Existence and Global Uniform Asymptotic Stability of Almost Periodic Solutions for Cellular Neural Networks with Discrete and Distributed Delays

Definition 1 (see [11].)If there is a constant M = M(σ, φ) for (σ, φ) ∈ Ω⊆R × C such that the solution x(t, σ, φ) of (5) through (σ, φ) when t ≥ σ − r satisfies |x(t, σ, φ)| < M, then the solution x(t, σ, φ) is bound.

Definition 2 (see [11].)Lyapunov functionals V(t, φ) : R × C → R, C = C([−r, 0], Rⁿ). Suppose that the solution of (5) through (σ, φ) is x(t, σ, φ), x_t(σ, φ) is defined as x(t + θ, σ, φ),  θ ∈ [−r, 0]. The total derivative is defined as follows:

Then $$ is the right derivative of functionals V(t, φ) along (5).

Lemma 3 (see [11].)Lagging-type almost periodic differential equation (5) has an asymptotically almost periodic solution $$, which satisfies $$ or H = +∞ for all t ≥ 0 defined in R₊; then (5) has an almost periodic solution.

Lemma 4 (see [11].)There is a continuous V functional of V(t, φ, ψ) for $$ such that

• (Ha)

  u(|φ − ψ|) ≤ V(t, φ, ψ) ≤ v(|φ − ψ|),

• (Hb)

  |V(t, φ₁, ψ₁) − V(t, φ₂, ψ₂)| ≤ k(|φ₁ − φ₂| + |ψ₁ − ψ₂|),

• (Hc)

  $$,

where a is a positive constant, u(s) and v(s) are continuous and non-decreasing, when s → 0, u(s) → 0, and k is a positive constant. At this time, if (5) has a bounded solution x(t, σ, φ) such that |x(t, σ, φ)| ≤ H₁, where t ≥ σ ≥ 0,  H > H₁ > 0, then (5) in C_H has a unique almost periodic solution which is global uniform asymptotic stability.

Theorem 5. Assume that (H2.1)–(H2.4) hold; then all solutions of system (3) are bounded.

Proof. Let |f_j(x_j)| ≤ P_j, |g_j(x_j)| ≤ Q_j, |h_j(x_j)| ≤ R_j and set $$$$. From system (3) and the assumption (H2.3), we get

for i = 1,2, …, n. By (11), we obtain

This shows that

This completes the proof of the theorem.

Theorem 6. Assume that (H2.1)–(H2.4) hold, and suppose further that

Then, in system (3), there exists an almost periodic solution, which is global uniform asymptotic stable.

Proof. From the condition (H2.1), we rewrite system (3) as follows:

The product system of the system (15) is the following form: In order to apply Lemma 4, we construct the Lyapunov functionals about the product system (16) as follows:

For convenience sake, we denote

where

Using (H2.1) and $$, by the triangle inequality, we have

Calculating the upright derivative of V₁(t) along system (16) as follows:

Similarly, we calculate the upright derivatives of V₂(t) and V₃(t) along system (16), respectively, as follows:

Note that

Combining with (21) and (22) and the assumptions of Theorem 5, we get

By Theorem 5 and Lemmas 3 and 4, there exists an almost periodic solution of system (3), which is global uniform asymptotic stable. This completes the proof of Theorem 6.

Example 1. Consider the following cellular neural network which consists of two neurons:

where

We select the functions f_j(x) = g_j(x) = h_j(x) = sh(x)/ch(x) and the kernel functions k_ij(s) = e^−s. Then, $$. Because the periods of sin πt and cos  t are 2 and 2π, respectively. The quotient of 2π and 2 is irrational. Then system (25) is an almost periodic system. In addition, a₀ = 1.5 and

From P_j = Q_j = R_j = 1, we have $$; then we get $$.

It is easy for us to verify that the conditions (H2.1)–(H2.4) in Theorem 5 hold. Therefore, in the system (25), there exists an almost periodic solution, which is global uniform asymptotic stable.