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Existence and global attractivity of periodic solutions in a higher order difference equation

Chuanxi Qian;  Justin Smith — 2018

Archivum Mathematicum

Consider the following higher order difference equation x ( n + 1 ) = f ( n , x ( n ) ) + g ( n , x ( n - k ) ) , n = 0 , 1 , ⋯ where f ( n , x ) and g ( n , x ) : { 0 , 1 , ⋯ } × [ 0 , ∞ ) → [ 0 , ∞ ) are continuous functions in x and periodic functions in n with period p , and k is a nonnegative integer. We show the existence of a periodic solution { x ˜ ( n ) } under certain conditions, and then establish a sufficient condition for { x ˜ ( n ) } to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also given.

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