On the Nonautonomous Equation x n + 1 = max { A n x n ,   B n x n − 1 }

We study the boundedness and the periodic character of the solutions of the nonautonomous equation x n + 1 = max { A n x n , B n x n − 1 } ,               n = 0 , 1 , … https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745854/a0661dc5-cfc8-48ab-844c-916895f230b9/content/eq248.tif"/> where { A n } n = 0 ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745854/a0661dc5-cfc8-48ab-844c-916895f230b9/content/eq249.tif"/> and { B n } n = 0 ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745854/a0661dc5-cfc8-48ab-844c-916895f230b9/content/eq250.tif"/> are sequences of negative real numbers, and where the initial conditions x _–1 and x ₀ are nonzero real numbers. We also investigate the equation x n + 1 = max { 1 x n , A n x n − 1 }             ,               n = − 1 , 0 , … https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745854/a0661dc5-cfc8-48ab-844c-916895f230b9/content/eq251.tif"/> where { A n } n = 0 ∞   https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203745854/a0661dc5-cfc8-48ab-844c-916895f230b9/content/eq252.tif"/> is a sequence of positive real numbers of period 2, and where the initial conditions x _–1 and x ₀ are positive real numbers.