Some Nonlinear Weakly Singular Integral Inequalities with Two Variables and Applications

Abstract

Some nonlinear weakly singular integral inequalities in two variables which generalize some known results are discussed. The results can be used as powerful tools in the analysis of certain classes of differential equations, integral equations, and evolution equations. An example is presented to show boundedness of solution of a differential equation here.

1. Introduction

Various singular integral inequalities play an important role in the development of the theory of differential equations, functional differential equations, and integral equations. For example, Henry [1] proposed a linear integral inequality with singular kernel to investigate some qualitative properties for a parabolic differential equation, and Sano and Kunimatsu [2] gave a modified version of Henry type inequality. However, such results are expressed by a complicated power series which are sometimes inconvenient for their applications. To avoid the shortcoming of these results, Medveď [3] presented a new method to discuss nonlinear singular integral inequalities of Henry type and their Bihari version as follows:

(1.1)

and the estimates of solutions are given, respectively. In [4], Medveď also generalized his results to an analogue of the Wendroff inequalities for functions in two variables. From then on, more attention has been paid to such inequalities with singular kernel (see [5–9]). In particular, Ma and Yang [8] used a modification of Medveď method to obtain pointwise explicit bounds on solutions of more general weakly singular integral inequalities of the Volterra type, and later Ma and Pečarić [9] used this method to study nonlinear inequalities of Henry type. Recently, Cheung et al. [10] applied the modified Medveď method to investigate some new weakly singular integral inequalities of Wendroff type and applications to fractional differential and integral equations.

In this paper, motivated mainly by the work of Ma et al. [8, 9] and Cheung et al. [10], we discuss more general form of nonlinear weakly singular integral inequality of Wendroff type for functions in two variables

(1.2)

Our results can generalize some known results and be used more effectively to study the qualitative properties of the solutions of certain partial differential and integral equations. Moreover, an example is presented to show the usefulness of our results.

2. Main Result

In what follows, denotes the set of real numbers, and . denotes the collection of continuous functions from the set to the set . and denote the first-order partial derivatives of with respect to and , respectively.

Before giving our result, we cite the following definition and lemmas.

Definition 2.1 (see [8]).

Let be an ordered parameter group of nonnegative real numbers. The group is said to belong to the first-class distribution and is denoted by if conditions , , and are satisfied; it is said to belong to the second-class distribution and is denoted by if conditions , and are satisfied.

Lemma 2.2 (see [8]).

Let α, , , and be positive constants. Then,

(2.1)

where () is well-known -function and .

Lemma 2.3 (see [8]).

Suppose that the positive constants α, , , , and satisfy the following conditions:

(1)if , ;

(2)if , .

Then, for ,

(2.2)

are valid.

Assume that

(A₁) and ;

(A₂) is nondecreasing and .

Let and .

Theorem 2.4.

Under assumptions (A₁) and (A₂), if satisfies (1.2), then

• (1) for ,

  

  (2.3)

for and , where

(2.4)

is the inverse of ,

(2.5)

and are chosen such that

(2.6)

• (2) for ,

  

  (2.7)

for and , where

(2.8)

is the inverse of ,

(2.9)

and are chosen such that

(2.10)

Proof.

With the definition of and , clearly, and are nonnegative and nondecreasing in and . Furthermore, and . From (1.2), we have

(2.11)

Next, for convenience, we introduce indices , . Denote that if , then let and ; if , then let and . Then holds for .

Using the Hölder inequality with indices , to (2.11), we get

(2.12)

By

(2.13)

from (2.12) and Lemma 2.2, we have

(2.14)

where

(2.15)

and is given in Lemma 2.3 for .

Since and (), then and are also nondecreasing in and . Taking any arbitrary and with , , we obtain

(2.16)

for , . Denote

(2.17)

and let

(2.18)

Then, or . Meanwhile, , and is nondecreasing in and . Considering

(2.19)

we have

(2.20)

where we apply the fact that is nondecreasing in . Integrating both sides of the above inequality from 0 to , we obtain

(2.21)

for , , where

(2.22)

From assumption (A₂), is strictly increasing so its inverse is continuous and increasing in its corresponding domain. Replacing and by and , we have

(2.23)

Since and are arbitrary, we replace and by and , respectively, and get

(2.24)

for and . The above inequality can be rewritten as

(2.25)

Therefore, we have

(2.26)

for and .

Finally, considering two situations for and using parameters α, β, to denote , , , and in the above inequality, we can obtain the estimations, respectively. we omit the details here.

Remark 2.5.

Medveď[4, Theorem 2.2]investigated the special case(, )of inequality ( 1.2 ) under the assumption that "satisfies the condition." However, in our result, thecondition is eliminated. If  we takeand, then we can obtain the result of linear case[4, Theorem 2.4].

Remark 2.6.

Let, thenor. Therefore, if we take, the formula(2.6)in[10]is the special case of inequality(1.2),and we can obtain more concise results than(2.7)and(2.9)in[10]. Moreover, here the conditionalso can be eliminated.

Remark 2.7.

When does not belong to or , there are some technical problems which we do not discuss here.

3. Some Corollaries

Corollary 3.1.

Let functions , , be defined as in Theorem 2.4, and let be a constant with . Suppose that

(3.1)

Then,

• (1) for ,

if ,

(3.2)

if ,

(3.3)

for , , where , , are defined as in Theorem 2.4,

• (2) for ,

if ,

(3.4)

if ,

(3.5)

for , , where , , are defined as in Theorem 2.4.

Proof.

Clearly, inequality (3.1) is the special case of (1.2). Taking , we can get (3.1).

(i)If ,

(3.6)

(ii)If ,

(3.7)

Therefore, the positive numbers and in (2.6) and (2.10) can be taken as , and the results can be obtained by simple computation. We omit the details.

Corollary 3.2.

Let functions , , be defined as in Theorem 2.4. Suppose that and satisfies

(3.8)

Then,

• (i) if ,

  

  (3.9)

for and , where

(3.10)

, , , are defined as in Theorem 2.4, and are chosen such that

(3.11)

• (ii) if ,

  

  (3.12)

for and , where

(3.13)

, , , are defined as in Theorem 2.4, and are chosen such that

(3.14)

Proof.

By the two mentioned lemmas, it follows from (3.8) that

(3.15)

where and

(3.16)

(i)For ,

applying Corollary 3.1 to (3.15), we have

(3.17)

Letting

(3.18)

we get

(3.19)

Since inequality (3.19) is similar to (2.12), we can repeat the procedure of proof in Theorem 2.4 and get (3.9).

(ii)As for the case that , the proof is similar to the argument in the proof of case (i) with suitable modification. We omit the details.

Remark 3.3.

When , or , , we can get the results which are similar to that in Corollary 3.2 and omit them here.

4. Application

In this section, we will apply our result to discuss the boundedness of certain partial integral equation with weakly singular kernel.

Suppose that satisfies the inequality as follow:

(4.1)

for , . Then, (4.1) is the special case of inequality (1.2) that is,

(4.2)

Obviously, . Letting , , we have

(4.3)

Applying (2.3) in Theorem 2.4, we get for ,

(4.4)

which implies that in (4.1) is bounded.