Abstract: By using Leray-Schauder degree theory and method of upper and lower solutions, the author  study the existence and multiplicity of positive solutions for the elastic beam equation with nonhomogeneous boundary conditions.  When the nonlinear term f satisfies suitable conditions, the author prove that there is a positive number b*, such that there are at least two positive solutions to the problem when 0<b<b*, there is exactly one positive solution to the problem when b=b*, there is no positive solution to the problem when b>b*.

Key words: positive solution, nonhomogeneous, Leray-Schauder degree, method of upper and lower solutions