Asymptotic Stability Results for Nonlinear Fractional Difference Equations

Definition 2.1 (see [14].)Let ν > 0. The ν-th fractional sum x is defined by

where f is defined for s = a mod (1) and Δ^−νf is defined for t = (a + ν) mod (1), and t^(ν) = Γ(t + 1)/Γ(t − ν + 1). The fractional sum Δ^−ν maps functions defined on N_a to functions defined on N_a+ν.

Definition 2.2 (see [14].)Let μ > 0 and m − 1 < μ < m, where m denotes a positive integer, m = ⌈μ⌉, ⌈·⌉ ceiling of number. Set ν = m − μ. The μ-th fractional difference is defined as

Theorem 2.3 (see [15].)Let f be a real-value function defined on N_a and μ, ν > 0, then the following equalities hold:

• (i)

  Δ^−ν[Δ^−μf(t)] = Δ^−(μ+ν)f(t) = Δ^−μ[Δ^−νf(t)];

• (ii)

  $$.

Lemma 2.4 (see [15].)Let μ ≠ 1 and assume μ + ν + 1 is not a nonpositive integer, then

Lemma 2.5 (see [15].)Assume that the following factorial functions are well defined:

• (i)

  If 0 < α < 1, then t^(αγ) ≥ (t^(γ)) ^α;

• (ii)

  t^(β+γ) = (t − γ) ^(β)t^(γ).

Lemma 2.6 (see [13].)Let μ > 0 be noninteger, m = ⌈μ⌉, ⌈·⌉, ν = m − μ, thus one has

Lemma 2.7. The equivalent fractional Taylor’s difference formula of (1.1) is

Proof. Apply the Δ^−α operator to each side of the first formula of (1.1) to obtain

Apply Theorem 2.3 to the left-hand side of (2.6) to obtain

So, applying Definition 2.1 to the right-hand side of (2.6), for t ∈ N_α we obtain (2.5). The recursive iteration to this Taylor’s difference formula implies that (2.5) represents the unique solution of the IVP (1.1). This completes the proof.

Lemma 2.8 (see [4], 1.5.15.)The quotient expansion of two gamma functions at infinityis

Corollary 2.9. One has

Proof. According to Lemma 2.8,

Then, t^(−β) > (t + α) ^(−β) for α, β, t > 0. This completes the proof.

Definition 2.10. The solution x = φ(t) of the IVP (1.1) is said to be

• (i)

  stable if for any ɛ > 0 and t₀ ∈ R⁺, there exists a δ = δ(t₀, ɛ) > 0 such that

for |x₀ − φ(t₀)| ≤ δ(t₀, ɛ) and all t ≥ t₀;

• (ii)

  attractive if there exists η(t₀) > 0 such that ∥x₀∥≤η implies

• (iii)

  asymptotically stable if it is stable and attractive.

The space $$ is the set of real sequences defined on the set of positive integers where any individual sequence is bounded with respect to the usual supremum norm. It is well known that under the supremum norm $$ is a Banach space [26].

Definition 2.11 (see [27].)A set Ω of sequences in $$ is uniformly Cauchy (or equi-Cauchy), if for every ɛ > 0, there exists an integer N such that |x(i) − x(j)| < ɛ, whenever i, j > N for any x = {x(n)} in Ω.

Theorem 2.12 (see [27], discrete Arzela-Ascoli’s theorem.)A bounded, uniformly Cauchy subset Ω of $$ is relatively compact.

Theorem 2.13 (see [20], Krasnoselskii’s fixed point theorem.)Let S be a nonempty, closed, convex, and bounded subset of the Banach space X and let A : X → X and B : S → X be two operators such that

• (a)

  A is a contraction with constant L < 1,

• (b)

  B is continuous, BS resides in a compact subset of X,

• (c)

  [x = Ax + By,   y ∈ S]⇒x ∈ S.Then the operator equation Ax + Bx = x has a solution in S.

Lemma 3.1. Assume that the following condition is satisfied:

(H₁) there exist constants β₁ ∈ (α, 1) and L₁ ≥ 0 such that

Then the operator B is continuous and BS₁ is a compact subset of R for $$, where γ₁ = (−1/2)(α − β₁), and n₁ ∈ N satisfies that

Proof. For t ∈ N_α, apply Lemma 2.8 and γ₁ > 0,

and we have that $$ as t → ∞, then there exists a n₁ ∈ N such that inequality (3.4) holds, which implies that the set S₁ exists.

We firstly show that B maps S₁ in S₁.

It is easy to know that S₁ is a closed, bounded, and convex subset of R.

Apply condition (H₁), Lemma 2.5, Corollary 2.9 and (3.4), for $$, we have

which implies that BS₁ ⊂ S₁ for $$.

Nextly, we show that B is continuous on S₁.

Let ɛ > 0 be given then there exist T₁ ∈ N and T₁ ≥ n₁ such that $$ implies that

Let {x_n} be a sequence such that x_n → x. For t ∈ {α + n₁, α + n₁ + 1, …, α + T₁ − 1}, applying the continuity of f and Lemma 2.6, we have

For $$,

Thus, for all $$, we have

which implies that B is continuous.

Lastly, we show that BS₁ is relatively compact.

Let $$ and t₂ > t₁, thus we have

Thus, {Bx : x ∈ S₁} is a bounded and uniformly Cauchy subset by Definition 2.11, and BS₁ is relatively compact by means of Theorem 2.12. This completes the proof.

Lemma 3.2. Assume that condition (H₁) holds, then a solution of (1.1) is in S₁ for $$.

Proof. Notice if that x(t) is a fixed point of P, then it is a solution of (1.1). To prove this, it remains to show that, for fixed y ∈ S₁, x = Ax + By⇒x ∈ S₁ holds.

If x = Ax + By, applying condition (H₁) and (3.4), for $$, we have

Thus, x(t) ∈ S₁ for $$. According to Theorem 2.13 and Lemma 3.1, there exists a x ∈ S₁ such that x = Ax + Bx, that is, P has a fixed point in S₁ which is a solution of (1.1) for $$. This completes the proof.

Theorem 3.3. Assume that condition (H₁) holds, then the solutions of (1.1) is attractive.

Proof. By Lemma 3.2, the solutions of (1.1) exist and are in S₁. All functions x(t) in S₁ tend to 0 as t → ∞. Then the solutions of (1.1) tend to zero as t → ∞. This completes the proof.

Theorem 3.4. Assume that the following condition is satisfied:

(H₂) there exist constants β₂ ∈ (α, 1) and L₂ ≥ 0 such that

Then the solutions of (1.1) are stable provided that

Proof. Let x(t) be a solution of (1.1), and let $$ be a solution of (1.1) satisfying the initial value condition $$. For t ∈ N_α, applying condition (H₂), we have

which implies that

For any given ɛ > 0, let δ = ((1 − c)/α)ɛ, $$ follows that $$, which yields that the solutions of (1.1) are stable. This completes the proof.

Theorem 3.5. Assume that conditions (H₁) and (H₂) hold, then the solutions of (1.1) are asymptotically stable provided that (3.14) holds.

Theorem 3.5 is the simple consequence of Theorems 3.3 and 3.4.

Theorem 3.6. Assume that the following condition is satisfied:

(H₃) there exist constants β₃ ∈ (α, (1/2)(1 + α)), γ₂ = (1/2)(1 − α), and L₃ ≥ 0 such that

Then the solutions of (1.1) is attractive.

Proof. Set

where n₂ ∈ N satisfies that

We first prove condition (c) of Theorem 2.13, that is, for fixed y ∈ S₂ and for all x ∈ R, x = Ax + By⇒x ∈ S₂ holds.

If x = Ax + By, applying condition (H₃) and (3.19), for $$, we have

Thus, condition (c) of Theorem 2.13 holds.

The proof of condition (b) of Theorem 2.13 is similar to that of Lemma 3.1, and we omit it. Therefore, P has a fixed point in S₂ by using Theorem 2.13, that is, the IVP (1.1) has a solution in S₂. Moreover, all functions in S₂ tend to 0 as t → ∞, then the solution of (1.1) tends to zero as t → ∞, which shows that the zero solution of (1.1) is attractive. This completes the proof.

Theorem 3.7. Assume that conditions (H₂) and (H₃) hold, then the solutions of (1.1) are asymptotically stable provided that (3.14) holds.

Theorem 3.8. Assume that the following condition is satisfied:

(H₄) there exist constants η ∈ (0,1),   β₄ ∈ (α, (2 + αη)/(2 + η)), and L₄ ≥ 0 such that

Then the solutions of (1.1) is attractive.

Proof. Set

where γ₃ = (1/2)(β₄ − α), and n₃ ∈ N satisfies that

Here we only prove that condition (c) of Theorem 2.13 holds, and the remaining part of the proof is similar to that of Theorem 3.6.

Since η ∈ (0,1),   β₄ ∈ (α, (2 + αη)/(2 + η)), and γ₃ = (1/2)(β₄ − α), then γ₃, γ₃η, α + γ₃ ∈ (0,1),   β₄ + γ₃η ∈ (α, 1).

If x = Ax + By, applying condition (H₄), Lemma 2.5 and (3.23), for $$, we have

Thus, condition (c) of Theorem 2.13 holds. This completes the proof.

Example 4.1. Consider

where f(t, x(t)) = 0.2t^(−0.75)sin (x(t)), t ∈ N_0.5.

Since

thisimplies that condition (H₁) holds.

In addition,

Thus, condition (H₂) is satisfied.

Moreover, from L₂ = 0.2, α = 0.5, and β₂ = 0.75, we have

which implies that inequality (3.14) holds.

Thus the solutions of (4.1) are asymptotically stable by Theorem 3.5.

Example 4.2. Consider

where f(t, x(t)) = 0.2(t + 1) ^(−0.6)x(t), t ∈ N_0.5.

Since β₃ = 0.6,   α = 0.5, we have that β₃ ∈ (α, (1/2)(1 + α)), γ₂ = 0.25 and

which implies that condition (H₃) is satisfied.

Meanwhile,

which implies that condition (H₂) is satisfied.

From L₂ = 0.2, α = 0.5, and β₂ = 0.6, we have

which implies that inequality (3.14) holds.

Thus the solutions of (4.5) are asymptotically stable by Theorem 3.7.

Example 4.3. Consider

where f(t, x(t)) = (t + 1) ^(−0.6)x^1/3(t), t ∈ N_0.5.

Since α = 0.5,   β₄ = 0.6,   η = 1/3, we have that η ∈ (0,1),   β₄ ∈ (α, (2 + αη)/(2 + η)) and

then condition (H₄) is satisfied.

The solutions of (4.9) are attractive by Theorem 3.8.