Some New Results on Composition-Delay Equations with Asymptotically Periodic Solutions

Abstract

The purpose of this thesis is to convey several new results in the field of piecewise difference equations, paying particular attention to higher order equations with asymptotically periodic solutions. A study of solutions to the equation $$y_n = \min \{ y_{n-k_1}-y_{n-m_1} , y_{n-k_2}-y_{n-m_2} \}$$ is presented. Related results are then obtained for a class of difference equations satisfying certain symmetry and monotonicity conditions. In particular we consider equations of the form $$ y_n = \min \{ f(y_{n-k_1},y_{n-m_1} ),f( y_{n-k_2} , y_{n-m_2} ) \}, $$ where $f(u,v) = h(u,v)/v$ for $h$ symmetric in $u$ and $v,$ and $f$ satisfies monotonicity conditions. The results are then extended to the form $$ y_n = \min \{ f(y_{n-k_1},y_{n-m_1} ),f( y_{n-k_2} , y_{n-m_2}),\dots ,f( y_{n-k_L} , y_{n-m_L} ) \} .$$