Some Nonlinear Weakly Singular Integral Inequalities with Two Variables and Applications
© Hong Wang and Kelong Zheng. 2010
Received: 23 October 2010
Accepted: 22 December 2010
Published: 29 December 2010
Some nonlinear weakly singular integral inequalities in two variables which generalize some known results are discussed. The results can be used as powerful tools in the analysis of certain classes of differential equations, integral equations, and evolution equations. An example is presented to show boundedness of solution of a differential equation here.
and the estimates of solutions are given, respectively. In , Medveď also generalized his results to an analogue of the Wendroff inequalities for functions in two variables. From then on, more attention has been paid to such inequalities with singular kernel (see [5–9]). In particular, Ma and Yang  used a modification of Medveď method to obtain pointwise explicit bounds on solutions of more general weakly singular integral inequalities of the Volterra type, and later Ma and Pečarić  used this method to study nonlinear inequalities of Henry type. Recently, Cheung et al.  applied the modified Medveď method to investigate some new weakly singular integral inequalities of Wendroff type and applications to fractional differential and integral equations.
Our results can generalize some known results and be used more effectively to study the qualitative properties of the solutions of certain partial differential and integral equations. Moreover, an example is presented to show the usefulness of our results.
2. Main Result
In what follows, denotes the set of real numbers, and . denotes the collection of continuous functions from the set to the set . and denote the first-order partial derivatives of with respect to and , respectively.
Before giving our result, we cite the following definition and lemmas.
Definition 2.1 (see ).
Let be an ordered parameter group of nonnegative real numbers. The group is said to belong to the first-class distribution and is denoted by if conditions , , and are satisfied; it is said to belong to the second-class distribution and is denoted by if conditions , and are satisfied.
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Medveď[4, Theorem 2.2]investigated the special case( , )of inequality ( 1.2 ) under the assumption that " satisfies the condition ." However, in our result, the condition is eliminated. If we take and , then we can obtain the result of linear case[4, Theorem 2.4].
Let , then or . Therefore, if we take , the formula(2.6)inis the special case of inequality(1.2),and we can obtain more concise results than(2.7)and(2.9)in. Moreover, here the condition also can be eliminated.
3. Some Corollaries
Since inequality (3.19) is similar to (2.12), we can repeat the procedure of proof in Theorem 2.4 and get (3.9).
In this section, we will apply our result to discuss the boundedness of certain partial integral equation with weakly singular kernel.
This work is supported by Scientific Research Fund of Sichuan Provincial Education Department (no. 09ZC109). The authors are very grateful to the referees for their helpful comments and valuable suggestions.
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