Abstract: A corollary and a special continuation theorem are given in this paper.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 spacial 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.

Key words: Topological degree , Continuation theorem ,  Boundary value problem ,  Rayleigh equation ,  Periodic solution ,  Positive periodic solution