Journal of Applied Mathematics

Volume 2013 (2013), Article ID 853476, 10 pages

http://dx.doi.org/10.1155/2013/853476

Research Article

A Generalized Nonlinear Gronwall-Bellman Inequality with Maxima in Two Variables

Yong Yan

Department of Mathematics, Sichuan University for Nationalities, Kangding, Sichuan 626001, China

Received 15 November 2012; Accepted 20 January 2013

Academic Editor: Jitao Sun

Copyright © 2013 Yong Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper deals with a generalized form of nonlinear retarded Gronwall-Bellman type integral inequality in which the maximum of the unknown function of two variables is involved. This form includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals. Requiring neither monotonicity nor separability of given functions, we apply a technique of monotonization to estimate the unknown function. Our result can be used to weaken conditions for some known results. We apply our result to a boundary value problem of a partial differential equation with maxima for uniqueness.

1. Introduction

The Gronwall-Bellman inequality [1, 2] plays an important role in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equations. There can be found a lot of its generalizations in various cases from literatures (see, e.g., [3–18]). In 1956, Bihari [3] discussed the integral inequality

where is a constant, is a continuous and nonnegative function, and is a continuous and nondecreasing positive function. Replacing by a function in (1), Lipovan [4] investigated the retarded integral inequality

Their results were further generalized by Agarwal et al. [5] to the inequality

where the constant is replaced with a function , ’s are continuously differentiable and nondecreasing functions, and ’s are continuous and nondecreasing positive functions such that

that is, each ratio is also nondecreasing on , called in [6] that is stronger nondecreasing than . On the basis of this work, Wang [7] considered the inequality of two variables

where the functions , , and are not required to be monotone, and those ’s are not required to be stronger monotone than the one after the next as shown in (4). This inequality belongs to both the case of multivariables, to which great attentions [7–11] have been paid, and to the case that the left-hand side is a composition of the unknown function with a known function, in which Ou-Iang's idea [19] was applied [11–14]. He applied a technique of monotonization to construct a sequence of functions, made each function possess stronger monotonization than the previous one, and gave an estimate for the unknown function .

On the other aspect, many problems in the control theory can be modeled in the form of differential equations with the maxima of the unknown function [20–22]. In connection with the development of the theory of differential equations with maxima (see, e.g., [20, 21, 23]) and partial differential equations with maxima [24, 25], a new type of integral inequalities with maxima is required, respectively. There have been given some results for integral inequalities containing the maxima of the unknown function [23, 26–28]. Concretely, in 2012, Bohner et al. [26] discussed the following system of integral inequalities:

where , ’s, ’s, , and are nonnegative continuous functions and ’s are nonnegative continuously differentiable and nondecreasing functions. They required that , is on and increasing such that for , and satisfies the following: (i) is an increasing function, and (ii) for all and . Bainov and Hristova [23] considered the following system:

where is nonnegative and nondecreasing in both of its arguments, , , and are continuous and nonnegative functions, and .

In this paper, we consider the system of integral inequalities as follows:

where , ’s, ’s, and are continuous and nonnegative functions, ’s and ’s are nonnegative continuously differentiable and nondecreasing functions, and . As required in previous works [27–29], we suppose that , , is constant. In this paper, we require neither monotonicity of , 's, 's, and nor . We monotonize those ’s to make a sequence of functions in which each one possesses stronger monotonicity than the previous one so as to give an estimation for the unknown function. We can use our result to discuss inequalities (6) and (7), giving the stronger results under weaker conditions. We finally apply the obtained result to a boundary value problem of a partial differential equation with maxima for uniqueness.

2. Main Result

Consider system (8) of integral inequalities with and in . Let , . Suppose that

(H₁) and , , are nondecreasing such that on ,   on and ; (H₂) all ’s are continuous and nonnegative functions on ;(H₃) and are continuous, and is strictly increasing such that ; (H₄) all ’s () are continuous on and positive on ; (H₅) is a continuous and nonnegative function on .

For those ’s given in (), define , , inductively by

for and

for , where for , if or if for , and be a given very small constant.

Theorem 1. Suppose that hold, for all and satisfies the system (8) of integral inequalities. Then,

for all , where

is the inverse of the function

is a given constant, is defined just before the theorem, and is defined recursively by

for , and are chosen such that

for .

For the special choice that , , , , , , , , and , where is a nonnegative continuously differentiable and nondecreasing function, Theorem 1 gives an estimate for the unknown in the system (7). we require neither the monotonicity of nor the monotonicity of . Obviously, Lemma 2 and Theorem 1 are applicable to more general forms than Corollary 2.3.4 in [23]. Even if is enlarged to such that (8) is changed into the form of in [29], where , our theorem gives a better estimate. For example, the system of inequalities

implies that

by enlarging to . Applying Theorem 1, we obtain

On the other hand, Theorem 2.2 of [29] gives from (17) that

Clearly, (18) is sharper than (19) for large and .

In order to prove Theorem 1, we need the following lemma.

Lemma 2. Suppose that

(C1) and are nondecreasing such that on and on and ;(C2), for ; (C3) all ’s are continuous and nondecreasing on and positive on such that ; (C4) is continuously differentiable in and , nonnegative on , and for all .

If satisfies the system of inequalities as follows:

then

for all , where is the inverse of the function

is a given constant, and is defined recursively by

for , and are chosen such that

for .

Proof. From (23), we see that is nondecreasing on , , and for . It implies from (20) that

for all . Let

Clearly, is nondecreasing in . Then, we have

From (25), (27), and (28) and the definition of on , we get

Applying Theorem  1 of [7] to the case that , , , and , , we obtain (21) from (28). This completes the proof.

Proof of Theorem 1. First of all, we monotonize some given functions , , , and in the system (8) of integral inequalities. Let

From (13), we see that the function is strictly increasing, and therefore its inverse is well defined, continuous, and increasing in its domain. The sequence , defined by , consists of nondecreasing nonnegative functions on and satisfies

Moreover,

because the ratios , , are all nondecreasing. Furthermore, let

which is nondecreasing in and for each fixed and and satisfies for all . The monotonicity of implies that

for . From (8) and the definition of , we obtain

Concerning (34), we consider the auxiliary system of inequalities

where and are chosen arbitrarily, and claim

for all , , where

, is defined inductively by

for , and are chosen such that

for .

Notice that we may take and . In fact, the monotonicity that and are both nondecreasing in and for fixed , . Furthermore, it is easy to check that , for . If , are replaced with , , respectively, on the left side of (39), we get from (15) that

Thus, it means that we can take , .

Now, we prove (36) by induction. From (33), (35), and the definitions of , , and , we obtain

for all , where and are chosen arbitrarily. Since and , we have . Define a function by

Clearly, is nondecreasing in . By (41) and the definition of , we have

Then noting that is nondecreasing and is strictly increasing, from (43), we obtain

It follows from (43), (44), and the definition of that

In order to demonstrate the basic condition of monotonicity, let , which is clearly a continuous and nondecreasing function on . Thus, each is continuous and nondecreasing on and satisfies for . Moreover, since , is also continuous and nondecreasing on and positive on , implying that , for . By Lemma 2 and (45),

for and . It follows from (43) and (46) that

for and . This proves the claimed (36).

Taking , , and in (36), we have

for all , . It is easy to verify . Thus, (48) can be written as

Since are arbitrary, replacing and with and , respectively, we get

for all . This completes the proof.

3. Applications

In this section, we apply our result to prove the boundedness of solutions for a differential equation with the maxima.

Consider a system of partial differential equations with maxima

where , , are nondecreasing such that , , and ( is a positive constant) for , , and , satisfy that and , for all .

Equation (51) is more general than the equation considered in Section 2.4 of [23]. The following result gives an estimate for its solutions.

Corollary 3. Suppose that functions and in (51) satisfy

where and , . Then, any solution of (51) has the estimate

for all , where

and , are given as in Theorem 1, and constants , are given arbitrarily.

Proof. From (51), we obtain

From (52) and (55), we get

Set for . Noting that , from (56), we get

Applying Theorem 1 to specified , , , , and , , , , and , we obtain (53) from (57).

Next, we discuss the uniqueness of solutions for system (51).

Corollary 4. Suppose that and

for all and all , where and are both nondecreasing such that , for , is also nondecreasing, and , . Then, system (51) has at most one solution on .

Proof. . From (51), we get

Assume that (59) has two different solutions and . From the equivalent integral equation system (55), we have

for all . The continuity of the function implies that for any fixed points and there exists a point such that the inequality holds, and therefore

Hence,

Let

Because , from (62), we obtain

Applying Theorem 1 to specified , , , , , , , , and , from (64), we obtain

for all , where

By the definition of and properties of , noting that , we obtain

Since is finite on a finite interval, and , by (67), we obtain

Thus, we obtain from (68), (69), and (70) immediately. Similarly, noting that is finite on finite interval, and , from (69), we obtain

Thus, we conclude from (65), (69), and (71) that , which implies that , for all , where , are given as in Theorem 1. The uniqueness is proved.

Acknowledgments

The author is grateful to the reviewer for his valuable suggestions. This research was supported by Scientific Research Fund of Sichuan Provincial Education Department of China.

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