Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 8 (2015), 142–152
Research Article
Uniqueness and global exponential stability of
almost periodic solution for Hematopoiesis model on
time scales
Zhijian Yao
Department of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601,China.
Communicated by C. Park
Abstract
This paper deals with almost periodic Hematopoiesis dynamic equation on time scales. By applying a
novel method based on the fixed point theorem of decreasing operator, we establish sufficient conditions
for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence which
converges to the almost periodic positive solution. Moreover, we investigate global exponential stability of
c
the almost periodic positive solution by means of Gronwall inequality. 2015
All rights reserved.
Keywords: Hematopoiesis model on time scales, almost periodic solution, global exponential stability,
fixed point theorem of decreasing operator, exponential dichotomy.
2010 MSC: 34K14, 34K20.
1. Introduction
In 1977, Mackey and Glass [15] investigated the Hematopoiesis model
x0 (t) = −ax(t) +
β
1+
xN (t
− τ)
,
which described the production of blood cells. Gyori and Ladas [6] have investigated the global attractively
of positive equilibrium for this model. Moreover, the above model and some generalized models have been
investigated by many authors, see [4, 7, 11, 12, 16, 17, 18]. Due to the various seasonal effects of the
environmental factors in real life situation (e.g., seasonal effects of weather, food supplies, mating habits,
Email address: zhijianyao@126.com (Zhijian Yao)
Received 2014-11-20
Zhijian Yao, J. Nonlinear Sci. Appl. 8 (2015), 142–152
143
harvesting, etc.), it is rational and practical to study the biological system with periodic coefficients or
almost periodic coefficients. Some authors [17, 18] have studied nonautonomous differential equations with
periodic coefficients of the above model.
As we know, in the real world, some processes vary continuously while others vary discretely. These
processes can be modeled by differential equations and difference equations, respectively. However, there
are also many processes that vary both continuously and discretely. The theory of time scale calculus and
dynamic equations on time scales provides us with a powerful tool for solving such mixed processes. The
calculus on time scales (see [1, 2] and references cited therein) was initiated by Stefan Hilger in his 1988
Ph.D. dissertation [9] in order to unify continuous and discrete analysis, and it has a tremendous potential
for applications and has recently received great attention. The two main features of the calculus on time
scales are unification and extension.
The existence and stability of periodic solution or almost periodic solution for differential equations and
difference equations are very basic and important problems. It is natural to ask whether we can explore
such existence and stability problems in a unified way and offer more general conclusions. The study of
dynamic equations on time scales can unify and extend the fields of differential and difference equations.
Motivated by the above facts, in this paper, we investigate the following nonautonomous almost periodic
Hematopoiesis dynamic equation on time scales
∆
x (t) = −a(t)x(t) +
m
X
i=1
1+
βi (t)
.
(t − τi (t))
xNi
(1.1)
Almost periodicity is closer to the reality in biological systems [5, 10]. However, to our knowledge, no
papers deal with the existence and global exponential stability of unique almost periodic positive solution
for the above model (1.1) on time scales.
In this paper, we aim to establish sufficient conditions that guarantee the existence of unique almost
periodic positive solution of model (1.1) on time scales. The technique used in this paper is different from
the usual methods employed to solve almost periodic cases such as the contraction mapping principle and
Liapunov functional. Our method is based on the fixed point theorem of decreasing operator. Moreover,
we also investigate global exponential stability of almost periodic positive solution by means of Gronwall
inequality. The results of this paper are new and more valuable in applications, which complement and
extend the previously obtained results in [4, 6, 7, 11, 12, 16, 17, 18]. Our study reveals that it is unnecessary
to prove results for differential equations and separately again for difference equations. We can unify such
existence and stability problems in the framework of dynamic equations on time scales.
2. Preliminaries
In this section, we present some basic definitions and preliminary results from the calculus on time scales
and almost periodic functions. For more details, see [1, 2, 13, 14].
The symbol T denotes a time scale, which is a nonempty closed subset of R. Some examples of such
time scales are
R,
Z,
[
[2k, 2k + 1],
k∈Z
[ [
{k +
k∈Z n∈N
1
}.
n
Definition 2.1. The forward and backward jump operators σ, ρ: T → T and the graininess µ: T → R+ are
defined, respectively, by
σ(t) = inf {s ∈ T : s > t} , ρ(t) = sup {s ∈ T : s < t} , µ(t) = σ(t) − t.
A point t ∈ T is called left-dense if t > inf T and ρ(t) = t, left-scattered if ρ(t) < t, right-dense if
t < sup T and σ(t) = t, and right-scattered if σ(t) > t.
Zhijian Yao, J. Nonlinear Sci. Appl. 8 (2015), 142–152
144
If T has a left-scattered maximum m, define Tk = T − {m}; otherwise, set Tk = T.
If T has a right-scattered minimum m, define Tk = T − {m}; otherwise, set Tk = T.
Definition 2.2. A function f : T → R is right-dense continuous provided it is continuous at right-dense
points in T and its left-side limits exist (finite) at left-dense points in T. If f is continuous at each right-dense
point and each left-dense point, then f is said to be a continuous function on T.
Definition 2.3. For f : T → R, we define f ∆ (t) to be the number (if it exists) with the property that for
any given
U of t such that
ε > 0, there exists∆a neighborhood
(f (σ(t)) − f (s)) − f (t) (σ(t) − s) < ε |σ(t) − s| for all s ∈ U .
We call f ∆ (t) the delta (or Hilger) derivative of f at t.
If F ∆ (t) = f (t), then we define the delta integral by
Z t
f (s)∆s = F (t) − F (r) f or t, r ∈ T.
r
Definition 2.4. A function p : T → R is called regressive provided 1 + µ(t)p(t) 6= 0 for all t ∈ T. The set
of all regressive and rd-continuous functions p : T → R will be denoted by < = <(T, R). We define the set
<+ = <+ (T, R) = {p ∈ < : 1 + µ(t)p(t) > 0, ∀t ∈ T}.
Definition 2.5. If p is a regressive function, then the generalized exponential function ep is defined as the
unique solution of the initial value problem y ∆ = p(t)y, y(s) = 1, where s ∈ T.
An explicit formula for epn(t, s) is given by o
Rt
ep (t, s) = exp s ξµ(τ ) (p(τ )) ∆τ
for all s, t ∈ T
with
Log(1+hz)
if h 6= 0,
,
h
ξh (z) =
if h = 0.
z,
Definition 2.6. Let p, q : T → R are two regressive functions, define
p ⊕ q = p + q + µpq, p = −
p
, p q = p ⊕ (q).
1 + µp
Lemma 2.7. Assume that p, q : T → R are two regressive functions, then
(i) e0 (t, s) ≡ 1, ep (t, t) ≡ 1 ;
(ii) ep (σ(t), s) = (1 + µ(t)p(t)) ep (t, s);
(iii)
1
ep (t,s)
= ep (t, s), ep (t, s) =
1
ep (s,t)
= ep (s, t);
(iv) ep (t, s)ep (s, r) = ep (t, r), ep (t, s)eq (t, s) = ep⊕q (t, s);
(v) (ep (t, s))∆ = pep (t, s);
(vi) If a, b, c ∈ T, then
Rb
a
p(s)ep (c, σ(s))∆s = ep (c, a) − ep (c, b).
Definition 2.8. ([13]) Let Γ be a collection of sets which is constructed by subsets of R. A time scale T
is called an almost periodic time scale with respect to Γ, if
(
)
\
∗
Γ = ±τ ∈
Λ : t ± τ ∈ T, ∀t ∈ T 6= Ø
Λ∈Γ
and Γ∗ is called the smallest almost periodic set of T.
Zhijian Yao, J. Nonlinear Sci. Appl. 8 (2015), 142–152
145
Definition 2.9. ([13]) Let T be an almost periodic time scale with respect to Γ. A function f (t) ∈ C (T, Rn )
is called almost periodic if for any given ε > 0, the set E(f, ε) = {τ ∈ Γ∗ : |f (t + τ ) − f (t)| < ε, ∀t ∈ T} is
relatively dense in T; that is, for any given ε > 0, there exists a real number l = l(ε) > 0 such that each
interval of length l contains at least one τ = τ (ε) ∈ E(f, ε) satisfying |f (t + τ ) − f (t)| < ε, ∀t ∈ T.
The set E(f, ε) is called ε-translation set of f (t), τ is called ε-translation number of f (t), and l(ε) is called
contain interval length of E(f, ε).
Remark 2.10. If Γ = {R} and T = R , then Γ∗ = R, in this case, Definition 2.9 is equivalent to the definition
of almost periodic function in [5]. If Γ = {Z} and T = Z , then Γ∗ = Z, in this case, Definition 2.9 is
equivalent to the definition of almost periodic sequence in [3].
Lemma 2.11. ([13]) Let f ∈ C (T, Rn ) be an almost periodic function, then f (t) is bounded on T.
Lemma 2.12. ([13]) If f, g ∈ C (T, Rn ) are almost periodic, then f + g and f g are almost periodic.
Lemma 2.13. ([13]) If f (t) is almost periodic and G(·) is uniformly continuous defined on the value field
of f (t), then G ◦ f is almost periodic.
Lemma 2.14. ([13]) If f (t) ∈ CR (T, Rn ) is almost periodic, then F (t) is almost periodic if and only if F (t)
t
is bounded on T , where F (t) = 0 f (s)∆s.
Definition 2.15. ([13, 19])
The linear system
Let Q(t) be n × n rd-continuous matrix function on T.
x∆ (t) = Q(t)x(t),
t∈T
(2.1)
is said to admit an exponential dichotomy on T if there exist positive constants k, α, projection P and the
fundamental solution matrix X(t) of (2.1) satisfying
X(t)P X −1 (σ(s)) ≤ keα (t, σ(s))
for t ≥ σ(s),
X(t)(I − P )X −1 (σ(s)) ≤ keα (σ(s), t)
for t ≤ σ(s),
s, t ∈ T,
s, t ∈ T.
Consider almost periodic system
x∆ (t) = Q(t)x(t) + g(t),
t ∈ T,
(2.2)
where Q(t) is an almost periodic matrix function, g(t) is an almost periodic vector function.
Lemma 2.16. ([13, 14]) If the linear system (2.1) admits an exponential dichotomy, then the almost
periodic system (2.2) has a unique almost periodic solution x(t) as follows
Z t
Z +∞
−1
x(t) =
X(t)P X (σ(s))g(s)∆s −
X(t)(I − P )X −1 (σ(s))g(s)∆s.
−∞
t
Lemma 2.17. ([1]) Let Q(t) be a regressive n × n matrix-valued function on T. Let t0 ∈ T and x0 ∈ Rn ,
then the initial value problem
x(t0 ) = x0
x∆ (t) = Q(t)x(t),
has a unique solution x(t) as follows
x(t) = eQ (t, t0 )x0 .
+
Lemma
2.18. ([13]) Let ci (t) be almost periodic function on T, where ci (t) > 0, −ci (t) ∈ < , ∀t ∈ T and
min inf ci (t) > 0.
1≤i≤n
t∈T
Then the linear system
x∆ (t) = diag (−c1 (t), −c2 (t), · · ·, −cn (t)) x(t)
admits an exponential dichotomy on T.
Zhijian Yao, J. Nonlinear Sci. Appl. 8 (2015), 142–152
146
By Lemma 2.17, we can get
Lemma 2.19.
Let −C = diag (−c1 (t), −c2 (t), · · ·, −cn (t)), then X(t) = e−C (t, t0 ) is a fundamental
solution matrix of the linear system x∆ (t) = diag (−c1 (t), −c2 (t), · · ·, −cn (t)) x(t).
Definition 2.20. Let X be a Banach space and P be a closed,nonempty subset of X, P is called a cone if
(i)x ∈ P, λ ≥ 0 implies λx ∈ P ; (ii)x ∈ P, −x ∈ P implies x = θ. ( θ is zero element).
Every cone P ⊂ X induces an ordering in X, we define ’≤’ with respect to P by x ≤ y if and only if
y − x ∈ P.
Definition 2.21. A cone P of X is called normal cone if there exists a positive constant σ, such that
||x + y|| ≥ σ for any x, y ∈ P ,||x|| = ||y|| = 1.
Definition 2.22. Let P be a cone of X and A : P → P an operator. A is called decreasing if θ ≤ x ≤ y
implies Ax ≥ Ay.
The following fixed point theorem of decreasing operator (see [8]) is an important tool in our proofs.
Lemma 2.23. ([8]) Suppose that
(i) P is normal cone of Banach space X, operator A : P → P is decreasing;
(ii) Aθ > θ, A2 θ ≥ ε0 Aθ, where ε0 > 0;
(iii) For ∀0 < c < d < 1, there exists η = η(c, d) > 0 such that
A(λx) ≤ [λ(1 + η)]−1 Ax
for ∀c ≤ λ ≤ d and θ < x ≤ Aθ.
Then, the operator A has a unique positive fixed point x∗ > θ. Moreover, kxk − x∗ k → 0, (k → ∞) , where
xk = Axk−1 (k = 1, 2, · · ·) for any initial x0 ∈ P .
Remark 2.24. In Lemma 2.23, the operator A does not need continuity and compactness.
3. Existence of the unique almost periodic positive solution
In this paper, we use notations: for any bounded function f (t), we denote f¯ = sup f (t), f = inf f (t).
t∈T
t∈T
Throughout this paper, we assume that the bounded almost periodic functions a(t), βi (t), τi (t) satisfy
0 < a ≤ a(t) ≤ ā, 0 < βi ≤ βi (t) ≤ βi , 0 < τi ≤ τi (t) ≤ τi , −a(t) ∈ <+ and Ni > 0 (i = 1, 2, · · ·, m).
Due to biological significance, we restrict our attention to positive solutions of equation (1.1). The initial
condition associated with equation (1.1) is given by
x(t; φ) = φ(t) > 0 for t ∈ [−τ ∗ , 0]T , τ ∗ = max {τi }.
1≤i≤m
Let X = {w(t)|w ∈ C(T, R), w(t) is almost periodic function} with the norm kwk = sup |w(t)| , then X is
t∈T
Banach space.
For w(t) ∈ X , we consider equation
x∆ (t) = −a(t)x(t) +
m
X
i=1
βi (t)
.
1 + wNi (t − τi (t))
(3.1)
Since inf a(t) = a > 0, then from Lemma 2.18 we know that the linear equation x∆ (t) = −a(t)x(t) admits
t∈T
exponential dichotomy on T.
Hence, by Lemma 2.16, we know that equation (3.1) has exactly one almost periodic solution:
Z t
m
X
βi (s)
xw (t) =
e−a (t, σ(s))
∆s.
N
1 + w i (s − τi (s))
−∞
i=1
We define operator
A:X→X ,
Z t
m
X
(Aw)(t) =
e−a (t, σ(s))
−∞
i=1
βi (s)
∆s,
1 + wNi (s − τi (s))
w ∈ X.
Zhijian Yao, J. Nonlinear Sci. Appl. 8 (2015), 142–152
147
Obviously, w(t) is the almost periodic solution of equation (1.1) if and only if w is the fixed point of the
operator A.
Define a cone Ω = {w|w ∈ X, w(t) ≥ 0, t ∈ T}.
m
1X
Let M =
βi .
a
i=1
We make assumptions:
(C1 ) 0 < Ni ≤ 1;
(Ni − 1)M Ni ≤ 1.
(C2 ) Ni > 1,
Theorem 3.1. Assume that for any i ∈ {1, 2, · · ·, m}, (C1 ) or (C2 ) holds, then equation (1.1) has a unique
almost periodic positive solution w∗ (t). Moreover, kwk − w∗ k → 0, (k → ∞), wk = Awk−1 (k = 1, 2, · · ·)
for any initial w0 ∈ Ω.
Proof. Firstly, we prove that AΩ ⊂ Ω.
For ∀w ∈ Ω , then
Z t
m
X
e−a (t, σ(s))
(Aw)(t) =
−∞
i=1
1+
βi (s)
∆s > 0.
(s − τi (s))
w Ni
In addition, for ∀w ∈ Ω , we know that equation (3.1) has exactly one almost periodic solution
Z t
m
X
βi (s)
∆s.
xw (t) =
e−a (t, σ(s))
N
1 + w i (s − τi (s))
−∞
i=1
Since xw (t) is almost periodic, then (Aw)(t) is almost periodic.
This, together with (3.2), implies Aw ∈ Ω. So we have AΩ ⊂ Ω.
It is clear that Ω is normal cone, A : Ω → Ω is decreasing operator.
Now, we will show that condition (ii) of Lemma 2.23 is satisfied.
Note that
Z
t
e−a (t, σ(s))
(Aθ)(t) =
−∞
m
X
Z
t
βi (s)∆s ≤
e−a (t, σ(s))
−∞
i=1
m
X
βi ∆s
i=1
m
=
1X
βi = M,
a
i=1
and
t
Z
(Aθ)(t) =
e−a (t, σ(s))
−∞
Z
βi (s)∆s
i=1
t
≥
m
X
e−ā (t, σ(s))
−∞
m
X
i=1
m
1X
βi ∆s =
βi > 0,
ā
i=1
which implies Aθ > θ.
Again, we have
(A2 θ)(t) =
t
Z
e−a (t, σ(s))
−∞
t
i=1
Z
≥
m
X
1
≥
1+B
Z
t
−∞
m
X
βi (s)
∆s
1 + M Ni
i=1
m
X
e−a (t, σ(s))
βi (s)∆s = ε0 (Aθ)(t),
e−a (t, σ(s))
−∞
βi (s)
∆s
1 + (Aθ)Ni (s − τi (s))
i=1
(3.2)
Zhijian Yao, J. Nonlinear Sci. Appl. 8 (2015), 142–152
this implies A2 θ ≥ ε0 Aθ, here ε0 =
1
1+B ,
148
N
M i .
B = max
1≤i≤m
Finally, we show that condition (iii) of Lemma 2.23 is satisfied.
Let ∀0 < c < d < 1, for ∀c ≤ λ ≤ d and θ < x ≤ Aθ, we have 0 < ||x|| ≤ ||Aθ|| ≤ M .
Z
t
A(λx)(t) =
e−a (t, σ(s))
−∞
Z
i=1
t
e−a (t, σ(s))
=
−∞
Note that
m
X
m
X
i=1
≤λ
∆s
βi (s)
1 + xNi (s − τi (s))
∆s.
1 + xNi (s − τi (s)) 1 + λNi xNi (s − τi (s))
λNi − 1
1+
1 + λNi M Ni
So we obtain
t
Z
A(λx)(t) ≤
e−a (t, σ(s))
−∞
1
=
λ
fi (t) =
− τi (s))
1 + xNi (s − τi (s))
λNi − 1
−Ni
=
λ
1
+
1 + λNi xNi (s − τi (s))
1 + λNi xNi (s − τi (s))
−Ni
Let
1+
βi (s)
N
i
λ xNi (s
M Ni )t
(1 +
,
1 + tNi M Ni
=
m
X
i=1
t
Z
e−a (t, σ(s))
−∞
1 + M Ni
.
1 + λNi M Ni
1 + M Ni
βi (s)
∆s
1 + xNi (s − τi (s)) 1 + λNi M Ni
m
X
i=1
(1 + M Ni )λ
βi (s)
∆s.
1 + xNi (s − τi (s)) 1 + λNi M Ni
we have
0
fi (t) =
1 + M Ni
1 + (1 − Ni )tNi M Ni
(1 + tNi M Ni )2
.
0
Since (C1 ) 0 < Ni ≤ 1 or (C2 ) Ni > 1, (Ni − 1)M Ni ≤ 1 (i = 1, 2, · · ·, m) holds, then we know fi (t) > 0 for
0 < t < 1, so we have 0 = fi (0) < fi (c) ≤ fi (λ) ≤ fi (d) < fi (1) = 1.
Hence we get
1
λ
Z
1
≤
λ
Z
A(λx)(t) ≤
≤
=
e−a (t, σ(s))
−∞
m
X
i=1
t
e−a (t, σ(s))
−∞
1
g(d)
λ
m
X
i=1
Z
t
e−a (t, σ(s))
−∞
βi (s)
fi (λ)∆s
1 + xNi (s − τi (s))
1+
m
X
i=1
βi (s)
fi (d)∆s
(s − τi (s))
xNi
βi (s)
∆s
1 + xNi (s − τi (s))
1
1
1
1
1
(Ax)(t) = ·
g(d)(Ax)(t) = ·
(Ax)(t),
λ
λ 1+ 1 −1
λ 1 + η(d)
g(d)
here g(d) = max {fi (d)} ,
1≤i≤m
t
0 < g(d) < 1, η = η(d) =
1
g(d)
− 1 > 0.
By Lemma 2.23, we know the operator A has a unique positive fixed point w∗ > θ , which means equation
(1.1) has a unique almost periodic positive solution w∗ (t). Moreover, kwk − w∗ k → 0, (k → ∞), wk =
Awk−1 (k = 1, 2, · · · ) for any initial w0 ∈ Ω. The proof of Theorem 3.1 is completed.
Zhijian Yao, J. Nonlinear Sci. Appl. 8 (2015), 142–152
149
Remark 3.2. Theorem 3.1 of this paper not only gives sufficient conditions for the existence of unique almost
periodic positive solution, but also gives iterative sequence {wk (t)}, which converges to the almost periodic
positive solution w∗ (t).
Remark 3.3. From the above proof, we have
w∗ (t)
=
(Aw∗ )(t)
t
Z
e−a (t, σ(s))
=
m
X
−∞
t
Z
≤
e−a (t, σ(s))
−∞
i=1
m
X
Z
1+
βi (s)
N
∗
w i (s −
τi (s))
m
X
t
βi (s)∆s ≤
e−a (t, σ(s))
−∞
i=1
∆s
βi ∆s
i=1
m
1X
=
βi = M.
a
i=1
On the other hand, we also have
w∗ (t)
=
(Aw∗ )(t)
Z
t
=
e−a (t, σ(s))
−∞
Z
t
≥
e−a (t, σ(s))
−∞
So we get
m
X
i=1
1
≥
1+B
Z
1
≥
1+B
Z
i=1
1+
βi (s)
N
∗
w i (s −
τi (s))
∆s
βi (s)
∆s
1 + M Ni
t
e−a (t, σ(s))
m
X
−∞
i=1
t
m
X
e−ā (t, σ(s))
−∞
m
X
i=1
βi (s)∆s
m
X
1
βi ∆s =
βi .
ā(1 + B)
i=1
m
X
1
βi ≤ w∗ (t) ≤ M ,
ā(1 + B)
i=1
here B = max
1≤i≤m
M Ni .
4. Global exponential stability of almost periodic positive solution
Theorem 4.1. Assume that Ni ≥ 1,
(Ni − 1)M Ni ≤ 1, (i = 1, 2, · · ·, m) and a >
m
X
βi Ni . Then
i=1
equation (1.1) has a unique globally exponentially stable almost periodic positive solution.
Proof. Since the condition Ni ≥ 1, (Ni − 1)M Ni ≤ 1, (i = 1, 2, · · ·, m) is satisfied, then by Theorem 3.1 we
m
X
1
know equation (1.1) has a unique almost periodic positive solution w∗ (t), and
βi ≤ w∗ (t) ≤ M .
ā(1 + B)
i=1
Let ψ(t) be the initial function of w∗ (t), w∗ (t; ψ) = ψ(t) for t ∈ [−τ ∗ , 0]T . Now we prove w∗ (t) is globally
exponentially stable.
t ∈ [−τ ∗ , 0]T .
Suppose x(t) is arbitrary solution of equation (1.1) with initial function x(t; φ) = φ(t) > 0,
Zhijian Yao, J. Nonlinear Sci. Appl. 8 (2015), 142–152
Let
150
y(t) = x(t) − w∗ (t), then we have
y ∆ (t) = (x(t) − w∗ (t))∆
!
m
m
X
X
βi (t)
βi (t)
∗
= −a(t)x(t) +
− −a(t)w (t) +
1 + xNi (t − τi (t))
1 + w∗ Ni (t − τi (t))
i=1
i=1
m
m
X
X
βi (t)
βi (t)
−
= −a(t) (x(t) − w∗ (t)) +
.
N
∗
1 + xNi (t − τi (t))
1 + w i (t − τi (t))
i=1
i=1
Let
m
X
h(t) =
i=1
(4.1)
m
X
βi (t)
βi (t)
,
−
N
N
∗
i
1 + x (t − τi (t))
1 + w i (t − τi (t))
i=1
then it follows from (4.1) that
y ∆ (t) = −a(t)y(t) + h(t).
(4.2)
From (4.2), we know that y(t) can be expressed as follows
Z
t
y(t) = e−a (t, t0 )y(t0 ) +
t0
e−a (t, s)h(s)∆s, (t ≥ t0 ), t0 ∈ [−τ ∗ , 0]T .
(4.3)
Thus, (4.3) implies that
Z
t
y(t) = e−a (t, t0 ) (φ(t0 ) − ψ(t0 )) +
e−a (t, s)h(s)∆s.
(4.4)
t0
Note that
m
X
1
1
|h(t)| = −
βi (t)
1 + xNi (t − τi (t)) 1 + w∗ Ni (t − τi (t)) i=1
m
X
1
1
.
≤
βi (t) −
1 + xNi (t − τi (t)) 1 + w∗ Ni (t − τ (t)) i
i=1
By the mean value theorem, we have
1
1
1 + xNi (t − τi (t)) − 1 + w∗ Ni (t − τ (t)) i
N ξ Ni −1
i
∗
[x
(t
−
τ
(t))
−
w
(t
−
τ
(t))]
= −
i
i
(1 + ξ Ni )2
=
Ni ξ Ni −1
(1 +
ξ Ni )2
|x (t − τi (t)) − w∗ (t − τi (t))| ,
in which ξ lies between x (t − τi (t)) and w∗ (t − τi (t)).
Note that the function gi (x) =
Ni xNi −1
(1 + xNi )2
< Ni for ∀x ∈ (0, +∞) and Ni ≥ 1, (i = 1, 2, · · ·, m).
Thus we have
Ni ξ Ni −1
(1 + ξ Ni )2
< Ni
(4.5)
for Ni ≥ 1.
(4.6)
Zhijian Yao, J. Nonlinear Sci. Appl. 8 (2015), 142–152
151
From (4.6), we get
1
1
∗
1 + xNi (t − τi (t)) − 1 + w∗ Ni (t − τ (t)) < Ni |x (t − τi (t)) − w (t − τi (t))| .
i
(4.7)
Hence, by (4.5) and (4.7), we get
|h(t)| <
m
X
∗
∗
βi (t)Ni |x (t − τi (t)) − w (t − τi (t))| ≤ kx − w k
i=1
m
X
βi Ni .
i=1
It follows that
∗
kh(t)k ≤ kx − w k
m
X
βi Ni = kyk
i=1
m
X
βi Ni .
i=1
Take norm at both sides of (4.4), we obtain
Z
t
ky(t)k ≤ e−a (t, t0 ) kφ − ψk +
e−a (t, s) kh(s)k ∆s
t0
Z t
m
X
βi Ni ∆s.
≤ e−a (t, t0 ) kφ − ψk +
e−a (t, s) kyk
t0
(4.8)
i=1
From (4.8), we get
ky(t)k
≤ kφ − ψk +
e−a (t, t0 )
Z
t
t0
m
X
kyk
βi Ni ∆s.
e−a (s, t0 )
i=1
By Gronwall inequality (see [1]), we obtain
m
X
ky(t)k
≤ kφ − ψk eγ (t, t0 ),
here γ =
βi Ni .
e−a (t, t0 )
i=1
Hence we get
ky(t)k ≤ kφ − ψk eγ (t, t0 )e−a (t, t0 )
≤ kφ − ψk eγ (t, t0 )e−a (t, t0 )
= kφ − ψk e−(a−γ) (t, t0 ).
That is kx(t) − w∗ (t)k ≤ kφ − ψk e−(a−γ) (t, t0 ), here a > γ, which means w∗ (t) is globally exponentially
stable. The proof of Theorem 4.1 is completed.
Remark 4.2. As mentioned in the introduction, one of our principal aims is to unify the existence and stability
of almost periodic solution for some differential equations and their corresponding discrete analogues.
If T = R and T = Z, then equation (1.1) reduces to
x0 (t) = −a(t)x(t) +
m
X
i=1
βi (t)
,t ∈ R
1 + xNi (t − τi (t))
and
x(k + 1) − x(k) = −a(k)x(k) +
m
X
i=1
βi (k)
, k ∈ Z,
1 + xNi (k − τi (k))
respectively.
Our studies unify differential equations and difference equations. For the existence and stability of
almost periodic solution of differential equations and difference equations, it is unnecessary to prove results
for differential equations and separately again for their discrete analogues (difference equations). We can
unify such problems in the framework of dynamic equations on time scales.
Zhijian Yao, J. Nonlinear Sci. Appl. 8 (2015), 142–152
152
Acknowledgements This work is supported by Natural Science Foundation of Education Department of
Anhui Province (KJ2014A043).
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