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Journal of Applied Mathematics

Volume 2014 (2014), Article ID 763761, 9 pages

http://dx.doi.org/10.1155/2014/763761

## Estimation of Unknown Functions of Iterative Difference Inequalities and Their Applications

^{1}School of Computer and Information Engineering, Hechi University, Guangxi, Yizhou 546300, China^{2}School of Mathematics and Statistics, Hechi University, Guangxi, Yizhou 546300, China^{3}Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia

Received 28 March 2014; Accepted 10 July 2014; Published 22 July 2014

Academic Editor: Youjun Deng

Copyright © 2014 Ricai Luo et al. 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 some new nonlinear retarded finite difference inequalities. The results that we propose here can be used as tools in the theory of certain new classes of finite difference equations in various difference situations. We also give applications of the inequalities to show the usefulness of our results.

#### 1. Introduction

An integral inequality that provides an explicit bound to the unknown function furnishes a handy tool to investigate qualitative properties of solutions of differential and integral equations. One of the best known and widely used inequalities in the study of nonlinear differential equations is Gronwall-Bellman inequality [1, 2], which can be stated as follows. If and are nonnegative continuous functions on an interval satisfying for some constant , then Being an important tool in the study of 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 [3–5]. Some recent works can be found in [6–12] and the references therein. Along with the development of the theory of integral inequalities and the theory of difference equations, more and more attentions are paid to discrete versions of Gronwall-type inequalities; see [13–36] and the references cited therein.

Sugiyama [13] established the most precise and complete discrete analogue of the Gronwall inequality [1] in the following form. Let and be nonnegative functions defined for , and suppose that for every . If where is the set of points (), is a given integer, and is a nonnegative constant, then Pachpatte [15] established some generalized discrete analogue of the Gronwall inequality in the following form. Let be a positive and monotone nondecreasing function on , and let be nonnegative functions on . If satisfies then where

Lemma 1 (see [16]). *Suppose that is a nonnegative constant and , , , , and are nonnegative functions defined on , for all . If satisfies the inequality
**
then
**
where
**
in which
**
and for all .*

Lemma 2 (see [14, 18]). *Let be a real-valued function defined for , and monotone nondecreasing with respect to for any fixed . Let be a real-valued function defined for such that
**
Let be a solution of
**
such that . Then
*

Pachpatte [18, 19] also established some difference inequalities of product form as follows. Let be nonnegative functions defined on and let be a nonnegative constant. Let be a nonnegative function defined for , and monotone nondecreasing with respect to for any fixed . If satisfies then where is defined by (7), and is a solution of Let be nonnegative functions defined for and let be a nonnegative constant. Let be a nonnegative function defined for , and monotone nondecreasing with respect to for any fixed . If satisfies then where is defined by (7), and is a solution of the difference equation

Motivated by the results given in [16, 18, 19], in this paper, we discuss new nonlinear finite difference inequalities: Our inequalities can be used as tools in the study of certain classes of finite difference equations. We also present some immediate applications to show the importance of our results to study the various problems in the theory of finite difference equations.

#### 2. Main Results

Throughout this paper, let , . Let and . For function , , we define the operator by . Obviously, the linear difference equation with the initial condition has the solution . For convenience, in the sequel we complementarily define that and .

Theorem 3. *Let be a constant, positive functions defined on , a monotone increasing function, and a monotone decreasing function. Let be a nonnegative function defined on such that
*(i)*Suppose . If , then
*(ii)*Suppose . Then
*

*Proof. *(i) We apply mean value theorem for differentiation to the function
and then there exists between and such that
Because is monotone increasing and is monotone decreasing and , we see that and . So for all values of between and we have
From (22) and (27), we have
Since , from (26) and (28) we have
Taking in (29) and summing up over from to , we obtain
From (30), we obtain our required estimation (23).

(ii) Now by following the same steps as in the proof of (i) before (29) we have
because . Taking in (31) and summing up over from to , we obtain
From (32), we obtain our required estimation (24).

Theorem 4. *Let be a positive and monotone nondecreasing function defined on and nonnegative functions defined on . If satisfies
**
then
**
where
**
in which
**
and for all .*

*Proof. *Fix , where is chosen arbitrarily, since is a nonnegative and monotone nondecreasing function, from (33), we have
Define a function by the right-hand side of (37). Then is a positive and monotone nondecreasing function defined on . We have
Using the definitions of the operator and , we obtain
Let
Then
It follows that
Adding to both sides of the above inequality we have
Put
and then , and
We see that the inequality
Define a function
Multiplying by to both sides of (46) we obtain
Let , , , and . Because is monotone increasing, is monotone decreasing and ; applying Theorem 3 to (48) we obtain
where , are used. Define a function of the right-hand side of (49). Substituting (49) in (43) we obtain
Performing the same derivation as in (46)–(49), we obtain from (50) that
Define a function of the right-hand side of (51). Substituting (51) in (39) we obtain
Using (38), from (52) it follows that
Since is arbitrary, from (53), we get the required estimate (35).

Theorem 5. *Let be nonnegative functions defined for and a nonnegative constant. Let be a real-valued function defined for , , and monotone nondecreasing with respect to for any fixed . If satisfies (21), then
**
where
**
in which
**
and is a solution of the difference equation
**
where
**
in which
**
and for all .*

*Proof. *We first assume that and define a function by the right-hand side of (21). Then is a positive and monotone nondecreasing function defined on . We have
Using the definitions of the operator and , we obtain
From (61) it follows that the inequality
holds for all . Setting in (62) and substituting , successively, we get
Define a function by
Then and
Using (64), the inequality (63) can be written as
Since is positive and monotone nondecreasing for , satisfy the conditions in Theorem 4. Now an application of Theorem 4 to (66) yields
where
in which
Since is monotone nondecreasing with respect to for any fixed , from (65) and (67), we have
Now as a suitable application of Lemma 2, we obtain
where is a solution of (57). Using (60), (67), and (71), we obtain our required estimation (54).

If is nonnegative, we can carry out the above procedure with instead of where is an arbitrary small number. Letting , we obtain (54).

#### 3. Application to Finite Difference Equations

In this section, we consider the following difference equation: where , , are real-valued functions defined, respectively, on , , , is as defined in Theorem 5, and is a constant. We assume that where , , are as defined in Theorem 5. Using the definitions of the operator , from (72), we see that the inequality holds for all . It follows that From (76), we have Using the conditions (74), we obtain Now an application of Theorem 5 to (78) yields the estimation of the difference equation (72), that is, where in which and is a solution of the difference equation where in which and for all .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research was supported by National Natural Science Foundation of China (Project no. 11161018,11171079), the NSF of Guangxi Zhuang Autonomous Region (no. 2012GXNSFAA053009), and the SRF of the Education Department of Guangxi Zhuang Autonomous Region (nos. 201106LX599 and 201106LX591).

#### 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: Second Series*, vol. 20, no. 4, pp. 292–296, 1919. View at Publisher · View at Google Scholar · View at MathSciNet - R. Bellman, “The stability of solutions of linear differential equations,”
*Duke Mathematical Journal*, vol. 10, pp. 643–647, 1943. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. S. Mitrinović, J. E. Pečarić, and A. M. Fink,
*Inequalities Involving Functions and Their Integrals and Derivatives*, vol. 53 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. View at Publisher · View at Google Scholar · View at MathSciNet - D. Bainov and P. Simeonov,
*Integral Inequalities and Applications*, vol. 57 of*Mathematics and its Applications (East European Series)*, Kluwer Academic, Dordrecht, The Netherlands, 1992. View at Publisher · View at Google Scholar · View at MathSciNet - B. G. Pachpatte,
*Inequalities for Differential and Integral Equations*, vol. 197, Academic Press, New York, NY, USA, 1998. View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - O. Lipovan, “Integral inequalities for retarded Volterra equations,”
*Journal of Mathematical Analysis and Applications*, vol. 322, no. 1, pp. 349–358, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - W. Cheung, “Some new nonlinear inequalities and applications to boundary value problems,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 64, no. 9, pp. 2112–2128, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - R. P. Agarwal, C. S. Ryoo, and Y. Kim, “New integral inequalities for iterated integrals with applications,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 24385, 18 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - 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. View at Publisher · View at Google Scholar · View at MathSciNet - S. Sugiyama, “On the stability problems of difference equations,”
*Bulletin of the Science and Engineering Research Laboratory, Waseda University*, vol. 45, pp. 140–144, 1969. View at Google Scholar - B. G. Pachpatte, “Finite-difference inequalities and an extension of Lyapunov's method,”
*The Michigan Mathematical Journal*, vol. 18, pp. 385–391, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. G. Pachpatte, “On some new discrete inequalities and their applications,”
*Proceedings of the National Academy of Sciences, India*, vol. 46, no. 4, pp. 255–262, 1976. View at Google Scholar · View at MathSciNet - B. G. Pachpatte, “On discrete inequalities related to Gronwall's inequality,”
*Proceedings of the Indian Academy of Science A*, vol. 85, no. 1, pp. 26–40, 1977. View at Publisher · View at Google Scholar - B. G. Pachpatte, “Finite difference inequalities and discrete time control systems,”
*Indian Journal of Pure and Applied Mathematics*, vol. 9, no. 12, pp. 1282–1290, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. G. Pachpatte, “Comparison theorems related to a certain inequality used in the theory of differential equations,”
*Soochow Journal of Mathematics*, vol. 22, no. 3, pp. 383–394, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. G. Pachpatte, “Inequalities applicable in the theory of finite difference equations,”
*Journal of Mathematical Analysis and Applications*, vol. 222, no. 2, pp. 438–459, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - E. Yang, “A new nonlinear discrete inequality and its application,”
*Annals of Differential Equations: Weifen Fangcheng Niankan*, vol. 17, no. 3, pp. 261–267, 2001. View at Google Scholar · View at MathSciNet - 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, 2001. View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. G. Pachpatte,
*Integral and Finite Difference Inequalities and Applications*, vol. 205 of*North-Holland Mathematics Studies*, Elsevier Science, Amsterdam, The Netherlands, 2006. View at MathSciNet - W. Sheng and W. N. Li, “Bounds on certain nonlinear discrete inequalities,”
*Journal of Mathematical Inequalities*, vol. 2, no. 2, pp. 279–286, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. J. Cho, S. S. Dragomir, and Y. Kim, “On some integral inequalities with iterated integrals,”
*Journal of the Korean Mathematical Society*, vol. 43, no. 3, pp. 563–578, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - W. Wang, “A generalized sum-difference inequality and applications to partial difference equations,”
*Advances in Difference Equations*, vol. 2008, article 12, Article ID 695495, 2008. View at Google Scholar · View at MathSciNet - W.-S. Wang, “Estimation on certain nonlinear discrete inequality and applications to boundary value problem,”
*Advances in Difference Equations*, vol. 2009, Article ID 708587, 8 pages, 2009. View at Publisher · View at Google Scholar - K.-L. Zheng, S.-M. Zhong, and M. Ye, “Discrete nonlinear inequalities in time control systems,” in
*Proceedings of the International Conference on Apperceiving Computing and Intelligence Analysis (ICACIA '09)*, pp. 403–406, Chengdu, China, October 2009. View at Publisher · View at Google Scholar · View at Scopus - W. S. Wang, Z. Li, and W. S. Cheung, “Some new nonlinear retarded sum-difference inequalities with applications,”
*Advances in Difference Equations*, vol. 2011, no. 1, article 41, 2011. View at Google Scholar · View at MathSciNet - H. Zhou, D. Huang, W.-S. Wang, and J.-X. Xu, “Some new difference inequalities and an application to discrete-time control systems,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 214609, 14 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Q. He, T. Sun, and H. Xi, “Dynamics of a family of nonlinear delay difference equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 456530, 4 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Z.-X. Chen and K. H. Shon, “Fixed points of meromorphic solutions for some difference equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 496096, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - W. S. Wang and S. Wu, “Some difference inequalities for iterated sums with applications,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 804152, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Qin and W. Wang, “A generalized nonlinear sum-difference inequality of product form,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 247585, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet