Journal of Applied Mathematics

Volume 2013 (2013), Article ID 247585, 7 pages

http://dx.doi.org/10.1155/2013/247585

Research Article

A Generalized Nonlinear Sum-Difference Inequality of Product Form

YongZhou Qin and Wu-Sheng Wang

School of Mathematics and Statistics, Hechi University, Guangxi, Yizhou 546300, China

Received 29 August 2013; Accepted 1 November 2013

Academic Editor: Mehmet Sezer

Copyright © 2013 YongZhou Qin and Wu-Sheng Wang. 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.

We establish a generalized nonlinear discrete inequality of product form, which includes both nonconstant terms outside the sums and composite functions of nonlinear function and unknown function without assumption of monotonicity. Upper bound estimations of unknown functions are given by technique of change of variable, amplification method, difference and summation, inverse function, and the dialectical relationship between constants and variables. Using our result we can solve both the discrete inequality in Pachpatte (1995). Our result can be used as tools in the study of difference equations of product form.

1. Introduction

Being an important tool in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential equations and integral equations, various generalizations of Gronwall inequalities [1, 2] and their applications have attracted great interests of many mathematicians (such as [3–6]). Some recent works can be found, for example, in [7–10] and some references therein. Along with the development of the theory of integral inequalities and the theory of difference equations, more attention is paid to some discrete versions of Gronwall-Bellman type inequalities (such as [3, 4, 11–13]). Some recent works can be found, for example, in [14–24] and some references therein.

Pachpatte [4] obtained the explicit bound to the unknown function of the following sum-difference inequality:

Pachpatte [3] obtained the estimation of the unknown function of the following inequality:

Then, the estimation can be used to study the boundedness, asymptotic behavior, and slow growth of the solutions of the sum-difference equation:

However, the bound given on such inequalities in [3, 4] is not directly applicable in the study of certain sum-difference equations. It is desirable to establish new inequalities of the above type, which can be used more effectively in the study of certain classes of sum-difference equations of product form.

In this paper, we establish a new integral inequality of product form

where,,  may not be monotone. For,, we employ a technique of monotonization to construct two functions; the second possesses stronger monotonicity than the first. We can demonstrate that inequalities (1) and (2), considered in [3, 4], respectively, can also be solved with our result. Finally, we expound that we can give estimation of solutions of a class of sum-difference equations of product form.

2. Main Result and Proof

In this section, we proceed to solve the discrete inequality (4) and present explicit bounds on the embedded unknown function. Let,, and  . For function, its difference is defined by. Obviously, the linear difference equationwith the initial conditionhas the solution. For convenience, in the sequel we complementarily define that.

First of all, we monotonize some given functions,,in the sum; let

whereandare all nondecreasing in  and satisfy

Let

whereis nondecreasing in  andis also nondecreasing inand satisfies

where,denote the inverse function of,, respectively.

Theorem 1. Let,be nonnegative and given functions on. Suppose thatis a nonnegative and unknown function. Then, the discrete inequality (4) gives

where,,are defined by (9), (10), and (11), respectively,,,denote the inverse functions of,,, respectively,

andis the largest natural number such that

Proof. Using (5), (6), (7), and (8), we observe that

whereis chosen arbitrarily. Letdenote the function on the right-hand side of (15), namely,

which is a nonnegative and nondecreasing function onwith. Then (4) is equivalent to

Using the difference formula

and the monotonicity ofand, from (16) and (17), we observe that

for all. From (19), we have

On the other hand, by the mean value theorem for integrals, for arbitrarily given integers,, there existsin the open intervalsuch that

whereis defined by (9). From (20) and (21), we have

for all. By settingin (22) and substitutingsuccessively, we obtain

Letdenote the function on the right-hand side of (23); namely,

Then,is a nonnegative and nondecreasing function on, and (23) is equivalent to

From (24), we obtain

From (26), we have

Once again, performing the same procedure as in (21), (22), and (23), (27) gives

whereis defined by (10). Letdenote the function on the right-hand side of (28); namely,

Then,is a nonnegative and nondecreasing function on, and (28) is equivalent to

From (29) and (30), we obtain

for all. From (31), we have

Once again, performing the same procedure as in (21), (22), and (23), (32) gives

Using (17), (25), and (30), from (33) we have

As, (34) yields

Since, andis chosen arbitrarily in (35), the estimation (12) is derived. This completes the proof of Theorem 1.

3. Application

We consider a sum-difference equation of product form

From (36), we have

Let,,,, andin (37); then (37) is the inequality of the form (4). Applying our result we get the estimation of solution of the sum-difference equations of product form (36).

Acknowledgments

This research was supported by National Natural Science Foundation of China (Project no. 11161018) and Guangxi Natural Science Foundation (2012GXNSFAA053009).

References

T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,” Annals of Mathematics, vol. 20, no. 4, pp. 292–296, 1919.

R. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal, vol. 10, pp. 643–647, 1943.

B. G. Pachpatte, “On a new inequality suggested by the study of certain epidemic models,” Journal of Mathematical Analysis and Applications, vol. 195, no. 3, pp. 638–644, 1995.

B. G. Pachpatte, “On some new inequalities related to certain inequalities in the theory of differential equations,” Journal of Mathematical Analysis and Applications, vol. 189, no. 1, pp. 128–144, 1995.

R. P. Agarwal, S. Deng, and W. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,” Applied Mathematics and Computation, vol. 165, no. 3, pp. 599–612, 2005.

W.-S. Cheung, “Some new nonlinear inequalities and applications to boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 9, pp. 2112–2128, 2006.

W.-S. Wang, “A generalized retarded Gronwall-like inequality in two variables and applications to BVP,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 144–154, 2007.

R. P. Agarwal, C. S. Ryoo, and Y.-H. Kim, “New integral inequalities for iterated integrals with applications,” Journal of Inequalities and Applications, vol. 2007, Article ID 24385, 18 pages, 2007.

A. Abdeldaim and M. Yakout, “On some new integral inequalities of Gronwall-Bellman-Pachpatte type,” Applied Mathematics and Computation, vol. 217, no. 20, pp. 7887–7899, 2011.

W. S. Wang, D. Huang, and X. Li, “Generalized retarded nonlinear integral inequalities involving iterated integrals and an application,” Journal of Inequalities and Applications, vol. 2013, article 376, 2013.

T. E. Hull and W. A. J. Luxemburg, “Numerical methods and existence theorems for ordinary differential equations,” Numerische Mathematik, vol. 2, no. 1, pp. 30–41, 1960.

D. Willett and J. S. W. Wong, “On the discrete analogues of some generalizations of Gronwall's inequality,” vol. 69, pp. 362–367, 1965.

B. G. Pachpatte and S. G. Deo, “Stability of discrete-time systems with retarded argument,” Utilitas Mathematica, vol. 4, pp. 15–33, 1973.

E. Yang, “A new nonlinear discrete inequality and its application,” Annals of Differential Equations, vol. 17, no. 3, pp. 261–267, 2001.

B. G. Pachpatte, “On some fundamental integral inequalities and their discrete analogues,” Journal of Inequalities in Pure and Applied Mathematics, vol. 2, no. 2, article 15, 13 pages, 2001.

F. W. Meng and W. N. Li, “On some new nonlinear discrete inequalities and their applications,” Journal of Computational and Applied Mathematics, vol. 158, no. 2, pp. 407–417, 2003.

W.-S. Cheung and J. Ren, “Discrete non-linear inequalities and applications to boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 708–724, 2006.

B. G. Pachpatte, Integral and Finite Difference Inequalities and Applications, vol. 205, Elsevier Science, Amsterdam, The Netherlands, 2006.

W. Sheng and W. N. Li, “Bounds on certain nonlinear discrete inequalities,” Journal of Mathematical Inequalities, vol. 2, no. 2, pp. 279–286, 2008.

Q.-H. Ma and W.-S. Cheung, “Some new nonlinear difference inequalities and their applications,” Journal of Computational and Applied Mathematics, vol. 202, no. 2, pp. 339–351, 2007.

W. S. Wang, “A generalized sum-difference inequality and applications to partial Difference equations,” Advances in Difference Equations, vol. 2008, article 12, 2008.

W. S. Wang, “Estimation on certain nonlinear discrete inequality and applications to boundary value problem,” Advances in Difference Equations, vol. 2009, Article ID 708587, 2009.

W. S. Wang, Z. Li, and W. S. Cheung, “Some new nonlinear retarded sum-difference inequalities with applications,” Advances in Difference Equations, vol. 2011, article 41, 2011.

H. Zhou, D. Huang, and W. S. Wang, “Some new difference inequalities and an application to discrete-time control systems,” Journal of Applied Mathematics, vol. 2012, Article ID 214609, 14 pages, 2012.