Global Asymptotic Stability of Two Families of Nonlinear Difference Equations
Author(s): XI Hong-jian, SUN Tai-xiang, ZHAO Jin-feng
Pages: 93-95
Year: 2006 Issue: 2
Journal: GUANGXI SCIENCES
Keyword: difference equation; equilibrium; global asymptotic stability;
Abstract: Two families of difference equations are discussed. They are the form xn+1=∑i∈Zk-{j,s,t}xn-i+xrn-t+xn-jxmn-s+A/∑i∈Zk-{j,s,t}xn-i+xmn-s+xn-jxrn-t+A,n=0,1,...,where k∈{2,3,...},j,s,t∈Zk≡{0,1,...,k} with s≠t and j( ){s,t},A,r,m∈[0,+∞) and the initial values x-k,x-k+1,...,x0∈(0,+∞),and the form xn+1=∑i∈Zk-{j0,j1,...,js}xn-i+xn-j0xn-j1...xn-js+1/∑i∈Zk-{j0,j1,...,js-1}xn-i+xn-j0xn-j1...xn-js-1,n=0,1,...,where k∈{1,2,3,...},1≤s≤k,{j0...,js}( )Zk with ji≠jl for i≠l and the initial values x-k,x-k+1,...,x0∈(0,+∞).For these difference equations,it is proved that the unique equilibrium =1 is globally asymptotically stable,which includes the corresponding results of the references [3～5,7].
Pages: 93-95
Year: 2006 Issue: 2
Journal: GUANGXI SCIENCES
Keyword: difference equation; equilibrium; global asymptotic stability;
Abstract: Two families of difference equations are discussed. They are the form xn+1=∑i∈Zk-{j,s,t}xn-i+xrn-t+xn-jxmn-s+A/∑i∈Zk-{j,s,t}xn-i+xmn-s+xn-jxrn-t+A,n=0,1,...,where k∈{2,3,...},j,s,t∈Zk≡{0,1,...,k} with s≠t and j( ){s,t},A,r,m∈[0,+∞) and the initial values x-k,x-k+1,...,x0∈(0,+∞),and the form xn+1=∑i∈Zk-{j0,j1,...,js}xn-i+xn-j0xn-j1...xn-js+1/∑i∈Zk-{j0,j1,...,js-1}xn-i+xn-j0xn-j1...xn-js-1,n=0,1,...,where k∈{1,2,3,...},1≤s≤k,{j0...,js}( )Zk with ji≠jl for i≠l and the initial values x-k,x-k+1,...,x0∈(0,+∞).For these difference equations,it is proved that the unique equilibrium =1 is globally asymptotically stable,which includes the corresponding results of the references [3～5,7].
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