Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function

Wen-Xiu Ma
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Wen-Xiu Ma: Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Mathematics, 2020, vol. 8, issue 5, 1-7

Abstract: Let k , l be two integers with k ≥ 0 and l ≥ 2 , c a real number greater than or equal to 1, and f a multivariable function satisfying f ( w 1 , w 2 , w 3 , ? , w l ) ≥ 0 when w 1 , w 2 ≥ 0 . We consider an arbitrary order nonlinear difference equation with the indicated function f : 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 , n ≥ 0 , where initial values z − k , z − k + 1 , ? , z 0 are positive and w i , i ≥ 3 , are arbitrary functions of z j , n − k ≤ j ≤ n . We classify its solutions into three types with different asymptotic behaviors, and verify the global asymptotic stability of its positive equilibrium solution z ¯ = c .

Keywords: difference equation; positive equilibrium; oscillatory solution; strong negative feedback; global asymptotic stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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