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The existence and finite-time stability (FTS) of equilibrium point (EP) for a kind of inertial neural networks (INNS) with varying-time delays is studied. Firstly, by adopting the degree theory and the maximum-valued method, a sufficient condition in the existence of EP is attained. Then by adopting the maximum-valued approach and the figure analysis approach, without adopting the matrix measure theory, linear matrix inequality (LMI), and FTS theorems, a sufficient condition in the FTS of EP for the discussed INNS is proposed.

In this article, we mainly focus on the finite-time synchronization of delayed multinonidentical coupled complex dynamical networks. By applying the Zero-point theorem, novel differential inequalities, and designing three novel controllers, we obtain three new criteria to assure the finite-time synchronization between the drive system and the response system. The inequalities occurred in this paper are absolutely different from those in other papers. And the controllers provided here are fully novel. We also illustrate the theoretical results through some examples.

In this paper, we are interested in the finite-time synchronization of complex-valued BAM neural networks with time delays. Without applying Lyapunov–Krasovskii functional theory, finite-time convergence theorem, graph-theoretic method, the theory of complex functions or the integral inequality method, by using the quadratic analysis approach, inequality techniques and designing two classes of novel controllers, two novel sufficient conditions are achieved to guarantee finite-time synchronization between the master system and the slave system. The quadratic analysis method used in our paper is a different study approach of finite-time synchronization from those in existing papers. Therefore the controllers designed in our paper are fully novel.

Abstract In the paper, we are concerned with the existence of periodic solutions for a coupled Rayleigh system. By combining graph theory with coincidence degree theory as well as Lyapunov function method, two new sufficient conditions on the existence of periodic solutions for the coupled Rayleigh system are established. Our results on the existence of periodic solutions for the coupled Rayleigh system improve those obtained in the existing literature for coupled Rayleigh system. Hence, our results are new and complementary to the existing papers. First part title: Introduction Second part title: Preliminaries Third part title: The existence of periodic solutions Fourth part title: Numerical test Five part title: Conclusion

In this paper, we consider the finite-time synchronization for drive-response BAM neural networks with time-varying delays. Instead of using the finite-time stability theorem and integral inequality method, by using the maximum-value method, two new criteria are obtained to ensure the finite-time synchronization for the considered drive-response systems. The inequalities in our paper, applied to obtaining the maximum-valued and designing the novel controllers, are different from those in existing papers.

In this paper, we are concerned with global asymptotic synchronization of Duffing-type oscillator system. Without using matrix measure theory, graph theory and LMI method, which are recently widely applied to investigating global exponential/asymptotic synchronization for dynamical systems and complex networks, four novel sufficient conditions on global asymptotic synchronization for above system are acquired on the basis of constant variation method, integral factor method and integral inequality skills.

Abstract In this paper, by combining graph theory with coincidence degree theory as well as Lyapunov functional method, sufficient conditions to guarantee the existence and global exponential stability of periodic solutions of the complex-valued neural networks of neutral type are established. In our results, the assumption on the boundedness for the activation function in (Gao and Du in Discrete Dyn. Nat. Soc. 2016:Article ID 1267954, 2016) is removed and the other inequality conditions in (Gao and Du in Discrete Dyn. Nat. Soc. 2016:Article ID 1267954, 2016) are replaced with new inequalities.

Using a fixed point theorem in cones, we obtain conditions that guarantee the existence and attractivity of the unique positive periodic solution for a discrete Lasota-Wazewska model.

The technique of approximate partition of unity, the way of Fourier series, and inequality technique are used to construct a neural network with two weights and with sigmoidal functions. Furthermore by using inequality technique, we prove that the neural network with two weights can more precisely approximate any nonlinear continuous function than BP neural network constructed in (Chen et al., 2012).

By using continuation theorem of coincidence degree theory, sufficient conditions of the existence of positive periodic solutions are obtained for a generalized predator-prey system with diffusion and delays. In this paper, we construct a V-function to make the prior estimation for periodic solutions, which makes the discussion more concise. Moreover, to compute the mapping's topological degree, a polynomial function matrix is constructed straightforwardly as a homotopic mapping for the generalized one, which improves the methods of computation on topological degree for a generalized mapping.

We investigate first the existence of periodic solution in general Cohen-Grossberg BAM neural networks with multiple time-varying delays by means of using degree theory. Then using the existence result of periodic solution and constructing a Lyapunov functional, we discuss global exponential stability of periodic solution for the above neural networks. Our result on global exponential stability of periodic solution is different from the existing results. In our result, the hypothesis for monotonicity ineqiality conditions in the works of Xia (2010) Chen and Cao (2007) on the behaved functions is removed and the assumption for boundedness in the works of Zhang et al. (2011) and Li et al. (2009) is also removed. We just require that the behaved functions satisfy sign conditions and activation functions are globally Lipschitz continuous.