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We are concerned with the dynamic properties for the class of differential equation x + εp(t)x =q(t)x β. Dynamic properties include existence, nonexistence, stability and instability. Here β> 0٠β ≠ l,٠ε>0 is a small parameter and p,g are continuous T-periodic functions. Our results are proved using the averaging method and the third approximation method. Compared with the existing literature，the periodic solutions obtained in this paper are obtained. [ABSTRACT FROM AUTHOR]

In the recommendation system. the inner product interaction of a single learning matrix de composition or the use of deep neural networks to capture user interaction with the project is not sufficient to effectively learn the potential characteristics of users and projects. In view of this problem. this paper proposes a collaborative filtering algorithm based on deep learning feature representation (DLFeaCF) based on display feedback and implicit feedback. The model first learns the inner product and outer product interaction between the user and the project. Then, based on the inner product, it uses the nonlinear interactive learning ability of the multi-layer perceptron (MLP) to obtain the two aspects from implicit mapping and feature mapping: on the basis of the outer product, CNN learning is used to capture the local features of the user and the project finally, combine the features in the fusion layer and obtain the prediction score, The experiments on the real MovieLens dataset show that the DLFeaCF model can achieve better recommendation performance. [ABSTRACT FROM AUTHOR]