Xu, Daoyi; Zhao, Hongyong Invariant set and attractivity of nonlinear differential equations with delays. (English) Zbl 1023.34066 Appl. Math. Lett. 15, No. 3, 321-325 (2002). The authors discuss the invariant set, the attracting set, and the basin of attraction of nonlinear and nonautonomous delay differential equations \[ \dot y(t) =-Ay(t)+F(t,y_t)+q(t), \quad A =\text{diag} \{a_i \}. \] Such equation can be seen as a generalization of the neural network models. Some sufficient conditions are obtained for the existence of an attracting set and the basin of attraction. The approach is to employ the variation of constants formula. Two examples are given to explain the results. Reviewer: Rong Yuan (Beijing) Cited in 13 Documents MSC: 34K19 Invariant manifolds of functional-differential equations 34K20 Stability theory of functional-differential equations 92B20 Neural networks for/in biological studies, artificial life and related topics Keywords:attracting set; attraction basin; delay differential equation; accretive operator; almost-periodic function; semigroup PDFBibTeX XMLCite \textit{D. Xu} and \textit{H. Zhao}, Appl. Math. Lett. 15, No. 3, 321--325 (2002; Zbl 1023.34066) Full Text: DOI References: [1] Sawano, K., Positively invariant sets for functional differential equations with infinite delay, Tôhoku Math. J., 32, 2, 557-566 (1980) · Zbl 0471.34056 [2] Seifert, G., Positively invariant closed sets for systems of delay differential equations, J. Differential Equations, 22, 292-304 (1976) · Zbl 0332.34068 [3] Bates, P.; Lu, K.; Zeng, C., Existence and persistence of invariant manifold for semiflows in Banach space, Memoirs of Amer. Math. Soc., 645 (1998) [4] Siljak, D. D., Large-Scale Dynamic System: Stability and Structure (1978), Elsevier North-Holland: Elsevier North-Holland New York · Zbl 0382.93003 [5] Lakshmikantham, V.; Leela, S., Differential and Integral Inequalities, I (1969), Academic Press: Academic Press New York · Zbl 0177.12403 [6] Michel, A. N.; Jay, A. F.; Wolfgand, P., Qualitative analysis of neural networks, IEEE Transactions on Circuits and Systems, 36, 229-243 (1989) · Zbl 0672.94015 [7] Razgulin, A. V., The attractor of the delayed functional-differential diffusion equation. Numerical methods in mathematical physics, Comput. Math. Model, 8, 181-186 (1997) [8] Kolmanovskii, V. B.; Nosov, V. R., Stability of Functional Differential Equations (1986), Academic Press: Academic Press Orlando, FL · Zbl 0593.34070 [9] Xu, D. Y.; Xu, A., Domain of attraction of nonlinear functional difference systems, Chinese Science Bulletin, 43, 1828-1830 (1998) [10] Xu, D. Y.; Li, S. Y.; Pu, Z. L.; Guo, Q. Y., Domain of attraction of nonlinear discrete systems with delays, Computers Math. Applic., 38, 5/6, 155-162 (1999) · Zbl 0939.39013 [11] Xu, D. Y., Integro-differential equations and delay integral inequalities, Tôhoku Math. J., 44, 365-378 (1992) · Zbl 0760.34059 [12] Gopalsamy, K.; He, X., Stability in asymmetric Hopfield nets with transmission delays, Phys. D, 76, 344-358 (1994) · Zbl 0815.92001 [13] Hopfield, J. J., Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci. U.S.A., 81, 3088-3092 (1984) · Zbl 1371.92015 [14] Lasalle, J. P., The Stability of Dynamical System (1976), SIAM: SIAM Philadelphia · Zbl 0364.93002 [15] Horn, R. A.; Johnson, C. R., Matrix Analysis (1991), World Publishing: World Publishing Beijing · Zbl 0729.15001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.