Global Asymptotic Stability Of Two Types Of Differential Equations

In this paper we investigate the oscillation, the global attractivity, and the global asymptotic stability of all positive solutions of the two kinds of nonlinear difference equations and x_n+1=e^-bx_n2+c, and give sufficient conditions for the global asymptotic stability of the two kinds of equations. This paper consists of four parts.In chapter one, we introduce the historical background, the progress of the difference equations.In chapter two, we introduce the related concepts and briefly narrate predecessor's results.In chapter three, we detaily discuss the oscillation, the periodicity and the global asymptotic stability of all positive solutions of the non-linear difference equation where, a,b∈[0,∞),A∈R⁺,x_-k,…,x_-1,x₀∈(0,∞), p is arbitrary real number, k is a positive integer. And we prove the unique positive equilibrium of the equation is an attractor of it and give a sufficient condition of the oscillation of the equation.In chapter four, we discuss the equation x_n+1=e^-bx_n2+c,where b∈(0,∞),c∈R and prove the global asymptotic stability of all positive solutions of the equation.