#### Abstract

Some new Volterra-Fredholm-type discrete inequalities in two independent variables are established, which provide a handy tool in the study of qualitative and quantitative properties of solutions of certain difference equations. The established results extend some known results in the literature.

#### 1. Introduction

In the research of solutions of certain differential and difference equations, if the solutions are unknown, then it is necessary to study their qualitative and quantitative properties such as boundedness, uniqueness, and continuous dependence on initial data. The Gronwall-Bellman inequality [1, 2] and its various generalizations which provide explicit bounds play a fundamental role in the research of this domain. Many such generalized inequalities (e.g., see [3–30] and the references therein) have been established in the literature including the known Ou-Liang's inequality [3]. In [8], Ma generalized the discrete version of Ou-Liang's inequality in two variables to Volterra-Fredholm form for the first time, which has proved to be very useful in the study of qualitative as well as quantitative properties of solutions of certain Volterra-Fredholm-type difference equations. But since then, few results on Volterra-Fredholm-type discrete inequalities have been established. Recent results in this direction include the work of Ma [9] to our knowledge. We notice, in the analysis of some certain Volterra-Fredholm-type difference equations with more complicated forms, that the bounds provided by the earlier inequalities are inadequate and it is necessary to seek some new Volterra-Fredholm-type discrete inequalities in order to obtain a diversity of desired results.

Our aim in this paper is to establish some new generalized Volterra-Fredholm-type discrete inequalities, which extend Ma's work in [9], and provide new bounds for unknown functions lying in these inequalities. We will illustrate the usefulness of the established results by applying them to study the boundedness, uniqueness, and continuous dependence on initial data of solutions of certain more complicated Volterra-Fredholm-type difference equations.

Throughout this paper, denotes the set of real numbers , and denotes the set of integers, while denotes the set of nonnegative integers. Let , where and are two constants. are two constants, and , are all constants. If is a lattice, then we denote the set of all -valued functions on by and denote the set of all -valued functions on by . Finally, for a function , we have provided .

#### 2. Main Results

Lemma 2.1 (see [15]). *Assume that ,, and then for any *

Lemma 2.2. *Suppose that , is a constant. If is nondecreasing in the third variable, then, for ,
**
implies that
*

Lemma 2.3. *Suppose that . If is nondecreasing in the first variable, then, for ,
**
implies that
*

*Remark 2.4. *Lemma 2.3 is a direct variation of [19, Lemma 2.5()], and we note here.

Theorem 2.5. *Suppose that , , , , with nondecreasing in the last two variables. are nonnegative constants with , while are nonnegative constants with . If, for , satisfies
**
then
**
provided that , where
*

*Proof. *Denote
Then, we have
and, furthermore, from Lemma 2.1 we have
So
where , and is defined in (2.8). Then, using that is nondecreasing in every variable, we obtain
where is defined in (2.11).

Since are nondecreasing in the last two variables, then is also nondecreasing in the last two variables, and by a suitable application of Lemma 2.2 we obtain
where is defined in (2.10). Furthermore, considering the definition of and (2.17) we have
where is defined in (2.9). Then,
Combining (2.17) and (2.19) we deduce
Then, combining (2.13) and (2.20), we obtain the desired result.

Corollary 2.6. *Suppose that with nondecreasing in every variable. . are defined as in Theorem 2.5. If, for , satisfies
**
then
**
provided that , where
*

The proof of Corollary 2.6 can be completed by setting ,,, ,,,,,,,,,,,,,,, in Theorem 2.5.

Corollary 2.7. *Suppose that are defined as in Theorem 2.5. If, for , satisfies
**
then
**
provided that , where
*

Theorem 2.8. *Suppose that , are defined as in Theorem 2.5. Furthermore, assume that is nondecreasing in the first variable. If, for , satisfies
**
then
**
provided that , where
*

*Proof. *Denote
Then, we have
Obviously is nondecreasing in the first variable. So by Lemma 2.3 we obtain
where . Define
Then,
and, furthermore, by (2.39) and Lemma 2.1 we have
where , and are defined in (2.29)–(2.31) respectively.

Similar to the process of (2.15)–(2.20) we deduce
where are defined in (2.32) and (2.33).

Combining (2.39) and (2.41), we get the desired result.

*Remark 2.9. *If we set ,,,,,,,,,,,,,,,= ,,,,,,,, or set in Theorem 2.8, then immediately we get two corollaries which are similar to Corollaries 2.6 and 2.7, and we omit the details for them.

Theorem 2.10. *Suppose that are defined as in Theorem 2.5. , satisfies for . If, for , satisfies
**
then
**
provided that , where
*

*Proof. *Denote
Then
and, furthermore, from Lemma 2.1 we have
where = and are defined in (2.44) and (2.46) respectively.

Similar to the process of (2.15)–(2.20) we deduce
where are defined in (2.45) and (2.47) respectively.

Combining (2.50) and (2.52), we get the desired result.

Theorem 2.11. *Suppose that , are defined as in Theorem 2.5. Furthermore, assume is nondecreasing in the first variable. ,, are defined as in Theorem 2.10. If, for , satisfies
**
then
**
provided that , where
*