In the theory of differential and integral equations, Gronwall’s lemma has been widely used in various applications since its first appearance in the article by Bellman in 1943. In this article the author gave a fundamental lemma, known as Gronwall-Bellman Lemma, which is used to study the stability and asymptotic behavior of solutions of differential equations. Gronwall’s lemma has seen several generalizations to various forms [3, 11, 28].

The literature on these inequalities and their applications is vast; see [1, 2, 4] and the references given therein [19, 20, 23]. In addition, as the theory of calculus on time scales has developed over the last few years, some Gronwall-type integral inequalities on time scales have been established by many authors [17, 21, 24].

In this work, we shall study some fractional integral inequalities which can be used to prove the uniqueness of solutions for differential and integral equations of fractional-order. These inequalities are similar to Bellman-Gronwall type inequalities which play a fundamental role in the qualitative as well as quantitative study of differential equations.

Let L₁ = L₁[a, b] be the class of Lebesgue integrable functions on [a, b] with the standard norm.

Now, we shall introduce the definitions of the fractional-order integral operators (see [18, 25, 26, 27]). Let β be a positive real number.

Definition 1.1

The left-sided fractional integral of order β of the function f is defined on [a, b] by

and when a = 0, we have I0+βf(t), t>0.

Definition 1.2

The right-sided fractional integral of order β of the function f is defined on [a, b] by

and when b = 0, we have I0-βf(t), t<0.

For further properties of fractional calculus see [18, 25, 26, 27].

The existence of solutions of nonlinear quadratic integral equations and some properties of their solutions have been well studied recently (see [5, 6, 7, 8, 9, 10] and [12, 13, 14, 15, 16]).

Let α, β > 0. The existence of solutions of the quadratic integral equation of arbitrary (fractional) orders α and β

have been studied in [16] and [14] under the following assumptions.

• The functions f, g : [a, b] × R₊ → R₊ satisfy the Carathèodory condition (i.e. are measurable in t for all x ∈ R₊ and continuous in x for all t ∈ [a, b]). Moreover, there exist two functions m₁,m₂ ∈ L₁, and a positive constant k such that

  ∣f(t,x)∣≤m1(t)+k x(t),         ∣g(t,x)∣≤m2(t),   ∀(t,x)∈[a,b]×R+.

• There exists a positive constant M₂ such that Ia+γm2(t)≤M2, γ <β.

• There exists a positive constant M₁ such that Ia+σm1(t)≤M1, σ<α.

• h ∈ L₁.

Now, consider the nonlinear quadratic integral inequality of fractional orders

Theorem 2.1

Let assumptions (i)–(iv) be satisfied. Let x(t) ∈ L₁and satisfies inequality(2.2)for almost all t ∈ [a, b]. IfkM2bα+β-γΓ(β-γ+1)Γ(α+1)<1, then

Remark 2.2

If we replace assumptions (ii) and (iii) by

• (ii*) There exists a positive constant M₂ such that Ia+βm2(t)≤M2;

• (iii*) There exists a positive constant M₁ such that Ia+αm1(t)≤M1,

then we can obtain the following result.

Theorem 2.3

Let assumptions (i), (ii*), (iii*) and(iv) be satisfied. Let x(t) ∈ L₁satisfies the inequality(2.2)for almost all t ∈ [a, b]. Ifk M2 bα+βΓ(β+1)Γ(α+1)<1, then

Inequality (2.2) involves many integral inequalities of fractional order. So, some particular cases can be obtained as follows.

Corollary 2.4

Let f, g : [a, b] × R₊ → R₊are continuous functions, and h ∈ L₁. If x(t) ∈ L₁and satisfies the inequality (2.2) for almost all t ∈ [a, b]. Then

whereM1=sup∀t∈[a,b]∣g(t,x)∣,M2=sup∀t∈[a,b]∣f(t,x)∣.

Corollary 2.5

Let assumptions (i)–(iv) (with α = β, f = g) be satisfied. Let x(t) ∈ L₁and satisfies the inequality

for almost all t ∈ [a, b]. IfkM2b2α-γΓ(α-γ+1)Γ(α+1)<1, then

Corollary 2.6

Let assumptions (i)–(iv) (with α = β, f = g and h = 0) be satisfied. Let x(t) ∈ L₁and satisfies the inequality

for almost all t ∈ [a, b]. IfkM2b2α-γΓ(α-γ+1)Γ(α+1)<1, then

Letting α, β → 1, then we have

Corollary 2.7

Let h(t), f(t, x) and g(t, x) satisfy the assumptions of Theorem 2.3. Let x(t) ∈ L₁and satisfies the quadratic integral inequality

for almost all t ∈ [a, b]. If kM₂b² < 1, then

Letting β → 0, then we have

Corollary 2.8

Let h(t), f(t, x) and g(t, x) satisfy the assumptions of Theorem 2.3. Let x(t) ∈ L₁and satisfies the quadratic integral inequality

for almost all t ∈ [a, b]. Ifk.M2bαΓ(α+1)<1 ∀t∈[a,b], then

Corollary 2.9

Let h(t), f(t, x) satisfy assumptions of Corollary 2.8 and g(t, x) = 1. Let x(t) ∈ L₁and satisfies the inequality

for almost all t ∈ [a, b]. IfkbαΓ(α+1)<1, then

Letting f(t, x(t)) = m(t)x(t), β → 0 and g(t, x(t)) = 1, then we have the following result.

Corollary 2.10

Let h(t),m(t) ∈ L₁and m(t) > 0. Let x(t) ∈ L₁and satisfies the inequality

for almost all t ∈ [a, b]. If Mb^α < Γ(α + 1), where M is a positive constant such thatsup∀t∈[a,b]∣m(t)∣=M, then

When m(t) is continuous, then we get the following corollary.

Corollary 2.11

Let h(t) ∈ L₁, m(t) > 0 and m(t) is continuous on [a, b]. Let x(t) ∈ L₁and satisfies(2.5)for almost all t ∈ [a, b]. If b^αM <Γ(α + 1), then

wheresupt∈[a,b]∣m(t)∣=M.

Some special cases will be considered, when m(t) = K,K ≠ 0.

Corollary 2.12

Let h(t) ∈ L₁and x(t) ∈ L₁satisfying the inequality

for almost all t ∈ [a, b]. If b^αK <Γ(α + 1), then

Corollary 2.13

Let 0 ≤ β ≤ α,A ≥ 0 and let x(t) ∈ L₁satisfying the inequality

for almost all t ∈ [0, b]. If b^αK <Γ(α + 1), then

where C depends only on K, α and b.

For h(t) = 0, we have the following corollary.

Corollary 2.14

Let x(t) ∈ L₁and x(t) > 0 satisfying

for almost all t ∈ [a, b]. If b^αK <Γ(α + 1), then

Now, if we reverse the inequality (2.5) we shall obtain the next lemma.

Lemma 2.15

Let h(t), k(t) ∈ L₁and k(t) > 0. Let x(t) ∈ L₁and satisfies the inequality

for almost all t ∈ [a, b]. If there exist a function m ∈ L₁and a positive constant M such thatIb-βm(t)≤M, β < α. Moreover, Mb^α−β < Γ(α − β + 1). Then

Corollary 2.16

Let h(t) ∈ L₁, k(t) > 0 and k(t) is continuous on [a, b]. Let x(t) ∈ L₁and satisfies(2.6)for almost all t ∈ [a, b]. If b^αM <Γ(α + 1), then

wheresupt∈[a,b]∣k(t)∣=M.

Corollary 2.17

Let h(t) ∈ L₁and x(t) ∈ L₁satisfying the inequality

for almost all t ∈ [a, b]. If b^αk < Γ(α + 1), then

Corollary 2.18

Let 0 ≤ β ≤ α,A ≥ 0 and let x(t) ∈ L₁satisfying the inequality

for almost all t ∈ [0, b]. If b^αk < Γ(α + 1), then

where C depends only on k,α and b.

Corollary 2.19

Let x(t) ∈ L₁and satisfies the inequality

for almost all t ∈ [a, b]. If b^αk < Γ(α + 1), then

Remark 2.20

Clearly, we can obtain similar results if we replace the left-sided fractional-order integralIa+α by the right-sided fractional-order integralIb-α in inequality (2.2).

Over the years integral inequalities have become an important tool in the analysis of various differential and integral equations. These inequalities are useful in investigating the asymptotic behavior and the stability on the solutions of integral equations. Pachpatte [22] gave a new integral inequality and studied the boundedness, asymptotic behavior and growth of the solutions of an integral equation using the inequality. We introduce this inequality as follows.

Theorem 3.1.([22])

Let u, f, g be real-valued nonnegative continuous functions defined on R⁺, and c₁, c₂be nonnegative constants. If

andc1c2∫0tR(s)Q(s)ds<1for all t ∈ R⁺, then

where

Now consider the quadratic inequality of fractional order

Theorem 3.2

Let assumptions (i), (ii*), (iii*) and (iv) be satisfied. Let x(t) ∈ L₁satisfies the inequality(3.1)for almost all t ∈ [a, b]. Ifk(c2+M1)bαΓ(α+1)<1, then