Asymptotic Stability of Impulsive Cellular Neural Networks with Infinite Delays via Fixed Point Theory

Definition 1. The trivial equilibrium x = 0 is said to be stable, if, for any ε > 0, there exists δ > 0 such that for any initial condition φ(s) ∈ C[[ϑ, 0], Rⁿ] satisfying  |φ| < δ:

Definition 2. The trivial equilibrium x = 0 is said to be asymptotically stable if the trivial equilibrium x = 0 is stable, and for any initial condition φ(s) ∈ C[[ϑ, 0], Rⁿ], lim _t→∞∥x(t; s, φ)∥ = 0 holds.

Theorem 3 (see [30].)Let Υ be a contraction operator on a complete metric space Θ, then there exists a unique point ζ ∈ Θ for which Υ(ζ) = ζ.

Theorem 4. Assume that conditions (A1)–(A3) hold. Provided that

• (i)

  there exists a constant μ such that inf _k=1,2,…{t_k − t_k−1} ≥ μ,

• (ii)

  there exist constants p_i such that p_ik ≤ p_iμ for i ∈ 𝒩 and k = 1,2, …,

• (iii)

  $$,

• (iv)

  $$, where $$,

then the trivial equilibrium x = 0 is asymptotically stable.

Proof. Multiplying both sides of (1) with $$ gives, for t > 0 and t ≠ t_k,

which yields after integrating from t_k−1 + ε (ε > 0) to t ∈ (t_k−1, t_k) (k = 1,2, …)

Letting ε → 0 in (11), we have

for t ∈ (t_k−1, t_k) (k = 1,2, …). Setting t = t_k − ε (ε > 0) in (12), we get which generates by letting ε → 0

Noting x_i(t_k − 0) = x_i(t_k), (14) can be rearranged as

Combining (12) and (15), we reach that

is true for t ∈ (t_k−1, t_k] (k = 1,2, …). Further, holds for t ∈ (t_k−1, t_k] (k = 1,2, ⋯). Hence, which produces, for t > 0,

Note x_i(0) = φ_i(0) in (19). We then define the following operator π acting on ℋ, for $$:

where π(y_i)(t) : [ϑ, ∞) → R (i ∈ 𝒩) obeys the rules as follows: on t ≥ 0 and π(y_i)(s) = φ_i(s) on s ∈ [ϑ, 0].

The subsequent part is the application of the contraction mapping principle, which can be divided into two steps.

Step 1. We need to prove π(ℋ) ⊂ ℋ. Choosing y_i(t) ∈ ℋ_i (i ∈ 𝒩), it is necessary to testify π(y_i)(t) ⊂ ℋ_i.

First, since π(y_i)(s) = φ_i(s) on s ∈ [ϑ, 0] and φ_i(s) ∈ C[[ϑ, 0], R], we know π(y_i)(s) is continuous on s ∈ [ϑ, 0]. For a fixed time t > 0, it follows from (21) that

where

Owing to y_i(t) ∈ ℋ_i, we see that y_i(t) is continuous on t ≠ t_k (k = 1,2, …); moreover, $$ and $$ exist, and $$.

Consequently, when t ≠ t_k (k = 1,2, …) in (22), it is easy to find that Q_i → 0 as r → 0 for i = 1, …, 4, and so π(y_i)(t) is continuous on the fixed time t ≠ t_k (k = 1,2, …).

On the other hand, as t = t_k (k = 1,2, …) in (22), it is not difficult to find that Q_i → 0 as r → 0 for i = 1,2, 3. Furthermore, if letting r < 0 be small enough, we derive

which implies $$ as t = t_k. While letting  r > 0  tend to zero gives which yields $$ as t = t_k.

According to the above discussion, we find that π(y_i)(t) : [ϑ, ∞) → R is continuous on t ≠ t_k (k = 1,2, …); moreover, $$ and $$ exist; in addition, $$.

Next, we will prove π(y_i)(t) → 0 as t → ∞. For convenience, denote

where $$, $$, $$, and$$.

Due to y_j(t) ∈ ℋ_j (j ∈ 𝒩), we know lim _t→∞y_j(t) = 0. Then for any ε > 0, there exists a T_j > 0 such that t ≥ T_j implies |y_j(t)  | < ε. Choose T^* = max _j∈𝒩{T_j}. It is derived from (A1) that, for t ≥ T^*,

Moreover, as $$, we can find a $$ for the given ε such that $$ implies $$, which leads to namely,

On the other hand, since t − τ_j(t) → ∞ as t → ∞, we get lim _t→∞y_j(t − τ_j(t)) = 0. Then for any ε > 0, there also exists a $$ such that $$ implies |y_j(s − τ_j(s))| < ε. Select $$. It follows from (A2) that

which results in

Furthermore, from (A3), we know that |I_ik(y_i(t_k))| ≤ p_ik|y_i(t_k)|. So

As y_i(t) ∈ ℋ_i, we have lim _t→∞y_i(t) = 0. Then for any ε > 0, there exists a nonimpulsive point T_i > 0 such that s ≥ T_i implies |y_i(s)| < ε. It then follows from conditions (i) and (ii) that

which produces

From (30), (32), and (35), we deduce π(y_i)(t) → 0 as t → ∞ for i ∈ 𝒩. We therefore conclude that π(y_i)(t) ⊂ ℋ_i (i ∈ 𝒩) which means π(ℋ) ⊂ ℋ.

Step 2. We need to prove π is contractive. For $$ and $$, we estimate

where $$,$$.

Note

It hence follows from (37) that

which implies

Therefore,

In view of condition (iii), we see π is a contraction mapping, and, thus there exists a unique fixed point $$ of π in ℋ which means the transposition of $$ is the vector-valued solution to (1)–(3) and its norm tends to zero as t → ∞.

To obtain the asymptotic stability, we still need to prove that the trivial equilibrium x = 0 is stable. For any ε > 0, from condition (iv), we can find δ satisfying 0 < δ < ε such that $$. Let |φ| < δ. According to what has been discussed above, we know that there exists a unique solution $$ to (1)–(3); moreover,

here $$, $$, $$, and $$.

Suppose there exists t^* > 0 such that ∥x(t^*; s, φ)∥ = ε and ∥x(t; s, φ)∥ < ε as 0 ≤ t < t^*. It follows from (41) that

As we obtain |x_i(t^*)| < δ + λ_iε.

So $$. This contradicts the assumption of ∥x(t^*; s, φ)∥ = ε. Therefore, ∥x(t; s, φ)∥ < ε holds for all t ≥ 0. This completes the proof.

Corollary 5. Assume that conditions (A1)–(A3) hold. Provided that

• (i)

  inf _k=1,2,…{t_k − t_k−1} ≥ 1,

• (ii)

  there exist constants p_i such that p_ik ≤ p_i for i ∈ 𝒩 and k = 1,2, …,

• (iii)

  $$,

• (iv)

  $$, where $$,

then the trivial equilibrium x = 0 is asymptotically stable.

Proof. Corollary 5 is a direct conclusion by letting μ = 1 in Theorem 4.

Remark 6. In Theorem 4, we can see it is the fixed point theory that deals with the existence and uniqueness of solution and the asymptotic analysis of trivial equilibrium at the same time, while Lyapunov method fails to do this.

Remark 7. The presented sufficient conditions in Theorems 4 and Corollary 5 do not require even the boundedness and differentiability of delays, let alone the monotone decreasing behavior of delays which is necessary in some relevant works.

Theorem 8. Assume that conditions (A1)-(A2) hold. Provided that

• (i)

  $$,

• (ii)

  $$, where $$,

then the trivial equilibrium x = 0 is asymptotically stable.