An Extension to Nonlinear SumDifference Inequality and Applications
 WuSheng Wang^{1} and
 Xiaoliang Zhou^{2}Email author
DOI: 10.1155/2009/486895
© W.S.Wang and X. Zhou 2009
Received: 31 March 2009
Accepted: 17 May 2009
Published: 25 June 2009
Abstract
We establish a general form of sumdifference inequality in two variables, which includes both more than two distinct nonlinear sums without an assumption of monotonicity and a nonconstant term outside the sums. We employ a technique of monotonization and use a property of stronger monotonicity to give an estimate for the unknown function. Our result enables us to solve those discrete inequalities considered in the work of W.S. Cheung (2006). Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence.
1. Introduction
where , , and are all constants, , and are both nonnegative realvalued functions defined on a lattice in , and is a continuous nondecreasing function satisfying for all .
where . In (1.7) we replace the constant , the functions , , , and in (1.6) with a function , more general functions , and , respectively. Moreover, we consider more than two nonlinear terms and do not require the monotonicity of every . We employ a technique of monotonization to construct a sequence of functions which possesses stronger monotonicity than the previous one. Unlike the work in [26] for two sum terms, the maximal regions of validity for our estimate of the unknown function are decided by boundaries of more than two planar regions. Thus we have to consider the inclusion of those regions and find common regions. We demonstrate that inequalities (1.6) and other inequalities considered in [26] can also be solved with our result. Furthermore, we apply our result to boundary value problems of a partial difference equation for boundedness, uniqueness, and continuous dependence.
2. Main Result
Throughout this paper, let , , and , are given nonnegative integers. For any integers , let , , and . Define , and let denote the sublattice in for any .
For functions , their firstorder differences are defined by and . Obviously, the linear difference equation with the initial condition has the solution . In the sequel, for convenience, we complementarily define that .

(H_{1}) is a strictly increasing continuous function on satisfying that and for all .

(H_{2}) All are continuous and positive functions on .

(H_{3}) on .

(H_{4}) All are nonnegative functions on .
which is nondecreasing in and for each fixed and and satisfies for all .
Theorem 2.1.
Remark 2.2.
The comparison between the both sides implies that (2.8) is equivalent to the condition given in the definition of in Theorem 2.1 with .
Remark 2.3.
If we choose , , , with , and and restrict to be a constant in (1.7), then we can apply Theorem 2.1 to inequality (1.6) discussed in [26].
3. Proof of Theorem
Thus, it means that we can take . Moreover, , .
In the following, we will use mathematical induction to prove (3.5).
that is, (3.5) is true for .
We further claim that the term is the same as , defined in (3.6), . For convenience, let . Obviously, it is that .
for all .
Considering continuities of and for as well as of in and letting , we obtain (2.4). This completes the proof.
We remark that , lie on the boundary of the lattice . In particular, (2.4) is true for all when every ( ) satisfies . Therefore, we may take , .
4. Applications to a Difference Equation
for given functions and ( ) satisfying for , and functions and satisfy that . Obviously, (4.1) is a generalization of the BVP problem considered by [26, Section 3], and the theorems of [26] are not able to solve it. In the following we first apply our main result to the discussion of boundedness of (4.1).
Corollary 4.1.
Proof.
Let . Since , (4.6) is of the form (1.6). Applying our Theorem 2.1 to inequality (4.6), we obtain the estimate of as given in this corollary.
for all , then every solution of BVP (4.1) is bounded on .
Next, we discuss the uniqueness of solutions for BVP (4.1).
Corollary 4.2.
for and , where as assumed in the beginning of Section 2 with natural numbers and are both nonnegative functions defined on the lattice , are both nondecreasing with the nondecreasing ratio such that , for all and for and is strictly increasing odd function satisfying for . Then BVP (4.1) has at most one solution on .
Proof.
Thus we conclude from (2.4) that , implying that for all since is strictly increasing. It proves the uniqueness.
Remark 4.3.
If or in (4.8), the conclusion of the Corollary 4.2 also can be obtained.
where is strictly increasing odd function satisfying for , , and are functions satisfying .
Corollary 4.4.
are all sufficiently small. Then solution of BVP (4.14) is sufficiently close to the solution of BVP (4.1).
Proof.
By (4.10) we see that ( ) as . It follows from (4.18) that and hence depends continuously on and .
Remark 4.5.
Our requirement of the small difference in Corollary 4.4 is stronger than the condition (iii) in [26, Theorem ], but it is easier to check than the condition of them.
Declarations
Acknowledgments
The authors thank Professor Weinian Zhang (Sichuan University) for his valuable discussion. The authors also thank the referees for their helpful comments and suggestions. This work is supported by the Natural Science Foundation of Guangxi Autonomous Region (200991265), by Scientific Research Foundation of the Education Department of Guangxi Autonomous Region of China (200707MS112) and by Foundation of Natural Science of Hechi University (2006AN001) and Key Discipline of Applied Mathematics of Hechi University of China (200725).
Authors’ Affiliations
References
 Gronwall TH: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Annals of Mathematics 1919,20(4):292296. 10.2307/1967124MATHMathSciNetView ArticleGoogle Scholar
 Bellman R: The stability of solutions of linear differential equations. Duke Mathematical Journal 1943, 10: 643647. 10.1215/S0012709443010592MATHMathSciNetView ArticleGoogle Scholar
 Agarwal RP, Deng S, Zhang W: Generalization of a retarded Gronwalllike inequality and its applications. Applied Mathematics and Computation 2005,165(3):599612. 10.1016/j.amc.2004.04.067MATHMathSciNetView ArticleGoogle Scholar
 Baĭnov D, Simeonov P: Integral Inequalities and Applications, Mathematics and Its Applications (East European Series). Volume 57. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+245.Google Scholar
 Cheung WS, Ma QH: On certain new GronwallOuIang type integral inequalities in two variables and their applications. Journal of Inequalities and Applications 2005,10(4):347361.MathSciNetGoogle Scholar
 Lipovan O: Integral inequalities for retarded Volterra equations. Journal of Mathematical Analysis and Applications 2006,322(1):349358. 10.1016/j.jmaa.2005.08.097MATHMathSciNetView ArticleGoogle Scholar
 Ma QH, Yang EH: On some new nonlinear delay integral inequalities. Journal of Mathematical Analysis and Applications 2000,252(2):864878. 10.1006/jmaa.2000.7134MATHMathSciNetView ArticleGoogle Scholar
 Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications (East European Series). Volume 53. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+587.View ArticleGoogle Scholar
 Pachpatte BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering. Volume 197. Academic Press, San Diego, Calif, USA; 1998:x+611.Google Scholar
 Wang WS: A generalized retarded Gronwalllike inequality in two variables and applications to BVP. Applied Mathematics and Computation 2007,191(1):144154. 10.1016/j.amc.2007.02.099MATHMathSciNetView ArticleGoogle Scholar
 Wang WS: A generalized sumdifference inequality and applications to partial difference equations. Advances in Difference Equations 2008, 2008:12.Google Scholar
 Wang WS, Shen CX: On a generalized retarded integral inequality with two variables. Journal of Inequalities and Applications 2008, 2008:9.Google Scholar
 Wang WS: Estimation on certain nonlinear discrete inequality and applications to boundary value problem. Advances in Difference Equations 2009, 2009:8.Google Scholar
 Zhang W, Deng S: Projected GronwallBellman's inequality for integrable functions. Mathematical and Computer Modelling 2001,34(34):393402. 10.1016/S08957177(01)00070XMATHMathSciNetView ArticleGoogle Scholar
 Zheng K, Wu Y, Deng S: Nonlinear integral inequalities in two independent variables and their applications. Journal of Inequalities and Applications 2007, 2007:11.Google Scholar
 Hull TE, Luxemburg WAJ: Numerical methods and existence theorems for ordinary differential equations. Numerische Mathematik 1960, 2: 3041. 10.1007/BF01386206MATHMathSciNetView ArticleGoogle Scholar
 Pachpatte BG, Deo SG: Stability of discretetime systems with retarded argument. Utilitas Mathematica 1973, 4: 1533.MATHMathSciNetGoogle Scholar
 Willett D, Wong JSW: On the discrete analogues of some generalizations of Gronwall's inequality. Monatshefte für Mathematik 1965, 69: 362367. 10.1007/BF01297622MATHMathSciNetView ArticleGoogle Scholar
 Pachpatte BG: On some fundamental integral inequalities and their discrete analogues. Journal of Inequalities in Pure and Applied Mathematics 2001,2(2, article 15):113.Google Scholar
 Dafermos CM: The second law of thermodynamics and stability. Archive for Rational Mechanics and Analysis 1979,70(2):167179.MATHMathSciNetView ArticleGoogle Scholar
 Pang PYH, Agarwal RP: On an integral inequality and its discrete analogue. Journal of Mathematical Analysis and Applications 1995,194(2):569577. 10.1006/jmaa.1995.1318MATHMathSciNetView ArticleGoogle Scholar
 Pachpatte BG: On some new inequalities related to certain inequalities in the theory of differential equations. Journal of Mathematical Analysis and Applications 1995,189(1):128144. 10.1006/jmaa.1995.1008MATHMathSciNetView ArticleGoogle Scholar
 Pachpatte BG: Inequalities for Finite Difference Equations, Monographs and Textbooks in Pure and Applied Mathematics. Volume 247. Marcel Dekker, New York, NY, USA; 2002:x+514.Google Scholar
 Pachpatte BG: Integral and Finite Difference Inequalities and Applications, NorthHolland Mathematics Studies. Volume 205. Elsevier Science, Amsterdam, The Netherlands; 2006:x+309.Google Scholar
 Cheung WS: Some discrete nonlinear inequalities and applications to boundary value problems for difference equations. Journal of Difference Equations and Applications 2004,10(2):213223. 10.1080/10236190310001604238MATHMathSciNetView ArticleGoogle Scholar
 Cheung WS, Ren J: Discrete nonlinear inequalities and applications to boundary value problems. Journal of Mathematical Analysis and Applications 2006,319(2):708724. 10.1016/j.jmaa.2005.06.064MATHMathSciNetView ArticleGoogle Scholar
 Pinto M: Integral inequalities of Biharitype and applications. Funkcialaj Ekvacioj 1990, 33: 387430.MATHMathSciNetGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. 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.