenyuan, Sun Jingxian and so on, have proud works in various fields of nonlinear functional annalysis.In this paper, we investigate the multiple solutions for the fourth-order m-point boundary value problem:where 0＜a_i＜1,i=1,2,…,m-2,0＜η₁＜η₂＜…＜η_m-2＜1 are contants,(?)＜1,m≥3, and f∈C(R×R,R). With some conditions given, by means of the theory of the fixed point index in a cone and the Leray - Schauder degree, we obtain that there are at least 6 different nontrivial solutions for the boundary value problem above. Moreover, if f is odd, we obtain that there are at least 8 different nontrivial solutions for the boundary value problem above.