| Abstract: | 設{ } n c 是正實數序列，G : [0,∞) → [0,∞) , G′ : (0,∞) → (0, ∞) 都是連續使得G(0) = 0. 我們考慮差分方程式 ⎩ ⎨ ⎧ ≥ = − + + 0 for some (1.2) (1.1) 1 1 a n a a c G(|a |) n n n n n 和 ⎩ ⎨ ⎧ ≥ = + + + 0 for all (1.4) (1.3) 1 1 a n a a c G(|a |) n n n n n 很明顯地， ≡ 0 n a 是一明顯解，我們想要考慮一族的函數G (例如 ,G(x) = x p p > 0 G(x) = ex −1) 問題： (1) 找{ } n c 的充分必要條件使得≡ 0 n a 是(1.1)和(1.2)的唯一解( (1.3)跟(1.4)) (2) 當0 < l ≤ ∞ ,在{ } n c 上找一條件使得存在一唯一的{ } n a 滿足(1.1)、(1.2)及最終條件 a l n lim = ，並且在這情況下，我們能估計| a - l | n 或是a ( l = ∞) n 當的漸近狀態 (3) 類似 (2), 當 0 ≤ l < ∞ 對應於(1.3)和(1.4) Let { } be a positive sequence and : [0, ) [0, ) , : (0, ) (0, ) 1 1 1 1 = > = − ≡ ⎩ ⎨ ⎧ ≥ = + ⎩ ⎨ ⎧ ≥ = − = ∞ → ∞ ′ ∞ → ∞ + + + + p x n n n n n n n n n n n n G x x p G x e G a a n a a c G(|a |) a n a a c G(|a |) G c G G (3) Similar to (2), for 0 ,under (1.3)and(1.4) we estimate | - | or the asymptotic behavior of ( hen ) of (1.1)and(1.2)with the last condition lim .Moreover , in this case , (2) For 0 , find a condition on { } such that there is a unique solution { } is the unique solution of (1.1) and (1.2)(respectively(1.3),(1.4)) (1) Find a sufficent and necessary condition on { } such that 0 Question : |
