Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
Research Article
Almost automorphic functions on time scales and
almost automorphic solutions to shunting inhibitory
cellular neural networks on time scales
Yongkun Li∗, Bing Li, Xiaofang Meng
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, People’s Republic of China.
Communicated by J. J. Nieto
Abstract
In this paper, we first propose a new definition of almost automorphic functions on almost periodic time
scales and study some of their basic properties. Then we prove a result ensuring the existence of an almost
automorphic solution for both the linear nonhomogeneous dynamic equation on time scales and its associated
homogeneous equation, assuming that the latter admits an exponential dichotomy. Finally, as an application
of our results, we establish the existence and global exponential stability of almost automorphic solutions to
a class of shunting inhibitory cellular neural networks with time-varying delays on time scales. Our results
about the shunting inhibitory cellular neural networks with time-varying delays on time scales are new both
for the case of differential equations (the time scale T = R) and difference equations (the time scale T = Z).
c
2015
All rights reserved.
Keywords: Time scales, Almost automorphic functions, Dynamic equations, Exponential dichotomy.
2010 MSC: 34N05, 34K14, 43A60, 92B20.
1. Introduction
The theory of time scales was initiated by Hilger [17] in his Ph.D. thesis in 1988, which unifies the
continuous and discrete cases. The theory of dynamic equations on time scales contains, links and extends
the classical theory of differential and difference equations. In the recent years, there has been an increasing
interest in studying the existence of periodic solutions and almost periodic solutions of various dynamic
∗
Corresponding author
Email addresses: yklie@ynu.edu.cn (Yongkun Li), bli123@126.com (Bing Li), xfmeng@126.com (Xiaofang Meng)
Received 2015-07-01
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
1191
equations on time scales; we refer the reader to the papers [1, 2, 3, 8, 12, 14, 15, 16, 18, 22, 27, 29, 30, 33,
34, 35, 38, 40, 41, 43, 45, 46, 47, 48, 50].
The concept of almost automorphy was introduced in the literature by S. Bochner in 1955 in the context
of differential geometry [5] (see also Bochner [6, 7]). Since then, this concept has been extended in various
directions (see, e.g., [4, 11]).
Remark 1.1. Although every almost periodic function is almost automorphic, the converse is not true.
In order to study almost periodic, pseudo almost periodic and almost automorphic dynamic equations
on time scales, the following concept of almost periodic time scales was proposed in [24].
Definition 1.2 ([24]). A time scale T is called almost periodic if
Π = τ ∈ R : t ± τ ∈ T, ∀t ∈ T 6= {0}.
Based on Definition 1.2, almost periodic functions [24], pseudo almost periodic functions [25], almost
automorphic functions [31, 39] and weighted piecewise pseudo almost automorphic functions [36] on time
scales were defined successfully. For example, the authors of [39] proposed the following concept of almost
automorphic functions on time scales.
Definition 1.3 ([39]). Let T be an almost periodic time scale and X be a Banach space. A bounded
0
continuous function f : T → X is said to be almost automorphic if for every sequence (sn ) ⊂ Π there exists
0
a subsequence (sn ) ⊂ (sn ) such that
lim f (t + sn ) = f¯(t)
n→∞
is well defined for each t ∈ T, and
lim f¯(t − sn ) = f (t)
n→∞
for each t ∈ T.
A continuous function f : T × X → X is said to be almost automorphic if f (t, x) is almost automorphic
in t ∈ T uniformly in x ∈ B, where B is any bounded subset of X.
The concept of almost periodic time scales in sense of Definition 1.2 is very restrictive, in the sense that
it is a kind of periodic time scale (see [18]). This excludes many interesting time scales. Therefore, it is a
challenging and important problem in theory and applications to find a new concept of almost periodic time
scales. Recently, some new types of almost periodic time scales were introduced in [20, 28, 37]. However,
there has been no definition of almost automorphic functions on these new almost periodic time scales yet.
Motivated by the above discussions, the main aim of this paper is to propose a new definition of almost
automorphic functions on almost periodic time scales and study the existence of an almost automorphic
solution for both the linear nonhomogeneous dynamic equation on time scales and its associated homogeneous equation. As an application of our results, the existence and global exponential stability of almost
automorphic solutions to a class of shunting inhibitory cellular neural networks with time-varying delays on
time scales is established.
The organization of this paper is as follows: In Section 2, we introduce some notations and definitions
and state some preliminary results which are needed in later sections. In Section 3, we first introduce a
new definition of almost automorphic functions on time scales, then we discuss some of their properties.
Finally, we propose two open problems. In Section 4, we prove a result ensuring the existence of an almost
automorphic solution for both the linear nonhomogeneous dynamic equation on time scales and its associated homogeneous equation, assuming that the associated homogeneous equation admits an exponential
dichotomy. In Section 5, as an application of our results, we establish the existence and global exponential
stability of almost automorphic solutions to a class of shunting inhibitory cellular neural networks with timevarying delays on time scales. In Section 6, we give examples to illustrate the feasibility and effectiveness of
the results obtained in Section 5. Finally, we draw a conclusion in Section 7.
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
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2. Preliminaries
In this section, we shall first recall some fundamental definitions and lemmas which are used in what
follows.
A time scale T is an arbitrary nonempty closed subset of the real set R with the topology and ordering
inherited from R. The forward jump operator σ : T → T is defined by σ(t) = inf s ∈ T,s > t for all
t ∈ T, while the backward jump operator ρ : T → T is defined by ρ(t) = sup s ∈ T, s < t for all t ∈ T.
Finally, the graininess function µ : T → [0, ∞) is defined by 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. If T has a left-scattered maximum m, then Tk = T \ {m}; otherwise Tk = T.
If T has a right-scattered minimum m, then Tk = T \ {m}; otherwise Tk = T. A function f : T → R is
rd-continuous provided it is continuous at right-dense points in T and its left-side limits exist at left-dense
points in T. The set of all rd-continuous functions f : T → R will be denoted by Crd = Crd (T) = Crd (T, R).
A function r : T → R is called regressive if 1 + µ(t)r(t) 6= 0 for all t ∈ Tk . The set of all regressive and
right-dense continuous functions r : T → R will be denoted by R = R(T) = R(T, R). We define the set
R+ = R+ (T, R) = {r ∈ R : 1 + µ(t)r(t) > 0, ∀t ∈ T}. Let A be an m × n-matrix-valued function on T.
We say that A is rd-continuous on T if each entry of A is rd-continuous on T. We denote the class of all
rd-continuous m×n matrix-valued functions on T by Crd = Crd (T) = Crd (T, Rm×n ). An n×n-matrix-valued
function A on a time scale T is called regressive (with respect to T) provided (1 + µ(t)A(t)) is invertible for
all t ∈ Tk . The set of all regressive and rd-continuous functions is denoted by R = R(T) = R(T, Rn×n ). For
more knowledge of time scales, one can refer to [9, 10].
Definition 2.1 ([23]). Let x ∈ Rn and A(t) be an n × n rd-continuous matrix on T. The linear system
x∆ (t) = A(t)x(t), t ∈ T
(2.1)
is said to admit an exponential dichotomy on T if there exist positive constants k, α, a projection P and a
fundamental solution matrix X(t) of (2.1), satisfying
|X(t)P X −1 (σ(s))| ≤ keα (t, σ(s)), s, t ∈ T, t ≥ σ(s),
|X(t)(I − P )X −1 (σ(s))| ≤ keα (σ(s), t), s, t ∈ T, t ≤ σ(s),
where | · | is a matrix norm on T, that is, if A = (aij )n×n then we can take |A| = (
n P
n
P
1
|aij |2 ) 2 ).
i=1 j=1
Definition 2.2 ([20]). Let T1 and T2 be two time scales. We define
dist(T1 , T2 ) = max{sup dist(t, T2 ) , sup dist(t, T1 ) }
t∈T1
t∈T2
where dist(t, T2 ) = inf {|t − s|}, dist(t, T1 ) = inf {|t − s|}. Let τ ∈ R and T be a time scale. We define
s∈T2
s∈T1
dist(T, Tτ ) = max{sup dist(t, Tτ ) , sup dist(t, T) },
t∈T
t∈Tτ
where Tτ := T ∩ {T − τ } = T ∩ {t − τ : ∀t ∈ T}, dist(t, Tτ ) = inf {|t − s|}, dist(t, T) = inf {|t − s|}.
s∈Tτ
s∈T
Definition 2.3 ([20]). A time scale T is called an almost periodic time scale if for every ε > 0 there exists a
constant l(ε) > 0 such that each interval of length l(ε) contains a τ (ε) such that Tτ 6= ∅ and dist(T, Tτ ) < ε,
that is, for any ε > 0, the set Π(T, ε) = {τ ∈ R, dist(T, Tτ ) < ε} is relatively dense. τ is called the
ε-translation number of T.
Obviously, if T is an almost periodic time scale, then inf T = −∞ and sup T = +∞; if T is a periodic
time scale (see [20]), then dist(T, Tτ ) = 0, that is T = Tτ .
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
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Remark 2.4. One can easily see that if a time scale is an almost periodic time scale under Definition 1.2,
then it is also an almost periodic time scale under Definition 2.3.
Lemma 2.5 ([20]). Let T be an almost periodic time scale under Definition 2.3. Then
(i) if τ ∈ Π(T, ε), then t + τ ∈ T for all t ∈ Tτ ;
(ii) if ε1 < ε2 , then Π(T, ε1 ) ⊂ Π(T, ε2 );
(iii) if τ ∈ Π(T, ε), then −τ ∈ Π(T, ε) and dist(Tτ , T) = dist(T−τ , T);
(iv) if τ1 , τ2 ∈ Π(T, ε), then τ1 + τ2 ∈ Π(T, 2ε).
3. Almost periodic time scales and almost automorphic functions
In this section, we first introduce a new concept of almost periodic time scales and a new definition
of almost automorphic functions on time scales, then we discuss some of their properties. The following
definition is a slightly modified version of Definition 2 in [28].
Definition 3.1. A time scale T is called almost periodic if
e=
(i) Π := {τ ∈ R : Tτ 6= ∅} =
6 {0} and T
6 ∅,
(ii) if τ1 , τ2 ∈ Π, then τ1 ± τ2 ∈ Π,
e = T Tτ .
where Tτ = T ∩ {T − τ } = T ∩ {t − τ : ∀t ∈ T} and T
τ ∈Π
e
It is obvious that if τ ∈ Π, then ±τ ∈ Π and t ± τ ∈ T for all t ∈ T.
Remark 3.2. Noticing the fact that Π = {τ ∈ R : Tτ 6= ∅} ⊃ Π(T, ε), from Lemma 2.5, one can easily see
that if a time scale is an almost periodic time scale under Definition 2.3, then it is also an almost periodic
time scale under Definition 3.1.
Definition 3.3. Let T be an almost periodic time scale and X be a Banach space. A bounded rd-continuous
0
function f : T → X is said to be almost automorphic if for every sequence (sn ) ⊂ Π there exists a subsequence
0
(sn ) ⊂ (sn ) such that
lim f (t + sn ) = f¯(t)
n→∞
e and
is well defined for each t ∈ T,
lim f¯(t − sn ) = f (t)
n→∞
e
for each t ∈ T.
Denote by AA(T, X) the set of all almost automorphic functions on the time scale T.
Remark 3.4. From Definition 3.3, one can easily see that if a function f : T → X is almost automorphic
under Definition 1.3, then it is also almost automorphic under Definition 3.3.
Lemma 3.5. Let T be an almost periodic time scale and suppose f, f1 , f2 ∈ AA(T, X). The following
assertions hold:
(i) f1 + f2 ∈ AA(T, X);
(ii) αf ∈ AA(T, X) for any constant α ∈ R;
e
(iii) fc (t) ≡ f (c + t) ∈ AA(T, X) for each fixed c ∈ T.
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
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0
0
Proof. (i) Let f1 , f2 ∈ AA(T, X). Then for every sequence (sn ) ⊂ Π there exists a subsequence (sn ) ⊂ (sn )
such that
lim f1 (t + sn ) = f¯1 (t) and lim f2 (t + sn ) = f¯2 (t)
n→∞
n→∞
e and
are well defined for each t ∈ T,
lim f¯1 (t − sn ) = f1 (t) and
n→∞
lim f¯2 (t − sn ) = f2 (t)
n→∞
e Therefore, we obtain that
for each t ∈ T.
lim (f1 + f2 )(t + sn ) = lim f1 (t + sn ) + f2 (t + sn ) = f¯1 (t) + f¯2 (t)
n→∞
n→∞
e and
is well defined for each t ∈ T,
lim (f¯1 + f¯2 )(t − sn ) = lim f¯1 (t − sn ) + f¯2 (t − sn ) = f1 (t) + f2 (t)
n→∞
n→∞
e
for each t ∈ T.
0
0
(ii) Since f ∈ AA(T, X), it follows that for every sequence (sn ) ⊂ Π there exists a subsequence (sn ) ⊂ (sn )
such that
lim (αf )(t + sn ) = lim αf (t + sn ) = αf¯(t) = (αf¯)(t)
n→∞
n→∞
e and
is well defined for each t ∈ T,
lim (αf¯)(t − sn ) = lim αf¯(t − sn ) = αf (t) = (αf )(t)
n→∞
n→∞
e
for each t ∈ T.
e that for every sequence (s0 ) ⊂ Π there exists a subsequence
(iii) It follows from f ∈ AA(T, X), c ∈ T
n
0
(sn ) ⊂ (sn ) such that
lim fc (t + sn ) = lim f ((c + t) + sn ) = f¯(c + t) = f¯c (t)
n→∞
n→∞
e and
is well defined for each t ∈ T,
lim f¯c (t − sn ) = lim f¯((c + t) − sn ) = f (c + t) = fc (t)
n→∞
n→∞
e The proof is completed.
for each t ∈ T.
Lemma 3.6. Let T be an almost periodic time scale. If the functions f, φ : T → X are almost automorphic,
then the function φf : T → X defined by (φf )(t) = φ(t)f (t) is also almost automorphic.
Proof. Both functions φ and f are bounded since they are almost automorphic. We denote K1 = sup kφ(t)k.
0
00
t∈T
0
00
Given a sequence (sn ) ⊂ Π, there exists a subsequence (sn ) ⊂ (sn ) such that lim φ(t + sn ) = φ̄(t)
n→∞
e and lim φ̄(t − s00 ) = φ(t) for each t ∈ T.
e Since f is almost automorphic,
is well defined for each t ∈ T
n
n→∞
00
e and
there exists a subsequence (sn ) ⊂ (sn ) such that lim f (t + sn ) = f¯(t) is well defined for each t ∈ T
n→∞
e Now, we have
lim f¯(t − sn ) = f (t) for each t ∈ T.
n→∞
kφ(t + sn )f (t + sn ) − φ̄(t)f¯(t)k ≤ kφ(t + sn )f (t + sn ) − φ(t + sn )f¯(t)k + kφ(t + sn )f¯(t) − φ̄(t)f¯(t)k
≤ K1 kf (t + sn ) − f¯(t)k + K2 kφ(t + sn ) − φ̄(t)k
≤ (K1 + K2 )ε
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
1195
for n sufficiently large, where K2 = sup kf¯(t)k < ∞. Thus, we obtain
t∈T
lim φ(t + sn )f (t + sn ) = φ̄(t)f¯(t)
n→∞
e
for each t ∈ T.
It is also easy to check that
lim φ̄(t − sn )f¯(t − sn ) = φ(t)f (t)
n→∞
e The proof is now complete.
for each t ∈ T.
Lemma 3.7. Let T be an almost periodic time scale and (fn ) be a sequence of almost automorphic functions
e Then f is an almost automorphic function.
such that lim fn (t) = f (t) converges uniformly for each t ∈ T.
n→∞
0
Proof. Consider a sequence (sn ) ⊂ Π. As in the standard case of almost automorphic functions, the approach
0
(1)
follows across the diagonal procedure. Since f1 ∈ AA(T, X), there exists a subsequence (sn ) ⊂ (sn ) such
that
¯
lim f1 (t + s(1)
n ) = f1 (t)
n→∞
e and
is well defined for each t ∈ T,
lim f¯1 (t − s(1)
n ) = f1 (t)
n→∞
(1)
e Since f2 ∈ AA(T, X), there exists a subsequence (s(2)
for each t ∈ T.
n ) ⊂ (sn ) such that
¯
lim f2 (t + s(2)
n ) = f2 (t)
n→∞
e and
is well defined for each t ∈ T,
lim f¯2 (t − s(2)
n ) = f2 (t)
n→∞
e Following this procedure, we can construct a subsequence (sn(n) ) ⊂ (s0 ) such that
for each t ∈ T.
n
lim fi (t + sn(n) ) = f¯i (t)
n→∞
(3.1)
e and all i = 1, 2, . . ..
for each t ∈ T
Note that
(n)
(n)
(n)
¯
kf¯i (t) − f¯j (t)k ≤ kf¯i (t) − fi (t + s(n)
n )k + kfi (t + sn ) − fj (t + sn )k + kfj (t + sn ) − fj (t)k.
(3.2)
By the uniform convergence of (fn ), for any ε > 0 we can find N = N (ε) ∈ N sufficiently large such that for
all i, j > N , we have
(n)
kfi (t + s(n)
(3.3)
n ) − fj (t + sn )k < ε
e and all n ≥ 1.
for all t ∈ T
Hence, taking i, j sufficiently large in (3.2) and using (3.1) and (3.3), we can conclude that (f¯i (t)) is a
Cauchy sequence. Since X is a Banach space, it follows that (f¯i (t)) is a sequence which converges pointwise
on X. Let f¯(t) be the limit of (f¯i (t)). Then for each i = 1, 2, . . . we have
(n)
(n)
(n)
¯
¯
¯
¯
kf (t + s(n)
n ) − f (t)k ≤ kf (t + sn ) − fi (t + sn )k + kfi (t + sn ) − fi (t)k + kfi (t) − f (t)k.
(3.4)
Thus, for i sufficiently large, by (3.4) and using the almost automorphicity of fi and the convergence of the
functions fi and f¯i , we obtain
¯
lim f (t + s(n)
n ) = f (t)
n→∞
e Similarly, we can get
for each t ∈ T.
lim f¯(t − s(n)
n ) = f (t)
n→∞
e This completes the proof.
for each t ∈ T.
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
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Lemma 3.8. Let T be an almost periodic time scale and X, Y be Banach spaces. If f : T → X is an almost
automorphic function and ϕ : X → Y is a continuous function, then ϕ ◦ f : T → Y is an almost automorphic
function.
0
0
Proof. Since f ∈ AA(T, X), for every sequence (sn ) ⊂ Π there exists a subsequence (sn ) ⊂ (sn ) such that
e and lim f¯(t − sn ) = f (t) for each t ∈ T.
e
lim f (t + sn ) = f¯(t) is well defined for each t ∈ T
n→∞
n→∞
By the continuity of the function ϕ, it follows that
lim ϕ(f (t + sn )) = ϕ lim f (t + sn ) = (ϕ ◦ f¯)(t).
n→∞
n→∞
On the other hand, we can get
lim ϕ(f¯(t − sn )) = ϕ lim f¯(t − sn ) = (ϕ ◦ f )(t)
n→∞
n→∞
e Thus, ϕ ◦ f ∈ AA(T, Y). The proof is complete.
for each t ∈ T.
Definition 3.9. Let T be an almost periodic time scale and X be a Banach space. A bounded rd-continuous
0
function f : T × X → X is said to be almost automorphic if for any sequence (sn ) ⊂ Π there exists a
0
subsequence (sn ) ⊂ (sn ) such that
lim f (t + sn , x) = f¯(t, x)
n→∞
e t + sn ∈ T, x ∈ X, and
is well defined for each t ∈ T,
lim f¯(t − sn , x) = f (t, x)
n→∞
e x ∈ X.
for each t ∈ T,
Denote by AA(T × X, X) the set of all such functions.
Remark 3.10. From Definition 3.9, one can easily see that if f : T × X → X is an almost automorphic
function under Definition 1.3, then it is also an almost automorphic function under Definition 3.9.
e If
Lemma 3.11. Let f ∈ AA(T × X, X) satisfy the Lipschitz condition in x ∈ X uniformly in t ∈ T.
ϕ ∈ AA(T, X), then f (t, ϕ(t)) is almost automorphic.
0
0
Proof. For any sequence (sn ) ⊂ Π, there exists a subsequence (sn ) ⊂ (sn ) such that
lim f (t + sn , x) = f¯(t, x)
n→∞
e x ∈ X, and
is well defined for each t ∈ T,
lim f¯(t − sn , x) = f (t, x)
n→∞
e x ∈ X. Since ϕ ∈ AA(T, X), there exists a subsequence (τn ) ⊂ (sn ) such that
for each t ∈ T,
lim ϕ(t + τn ) = ϕ̄(t)
n→∞
e and
is well defined for each t ∈ T,
lim ϕ̄(t − τn ) = ϕ(t)
n→∞
e Since f satisfies the Lipschitz condition in x ∈ X uniformly in t ∈ T,
e there exists a positive
for each t ∈ T.
constant L such that
kf (t + τn , ϕ(t + τn )) − f¯(t, ϕ̄(t))k ≤ kf (t + τn , ϕ(t + τn )) − f (t + τn , ϕ̄(t))k + kf (t + τn , ϕ̄(t)) − f¯(t, ϕ̄(t))k
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
1197
≤ Lkϕ(t + τn ) − ϕ̄(t)k + kf (t + τn , ϕ̄(t)) − f¯(t, ϕ̄(t))k → 0, n → ∞
and
kf¯(t − τn , ϕ̄(t − τn )) − f (t, ϕ(t))k ≤ kf¯(t − τn , ϕ̄(t − τn )) − f¯(t − τn , ϕ(t))k + kf¯(t − τn , ϕ(t)) − f (t, ϕ(t))k
≤ Lkϕ̄(t − τn ) − ϕ(t)k + kf¯(t − τn , ϕ(t)) − f (t, ϕ(t))k → 0, n → ∞.
0
0
Hence, for any sequence (sn ) ⊂ Π there exists a subsequence (τn ) ⊂ (sn ) such that
lim f (t + τn , ϕ(t + τn )) = f¯(t, ϕ̄(t))
n→∞
e and
is well defined for each t ∈ T,
lim f¯(t − τn , ϕ̄(t − τn )) = f (t, ϕ(t))
n→∞
e that is, f (t, ϕ(t)) is almost automorphic. This completes the proof.
for each t ∈ T,
Definition 3.12. Let T be an almost periodic time scale. The graininess function µ : T → R+ is said to be
0
0
almost automorphic if for every sequence (sn ) ⊂ Π, there exists a subsequence (sn ) ⊂ (sn ) such that
lim µ(t + sn ) = µ̄(t)
n→∞
e and
is well defined for each t ∈ T,
lim µ̄(t − sn ) = µ(t)
n→∞
e
for each t ∈ T.
In the following, in order to make the graininess function µ have a better property, we will use Definition
2.3 as the definition of almost periodic time scales.
From Corollary 15, Theorem 19 in [20] and Definition 3.12, we can obtain
Lemma 3.13. Let T be an almost periodic time scale under Definition 2.3. Then the graininess function µ
is almost automorphic.
Open problem 1. Let T be an almost periodic time scale under Definition 3.1. Does it follow that the
graininess function µ is almost automorphic?
Open problem 2. Let the graininess function µ of T be almost automorphic. Does it follow that T is an
almost periodic time scale under Definition 3.1?
4. Automorphic solutions to linear dynamic equations
Consider the linear nonhomogeneous dynamic equation on time scales
x∆ (t) = A(t)x(t) + f (t), t ∈ T,
(4.1)
where A : T → Rn×n and f : T → Rn , and its associated homogeneous equation
x∆ (t) = A(t)x(t), t ∈ T.
(4.2)
Throughout this section, we restrict our discussions to almost periodic time scales under Definition 2.3 and
we assume that A(t) is almost automorphic on T, which means that each entry of the matrix A(t) is almost
automorphic.
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
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Lemma 4.1. Let T be an almost periodic time scale, A(t) ∈ R(T, Rn×n ) be almost automorphic and nonsingular on T and the sets {A−1 (t)}t∈T and {(I + µ(t)A(t))−1 }t∈T be bounded on T. Then, A−1 (t) and
(I + µ(t)A(t))−1 are almost automorphic on T. Moreover, suppose that f ∈ AA(T, Rn ) and (4.2) admits an
exponential dichotomy. Then (4.1) has a solution x(t) ∈ AA(T, Rn ), and x(t) is given by
Z +∞
Z t
X(t)(I − P )X −1 (σ(s))f (s)∆s,
(4.3)
X(t)P X −1 (σ(s))f (s)∆s −
x(t) =
−∞
t
where X(t) is the fundamental solution matrix of (4.2).
Proof. We divide the proof into several steps.
Step 1. A−1 (t) is almost automorphic on T.
0
Consider a sequence (sn ) ⊂ Π. Since A(t) is almost automorphic on time scales, there exists a subse0
quence (sn ) ⊂ (sn ) such that
lim A(t + sn ) = Ā(t)
n→∞
e and
is well defined for each t ∈ T,
lim Ā(t − sn ) = A(t)
n→∞
e
for each t ∈ T.
e define An = A(t + sn ), n ∈ N. By hypothesis, the set {A−1 }n∈N is bounded, that is, there
Given t ∈ T,
n
−1 − A−1 = A−1 (A − A )A−1 , it
exists a positive constant M such that |A−1
|
<
M
.
From
the
identity
A
m
n
n
n
m
n
m
e
follows that {An } and {A−1
n } are Cauchy sequences. Hence, for each t ∈ T fixed, there exists a matrix S
such that
A−1
n (t) → S(t), n → ∞.
Then we have
−1
lim An A−1
= I,
n = ĀĀ
n→∞
e
where I denotes the identity matrix, and we obtain that Ā(t) is invertible and Ā−1 (t) = S(t) for each t ∈ T.
Since the map A → A−1 is continuous on the set of nonsingular matrices, we infer that
lim A−1 (t + sn ) = Ā−1 (t)
n→∞
e Similarly, we can get that
is well defined for each t ∈ T.
lim Ā−1 (t − sn ) = A−1 (t)
n→∞
e
for each t ∈ T.
Step 2. (I + µ(t)A(t))−1 is almost automorphic on T.
0
Since A(t) and µ(t) are almost automorphic functions, for every sequence (sn ) ⊂ Π there exists a
0
subsequence (sn ) ⊂ (sn ) such that
lim A(t + sn ) = Ā(t)
n→∞
and
lim µ(t + sn ) = µ̄(t)
n→∞
e and
are well defined for each t ∈ T,
lim Ā(t − sn ) = A(t)
n→∞
and
lim µ̄(t − sn ) = µ(t)
n→∞
e Therefore, we have
for each t ∈ T.
lim I + A(t + sn )µ(t + sn ) = I + Ā(t)µ̄(t)
n→∞
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
1199
e and
for each t ∈ T,
lim I + Ā(t − sn )µ̄(t − sn ) = I + A(t)µ(t)
n→∞
e Hence, (I + A(t)µ(t)) is almost automorphic on T. In addition, since A(t) is a regressive
for each t ∈ T.
matrix, (I +A(t)µ(t)) is nonsingular on T. By hypothesis, the set {(I +A(t)µ(t))−1 }t∈T is bounded. Similarly
as in the proof of Step 1, we obtain that (I + A(t)µ(t))−1 is almost automorphic on T.
Step 3. The system (4.1) has an automorphic solution.
Since the linear system (4.2) admits an exponential dichotomy, the system (4.1) has a bounded solution
x(t) given by
Z +∞
Z t
−1
X(t)(I − P )X −1 (σ(s))f (s)∆s.
X(t)P X (σ(s))f (s)∆s −
x(t) =
−∞
t
0
Since A(t), µ(t) and f (t) are almost automorphic functions, for every sequence (sn ) ⊂ Π there exists a
0
subsequence (sn ) ⊂ (sn ) such that
lim A(t + sn ) = Ā(t),
n→∞
lim µ(t + sn ) = µ̄(t)
and
lim µ̄(t − sn ) = µ(t)
and
n→∞
lim f (t + sn ) = f¯(t)
n→∞
e and
are well defined for each t ∈ T,
lim Ā(t − sn ) = A(t),
n→∞
n→∞
e
for each t ∈ T.
We let
Z
t
lim f¯(t − sn ) = f (t)
n→∞
X(t)P X −1 (σ(s))f (s)∆s
B(t) =
−∞
and
Z
t
B̄(t) =
M (t, s)f¯(s)∆s,
−∞
where M (t, s) = lim X(t + sn )P X −1 (σ(s + sn )).
n→∞
Then, we obtain
Z
Z t+sn
−1
X(t + sn )P X (σ(s))f (s)∆s −
kB(t + sn ) − B̄(t)k = t
−∞
−∞
M (t, s)f¯(s)∆s
Z t
Z t
−1
=
X(t + sn )P X (σ(s + sn ))f (s + sn )∆s −
M (t, s)f¯(s)∆s
−∞
−∞
Z t
≤
X(t + sn )P X −1 (σ(s + sn ))f (s + sn )∆s
−∞
Z t
−1
¯
−
X(t + sn )P X (σ(s + sn ))f (s)∆s
−∞
Z t
Z t
+
X(t + sn )P X −1 (σ(s + sn ))f¯(s)∆s −
M (t, s)f¯(s)∆s
−∞
−∞
Z t
=
X(t + sn )P X −1 (σ(s + sn )) f (s + sn ) − f¯(s) ∆s
−∞
Z t
+
X(t + sn )P X −1 (σ(s + sn )) − M (t, s) f¯(s)∆s.
(4.4)
−∞
In addition, since the function f is almost automorphic, we have that f¯ is a bounded function. Passing to
the limit in (4.4), we get
lim B(t + sn ) = B̄(t)
(4.5)
n→∞
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
1200
e Analogously, one can prove that
for each t ∈ T.
lim B̄(t − sn ) = B(t)
(4.6)
n→∞
e
for each t ∈ T.
On the other hand, let
Z
C(t) =
+∞
X(t)(I − P )X −1 (σ(s))f (s)∆s
t
and
Z
t
C̄(t) =
N (t, s)f¯(s)∆s,
−∞
where N (t, s) = lim X(t + sn )(I −
n→∞
P )X −1 (σ(s
+ sn )). Then, similar to the proofs of (4.5) and (4.6), we
0
0
can show that given a sequence (sn ) ⊂ Π, there exists a subsequence (sn ) ⊂ (sn ) such that
lim C(t + sn ) = C̄(t)
n→∞
e and
for each t ∈ T,
lim C̄(t − sn ) = C(t)
n→∞
e
for each t ∈ T.
Finally, we define x̄(t) = B̄(t) − C̄(t). Then, by the definition of x from (4.3), one can prove that given
0
0
a sequence (sn ) ⊂ Π, there exists a subsequence (sn ) ⊂ (sn ) such that
lim x(t + sn ) = x̄(t)
n→∞
e and
is well defined for each t ∈ T,
lim x̄(t − sn ) = x(t)
n→∞
e Hence, x(t) is an almost automorphic solution of system (4.1). This completes the proof.
for each t ∈ T.
Similar to the proof of Lemma 2.15 in [23], one can easily show that
e i = 1, 2, . . . , n and
Lemma 4.2. Let ci (t) be almost automorphic on T, where ci (t) > 0, −ci (t) ∈ R+ , t ∈ T,
min {inf ci (t)} = m
e > 0. Then the linear system
1≤i≤n t∈T
e
x∆ (t) = diag(−c1 (t), −c2 (t), . . . , −cn (t))x(t)
admits an exponential dichotomy on T.
5. An application
In the last forty years, shunting inhibitory cellular neural networks (SICNNs) have been extensively
applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image
processing. Hence, they have been the object of intensive analysis by numerous authors in recent years.
Many important results on the dynamical behaviors of SICNNs have been established and successfully
applied to signal processing, pattern recognition, associative memories, and so on. We refer the reader to
[13, 19, 21, 23, 26, 32, 42, 44, 49] and the references cited therein. However, to the best of our knowledge,
there is no paper published on the existence of almost automorphic solutions to SICNNs governed by
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
1201
differential or difference equations. So, our main purpose in this section is to study the existence of almost
automorphic solutions to the following SICNN with time-varying delays on time scales
X
kl
Bij
(t)f (xkl (t − τkl (t)))xij (t)
x∆
ij (t) = −aij (t)xij (t) −
Ckl ∈Nr (i,j)
X
−
kl
Cij
(t)
Ckl ∈Np (i,j)
Z
t
g(xkl (u))∆uxij (t) + Lij (t),
(5.1)
t−δkl (t)
where t ∈ T, T is an almost periodic time scale in the sense of Definition 2.3, i = 1, 2, . . . , m, j = 1, 2, . . . , n,
Cij denotes the cell at the (i, j) position of the lattice, the r-neighborhood Nr (i, j) of Cij is given by
Nr (i, j) = {Ckl : max(|k − i|, |l − j|) ≤ r, 1 ≤ k ≤ m, 1 ≤ l ≤ n},
Np (i, j) is similarly specified, xij is the activity of the cell Cij , Lij (t) is the external input to Cij , aij (t) > 0
kl (t) ≥ 0 and C kl (t) ≥ 0 are the connections or coupling
represents the passive decay rate of the cell activity, Bij
ij
strengths of postsynaptic activity of the cells in Nr (i, j) and Np (i, j) transmitted to cell Cij depending upon
variable delays and continuously distributed delays, respectively, and the activity functions f (·) and g(·) are
continuous functions representing the output or firing rate of the cell Ckl , τkl (t) and δkl (t) are transmission
delays at time t and satisfy t − τkl (t) ∈ T and t − δkl (t) ∈ T for t ∈ T, k = 1, 2, . . . , m, l = 1, 2, . . . , n.
The initial condition associated with system (5.1) is of the form
where θ = max
xij (s) = ϕij (s), s ∈ [−θ, 0]T , i = 1, 2, . . . , m, j = 1, 2, . . . , n,
max
τkl ,
max
δkl , and ϕij (·) denotes a real-value bounded rd-continuous
1≤k≤m,1≤l≤n
1≤k≤m,1≤l≤n
function defined on [−θ, 0]T .
Throughout this section, we let [a, b]T = {t|t ∈ [a, b] ∩ T} and we restrict our discussions to almost
periodic time scales in the sense of Definition 2.3. For convenience, for an almost automorphic function
f : T → R, denote f = inf f (t), f = sup f (t). We denote by R the set of real numbers, and by R+ the set of
t∈T
t∈T
positive real numbers.
Set
x = {xij (t)} = (x11 (t), . . . , x1n (t), . . . , xi1 (t), . . . , xin (t), . . . , xm1 (t) . . . , xmn (t)) ∈ Rm×n .
For all x = {xij (t)} ∈ Rm×n , we define the norm kx(t)k = maxi,j |xij (t)| and kxk = sup kx(t)k.
t∈T
Let B = {ϕ|ϕ = {ϕij (t)} = (ϕ11 (t), . . . , ϕ1n (t), . . . , ϕi1 (t), . . . , ϕin (t), . . . , ϕm1 (t) . . . , ϕmn (t))}, where ϕ
is an almost automorphic function on T. For all ϕ ∈ B, if we define the norm kϕkB = sup kϕ(t)k, then B is
t∈T
a Banach space.
Definition 5.1. The almost automorphic solution x∗ (t) = {x∗ij (t)} of system (5.1) with initial value ϕ∗ =
{ϕ∗ij (t)} is said to be globally exponentially stable if there exist positive constants λ with λ ∈ R+ and
M > 1 such that every solution x(t) = {xij (t)} of system (5.1) with initial value ϕ = {ϕij (t)} satisfies
kx − x∗ kB ≤ M eλ (t, t0 )kϕ − ϕ∗ kB , ∀t ∈ [t0 , +∞)T , t0 ∈ T.
Theorem 5.2. Suppose that
(H1 ) for ij ∈ Λ = {11, 12, . . . , 1n, . . . , m1, m2, . . . , mn}, −aij ∈ R+ , where R+ denotes the set of positively
kl , C kl , τ , L ∈ B;
regressive functions from T to R, aij , Bij
ij
ij kl
(H2 ) functions f, g ∈ C(R, R) and there exist positive constants Lf , Lg , M f , M g such that for all u, v ∈ R,
|f (u) − f (v)| ≤ Lf |u − v|, f (0) = 0, |f (u)| ≤ M f ,
|g(u) − g(v)| ≤ Lg |u − v|, g(0) = 0, |g(u)| ≤ M g ;
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
1202
(H3 ) there exists a constant ρ such that
P
max
Ckl ∈Nr (i,j)
kl M f ρ2 +
Bij
P
Ckl ∈Np (i,j)
kl M g δ ρ2 + L
Cij
ij kl
<ρ
aij
i,j
and
P
max
Ckl ∈Nr (i,j)
kl (M f + Lf )ρ +
Bij
P
Ckl ∈Np (i,j)
kl (M g + Lg )δ ρ
Cij
kl < 1.
aij
i,j
Then, system (5.1) has a unique almost automorphic solution in E = {ϕ ∈ B : kϕkB ≤ ρ}.
Proof. For any given ϕ ∈ B, we consider the following system
X
kl
Bij
(t)f (ϕkl (t − τkl (t)))ϕij (t)
x∆
ij (t) = −aij (t)xij (t) −
Ckl ∈Nr (i,j)
X
−
kl
Cij
(t)
g(ϕkl (u))∆uϕij (t) + Lij (t), ij = 11, 12, . . . , mn.
min
(5.2)
t−δkl (t)
Ckl ∈Np (i,j)
Since
t
Z
{inf aij (t)} > 0 and −aij ∈ R+ , it follows from Lemma 4.2 that the linear system
1≤i≤m;1≤j≤n t∈T
x∆
ij (t) = −aij (t)xij (t), ij = 11, 12, . . . , mn
admits an exponential dichotomy on T. Thus, by Lemma 4.1, we obtain that system (5.2) has exactly one
almost automorphic solution as follows
Z t
X
ϕ
ϕ
kl
x (t) := {xij (t)} =
e−aij (t, σ(s)) −
Bij
(s)f (ϕkl (s − τkl (s)))ϕij (s)
−∞
X
−
kl
Cij
(s)
Z
Ckl ∈Nr (i,j)
s
g(ϕkl (u))∆uϕij (s) + Lij (s) ∆s .
t−δkl (s)
Ckl ∈Np (i,j)
Now, we define the operator T : B → B by setting
T (ϕ(t)) = xϕ (t), ∀ϕ ∈ B.
First, we will check that for any ϕ ∈ E, T ϕ ∈ E. For any ϕ ∈ E, we have
Z t
X
kl
kT (ϕ)kB = sup max e−aij (t, σ(s)) −
Bij
(s)f (ϕkl (s − τkl (s)))ϕij (s)
t∈T
i,j
−
X
−∞
Ckl ∈Nr (i,j)
kl
Cij
(s)
Z
t−δkl (s)
Ckl ∈Np (i,j)
t
Z
≤ sup max i,j
t∈T
−∞
X
+
kl
Cij
Ckl ∈Np (i,j)
Z
t
≤ sup max
t∈T
i,j
s
−∞
g(ϕkl (u))∆uϕij (s) + Lij (s) ∆s
e−aij (t, σ(s))
X
kl f (ϕ (s − τ (s)))ϕ (s)
Bij
ij
kl
kl
Ckl ∈Nr (i,j)
s
Lij
g(ϕkl (u))∆uϕij (s) ∆s + max
i,j aij
t−δkl (s)
Z
e−aij (t, σ(s))
X
Ckl ∈Nr (i,j)
kl M f |ϕ (s − τ (s))||ϕ (s)|
Bij
ij
kl
kl
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
X
+
kl M g
Cij
Ckl ∈Np (i,j)
t
Z
t∈T
+
X
s
−∞
|ϕkl (u)|∆u|ϕij (s)| ∆s + max
i,j
t−δkl (s)
X
kl M g δ ρ2 ∆s + max
Cij
kl
i,j
kl M f ρ2 +
Bij
P
≤ max
Lij
aij
kl M f ρ2
Bij
Ckl ∈Nr (i,j)
Ckl ∈Np (i,j)
e−aij (t, σ(s))
≤ sup max
i,j
Z
1203
Ckl ∈Nr (i,j)
Lij
aij
P
Ckl ∈Np (i,j)
kl M g δ ρ2 + L
Cij
ij kl
≤ ρ.
aij
i,j
Therefore, we have that kT (ϕ)kB ≤ ρ. This implies that T is a self-mapping from E to E. Next, we prove
that T is a contraction mapping on E. In fact, for any ϕ, ψ ∈ E, we can get
kT (ϕ) − T (ψ)kB = sup kT (ϕ)(t) − T (ψ)(t)k
t∈T
Z t
e−aij (t, σ(s))
= sup max i,j
t∈T
−∞
X
kl
Bij
(s) f (ϕkl (s − τkl (s)))ϕij (s)
Ckl ∈Nr (i,j)
X
− f (ψkl (s − τkl (s)))ψij (s) +
kl
Cij
(s)
s
−
g(ψkl (u))∆uψij (s)
t−δkl (s)
Z
t
t∈T
+
i,j
−∞
X
C kl ij
i,j
t
−∞
g(ϕkl (u))∆u|ϕij (s) − ψij (s)| ∆s
e−aij (t, σ(s))
kl Cij
Z
Z
t
t∈T
i,j
X
+
t
+ sup max
i,j
−∞
s
t
≤ sup max
t∈T
i,j
−∞
M |ϕkl (u)|∆u|ϕij (s) − ψij (s)| ∆s
e−aij (t, σ(s))
Ckl ∈Np (i,j)
Z
kl M f |ϕ (s − τ (s))||ϕ (s) − ψ (s)|
Bij
ij
ij
kl
kl
g
X
× |ψij (s)| +
X
t−δkl (s)
Ckl ∈Np (i,j)
t∈T
g(ϕkl (u)) − g(ψkl (u)) ∆u |ψij (s)|
Ckl ∈Nr (i,j)
Z
kl
Cij
Z
s
e−aij (t, σ(s))
−∞
kl f (ϕ (s − τ (s))) − f (ψ (s − τ (s)))
Bij
kl
kl
kl
kl
t−δkl (s)
Ckl ∈Np (i,j)
≤ sup max
X
Ckl ∈Nr (i,j)
X
× |ψij (s)| +
kl f (ϕ (s − τ (s)))|ϕ (s) − ψ (s)|
Bij
ij
ij
kl
kl
t−δkl (s)
+ sup max
t∈T
X
Ckl ∈Nr (i,j)
s
Z
Ckl ∈Np (i,j)
Z
g(ϕkl (u))∆uϕij (s)
∆s
e−aij (t, σ(s))
≤ sup max
s
t−δkl (s)
Ckl ∈Np (i,j)
Z
Z
X
kl Lf |ϕ (s − τ (s)) − ψ (s − τ (s))|
Bij
kl
kl
kl
kl
Ckl ∈Nr (i,j)
kl
Cij
Z
s
L |ϕkl (u) − ψkl (u)|∆u |ψij (s)|
g
t−δkl (s)
e−aij (t, σ(s))
X
Ckl ∈Nr (i,j)
kl M f ρ
Bij
+
X
Ckl ∈Np (i,j)
kl M g δ ρ
Cij
kl
∆s kϕ − ψkB
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
Z
t
+ sup max
i,j
t∈T
−∞
P
≤ max
Ckl ∈Nr (i,j)
e−aij (t, σ(s))
X
kl Lf ρ
Bij
+
Ckl ∈Nr (i,j)
kl (M f + Lf )ρ +
Bij
1204
X
∆s kϕ − ψkB
Ckl ∈Np (i,j)
P
Ckl ∈Np (i,j)
kl (M g + Lg )δ ρ
Cij
kl kϕ − ψkB .
aij
i,j
kl Lg δ ρ
Cij
kl
By (H3 ), we have that kT (ϕ) − T (ψ)kB < kϕ − ψkB . Hence, T is a contraction mapping from E to E.
Therefore, T has a fixed point in E, that is, (5.1) has a unique almost automorphic solution in E. This
completes the proof.
Theorem 5.3. Assume that (H1 )-(H3 ) hold. Then system (5.1) has a unique almost automorphic solution
which is globally exponentially stable.
Proof. From Theorem 5.2, we see that system (5.1) has at least one almost automorphic solution
x∗ (t) = {x∗ij (t)} with the initial condition ϕ∗ (t) = {ϕ∗ij (t)}. Suppose that {x(t)} = {(x∗ij (t)} is an arbitrary solution with the initial condition ϕ(t) = {ϕij (t)}. Set y(t) = x(t) − x∗ (t). Then it follows from
system (5.1) that
X
kl
∆
Bij
(t) f (xkl (t − τkl (t)))xij (t)
yij
(t) = −aij (t)yij (t) −
Ckl ∈Nr (i,j)
−
f (x∗kl (t
−
τkl (t)))x∗ij (t)
X
−
kl
Cij
(t)
Z
t
g(xkl (u))∆uxij (t)
t−δkl (t)
Ckl ∈Np (i,j)
Z
t
g(x∗kl (u))∆ux∗ij (t)
−
(5.3)
t−δkl (t)
for i = 1, 2, . . . , m, j = 1, 2, . . . , n, and the initial condition of (5.3) is
ψij (s) = ϕij (s) − ϕ∗ij (s), s ∈ [−θ, 0]T , ij = 11, 12, . . . , mn.
Then, it follows from (5.3) that for i = 1, 2, . . . , m, j = 1, 2, . . . , n and t ≥ t0 , we have
Z t
X
kl
yij (t) = yij (t0 )e−aij (t, t0 ) −
e−aij (t, σ(s))
Bij
(s) f (xkl (s − τkl (s)))xij (s)
t0
−f (x∗kl (s
−
Ckl ∈Nr (i,j)
τkl (s)))x∗ij (s)
X
+
Ckl ∈Np (i,j)
Z
s
Z
s
kl
Cij
(s)
g(xkl (u))∆uxij (s)
t−δkl (s)
∆s.
g(x∗kl (u))∆ux∗ij (s)
−
(5.4)
s−δkl (s)
Let Sij be defined as follows:
Sij (ω) = aij − ω − exp(ω sup µ(s))Wij (ω), i = 1, 2, . . . , m, j = 1, 2, . . . , n,
s∈T
where
Wij (ω) =
X
kl M f ρ +
Bij
Ckl ∈Nr (i,j)
+
X
X
kl (M g + Lg )δ ρ exp ωδ
Cij
kl
kl
Ckl ∈Np (i,j)
kl L ρ exp ωτ
Bij
, i = 1, 2, . . . , m, j = 1, 2, . . . , n.
kl
f
Ckl ∈Nr (i,j)
By (H3 ), we get
Sij (0) = aij − Wij (0) > 0, i = 1, 2, . . . , m, j = 1, 2, . . . , n.
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
1205
Since Sij is continuous on [0, +∞) and Sij (ω) → −∞, as ω → +∞, there exist ξij > 0 such that
Sij (ξij ) = 0 and Sij (ω) > 0 for ω ∈ (0, ξij ), i = 1, 2, . . . , m, j = 1, 2, . . . , n. Take a =
min
ξij .
1≤i≤m,1≤j≤n
We have Sij (a) ≥ 0, i = 1, 2, . . . , m, j = 1, 2, . . . , n, so we can choose a positive constant 0 < λ <
min a,
min
{aij } such that
1≤i≤m,1≤j≤n
Sij (λ) > 0,
i = 1, 2, . . . , m, j = 1, 2, . . . , n,
which implies that
exp(λ sup µ(s))Wij (λ)
s∈T
< 1, i = 1, 2, . . . , m, j = 1, 2, . . . , n.
aij − λ
Let
aij
M=
max
1≤i≤m,1≤j≤n
(5.5)
Wij (0)
.
By (H3 ) and (5.5), we have M > 1.
Moreover, we have eλ (t, t0 ) > 1, where t ≤ t0 . Hence, it is obvious that
ky(t)kB ≤ M eλ (t, t0 )kψkB ,
∀t ∈ [t0 − θ, t0 ]T ,
where λ ∈ R+ . In the following, we will show that
ky(t)kB ≤ M eλ (t, t0 )kψkB ,
∀t ∈ (t0 , +∞)T .
(5.6)
To prove (5.6), we first show that for any p > 1, the following inequality holds:
ky(t)kB < pM eλ (t, t0 )kψkB ,
∀t ∈ (t0 , +∞)T ,
(5.7)
which implies that, for i = 1, 2, . . . , m, j = 1, 2, . . . , n, we have
|yij (t)| < pM eλ (t, t0 )kψkB ,
∀t ∈ (t0 , +∞)T .
(5.8)
If (5.8) is not true, then there exists t1 ∈ (t0 , +∞)T such that
|yij (t1 )| ≥ pM eλ (t1 , t0 )kψkB and |yij (t)| < pM eλ (t, t0 )kψkB ,
∀t ∈ (t0 , t1 )T .
Therefore, there must exist a constant c ≥ 1 such that
|yij (t1 )| = cpM eλ (t1 , t0 )kψkB and |yij (t)| < cpM eλ (t, t0 )kψkB ,
Note that, in view of (5.4), we have that
Z
|yij (t1 )| = yij (t0 )e−aij (t1 , t0 ) −
t1
e−aij (t1 , σ(s))
t0
s
−
kl
Bij
(s) f (xkl (s − τkl (s)))xij (s)
Ckl ∈Nr (i,j)
Z s
kl
Cij (s)
g(xkl (u))∆uxij (s)
s−δkl (s)
Ckl ∈Np (i,j)
X
−f (x∗kl (s − τkl (s)))x∗ij (s) +
Z
X
∀t ∈ (t0 , t1 )T .
∆s
g(x∗kl (u))∆ux∗ij (s)
s−δkl (s)
Z
≤ e−aij (t1 , t0 )kψkB +
t1
t0
e−aij (t1 , σ(s))
X
Ckl ∈Nr (i,j)
kl f (x (s − τ (s)))x (s)
Bij
ij
kl
kl
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
Z
kl Cij
X
−f (x∗kl (s − τkl (s)))x∗ij (s) +
s
−
s−δkl (s)
g(x∗kl (u))∆ux∗ij (s)
Z
≤ e−aij (t1 , t0 )kψkB +
X
+
kl
Cij
Ckl ∈Np (i,j)
X
+
Z
t1
g(xkl (u))∆uxij (s)
s−δkl (s)
Ckl ∈Np (i,j)
Z
s
∆s
e−aij (t1 , σ(s))
t0
X
kl M f |x (s − τ (s))||y (s)|
Bij
ij
kl
kl
Ckl ∈Nr (i,j)
s
M g |xkl (u)|∆u|yij (s)|
s−δkl (s)
kl Lf |y (s − τ (s))||x∗ (s)|
Bij
kl
kl
ij
Ckl ∈Nr (i,j)
X
+
kl
Cij
Ckl ∈Np (i,j)
Z
s
∗
L |ykl (u)|∆u |xij (s)| ∆s
g
s−δkl (s)
Z t1
e−aij ⊕λ (t1 , σ(s))
≤ e−aij (t1 , t0 )kψkB + cpM eλ (t1 , t0 )kψkB t0
n
X
X
kl M f ρe (σ(s), s) +
kl M g δ ρe (σ(s), s − δ (s))
×
Bij
Cij
λ
kl λ
kl
Ckl ∈Nr (i,j)
X
+
Ckl ∈Np (i,j)
f
kl L ρe (σ(s), s − τ (s))
Bij
λ
kl
Ckl ∈Nr (i,j)
X
+
kl Lg δ ρe (σ(s), s
Cij
kl λ
Ckl ∈Np (i,j)
o
− δkl (s)) ∆s
Z t1
≤ e−aij (t1 , t0 )kψkB + cpM eλ (t1 , t0 )kψkB e−aij ⊕λ (t1 , σ(s))
t0
n
X
kl M f ρ exp λ sup µ(s)
×
Bij
s∈T
Ckl ∈Nr (i,j)
X
+
kl M g δ ρ exp λ(δ + sup µ(s))
Cij
kl
kl
s∈T
Ckl ∈Np (i,j)
X
+
kl Lf ρ exp λ(τ + sup µ(s))
Bij
kl
s∈T
Ckl ∈Nr (i,j)
X
+
kl Lg δ ρ exp
Cij
kl
Ckl ∈Np (i,j)
o
λ(δkl + sup µ(s)) ∆s
s∈T
X
h
1
kl M f ρ
e−aij ⊕λ (t1 , t0 ) + exp λ sup µ(s)
Bij
cpM
s∈T
Ckl ∈Nr (i,j)
X
i
kl (M g + Lg )δ ρ exp λδ
kl Lf ρ exp λτ
Cij
Bij
kl
kl +
kl
= cpM eλ (t1 , t0 )kψkB
X
+
Ckl ∈Np (i,j)
Z
Ckl ∈Nr (i,j)
t1
×
e−aij ⊕λ (t1 , σ(s))∆s
X
h
1
kl M f ρ
< cpM eλ (t1 , t0 )kψkB
e−(aij −λ) (t1 , t0 ) + exp λ sup µ(s)
Bij
M
s∈T
Ckl ∈Nr (i,j)
X
X
i
kl (M g + Lg )δ ρ exp λδ
kl Lf ρ exp λτ
+
Cij
+
B
kl
kl
kl
ij
t0
Ckl ∈Np (i,j)
Ckl ∈Nr (i,j)
1206
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
×
Z
1
−(aij − λ)
t1
t0
(−(aij − λ))e−(aij −λ) (t1 , σ(s))∆s
= cpM eλ (t1 , t0 )kψkB
+
X
1207
1
−
M
g
exp λ sup µ(s) s∈T
aij − λ
g
X
Ckl ∈Nr (i,j)
Ckl ∈Np (i,j)
kl Lf ρ exp λτ
Bij
kl
X
kl (M + L )δ ρ exp λδ
Cij
kl
kl +
kl M f ρ
Bij
Ckl ∈Nr (i,j)
×e−(aij −λ) (t1 , t0 ) +
+
X
exp λ sup µ(s) kl (M g
Cij
s∈T
aij − λ
X
kl M f ρ
Bij
Ckl ∈Nr (i,j)
g
X
+ L )δkl ρ exp λδkl +
Ckl ∈Np (i,j)
kl Lf ρ exp
Bij
λτkl
Ckl ∈Nr (i,j)
< cpM eλ (t1 , t0 )kψkB ,
which is a contradiction. Therefore, (5.7) holds. Letting p → 1, we obtain that (5.6) holds. Hence, we have
that
ky(t)kB < M eλ (t, t0 )kψkB , ∀t ∈ (t0 , +∞)T ,
which means that the almost automorphic solution of system (5.1) is globally exponentially stable. The
proof is complete.
Remark 5.4. Theorems 5.2 and 5.3 are new even for the cases of differential equations (T = R) and difference
equations (T = Z).
6. Examples
In this section, we will give examples to illustrate the feasibility and effectiveness of the results obtained
in Section 5.
Consider the following SICNNs with time-varying delays:
X
kl
x∆
Bij
(t)f (xkl (t − τkl (t)))xij (t)
ij (t) = −aij (t)xij (t) −
Ckl ∈Nr (i,j)
−
X
Ckl ∈Np (i,j)
kl
Cij
(t)
Z
t
g(xkl (u))∆uxij (t) + Lij (t),
(6.1)
t−δkl (t)
where i = 1, 2, 3, j = 1, 2, 3, r = p = 1, f (x) = 0.05| cos x|, g(x) =
|x|
20 ,
t ∈ T.
Example 6.1. If T = R, then µ(t) ≡ 0. Take
√

 

a11 (t) a12 (t) a13 (t)
3 + | sin t|
4 + | cos 2t| 2 + | sin t|
 a21 (t) a22 (t) a23 (t)  =  2 + | cos( 1 t)| 5 + | sin 2t| 1 + | cos 2t|  ,
3
a31 (t) a32 (t) a33 (t)
4 + | cos 2t|
3 + | sin 2t|
2 + | cos t|

 
 

B11 (t) B12 (t) B13 (t)
C11 (t) C12 (t) C13 (t)
0.1| cos t| 0.2| cos 2t| 0.1| sin t|
 B21 (t) B22 (t) B23 (t)  =  C21 (t) C22 (t) C23 (t)  =  0.1| sin 2t| 0.1| cos t| 0.2| cos 2t|  ,
B31 (t) B32 (t) B33 (t)
C31 (t) C32 (t) C33 (t)
0.1| cos t| 0.2| cos 2t| 0.1| cos t|

 

L11 (t) L12 (t) L13 (t)
0.1 sin t
cos t
0.2 sin t
 L21 (t) L22 (t) L23 (t)  =  0.2 sin t 0.4 sin t 0.1 sin t  ,
L31 (t) L32 (t) L33 (t)
0.1 sin t 0.3 sin t 0.1 cos t
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
1208

 

δ11 (t) δ12 (t) δ13 (t)
0.5 + 0.5 sin t
cos t
0.7 + 0.2 sin t
 δ21 (t) δ22 (t) δ23 (t)  =  0.8 + 0.2 sin t 0.2 + 0.4 sin t 0.7 + 0.2 sin t  .
δ31 (t) δ32 (t) δ33 (t)
0.9 + 0.1 sin t 0.6 + 0.3 sin t 0.4 + 0.4 cos t
Obviously, (H1 ) holds. Clearly, we have
X
M f = M g = 0.05, Lf = Lg = 0.05,
X
kl =
B12
Ckl ∈N1 (1,2)
X
Ckl ∈N1 (1,2)
X
kl =
B21
X
kl =
B23
X
kl =
B32
Ckl ∈N1 (3,2)
i,j
P
Ckl ∈Nr (i,j)
Lij
aij
X
kl = 0.8,
C32
Ckl ∈N1 (3,3)
X
kl = 0.5,
C31
Ckl ∈N1 (3,1)
X
kl =
B33
kl = 0.6
C33
Ckl ∈N1 (3,3)
= 0.25. Take ρ = 1. Then
kl M f ρ2 +
Bij
P
Ckl ∈Np (i,j)
kl M g ρ2 δ + L
Cij
ij kl
aij
i,j
kl = 1.2,
C22
Ckl ∈N1 (2,2)
kl =
B31
Ckl ∈N1 (3,1)
Ckl ∈N1 (3,2)
and we can easily check that max
max
kl = 1,
C23
Ckl ∈N1 (2,3)
Ckl ∈N1 (2,3)
X
X
kl =
B22
Ckl ∈N1 (2,2)
X
kl = 0.6,
C13
Ckl ∈N1 (1,3)
X
kl = 0.8,
C21
X
kl =
B13
Ckl ∈N1 (1,3)
Ckl ∈N1 (2,1)
Ckl ∈N1 (2,1)
X
X
kl = 1,
C12
kl = 0.5,
C11
Ckl ∈N1 (1,1)
Ckl ∈N1 (1,1)
X
X
kl =
B11
≈ 0.2669 < ρ = 1
and
P
max
Ckl ∈Nr (i,j)
i,j
kl (M f + Lf )ρ +
Bij
P
Ckl ∈Np (i,j)
aij
kl (M g + Lg )ρδ
Cij
kl ≈ 0.5337 < 1.
The combination of the above two inequalities means that (H3 ) is satisfied for ρ = 1.
Therefore, we have shown that assumptions (H1 )-(H3 ) are satisfied. By Theorem 5.2, system (6.1) has
exactly one almost automorphic solution in E = {ϕ ∈ B : kϕkB ≤ ρ}. Moreover, by Theorem 5.3 this
solution is globally exponentially stable.
Example 6.2. If T = Z,

a11 (t) a12 (t)
 a21 (t) a22 (t)
a31 (t) a32 (t)

then µ(t) ≡ 1. Take
√
 

a13 (t)
0.3 + 0.1| sin t|
0.4 + 0.1| cos 2t| 0.2 + 0.1| sin t|
a23 (t)  =  0.2 + 0.2| cos( 31 t)| 0.5 + 0.1| sin 2t| 0.1 + 0.1| cos 2t|  ,
a33 (t)
0.4 + 0.1| cos 2t|
0.3 + 0.2| sin 2t|
0.2 + 0.1| cos t|
 
 

B11 (t) B12 (t) B13 (t)
C11 (t) C12 (t) C13 (t)
0.02| cos t| 0.02| cos 2t| 0.03| sin t|
 B21 (t) B22 (t) B23 (t)  =  C21 (t) C22 (t) C23 (t)  =  0.01| sin 2t| 0.01| cos t| 0.02| cos 2t|  ,
B31 (t) B32 (t) B33 (t)
C31 (t) C32 (t) C33 (t)
0.03| cos t| 0.02| cos 2t| 0.01| cos t|

 

L11 (t) L12 (t) L13 (t)
0.01 sin t 0.01 cos t 0.02 sin t
 L21 (t) L22 (t) L23 (t)  =  0.02 sin t 0.04 sin t 0.03 sin t  ,
L31 (t) L32 (t) L33 (t)
0.01 sin t 0.03 sin t 0.02 cos t

 

δ11 (t) δ12 (t) δ13 (t)
sin t
0.9 cos t
0.2 + 0.5 sin t
 δ21 (t) δ22 (t) δ23 (t)  =  0.2 + 0.8 sin t 0.4 + 0.6 sin t 0.1 + 0.9 sin t  .
δ31 (t) δ32 (t) δ33 (t)
sin t
sin t
0.7 + 0.2 cos t
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
1209
Obviously, (H1 ) holds. Clearly, we have
X
M f = M g = 0.05, Lf = Lg = 0.05,
Ckl ∈N1 (1,1)
X
X
kl =
B12
X
kl = 0.11,
C12
Ckl ∈N1 (1,2)
Ckl ∈N1 (1,2)
Ckl ∈N1 (1,3)
X
X
X
kl =
B21
kl = 0.11,
C21
Ckl ∈N1 (2,1)
Ckl ∈N1 (2,1)
Ckl ∈N1 (2,2)
X
X
X
kl =
B23
Ckl ∈N1 (2,3)
Ckl ∈N1 (2,3)
X
X
kl =
B32
and we can easily check that max
i,j
P
max
Ckl ∈Nr (i,j)
Lij
aij
X
kl = 0.1,
C32
Ckl ∈N1 (3,3)
X
kl =
B13
kl = 0.08,
C13
Ckl ∈N1 (1,3)
X
kl =
B22
kl = 0.17,
C22
Ckl ∈N1 (2,2)
X
kl =
B31
kl = 0.07,
C31
Ckl ∈N1 (3,1)
X
kl =
B33
kl = 0.06
C33
Ckl ∈N1 (3,3)
= 0.15. Take ρ = 1, then
kl M f ρ2 +
Bij
P
Ckl ∈Np (i,j)
kl M g ρ2 δ + L
Cij
ij kl
aij
i,j
kl = 0.06,
C11
Ckl ∈N1 (1,1)
Ckl ∈N1 (3,1)
Ckl ∈N1 (3,2)
Ckl ∈N1 (3,2)
kl = 0.11,
C23
X
kl =
B11
≈ 0.1904 < ρ = 1
and
P
max
Ckl ∈Nr (i,j)
i,j
kl (M f + Lf )ρ +
Bij
P
Ckl ∈Np (i,j)
aij
kl (M g + Lg )ρδ
Cij
kl ≈ 0.3657 < 1.
The combination of the above two inequalities means that (H3 ) is satisfied for ρ = 1.
Therefore, we have shown that assumptions (H1 )-(H3 ) are satisfied. By Theorem 5.2, system (6.1) has
exactly one almost automorphic solution in E = {ϕ ∈ B : kϕkB ≤ ρ}. Moreover, by Theorem 5.3 this
solution is globally exponentially stable.
7. Conclusion
In this paper, we propose a new concept of almost automorphic functions on almost periodic time scales
and study some basic properties. We state two open problems concerning the relationship between the
algebraic operation property of elements of a time scale and the analytical property of the time scale.
Then, based on these concepts and results, we establish the existence of an almost automorphic solution for
both the linear nonhomogeneous dynamic equation on time scales and its associated homogeneous equation.
Finally, as an application of the results, we study the existence and global exponential stability of almost
automorphic solutions to a class of shunting inhibitory cellular neural networks with time-varying delays on
time scales.
Acknowledgements:
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under
Grant 11361072.
Y. K. Li, B. Li, X. F. Meng, J. Nonlinear Sci. Appl. 8 (2015), 1190–1211
1210
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