Dynamics of a Family of Nonlinear Delay Difference Equations

Abstract

We study the global asymptotic stability of the following difference equation: $x_{n+\mathrm{1}}=f(x_{n-k_{\mathrm{1}}},x_{n-k_{\mathrm{2}}},\dots ,x_{n-k_s};x_{n-m_{\mathrm{1}}},x_{n-m_{\mathrm{2}}},\dots ,x_{n-m_t}),n=\mathrm{0,1},\dots ,$ where and with $\{k_{\mathrm{1}},k_{\mathrm{2}},\dots ,k_s\}{\bigcap }_{\mathrm{‍}}\{m_{\mathrm{1}},m_{\mathrm{2}},\dots ,m_t\}=\mathrm{\varnothing },$ the initial values are positive, and $f\in C(E^{s+t},(\mathrm{0},+\mathrm{\infty }))$ with $E\in \{(\mathrm{0},+\mathrm{\infty }),[\mathrm{0},+\mathrm{\infty })\}$. We give sufficient conditions under which the unique positive equilibrium $\stackrel{-}{x}$ of that equation is globally asymptotically stable.