Abstract: This paper considers a class of the fourth-order boundary value problems with nonlinear boundary conditions{u⁽⁴⁾(t)=f(t,u(t)), t∈(0,1),u(0)=u″(0)=u(1)=0,u'(1)+C(u(1))u(1)=0,where f:［0,1］×R→［0,∞)satisfies L¹-Carathéodory conditions, C:［0,∞)→［0,∞)is continuous. The existence and multiplicity of the above problems are obtained by analyzing the properties of Green function and using Leggett-Williams fixed point theorem. Finally, an example is given to verify the validity of the obtained theorem.

Key words: cone, Leggett-Williams fixed point theorem, nonlinear boundary condition, positive solution