Positive Periodic Solutions of Nicholson-Type Delay Systems with Nonlinear Density-Dependent Mortality Terms

and Applied Analysis 3 We also assume that aij , bij , cik, γik : R → 0, ∞ and τik : R → 0, ∞ are all ω-periodic functions, ri max1≤k≤l{τ ik}, and i, j 1, 2, k 1, . . . , l. Set Ai 2 ∫ω 0 aii t bii t dt, Bi l ∑ j 1 ∫ω 0 cij t dt, γ i max 1≤j≤l { γ ij } , γ− i min 1≤j≤l { γ− ij } , D1 ∫ω 0 a12 t dt, D2 ∫ω 0 a21 t dt, Ci ∫ω 0 aii t dt, i 1, 2. 1.6 Let R R be the set of all nonnegative real vectors; we will use x x1, x2, . . . , xn T ∈ R to denote a column vector, in which the symbol T denotes the transpose of a vector. We let |x| denote the absolute-value vector given by |x| |x1|, |x2|, . . . , |xn| T and define ||x|| max1≤i≤n|xi|. For matrix A aij n×n, A denotes the transpose of A. A matrix or vector A ≥ 0 means that all entries of A are greater than or equal to zero. A > 0 can be defined similarly. For matrices or vectors A and B, A ≥ B resp. A > B means that A − B ≥ 0 resp. A − B > 0 . We also define the derivative and integral of vector function x t x1 t , x2 t T as x′ x′ 1 t , x ′ 2 t T and ∫ω 0 x t dt ∫ω 0 x1 t dt, ∫ω 0 x2 t dt T . The organization of this paper is as follows. In the next section, some sufficient conditions for the existence of the positive periodic solutions of model 1.3 are given by using the method of coincidence degree. In Section 3, an example and numerical simulation are given to illustrate our results obtained in the previous section. 2. Existence of Positive Periodic Solutions In order to study the existence of positive periodic solutions, we first introduce the continuation theorem as follows. Lemma 2.1 continuation theorem 14 . Let X and Z be two Banach spaces. Suppose that L : D L ⊂ X → Z is a Fredholm operator with index zero and Ñ : X → Z is L -compact on Ω, where Ω is an open subset of X. Moreover, assume that all the following conditions are satisfied: 1 Lx/ λÑx, for all x ∈ ∂Ω ∩D L , λ ∈ 0, 1 ; 2 Ñx / ∈ ImL, for all x ∈ ∂Ω ∩ KerL; 3 the Brouwer degree deg { QÑ,Ω ∩ KerL, 0 } / 0. 2.1 Then equation Lx Ñx has at least one solution in domL ∩Ω. Our main result is given in the following theorem. 4 Abstract and Applied Analysis Theorem 2.2. Suppose Ci > 2Di, ln 2Bi Ai > Ai, i 1, 2, 2.2 l ∑ j 1 c 1j a11γ − 1je a 12 a11 < 1, l ∑ j 1 c 2j a22γ − 2je a 21 a22 < 1. 2.3 Then 1.3 has a positive ω-periodic solution. Proof. Set N t N1 t ,N2 t T andNi t ei t i 1, 2 . Then 1.3 can be rewritten as x′ 1 t − a11 t b11 t ex1 t a12 t e2 t −x1 t b12 t ex2 t l ∑ j 1 c1j t e1 t−τ1j t −x1 t −γ1j t e x1 t−τ1j t : Δ1 x, t , x′ 2 t − a22 t b22 t ex2 t a21 t e1 t −x2 t b21 t ex1 t l ∑ j 1 c2j t e2 t−τ2j t −x2 t −γ2j t e x2 t−τ2j t : Δ2 x, t . 2.4 As usual, let X Z {x x1 t , x2 t T ∈ C R,R2 : x t ω x t for all t ∈ R} be Banach spaces equipped with the supremum norm || · ||. For any x ∈ X, because of periodicity, it is easy to see that Δ x, · Δ1 x, · ,Δ2 x, · T ∈ C R,R2 is ω-periodic. Let L : D L { x ∈ X : x ∈ C1 ( R,R2 )} x −→ x′ x′ 1, x′ 2 )T ∈ Z, P : X x −→ ( 1 ω ∫ω 0 x1 s ds, 1 ω ∫ω 0 x2 s ds )T ∈ X, Q : Z z −→ ( 1 ω ∫ω 0 z1 s ds, 1 ω ∫ω 0 z2 s ds )T ∈ Z, Ñ : X x −→ Δ x, · ∈ Z. 2.5 It is easy to see that ImL { x | x ∈ Z, ∫ω 0 x s ds 0, 0 T } , KerL R2, ImP KerL, KerQ ImL. 2.6 Abstract and Applied Analysis 5 Thus, the operator L is a Fredholm operator with index zero. Furthermore, denoting by L−1 P : ImL → D L ∩ KerP the inverse of L|D L ∩KerP , we have L−1 P y t − 1 ω ∫ω 0 ∫ t 0 y s dsdt ∫ t 0 y s ds ( − 1 ω ∫ω 0 ∫ t 0 y1 s dsdt ∫ t 0 y1 s ds,− 1 ω ∫ω 0 ∫ t 0 y2 s dsdt ∫ t 0 y2 s ds )T . 2.7and Applied Analysis 5 Thus, the operator L is a Fredholm operator with index zero. Furthermore, denoting by L−1 P : ImL → D L ∩ KerP the inverse of L|D L ∩KerP , we have L−1 P y t − 1 ω ∫ω 0 ∫ t 0 y s dsdt ∫ t 0 y s ds ( − 1 ω ∫ω 0 ∫ t 0 y1 s dsdt ∫ t 0 y1 s ds,− 1 ω ∫ω 0 ∫ t 0 y2 s dsdt ∫ t 0 y2 s ds )T . 2.7