Global asymptotic stability in a perturbed higher-order linear difference equation.
The author establishes sufficient conditions for the global asymptotic stability of the zero solution of the following perturbed nonautonomous linear difference equation \[ x(n+1)= \sum_{i=0}^kp_i(n)x(n-i)+f(n,x(n),x(n-1),\dots,x(n-l))\tag{*} \] where \(k,l\in \mathbb{N}\), the coefficients \(p_i(n)\,(n\in \mathbb{N},0\leq i\leq k)\,\) are real numbers and \(f:\mathbb{N}\times \mathbb{R}^{l+1}\rightarrow \mathbb{R}\) satisfies the condition \[ \left| f(n,x_0,x_1,\dots,x_l)\right| \leq q_ 0{\leq i\leq l}\;{\max }\left| x_i\right| \] for \(n\in \mathbb{N}\) and \(x_i\in \mathbb{R}\), \(0\leq i\leq l,\) and \(q\) is a nonnegative constant . For every fixed \(n_0\in \mathbb{N},\) let \(\{u(n,n_0)\}_{n_0-k}^\infty\), be the solution of (*) with initial conditions \(u(n,n_0)=0,\) for \(n_0-k\leq n\leq n_0-1\,\) and \(u(n_0,n_0)=1.\) The main result of this paper is the following: Theorem. Suppose that for \(n=0,1,\dots\) and \(\,0\leq i\leq k\) the conditions \(\left| p_i(n)\right| \leq K,\) and \[ \sum_{j=0}^{n-1}\left| u(n,j+1)\right| \leq L,\tag{**} \] hold, where \(K,L>0\) are constants. If \(q<L^{-1},\) then the zero solution of (*) is globally asymptotically stable. Several cases, where the condition (**) is satisfied, are considered .
scientific article; zbMATH DE number 2082226
scientific article