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

Wu-Sheng Wang

Author Affiliations

Department of Mathematics, Hechi University, Guangxi, Yizhou 546300, China

Advances in Difference Equations 2009, 2009:708587  doi:10.1155/2009/708587

 Received: 1 November 2008 Accepted: 14 January 2009 Published: 9 February 2009

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.

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 [38]). 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 [911]). Some recent works can be found, for example, in [1217] 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,

(11)

then for all natural numbers,

(12)

Sheng and Li [16] considered the inequalities

(13)

where for .

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

(14)

(15)

(16)

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

(21)

then

(22)

Proof.

Define a function by

(23)

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

(24)

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

(25)

where

(26)

Proof.

Define a function by

(27)

From (1.4), we have

(28)

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

(29)

(210)

It follows from (2.9) and (2.10) that

(211)

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

(212)

where , and satisfies the inequality (1.5). Then

(213)

where

(214)

(215)

Proof.

Define a function by

(216)

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

(217)

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

(218)

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

(219)

where

(220)

(221)

Proof.

Define a function by

(222)

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

(223)

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

(224)

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:

(31)

satisfies

(32)

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

(33)

where

(34)

Proof.

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

(35)

It follows from (3.2) and (3.5) that

(36)

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.

### Acknowledgments

This work is supported by Scientific Research Foundation of the Education Department Guangxi Province of China (200707MS112) and by Foundation of Natural Science and Key Discipline of Applied Mathematics of Hechi University of China.

### References

1. Bellman, R: The stability of solutions of linear differential equations. Duke Mathematical Journal. 10(4), 643–647 (1943). Publisher Full Text

2. Gronwall, TH: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Annals of Mathematics. 20(4), 292–296 (1919). Publisher Full Text

3. Agarwal, RP, Deng, S, Zhang, W: Generalization of a retarded Gronwall-like inequality and its applications. Applied Mathematics and Computation. 165(3), 599–612 (2005). Publisher Full Text

4. Lipovan, O: Integral inequalities for retarded Volterra equations. Journal of Mathematical Analysis and Applications. 322(1), 349–358 (2006). Publisher Full Text

5. Ma, Q-H, Yang, E-H: On some new nonlinear delay integral inequalities. Journal of Mathematical Analysis and Applications. 252(2), 864–878 (2000). Publisher Full Text

6. Pachpatte, BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering,p. x+611. Academic Press, San Diego, Calif, USA (1998)

7. Wang, W-S: A generalized sum-difference inequality and applications to partial difference equations. Advances in Difference Equations. 2008, (2008)

8. Zhang, W, Deng, S: Projected Gronwall-Bellman's inequality for integrable functions. Mathematical and Computer Modelling. 34(3-4), 393–402 (2001). Publisher Full Text

9. Hull, TE, Luxemburg, WAJ: Numerical methods and existence theorems for ordinary differential equations. Numerische Mathematik. 2(1), 30–41 (1960). Publisher Full Text

10. Pachpatte, BG, Deo, SG: Stability of discrete-time systems with retarded argument. Utilitas Mathematica. 4, 15–33 (1973)

11. Willett, D, Wong, JSW: On the discrete analogues of some generalizations of Gronwall's inequality. Monatshefte für Mathematik. 69, 362–367 (1965). PubMed Abstract | Publisher Full Text

12. Cheung, W-S, Ren, J: Discrete non-linear inequalities and applications to boundary value problems. Journal of Mathematical Analysis and Applications. 319(2), 708–724 (2006). Publisher Full Text

13. Li, WN, Sheng, W: Some nonlinear integral inequalities on time scales. Journal of Inequalities and Applications. 2007, (2007)

14. Pachpatte, BG: On some new inequalities related to certain inequalities in the theory of differential equations. Journal of Mathematical Analysis and Applications. 189(1), 128–144 (1995). Publisher Full Text

15. Pang, PYH, Agarwal, RP: On an integral inequality and its discrete analogue. Journal of Mathematical Analysis and Applications. 194(2), 569–577 (1995). Publisher Full Text

16. Sheng, W, Li, WN: Bounds on certain nonlinear discrete inequalities. Journal of Mathematical Inequalities. 2(2), 279–286 (2008)

17. Wang, W-S, Shen, C-X: On a generalized retarded integral inequality with two variables. Journal of Inequalities and Applications. 2008, (2008)

18. Mitrinović, DS: Analytic Inequalities, Die Grundlehren der Mathematischen Wissenschaften,p. xii+400. Springer, New York, NY, USA (1970)

19. Pachpatte, BG: On some fundamental finite difference inequalities. Tamkang Journal of Mathematics. 32(3), 217–223 (2001)