Ψ-Stability of Nonlinear Volterra Integro-Differential Systems with Time Delay

Definition 1 (see [4], [8].)The trivial solution of (3) ((4) or (5)) is said to be Ψ-stable on ℝ₊ if for every ε > 0 and any t₀ ∈ ℝ₊, there exists δ = δ(ε, t₀) > 0 such that any solution x(t) of (3) ((4) or (5)), which satisfies the inequality ∥Ψ(t₀)x(t₀)∥ < δ, exists and satisfies the inequality ∥Ψ(t)x(t)∥ < ε for all t ≥ t₀.

Definition 2 (see [4], [8].)The trivial solution of (3) ((4) or (5)) is said to be Ψ-uniformly stable on ℝ₊ if it is Ψ-stable on ℝ₊ and the previous δ is independent of t₀.

Lemma 3. Let h, k, p, q ∈ C(ℝ₊ × ℝ₊, ℝ₊) with (t, s) ↦ ∂_th(t, s), ∂_tk(t, s), ∂_tp(t, s), ∂_tq(t, s) ∈ C(ℝ₊ × ℝ₊, ℝ₊). Assume, in addition, that b ∈ C(ℝ₊, ℝ₊) and α ∈ C¹(ℝ₊, ℝ₊) are nondecreasing functions and α(t) ≤ t for t ≥ 0. If u ∈ C(ℝ₊, ℝ₊) satisfies

for t ≥ 0, and $$, then where $$, $$.

Proof. Let T ≥ 0 be fixed and denote

then u(t) ≤ b(t) + x(t),  and x is nondecreasing on ℝ₊. For t ∈ [0, T], by calculations we get the following: Suppose that b(0) > 0 (if b(0) = 0, carry out the following arguments with b(t) + ε instead of b(t), where ε > 0 is an arbitrary small constant, and subsequently pass to the limit as ε → 0 to complete the proof), then we get Let then, we have Multiplying the above inequality by e^q(t) = Q(t), we get Consider now the integral on the interval [0, t] to obtain so for 0 ≤ t ≤ T. Let t = T, since $$, then we have Since T ≥ 0 was arbitrarily chosen, considering u(t) ≤ b(t) + x(t), we get (8).

Lemma 4. Let h, k, p, q, b, α be as in Lemma 3. If u ∈ C(ℝ₊, ℝ₊) satisfies

for t ≥ 0, and $$, then where $$, $$.

Theorem 5. If there exist functions a(t, s), b(t, s) ∈ C(ℝ₊ × ℝ₊, ℝ₊) with (t, s) ↦ ∂_ta(t, s), ∂_tb(t, s) ∈ C(ℝ₊ × ℝ₊, ℝ₊) such that

for 0 ≤ s ≤ t and for all x ∈ ℝⁿ. Moreover, and |Ψ(t)x(α(t))|≤|Ψ(α(t))x(α(t))|, where L₁, L₂ are nonnegative constants. If α(t) = t − τ(t) is an increasing diffeomorphism of ℝ₊. Then, the trivial solution of system (3) is Ψ-uniformly stable on ℝ₊.

Proof. Suppose that x(t, t₀, x₀) : = x(t) is the unique solution of system (3) which satisfies x(t₀) = x₀, since

after performing the change of variables r = α(s) in the second integral, and α⁻¹ is the inverse of the diffeomorphism α then, it follows that this implies by Lemma 3 that so for every ε > 0, choose $$, then for ∥Ψ(t₀)x₀∥ < δ and for all 0 ≤ t₀ ≤ t < ∞. Hence, the conclusion of the theorem follows.

Theorem 6. Let all the conditions in Theorem 5 hold. Suppose further that there exist functions m(t, s), n(t, s) ∈ C(ℝ₊ × ℝ₊, ℝ₊) with (t, s) ↦ ∂_tm(t, s), ∂_tn(t, s) ∈ C(ℝ₊ × ℝ₊, ℝ₊) such that

for 0 ≤ s ≤ t and for all x ∈ ℝⁿ, moreover, where L₃ is a nonnegative constant. Then, the trivial solutions of systems (4) and (5) are Ψ-uniformly stable on ℝ₊.

Proof. For that system (4), suppose x(t, t₀, x₀): = x(t) is the unique solution of system (4) which satisfies x(t₀) = x₀, since

it follows that after performing the change of variables r = α(s)  (or r = α(u)) at some intermediate step, and α⁻¹ is the inverse of the diffeomorphism α. Denote This implies by Lemma 3 that for $$ and 0 ≤ t₀ ≤ t. So, for every ε > 0 and t₀ ≥ 0, let $$ be a constant and choose $$, then for ∥Ψ(t₀)x₀∥ < δ and for all 0 ≤ t₀ ≤ t < ∞. This proves that the trivial solution of system (4) is Ψ-uniformly stable on ℝ₊.

Using Lemma 4, the proof of system (5) is similar to that of system (4) and the details are left to the readers.

Remark 7. For Ψ_i = 1, i = 1,2, …, n, we obtain the theorems of classical stability and uniform stability.

Example 8. Consider the nonlinear differential system

In (33), f(t, x(t)) = (x₁(t), −x₂(t)) ^T, $$. Let $$, then a(t, s) = b(t, s) = e^−(t−s) for 0 ≤ s ≤ t ≤ ∞, it is easy to verify that L₁ = 2, L₂ = 1, and all the assumptions in Theorem 5 satisfied, so the trivial solution of system (33) is ψ-uniformly stable on ℝ₊.

Example 9. Consider the nonlinear Volterra integro-differential system as follows:

In (34), f(t, x(t)) = (x₁(t), −x₂(t)) ^T, g ≡ 0, $$, $$. Choose the same matrix function Ψ(t), then a(t, s) = n(t, s) = e^−(t−s), b(t, s) ≡ 0, m(t, s) = e^−2(t−s) for 0 ≤ s ≤ t ≤ ∞, it is easy to verify that L₁ = L₂ = 1, L₃ = 1/2, and all the assumptions in Theorem 6 are satisfied, so the trivial solution of system (34) is ψ-uniformly stable on ℝ₊.