Some New Volterra-Fredholm-Type Discrete Inequalities and Their Applications in the Theory of Difference Equations
Bin Zheng1and Qinghua Feng1,2
Academic Editor: Martin D. Schechter
Received21 Mar 2011
Accepted29 Jun 2011
Published29 Aug 2011
Abstract
Some new Volterra-Fredholm-type discrete inequalities in two independent variables are
established, which provide a handy tool in the study of qualitative and quantitative properties of solutions
of certain difference equations. The established results extend some known results in the literature.
1. Introduction
In the research of solutions of certain differential and difference equations, if the solutions are unknown, then it is necessary to study their qualitative and quantitative properties such as boundedness, uniqueness, and continuous dependence on initial data. The Gronwall-Bellman inequality [1, 2] and its various generalizations which provide explicit bounds play a fundamental role in the research of this domain. Many such generalized inequalities (e.g., see [3–30] and the references therein) have been established in the literature including the known Ou-Liang's inequality [3]. In [8], Ma generalized the discrete version of Ou-Liang's inequality in two variables to Volterra-Fredholm form for the first time, which has proved to be very useful in the study of qualitative as well as quantitative properties of solutions of certain Volterra-Fredholm-type difference equations. But since then, few results on Volterra-Fredholm-type discrete inequalities have been established. Recent results in this direction include the work of Ma [9] to our knowledge. We notice, in the analysis of some certain Volterra-Fredholm-type difference equations with more complicated forms, that the bounds provided by the earlier inequalities are inadequate and it is necessary to seek some new Volterra-Fredholm-type discrete inequalities in order to obtain a diversity of desired results.
Our aim in this paper is to establish some new generalized Volterra-Fredholm-type discrete inequalities, which extend Ma's work in [9], and provide new bounds for unknown functions lying in these inequalities. We will illustrate the usefulness of the established results by applying them to study the boundedness, uniqueness, and continuous dependence on initial data of solutions of certain more complicated Volterra-Fredholm-type difference equations.
Throughout this paper, denotes the set of real numbers , and denotes the set of integers, while denotes the set of nonnegative integers. Let , where and are two constants. are two constants, and , are all constants. If is a lattice, then we denote the set of all -valued functions on by and denote the set of all -valued functions on by . Finally, for a function , we have provided .
2. Main Results
Lemma 2.1 (see [15]). Assume that ,, and then for any
Lemma 2.2. Suppose that , is a constant. If is nondecreasing in the third variable, then, for ,
implies that
Lemma 2.3. Suppose that . If is nondecreasing in the first variable, then, for ,
implies that
Remark 2.4. Lemma 2.3 is a direct variation of [19, Lemma 2.5()], and we note here.
Theorem 2.5. Suppose that , , , , with nondecreasing in the last two variables. are nonnegative constants with , while are nonnegative constants with . If, for , satisfies
then
provided that , where
Proof. Denote
Then, we have
and, furthermore, from Lemma 2.1 we have
So
where , and is defined in (2.8). Then, using that is nondecreasing in every variable, we obtain
where is defined in (2.11). Since are nondecreasing in the last two variables, then is also nondecreasing in the last two variables, and by a suitable application of Lemma 2.2 we obtain
where is defined in (2.10). Furthermore, considering the definition of and (2.17) we have
where is defined in (2.9). Then,
Combining (2.17) and (2.19) we deduce
Then, combining (2.13) and (2.20), we obtain the desired result.
Corollary 2.6. Suppose that with nondecreasing in every variable. . are defined as in Theorem 2.5. If, for , satisfies
then
provided that , where
The proof of Corollary 2.6 can be completed by setting ,,, ,,,,,,,,,,,,,,, in Theorem 2.5.
Corollary 2.7. Suppose that are defined as in Theorem 2.5. If, for , satisfies
then
provided that , where
Theorem 2.8. Suppose that , are defined as in Theorem 2.5. Furthermore, assume that is nondecreasing in the first variable. If, for , satisfies
then
provided that , where
Proof. Denote
Then, we have
Obviously is nondecreasing in the first variable. So by Lemma 2.3 we obtain
where . Define
Then,
and, furthermore, by (2.39) and Lemma 2.1 we have
where , and are defined in (2.29)–(2.31) respectively. Similar to the process of (2.15)–(2.20) we deduce
where are defined in (2.32) and (2.33). Combining (2.39) and (2.41), we get the desired result.
Remark 2.9. If we set ,,,,,,,,,,,,,,,= ,,,,,,,, or set in Theorem 2.8, then immediately we get two corollaries which are similar to Corollaries 2.6 and 2.7, and we omit the details for them.
Theorem 2.10. Suppose that are defined as in Theorem 2.5. , satisfies for . If, for , satisfies
then
provided that , where
Proof. Denote
Then
and, furthermore, from Lemma 2.1 we have
where = and are defined in (2.44) and (2.46) respectively. Similar to the process of (2.15)–(2.20) we deduce
where are defined in (2.45) and (2.47) respectively. Combining (2.50) and (2.52), we get the desired result.
Theorem 2.11. Suppose that , are defined as in Theorem 2.5. Furthermore, assume is nondecreasing in the first variable. ,, are defined as in Theorem 2.10. If, for , satisfies
then
provided that , where
The proof for Theorem 2.11 is similar to the combination of Theorems 2.8 and 2.10, and we omit the details here.
Remark 2.12. If we take ,,, and ,, in Corollary 2.6, then Corollary 2.6 reduces to [9, Theorem 2.5]. If furthermore , then Corollary 2.6 reduces to [9, Theorem 2.1]. If we take ,, and , ,, in Theorem 2.10, then Theorem 2.10 reduces to [9, Theorem 2.7]. If furthermore , then Theorem 2.10 reduces to [9, Theorem 2.6].
3. Applications
In this section, we will present some applications for the established results above and show that they are useful in the study of boundedness, uniqueness, and continuous dependence of solutions of certain difference equations.
Example 3.1. Consider the following Volterra-Fredholm sum-difference equation:
where , is an odd number, ,, are two integers defined the same as in Theorem 2.5.
Theorem 3.2. Suppose that is a solution of (3.1), and , ,,, where are nonnegative constants satisfying , and are nondecreasing in the last two variables; then one has
provided that , where
Proof. From (3.1) we have
Then a suitable application of Theorem 2.5 (with ) to (3.4) yields the desired result.
The following theorem deals with the uniqueness of the solutions of (3.1).
Theorem 3.3. Suppose that ,, hold for , where , with nondecreasing in the last two variables, and , where , and , then (3.1) has at most one solution.
Proof. Suppose that are two solutions of (3.1). Then,
Treat as one variable, and a suitable application of Corollary 2.7 yields , which implies that . Since is an odd number, then we have , and the proof is complete.
Finally we study the continuous dependence of the solutions of (3.1) on the functions .
Theorem 3.4. Suppose that is a solution of (3.1), , , hold for , where , with nondecreasing in the last two variables, and, furthermore,
where is a constant, and is the solution of the following difference equation:
where ; then one has
provided that , where
Proof. From (3.1) and (3.7) we deduce
Then a suitable application of Corollary 2.7 yields the desired result.
Remark 3.5. We note that the results in [1–30] are not available here to establish the analysis above.
Acknowledgment
The authors thank the referees very much for their careful comments and valuable suggestions on this paper.
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.
X. Q. Zhao, Q. X. Zhao, and F. W. Meng, “On some new nonlinear discrete inequalities and their applications,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 2, article 52, pp. 1–9, 2006.
E. H. Yang, “On some nonlinear integral and discrete inequalities related to Ou-Iang's inequality,” Acta Mathematica Sinica, vol. 14, no. 3, pp. 353–360, 1998.
Q. H. Ma, “Some new nonlinear Volterra-Fredholm-type discrete inequalities and their applications,” Journal of Computational and Applied Mathematics, vol. 216, no. 2, pp. 451–466, 2008.
Q. H. Ma, “Estimates on some power nonlinear Volterra-Fredholm type discrete inequalities and their applications,” Journal of Computational and Applied Mathematics, vol. 233, no. 9, pp. 2170–2180, 2010.
Q. H. Ma and J. Pečarić, “Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 2, pp. 393–407, 2008.
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.
B. G. Pachpatte, “On some new inequalities related to a certain inequality arising in the theory of differential equations,” Journal of Mathematical Analysis and Applications, vol. 251, no. 2, pp. 736–751, 2000.
W. S. Cheung, Q. H. Ma, and J. Pečarić, “Some discrete nonlinear inequalities and applications to difference equations,” Acta Mathematica Scientia, vol. 28, no. 2, pp. 417–430, 2008.
S. F. Deng, “Nonlinear discrete inequalities with two variables and their applications,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2217–2225, 2010.
F. C. Jiang and F. W. Meng, “Explicit bounds on some new nonlinear integral inequalities with delay,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 479–486, 2007.
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.
P. Y. H. Pang and R. P. Agarwal, “On an integral inequality and its discrete analogue,” Journal of Mathematical Analysis and Applications, vol. 194, no. 2, pp. 569–577, 1995.
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.
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, “Some discrete nonlinear inequalities and applications to boundary value problems for difference equations,” Journal of Difference Equations and Applications, vol. 10, no. 2, pp. 213–223, 2004.
F. W. Meng and D. H. Ji, “On some new nonlinear discrete inequalities and their applications,” Journal of Computational and Applied Mathematics, vol. 208, no. 2, pp. 425–433, 2007.
W. S. Cheung and Q. H. Ma, “On certain new Gronwall-Ou-Iang type integral inequalities in two variables and their applications,” Journal of Inequalities and Applications, vol. 2005, no. 4, pp. 347–361, 2005.
W. S. Cheung and J. L. 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.
A. Gallo and A. M. Piccirillo, “About some new generalizations of Bellman-Bihari results for integro-functional inequalities with discontinuous functions and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e2276–e2287, 2009.
F. W. Meng and W. N. Li, “On some new integral inequalities and their applications,” Applied Mathematics and Computation, vol. 148, no. 2, pp. 381–392, 2004.
R. Xu and Y. G. Sun, “On retarded integral inequalities in two independent variables and their applications,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1260–1266, 2006.
Z. L. Yuan, X. W. Yuan, and F. W. Meng, “Some new delay integral inequalities and their applications,” Applied Mathematics and Computation, vol. 208, no. 1, pp. 231–237, 2009.
R. P. Agarwal, S. F. Deng, and W. N. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,” Applied Mathematics and Computation, vol. 165, no. 3, pp. 599–612, 2005.