The global attractivity of the rational difference equation 𝑦_{𝑛}=𝐴+(\frac{𝑦_{𝑛-𝑘}}𝑦_{𝑛-𝑚})^{𝑝}

This paper studies the behavior of positive solutions of the recursive equation y n = A + ( y n − k y n − m ) p , n = 0 , 1 , 2 , … , \begin{eqnarray} y_n=A+\left (\frac {y_{n-k}}{y_{n-m}}\right )^p,\quad n=0,1,2,\ldots , \nonumber \end{eqnarray} with y − s , y − s + 1 , … , y − 1 ∈ ( 0 , ∞ ) y_{-s},y_{-s+1}, \ldots , y_{-1} \in (0, \infty ) and k , m ∈ { 1 , 2 , 3 , 4 , … } k,m \in \{1,2,3,4,\ldots \} , where s = max { k , m } s=\max \{k,m\} . We prove that if g c d ( k , m ) = 1 \mathrm {gcd}(k,m) = 1 , and p ≤ min { 1 , ( A + 1 ) / 2 } p\leq \min \{1,(A+1)/2\} , then y n y_n tends to A + 1 A+1 . This complements several results in the recent literature, including the main result in K. S.  Berenhaut, J. D. Foley and S. Stević, The global attractivity of the rational difference equation y n = 1 + y n − k y n − m y_{n}=1+\frac {y_{n-k}}{y_{n-m}} , Proc. Amer. Math. Soc. , 135 (2007) 1133–1140.