Bounds on some new weakly singular Wendroff-type integral inequalities and applications
© Zheng; licensee Springer. 2013
Received: 21 December 2012
Accepted: 17 March 2013
Published: 8 April 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.
Keywordsintegral inequalities weakly singular Wendroff type boundedness
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 ).
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)
Lemma 2.2 (see )
where (, ) is the well-known B-function and .
The basic assumptions for inequality (2.1) are as follows:
(A1) , and such that (, );
(A2) and ;
(A3) is nondecreasing and .
Let and .
where and are given by (2.3) for .
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.
for and , where p, q, and () are defined as in Theorem 2.1.
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.
for and , where p, q, and () are defined as in Theorem 2.1.
Similar to the computation in Corollary 2.1, the estimate (2.17) holds. □
Note that the nonsingular case of the above integral inequality was studied by Pachpatte .
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.
which implies that in (3.1) is bounded.
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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