Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function - Wen-Xiu Ma (Mathematics, 2020)
Let <semantics> k , l </semantics> be two integers with <semantics> k ≥ 0 </semantics> and <semantics> l ≥ 2 </semantics> , c a real number greater than or equal to 1, and f a multivariable function satisfying <semantics> f ( w 1 , w 2 , w 3 , ⋯ , w l ) ≥ 0 </semantics> when <semantics> w 1 , w 2 ≥ 0 </semantics> . We consider an arbitrary order nonlinear difference equation with the indicated function f: <semantics> z n + 1 = c ( z n + z n − k ) + ( c − 1 ) z n z n − k + c f ( z n , z n − k , w 3 , ⋯ , w l ) z n z n − k + f ( z n , z n − k , w 3 , ⋯ , w l ) + c , <mspace width="4pt"></mspace> n ≥ 0 , </semantics> where initial values <semantics> z − k , z − k + 1 , ⋯ , z 0 </semantics> are positive and <semantics> w i , <mspace width="4pt"></mspace> i ≥ 3 </semantics> , are arbitrary functions of <semantics> z j , <mspace width="4pt"></mspace> n − k ≤ j ≤ n </semantics> . We classify its solutions into three types with different asymptotic behaviors, and verify the global asymptotic stability of its positive equilibrium solution <semantics> <mover accent="true"> z ¯ </mover> = c </semantics> .