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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 804152, 9 pages
Some Difference Inequalities for Iterated Sums with Applications
1School of Mathematics and Statistics, Hechi University, Yizhou, Guangxi 546300, China
2Department of Mathematics and Computer Science, Longyan University, Longyan, Fujian 364012, China
Received 4 September 2013; Revised 13 November 2013; Accepted 28 November 2013
Academic Editor: Jaume Giné
Copyright © 2013 Wu-Sheng Wang and Shanhe Wu. 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.
The main objective of this paper is to establish two new nonlinear sum-difference inequalities with multiple iterated sums. Under several practical assumptions, the inequalities are solved through rigorous analysis, and explicit bounds for the unknown functions are given clearly. These new inequalities can be used as handy tools in the study of the estimation of solutions of difference 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
It has become one of the very few classic and most influential results in the theory and applications of inequalities. Because of its fundamental importance, over the years, many generalizations and analogous results of (2) have been established, such as [3–14].
Among these references, Baĭnov and Simeonov [4, P. 107] considered the following interesting Gronwall-type inequality: Kim  considered analogous Gronwall-type integral inequalities involving iterated integrals,
In 2011, Abdeldaim and Yakout  studied some new integral inequalities of Gronwall-Bellman-Pachpatte type such as
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; for detailed information, please refer to the literatures [15–35]. For instance, Pachpatte  considered the following discrete inequality:
In 2006, Cheung and Ren  studied
Later, Zheng et al.  discussed the following discrete inequality:
In 2012, Zhou et al.  studied the following inequalities:
However, the above results are not applicable to some certain inequalities with multiple iterated sums. Hence, it is desirable to consider more general difference inequalities of these extended types. They can be used in the study of certain classes of difference equations or applied in many practical engineering problems.
All the assumptions on (10) and (11) are given in the next sections. The inequalities (10) and (11) consist of multiple iterated sums. Under several practical assumptions, the inequalities are solved through rigorous analysis, and explicit bounds for the unknown functions are given clearly. Further, the derived results are applied to study the estimation of solutions of difference equations.
2. Main Results
Throughout this paper, let and . For function , its difference is defined by . Obviously, the linear difference equation with the initial condition has the solution . For convenience, in the sequel we complementarily define that .
Lemma 1. Let , and be real-valued nonnegative functions defined on and satisfy the inequality where is a nonnegative constant. Then,
Theorem 2. Let and be nonnegative functions defined on with nondecreasing on . Moreover, let , , be nonnegative functions for and nondecreasing in for fixed . If and is not equal to 1, then the discrete inequality (10) gives where can be successively determined from the formulas for , , where is the inverse functions of , , is chosen arbitrarily, and is the largest natural number such that
Remark 4. We can obtain using MATLAB program: firstly let , when ; let , when ; let , and so on until . If , for all , then .
Proof. Fix , where is chosen arbitrarily and is defined by (20). For , from (10) we have
Now we introduce the functions
For and , then , , are all positive and nondecreasing functions on with , , and the inequalities (22) and (23) imply that
From (22), we observe that
We claim that for , .
Now we prove (26) and (27) by induction. Obviously, (26) is true for by (26). We make the inductive assumption that (26) is true for . By the inductive assumption and (24), from (23) we obtain
It actually proves (26) by induction. From (23) and (26), we have
It proves (27). From (27), we have
On the other hand, by the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that
By setting in (31) and substituting , successively, we obtain
Let denote the right-hand side of (32), which is a positive and nondecreasing function on with
Then, (32) is equivalent to
By the definition of , we obtain
From (34) and (35), we get
Once again, performing the similar procedure from (30) to (32), (36) gives where is defined by (19). By combining (33), (34), and (37), we can obtain that where is defined by (17). Applying Lemma 1 to (26) for , we have where is defined by (18). From (24) and (39), we have
Since is arbitrary, from (40), we get the required estimate where is defined by (20). Theorem 2 is proved.
Theorem 5. Let , , and be nonnegative functions defined on with and nondecreasing on , and let , , be nonnegative functions for , , which are nondecreasing in for fixed . Suppose that is a nondecreasing continuous function on with for . The inequality (11) implies that where is the inverse function of , and is the largest natural number such that
Remark 6. We can obtain using Matlab program similar to Remark 4.
Proof. Let the function be positive. Fix , where is chosen arbitrarily and is defined by (45). For , from (11) we have
We denote the right-hand side of (46) by for . Then , the function is positive and nondecreasing in , , and
Define a function by for all . From (47) and (48), we have
From (48), we have
From (50), we get where for all .
Continuing in this way, we obtain where
From (54) and the inequality , we have
We define the functions , which are nondecreasing and continuously differentiable on with , on .
On the other hand, by the formula of partial integration, we have
By the monotonicity of , , from (56) we have
By the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that
From (55), (57), and (58), we have
Next, from the inequalities (53) and (59), we have
Once again, applying the same procedure from (56) to (59) to the inequality (60), we get
Proceeding in this way, we obtain
Using the inequalities (49) and (62), we have where is defined by (44).
Once again, using the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that where is defined by (43). Using (63), (64), and , we obtain
In (65), let ; we have
Due to the randomness of , (42) is achieved immediately from (66).
In this section, we apply Theorem 5 to the following difference equation:
Corollary 7. Assume that is defined on , and there exist nonnegative functions , , such that where the function is as in Theorem 5. If is any solution of the problem (67), then where the functions , are as in Theorem 5, and is the largest natural number such that
Proof. The difference equation (67) is equivalent to
Using (68), from (72), we have
Notice that, by the assumption, all functions in (73) satisfy the conditions of Theorem 5. Applying Theorem 5 to the inequality (73), (69) is immediately derived. This completes the proof of Corollary 7.
Conflict of Interests
The authors declare that they have no competing interests.
This research was supported, in part, by the National Natural Science Foundation of China under Grant 11161018 by the Guangxi Natural Science Foundation of China under Grant 2012GXNSFAA053009, and by the Natural Science Foundation of Fujian province of China under Grant 2012J01014. The authors are grateful to the anonymous referees for their careful comments and valuable suggestions on this paper.
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