Proof Because R 02 > 1 and R 01 < 1, there exists another boundary equilibrium P 2 ∗ for system (1).
If (tau=0) and (A_{0}neq0), then there exists a unique boundary order-1 limit cycle ((x^{T}(t), 0)) for model (2.3).
Furthermore, if R 01 > 1 and R 02 < 1, there exists a nontrivial boundary equilibrium P 1 ∗ = ( S 1 1 ∗, I 1 1 ∗, S 2 1 ∗, 0 ), where.
If R 02 > 1 and R 01 < 1, there exists another nontrivial boundary equilibrium P 2 ∗ = ( S 1 2 ∗, 0, S 2 2 ∗, I 2 2 ∗ ), where.
There exists δ > 0 (which depends on L ˜ ) such that if complex numbers { λ n, α n } n ≥ 1 satisfy the condition Ω < δ, then there exists a unique boundary value problem L ( q ( x ), T ) with q ( x ) ∈ L 2 ( 0, T ), for which the numbers { λ n, α n } n ≥ 1 are the spectral data, and ∥ q − q ˜ ∥ L 2 ( 0, T ) < C Ω, (93).
By the Julia-Carathédory theorem [13] (see also [7]) and the Wolff lemma [11], if (fin H mathbb{D,mathbb{D}})) with no interior fixed point, then there exists a unique regular boundary fixed point ξ such that (f'(xi in 0,1]); and if (fin H mathbb {D,mathbb{D}})) with an interior fixed point, then (f'(xi >1) for any boundary fixed point (xiinpartialmathbb{D}).
Therefore, model (6.11) has a periodic solution, denoted by (x^{T}(t)) and (x^{T}(t)= 1-theta V_{L}exp(bt)) with period T, which means that for model (2.2) there exists a boundary order-1 limit cycle ((x^{T}(t),0)).
Being different from other ordinary parameter identification problems in parabolic equations, in our mathematical model there exists degeneracy on the lateral boundaries of the domain, which may cause the corresponding boundary conditions to go missing.
If (Khat{G}<1), then there exists the unique solution for boundary value problem (1 - 2).
By Theorem 3.3, there exists (lambda ^>0) such that boundary value problem (4.1) has at least one positive solution provided (lambda in (0,lambda ^)).
