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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 937908, 12 pages
A New Fractional Integral Inequality with Singularity and Its Application
1Department of Building Environment and Services Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China
2Department of Mathematical Sciences, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China
Received 3 March 2012; Revised 30 April 2012; Accepted 30 April 2012
Academic Editor: Bashir Ahmad
Copyright © 2012 Qiong-Xiang Kong and Xiao-Li Ding. 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.
We prove an integral inequality with singularity, which complements some known results. This inequality enables us to study the dependence of the solution on the initial condition to a fractional differential equation in the weighted space.
Integral inequalities provide an excellent tool for the properties of solutions to differential equations, such as boundedness, existence, uniqueness, and stability (e.g., see [1–10]). For this reason, the study of integral inequalities has been emphasized by many authors. For example, in 1919, Gronwall in  proved a remarkable inequality which can be described by the following.
Suppose that satisfies the relation where all the functions involved are continuous on the interval , , and . Consider The inequality has attracted and continues to attract considerable attention in the literature. In 2007, Ye et al.  reported an integral inequality with singular kernel. The inequality can be stated as follows.
If , is a nonnegative and locally integrable on , is a nonnegative, nondecreasing continuous function on , and , where , is a positive constant. Further suppose that is nonnegative and locally integrable on with Then Besides the above-mentioned inequalities, there are still many inequalities (e.g., see [13–15]).
But in the analysis of the dependence of the solution on the initial condition of a fractional differential equation in the weighted space, the bounds provided by the existing inequalities are not adequate. So it is natural and necessary to seek new inequality in order to obtain our desired results. In this paper, we present a new integral inequality, and then apply our inequality to investigate the dependence of the solution on the initial condition of a fractional differential equations in the weighted space.
2. An Integral Inequality
In this section, our main aim is to establish an integral inequality with singularity. Before proceeding, we give some useful definitions and lemmas.
Lemma 2.3 (see ). Let , . Then the quotient expansion of two gamma functions at infinity can be represented as follows:
Lemma 2.4. Let , . Then one has
Proof. By Lemma 2.3, we have , which proves that as . The proof of this lemma is completed.
Based on Lemma 2.4, we can define a function.
Definition 2.5. Let , . Then the following definition: is well defined, where is a positive constant, and .
Proof. We only need to show that the series in (2.4) is uniformly convergent for . By Lemma 2.4, we know that as . Since , as . This implies that the series in (2.4) is uniformly convergent for . It follows that the definition is well defined.
Lemma 2.6. Let , and . Then one has
Proof. Making the substitution and combining the relation (2.1), we obtain The proof of this lemma is completed.
Now we can state the integral inequality.
Theorem 2.7. Let , , , , and let be a nonnegative, nondecreasing continuous function on , , where , is a positive constant. Further suppose that is nonnegative and is locally integrable on with Then one has
Proof. For convenience, we define an operator
Then (2.7) can be rewritten in the form
Since and are nonnegative, it is easy to induce that
Let us prove that the following relation
holds for any , where , and as for each in .
Obviously, inequality (2.12) is valid for , due to . Suppose that the inequality is satisfied for any fixed . Let us verify that it is also satisfied for . We first prove the case . According to the induction hypothesis and Lemma 2.6, we have which is estimated with the help of So, for the case , inequality (2.12) is true for any . Now we prove the case . Similarly, according to the induction hypothesis and Lemma 2.6, we get which is calculated with the help of So, for the case , inequality (2.12) is true for any . Based on this analysis, we conclude that inequality (2.12) holds for any .
Next, we show that as . Now, we go back to inequality (2.12). For the case , we denote , where . Note that Since , by Lemma 2.4, we obtain as . This implies that as . It follows that as for the case . For the case , we denote , where . Note that Using the same arguments as above, we know that as . It follows that as for the case . So, it has as for the two cases and . This, together with (2.11), leads to .
Finally, we show that where , , .
Obviously, inequality (2.19) is true for . Suppose that the inequality is satisfied for any fixed . Let us verify that it is also satisfied for . According to the induction hypothesis and Lemma 2.6, we obtain This proves that inequality (2.19) is satisfied for any . In other words, we have proved that where , , . By virtue of Definition 2.5, we can arrive at inequality (2.8) and the proof of this theorem is completed.
Corollary 2.8. Let , , , . And suppose that is nonnegative and is locally integrable on () with Then one has
Corollary 2.9. Let , , be a nonnegative, nondecreasing continuous function on , , where , is a positive constant. And suppose that is nonnegative and locally integrable on with Then one has
In this section, we will apply our established result to study the dependence of the solution on the initial condition of a fractional differential equation with the Riemann-Liouville derivative.
For the reader's convenience, we first recall several definitions of the Reimann-Liouville integral and derivative. From now on, we assume that is a finite positive constant, that is, .
Now we consider the following initial value problem of the form
With regard to problem (3.3), the existence and uniqueness of the solution can be found in the book by Kilbas et al. . For the completeness of this paper, we give the existence and uniqueness of the solution to (3.3) in the weighted space . The space consists of all functions such that , which turns out to be a Banach space when endowed with the norm .
Theorem 3.3 (see ). Let , and be a function such that for any , . Further assume that for any , , the following inequality holds, where is a constant. Then there exists a unique solution to problem (3.3) in the space .
Theorem 3.4. Let , and be a function such that for any , . Further assume that for any , , the following inequality holds, where is a constant. Assume that and are the solutions of problem (3.3) and respectively. Then, for , one has where are positive constants such that
Proof. The proof is rather technical. We first prove the case and . Since and are the solutions of (3.3) and (3.6), we have
Subtracting (3.10) from (3.9) and using the Lipschitz condition (3.5), we obtain
Taking into account that , we multiply at both sides of inequality (3.11) by to get
Denote . Then, (3.12) can be written as
Putting , , , , , we see that , , , , and is a nondecreasing continuous function due to . So, applying Theorem 2.7 to (3.13), we obtain
Next, we prove the case and . Notice that the Lipschitz condition (3.5) holds for each in . Since and , we can always choose two positive constants , such that Condition (3.15) means that and . That is to say, if the Lipschitz condition (3.5) holds for each in , then we can always choose two positive constants , such that the following condition holds for each in .
Subtracting (3.10) from (3.9) and using condition (3.16), we obtain Multiplying on both sides of (3.17), we get where is defined as before. Now, putting , , , , , we see that , , , , is a nondecreasing continuous function due to . So, applying Theorem 2.7 to (3.18), we have
Finally, we prove the case and . Since and , we can always choose two positive constants , such that Condition (3.20) means that and . Using the same arguments as above, we can obtain that where is defined as before. So the conclusion of this theorem is true.
This work was supported by the National Natural Science Foundation of China (11071192) and the International Science and Technology (S&T) Cooperation Program of China (2010DFA14700).
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