Abstract
In this work, we derive the solution formulas and study their behaviors for the difference equations x n + 1 = α x n x n − 3 / − β x n − 3 + γ x n − 2 , n ∈ ℕ 0 and x n + 1 = α x n x n − 3 / β x n − 3 − γ x n − 2 , n ∈ ℕ 0 with real initials and positive parameters. We show that there exist periodic solutions for the second equation under certain conditions when β 2 < 4 α γ . Finally, we give some illustrative examples.
Highlights
In [1–5], the first author ([1] together with Kamal) solved and studied the solutions for the difference equations xn+1 xnxn− 1 xn − xn−, xn+1 − xnxn− 1 xn + xn− b + axn− 3 cxn− 1xn− (1) xn+1axnxn− 1, ± bxn− 1 + cxn− 2 xn+1
Suppose xn∞ n − 3 is an admissible solution of equation (6). e solution of equation (6) is
Assume that xn∞ n − 3 is an admissible solution of equation (6). en, the following hold: (1) If a + b > 1, the solution xn∞ n − 3 converges to zero
Summary
Motivated by [27], we shall solve, find the forbidden set, and study (∀ positive real values of α, β, c ) global behavior of the admissible solutions for equations (3) and (4) where Suppose xn∞ n − 3 is an admissible solution of equation (6). (3) If a > b + 1, the solution xn∞ n − 3 converges to zero.
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