Gronwall-Bellman Type Inequalities and Their Applications to Fractional Differential Equations

Lemma 1 (Jensen′s inequality). Let n ∈ N, and let a₁, …, a_n be nonnegative real numbers. Then, for r > 1,

Lemma 2. Let I = [t₀, T) ⊂ R, k(t), b(t), p(t) ∈ C(I, R⁺), (R⁺ = [0, ∞), T ≤ ∞). If u(t) ∈ C(I, R⁺), and

where 0 ≤ γ < 1. Then, for t ∈ I, one has where $$.

Proof. Given t₀ < T₀ < T, for t ∈ [t₀, T₀],

Define a function $$, t ∈ [t₀, T₀]; then z(t₀) = A(T₀), u(t) ≤ z(t), z(t) is positive and nondecreasing for t ∈ I, and Let $$, and C(t₀) = z(t₀) = A(T₀); we obtain which implies that And so By the arbitrary of T₀ ∈ I, we obtain the inequality (3). The proof is complete.

Lemma 3. Let k(t), b(t), p(t) ∈ C(I, R⁺), φ(t) ∈ C([t₀ − r, t₀], R⁺), and k(t₀) = φ(t₀). If u(t) ∈ C(I, R⁺) and

where 0 ≤ γ < 1 and r ≥ 0 is a real constant, then where A(t) is defined as that in Lemma 2.

Proof. For t ∈ [t₀, t₀ + r], we have

By Gronwall inequality, we have the inequality (11). We prove that (10) holds for t ∈ [t₀ + r, T) now. Given that T₀ ∈ [t₀ + r, T) and for t ∈ [t₀ + r, T₀], we get Define a function $$, t ∈ [t₀ + r, T₀]; then z(t₀) = A(T₀), u(t) ≤ z(t), z(t) is positive and nondecreasing for t ∈ [t₀ + r, T₀], and As that in the proof of Lemma 2, we obtain And then By the arbitrary of T₀ ∈ (t₀ + r, T), we obtain the inequality (10). The proof is complete.

Theorem 4. Let a(t), b(t),  p(t) ∈ C(I, R⁺). If u(t) ∈ C(I, R⁺) and

where β > 0 and 0 < γ < 1 are constants, then the following assertions hold.

• (i)

  Suppose that β > 1/2. Then

  $$ ()

•  

  where $$, K₁ = 2Γ(2β − 1)/4^β−1, and R₁(t) = e^2(γ−1)t.

• (ii)

  Suppose that β ∈ (0, 1/2], q = (1 + β)/β, and p = 1 + β. Then

  $$ ()

•  

  where $$, $$ and R₂(t) = e^q(γ−1)t.

Proof. (i) Using the Cauchy-Schwarz inequality, we obtain

Using Lemma 1, we obtain Let $$; we get Using Lemma 2 and noticing that A₁(t) is nondecreasing, we get by the relationship of v(t) and u(t), the first inequality (18) holds.

(ii) By the hypothesis, we get (1/p) + (1/q) = 1. Using Hölder inequality, we obtain

Using Lemma 1, we obtain Let $$, we get Using Lemma 2 and noticing that A₂(t) is nondecreasing, we get and by the relation of u(t) and v(t), (19) holds. The proof is complete.

Theorem 5. Let a(t), b(t), p(t) ∈ C(I, R⁺), and a(t₀) = φ(t₀). If u(t) ∈ C([t₀ − r, T), R⁺) with

where β > 0 and 0 < γ < 1 are constants, then the following assertions hold.

• (i)

  Suppose that β > 1/2. Then

  $$ ()
  where A₁(t), K₁, and R₁(s) are defined as those in Theorem 4.

• (ii)

  Suppose that β ∈ (0, 1/2], q = (1 + β)/β, and p = 1 + β. Then,

  $$ ()
  where A₂(t), K₂, and R₂(s) are defined as those in Theorem 4.

Proof. (i) Using the Cauchy-Schwarz inequality by (28), we obtain

Using Lemma 1, we obtain Let $$, we get Using Lemma 3, we get the first inequality of (29) and the second inequality of (29) is easily obtained.

(ii) By the hypothesis, we get (1/p) + (1/q) = 1. Using Hölder inequality, we obtain

Using Lemma 1, we obtain Let $$; we get Using Lemma 2, we get the first inequality of (30) and the second inequality of (30) is easily obtained. The proof is complete.

Theorem 6. Let a(t), b(t), p(t) ∈ C(I, R⁺), and a(t₀) = φ(t₀). If u(t) ∈ C(I, R⁺) and

where β > 0, then the following assertions hold.

• (i)

  Suppose that β > 1/2. Then

  $$ ()
  where A₁(t) and K₁ are defined as those in Theorem 4.

• (ii)

  Suppose that β ∈ (0, 1/2], q = (1 + β)/β and p = 1 + β. Then

  $$ ()
  where A₂(t) and K₂ are defined as those in Theorem 4.

Remark 7. In [6, Theorem 1.2.4], a(t) is continuously differentiable, but in Theorem 6, a(t) is only continuous in the interval I ⊂ R⁺, so the methods of [6, Theorem 1.2.4] are invalid for Theorem 6. In [7, Theorem 1], Ye et al. also considered the similar integral inequalities using an iterative method, but we use different methods differing from the previously mentioned two papers.

Theorem 8. Suppose that |h(t)x(t) + f(t, x)| ≤ b(t) | x(t)| + p(t) | x(t)|^γ, where b(t), p(t) ∈ C(I, R⁺), 0 < γ < 1 is real number. If x(t) is any solution of the initial value problem (40), then the following estimations hold.

• (i)

  Suppose that β > 1/2. Then

  $$ ()
  where $$.

• (ii)

  Suppose that β ∈ (0, 1/2], q = (1 + β)/β, and p = 1 + β. Then

  $$ ()
  where $$. Notice that K₁, K₂, R₁(s), and R₂(s) are the same as those in Theorem 4, m = −[−β].

Proof. By (41), it is easy to derive that

Using Theorem 4, we get the desired conclusion. This proves the theorem.

Theorem 9. Suppose that |h(t)x(t) + f(t, x)| ≤ b(t) | x(t)| + p(t) | x(t)|^γ, where b(t), p(t) ∈ C(I, R⁺), and 0 < γ < 1 is real number. If x(t) is any solution of the initial value problem (46), then the following estimations hold.

• (i)

  Suppose that β > 1/2. Then

  $$ ()

• (ii)

  Suppose that β ∈ (0, 1/2], q = (1 + β)/β and p = 1 + β. Then

  $$ ()

Notice that B₁(t), B₂(t), K₁, K₂, R₁(s), and R₂(s) are the same as those in Theorem 8, m = −[−β].