Theory and Modern Applications

# Some Nonlinear Integral Inequalities in Two Independent Variables

## Abstract

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.

## 1. Introduction

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 [1â€“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 [11]).

Let and . Then

Lemma 2.2 (see [7]).

Let be nonnegative and continuous functions defined for

1. (i)

Assume that is nondecreasing for . If

(2.1)

for , then

(2.2)

for .

2. (ii)

Assume that is nonincreasing for . If

(2.3)

for , then

(2.4)

for .

Next, we establish our main results.

Theorem 2.3.

Let and .

1. (i)

If

(E1)

then

(2.5)

where

(2.6)
(2.7)
2. (ii)

If

(Ex20321)

then

(2.5x2032)

where

(2.6x2032)

and is defined by (2.7).

Proof.

We only give the proof of (i). The proof of (ii) can be completed by following the proof of (i).

1. (i)

Define a function by

(2.8)

Then can be restated as

(2.9)

Using Lemma 2.1, from (2.9), we easily obtain

(2.10)
(2.11)

Combining (2.8), (2.9), and (2.11), we have

(2.12)

where and are defined by (2.6) and (2.7), respectively. Obviously, is nonnegative, continuous, nondecreasing in , and nonincreasing in for .

Firstly, we assume that for . From (2.12), we easily observe that

(2.13)

Letting

(2.14)

we easily see that is nonincreasing in , and

(2.15)

Therefore,

(2.16)

Treating , , fixed in (2.16), dividing both sides of (2.16) by , setting , and integrating the resulting inequality from 0 to , we have

(2.17)

It follows from (2.15) and (2.17) that

(2.18)

Therefore, the desired inequality (2.5) follows from (2.10) and (2.18).

If is nonnegative, we carry out the above procedure with instead of , where is an arbitrary small constant, and subsequently pass to the limit as to obtain (2.5). This completes the proof.

Theorem 2.4.

Assume that , and . Let , and

(2.19)

for , where .

1. (i)

If

(E2)

then

(2.20)

where

(2.21)
(2.22)
2. (ii)

If

(Ex20322)

then

(2.20x2032)

where

(2.21x2032)

and is defined by (2.22).

Proof.

We only prove the part (i). The proof of (ii) can be completed by following the proof of (i).

1. (i)

Define a function by

(2.23)

Then, as in the proof of Theorem 2.3, we obtain (2.9)â€“(2.11). Therefore, we have

(2.24)
(2.25)

It follows from (2.23)â€“(2.25) that

(2.26)

where and are defined by (2.21) and (2.22), respectively.

It is obvious that is nonnegative, continuous, nondecreasing in , and nonincreasing in for . By following the proof of Theorem 2.3, from (2.26), we have

(2.27)

Combining (2.10) and (2.27), we obtain the desired inequality (2.20). The proof is complete.

Theorem 2.5.

Let , and be the same as in Theorem 2.4, and .

1. (i)

Assume that is nondecreasing in , and the condition (2.19) holds. If

(E3)

then

(2.28)

where

(2.29)
(2.30)
(2.31)
2. (ii)

Assume that is nonincreasing in , and the condition (2.19) holds. If

(Ex20323)

then

(2.28x2032)

where

(2.29x2032)
(2.30x2032)
(2.31x2032)

Proof. (i) Define a function by

(2.32)

where

(2.33)

Then can be restated as

(2.34)

Noting the assumption that is nondecreasing in , we easily see that is a nonnegative and nondecreasing function in Therefore, treating , fixed in (2.34) and using part (i) of Lemma 2.2 to (2.34), we get

(2.35)

that is,

(2.36)

where is defined by (2.29). Using Lemma 2.1, from (2.36) we have

(2.37)
(2.38)

Combining (2.33) and (2.38), and noting the hypotheses (2.19), we obtain

(2.39)

where and are defined by (2.30) and (2.31), respectively.

It is obvious that is nonnegative, continuous, nondecreasing in and nonincreasing in for . By following the proof of Theorem 2.3, from (2.39), we obtain

(2.40)

Obviously, the desired inequality (2.28) follows from (2.37) and (2.40).

1. (ii)

Noting the assumption that is nonincreasing in and using the part (ii) of Lemma 2.2, we can complete the proof by following the proof of (i) with suitable changes. Therefore, the details are omitted here.

By using the ideas of the proofs of Theorems 2.5 and 2.3, we easily prove the following theorem.

Theorem 2.6.

Let , and .

1. (i)

Assume that is nondecreasing in . If

(E4)

then

(2.41)

where

(2.42)

and is defined by (2.29).

2. (ii)

Assume that is nonincreasing in . If

(Ex20324)

then

(2.40x2032)

where

(2.41x2032)

and is defined by (25').

Remark 2.7.

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.

Corollary 2.8.

Let and .

1. (i)

If

(E5)

then

(2.43)

where

(2.44)
(2.45)
2. (ii)

If

(Ex20325)

then

(2.42x2032)

where

(2.43x2032)

and is defined by (2.45).

Corollary 2.9.

Let and .

1. (i)

If

(E6)

then

(2.46)

where

(2.47)
2. (ii)

If

(Ex20326)

then

(2.45x2032)

where

(2.46x2032)

Remark 2.10.

If we add to the assumptions of [7, Theorems 2.2â€“2.4], then we easily see that [7, Theorems 2.2â€“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 [7] 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.

Example 3.1.

Consider the partial differential equation:

(3.1)

where , and is a real constant, and is a constant.

Suppose that

(3.2)

where and for , and is a constant. Let be a solution of (3.1) for ; then

(3.3)

where

(3.4)

In fact, if is a solution of (3.1), then it can be written as (see [1, page 80])

(3.5)

for .

It follows from (3.2) and (3.5) that

(3.6)

Now, a suitable application of part (ii) of Theorem 2.3 to (3.6) yields the required estimate in (3.3).

## References

1. BaÄ­nov 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.

2. Hacia L: On some integral inequalities and their applications. Journal of Mathematical Analysis and Applications 1997,206(2):611-622. 10.1006/jmaa.1997.5258

3. Li WN, Sheng W: On some nonlinear integral inequalities with an advanced argument. Communications in Mathematical Analysis 2006,1(1):12-20.

4. 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:1019600715281

5. MÃ¡tÃ© 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-X

6. 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-X

7. 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/

8. Pachpatte BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering. Volume 197. Academic Press, San Diego, Calif, USA; 1998:x+611.

9. Pachpatte BG: Explicit bounds on certain integral inequalities. Journal of Mathematical Analysis and Applications 2002,267(1):48-61. 10.1006/jmaa.2001.7743

10. 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.

11. MitrinoviÄ‡ DS: Analytic Inequalities. Springer, New York, NY, USA; 1970:xii+400.

## Acknowledgment

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).

## Author information

Authors

### Corresponding author

Correspondence to WeiNian Li.

## Rights and permissions

Reprints and permissions

Li, W. Some Nonlinear Integral Inequalities in Two Independent Variables. Adv Differ Equ 2010, 984141 (2010). https://doi.org/10.1155/2010/984141