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Global stability for a delay difference equation. (English) Zbl 1181.39016

The authors consider a delay difference equation of the form
\[ x_{n+1}=\frac{\sum^s_{i=1}f_{2i-1}(X[n])f_{2i}(X[n])+ \sum^{2t}_{i=1}g_{i}(X[n])+h(X[n])+(s-t)} {\sum^{2s}_{i=1}f_{i}(X[n])+\sum^{t}_{i=1}g_{2i-1}(X[n])g_{2i}(X[n])+h(X[n])}, \]
where \(X[n]=(x_n,\dots,x_{n-k})\), \(f_i\), \(g_j\in C((0,\infty)^{k+1},(0,\infty))\) for \(i\in\{1,2,\dots,2s\}\) and \({j\in\{1,2,\dots,2t\}}\), \(h\in C((0,\infty)^{k+1},[0,\infty))\), \(k\in\{1,2,\dots\}\) and the initial values \(x_{-k}, x_{-k+l},\dots,x_0\in (0,\infty)\).
The following theorem on the global stability is proved. Let \(u^*=\max\{u, 1/u\}\) for any \(u\in \mathbb R_+\). If \([f_i(u_0,u_1,\dots,u_k)]^*\leq \max\{u_0^*,u_1^*,\dots,u_k^*\}\) and \([g_j(u_0,u_1,\dots,u_k)]^*\leq \max\{u_0^*,u_1^*,\dots,u_k^*\}\) for \(i=1,2,\dots,2s\) and \(j=1,2,\dots,2t\), then \(\bar{x}=1\) is a unique positive equilibrium of the equation which is globally asymptotically stable.

MSC:

39A30 Stability theory for difference equations
39A20 Multiplicative and other generalized difference equations
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References:

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