Some Nonlinear Gronwall-Bellman-Gamidov Integral Inequalities and Their Weakly Singular Analogues with Applications

Abstract

Some Gronwall-Bellman-Gamidov type integral inequalities with power nonlinearity and their weakly singular analogues are established, which can give the explicit bound on solution of a class of nonlinear fractional integral equations. An example is presented to show the application for the qualitative study of solutions of a fractional integral equation with the Riemann-Liouville fractional operator.

1. Introduction

Integral inequalities, which provide explicit bounds on unknown functions, play a fundamental role in the development of the theory of linear and nonlinear differential equation and integral equation. One of the best known and widely used inequalities is the so-called Gronwall-Bellman integral inequality. In view of the important applications of the Gronwall-Bellman inequality, in the past few years, Pachpatte [1–3] established a number of new generalizations of such inequality which can be used as powerful tools in the study of certain new classes of differential and integral equations. Meanwhile, many authors have researched various generalizations of the Gronwall-Bellman inequality; for example, we refer the reader to [4–12].

In [6], Baǐnov and Simeonov discussed the following useful integral inequality: 𝑢(𝑡)≤𝑐+^𝑡₀𝑓(𝑠)𝑢(𝑠)𝑑𝑠+^𝑇₀𝑔(𝑠)𝑢(𝑠)𝑑𝑠, (1) which came from the study of the boundary value problem for higher order differential equations by Gamidov [13] and was extended by Pachpatte [2] as follows: 𝑢(𝑡)≤𝑐+^𝑡₀𝑓(𝑡,𝑠)𝑢(𝑠)𝑑𝑠+^𝑇₀𝑔(𝑡,𝑠)𝑢(𝑠)𝑑𝑠. (2)

Remark 1. It should be noted that the derived result in [2] is not right. In the proof, it involves the definition of 𝑧(𝛼), which was treated as a constant by mistake. For example, consider the following integral equation: 𝑢(𝑡)=1+¹₀𝑡𝑠𝑢(𝑠)𝑑𝑠, (3) in which 𝑐=1, 𝑓(𝑡,𝑠)=0, 𝑔(𝑡,𝑠)=𝑡𝑠, and 𝑇=1. We can obtain the solution of the equation above; that is, 𝑢(𝑡)=(3/4)𝑡+1. According to the formula of its upper bound reported in [2], one gets that 𝑢(𝑡)=(3/4)𝑡+1≤1/(1−(1/2)𝑡) for 𝑡∈[0,1]. Clearly, the result does not hold for 𝑡∈[0,1]. Hence, a revised one will be provided in later section.

On the other hand, Zheng [14] also established a weakly singular version of the Gronwall-Bellman-Gamidov inequality as follows: 𝑢(𝑡)1≤𝐶+Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑢(𝑠)+1𝑑𝑠Γ(𝛼)^𝑇₀(𝑇−𝑠)^𝛼−1𝑓(𝑠)𝑢(𝑠)𝑑𝑠, (4) and discussed its application in a fractional integral equation with the modified Riemann-Liouville derivative. As for weakly singular inequalities and their applications, more results can be found (e.g., see [15–23] and the references therein).

In this paper, motivated by the work in [2, 6, 14], we consider a Gronwall-Bellman-Gamidov integral inequality with power nonlinearity, 𝑢^𝑚(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝑓(𝑠)𝑢^𝑛(𝑠)(𝑡)𝑑𝑠+𝑐^𝑇₀𝑔(𝑠)𝑢^𝑟(𝑠)𝑑𝑠, (5) and its weakly singular analogue 𝑢^𝑚(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝑡^𝛼1−𝑠^{𝛼1𝛽1−1}𝑠^𝛾1−1𝑓(𝑠)𝑢^𝑛(𝑠)(𝑡)𝑑𝑠+𝑐^𝑇₀𝑇^𝛼2−𝑠^{𝛼2𝛽2−1}𝑠^𝛾2−1𝑔(𝑠)𝑢^𝑟(𝑠)𝑑𝑠, (6) where 𝑚≥𝑛≥0, 𝑚≥𝑟≥0, and [𝛼_𝑖,𝛽_𝑖,𝛾_𝑖] (𝑖=1,2) is the ordered parameter group. The presented inequalities can be used as a handy tool in the qualitative as well as quantitative analysis of solutions of certain fractional differential equation and integral equation. Furthermore, an application of our result to certain fractional integral equation with the Riemann-Liouville (R-L) fractional operator is also involved.

2. Nonlinear Gronwall-Bellman-Gamidov Inequalities

Throughout this paper, 𝑅 denotes the set of real numbers, 𝑅₊=[0,+∞), 𝑅₀=(0,+∞), and 𝐼=[0,𝑇] (𝑇≥0 is a constant). 𝐶(𝑋,𝑌) denotes the collection of continuous functions from 𝑋 to 𝑌.

We firstly give some lemmas, which will be used in the proof of the main results.

Lemma 2 (see [10]). Let 𝑎≥0, 𝑚≥𝑛≥0, and 𝑚≠0. Then 𝑎^𝑛/𝑚≤𝑛𝑚𝐾^{(𝑛−𝑚)/𝑚}𝑎+𝑚−𝑛𝑚𝐾^𝑛/𝑚 (7) for any 𝐾>0.

Lemma 3. Suppose 𝑢(𝑡), 𝑚(𝑡), 𝑛(𝑡), and 𝑙(𝑡)∈𝐶(𝐼,𝑅₊). If 𝑢(𝑡)≤𝑚(𝑡)+𝑛(𝑡)^𝑇₀𝑙(𝑠)𝑢(𝑠)𝑑𝑠, (8) then 𝑢(𝑡)(𝑡)+𝑛(𝑡)∫≤𝑚^𝑇0𝑚(𝑠)𝑙(𝑠)𝑑𝑠1−∫^𝑇₀𝑛(𝑠)𝑙(𝑠),𝑑𝑠 (9) for 𝑡∈𝐼, provided that ∫^𝑇₀𝑛(𝑠)𝑙(𝑠)𝑑𝑠<1.

Proof. Let 𝑘=∫^𝑇₀𝑙(𝑠)𝑢(𝑠)𝑑𝑠. Obviously, 𝑘 is a constant. It follows from (8) that 𝑢(𝑡)≤𝑚(𝑡)+𝑛(𝑡)𝑘, (10) which yields 𝑙(𝑡)𝑢(𝑡)≤𝑚(𝑡)𝑙(𝑡)+𝑛(𝑡)𝑙(𝑡)𝑘. (11)
Integrating (11) with respect to 𝑡 from 0 to 𝑇, we have 𝑘=^𝑇₀𝑙(𝑠)𝑢(𝑠)𝑑𝑠≤^𝑇₀𝑚(𝑠)𝑙(𝑠)𝑑𝑠+𝑘^𝑇₀𝑛(𝑠)𝑙(𝑠)𝑑𝑠. (12)
It is easy to observe that ∫𝑘≤^𝑇0𝑚(𝑠)𝑙(𝑠)𝑑𝑠1−∫^𝑇₀𝑛(𝑠)𝑙(𝑠),𝑑𝑠 (13)
provided that ∫^𝑇₀𝑛(𝑠)𝑙(𝑠)𝑑𝑠<1. Substituting the inequality above into (10), we get (9).

Lemma 4. Suppose 𝑢(𝑡), 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑓(𝑡), and 𝑔(𝑡)∈𝐶(𝐼,𝑅₊). If 𝑎(𝑡), 𝑏(𝑡), and 𝑐(𝑡) are nondecreasing and 𝑢(𝑡) satisfies 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝑓(𝑠)𝑢(𝑠)𝑑𝑠+𝑐(𝑡)^𝑇₀𝑔(𝑠)𝑢(𝑠)𝑑𝑠, (14) then 𝑢(𝑡)≤𝑎(𝑡)(𝑡)×∫+𝑐_𝑇0𝑎(𝑠)𝑔(𝑠)exp𝑏(𝑠)∫^𝑠₀𝑓(𝜎)𝑑𝜎𝑑𝑠1−∫^𝑇₀𝑐(𝑠)𝑔(𝑠)exp𝑏(𝑠)∫^𝑠₀𝑓(𝜎)𝑏(𝑡)𝑑𝜎𝑑𝑠×exp^𝑡₀𝑓(𝜎)𝑑𝜎, (15) for 𝑡∈𝐼, provided that ^𝑇₀𝑐(𝑠)𝑔(𝑠)exp𝑏(𝑠)^𝑠₀𝑓(𝜎)𝑑𝜎𝑑𝑠<1. (16)

Proof. Fix any 𝑇^∗, 0<𝑇^∗<𝑇; then for 0≤𝑡≤𝑇^∗, we have 𝑢(𝑡)≤𝑎𝑇^∗+𝑏𝑇^∗𝑡₀𝑓(𝑠)𝑢(𝑠)𝑇𝑑𝑠+𝑐^∗𝑇₀𝑔(𝑠)𝑢(𝑠)𝑑𝑠, (17) since 𝑎(𝑡), 𝑏(𝑡), and 𝑐(𝑡) are nondecreasing. Define 𝑧(𝑡) by the right side of (17); then 𝑢(𝑡)≤𝑧(𝑡), 𝑧(0)=𝑎𝑇^∗+𝑐𝑇^∗𝑇₀𝑔(𝑠)𝑢(𝑠)𝑑𝑠∶=𝒵0,𝑇^∗,𝑧^′(𝑡)=𝑏𝑇^∗𝑓(𝑡)𝑢(𝑡)≤𝑏𝑇^∗𝑓(𝑡)𝑧(𝑡) (18)(19) for 0≤𝑡≤𝑇^∗. From (19), we have 𝑧^′(𝑡)𝑧(𝑡)≤𝑏𝑇^∗𝑓(𝑡). (20)
Letting 𝑡=𝜎 in (20) and integrating it with respect to 𝜎 from 0 to 𝑇^∗, we get 𝑧𝑇^∗≤𝑧(0)exp𝑏𝑇^∗𝑇∗₀𝑓(𝜎)𝑑𝜎=𝒵0,𝑇^∗exp𝑏𝑇^∗𝑇∗₀𝑓(𝜎)𝑑𝜎. (21)
Since 𝑇^∗ is arbitrary, from (21) with 𝑇^∗ replaced by 𝑡 and 𝑢(𝑡)≤𝑧(𝑡), we have 𝑢(𝑡)≤𝑧(𝑡)≤𝑧(0)exp𝑏(𝑡)^𝑡₀𝑓(𝜎)()𝑏(𝑡)𝑑𝜎=𝒵0,𝑡exp^𝑡₀𝑓(𝜎)𝑑𝜎, (22) where 𝑧(0)=𝒵(0,𝑡)=𝑎(𝑡)+𝑐(𝑡)^𝑇₀𝑔(𝑠)𝑢(𝑠)𝑑𝑠. (23)
According to (22), it follows from (23) that 𝒵(0,𝑡)=𝑎(𝑡)+𝑐(𝑡)^𝑇₀𝑔(𝑠)𝑢(𝑠)(𝑡)(𝑡)𝑑𝑠≤𝑎+𝑐^𝑇₀𝑔(𝑠)𝒵()𝑏(𝑠)0,𝑠×exp^𝑠₀𝑓(𝜎)𝑑𝜎𝑑𝑠. (24)
Applying Lemma 3, we have 𝒵()(𝑡)(𝑡)∫0,𝑡≤𝑎+𝑐^𝑇0𝑎(𝑠)𝑔(𝑠)exp𝑏(𝑠)∫^𝑠₀𝑓(𝜎)𝑑𝜎𝑑𝑠1−∫^𝑇₀𝑐(𝑠)𝑔(𝑠)exp𝑏(𝑠)∫^𝑠₀𝑓(𝜎).𝑑𝜎𝑑𝑠 (25)
Substituting the inequality above into (20), we can get (13). The proof is complete.

Remark 5. Even if 𝑎(𝑡), 𝑏(𝑡), and 𝑐(𝑡) are not nondecreasing, the result also holds, since we can replace it by 𝑎(𝑡)=max_{0≤𝑠≤𝑡}𝑎(𝑠), 𝑏(𝑡)=max_{0≤𝑠≤𝑡}𝑏(𝑠), and 𝑐(𝑡)=max_{0≤𝑠≤𝑡}𝑐(𝑠).

Remark 6. Pachpatte [24] also discussed inequality (14) and derived a slightly complicated bound, but the formula of bound of 𝑢(𝑡) in our lemma is quite simple and can be extended easily.

Theorem 7. Let 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑓(𝑡), and 𝑔(𝑡) be defined as in Lemma 4. Suppose that 𝑢(𝑡)∈𝐶(𝐼,𝑅₊) satisfies (5). If ^𝑇₀𝓒(𝑠)𝓖(𝑠)exp𝓑(𝑠)^𝑠₀𝓕(𝜎)𝑑𝜎𝑑𝑠<1, (26) then 𝑢(𝑡)≤𝑎(𝑡)+𝓐(𝑡)(𝑡)×+𝓒_𝑇0𝓐(𝑠)𝓖(𝑠)×exp𝓑(𝑠)^𝑠₀𝓕(𝜎)×𝑑𝜎𝑑𝑠1−^𝑇₀𝓒(𝑠)𝓖(𝑠)×exp𝓑(𝑠)^𝑠₀𝓕(𝜎)𝑑𝜎𝑑𝑠⁻¹×exp𝓑(𝑡)^𝑡₀𝓕(𝜎)𝑑𝜎^1/𝑚, (27)
for 𝑡∈𝐼, where 𝑚≥𝑛≥0, 𝑚≥𝑟≥0, 𝑚, 𝑛, and 𝑟 are constants, and 𝓐(𝑡)=𝑏(𝑡)^𝑡₀𝑓(𝑠𝑛)𝑚𝐾₁^{(𝑛−𝑚)/𝑚}(𝑠)𝑎(𝑠)+𝑚−𝑛𝑚𝐾₁^𝑛/𝑚(𝑠(𝑡))𝑑𝑠+𝑐^𝑇₀𝑔(𝑠𝑟)𝑚𝐾₂^{(𝑟−𝑚)/𝑚}(𝑠)𝑎(𝑠)+𝑚−𝑟𝑚𝐾₂^𝑟/𝑚(𝑠𝓑(𝑡)=𝑛)𝑑𝑠,𝑚𝑏(𝑡)(𝑡)=𝑟,𝓒𝑚𝑐(𝑡),𝓕(𝑡)=𝑓(𝑡)𝐾₁^{(𝑛−𝑚)/𝑚}(𝑡),𝓖(𝑡)=𝑔(𝑡)𝐾₂^{(𝑟−𝑚)/𝑚}(𝑡), (28) for any 𝐾_𝑖(𝑡)∈𝐶(𝐼,𝑅₀) (𝑖=1,2).

Proof. Letting 𝑣(𝑡)=𝑏(𝑡)^𝑡₀𝑓(𝑠)𝑢^𝑛(𝑠)𝑑𝑠+𝑐(𝑡)^𝑇₀𝑔(𝑠)𝑢^𝑟(𝑠)𝑑𝑠, (29) from (5), we have 𝑢^𝑚(𝑡)≤𝑎(𝑡)+𝑣(𝑡), (30) or 𝑢(𝑡)≤(𝑎(𝑡)+𝑣(𝑡))^1/𝑚. (31)
Applying Lemma 2, for any 𝐾_𝑖(𝑡)∈𝐶(𝐼,𝑅₀) (𝑖=1,2), we get 𝑢^𝑛(𝑡)≤(𝑎(𝑡)+𝑣(𝑡))^𝑛/𝑚≤𝑛𝑚𝐾₁^{(𝑛−𝑚)/𝑚}(𝑡)(𝑎(𝑡)+𝑣(𝑡))+𝑚−𝑛𝑚𝐾₁^𝑛/𝑚(𝑡),𝑢^𝑟(𝑡)≤(𝑎(𝑡)+𝑣(𝑡))^𝑟/𝑚≤𝑟𝑚𝐾₂^{(𝑟−𝑚)/𝑚}(𝑡)(𝑎(𝑡)+𝑣(𝑡))+𝑚−𝑟𝑚𝐾₂^𝑟/𝑚(𝑡). (32)Substituting (32) into (29), we get 𝑣(𝑡)≤𝑏(𝑡)^𝑡₀𝑓(𝑠𝑛)𝑚𝐾₁^{(𝑛−𝑚)/𝑚}(𝑠𝑎(𝑠)(𝑠+)(+𝑣))𝑚−𝑛𝑚𝐾₁^𝑛/𝑚(𝑠(𝑡))𝑑𝑠+𝑐^𝑇₀𝑔(𝑠𝑟)𝑚𝐾₂^{(𝑟−𝑚)/𝑚}(𝑠𝑎(𝑠)(𝑠+)(+𝑣))𝑚−𝑟𝑚𝐾₂^𝑟/𝑚(𝑠(𝑡))𝑑𝑠=𝑏^𝑡₀𝑓(𝑠𝑛)𝑚𝐾₁^{(𝑛−𝑚)/𝑚}(𝑠)𝑎(𝑠)+𝑚−𝑛𝑚𝐾₁^𝑛/𝑚(𝑠(𝑡))𝑑𝑠+𝑐^𝑇₀𝑔(𝑠𝑟)𝑚𝐾₂^{(𝑟−𝑚)/𝑚}(𝑠)𝑎(𝑠)+𝑚−𝑟𝑚𝐾₂^𝑟/𝑚(𝑠+𝑛)𝑑𝑠𝑚𝑏(𝑡)^𝑡₀𝑓(𝑠)𝐾₁^{(𝑛−𝑚)/𝑚}(𝑠)𝑣(𝑠)+𝑟𝑑𝑠𝑚𝑐(𝑡)^𝑇₀𝑔(𝑠)𝐾₂^{(𝑟−𝑚)/𝑚}(𝑠)𝑣(𝑠)(𝑡)(𝑡)𝑑𝑠=𝓐+𝓑^𝑡₀𝓕(𝑠)𝑣(𝑠)(𝑡)𝑑𝑠+𝓒^𝑇₀𝓖(𝑠)𝑣(𝑠)𝑑𝑠, (33) which is similar to (14), where 𝓐(𝑡), 𝓑(𝑡), 𝓒(𝑡), 𝓕(𝑡), and 𝓖(𝑡) are defined as in (28). Clearly, 𝓐(𝑡), 𝓑(𝑡), 𝓒(𝑡), 𝓕(𝑡), 𝓖(𝑡)∈𝐶(𝐼,𝑅₊) and 𝓐(𝑡), 𝓑(𝑡), 𝓒(𝑡) are nondecreasing since 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡) are nondecreasing, respectively.
Applying Lemma 4 to (33), we have 𝑣(𝑡)≤𝓐(𝑡)(𝑡)×∫+𝓒_𝑇0𝓐(𝑠)𝓖(𝑠)exp𝓑(𝑠)∫^𝑠₀𝓕(𝜎)𝑑𝜎𝑑𝑠1−∫^𝑇₀𝓒(𝑠)𝓖(𝑠)exp𝓑(𝑠)∫^𝑠₀𝓕(𝜎)𝓑(𝑡)𝑑𝜎𝑑𝑠×exp^𝑡₀𝓕(𝜎)𝑑𝜎. (34) From (31) and (34), we get (27).

When 𝑚=2, 𝑛=𝑟=1 in Theorem 7, a Gronwall-Bellman-Pachpatte-Ou-Iang type inequality is obtained as follows.

Corollary 8. Let 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑓(𝑡), and 𝑔(𝑡) be defined as in Lemma 4. Suppose that 𝑢(𝑡)∈𝐶(𝐼,𝑅₊) satisfies 𝑢²(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝑓(𝑠)𝑢(𝑠)𝑑𝑠+𝑐(𝑡)^𝑇₀𝑔(𝑠)𝑢(𝑠)𝑑𝑠. (35) If ^𝑇₀12𝑐(𝑠)𝐺(𝑠)1exp2𝑏(𝑠)^𝑠₀𝐹(𝜎)𝑑𝜎𝑑𝑠<1, (36) then 𝑢(𝑡)≤𝑎(𝑡)+𝐴(𝑡)+12𝑐(𝑡)×^𝑇₀𝐴(𝑠)𝐺(𝑠)1×exp2𝑏(𝑠)^𝑠₀𝐹(𝜎)×𝑑𝜎𝑑𝑠1−^𝑇₀12𝑐(𝑠)𝐺(𝑠)1×exp2𝑏(𝑠)^𝑠₀𝐹(𝜎)𝑑𝜎𝑑𝑠⁻¹1×exp2𝑏(𝑡)^𝑡₀𝐹(𝜎)𝑑𝜎^1/2, (37) for 𝑡∈𝐼, where 𝐴(𝑡)=𝑏(𝑡)^𝑡₀12𝑓(𝑠)𝐾₁^−1/2(𝑠)𝑎(𝑠)+𝐾₁^1/2(𝑠(𝑡))𝑑𝑠+𝑐^𝑇₀12𝑔(𝑠)𝐾₂^−1/2(𝑠)𝑎(𝑠)+𝐾₂^1/2(𝑠𝐹(𝑡)(𝑡)𝐾)𝑑𝑠,=𝑓₁^−1/2(𝑡),𝐺(𝑡)=𝑔(𝑡)𝐾₂^−1/2(𝑡), (38) for any 𝐾_𝑖(𝑡)∈𝐶(𝐼,𝑅₀) (𝑖=1,2).

When 𝑚=𝑛=1, 0≤𝑟≤1 in Theorem 7, we can also get an interesting result as follows.

Corollary 9. Let 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑓(𝑡), and 𝑔(𝑡) be defined as in Lemma 4. Suppose that 𝑢(𝑡)∈𝐶(𝐼,𝑅₊) satisfies 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝑓(𝑠)𝑢(𝑠)𝑑𝑠+𝑐(𝑡)^𝑇₀𝑔(𝑠)𝑢^𝑟(𝑠)𝑑𝑠. (39) If 𝑟𝑐(^𝑇₀𝑠)𝐺(𝑠)exp𝑏(𝑠)^𝑠₀𝑓(𝜎)𝑑𝜎𝑑𝑠<1, (40) then 𝑢(𝑡)≤𝑎(𝑡)+𝐴(𝑡)(𝑡)×+𝑟𝑐_𝑇0𝐴(𝑠)𝐺(𝑠)exp𝑏(𝑠)^𝑠₀𝑓(𝜎)×𝑑𝜎𝑑𝑠1−^𝑇₀𝑟𝑐(𝑠)𝐺(𝑠)×exp𝑏(𝑠)^𝑠₀𝑓(𝜎)𝑑𝜎𝑑𝑠⁻¹×exp𝑏(𝑡)^𝑡₀𝑓(𝜎)𝑑𝜎, (41) for 𝑡∈𝐼, where 0≤𝑟≤1 and 𝐴(𝑡)=𝑏(𝑡)^𝑡₀𝑓(𝑠)𝑎(𝑠)(𝑡)𝑑𝑠+𝑐^𝑇₀𝑔(𝑠)𝑟𝐾^𝑟−1(𝑠)𝑎(𝑠)+(1−𝑟)𝐾^𝑟(𝑠)𝑑𝑠,𝐺(𝑡)=𝑔(𝑡)𝐾^𝑟−1(𝑡), (42) for any 𝐾(𝑡)∈𝐶(𝐼,𝑅₀).

3. Nonlinear Weakly Singular Integral Inequalities

Lemma 10 (discrete Jensen inequality). Let 𝐴₁,𝐴₂,…,𝐴_𝑛 be nonnegative real numbers and 𝑟>1 a real number. Then (𝐴₁+𝐴₂+⋯+𝐴_𝑛)^𝑟≤𝑛^𝑟−1𝐴^𝑟₁+𝐴^𝑟₂+⋯+𝐴^𝑟_𝑛. (43)

Lemma 11 (see [16]). Let 𝛼, 𝛽, 𝛾, and 𝑝 be positive constants. Then ^𝑡₀𝑡^𝛼−𝑠^{𝛼𝑝(𝛽−1)}𝑠^{𝑝(𝛾−1)}=𝑡𝑑𝑠_𝜃𝛼𝐵𝑝𝛾−1+1𝛼,𝑝𝛽−1+1,𝑡∈𝑅₊, (44) where 𝐵[𝜉,𝜂]=∫¹₀𝑠^𝜉−1(1−𝑠)^𝜂−1𝑑𝑠 (Re𝜉>0, Re𝜂>0) is the well-known 𝐵-function and 𝜃=𝑝[𝛼(𝛽−1)+𝛾−1]+1.

Assume that(𝐻)for the parameter group [𝛼_𝑖,𝛽_𝑖,𝛾_𝑖], 𝛼_𝑖∈(0,1], 𝛽_𝑖∈(0,1) and 𝛾_𝑖>1−1/𝑝 such that 1/𝑝+𝛼_𝑖(𝛽_𝑖−1)+𝛾_𝑖−1≥0, (𝑝>1, 𝑖=1,2).

Theorem 12. Under assumption (𝐻), let 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑓(𝑡), and 𝑔(𝑡) be defined as in Lemma 4. Suppose that 𝑢(𝑡)∈𝐶(𝐼,𝑅₊) satisfies (6). If ^𝑇₀𝒞(𝑠)𝒢(𝑠)expℬ(𝑠)^𝑠₀ℱ(𝜎)𝑑𝜎𝑑𝑠<1, (45) then 𝑤(𝑡)≤𝑎(𝑡)+𝒜(𝑡)(𝑡)×+𝒞_𝑡0𝒜(𝑠)𝒢(𝑠)expℬ(𝑠)^𝑠₀ℱ(𝜎)×𝑑𝜎𝑑𝑠1−^𝑡₀𝒞(𝑠)𝒢(𝑠)×expℬ(𝑠)^𝑠₀ℱ(𝜎)𝑑𝜎𝑑𝑠⁻¹×expℬ(𝑡)^𝑡₀ℱ(𝜎)𝑑𝜎^1/𝑞𝑚, (46) for 𝑡∈𝐼, where 𝑚≥𝑛≥0, 𝑚≥𝑟≥0, 𝑚, 𝑛, 𝑝, 𝑞, and 𝑟 are constants, 1/𝑝+1/𝑞=1, and 𝑀_𝑖=1𝛼_𝑖𝐵𝑝𝛾_𝑖−1+1𝛼_𝑖,𝑝𝛽_𝑖,𝜃−1+1_𝑖=𝑝𝛼_𝑖𝛽_𝑖−1+𝛾_𝑖𝑎(𝑡)−1+1,𝑖=1,2,=3^𝑞−1𝑎^𝑞(𝑡),𝑏̃(𝑡)=3^𝑞−1𝑏^𝑞(𝑡)(𝑀₁𝑡^𝜃1)^𝑞/𝑝,𝑐̃(𝑡)=3^𝑞−1𝑐^𝑞(𝑡)(𝑀₂𝑇^𝜃2)^𝑞/𝑝,𝒜(𝑡)=𝑏̃(𝑡)^𝑡₀𝑓^𝑞(𝑠)×𝑛𝑚𝐾₁^{(𝑛−𝑚)/𝑚}(𝑠)𝑎(𝑠)+𝑚−𝑛𝑚𝐾₁^𝑛/𝑚(𝑠+𝑐̃(𝑡))𝑑𝑠_𝑇0𝑔^𝑞(𝑠)×𝑟𝑚𝐾₂^{(𝑟−𝑚)/𝑚}(𝑠)𝑎(𝑠)+𝑚−𝑟𝑚𝐾₂^𝑟/𝑚(𝑠ℬ(𝑡)=𝑛)𝑑𝑠,𝑚𝑏̃(𝑡),𝒞(𝑡)=𝑟𝑚𝑐̃(𝑡),ℱ(𝑡)=𝑓^𝑞(𝑡)𝐾₁^{(𝑛−𝑚)/𝑚}(𝑡),𝒢(𝑡)=𝑔^𝑞(𝑡)𝐾₂^{(𝑟−𝑚)/𝑚}(𝑡), (47) for any 𝐾_𝑖(𝑡)∈𝐶(𝐼,𝑅₀) (𝑖=1,2).

Remark 13. When 𝑔(𝑡)=0, inequality (6) can be reduced to the case discussed by Ma and Pečarić [12]. But their result is based on the assumption that the ordered parameter group [𝛼_𝑖,𝛽_𝑖,𝛾_𝑖] (𝑖=1,2) obeys distribution I or II (for details, see [16]), which leads up to slightly complicated formula of bound on solutions.

Proof. From assumption (𝐻), using the Hölder inequality with indices 𝑝, 𝑞 to (6), we get 𝑢^𝑚(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝑡^𝛼1−𝑠^{𝛼1𝑝(𝛽1−1)}𝑠^{𝑝(𝛾1−1)}𝑑𝑠^1/𝑝×^𝑡₀𝑓^𝑞(𝑠)𝑢^𝑞𝑛(𝑠)𝑑𝑠^1/𝑞+𝑐(𝑡)^𝑇₀𝑇^𝛼2−𝑠^{𝛼2𝑝(𝛽2−1)}𝑠^{𝑝(𝛾2−1)}𝑑𝑠^1/𝑝×^𝑇₀𝑔^𝑞(𝑠)𝑢^𝑞𝑟(𝑠)𝑑𝑠^1/𝑞. (48)
Applying Lemma 10 to (48), we have 𝑢^𝑞𝑚(𝑡)≤3^𝑞−1𝑎^𝑞(𝑡)+3^𝑞−1𝑏^𝑞(𝑡)×^𝑡₀𝑡^𝛼1−𝑠^{𝛼1𝑝(𝛽1−1)}𝑠^{𝑝(𝛾1−1)}𝑑𝑠^𝑞/𝑝×^𝑡₀𝑓^𝑞(𝑠)𝑢^𝑞𝑛(𝑠)𝑑𝑠+3^𝑞−1𝑐^𝑞(𝑡)^𝑇₀𝑇^𝛼2−𝑠^{𝛼2𝑝(𝛽2−1)}𝑠^{𝑝(𝛾2−1)}𝑑𝑠^𝑞/𝑝×^𝑇₀𝑔^𝑞(𝑠)𝑢^𝑞𝑟(𝑠)𝑑𝑠=3^𝑞−1𝑎^𝑞(𝑡)+3^𝑞−1𝑏^𝑞(𝑡)𝑀₁𝑡^{𝜃1𝑞/𝑝}×^𝑡₀𝑓^𝑞(𝑠)𝑢^𝑞𝑛(𝑠)𝑑𝑠+3^𝑞−1𝑐^𝑞(𝑡)𝑀₂𝑇^{𝜃2𝑞/𝑝𝑇}₀𝑔^𝑞(𝑠)𝑢^𝑞𝑟(𝑠)𝑑𝑠, (49) where 𝑀_𝑖, 𝜃_𝑖 (𝑖=1,2) are given in (47).
Letting 𝑢^𝑞(𝑡)=𝑤(𝑡), we have 𝑤^𝑚(𝑡)≤𝑎(𝑡)+𝑏̃(𝑡)^𝑡₀𝑓^𝑞(𝑠)𝑤^𝑛(𝑠)+𝑐̃(𝑡)𝑑𝑠_𝑇0𝑔^𝑞(𝑠)𝑤^𝑟(𝑠)𝑑𝑠, (50) which is similar to inequality (5), where 𝑎(𝑡), 𝑏̃(𝑡), and 𝑐(̃𝑡) are also given in (47).
An application of Theorem 7 to the inequality above gives that 𝑤(𝑡)≤𝑎(𝑡)+𝒜(𝑡)(𝑡)×+𝒞_𝑇0𝒜(𝑠)𝒢(𝑠)×expℬ(𝑠)^𝑠₀ℱ(𝜎)×𝑑𝜎𝑑𝑠1−^𝑇₀𝒞(𝑠)𝒢(𝑠)×expℬ(𝑠)^𝑠₀ℱ(𝜎)𝑑𝜎𝑑𝑠⁻¹×expℬ(𝑡)^𝑡₀ℱ(𝜎)𝑑𝜎^1/𝑚 (51) holds for 𝑡∈𝐼, where 𝒜(𝑡), ℬ(𝑡), 𝒞(𝑡), ℱ(𝑡), and 𝒢(𝑡) are also given in (47). Since 𝑢(𝑡)=𝑤^1/𝑞(𝑡), we can get (46).

Similarly, if we take 𝑚=𝑛=1, 0≤𝑟≤1 in (5), the following result is obtained.

Corollary 14. Under assumption (𝐻), let 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑓(𝑡), and 𝑔(𝑡) be defined as in Lemma 4. Suppose that 𝑢(𝑡)∈𝐶(𝐼,𝑅₊) satisfies 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝑡^𝛼1−𝑠^{𝛼1𝛽1−1}𝑠^𝛾1−1𝑓(𝑠)𝑢(𝑠)(𝑡)𝑑𝑠+𝑐^𝑇₀𝑇^𝛼2−𝑠^{𝛼2𝛽2−1}𝑠^𝛾2−1𝑔(𝑠)𝑢^𝑟(𝑠)𝑑𝑠. (52) If ^𝑇₀𝑟𝑐̃(𝑠)𝔾(𝑠)exp𝑏̃(𝑠)^𝑠₀𝔽(𝜎)𝑑𝜎𝑑𝑠<1, (53) then 𝑢(𝑡)≤𝑎(𝑡)+𝔸(𝑡)𝑐̃(𝑡)×+𝑟_𝑇0𝔸(𝑠)𝔾(𝑠)exp𝑏̃(𝑠)^𝑠₀𝔽(𝜎)×𝑑𝜎𝑑𝑠1−^𝑇₀𝑟𝑐̃(𝑠)𝔾(𝑠)×exp𝑏̃(𝑠)^𝑠₀𝔽(𝜎)𝑑𝜎𝑑𝑠⁻¹×exp𝑏̃(𝑡)^𝑡₀𝔽(𝜎)𝑑𝜎^1/𝑞, (54) for 𝑡∈𝐼, where 0≤𝑟≤1, 𝑝 and 𝑞 are constants, 1/𝑝+1/𝑞=1, 𝑎(𝑡), 𝑏̃(𝑡), 𝑐(̃𝑡) are defined as in Theorem 12, and 𝔸(𝑡)=𝑏̃(𝑡)^𝑡₀𝑓^𝑞(𝑠)𝑎(𝑠)+𝑐̃(𝑡)𝑑𝑠_𝑇0𝑔^𝑞(𝑠)𝑟𝐾^𝑟−1(𝑠)𝑎(𝑠)+(1−𝑟)𝐾^𝑟(𝑠𝔽(𝑡))𝑑𝑠,=𝑓^𝑞(𝑡),𝔾(𝑡)=𝑔^𝑞(𝑡)𝐾^𝑟−1(𝑡), (55) for any 𝐾(𝑡)∈𝐶(𝐼,𝑅₀).

Remark 15. If we take 𝑚=2, 𝑛=𝑟=1, similar to Corollary 8, we can get a general Ou-Iang type singular inequality of (6). Here we leave the details to the reader.
When we take 𝛼₁=𝛼₂=1, 𝛾₁=𝛾₂=1 in (6), we also get the following result.

Corollary 16. Let 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑓(𝑡), and 𝑔(𝑡) be defined as in Lemma 4. Suppose that 𝑢(𝑡)∈𝐶(𝐼,𝑅₊) satisfies 𝑢^𝑚(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀(𝑡−𝑠)^𝛽1−1𝑓(𝑠)𝑢^𝑛(𝑠)(𝑡)𝑑𝑠+𝑐^𝑇₀(𝑇−𝑠)^𝛽2−1𝑔(𝑠)𝑢^𝑟(𝑠)𝑑𝑠. (56) If ^𝑇₀𝒞(𝑠)𝒢(𝑠)expℬ(𝑠)^𝑠₀ℱ(𝜎)𝑑𝜎𝑑𝑠<1, (57) then 𝑤(𝑡)≤𝑎(𝑡)+𝒜(𝑡)(𝑡)×+𝒞_𝑇0𝒜(𝑠)𝒢(𝑠)×expℬ(𝑠)^𝑠₀ℱ(𝜎)×𝑑𝜎𝑑𝑠1−^𝑇₀𝒞(𝑠)𝒢(𝑠)×expℬ(𝑠)^𝑠₀ℱ(𝜎)𝑑𝜎𝑑𝑠⁻¹×expℬ(𝑡)^𝑡₀ℱ(𝜎)𝑑𝜎^1/𝑞𝑚, (58) for 𝑡∈𝐼, where 𝑚, 𝑛,𝑟, 𝑝, 𝑞, 𝑎(𝑡), 𝑏̃(𝑡), 𝑐(̃𝑡), 𝒜(𝑡), ℬ(𝑡), 𝒞(𝑡), ℱ(𝑡), 𝒢(𝑡) are defined as in Theorem 12, 0≤𝛽_𝑖≤1 (𝑖=1,2), the choice of 𝑝 satisfies that 𝑝>1 and 1/𝑝+(𝛽_𝑖−1)>0, and 𝑀_𝑖, 𝜃_𝑖 are replaced by 𝑀_𝑖=𝐵1,𝑝𝛽_𝑖−1+1,𝜃_𝑖=𝑝𝛽_𝑖−1+1,𝑖=1,2. (59)

Remark 17. If we take 𝑚=𝑛=𝑟=1, 𝛽₁=𝛽₂=𝛽, 𝑎(𝑡)=𝐶, 𝑏(𝑡)=𝑐(𝑡)=1/Γ(𝛽), and 𝑓(𝑡)=𝑔(𝑡), inequality (6) becomes inequality (4). So, Zheng’s result [14] is the special case of our result.
Furthermore, if we take 𝑚=2, 𝑛=1, and 𝑟=1, the weakly singular case of the Gronwall-Bellman-Gamidov-Ou-Iang type inequality also can be obtained.

Corollary 18. Let 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑓(𝑡), and 𝑔(𝑡) be defined as in Lemma 4. Suppose that 𝑢(𝑡)∈𝐶(𝐼,𝑅₊) satisfies 𝑢²(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀(𝑡−𝑠)^𝛽1−1𝑓(𝑠)𝑢(𝑠)(𝑡)𝑑𝑠+𝑐^𝑇₀(𝑇−𝑠)^𝛽2−1𝑔(𝑠)𝑢(𝑠)𝑑𝑠. (60) If ^𝑇₀12𝑐̃(𝑠)𝐆(𝑠)1exp2𝑏̃(𝑠)^𝑠₀𝐅(𝜎)𝑑𝜎𝑑𝑠<1, (61) then 𝑢(𝑡)≤𝑎(𝑡)+𝐀(𝑡)+12𝑐̃(𝑡)×^𝑇₀𝐀(𝑠)𝐆(𝑠)1×exp2𝑏̃(𝑠)^𝑠₀𝐅(𝜎)×𝑑𝜎𝑑𝑠1−^𝑇₀12𝑐̃(𝑠)𝐆(𝑠)1×exp2𝑏̃(𝑠)^𝑠₀𝐅(𝜎)𝑑𝜎𝑑𝑠⁻¹1×exp2𝑏̃(𝑠)^𝑇₀𝐅(𝜎)𝑑𝜎^1/2𝑞, (62) for 𝑡∈𝐼, where 𝑎(𝑡), 𝑏̃(𝑡), and 𝑐(̃𝑡) are defined as in Theorem 12, 𝛽_𝑖, 𝑀_𝑖, 𝜃_𝑖, 𝑝, 𝑞 are defined as in Corollary 16, and 𝐀(𝑡)=𝑏̃(𝑡)^𝑡₀12𝑓^𝑞(𝑠)𝐾₁^−1/2(𝑠)𝑎(𝑠)+𝐾₁^1/2(𝑠+𝑐̃(𝑡))𝑑𝑠_𝑇012𝑔^𝑞(𝑠)𝐾₂^−1/2(𝑠)𝑎(𝑠)+𝐾₂^1/2(𝑠𝐅(𝑡))𝑑𝑠,=𝑓^𝑞(𝑡)𝐾₁^−1/2(𝑡),𝐆(𝑡)=𝑔^𝑞(𝑡)𝐾₂^−1/2(𝑡), (63) for any 𝐾_𝑖(𝑡)∈𝐶(𝐼,𝑅₀) (𝑖=1,2).

4. Applications

In this section, we give some applications of our result in the study of the boundedness of solutions of a fractional integral equation with the Riemann-Liouville (R-L) fractional operator.

Definition 19 (see [25]). The R-L fractional integral of order 𝛼 is defined by the following expression: 𝐼^𝛼𝑓(𝑡)=1Γ(1+𝛼)^𝑡₀𝑓(𝑠)(𝑑𝑠)^𝛼=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠. (64)
Consider the following fractional integral equation: 𝑢²(𝑡)=𝑎(𝑡)+𝐼^𝛼(𝐹((𝑡+1𝑡,𝑢)))Γ(𝛼)^𝑇₀(𝑇−𝑠)^𝛼−1𝐺(𝑠,𝑢(𝑠))𝑑𝑠,𝑡∈𝐼, (65) where 0<𝛼<1 and 𝐹,𝐺∈𝐶(𝑅×𝑅,𝑅).

Theorem 20. If 𝑎(𝑡)∈𝐶(𝐼,𝑅₊) is nondecreasing, |𝐹(𝑡,𝑢)|≤𝑓(𝑠)|𝑢| and 𝐺(𝑡,𝑢)≤𝑔(𝑠)|𝑢|, where 𝑓,𝑔∈𝐶(𝐼,𝑅₊). Under the condition ∫^𝑇₀(1/2)𝑐(̃𝑠)𝐆(𝑠)exp{(1/2)𝑏̃(𝑠)∫^𝑠₀𝐅(𝜎)𝑑𝜎}𝑑𝑠<1, the following estimate 𝑢(𝑡)≤𝑎(𝑡)+𝐀(𝑡)+12𝑐̃(𝑡)×^𝑇₀𝐀(𝑠)𝐆(𝑠)1×exp2𝑏̃(𝑠)^𝑠₀𝐅(𝜎)×𝑑𝜎𝑑𝑠1−^𝑇₀12𝑐̃(𝑠)𝐆(𝑠)1×exp2𝑏̃(𝑠)^𝑠₀𝐅(𝜎)𝑑𝜎𝑑𝑠⁻¹1×exp2𝑏̃(𝑠)^𝑡₀𝐅(𝜎)𝑑𝜎^1/2𝑞 (66) holds, where 𝑎(𝑡)=3^𝑞−1𝑎^𝑞(𝑡), 𝜃=𝑝(𝛼−1)+1, 𝑀=𝐵[1,𝑝(𝛼−1)+1], 𝑏̃(𝑡)=3^𝑞−1(1/Γ^𝑞(𝛼))(𝑀𝑡^𝜃)^𝑞/𝑝, and 𝑐(̃𝑡)=3^𝑞−1(1/Γ^𝑞(𝛼))(𝑀𝑇^𝜃)^𝑞/𝑝 and 𝐀(𝑡), 𝐅(𝑡), and 𝐆(𝑡) are defined as in Corollary 18.

Proof. According to Definition 19, from (65), we have 𝑢²(𝑡)(𝑡)+1=𝑎Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝐹((𝑠+1𝑠,𝑢))𝑑𝑠Γ(𝛼)^𝑇₀(𝑇−𝑠)^𝛼−1𝐺(𝑠,𝑢(𝑠))𝑑𝑠, (67) for 𝑡∈𝐼. Hence 𝑢²(𝑡(𝑡)+1)≤𝑎Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1|𝐹((𝑠+1𝑠,𝑢))|𝑑𝑠Γ(𝛼)^𝑇₀(𝑇−𝑠)^𝛼−1|𝐺((𝑠(𝑡)+1𝑠,𝑢))|𝑑𝑠≤𝑎Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠𝑢(𝑠+1)|)|𝑑𝑠Γ(𝛼)^𝑇₀(𝑇−𝑠)^𝛼−1𝑔(𝑠)|𝑢(𝑠)|𝑑𝑠. (68) Letting 𝛽₁=𝛽₂=𝛼, 𝑏(𝑡)=1/Γ(𝛼), and 𝑐(𝑡)=1/Γ(𝛼), and applying Corollary 18, we have 𝑎(𝑡)=3^𝑞−1𝑎^𝑞(𝑡),𝜃₁=𝜃₂()𝑀=𝑝𝛼−1+1∶=𝜃,₁=𝑀₂()𝑏̃(𝑡)=𝐵1,𝑝𝛼−1+1∶=𝑀,=3^𝑞−11Γ^𝑞(𝛼)𝑀𝑡^{𝜃𝑞/𝑝},𝑐̃(𝑡)=3^𝑞−11Γ^𝑞(𝛼)𝑀𝑇^{𝜃𝑞/𝑝}. (69) From (62), we get the desired estimate (66) which implies that 𝑢(𝑡) in (65) is bounded.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the referees for their helpful comments and valuable suggestions. This work is supported by the Doctoral Program Research Funds of Southwest University of Science and Technology (no. 11zx7129) and the Fundamental Research Funds for the Central Universities (no. skqy201324).