Some Nonlinear Integral Inequalities in Two Independent Variables
© Wei Nian Li. 2010
Received: 14 January 2010
Accepted: 29 May 2010
Published: 22 June 2010
We investigate some new nonlinear integral inequalities in two independent variables. The inequalities given here can be used as tools in the qualitative theory of certain nonlinear partial differential equations.
It is well known that the integral inequalities involving functions of one and more than one independent variables which provide explicit bounds on unknown functions play a fundamental role in the development of the theory of differential equations. In the past few years, a number of integral inequalities had been established by many scholars, which are motivated by certain applications. For details, we refer to literatures [1–10] and the references therein. In this paper we investigate some new nonlinear integral inequalities in two independent variables, which can be used as tools in the qualitative theory of certain partial differential equations.
2. Main Results
In what follows, denotes the set of real numbers and is the given subset of . The first-order partial derivatives of a defined for with respect to and are denoted by , and respectively. Throughout this paper, all the functions which appear in the inequalities are assumed to be real-valued and all the integrals involved exist on the respective domains of their definitions, denotes the class of all continuous functions defined on set with range in the set , and are constants, and .
We firstly introduce two lemmas, which are useful in our main results.
Lemma 2.1 (Bernoulli's inequality ).
Lemma 2.2 (see ).
Therefore, the desired inequality (2.5) follows from (2.10) and (2.18).
Combining (2.10) and (2.27), we obtain the desired inequality (2.20). The proof is complete.
By using the ideas of the proofs of Theorems 2.5 and 2.3, we easily prove the following theorem.
Noting that and are constants, and , we can obtain many special integral inequalities by using our main results. For example, let , and , respectively; from Theorem 2.3, we obtain the following corollaries.
If we add to the assumptions of [7, Theorems 2.2–2.4], then we easily see that [7, Theorems 2.2–2.4] are special cases of Theorems 2.3, 2.5, and 2.6, respectively. Therefore, our paper gives some extensions of the results of  in a sense.
3. An Application
In this section, using Theorem 2.3, we obtain the bound on the solution of a nonlinear differential equation.
Now, a suitable application of part (ii) of Theorem 2.3 to (3.6) yields the required estimate in (3.3).
This work is supported by the National Natural Science Foundation of China (10971018), the Natural Science Foundation of Shandong Province (ZR2009AM005), China Postdoctoral Science Foundation Funded Project (20080440633), Shanghai Postdoctoral Scientific Program (09R21415200), the Project of Science and Technology of the Education Department of Shandong Province (J08LI52), and the Doctoral Foundation of Binzhou University (2006Y01).
- Baĭnov D, Simeonov P: Integral Inequalities and Applications, Mathematics and Its Applications (East European Series). Volume 57. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+245.Google Scholar
- Hacia L: On some integral inequalities and their applications. Journal of Mathematical Analysis and Applications 1997,206(2):611-622. 10.1006/jmaa.1997.5258MathSciNetView ArticleMATHGoogle Scholar
- Li WN, Sheng W: On some nonlinear integral inequalities with an advanced argument. Communications in Mathematical Analysis 2006,1(1):12-20.MathSciNetMATHGoogle Scholar
- Ma Q-H, Yang E-H: Some new Gronwall-Bellman-Bihari type integral inequalities with delay. Periodica Mathematica Hungarica 2002,44(2):225-238. 10.1023/A:1019600715281MathSciNetView ArticleMATHGoogle Scholar
- Máté A, Nevai P:Sublinear perturbations of the differential equation and of the analogous difference equation. Journal of Differential Equations 1984,53(2):234-257. 10.1016/0022-0396(84)90041-XMathSciNetView ArticleMATHGoogle Scholar
- Meng FW, Li WN: On some new integral inequalities and their applications. Applied Mathematics and Computation 2004,148(2):381-392. 10.1016/S0096-3003(02)00855-XMathSciNetView ArticleMATHGoogle Scholar
- Pachpatte BG: On some fundamental integral inequalities and their discrete analogues. Journal of Inequalities in Pure and Applied Mathematics 2001., 2, Article 15: http://jipam.vu.edu.au/Google Scholar
- Pachpatte BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering. Volume 197. Academic Press, San Diego, Calif, USA; 1998:x+611.Google Scholar
- Pachpatte BG: Explicit bounds on certain integral inequalities. Journal of Mathematical Analysis and Applications 2002,267(1):48-61. 10.1006/jmaa.2001.7743MathSciNetView ArticleMATHGoogle Scholar
- Pachpatte BG: Integral and Finite Difference Inequalities and Applications, North-Holland Mathematics Studies. Volume 205. Elsevier Science B.V., Amsterdam, The Netherlands; 2006:x+309.Google Scholar
- Mitrinović DS: Analytic Inequalities. Springer, New York, NY, USA; 1970:xii+400.View ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.