Abstract: By constructing suitable Banach space,an existence theorem is established under a condition of linear growth for the third-order boundary value problem u′′′(t)+f(t,u(t),u′(t),u″(t))=0,0<t<1,u(O)=u′(0)=u′(1)=0, where the nonlinear term contains first and second derivatives of unknown function.In this theorem the nonlinear term f(t,u,v,w) may be singular at t=0 and t=1.The main ingredient is Leray-Schauder nonlinear alternative.

Key words:  singular ordinary differential equation, boundary value problem, existence theorem, nonlinear alternative