变分方法在时标上一类边值问题中的应用∗

摘要

In this paper, we are concerned with the existence of periodic solution to a nonau-tonomous second order boundary value problem on time scales T. By simultaneously utiliting the critical-point theorem and variational methods, we first transform the existence of solutions to the boundary value problem into the critical points of the associated functional by using variational methods. Some new results on the existence of at least one periodic solutions are established by means of the generalized mountain pass lemma. Our results are new even for the corresponding differential, difference equations as well as in general time scales. As an illustration, we verify the obtained results through an example.%本文研究时标T上一类非自治的二阶周期边值问题周期解的存在性。我们综合利用临界点理论和变分方法，先利用变分方法将研究边值问题解的存在性问题转化为研究一个算子临界点问题，再借助于广义山路引理得到所研究边值问题存在至少一个周期解，所得结果在相应的微分方程，差分方程以及通常的时标上都是新的，作为应用，给出了一个例子验证了所得结论。