Nonlinear Retarded Integral Inequalities with Two Variables and Applications

Integral inequalities that give explicit bounds on unknown functions provide a very useful and important device in the study of many qualitative as well as quantitative properties of solutions of partial differential equations, integral equations, and integrodifferential equation. One of the best known and widely used inequalities in the study of nonlinear differential equations is Gronwall inequality [1], which states that if and are nonnegative continuous functions on the interval satisfying

(1.1)

where is a nonnegative constant, then we have

(1.2)

Since the inequality (1.2) provides an explicit bound of the unknown function it furnishes a handy tool in the study of various properties of solutions of differential equations. Because of its fundamental importance, several generalizations and analogous results of Gronwall inequality [1, 2] and its applications have attracted great interests of many mathematicians (e.g., [3–5]). Some recent works can be found, for example, in [6–17] and some references therein. In 2005, Agarwal et al. [6] investigated the inequality

(1.3)

In 2006, Cheung [9] studied the inequality

(1.4)

for all , where is a constant.

In 2007, Wang [16] discussed the retarded integral inequality

(1.5)

for all .

In 2008, Agarwal et al. [7] discussed the retarded integral inequality

(1.6)

for all , where is a constant.

In 2009, Kim [12] obtained the explicit bound of the unknown function of the following inequality:

(1.7)

for all .

The purpose of the present paper is to establish some new nonlinear retarded integral inequalities of Gronwall-Bellman type with two variables. We can demonstrate that inequalities (1.4), (1.5), and (1.7), considered in [9, 12, 16], respectively, can also be solved with our results. We also apply our results to study the boundedness and uniqueness of the solutions of the boundary value problem of a partial differential equation.

Throughout this paper, denotes the set of real numbers, and are given numbers. , , are the subsets of and . For any , let denote the subset of . denotes the set of continuous differentiable functions of into .

Consider the following inequality:

(2.1)

Our inequality (2.1) not only includes a nonconstant term outside the integrals but also more than one distinct nonlinear integral without assumption of monotonicity. When , and , our inequality (2.1) reduces to (1.7) studied in [12]. When , and our inequality (2.1) reduces to (1.5) studied in [16].

Suppose that

is a strictly increasing continuous function on , ;

all are continuous functions on and positive on ;

on , and is nondecreasing in each variable;

and are nondecreasing such that and on , and on ;

is a constant;

all are nonnegative functions on .

Firstly, we technically consider a sequence of functions , which can be calculated recursively by

(2.2)

Moreover, we define the following functions:

(2.3)

(2.4)

Obviously, both and are strictly increasing and continuous functions. Letting denote inverse function, respectively, then both and are also continuous and increasing functions.

Let

(2.5)

(2.6)

Then and are nonnegative and nondecreasing in for each fixed and satisfy , , .

Theorem 2.1.

Suppose that hold and is a nonnegative function on satisfying (2.1). Then

(2.7)

for , where

(2.8)

is arbitrarily given on the boundary of the planar region

(2.9)

Corollary 2.2.

Let and be as defined in Theorem 2.1. Suppose that are constants. If

(2.10)

for all , then

(2.11)

for all , where and are defined by (2.5) and (2.6), and

(2.12)

denotes the inverse function of , and lies on the boundary of the planar region

(2.13)

Corollary 2.3.

Let and be as defined in Theorem 2.1. Supposing that

(2.14)

for all , then

(2.15)

for all , where

(2.16)

is defined by (2.5), , are as defined in (2.3) and (2.4), respectively, denote the inverse functions of and . lies on the boundary of the planar region

(2.17)

Theorem 2.4.

Suppose that hold and is a nonnegative function on satisfying

(2.18)

Then

(2.19)

for , where

(2.20)

for , and is arbitrarily given on the boundary of the planar region

(2.21)

Corollary 2.5.

Suppose that hold and is a nonnegative function on satisfying

(2.22)

where are nonnegative functions on . Then

(2.23)

for , where

(2.24)

for , and is arbitrarily given on the boundary of the planar region

(2.25)

Proof of Theorem 2.1.

Obviously, the sequence defined by in (2.2) is nondecreasing nonnegative functions and satisfies , . Moreover, the ratios , are all nondecreasing. From (2.1), (2.2) and (2.5), (2.6), we have

(3.1)

We first discuss the case that for all . Consider the auxiliary inequality

(3.2)

for all , where and are chosen arbitrarily. Let denote the function on the right-hand side of (3.2), which is a nonnegative and nondecreasing function on and . Then, we get the equivalent form of (3.2)

(3.3)

Since is nondecreasing and satisfies for . By the definition of , hypothesis , the monotonicity of and , and (3.3), we have

(3.4)

From (3.4), we have

(3.5)

Keeping fixed in (3.5), setting , integrating both sides of (3.5) with respect to from to , and using the definition of in (2.3), we have

(3.6)

for all , where

(3.7)

Let

(3.8)

From (3.6), we have

(3.9)

for all . We claim that the unknown function in (3.9) satisfies

(3.10)

for all , where

(3.11)

(3.12)

Now, we prove (3.10) by induction. For , let denote the function on the right-hand side of (3.9), which is a nonnegative and nondecreasing function on , and . Then we have

(3.13)

for all . From (3.13), we have

(3.14)

Keeping fixed in (3.14), setting , integrating both sides of (3.14) with respect to from to , and using the definition of in (2.4), we have

(3.15)

for all . Using , from (3.15), we obtain

(3.16)

for all . This proves that (3.10) is true for .

Next, we make the inductive assumption that (3.10) is true for . Now, we consider

(3.17)

for all . Let denote the nonnegative and nondecreasing function on the right-hand side of (3.17). Then and

(3.18)

Let

(3.19)

By (2.2), we see that each , , is a nondecreasing function. Then, we have

(3.20)

for all . Keeping fixed in (3.20), setting , integrating both sides of (3.20) with respect to from to , and using the definition of in (2.4), we have

(3.21)

for all . Let

(3.22)

(3.23)

Using (3.22) and (3.23), from (3.21), we have

(3.24)

It has the same form as (3.9). Let . Since , and are continuous, nondecreasing, and positive on , each is continuous, nondecreasing, and positive on . Moreover,

(3.25)

which are also continuous, nondecreasing, and positive on . Therefore, the inductive assumption for (3.9) can be used to (3.24), and then we have

(3.26)

for all , where

(3.27)

(3.28)

We note that

(3.29)

Thus, from (3.18), (3.22), (3.26), and (3.29), we have

(3.30)

for all . We can prove that the term of in (3.30) is just the same as defined in (3.12). Let . By (3.23), we have

(3.31)

Then using (3.28) and (3.29), we get

(3.32)

This proves that in (3.30) is just the same as defined in (3.12). Therefore, from (3.29), (3.30), and (3.32), we obtain

(3.33)

. The relations of (3.33) imply that in (3.26) and (3.28)

(3.34)

. Hence, (3.30) can be equivalently written as

(3.35)

for all The claim in (3.10) is proved by induction.

Therefore, by (3.3), (3.8), and (3.10), we have

(3.36)

for all . Hence, we obtain the estimation of the unknown function in the auxiliary inequality (3.2).

Letting , , from (3.36), we have

(3.37)

for all , . Since and and are arbitrarily chosen, this proves (2.7).

The remainder case is that for some . Let

(3.38)

where is an arbitrary small number. Obviously, , for all . Using the same arguments as above, where is replaced with , we get

(3.39)

for all . Letting , we obtain (2.7) because of continuity of in and continuity of , and for . This completes the proof.

The proofs of Corollary 2.2, 2.3, 2.5 and Theorem 2.4 are similar to the argument in the proofs of Theorem 2.1 with appropriate modification. We omit the details here.

Remark 3.1.

When and , Corollary 2.2 reduces to Theorem in [9].

Remark 3.2.

When and , Corollary 2.3 reduces to Theorem of Wang [16].

Remark 3.3.

When , Theorem 2.4 reduces to Theorem of Kim [12].

In this section, we apply our results to study the boundedness and uniqueness of the solutions of boundary value problem to a partial differential equation. We consider the partial differential equation with the initial boundary conditions:

(4.1)

(4.2)

for all , where is defined as in Section 2, , are defined in , is a continuous and strictly increasing odd function on , satisfying , for , , , , and are nondecreasing continuous functions, and the ratio is also nondecreasing, and for    .

In the following corollary, we firstly apply our result to discuss boundedness on the solution of problem (4.1).

Corollary 4.1.

Assume that is a continuous function for which there exist a constant , nonnegative functions , , such that

(4.3)

(4.4)

for all , and is nondecreasing in each variable. If is any solution of problem (4.1) with condition (4.2), then

(4.5)

for all , where

(4.6)

and are as defined in Theorem 2.1. lies on the boundary of the planar region

(4.7)

Proof.

It is easy to see that the solution of (4.1) satisfies the following equivalent integral equation:

(4.8)

By (4.3),(4.4), and (4.8), we have

(4.9)

Since , (4.9) is the form of (2.14). Applying Corollary 2.3 to inequality (4.9), using the relation

(4.10)

we obtain the estimation of as given in (4.5).

Corollary 4.1 gives a condition of boundedness for solutions, concretely. If there is a ,

(4.11)

for all . Then every solution of (4.1) is bounded on .

Next, we discuss the uniqueness of the solutions of (4.1).

Corollary 4.2.

Additionally, assume that

(4.12)

for and , where is defined as in Section 2, is a constant, , , are continuous nondecreasing with the nondecreasing ratio such that for all , and , , and is a strictly increasing odd function satisfying for all . Then, (4.1) has at most one solution on

Proof.

Let and be two solutions of (4.1). By (4.8) and (4.12), we have

(4.13)

for all , which is an inequality of the form (2.14), where is an arbitrary small number. Applying Corollary 2.2, we obtain an estimation of the difference in the form (4.5). Namely,

(4.14)

for all , where

(4.15)

and denotes the inverse function of .

Furthermore, by the definition of , we conclude that

(4.16)

Letting , it follows that

(4.17)

Thus, from (4.5), we deduce that , implying that , for all , since is strictly increasing. The uniqueness is proved.