Abstract
AbstractNon-autonomous first order difference equation of the form \(x_{n+1}=x_n+a_nf(x_n), n\in \mathbf{N}_0,\) is considered where \(f:\,\mathbf{R}\rightarrow \mathbf{R}\) is a continuous function satisfying the negative feedback assumption \(x f(x)<0, x\ne 0,\) and \(a_n\ge 0\) is a non-negative sequence. Sufficient conditions for the global asymptotic stability of the zero solution are derived in terms of the attractivity of the fixed point \(x_*=0\) under the iterations of distinct maps of the family of one-dimensional maps \(F_{\lambda }(x)=x+\lambda f(x), \lambda \ge 0.\) The principal motivation for consideration of the difference equation and the corresponding family of interval maps comes from a problem of asymptotic behavior in differential equations with piece-wise constant argument (DEPCA).KeywordsParametric family of interval mapsSuccessive iterations under distinct mapsGlobal attractivity of fixed pointsDEPCAReduction to difference equations
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