New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

Definition 1. The modified Riemann-Liouville derivative of order α is defined by the following expression:

Definition 2. The Riemann-Liouville fractional integral of order α on the interval [0, t] is defined by

Some important properties for the modified Riemann-Liouville derivative and fractional integral are listed as follows (the interval concerned below is always defined by [0, t]):

• (a)

  $$.

• (b)

  $$.

• (c)

  $$.

• (d)

  $$.

• (e)

  $$.

• (f)

  The modified Riemann-Liouville derivative for a constant is zero.

Lemma 3 (see [38].)Assume that a ≥ 0,   p ≥ q ≥ 0; and p ≠ 0, then, for any K > 0, one has

Lemma 4. Let α > 0, a,  b, u be continuous functions defined on t ≥ 0. Then for t ≥ 0,

Proof. By the properties (a), (b), and (c) we have the following observation:

Substituting t with τ, fulfilling fractional integral of order α for (6) with respect to τ from 0 to t, we deduce that

The desired result can be obtained subsequently.

Lemma 5. Suppose α > 0, the functions u, a, b, g, and h are nonnegative continuous functions defined on t ≥ 0, and a, b are nondecreasing, p, q, and r are constants with p ≥ q > 0,   p ≥ r > 0. If the following inequality holds:

where

Proof. Fix T ≥ 0, and let t ∈ [0, T]. Denote

Then we have

Since u,   g, and h are continuous, then there exists a constant M such that $$ for t ∈ [0, ε], where ε > 0. So for t ∈ [0, ε], we have $$. Then one can see v(0) = a(T). Furthermore,

By Lemma 3, we have

Letting t = T in (16) and considering T ≥ 0 is arbitrary, after substituting T with t, we get that

Combining (13) and (17), we get (10).

Remark 6. In Lemma 5, if a, b are not necessarily nondecreasing, then one can let $$ instead in the proof, and following in a similar process, obtain another explicit bound for u(t):

Remark 7. We note that if we take g(t) ≡ 1,   h(t) ≡ 0, and p = q = 1, then the inequality (9) in Lemma 5 reduces to the inequality in [39, Theorem 1]. So the present inequality is of more general form than that in [39]. Furthermore, the explicit bounds obtained for the function u(t) above are essentially different from that in [39].

Theorem 8. Suppose that α > 0, the functions u,   a,   b,   g,   p,   q, and r are defined as in Lemma 5, and p ≥ 1, m is a nonnegative continuous function defined on t ≥ 0. If the following inequality holds:

then we have

Proof. Denote z^p(t) by a(t) + $$. Then

Since a(t),   b(t) are both nondecreasing, then z(t) is also nondecreasing, and subsequently we can deduce that

Furthermore,

where $$ are defined in (22) and (23). Combining (26) and (28), we obtain the desired result.

Lemma 9. Suppose α > 0, the functions u,   a,   b, and g are defined as in Lemma 5, and ω is a nonnegative continuous function defined on t ≥ 0 being nondecreasing, and ω(r) > 0 for r > 0. Define $$, and assume G(v) < ∞ for v < ∞. If the following inequality holds:

then we have the following explicit estimate for u(t):

Proof. Fix T ≥ 0, and let t ∈ [0, T]. Denote

Since u, g, and ω are continuous, then there exists a constant M such that |g(t)ω(u(t))| ≤ M for t ∈ [0, ε], where ε > 0. So for t ∈ [0, ε], we have $$. Then one can see v(0) = a(T) + δ, and

which implies

That is,

Substituting t with τ, fulfilling fractional integral of order α for (35) with respect to τ from 0 to t, we deduce that

which implies

and furthermore,

Letting t = T in (38), we get that

Since T > 0 is arbitrary, substituting T with t in (39) and after letting δ → 0, we can obtain the desired result.

Theorem 10. Suppose α > 0, the functions u,   a,   b, and g are defined as in Lemma 5, m is a nonnegative continuous function defined on t ≥ 0, and ω is defined as in Lemma 9, and furthermore, assume that ω is submultiplicative; that is, ω(αβ) ≤ ω(α)ω(β),   α,   β ≥ 0. Define $$, and assume G(v) < ∞ for v < ∞. If the following inequality holds:

then we have the following explicit estimate for u(t):

Proof. Let $$.

Then we have

Since z(t) is nondecreasing, then furthermore we have

So

Since ω is submultiplicative, then furthermore we get that

Noticing that the structure of (45) is similar to that of the inequality (29), a suitable application of Lemma 9 to (45) yields that

Combining (43) and (46) we can obtain the desired result.

Theorem 11. Suppose that u(t) is a solution of the IVP (47). If |L(t, u, v)| ≤ g(t)|u|³ + |v|, and |M(t, u)| ≤ h(t)|u|³, where g, h are nonnegative continuous functions on [0, ∞), then we have the following estimate for u(t):

Proof. Similar to [28, Equation (5.1)], we can obtain the equivalent integral form of the IVP (47) as follows:

So

Then a suitable application of Lemma 5 to (50) (with α = 0.5,   p = q = r = 3) yields the desired result.

Remark 12. At the end of the proof of Theorem 11, if we apply the result of Remark 6 instead of Lemma 5 to (50), then we obtain the following estimate:

Remark 13. In Theorem 11, if we change the conditions by |L(t, u, v)| ≤ g(t) | u | +|v|, and |M(t, u)| ≤ h(t) | u|, where g,   h are nonnegative continuous functions on [0, ∞), then we can obtain the following estimate for u(t):

Theorem 14. If $$, $$, where g,   h are nonnegative continuous functions defined on [0, ∞), then the IVP (47) has at most one solution.

Proof. Suppose that the IVP (47) has two solutions u₁(t), u₂(t). Then similar to Theorem 11, we can obtain that

Furthermore,

which implies

Treating $$ as one whole function, a suitable application of Lemma 5 to (55) (with α = 0.5) yields $$, which implies u₁(t) ≡ u₂(t). So the proof is complete.

Theorem 15. Suppose that u(t) is the solution of the IVP (47), and $$ is the solution of following IVP of fractional differential equation:

where $$.

Proof. For the IVP (56), we have the following integral form:

So by (49) and (58), we deduce that

Furthermore,

A suitable application of Lemma 5 to (60) yields the desired result.

Remark 16. Theorem 15 indicates that the solution of the IVP (47) depends continuously on the initial value u(0) = C.

Example 17. Consider the following fractional integral equation:

Theorem 18. Suppose that u(t) is a solution of the fractional integral equation (61). If |L₁(t, u)| ≤ m(t) | u|³, |L₂(t, u, v)| ≤ g(t) | u | +|v|, and |M(t, u)| ≤ h(t) | u|, where m, g, and h are nonnegative continuous functions on [0, ∞), then we have the following estimate for u(t):

Proof. From (61), we have

Then a suitable application of Theorem 8 (with p = 3, q = r = 1) to (64) yields the desired estimate (62).

Theorem 19. For the fractional integral equation (62), if $$, $$, where m, g, and h are nonnegative continuous functions on [0, ∞), then the solution of (61) depends continuously on the initial value C.

Proof. Let $$ be the solution of the following fractional integral equation

A combination of (61) and (65) yields

Furthermore, we have

Then treating $$ as one whole function, applying Theorem 8 to (67), we get that

So the continuous dependence on the initial value C for the solution of (61) can be obtained from (68).

Remark 20. From the two examples presented above, one can see that the main results established in Section 2 are mainly used in the qualitative analysis as well as quantitative analysis of the solutions to some certain fractional differential or integral equations, such as the bound estimate, the number of the solutions, and the continuous dependence on the initial value for unknown solutions. On the other hand, by the variational iteration method and the homotopy perturbation method, approximate solutions for some fractional differential equations can be obtained (see the examples in [33–36], e.g.), while in few cases, the closed form of these approximate solutions can be obtained. So to this extent, we note that the starting point of establishing the main results in this paper is different from the variational iteration method and the homotopy perturbation method.