一类有理差分方程的全局渐近稳定性

Global Asymptotic Stability of a Class of Rational Difference Equation

研究差分方程xn+1=(fgh+f+g+h+a)/(fg+gh+hf+1+a)(n=0,1,…)的全局渐近稳定性,其中a∈(0,+∞),f=f(xn-r1,…,xn-rk)∈C((0,+∞)k,(0,+∞)),g=g(xn-m1,…,xn-ml)∈C((0,+∞)l,(0,+∞)),h=h(xn-s1,…,xn-sσ)∈C((0,+∞)σ,(0,+∞)),k,l,σ∈{1,2,…},0≤r1<…<rk,0≤m1<…<ml,0≤s1<…<sσ,并且初值为正实数.给出了该方程关于唯一正平衡点=x=1的全局稳定的充分条件,推广了参考文献[5]—[7]中的一些结果.

We study global asymptotic for positive solutions to the equation xn＋1=fgh＋f＋g＋h＋a/fg＋gh＋hf＋1＋a（n=0,1,…）,where a∈（1,＋∞）,f=f（x-r1,…,x-rk）∈C（（0,＋∞）^k,（0,＋∞））,g=g（xn-m1,…,xn-ml）∈C（（0,＋∞）^l,（0,＋∞））,h=h（xn-s1,…,xn-sσ）∈C（（0,＋∞）^σ,（0,＋∞））,with k,l,σ∈{1,2,…},0≤r1〈…〈rk,0≤m1〈…〈ml,0≤s1〈…〈sσ and the initial values are positive real numbers. Thesufficient conditions is given under which the unique equilibrium =↑x=1 = 1 of this equation is globally asymptotic stable, which inproves and the corresponding results obtained in the literature [5]-[7].