On the qualitative and quantitative analysis for two fourth–order difference equations

Our aim in the present paper is to derive the closed-form solutions for the two fourth-order difference equations x n + 1 = x n - 2 x n - 3 a x n + b x n - 3 , n ≥ 0 , and x n + 1 = x n - 2 x n - 3 - a x n + b x n - 3 , n ≥ 0 , with positive arbitrary real parameters a , b and arbitrary real initial conditions, as well as study the qualitative behaviors for each. For the first equation, we show that every admissible solution converges to a period-3 solution when a + b = 1 . For the second equation, we show that every admissible solution converges to zero if b > 2 when b 2 ≥ 4 a . When b 2 < 4 a , we show the existence of periodic solutions under certain conditions. We introduce the forbidden sets as well as provide some illustrative examples for the above-mentioned equations.