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 [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 [11]).

Let and . Then

Lemma 2.2 (see [7]).

Let be nonnegative and continuous functions defined for

• (i) Assume that is nondecreasing for . If

  

  (2.1)

  for , then

  

  (2.2)

  for .

• (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 .

• (i) If

  

  (E1)

  then

  

  (2.5)

  where

  

  (2.6)
  

  (2.7)

• (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).

• (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 .

• (i) If

  

  (E2)

  then

  

  (2.20)

  where

  

  (2.21)
  

  (2.22)

• (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).

• (i) Define a function by

  

  (2.23)

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

(2.24)

(2.25)

It follows from (2.23)–(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 .

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

  

  (E3)

  then

  

  (2.28)

  where

  

  (2.29)
  

  (2.30)
  

  (2.31)

• (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).

• (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 .

• (i) Assume that is nondecreasing in . If

  

  (E4)

  then

  

  (2.41)

  where

  

  (2.42)

  and is defined by (2.29).

• (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 .

• (i) If

  

  (E5)

  then

  

  (2.43)

  where

  

  (2.44)
  

  (2.45)

• (ii) If

  

  (Ex20325)

  then

  

  (2.42x2032)

  where

  

  (2.43x2032)

  and is defined by (2.45).

Corollary 2.9.

Let and .

• (i) If

  

  (E6)

  then

  

  (2.46)

  where

  

  (2.47)

• (ii) If

  

  (Ex20326)

  then

  

  (2.45x2032)

  where

  

  (2.46x2032)

Remark 2.10.

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 [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).