“Time Delays and Stimulus-Dependent Pattern Formation in Periodic Environments in Isolated Neurons”的注记

A Note on “Time Delays and Stimulus-Dependent Pattern Formation in Periodic Environments in Isolated Neurons”

指出了题目所给文献中的一个证明的错误,并利用Mawhin重合度理论中的连续性引理、Cauchy不等式等知识得出了所给文献中模型周期解的存在性的充分条件,这个条件与已有结果相比具有较少保守性.文中还利用Dini导数证明了该模型的周期解是唯一的,且是全局吸引的.最后举例说明了所得结果的有效性.

A mistake in the proof of some technical problems in the literature as shown in the title of this paper is pointed out. By employing the continuous theorem of Mawhin＇ s coincidence degree theory and the Cauchy inequality, sufficient conditions for the existence of the periodic solution to the model presented in the literature are obtained. These sufficient conditions are weaker than the ones in the existing literature. The unique and global attractiveness of the periodic solution are then proved by employing Dini＇ s derivative. An example is finally presented to demonstrate the effectiveness of the proposed results.