Journal of Applied Mathematics

Volume 2014 (2014), Article ID 218507, 9 pages

http://dx.doi.org/10.1155/2014/218507

Research Article

A Nonlinear Weakly Singular Retarded Henry-Gronwall Type Integral Inequality and Its Application

Yuanhua Lin,¹ Shanhe Wu,² and Wu-Sheng Wang¹

¹School of Mathematics and Statistics, Hechi University, Guangxi, Yizhou 546300, China

²Department of Mathematics and Computer Science, Longyan University, Longyan, Fujian 364012, China

Received 7 March 2014; Accepted 16 August 2014; Published 26 August 2014

Academic Editor: Angelo Ciaramella

Copyright © 2014 Yuanhua Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We establish a class of new nonlinear retarded weakly singular integral inequality. Under several practical assumptions, the inequality is solved by adopting novel analysis techniques, and explicit bounds for the unknown functions are given clearly. An application of our result to the fractional differential equations with delay is shown at the end of the paper.

1. Introduction

Integral inequalities play increasingly important roles in the study of existence, uniqueness, boundedness, oscillation, stability, invariant manifolds, and other qualitative properties of solutions of ordinary differential equations and integral equations. One of the best known and widely used inequalities in the study of nonlinear differential equations is Gronwall-Bellman inequality [1, 2], which can be stated as follows. If and are nonnegative continuous functions on an interval satisfying , , then , . Many papers are devoted to different generalizations of Bellman-Gronwall inequality. Very well-known generalization of Bellman-Gronwall inequality to the nonlinear case is the Bihari inequality [3]. In 1956, Bihari [3] discussed the integral inequality

where is a constant. In recent years, many researchers have devoted much effort to investigating weakly singular integral inequalities. For example, Henry [4] proposed a linear integral inequality with singular kernel to investigate some qualitative properties for a parabolic differential equation, and Sano and Kunimatsu [5] gave a modified version of Henry type inequality. However, such results are expressed by a complicated power series which are sometimes inconvenient for their applications. To avoid the shortcomings of these results, Medved’ [6] presented a new method to discuss nonlinear singular integral inequalities of Henry type and their Bihari version is as follows:

and the estimates of solutions are given. From then on, more attention has been paid to such inequalities with singular kernel; see [7–24] and the references cited therein. Ye and Gao [20] considered the integral inequality of Henry-Gronwall type with delay

and Henry-Gronwall type retarded integral inequality with singular kernel

In this paper, motivated by [6, 20], we discuss the nonlinear integral inequality of Henry-Gronwall type with delay

and Henry-Gronwall type nonlinear retarded integral inequality with singular kernel

2. Main Results

Throughout this paper, denotes the set of real numbers, . For convenience, before giving our main results, we cite some useful lemmas and definitions in the discussion of our proof as follows.

Definition 1 (see [6]). Let be a real number and . We say that a function satisfies a condition (), if

where is a continuous, nonnegative function.

Lemma 2 (discrete Jensen inequality [25]). Let be nonnegative real numbers, is real numbers, and is a natural number. Then

Lemma 3 (see [6]). (1) Let ; then

where is the gamma function.

(2) Let ; then

Proof. (1) Using a change of variables and successively, we have the estimate

Since , and .

(2) Using a change of variables and successively, we have the estimate

Since , , , and .

Theorem 4. Suppose that are nonnegative continuous functions on , is a nonnegative continuous function on , , and , , are constants. Suppose that the function satisfies the following conditions:

(1)) condition, that is, satisfies inequality (7);(2)subadditivity, that is, for all , .

If satisfies (5), then

where

Proof. Define a function by the right side of (5), that is,

Then , , and is a nonnegative, nondecreasing, and continuous function with , .

For , by the subadditivity satisfied by , we conclude

Letting in (18) and integrating both sides of inequality (18) from to , we obtain

where is chosen arbitrarily.

Define a function by the right side of (19), that is,

Then, the function is a nonnegative, nondecreasing, and continuous function with

Differentiating , we have

From (22), we obtain

Using (21), from (23) we obtain

where are defined by (14) and (15), respectively. From (24), we observe

Let in (25); we have

Since is chosen arbitrarily, from (26), we have the estimation

For , using the subadditivity of and monotony of , from (17) we have

Letting in (28) and integrating both sides of inequality (28) from to and using (27) we obtain

where , is seen as a constant, and is defined by (16).

Define a function by the right side of (29), that is,

Obviously, is a nonnegative, nondecreasing, and continuous function with

Differentiating , we have

From (33), we have

Using (31), from (34), we have

It follows that

In (36), let , and then we have

Since is chosen arbitrarily, from (32) and (37), we obtain the estimation

Noting that , from (27) and (38), we obtain our required estimations (13).

Remark 5. When . The estimations (13) in Theorem 4 are reduced to the corresponding estimations in [20].

Theorem 6. Suppose that satisfy the corresponding conditions in Theorem 4; is a constant. If satisfies (6), then the following assertions hold.

(1) Suppose . Then

where is defined by (14) in Theorem 4,

and is defined in (7) in Definition 1.

(2) Suppose that . Then

where

and .

Proof. First we will prove assertion (1). Suppose that . Using Cauchy-Schwarz inequality, we obtain from (6) that

Since satisfies () condition, using (7) in Definition 1 and (9) in Lemma 3, from (53) we derive that

for all . Using discrete Jensen inequality (8) with , , from (54) we obtain

Let and . From (55) we have

We observe that

is defined by (45). By the definitions of , and in (42), (43), and (44), from (56) we see

We observe that (58) have the same form as (5) and satisfy the corresponding conditions in Theorem 4. Applying Theorem 4 to (58), we obtain our required estimations (39).

(2) Now let us prove assertion (2). Suppose . Let ; then . Using Hölder inequality, from (6) we obtain

Since satisfies () condition, using (7) and (10), from (59) we derive

for all . Using Jensen inequality (8), from (60) we have

Let and . Then, we obtain from (61) that

We observe that

where is defined by (52). Using definitions of and in (49), (50), and (51), from (62) we have

We observe that (64) have the same form as (5) and satisfy the corresponding conditions in Theorem 4. Applying Theorem 4 to (64), we obtain our required estimations (46).

3. Application to Fractional Differential Equations (FDEs) with Delay

In this section, we apply our result to the following fractional differential equations (FDEs) with delay (see [20]):

where represents the Caputo fractional derivative of order , , and is as in Theorem 6.

Theorem 7. Suppose that

where are as in Theorem 6. Let . If is any solution of IVP (65), then the following estimates hold.

(1) Suppose . Then

where is defined by (14) in Theorem 4,

and are defined by (7) and (45), respectively.

(2) Suppose that . Then

where

and ; is defined by (52).

Proof. The solution of FDEs (65) can be written as (see [24])

When , from (71) we obtain

Applying Theorem 6 to (72), we obtain our required estimations (67) and (69).

Remark 8. When . Let ; we can obtain the estimations similar to (67) in Theorem 7.

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (no. 11161018), the Guangxi Natural Science Foundation (no. 2012GXNSFAA053009), the Scientific Research Foundation of the Education Department of Guangxi Province (no. LX2014330), and the Foundation of Scientific Research Project of Fujian Province Education Department of China (no. JK2012049). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

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