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在运用拓扑度的连续定理时,避免拓扑度的计算也就意味着最大化简化实际问题的处理过程.本文首先给出了一个特别的Manásevich-Mawhin连续定理和几个推论.相对于经典的Manásevich-Mawhin连续定理,在使用这个特别的连续定理及其推论处理实际问题时,我们能够避免计算拓扑度,且可以减少定理使用的条件.更重要的是,验证这个特别的连续定理的条件将更加容易和方便.其次,作为一个应用,本文应用上述特别的Manásevich-Mawhin连续定理研究了一般形式的Rayleigh型p-Lalacian泛函微分方程周期解和正周期解的存在性问题,获得了一些新的充分条件并推广和改进了一些已有的结果.“,”Avoiding the calculation of any topological degree also means to minimize the processing of practical problems when a continuation theorem of topological degree theory is used.In this paper,a special continuation theorem and several corollaries are given.Compared with the classical Manásevich-Mawhin continuation theorem,we can avoid calculating any topological degree and reduce the conditions of the theorem when using this special continuation theorem and its corollaries in applications.More importantly,the conditions for verifying this special continuation theorem will be easier and more convenient.As an application,we use this special continuation theorem and its corollary to study the existence of periodic solutions and positive periodic solutions for a generalized Rayleigh typep-Laplacian equation with deviating arguments and obtain some new sufficient conditions which generalize and improve the known results in the literatures.