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Thus, we have proved that the unique positive almost periodic solution ((overline{x}_{1}(t),overline{x}_{2}(t),ldots,overline{x}_{n+m}(t))^{T}) of system (1.4) is globally exponentially stable.

In [6], by using inverse-monotone operator theorems, the authors proved that the unique C 1 -smooth solution converges almost everywhere to the solution of the corresponding reduced problem as ε → 0 +.

with uniformly elliptic operator A = ∑ i, j = 1 n a i j ( x ) ∂ 2 ∂ x i ∂ x j + ∑ i = 1 n b i ( x ) ∂ ∂ x i + c ( x ) and c(x)≤ 0. It was proved that the unique solution of (1.4) tends to 0 monotonically and exponentially as t →+∞ provided that ∫ Ω φ ( x, y ) d y ≤ ρ < 1, x ∈ ∂ Ω. Parabolic equations with both nonlocal sources and nonlocal boundary conditions have been studied as well (see [9 12]).

Next we prove that is the unique solution of the variational inequality (3.6).

In the following, we prove that is the unique fixed point of in.

We further prove that, the unique solution of VI (3.3).

In particular, we prove that the unique solution (u_{lambda}(t)) of the problem is strongly increasing and depends continuously on the parameter λ.

Finally, by applying the theory of α-concave operators, we prove that the unique solution (u_{lambda}(t)) of problem (1.2) is strongly increasing and depends continuously on the parameter λ.

By Lemma 2.1, it is sufficient to prove that the unique positive equilibrium (u^) is globally attractive in (mathbb{C}_{[0,M+1]}setminus{0}), where mathbb{C}_{[0,M+1]}= bigl{ varphiinmathbb{C}^|0leqvarphi (theta,x) leq M+1, forall(theta,x in[-tau,0]timesoverline{Omega } bigr}.

The main contribution in this paper is to prove that the unique positive equilibrium point P = ( β − β δ + ( − 1 + α ) η β ϵ + γ η, γ ( − 1 + δ ) + ( − 1 + α ) ϵ β ϵ + γ η ). of the system (1.1) is globally asymptotically stable.

In addition, suppose that is upper (lower) -preserving, for some function, we prove that is the unique upper (lower) absorbing point with respect to the mapping.