On the rational difference equation (y_{{n+1}}={rac {lpha_{{0}}y_{{n}}+lpha_{{1}}y_{{n-p}}+lpha_{{2}}y_{{n-q}} +lpha_{{3}}y_{{n-r}}+lpha_{{4}}y_{{n-s}}}{eta_{{0}}y_{{n}}+eta_{{1}}y_{{n-p} }+eta_{{2}}y_{{n-q}}+eta_{{3}}y_{{n-r}}+eta_{{4}}y_{{n-s}}}})

摘要

In this paper, we examine and explore the boundedness, periodicity, and global stability of the positive solutions of the rational difference equation [ y_{{n+1} }={rac {lpha_{{0}}y_{{n}}+lpha_{{1}}y_{{n-p}}+lpha_{{2}}y_{{n-q}} +lpha_{{3}}y_{{ n-r}}+lpha_{{4}}y_{{n-s}}}{eta_{{0}}y_{{n}}+eta_{{1}}y_{{n-p} }+eta_{{2}}y_{{n-q} }+eta_{{3}}y_{{n-r}}+eta_{{4}}y_{{n-s}}}}, ] where the coefficients ({lpha_{i},eta_{i}in (0,infty ), i=0,1,2,3,4},) and (p,q,r), and (s) are positive integers. The initial conditions (y_{-s},...,y_{-r},..., y_{-q},..., y_{{-p }},..., y_{-1},y_{0}) are arbitrary positive real numbers such that (pqrs). Some numerical examples will be given to illustrate our result.