中文摘要：

在工程实际中，四阶两点边值问题u（4）=f（t,u（t）），t∈[0，1]用来描述弹性梁在垂直轴线外力作用下的形变．一端为固定铰支，一端为可动铰支的梁称为简支梁，它在两端点的位移与弯矩均为零，故其相应的边界条件为u（O）=u（1）=u＂（0）=u＂（1）=0．本文应用下降流不变集方法研究了一类简支梁方程，在非线性项f在0处渐近线性、∞处超二次的条件下，证明了方程存在一个正解．主要结果及其证明方法均不同于文献中的结果．

英文摘要：

In engineering, the fourth-order two-point boundary value problem u^（4） f（t,u（t））, t ∈[0, 1] is used to describe the deformation of an elastic beam un- der external vertical forces. A beam that has hinged connection at one end and roller connection in other end is called simply supported beam, and its correspond- ing equation satisfies the boundary condition u（0） = u（1） = u＂（0） = u＂（1） = 0 since its displacements and bending moments at both ends are equal to zero. In the paper, by using the descending flow invariant set method, it is proved that there exists a positive solution for a class of simply supported beam equations under the assumption that the nonlinear term f is asymptotically linear at 0 and superquadric at ∞ in u. The main result and its proof are quite different from those presented by other literature.