Existence of solutions for some third-order boundary-value problems

In this paper concerns the third-order boundary-value problem $$displaylines{ u'''(t)+ f(t, u(t),u'(t), u''(t))=0, quad 0 < t < 1, cr r_1 u(0) - r_2 u' (0)= r_3 u(1) + r_4 u'(1)= u''(0)=0. }$$ By placing certain restrictions on the nonlinear term f, we prove the existence of at least one solution to the boundary-value problem with the use of lower and upper solution method and of Schauder fixed-point theorem. The construction of lower or upper solutions is also presented.