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.
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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.
is nondecreasing for 
. If
, then
.
is nonincreasing for 
. If
, then
.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).
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)
by 
Then can be restated as
Using Lemma 2.1, from (2.9), we easily obtain
Combining (2.8), (2.9), and (2.11), we have
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
Letting
we easily see that 
is nonincreasing in 
, and
Therefore,
Treating 
, 
, fixed in (2.16), dividing both sides of (2.16) by 
, setting 
, and integrating the resulting inequality from 0 to 
, we have
It follows from (2.15) and (2.17) that
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
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).
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)
by 
Then, as in the proof of Theorem 2.3, we obtain (2.9)–(2.11). Therefore, we have
It follows from (2.23)–(2.25) that
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
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)
is nondecreasing in 
, and the condition (2.19) holds. If
is nonincreasing in 
, and the condition (2.19) holds. If
Proof. (i) Define a function
by
where
Then can be restated as
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
that is,
where 
is defined by (2.29). Using Lemma 2.1, from (2.36) we have
Combining (2.33) and (2.38), and noting the hypotheses (2.19), we obtain
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
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.
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').
is nondecreasing in 
. If
is defined by (2.29).
is nonincreasing in 
. If
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).
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:
where 
, and 
is a real constant, and 
is a constant.
Suppose that
where 
and 
for 
, and 
is a constant. Let 
be a solution of (3.1) for 
; then
where
In fact, if 
is a solution of (3.1), then it can be written as (see [1, page 80])
for 
.
It follows from (3.2) and (3.5) that
Now, a suitable application of part (ii) of Theorem 2.3 to (3.6) yields the required estimate in (3.3).
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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).
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Li, W. Some Nonlinear Integral Inequalities in Two Independent Variables. Adv Differ Equ 2010, 984141 (2010). https://doi.org/10.1155/2010/984141
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DOI: https://doi.org/10.1155/2010/984141