本文给出Mawhin连续定理的一个推论和一个特殊的连续定理.与经典的Mawhin连续定理相比,本文给出的特殊的连续定理在解决现实问题时可以避免计算任何拓扑度,并减少经典连续定理所使用的条件,且判断这个特殊连续定理的条件将变得更加简单和方便.值得注意的是,在使用拓扑度的连续定理时,避免计算拓扑度意味着可以极大地减少处理问题的过程.然后,本文利用这个特殊的连续定理,研究一类具有一般形式的二阶微分方程的边值问题,得到该微分方程解的存在性定理及推论.最后,作为上述定理的应用,研究一类带偏差变元的Rayleigh方程周期解和正周期解的存在性,得到一些新的充分条件,这些结果推广和改进了已有文献的结论.“,”In this paper we give a corollary to the Mawhin Continuation Theorem and a special continuation theorem.By comparing with the classical Mawhin Continuation Theorem,we can avoid calculating any topological degree and reduce the conditions of the theorem when using this special continuation theorem in applications.In particular,the conditions for verifying this special continuation theorem will be easier and more convenient.Notice that,avoiding the calculation of any topological degree means to minimize the processing of real world applications when a continuation theorem of topological degree theory is used.By using this special continuation theorem,a boundary value problem for a generalized second order differential equation is then studied and some theorems for the existence of solutions of the differential equation are obtained.As applications,the existence of periodic solutions and positive periodic solutions for a kind of Rayleigh equations with deviating arguments is investigated,and some new sufficient conditions which generalize and improve the known results in the literature are obtained.