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 ).
Let and . Then
Lemma 2.2 (see ).
- (i)Assume that is nondecreasing for . If(2.1)for , then(2.2)
- (ii)Assume that is nonincreasing for . If(2.3)for , then(2.4)
Next, we establish our main results.
and is defined by (2.7).
- (i)Define a function by(2.8)
where and are defined by (2.6) and (2.7), respectively. Obviously, is nonnegative, continuous, nondecreasing in , and nonincreasing in for .
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.
and is defined by (2.22).
- (i)Define a function by(2.23)
where and are defined by (2.21) and (2.22), respectively.
Combining (2.10) and (2.27), we obtain the desired inequality (2.20). The proof is complete.
- (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)
where and are defined by (2.30) and (2.31), respectively.
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.
- (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').
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.
and is defined by (2.45).
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.
where , and is a real constant, and is a constant.
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).
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