In this paper, we consider a class of delayed quaternion-valued cellular neural networks (DQVCNNs) with impulsive effects. By using a novel continuation theorem of coincidence degree theory, the existence of anti-periodic solutions for DQVCNNs is obtained with or without assuming that the activation functions are bounded. Furthermore, by constructing a suitable Lyapunov function, some sufficient conditions are derived to guarantee the global exponential stability of anti-periodic solutions for DQVCNNs. Our results are new and complementary to the known results even when DQVCNNs degenerate into real-valued or complex-valued neural networks. Finally, an example is given to illustrate the effectiveness of the obtained results. KEYWORDSanti-periodic solution, neural networks, quaternion, stabilitywhere q 0 , q 1 , q 2 , q 3 are real numbers and the elements i, j, and k obey the Hamilton's multiplication rules:In the past two decades, the theory of quaternion has been found a lot of applications in many fields such as such as attitude control, quantum mechanics, robotics, computer graphics, and so on. 26-30 Quaternion-valued neural networks (QVNNs), Math Meth Appl Sci. 2019;42:5-23.wileyonlinelibrary.com/journal/mma

This study examines the hitherto unexplored question of whether and how a firm's social performance influences the breadth of that firm's share ownership. We predict and find that firms with higher corporate social responsibility (CSR) ratings attract more institutional investors (especially long‐term, low‐stake and green institutional investors) and more individual investors. This finding is consistent with the notion that investors are more interested in firms with higher CSR ratings and thus prefer to hold stocks of such firms. We also find that firms with higher CSR ratings are associated with higher stock liquidity, lower cost of equity capital, more equity and debt issuance, and greater investment, and that sin stocks are associated with a lower investor base, which further corroborates our prediction. Our results are robust to potential endogeneity, the use of alternative model specifications, and an alternative proxy for CSR performance.

In this paper, we propose and analyze a delayed HIV-1 model with CTL immune response and virus waning. The two discrete delays stand for the time for infected cells to produce viruses after viral entry and for the time for CD8+ T cell immune response to emerge to control viral replication. We obtain the positiveness and boundedness of solutions and find the basic reproduction number R0. If R0 < 1, then the infection-free steady state is globally asymptotically stable and the infection is cleared from the T-cell population; whereas if R0 > 1, then the system is uniformly persistent and the viral concentration maintains at some constant level. The global dynamics when R0 > 1 is complicated. We establish the local stability of the infected steady state and show that Hopf bifurcation can occur. Both analytical and numerical results indicate that if, in the initial infection stage, the effect of delays on HIV-1 infection is ignored, then the risk of HIV-1 infection (if persists) will be underestimated. Moreover, the viral load differs from that without virus waning. These results highlight the important role of delays and virus waning on HIV-1 infection.

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