Long Time Behavior Of The Solutions Of Nonlinear Functional Differential Equations And Partial Functional Differential Equations

The main aim of this paper is to deal with the long time behavior of the solutions of nonlinear functional differential equations and partial functional differential equations.In chapter 1, the global attractor for strongly damped nonlinear wave equations with delay are studied. By using sectoral operator and the semigroup theory, combined with the spectral theory on matrices, we introduced a new norm in the phase space and conclude that this system is a dissipative system. Furthermore, we got the existence of the global attractor.In chapter 2, the robust stability of interval systems with time delay is studied. Making use of the relation between the time-delay systems and ordinary differential equations with complex coefficients, the authors obtain some sufficient conditions for the robust stability of interval systems with time-delay. Not only are these results easy to be verified, but also have less conservation than that of the similar result that is given in the relevant references.In chapter 3, we devote to the investigation of the global asymptotic stability for Hopfield neural networks with S-type distributed delayswhose special cases are the models with discrete delays and continuously distributed delays. Some sufficient conditions for the existence of a unique equilibrium and its global asymptotic stability are derived. These conditions need not the boundedness and differentiability of fj on R which are given in the relevant references. In the way of proof, the theory of topological degree is used to prove the existence of equilibrium, and the inequality technique is used to prove the global asymptotic stability.In chapter 4, Exponential stability of the Cohen-Grossberg neural networks with multiple time-varying delays are analyzed. By the theory of topological degree, sufficient conditions for the existence of the equilibrium are given, while by using inequality technique and M-matrix theory, sufficient conditions for the local and global exponential stability of the equilibrium are obtained. All results are established without assuming any symmetry of the connection matrices, and the boundedness, differentiability and monotonicity of the activation functions. The provide conditions are independent of delay.