Periodic Solution And Stability For Several Kinds Of Delayed Differential And Differece Equations

Based on fixed point theorem, coincidence degree theorem, the abstract con-tinuation theorem of k-set contractive operator and Lyapunov functional, periodic solution of several kinds of nonlinear delayed differential and difference equations and global exponential stability of bidirectional associated memory neural networks are studied. This dissertation is divided into six chapters.In Chapter one, a general introduction to the dissertation and some relevant background knowledge are given.In Chapter two, by Deimling fixed point index theorem and some analysis skill, periodic solutions of a kind of first order delayed differential equation is investigated and some sufficient conditions on the existence of periodic solution are obtained. Our results improve and expand the existing results.In Chapter three, based on Mawhin's continuation theorem?some analysis skill and inequality techniques, the existence of periodic solutions to two types of second order functional differential equation with complex deviating argument is studied. Some new results on existence of periodic solutions of second order neutral functional differential equation with deviating argument are obtained and some sufficient conditions on the existence and uniqueness of periodic solutions for a kind of second order Duffing-type differential equation with multiple deviating arguments are established.Our results extend the existing results.In Chapter four, by Manasevich-Mawhin continuation theorem and some anal-ysis skill, the existence of periodic solutions for a kind of Rayleigh type p-Laplacian equation with deviating argument and a kind of Rayleigh type multi-p-Laplacian equation with deviating argument is considered. Some new sufficient conditions on the existence of periodic solutions are presented. Then, combined Mawhin's continuation theorem and inequality techniques with a new priori estimate of a periodic solution, some new results on the existence of periodic solutions of non-autonomous Rayleigh type equation with varying delay are obtained.In Chapter five, the positive periodic solution for ratio-dependent n-species discrete time system is investigated. By using of inequality techniques, a sharp priori estimate of the periodic solutions is given. Based on the more precise priori estimate and the continuation theorem, an easily verifiable sufficient criterion of the existence of positive periodic solutions is established. Results obtained greatly improve the existing results. Then, based on an abstract continuous theorem of k-set contractive operator and some analysis skill, a neutral multi-delay logarithmic population model is studied, a new result is obtained for the existence of positive periodic solutions.In Chapter six, the existence, uniqueness, and global exponential stability of the equilibrium point of bidirectional associative memory (BAM) neural networks with time delays is studied. By applying Young's inequality and Holder's inequal-ity techniques together with the properties of monotonic continuous functions, without assuming that the activation functions and signal propagation functions are bounded, some new sufficient conditions for ascertaining global exponential stability, have been derived for BAM neural networks, improving and extending previous work. Then, the global exponential stability of discrete-time BAM neural networks with variable delays is dealt. A delay-dependent exponential stability criterion has been derived by means of a Lyapunov functional and inequality tech-niques. Unrealistic constraints on the delay functions have been removed. As this criterion has no extra constraints on the variable delay functions, it can be applied to quite general BAM neural networks with a broad range of time delay functions and is easily checked in practice.