- Research Article
- Open Access
An Extension to Nonlinear Sum-Difference Inequality and Applications
Advances in Difference Equations volume 2009, Article number: 486895 (2009)
We establish a general form of sum-difference 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.
Gronwall-Bellman inequality [1, 2] is a fundamental tool in the study of existence, uniqueness, boundedness, stability, invariant manifolds and other qualitative properties of solutions of differential equations and integral equation. There are a lot of papers investigating them such as [3–15]. Along with the development of the theory of integral inequalities and the theory of difference equations, more attentions are paid to some discrete versions of Bellman-Gronwall type inequalities (e.g., [16–18]). Starting from the basic form
discussed in , an interesting direction is to consider the inequality
a discrete version of Dafermos' inequality , where are nonnegative constants and are nonnegative functions defined on and , respectively. Pang and Agarwal  proved for (1.2) that for all . Another form of sum-difference inequality
was estimated by Pachpatte  as , where . Recently, Pachpatte [23, 24] discussed the inequalities of two variables
where is nondecreasing. In  another form of inequality of two variables
was discussed. Later, this result was generalized in  to the inequality
where , , and are all constants, , and are both nonnegative real-valued functions defined on a lattice in , and is a continuous nondecreasing function satisfying for all .
In this paper we establish a more general form of sum-difference inequality with positive integers ,
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  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  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 first-order 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 .
We give the following basic assumptions for the inequality (1.7).
(H1) is a strictly increasing continuous function on satisfying that and for all .
(H2) All are continuous and positive functions on .
(H3) on .
(H4) All are nonnegative functions on .
With given functions , and , we technically consider a sequence of functions , which can be calculated recursively by
For given constants and variable , we define
Obviously, is strictly increasing in and therefore the inverses are well defined, continuous, and increasing. Let
which is nondecreasing in and for each fixed and and satisfies for all .
Suppose that hold and is a nonnegative function on satisfying (1.7). Then, for , a sublattice in ,
where is determined recursively by
and is arbitrarily given on the boundary of the lattice
As explained in [3, Remark ], since different choices of in do not affect our results, we simply let denote when there is no confusion. For positive constants , let . Obviously, and . It follows that
that is, we obtain the same expression in (2.4) if we replace with . Moreover, by replacing with , the condition in the definition of in Theorem 2.1 reads
the left-hand side of which is equal to
and the right-hand side of which equals
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 .
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 .
3. Proof of Theorem
First of all, we monotonize some given functions , in the sums. Obviously, the sequence defined by in (2.1) consists of nondecreasing nonnegative functions and satisfies , for . Moreover,
as defined in  for comparison of monotonicity of functions , because every ratio is nondecreasing. By the definitions of functions , and , from (1.7) we get
Then, we discuss the case that for all . Because satisfies
it is positive and nondecreasing on . We consider the auxiliary inequality to (3.2), for all ,
where and are chosen arbitrarily, and claim that, for , a sublattice in ,
where is determined recursively by
, and is arbitrarily chosen on the boundary of the lattice
We note that , can be chosen appropriately such that
In fact, from the fact of being on the boundary of the lattice , we see that
Thus, it means that we can take . Moreover, , .
In the following, we will use mathematical induction to prove (3.5).
For , let . Then is nonnegative and nondecreasing in each variable on . From (3.4) we observe that
Moreover, we note that is nondecreasing and satisfies for and that . From (3.10) we have
On the other hand, by the Mean Value Theorem for integral and by the monotonicity of and , for arbitrarily given , there exists in the open interval such that
It follows from (3.11) and (3.12) that
Substituting with and summing both sides of (3.13) from to , we get, for all ,
We note from the definition of in (3.2) and the definition of in Section 2 that . By the monotonicity of and (3.10) we obtain
that is, (3.5) is true for .
Next, we make the inductive assumption that (3.5) is true for . Consider
for all . Let which is nonnegative and nondecreasing in each variable on . Then (3.16) is equivalent to
Since is nondecreasing and satisfies for and , from (3.17) we obtain, for all ,
On the other hand, by the Mean Value Theorem for integrals and by the monotonicity of and , for arbitrarily given there exists in the open interval such that
Therefore, it follows from (3.18) and (3.20) that
substituting with in (3.21) and summing both sides of (3.21) from to , we get, for all ,
where we note that . For convenience, let
From (3.17) and (3.22) we can get
the same form as (3.4) for , for all , where we note that for all . We are ready to use the inductive assumption for (3.24). In order to demonstrate the basic condition of monotonicity, let obviously which is a continuous and nondecreasing function on . Thus each is continuous and nondecreasing on and satisfies for . Moreover,
which is also continuous nondecreasing on and positive on . This implies that , for . Therefore, the inductive assumption for (3.5) can be used to (3.24) and we obtain, for all ,
where , is the inverse of (for ), is determined recursively by
and , are functions of such that lie on the boundary of the lattice
where denotes either if it converges or . Note that
Thus, from (3.17), (3.23), and (3.27), (3.26) can be equivalently written as
We further claim that the term is the same as , defined in (3.6), . For convenience, let . Obviously, it is that .
The remainder case is that for some . Let
where is an arbitrary small number. Obviously, for all . Using the same arguments as above and replacing with , we get
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
In this section we apply our result to the following boundary value problem (simply called BVP) for the partial difference equation:
where is defined as in the beginning of Section 2, is strictly increasing odd function satisfying for , satisfies
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  are not able to solve it. In the following we first apply our main result to the discussion of boundedness of (4.1).
All solutions of BVP (4.1) have the following estimation for all
where are given as in Theorem 2.1 and
Clearly, the difference equation of BVP (4.1) is equivalent to
It follows, by (4.2), that
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.
Corollary 4.1 gives a condition of boundedness for solutions. Concretely, if
for all , then every solution of BVP (4.1) is bounded on .
Next, we discuss the uniqueness of solutions for BVP (4.1).
Suppose additionally that
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 .
Assume that both and are solutions of BVP (4.1). From the equivalent form (4.5) of (4.1) we have
for all , which is an inequality of the form (1.7), where . Applying our Theorem 2.1 with the choice that , we obtain an estimate of the difference in the form (2.4), where because . Furthermore, by the definition of we see that
It follows that
since . Thus, by (4.10),
Similarly, we get and therefore
Thus we conclude from (2.4) that , implying that for all since is strictly increasing. It proves the uniqueness.
If or in (4.8), the conclusion of the Corollary 4.2 also can be obtained.
Finally, we discuss the continuous dependence of solutions of BVP (4.1) on the given functions , and . Consider a variation of BVP (4.1)
where is strictly increasing odd function satisfying for , , and are functions satisfying .
Let be a function as assumed in the beginning of Section 4 and satisfy (4.2) and (4.8) on the same lattice as assumed in Corollary 4.2. Suppose that the three differences
are all sufficiently small. Then solution of BVP (4.14) is sufficiently close to the solution of BVP (4.1).
By Corollary 4.2, the solution is unique. By the continuity and the strict monotonicity of , we suppose that
where is a small number. By the equivalent difference equation (4.5) and the inequality (4.8) we get
that is an inequality of the form (1.7). Applying Theorem 2.1 to (4.17), we obtain, for all , that
where are given as in Theorem 2.1,
By (4.10) we see that () as . It follows from (4.18) that and hence depends continuously on and .
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.
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):292-296. 10.2307/1967124
Bellman R: The stability of solutions of linear differential equations. Duke Mathematical Journal 1943, 10: 643-647. 10.1215/S0012-7094-43-01059-2
Agarwal RP, Deng S, Zhang W: Generalization of a retarded Gronwall-like inequality and its applications. Applied Mathematics and Computation 2005,165(3):599-612. 10.1016/j.amc.2004.04.067
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.
Cheung W-S, Ma Q-H: On certain new Gronwall-Ou-Iang type integral inequalities in two variables and their applications. Journal of Inequalities and Applications 2005,10(4):347-361.
Lipovan O: Integral inequalities for retarded Volterra equations. Journal of Mathematical Analysis and Applications 2006,322(1):349-358. 10.1016/j.jmaa.2005.08.097
Ma Q-H, Yang E-H: On some new nonlinear delay integral inequalities. Journal of Mathematical Analysis and Applications 2000,252(2):864-878. 10.1006/jmaa.2000.7134
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.
Pachpatte BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering. Volume 197. Academic Press, San Diego, Calif, USA; 1998:x+611.
Wang W-S: A generalized retarded Gronwall-like inequality in two variables and applications to BVP. Applied Mathematics and Computation 2007,191(1):144-154. 10.1016/j.amc.2007.02.099
Wang W-S: A generalized sum-difference inequality and applications to partial difference equations. Advances in Difference Equations 2008, 2008:-12.
Wang W-S, Shen C-X: On a generalized retarded integral inequality with two variables. Journal of Inequalities and Applications 2008, 2008:-9.
Wang W-S: Estimation on certain nonlinear discrete inequality and applications to boundary value problem. Advances in Difference Equations 2009, 2009:-8.
Zhang W, Deng S: Projected Gronwall-Bellman's inequality for integrable functions. Mathematical and Computer Modelling 2001,34(3-4):393-402. 10.1016/S0895-7177(01)00070-X
Zheng K, Wu Y, Deng S: Nonlinear integral inequalities in two independent variables and their applications. Journal of Inequalities and Applications 2007, 2007:-11.
Hull TE, Luxemburg WAJ: Numerical methods and existence theorems for ordinary differential equations. Numerische Mathematik 1960, 2: 30-41. 10.1007/BF01386206
Pachpatte BG, Deo SG: Stability of discrete-time systems with retarded argument. Utilitas Mathematica 1973, 4: 15-33.
Willett D, Wong JSW: On the discrete analogues of some generalizations of Gronwall's inequality. Monatshefte für Mathematik 1965, 69: 362-367. 10.1007/BF01297622
Pachpatte BG: On some fundamental integral inequalities and their discrete analogues. Journal of Inequalities in Pure and Applied Mathematics 2001,2(2, article 15):1-13.
Dafermos CM: The second law of thermodynamics and stability. Archive for Rational Mechanics and Analysis 1979,70(2):167-179.
Pang PYH, Agarwal RP: On an integral inequality and its discrete analogue. Journal of Mathematical Analysis and Applications 1995,194(2):569-577. 10.1006/jmaa.1995.1318
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):128-144. 10.1006/jmaa.1995.1008
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.
Pachpatte BG: Integral and Finite Difference Inequalities and Applications, North-Holland Mathematics Studies. Volume 205. Elsevier Science, Amsterdam, The Netherlands; 2006:x+309.
Cheung W-S: Some discrete nonlinear inequalities and applications to boundary value problems for difference equations. Journal of Difference Equations and Applications 2004,10(2):213-223. 10.1080/10236190310001604238
Cheung W-S, Ren J: Discrete non-linear inequalities and applications to boundary value problems. Journal of Mathematical Analysis and Applications 2006,319(2):708-724. 10.1016/j.jmaa.2005.06.064
Pinto M: Integral inequalities of Bihari-type and applications. Funkcialaj Ekvacioj 1990, 33: 387-430.
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 (2006A-N001) and Key Discipline of Applied Mathematics of Hechi University of China (200725).
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Wang, WS., Zhou, X. An Extension to Nonlinear Sum-Difference Inequality and Applications. Adv Differ Equ 2009, 486895 (2009). https://doi.org/10.1155/2009/486895
- Difference Equation
- Open Interval
- Invariant Manifold
- Nonnegative Function
- Discrete Version