Unique Positive Almost Periodic Solution for Discrete Nonlinear Delay Survival Red Blood Cells Model

and Applied Analysis 3 the hull of f . It is easy to see that if f n is almost periodic, then for all f1 ∈ H f , lim inf n→∞ f n inf n∈Z f n inf n∈Z f1 n , lim sup n→∞ f n sup n∈Z f n sup n∈Z f1 n . 1.7 Definition 1.2. Let f : Z × C → R. Then f is said to be almost periodic in n ∈ Z uniformly on compact set of C, if f n, · is continuous for each n ∈ Z, and for any > 0 and every compact setK ⊂ C, there is a constant l ,K > 0 such that in any interval of length l ,K there exists τ such that the inequality ∣∣f(n τ, φ) − f(n, φ)∣∣ < 1.8 is satisfied for all n ∈ Z and φ ∈ K. Definition 1.3. The positive solution N n to 1.1 , in the sense that N n > 0 for all n ≥ 0, is said to be global attractive, if for any φ ∈ C , lim n→∞ ( x ( n, 0, φ ) −N n ) 0. 1.9 The remainder of this paper is organized as follows. The main results are given in Section 2 while the proofs are left to Section 4. In Section 3, we will give some lemmas needed in the proofs of main results. 2. Main Results Theorem 2.1. Assume that 0 < δ∗ < 1 and P∗ > 0. Then, for any φ ∈ C , the solution x n, 0, φ of 1.1 satisfies u ≤ lim inf n→∞ x ( n, 0, φ ) ≤ lim sup n→∞ x ( n, 0, φ ) ≤ v, 2.1 where u, v is the limit of { un, vn } with u0 0 and vk P ∗ δ∗ e−q∗uk−1 , k 1, 2, . . . , uk P∗ δ∗ e−q ∗vk , k 1, 2, . . . . 2.2 The following corollary follows from 24, Theorems 1 and 2 . Corollary 2.2. Assume that the conditions of Theorem 2.1 are satisfied, and δ n , P n , q n and τ n are ω-periodic with ω > 0. Then 1.1 admits a positive ω-periodic solution. 4 Abstract and Applied Analysis Remark 2.3. When τ n ≡ ω, under the conditions of Corollary 2.2, Saker in 1, Theorem 2.1 proved that the conclusion of Corollary 2.2. Thus Corollary 2.2 extends Theorem 2.1 in 1 . Theorem 2.4. Assume that the conditions of Theorem 2.1 are satisfied and