An Extension to Nonlinear Sum-Difference Inequality and Applications

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

(1.1)

discussed in [19], an interesting direction is to consider the inequality

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

was discussed. Later, this result was generalized in [26] to the inequality

(1.6)

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 ,

(1.7)

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

(2.1)

For given constants and variable , we define

(2.2)

Obviously, is strictly increasing in and therefore the inverses are well defined, continuous, and increasing. Let

(2.3)

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 ,

(2.4)

where is determined recursively by

(2.5)

and is arbitrarily given on the boundary of the lattice

(2.6)

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

(2.7)

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

(2.8)

the left-hand side of which is equal to

(2.9)

and the right-hand side of which equals

(2.10)

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,

(3.1)

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

(3.2)

Then, we discuss the case that for all . Because satisfies

(3.3)

it is positive and nondecreasing on . We consider the auxiliary inequality to (3.2), for all ,

(3.4)

where and are chosen arbitrarily, and claim that, for , a sublattice in ,

(3.5)

where is determined recursively by

(3.6)

, and is arbitrarily chosen on the boundary of the lattice

(3.7)

We note that , can be chosen appropriately such that

(3.8)

In fact, from the fact of being on the boundary of the lattice , we see that

(3.9)

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

(3.10)

Moreover, we note that is nondecreasing and satisfies for and that . From (3.10) we have

(3.11)

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

(3.12)

It follows from (3.11) and (3.12) that

(3.13)

Substituting with and summing both sides of (3.13) from to , we get, for all ,

(3.14)

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

(3.15)

that is, (3.5) is true for .

Next, we make the inductive assumption that (3.5) is true for . Consider

(3.16)

for all . Let which is nonnegative and nondecreasing in each variable on . Then (3.16) is equivalent to

(3.17)

Since is nondecreasing and satisfies for and , from (3.17) we obtain, for all ,

(3.18)

where

(3.19)

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

(3.20)

Therefore, it follows from (3.18) and (3.20) that

(3.21)

substituting with in (3.21) and summing both sides of (3.21) from to , we get, for all ,

(3.22)

where we note that . For convenience, let

(3.23)

From (3.17) and (3.22) we can get

(3.24)

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,

(3.25)

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 ,

(3.26)

where , is the inverse of (for ), is determined recursively by

(3.27)

and , are functions of such that lie on the boundary of the lattice

(3.28)

where denotes either if it converges or . Note that

(3.29)

Thus, from (3.17), (3.23), and (3.27), (3.26) can be equivalently written as

(3.30)

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

(3.31)

where is an arbitrary small number. Obviously, for all . Using the same arguments as above and replacing with , we get

(3.32)

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:

(4.1)

where is defined as in the beginning of Section 2, is strictly increasing odd function satisfying for , satisfies

(4.2)

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

(4.3)

where are given as in Theorem 2.1 and

(4.4)

Proof.

Clearly, the difference equation of BVP (4.1) is equivalent to

(4.5)

It follows, by (4.2), that

(4.6)

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

(4.7)

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

(4.8)

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

(4.9)

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

(4.10)

It follows that

(4.11)

since . Thus, by (4.10),

(4.12)

Similarly, we get and therefore

(4.13)

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)

(4.14)

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

(4.15)

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

(4.16)

where is a small number. By the equivalent difference equation (4.5) and the inequality (4.8) we get

(4.17)

that is an inequality of the form (1.7). Applying Theorem 2.1 to (4.17), we obtain, for all , that

(4.18)

where are given as in Theorem 2.1,

(4.19)

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.