1. Introduction
Being an important tool in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equations, various generalizations of Gronwall inequalities [
1,
2] and their applications have attracted great interests of many mathematicians [
3–
5]. Some recent works can be found in [
6–
16] and references therein. Along with the development of the theory of integral inequalities and the theory of difference equations, more and more attentions are paid to discrete versions of Gronwall type inequalities [
17–
24]. For instance, Pachpatte [
17] considered the following discrete inequality:
(1.1)
In 2006, Cheung and Ren [
18] studied
(1.2)
Later, Zheng et al. [
24] discussed the following discrete inequality:
(1.3)
However, the above results are not applicable to inequalities that consist of multiple iterated sums, in particular those in which composite function of nonlinear function and unknown function is involved in each layer of iterated sums. Hence, it is desirable to consider more general difference inequalities of these extended types. They can be used in the study of certain classes of difference equations or applied in many practical engineering problems.
Motivated by the results given in [
7,
8,
11,
16–
19,
21], in this paper we discuss the following two types of inequalities:
(1.4)
(1.5)
for all
n ∈
N0. All the assumptions on (
1.4) and (
1.5) are given in the next sections. The inequalities (
1.5) consist of multiple iterated sums, and composite function of nonlinear functions and unknown function may be involved in each layer. Under several practical assumptions, the inequalities are solved through rigorous analysis, and explicit bounds for the unknown functions are given clearly. Further, the derived results are applied to the stability problem of a class of linear control systems with nonlinear perturbations.
2. Main Result
In this section, we proceed to solving the difference inequalities (1.4) and (1.5) and present explicit bounds on the embedded unknown functions. Throughout this paper, let N denote the set of all natural numbers, and N0 = [n0, K)∩N where n0 and K are two constants, satisfying K > n0.
The following theorem summarizes the result on the inequality (1.4).
Theorem 2.1. Let u(n) and a(n) be nonnegative functions defined on N0 with a(n) nondecreasing on N0. Moreover, let fi(n, s), i = 1,2, 3 be nonnegative functions for n0 ≤ s ≤ n ≤ K and nondecreasing in n for fixed s ∈ N0. Suppose that w(u) is a nondecreasing function on [0, ∞) with w(u) > 0 for u > 0. Then, the discrete inequality (1.4) gives
(2.1)
where
(2.2)
(2.3)
(2.4)
,
are the inverse functions of
W1,
W2, respectively, and
M1 is the largest natural number such that
(2.5)
Proof. Fix , where M is chosen arbitrarily and M1 is defined by (2.5). For n ∈ NM = [n0, M]∩N, from (1.4), we have
(2.6)
Denote the right-hand side of (
2.6) by
z1(
n), which is a positive and nondecreasing function on
NM with
z1(
n0) =
a(
M). Then, (
2.6) is equivalent to
(2.7)
From (
2.6) and (
2.7), we observe that
(2.8)
Furthermore, it follows from (
2.8) that
(2.9)
On the other hand, by the mean-value theorem for integrals, for arbitrarily given integers
n,
n + 1 ∈
NM, there exists
η in the open interval (
z1(
n),
z1(
n + 1)) such that
(2.10)
where
W1 is defined by (
2.4). By setting
n =
s in (
2.10) and substituting
s =
n0,
n0 + 1,
n0 + 2, …,
n − 1 successively, we obtain
(2.11)
Let
v1(
n) denote the right-hand side of (
2.11), which is a positive and nondecreasing function on
NM with
. Then, (
2.11) is equivalent to
(2.12)
By the definition of
v1, we obtain
(2.13)
Considering (
2.12), (
2.13) and the monotonicity properties of
w,
, and
z1, we get
(2.14)
for all
n ∈
NM. Once again, performing the same procedure as in (
2.10) and (
2.11), (
2.14) gives
(2.15)
for all
n ∈
NM, where
W2 is defined in (
2.3). In the sequel, (
2.7), (
2.12), and (
2.15) render to
(2.16)
Let
n =
M in (
2.16), then, we have
(2.17)
Noticing that
M is chosen arbitrarily, (
2.1) is directly induced by (
2.17). The proof of Theorem
2.1 is complete.
Now, we are in the position of solving the inequality (1.5).
Theorem 2.2. Let the functions u(n), a(n), fi(n, s), i = 1,2, 3, and φ(u) be the same as in Theorem 2.1. Suppose that wi(u), i = 1,2, 3 are nondecreasing functions on [0, ∞) with wi(u) > 0 for u > 0. If u(n) satisfies the discrete inequality (1.5), then
(2.18)
where
(2.19)
(2.20)
(2.21)
(2.22)
,
i = 1,2, 3 are the inverse functions of Φ
i,
i = 1,2, 3, respectively, and
M2 is the largest natural number such that
(2.23)
Proof. Fix , where M is chosen arbitrarily and M2 is given in (2.23). For n ∈ NM, from (1.5), we have
(2.24)
Let
z2(
n) represent the right-hand side of (
2.24), which is a positive and nondecreasing function on
with
z2(
n0) =
a(
M). Then, (
2.24) is equivalent to
(2.25)
Using (
2.24) and (
2.25), Δ
z2(
n): =
z2(
n + 1) −
z2(
n) can be estimated as follows:
(2.26)
Implying
(2.27)
for all
n ∈
NM. Performing the same derivation as in (
2.10) and (
2.11), we obtain from (
2.27) that
(2.28)
where Φ
1 is defined in (
2.20). Denote by
v2(
n) the right-hand side of (
2.28), which is a positive and nondecreasing function on
with
v2(
n0) = Φ
1(
z2(
n0))+
= Φ
1(
a(
M))+
. Then, (
2.28) is equivalent to
(2.29)
By the definition of
v2, we obtain
(2.30)
From (
2.29), (
2.30) and the monotonicity of
, and
z2, we get
(2.31)
for all
n ∈
NM. Similarly to (
2.28), it follows from (
2.31) that
(2.32)
for all
n ∈
NM, where Φ
2 is defined in (
2.21). Let
v3(
n) denote the right-hand side of (
2.32), which is a positive and nondecreasing function on
with
(2.33)
Then, (
2.32) is equivalent to
(2.34)
By the definition of
v3,
(2.35)
In consequence, (
2.34), (
2.35) and the monotonicity properties of
, and
v2 lead to
(2.36)
Similarly to (
2.28) and (
2.32), we obtain from (
2.36) that
(2.37)
where Φ
3 is defined in (
2.22).
Summarizing the results in (2.25), (2.29), (2.34), and (2.37), we can conclude that
(2.38)
for all
n ∈
NM. As
n =
M, (
2.38) yields
(2.39)
Since
M is chosen arbitrarily in (
2.39), the inequality (
2.18) is derived. This completes the proof of Theorem
2.2.
3. Applications
In this section, the result of Theorem
2.2 is applied to explore the asymptotic stability behavior of a class of discrete-time control systems [
17]
(3.1)
where
(3.2)
Control system (
3.1) can be regarded as the perturbation counterpart of the following closed-loop system:
(3.3)
The functions
x,
y,
θ,
σ are defined on
N →
Rr, the
r-dimensional vector space,
A(
n) is an
r ×
r matrix with det
A(
n) ≠ 0, and the functions
f and
k are defined on
N ×
Rr ×
Rr and
N ×
N ×
Rr, respectively. Moreover,
f and
k are supposed to meet the following constraints:
(3.4)
(3.5)
where
α > 0 is a constant,
gi,
i = 1,2, 3 are nonnegative real-valued functions defined on
N0 and
N0 ×
N0, respectively,
g2(
n,
s) and
g3(
n,
s) are nondecreasing in
n for fixed
s ∈
N0, and
wi(
u),
i = 1,2, 3 are positive and continuous functions defined on [0,
∞). The symbol |·| denotes norm on
Rr as well as a corresponding consistent matrix norm.
Corollary 3.1. Consider the discrete-time control systems (3.1) and (3.2), where the perturbation-related functions f and k satisfy the conditions (3.4) and (3.5). Assume that the fundamental solution matrix Y(n) of the linear system (3.3) satisfies
(3.6)
where
C > 0 is a constant. Then, any solutions of the control systems (
3.1) and (
3.2), denoted by
xσ(
n,
n0,
x0), can be estimated by
(3.7)
where
(3.8)
,
i = 4,5, 6 are the inverse functions of Φ
i,
i = 4,5, 6, respectively, and
M4 is the largest natural number such that
(3.9)
Proof. By using the variation of constants formula, any solution xσ(n, n0, x0) of (3.1) and (3.2) can be represented by
(3.10)
for all
n ∈
N0. Using the conditions (
3.4) and (
3.6) in (
3.10), we have
(3.11)
Further, using the relationships (
3.2), (
3.5), and (
3.11), we derive
(3.12)
for all
n ∈
N0. Let
u(
n) = |
xσ(
n,
n0,
x0) | exp (
αn), then, (
3.12) can be rewritten as
(3.13)
Let
a(
n) = |
x0 |
Cexp (
αn0),
f1(
n,
s) =
Cg1(
s)
eα(1+|
θ(
s)|),
f2(
n,
s) =
g2(
n,
s), and
f3(
n,
s) =
g3(
n,
s), then (
3.13) can be further estimated as follows:
(3.14)
for all
n ∈
N0. Notice that, by our assumption, all functions in (
3.14) satisfy the conditions of Theorem
2.2. Applying Theorem
2.2 to the inequality (
3.14), (
3.7) is immediately derived, where the relationship
u(
n) = |
xσ(
n,
n0,
x0) | exp (
αn) is adopted. This completes the proof of Corollary
3.1.
Based on Corollary 3.1 and one additional assumption, the next corollary gives the stability result of the control system (3.1) and (3.2).
Corollary 3.2. Under the assumptions of Corollary 3.1, if there exists a positive constant B such that
(3.15)
then the perturbed system (
3.1) and (
3.2) is exponentially asymptotically stable.
Proof. Under condition (3.15), (3.7) can be further estimated as follows:
(3.16)
The exponentially asymptotic stability of system (
3.1) and (
3.2) is directly implied.
Acknowledgments
This research was supported by National Natural Science Foundation of China (Project no. 11161018), the SERC Research Grant (Project no. 092 101 00558), Scientific Research Foundation of the Education Department of Guangxi Province of China (Project no. 201106LX599), and the Key Discipline of Applied Mathematics of Hechi University of China (200725).
