Global stability of a difference equation with maximum

We prove that every positive solution to the difference equationx"n=maxA"1x"n"-"1^@a^"^1,A"2x"n"-"2^@a^"^2,...,A"kx"n"-"k^@a^"^k,[email protected]?N"0,where [email protected]?N,A"i>0,@a"[email protected]?(0,1), i=1,...,k, converges to the following quantity maxA"1^1^@a^"^1^+^1,...,A"k^1^@a^"^k^+^1, confirming a quite recent conjecture of interest. We also prove another result on global convergence which concerns some cases when not all @a"i,i=1,...,k belong to the interval (0,1).

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