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An Extension to Nonlinear Sum-Difference Inequality and Applications
Advances in Difference Equations volume 2009, Article number: 486895 (2009)
Abstract
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.
1. Introduction
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 [19], an interesting direction is to consider the inequality

a discrete version of Dafermos' inequality [20], where are nonnegative constants and
are nonnegative functions defined on
and
, respectively. Pang and Agarwal [21] proved for (1.2) that
for all
. Another form of sum-difference inequality

was estimated by Pachpatte [22] as , where
. Recently, Pachpatte [23, 24] discussed the inequalities of two variables

where is nondecreasing. In [25] another form of inequality of two variables

was discussed. Later, this result was generalized in [26] 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 [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 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
.
Theorem 2.1.
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

Remark 2.2.
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
.
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
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 [27] 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
,

where

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 [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.
All solutions of BVP (4.1) have the following estimation for all

where are given as in Theorem 2.1 and

Proof.
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).
Corollary 4.2.
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
.
Proof.
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.
Remark 4.3.
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
.
Corollary 4.4.
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).
Proof.
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
.
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.
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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 (2006A-N001) and Key Discipline of Applied Mathematics of Hechi University of China (200725).
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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
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DOI: https://doi.org/10.1155/2009/486895
Keywords
- Difference Equation
- Open Interval
- Invariant Manifold
- Nonnegative Function
- Discrete Version