Bounds on some new weakly singular Wendroff-type integral inequalities and applications
Journal of Inequalities and Applications volume 2013, Article number: 159 (2013)
With a more concise condition for the ordered parameter groups, some nonlinear weakly singular integral inequalities of Wendroff type, which generalize some existing results, are established. Furthermore, application examples in the boundedness and uniqueness of the solution of a singular partial integral equation are given.
MSC:26A33, 34A08, 34A34, 45J05.
The Gronwall-Bellman integral inequality and its various nonlinear versions have made great achievements in the qualitative analysis for the solutions of differential and integral equations, as shown in [1, 2] and . However, qualitative properties and numerical analysis for the solutions of singular integral and differential equations depend on the study of corresponding integral inequalities, and the aforementioned inequalities are not directly applicable. In recent years, many researchers have devoted much effort to investigating weakly singular integral inequalities and their applications (see [4–12]). For example, the Volterra-type singular integro-differential equation
was discussed by McKee  when for the diffusion of discrete particles in a turbulent fluid; Henry  proposed a linear integral inequality with singular kernel to investigate some qualitative properties for a parabolic differential equation; Sano and Kunimatsu  gave a modified version of the Henry-type inequality; Ye et al.  used a generalized inequality to study the dependence of the solution for a fractional differential equation. In particular, Medveď  presented a new method to discuss nonlinear singular integral inequalities of Henry type and generalized his results to analogue Wendroff inequalities for functions in two variables. A slightly improved de-singularity approach has been used by Ma and Yang  to study a more general singular integral inequality as follows:
Later Ma and Pečarić  used this method to establish a priori bounds on solutions to the following nonlinear singular integral inequalities with power nonlinearity:
Bounds on solutions to inequalities (1.2) and (1.3) are established for the cases when the ordered parameter groups and obey distribution I or II (for details, see ).
With the development of the theory of inequalities for a two-dimensional case, more attention has also been paid to weakly singular integral inequalities in two variables and their applications to the partial differential equation with singular kernel. Upon the results in  and , Cheung and Ma  investigated some new weakly singular integral inequalities of Wendroff type
and their more general nonlinear version was presented by Wang and Zheng . It is to be noted that the ordered parameter groups in  and  make the application of inequalities more inconvenient. To overcome the weakness, in this paper we avoid the above-mentioned conditions and use another concise assumption to discuss some more general integral inequalities of Wendroff type. Our estimates are quite simple and the resulting formulas are similar to the classical Gronwall-Bihari inequalities. Furthermore, to show the applications of these inequalities, some examples are presented.
2 Main result
In what follows, R denotes the set of real numbers and . denotes the collection of continuous functions from the set X to the set Y. and denote the first-order partial derivatives of with respect to x and y, respectively.
Lemma 2.1 (Discrete Jensen inequality)
Let be nonnegative real numbers and be a real number. Then
Lemma 2.2 (see )
Let α, β, γ and p be positive constants. Then
where (, ) is the well-known B-function and .
Firstly, consider the following Wendroff-type integral inequality:
The basic assumptions for inequality (2.1) are as follows:
(A1) , and such that (, );
(A2) and ;
(A3) is nondecreasing and .
Let and .
Theorem 2.1 Under assumptions (A1), (A2) and (A3), if satisfies (2.1), then
for and , where
, is the inverse of ,
and are chosen such that
Proof By the definition of and , it is easy to see that and are nonnegative and nondecreasing in x and y. Moreover, and . From (2.1), it follows that
According to the assumption (A1), we choose suitable indices p, q. Applying the Hölder inequality with indices p, q to (2.6), we get
By Lemma 2.1, from (2.7), we have
where and are given by (2.3) for .
Since and (), and are also nondecreasing in x and y. Taking any arbitrary and with , we obtain
for , . Let
Then , namely . Meanwhile, . Considering
where we apply the fact that is nondecreasing in y, we have
Integrating both sides of the above inequality from 0 to x, we obtain
for , , where is given by (2.4). By virtue of the assumption (A3), is strictly increasing. So, its inverse is continuous and increasing in its corresponding domain. Replacing x and y by and , we have
Since and are arbitrary, we replace and by x and y, respectively, and get
for and . The above inequality can be rewritten as
Therefore, by , (2.2) holds for and . □
Remark 2.1 If we take , and , respectively, weakly singular kernel in (2.1) reduces to the formula in (1.4). Especially, it should be pointed out that if , and , the ordered parameter group is just I type defined in , and if , and , it is just II type defined in .
Remark 2.2 In , Medveď has investigated the Wendroff inequality with weakly singular kernel . However, his result holds under the condition ‘, ’ and the assumption that ‘ satisfies the condition (q)’ (see Definition 2.1 in ). In our result, the condition (q) is eliminated, and the ordered parameter groups can be discussed in more cases.
Corollary 2.1 Under assumptions (A1), (A2) and (A3), let , (). If satisfies
for and , where p, q, and () are defined as in Theorem 2.1.
Proof Let , then , that is, . It follows from (2.12) that
Clearly, satisfies the assumption (A3). By the definition of by (2.4), letting , we have
Substituting (2.14) and (2.15) into (2.2), we get
In view of , we can obtain (2.13). □
Remark 2.3 Let and ; we can get the interesting Henry-Ou-Iang type singular integral inequality in two variables. As for the concrete formula, we omit it here.
Now we turn to consider the case . In fact, (2.12) can be reduced to the corresponding linear version. Hence we only need to prove the following result.
Corollary 2.2 Under assumptions (A1), (A2) and (A3), if satisfies
for and , where p, q, and () are defined as in Theorem 2.1.
Proof In (2.17), also satisfies the assumption (A3). Here, we have
Similar to the computation in Corollary 2.1, the estimate (2.17) holds. □
Next, we consider the more complicated Wendroff-type integral inequality
Note that the nonsingular case of the above integral inequality was studied by Pachpatte .
Theorem 2.2 Under the assumption (A1), suppose that and satisfy the assumption (A3). If , and satisfies (2.19), then
for and , where p, q, and () are defined as in Theorem 2.1, is defined by
and are chosen such that
Proof As in the proof of Theorem 2.1, it follows from (2.19) that
from Lemma 2.1 and Lemma 2.2, we have
We introduce , as in the proof of Theorem 2.1, the above inequality can be rewritten as
Denoting the right-hand side of (2.25) by , we have
Similar to the proof of Theorem 2.1, we obtain (2.20). The details are omitted here. □
In this section, we present some examples to show applications in the boundedness and uniqueness of a certain partial integral equation with weakly singular kernel.
Example 1 Suppose satisfies the inequality as follows:
for , . Then (3.1) is the special case of inequality (2.1), that is,
From the condition , we have . From , we have . With the assumption (A1), let , then , . Hence
Using (2.2) in Theorem 2.1, we get for and ,
which implies that in (3.1) is bounded.
Consider the following linear singular integral equations:
where with . Here, ϵ is an arbitrary positive number, and and () satisfy the assumption (A1). From (3.4) and (3.5), we get
which is the formula of inequality (2.16). Applying Corollary 2.2 to (3.6), we have
where and () are defined as in Theorem 2.1. If , letting , we obtain the uniqueness of the global continuous solution of equation (3.4).
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The author declares that they have no competing interests.
This work is supported by the Doctoral Program Research Foundation of Southwest University of Science and Technology (No. 11zx7129). The author is very grateful to both reviewers for carefully reading this paper and for their comments.
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Zheng, K. Bounds on some new weakly singular Wendroff-type integral inequalities and applications. J Inequal Appl 2013, 159 (2013). https://doi.org/10.1186/1029-242X-2013-159