#### 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(H_{1}) and , , are nondecreasing such that on , on and ; (H_{2}) all ’s are continuous and nonnegative functions on ;(H_{3}) and are continuous, and is strictly increasing such that ; (H_{4}) all ’s () are continuous on and positive on ; (H_{5}) 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 . LetClearly, 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 byClearly, 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,