Estimation on Certain Nonlinear Discrete Inequality and Applications to Boundary Value Problem

Abstract

We investigate certain sum-difference inequalities in two variables which provide explicit bounds on unknown functions. Our result enables us to solve those discrete inequalities considered by Sheng and Li (2008). Furthermore, we apply our result to a boundary value problem of a partial difference equation for estimation.

1. Introduction

Various generalizations of the Gronwall inequality [1, 2] are fundamental tools in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equation. There are a lot of papers investigating them (such as [3–8]). Along with the development of the theory of integral inequalities and the theory of difference equations, more attentions are paid to some discrete versions of Gronwall-Bellman-type inequalities (such as [9–11]). Some recent works can be found, for example, in [12–17] and some references therein.

We first introduce two lemmas which are useful in our main result.

Lemma 1.1 (the Bernoulli inequality [18]).

Let and , then .

Lemma 1.2 ([19]).

Assume that are nonnegative functions and is nonincreasing for all natural numbers, if for all natural numbers,

(1.1)

then for all natural numbers,

(1.2)

Sheng and Li [16] considered the inequalities

(1.3)

where for .

In this paper, we investigate certain new nonlinear discrete inequalities in two variables:

(1.4)

(1.5)

(1.6)

where for .

Furthermore, we apply our result to a boundary value problem of a partial difference equation for estimation. Our paper gives, in some sense, an extension of a result of [16].

2. Main Result

Throughout this paper, let denote the set of all real numbers, let be the given subset of , and denote the set of nonnegative integers. For functions , their first-order differences are defined by , , and . We use the usual conventions that empty sums and products are taken to be 0 and 1, respectively. In what follows, we assume all functions which appear in the inequalities to be real-value, and are constants, and .

Lemma 2.1.

Assume that are nonnegative functions defined for , and is nonincreasing in each variable, if

(2.1)

then

(2.2)

Proof.

Define a function by

(2.3)

The function is nonincreasing in each variable, so is , we have

(2.4)

Using Lemma 1.2, the desired inequality (2.2) is obtained from (2.1), (2.3), and (2.4). This completes the proof of Lemma 2.1.

Theorem 2.2.

Suppose that and are nonnegative functions defined for , satisfies the inequality (1.4). Then

(2.5)

where

(2.6)

Proof.

Define a function by

(2.7)

From (1.4), we have

(2.8)

By applying Lemma 1.1, from (2.8), we obtain

(2.9)

(2.10)

It follows from (2.9) and (2.10) that

(2.11)

where we note the definitions of and in (2.6). From (2.6), we see is nonnegative and nonincreasing in each variable. By applying Lemma 2.1, the desired inequality (3.3) is obtained from (2.9) and (2.11). This completes the proof of Theorem 2.2.

Theorem 2.3.

Suppose that and are nonnegative functions defined for , satisfies

(2.12)

where , and satisfies the inequality (1.5). Then

(2.13)

where

(2.14)

(2.15)

Proof.

Define a function by

(2.16)

Then, as in the proof of Theorem 2.2, we have (2.8), (2.9), and (2.10). By (2.12),

(2.17)

It follows from (2.8), (2.9), (2.10), and (2.17) that

(2.18)

where we note the definitions of and in (2.14) and (2.15). From (2.14) we see is nonnegative and nonincreasing in each variable. By applying Lemma 2.1, the desired inequality (2.19) is obtained from (2.9) and (2.18). This completes the proof of Theorem 2.3.

Theorem 2.4.

Suppose that are the same as in Theorem 2.3, satisfies the inequality (1.6). Then

(2.19)

where

(2.20)

(2.21)

Proof.

Define a function by

(2.22)

Then, as in the proof of Theorem 2.2, we have (2.8), (2.9), and (2.10). By (2.12),

(2.23)

It follows from (2.8), (2.9), (2.10), and (2.23) that

(2.24)

where and are defined by (2.20) and (2.21), respectively. From (2.20), we see is nonnegative and nonincreasing in each variable. By applying Lemma 2.1, the desired inequality (2.19) is obtained from (2.9) and (2.24). This completes the proof of Theorem 2.4.

3. Applications to Boundary Value Problem

In this section, we apply our result to the following boundary value problem (simply called BVP) for the partial difference equation:

(3.1)

satisfies

(3.2)

where and are constants, , functions are given, and functions are nonincreasing. In what follows, we apply our main result to give an estimation of solutions of (3.1).

Corollary 3.1.

All solutions of BVP (3.1) have the estimate

(3.3)

where

(3.4)

Proof.

Clearly, the difference equation of BVP (3.1) is equivalent to

(3.5)

It follows from (3.2) and (3.5) that

(3.6)

Let . Equation (3.6) is of the form (1.4), here . Applying our Theorem 2.2 to inequality (3.6), we obtain the estimate of as given in Corollary 3.1.