By using the Wirtinger inequality and topology degree theory, we investigate the anti-periodic solutions problem for inertial competitive neutral-type neural networks and obtain the existence of anti-periodic solutions to the above system. Our results are completely new.

The being a wide range of applications of the Internet, social networks have become an effective and convenient platform for information communication, propagation and diffusion. Most of information exchange and spreading exist in social networks. The issue of information diffusion in social networks is getting more and more attention by government and individuals. The researchers investigated either empirical studies or focused on ordinary differential equation (ODE) models with only consideration of temporal dimension in most prior work. As is well known, partial differential equations (PDEs) can describe temporal and spatial patterns of information diffusion over online social networks; however, until now, results for understanding information propagation of social networks over both temporal and spatial dimensions are few. This paper is devoted to investigating a non-autonomous diffusive logistic model with Dirichlet boundary conditions to describe the process of information propagation in social networks. By constructing upper and lower solutions we obtain the dynamic behavior of the solution to the non-autonomous diffusive logistic model. Our results show that information diffusion is greatly affected by the diffusion coefficient \(d(t)\) and the intrinsic growth rate \(r(t)\).

This paper is devoted to investigating a class of neutral-type integral differential equations arising in an epidemic model. By using Mawhin’s continuation theorem and the properties of neutral-type operators, we obtain the existence conditions for positive periodic solutions of the considered neutral-type integral differential equation. Compared with previous results, the existence conditions in this paper are less restricted, thus extending the results of the existing literature. Finally, two examples are given to show the effectiveness and merits of the main results of this paper. Our results can be used to obtain the existence of a positive periodic solution to the corresponding non-neutral-type integral differential equation.