Non‐autonomous homogeneous rational difference equations of degree one: convergence and monotone solutions for second and third order case

Consider the non‐autonomous equations:x n + 1 = x nA nx n + B nx n − 1α nx n + β nx n − 1,α nx n + β nx n − 1 > 0 ,x n + 1 = x n − 1B nx n + D nx n − 2α nx n + γ nx n − 2,α nx n + γ nx n − 2 > 0 ,x n + 1 = x nB nx n − 1 + C nx n − 2β nx n − 1 + γ nx n − 2,β nx n − 1 + γ nx n − 2 > 0 , whereA n ≥ 0, B n ≥ 0, C n ≥ 0, α n ≥ 0, β n ≥ 0, γ n ≥ 0 ,n = 0 , 1 , 2 , … , and alsolim n → ∞A n = A ≥ 0, lim n → ∞B n = B ≥ 0, lim n → ∞C n = C ≥ 0, lim n → ∞α n = α > 0, lim n → ∞β n = β > 0, lim n → ∞γ n = γ > 0 .These are some non‐autonomous homogeneous rational difference equations of degree one. A reduction in order is consi‐dered. Convergence and monoton character of positive solutions are studied. Copyright © 2013 John Wiley & Sons, Ltd.