Let Ω be a bounded domain in R n, let a i j ( x, t ), b ( x, t ), c ( x, t ) be continuous functions in Q ¯, Q = Ω × ( 0, T ]. Suppose that b ( x, t ) < 0, c ( x, t ) is bounded and there exist positive constants λ 0 and Λ 0 such that λ 0 | ξ | 2 ≤ a i j ( x, t ) ξ i ξ j ≤ Λ 0 | ξ | 2, ∀ ξ ∈ R n.
As assumed in many references, such as [45], the activation function (f_{i}(cdot)) of neural network (1) is continuous, bounded, and there exist constants (lambda^_{i}) and (lambda^_{i}) such that lambda^_{i}leqfrac{f_{i}(a -f_{i}(b)}{a -f_{i}lambda^_{i}, qquad f_{i}(0)=0,quad a, bin mathbb{R}, aneq b, i=1,2,lambda^_{i}
The map Q : D → D is said to be β-convex-power condensing if Q is continuous, bounded and there exist x 0 ∈ D, n 0 ∈ N such that for every nonprecompact bounded subset B ⊂ D, we have β ( Q ( n 0, x 0 ) ( B ) ) < β ( B ). Obviously, if n 0 = 1, then a β-convex-power condensing operator is β-condensing.
(H0)* For any R > 0, f(I × K R × K R ) is bounded, and there exist two constants L1, L2> 0 with 2 L 1 + L 2 D ¯ < M such that α ( f ( I × B 1 × B 2 ) ) ≤ L 1 α ( B 1 ) + L 2 α ( B 2 ), for any B1, B2 ⊂ K R, where K R is defined as in (P0) and the condition (H1) or (H2), then the boundary value problem (1) and (3) has at least one positive solution.
Lemma 2 Assume that f ∈ C(I × K × K, K) satisfies the following condition (H0) For any R > 0, f (I × K R × K R ) is bounded, and there exist two constants L1, L2> 0 with L 1 + L 2 D ¯ < M 4 such that α ( f ( t, B 1, B 2 ) ) ≤ L 1 α ( B 1 ) + L 2 α ( B 2 ), for any t ∈ I and B1, B2 ⊂ K R, where K R is defined as in condition (P0).
Since is bounded, and, there exists a constant and a natural number such that for all, (2.27).
Assume that 1 2 g ( u ) u − G ( u ) is not bounded and there exists an ϵ > 0 such that ∫ Ω − a − ( x, t ) d x < ϵ.
Step 3. Claim that the sequence { x n } is bounded and there exists the limit lim n → ∞ ∥ x n − x 0 ∥ = c.
(HF) is measurable, is lsc for a.e., there exist such that with 2Vol and there exists such that for any bounded set, (Hk) is continuous, bounded and there exists such that.
end{aligned} Hence, (v(t)) is uniformly ultimately bounded and there exists a constant (M>0) such that (y(t)le M) and (z(t)le M) for sufficiently large t.
(H3) (k(a)) is bounded and there exists a maximum age (a^) for the virion production such that (k(a)>0) for (0< a< a^), (k(a)=0) for (ageq a^).
