Stability of Nonlinear Fractional Neutral Differential Difference Systems

Definition 1. The fractional order integral of a function $$ of order $$ is defined by

where Γ(·) is the gamma function.

Definition 2. For a function f given on the interval [t₀, ∞), the α order Riemann-Liouville fractional derivative of f is defined by

where $$.

Definition 3. For a function $$, the α order Caputo fractional derivative of f is defined by

Property 4. The following results are especially interesting.

• (i)

  For ν > −1, we have $$.

• (ii)

  When 0 < α < 1, we have

  $$ ()

• (iii)

  For α ∈ (0,1), T ≥ t₀, and $$, we have $$, $$.

Remark 5. From Property 4, if $$, α ∈ (0,1), then, for t ≥ t₀, we have the following.

• (i)

  f(t) ≥ f(t₀).

• (ii)

  In general, it is not true that f(t) is nondecreasing in t.

Definition 6 (see [21].)Suppose $$ is a subset of $$. Ones says the operator $$ is uniformly stable with respect to $$ if there are constants K, M such that, for any $$, σ ∈ [τ, +∞) and $$, the solution x(t, σ, φ, H) of (8) satisfies

Lemma 7 (see [21].)If $$ is uniformly stable with respect to $$, then there are positive constants a, b, c, d such that, for any $$, σ ∈ [τ, +∞), the solution x(t, σ, φ, h) of the equation

satisfies for all t ≥ σ. Furthermore, the constants a, b, c, d can be chosen so that for any s ∈ [σ, +∞) for all t ≥ s + r.

Definition 8 (see [22].)Suppose $$ is linear, continuous, and atomic at 0 and let $$. The operator $$ is said to be stable if the zero solution of the homogeneous difference equation,

is uniformly asymptotically stable.

Definition 9 (see [23].)The matrix $$ is Schur stable if the spectrum of the matrix lies in the open unit disc of the complex plane.

Remark 10. Let $$, where B is Schur stable. Then $$ is stable.

Definition 11. We say that the zero solution x = 0 of (17) is stable if, for any $$ and any ε > 0, there exists a δ = δ(t₀, ε) such that any solution x(t) = x(t, t₀, φ) of (17) with initial value φ at t₀, ∥φ∥ < δ, satisfies |x(t)| < ε for t ≥ t₀. It is asymptotically stable if it is stable and, for any $$ and any ε > 0, there exists a δ₀ = δ(t₀, ε), T(t₀, ε) > 0 such that ∥φ∥ < δ₀ implies |x(t)| ≤ ε, for t ≥ t₀ + T(t₀, ε); that is, lim⁡_t→+∞⁡x(t) = 0. It is uniformly stable if it is stable and δ = δ(ε) > 0 can be chosen independently of t₀. It is uniformly asymptotically stable if it is uniformly stable and there exists a δ₀ > 0; for any η > 0, there exists a T = T(η) > 0 such that ∥φ∥ < δ₀ implies |x(t)| < η for t > t₀ + T.

Theorem 12. Let $$, let B be Schur stable, and let x = 0 be an equilibrium point of system (17). Then the zero solution of system (17) is stable if and only if there exist a functional $$ and a continuous function u(s) with u(s) > 0 for s > 0 and u(0) = 0 such that the following conditions are satisfied.

• (1)

  V(t, 0) = 0.

• (2)

  $$.

• (3)

  For any given t₀ the functional V(t₀, ϕ) is continuous in ϕ at the point 0; that is, for any ε > 0 there exists δ > 0 such that the inequality ∥ϕ∥ < δ implies |V(t₀, ϕ) − V(t₀, 0)| = V(t₀, ϕ) < ε.

• (4)

  Along the solutions of system (17) the functional V(t, ϕ) satisfies V(t, x_t(t₀, φ)) ≤ V(t₀, φ) for t ≥ t₀.

Proof Sufficiency. Since the matrix B is Schur stable, there exist L ≥ 1 and 0 < ζ < 1 such that the inequality $$ holds for k ≥ 0.

For a given ε > 0  (ε < M), we set ε₁ = ε(1 − ζ)/L > 0. Since for a given t₀ functional V(t₀, ϕ) is continuous in ϕ at the point 0, there exists δ₁(ε, t₀) such that V(t₀, ϕ) < u(ε₁) for any $$, with ∥ϕ∥ < δ₁(ε, t₀). Here, we claim that δ₁(ε, t₀) ≤ ε₁. then, there exists an initial function $$ such that ∥φ∥ < δ₁(ε, t₀) and $$. On the one hand, for this initial function, we have $$. On the other hand, $$. The contradiction proves the desired inequality.

Now we take δ(ε, t₀) = δ₁(ε, t₀)/(1 + Lζ). For $$ with ∥φ∥ < δ(ε, t₀), we have

Next, we wish to show Assume by contradiction that there exists a t₁ ≥ t₀ for which Since and $$ is a continuous function of t, there exists $$ such that On the one hand, we have On the other hand, relation (20) provides the following inequality: The contradiction proves that inequality (22) is wrong and relation (21) is true. Then there exists a function z(t) with |z(t)| < ε₁, t ≥ t₀ such that

For a given t ≥ t₀, there must exist a positive integer number k such that t ∈ [t₀ + (k − 1)r, t₀ + kr]. Iterating equality (27) k − 1 times we obtain

Since t − kr ∈ [t₀ − r, t₀], and we obtain the following inequality: Therefore, the zero solution of system (17) is stable.

Necessity. Now, the zero solution of system (17) is stable, and we must prove that there exist a function u(s) and a functional V(t, ϕ) that satisfy the conditions (1)–(4).

Since the zero solution of system (17) is stable, for ε = M there exists δ(M, t₀) such that the inequality ∥φ∥ < δ(M, t₀) implies that |x(t, t₀, φ))| < M for t ≥ t₀. We define the function u(s) = s, $$ and the functional V(t, ϕ) as follows:

Since for φ = 0 the corresponding solution is trivial, x(t, t₀, 0) = 0, t ≥ t₀, we obtain V(t, 0) = 0. In the case where |x(s, t₀, φ)| < M, s ≥ t₀, we have In the other case where there exists T ≥ t₀ such that |x(T, t₀, φ)| ≥ M, the following inequality holds: Further, for a given t₀, the stability of the zero solution means that for any ε > 0 there exists δ₁ = δ(ε/(1 + Lζ), t₀) such that ∥φ∥ ≤ δ₁ implies Then Therefore, That is, for a fixed t₀ the functional V(t₀, φ) is continuous in φ at the point 0.

Finally, we need to show V(t, x_t(t₀, φ)) ≤ V(t₀, φ), t ≥ t₀. First, if |x(s, t₀, φ)| < M for s ≥ t₀,

Note that {s : s ≥ t}⊆{s : s ≥ t₀} for t ≥ t₀; then we have In the second case, there exists T ≥ t₀ such that |x(T, t₀, φ)| ≥ M; we have Therefore, we have Then, the proof is complete.

Remark 13. The functional (31) has only an academic value. Obviously, we cannot use such functionals in applications. The computation of practically useful Lyapunov functionals is a very difficult task.

Theorem 14. Let $$, B be Schur stable and let x = 0 be an equilibrium point of system (17). Suppose u(s) is a continuous function with u(s) > 0 for s > 0 and u(0) = 0. If there exists a continuous functional $$ such that the following conditions are satisfied:

• (1)

  V(t, 0) = 0,

• (2)

  $$,

• (3)

  along the solutions of the system (5) the functional V(t, ϕ) is continuously differentiable and satisfies

  $$ ()
  where β ∈ (0,1],

then the zero solution of system (17) is stable.

Proof. Note that the theorem conditions imply that of Theorem 12; therefore, the zero solution of system (17) is stable.

Theorem 15. Suppose that the assumptions in Theorem 14 are satisfied except replacing $$ by $$; then one has the same result for stability.

Proof. By using Property 4 we have

Since V(t₀, φ) ≥ 0, then $$. Then we can obtain the same result for stability.

Theorem 16. Let $$, let B be Schur stable, and let x = 0 be an equilibrium point of system (17). Suppose u(s), w(s) are continuous functions, u(s), w(s) are positive for s > 0, and u(0) = w(0) = 0. If there exists a continuously differentiable functional $$ such that the following conditions are satisfied:

• (1)

  V(t, 0) = 0,

• (2)

  $$,

• (3)

  $$, where β ∈ (0,1],

then the zero solution of system (17) is asymptotically stable.

Proof. Since the matrix B is Schur stable, there exist L ≥ 1 and 0 < ζ < 1 such that the inequality $$ holds for k ≥ 0.

Note that the conditions (1)–(3) of the theorem imply that of Theorem 14; then the zero solution of system (17) is stable; that is, for any ε > 0 and t₀, there exists δ(ε, t₀) > 0 such that for every initial function $$, with ∥φ∥ < δ(ε, t₀), the following inequality holds:

Now, let $$. Given t₀ and an initial function $$ with $$, we have

Next, we wish to show Suppose not, then there exist ε₀ > 0 and a sequence $$, t_k → ∞, as k → ∞ such that Without loss of generality we may assume that t_k+1 − t_k ≥ r for k ≥ 0. It follows from system (17) that Then, we have Since f takes bounded sets into bounded sets, there is a constant L₀ such that |f(t, x_t(t₀, φ))| ≤ L₀ for t ≥ t₀, $$. Then Hence, for any k ≥ 1, we have where θ = min⁡{r, [Γ(1 + α)ε₀/4L₀] ^1/α}. Let $$. Then there exists some $$ such that From the third condition of the theorem, we have That is, which contradicts the condition (2) of the theorem. Therefore, relation (45) is true. This means that there exists a function z(t) with lim⁡_t→∞⁡|z(t)| = 0 such that And given a positive value ε₁ < ε, there exists t₁ > t₀ such that Then there exists k₀ such that Lζ^kε < (1/2)ε₁ for k ≥ k₀. Thus, we have Therefore, we have lim⁡_t→∞⁡x(t, t₀, φ) = 0; that is, the zero solution of system (17) is asymptotically stable.

Theorem 17. Suppose that the assumptions in Theorem 16 are satisfied except replacing $$ by $$; then one has the same result for asymptotical stability.

Proof. By using Property 4 we have

Since V(t₀, φ) ≥ 0, then $$. Then we can obtain the same result for asymptotical stability.

Theorem 18. Suppose $$ in (17) is independent of t, $$ is continuous and maps bounded sets into bounded sets, and there exist continuous, nondecreasing nonnegative functions a(s), b(s), s ≥ 0, positive for s > 0, an open set S in $$, and a bounded open neighborhood U of zero in $$ such that V(ϕ) satisfies the following.

• (i)

  V(ϕ) > 0 on S, V(ϕ) = 0 on the boundary ∂S of S.

• (ii)

  0 belongs to the closure of S∩U.

• (iii)

  $$ on $$.

• (iv)

  $$ on $$, β ∈ (0,1].

Under these conditions, the zero solution of (17) is unstable. More specifically, each solution x = x(t, t₀, φ) of (17) with initial value φ in S∩U at $$ must reach the boundary of U in finite time.

Proof. Suppose φ₀ ∈ S∩U, $$. Then V(φ₀) > 0. From (iv), the solution x(t, t₀, φ₀) satisfies V(x_t) ≥ V(φ₀) as long as x_t ∈ S∩U. From (iii) and (iv), this implies

as long as x_t ∈ S∩U. Then this relation implies as long as x_t ∈ S∩U. Since U is bounded and V is bounded on U, there must be a t₁ such that $$. But hypothesis (i) implies that $$. This proves the last assertion of the theorem. Hypothesis (ii) implies that each neighborhood of zero contains a φ₀ in S∩U. Thus, x = 0 is unstable and the theorem is proved.

Theorem 19. Suppose $$ is uniformly stable with respect to $$, u(s), v(s), w(s) are continuous nondecreasing functions, u(s), v(s) are positive for s > 0, u(0) = v(0) = 0, and w(s) is nonnegative. If there exists a continuously differentiable functional $$ such that

• (i)

  $$,

• (ii)

  $$, β ∈ (0,1],

then the zero solution of (17) is uniformly stable. If, in addition, w(s) > 0 for s > 0, then it is uniformly asymptotically stable.

Proof. Suppose the constants b, c, d are defined as in Lemma 7, and x(t) = x(t, t₀, φ) is solution of (17)-(18). For any ε > 0, we can find a sufficiently small δ such that bδ < ε/2, v(δ) < u(ε/2(c + d)). Hence, for any initial time t₀ and any initial condition $$ with ∥φ∥ < δ, we have $$, and therefore V(t, x_t(t₀, φ)) ≤ V(t₀, φ) for any t ≥ t₀. This implies that

which implies that $$ for t ≥ t₀. Since $$ is uniformly stable, Lemma 7 implies Therefore, the zero solution is uniformly stable.

To prove uniform asymptotic stability, let ε₀∶ = ε = 1; choose δ₀∶ = δ(1) > 0 which correspond to uniform stability. Then, for any $$, ∥φ∥ < δ₀ implies

Next, for any η > 0, we wish to show that there is a T(δ₀, η) such that any solution x(t, t₀, φ) of (17) with ∥φ∥ < δ₀ satisfies ∥x_t(t₀, φ)∥ < η for t ≥ t₀ + T(δ₀, η). To do this, we show that there is a T(δ₀, η) and t₁ in [t₀, t₀ + T(δ₀, η)] such that $$, where δ = δ(η) is the above constant for uniform stability. The uniform stability then implies that ∥x_t(t₀, φ)∥ < η for t ≥ t₁ and, in particular, for t ≥ t₀ + T(δ₀, η).

For a, b, c, d as in Lemma 7, choose N = N(δ₀, η) so that

Let T(δ₀, η) = N + r + [v(δ₀)Γ(1 + β)/w(δ/2d)] ^1/β. Suppose there is a solution x = x(t, t₀, φ) of (17) with ∥φ∥ < δ₀ and ∥x_t(t₀, φ)∥ ≥ δ for t ≥ t₀. Taking s = t₀ + [v(δ₀)Γ(1 + β)/w(δ/2d)] ^1/β, t = s^′∶ = s + N + r in relation (12), we have Therefore, there exists a t^′ ∈ [s, s^′] such that $$. Then, by the condition (ii) of the theorem, we have and hence by Property 4 we conclude Then by using Property 4 and condition (i) of the theorem, we have As a result we obtain

This contradiction proves that there exists a t₁ ∈ [t₀, t₀ + T(δ₀, η)] such that $$. Thus, we have |x(t, t₀, φ)| < η, t ≥ t₀ + T(δ₀, η), whenever ∥φ∥ < δ₀. This proves the uniform asymptotic stability of the zero solution of (17).

Theorem 20. Suppose that the assumptions in Theorem 19 are satisfied except replacing $$ by $$; then one has the same result for uniform stability and uniform asymptotic stability.

Proof. By using Property 4 we have

Since V(t₀, φ) ≥ 0, then $$. Then we can obtain the same result for uniform stability and uniform asymptotic stability.

Proof. Let the Lyapunov candidate be $$, then, for 0 < β < 1, we have

Then, it follows from Theorem 14 that the equilibrium point x = 0 of system (70) is stable.