The Basins Of Attraction Of The Equilibrium Points For Two Classes Of Difference Equations

Difference equations (or recursive sequences) are considered as the discretization and numerical solutions of differertial equations and delay differential equations,which have great number of applicationsin economics, ecology, biology, physics, engineering, neural network, social sciences, etc.The investigation on difference equation is to discuss its eventually behavior of the solutions, including oscillation, boundedness and periodicity , global asymptotic stabilities and basins of attraction of the equilibrium points, etc. This paper mainly studies the basins of attraction of the equilibrium points for two classes of difference equations.In Chapter One ,we introduce briefly the historic background and the current situation of nonlinear difference equation and some known results about our theorems.In Chapter Two ,we study the positive solutions of the nonlinear difference equationwhere 0 < p < 1, and the initial values (x_-1,x₀)∈(0,+∞)×(0,+∞). We find out the set of all initial values such that the positive solutions of the equation are bounded, and show that every positive bounded solution of the equation converges to its positive equilibrium point. This answers one open problem proposed by M.R.S.Kulenovic and G.Ladas .In Chapter Three, we study the positive solutions of the nonlinear difference equationwhere the intial values (x_-1,x₀)∈(0,+∞)×(0,+∞), and q > 1 +p > 1. We find out the set of all initial values such that the positive solutions of the equation are bounded, and show that every positive bounded solution of the equation converges to its positive equilibrium point. This answers another open problem proposed by M.R.S.Kulenovic and G.Ladas .