本文应用奇异点理论,在g(x)为凹(凸)型函数时,给出周期系统(?)+a(t)g(x)=h(t)整体等价于Whitney意义下的尖点映射的结果.精确地说,算子Fx(t)=(?)+a(t)g(x(t))的奇异值集F(∑)为单连通超曲面并且将C[0,1]分成两个连通分支A1和A3,使得:(1)对周期为1的连续函数p(t)∈A1有唯一解.(2)对周期为1的连续函数p(t)∈A3恰有三个周期解.进一步,尖点集C的像集F(C)是C[0,1]中的,余维数等于2的子流形.对p∈F(C)有唯一解,而对p(t)∈F(∑)\F(C)恰有两个周期解.

Consider the differential equation (?) + a(t)g(x) = h(t), where a(t) and h(t) are 1-periodic functions such that a(f) does not change sign, and g is a concave-convex type function. By using the singularity method we obtain a complete geometric structure of 1-periodic solution, and the exact multiplicity results. More precisely, the image of singularities consists of codimensional 1 manifold that divides the C[0, 1] into two open sets A1,A3: (1) for h(t) A1, the equation has a unique periodic solution. (2) for h(t) ∈ A3, the equation has exactly three periodic solutions. (3) Moreover, the image F(C) of cusp singularities C is a codimensional 2 manifolds of ∈ [0,1] such that for h(t) ∈ F(C), the equation has a unique periodic solution, and for h(t) ∈ F(E)\F(C), the equation has exactly two periodic solutions.