Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 23, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE AND STABILITY OF ALMOST PERIODIC
SOLUTIONS FOR SICNNS WITH NEUTRAL TYPE DELAYS
QING-LONG LIU, HUI-SHENG DING
Abstract. This article concerns the shunting inhibitory cellular neural networks with neutral type delays. Under a weaker condition than the usual
Lipschitz condition, we establish the existence and stability of almost periodic solutions for SICNNs with neutral type delays. An example is given to
illustrate our main results.
1. Introduction
Since Bouzerdoum and Pinter [2, 3, 4] introduced and analyzed the shunting
inhibitory cellular neural networks (SICNNs), they have been extensively applied
in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision,
and image processing (cf. [10, 15] and references therein).
It is well known that studies on neural networks not only involve a discussion
of stability properties, but also involve many dynamic behaviors such as periodic
oscillatory behavior and almost periodic oscillatory properties. In applications, if
the various constituent components of the temporally nonuniform environment is
with incommensurable (nonintegral multiples) periods, then one has to consider
the environment to be almost periodic since there is no a priori reason to expect
the existence of periodic solutions. Therefore, if we consider the effects of the
environmental factors, almost periodicity is sometimes more realistic and more
general than periodicity. Also, as pointed out in [14, 11], compared with periodic
effects, almost periodic effects are more frequent in many real world applications.
In fact, this point of view is partially verified by a recent work [21], where the
authors proved that the “amount” of almost periodic functions (not periodic) is far
more than the “amount” of continuous periodic functions in the sense of category.
Thus, studying the existence of almost periodic solutions for differential equations
is natural and necessary.
Recently, many authors have studied the existence and stability of periodic solutions and almost periodic solutions for the following SICNNs:
X
kl
x0ij (t) = −aij xij (t) −
Cij
f [xkl (t − τ (t))]xij (t) + Lij (t),
Ckl ∈Nr (i,j)
2000 Mathematics Subject Classification. 34C25, 34K13, 34K25.
Key words and phrases. Almost periodic; shunting inhibitory cellular neural networks;
neutral type delays; stability.
c
2013
Texas State University - San Marcos.
Submitted November 4, 2013. Published January 10, 2014.
1
2
Q.-L. LIU, H.-S. DING
EJDE-2014/23
and its variants. We refer the reader to [1, 5, 6, 7, 8, 9, 12, 13, 16, 17, 18, 19] and
reference therein for some of recent developments on this topic.
Especially, in a very recent work, the authors in [17] investigated the existence
and stability of almost periodic solutions for the following SICNNs with neutral
type delays:
Z ∞
X
0
kl
Kij (u)f (xkl (t − u))duxij (t)
xij (t) = −aij (t)xij (t) −
Cij (t)
−
0
Ckl ∈Nr (i,j)
Z
∞
X
kl
Dij
(t)
Jij (u)g(x0kl (t
0
Dkl ∈Ns (i,j)
(1.1)
− u))duxij (t) + Lij (t),
i = 1, 2, . . . , m, j = 1, 2, . . . n, where m, n are two fixed positive integers, Cij is the
cell at the (i, j) position of the lattice, the r-neighborhood Nr (i, j) of Cij is defined
as follows:
Nr (i, j) = {Ckl : max |k − i|, |l − j| ≤ r, 1 ≤ k ≤ m, 1 ≤ l ≤ n},
and Ns (i, j) is defined similarly. Here xij (t) is the activity of cell Cij , Lij (t) is
the external input to Cij , the coefficient aij (t) is the passive decay rate of the
kl
cell activity, f, g are continuous activity functions of signal transmission, Cij
(t)
represents the connection or coupling strength of postsynaptic activity of the cell
kl
transmitted to the cell Cij , and Dij
(t) has a similar meaning.
In [17], the activation functions f and g satisfy the global Lipschitz conditions.
In this paper, as one will see, we allow for more general activity functions, i.e., we
will discuss the existence and stability of almost periodic solutions for the SICNNs
(1.1) under a weaker Lipschitz conditions on f and g.
Next, let us recall some basic notation and results about almost periodic functions. For more details, we refer the reader to [11, 14, 20].
Definition 1.1. A continuous function u: R → R is called almost periodic if for
each ε > 0 there exists l(ε) > 0 such that every interval I of length l(ε) contains a
number τ with the property that
|u(t + τ ) − u(t)| < ε.
We denote by AP (R) the set of all almost periodic functions from R to R, and
by AP 1 (R) the set of all continuously differentiable functions u : R → R satisfying
u, u0 ∈ AP (R).
Lemma 1.2. Let f, g ∈ AP (R) and k ∈ L1 (R+ ). Then the following assertions
hold:
(a) f + g ∈ AP (R) and f · g ∈ AP (R);
(b) the function t 7→ f (t − τ ) belongs to AP (R) for every τ ∈ R;
(c) F ∈ AP (R), where
Z
+∞
k(u)f (t − u)du,
F (t) =
t ∈ R.
0
(d) AP (R) is a Banach space under the norm kf k = supt∈R |f (t)|.
EJDE-2014/23
EXISTENCE AND STABILITY FOR SICNNS
3
2. Existence of almost periodic solution
For the rest of this article, we denote
J = {11, . . . , 1n, , . . . , m1, . . . , mn},
x(t) = {xij (t)} = (x11 (t), . . . , x1n (t), .., xm1 (t), . . . , xmn (t)),
X = {ϕ : ϕ = {ϕij }, ϕij , ϕ0ij ∈ AP (R)}.
For every ϕ ∈ X, we denote
kϕk = sup max{|ϕij (t)|},
t∈R ij∈J
0
kϕkX = max{kϕk, kϕ k} = max{sup max |ϕij (t)|, sup max |ϕ0ij (t)|}.
t∈R ij∈J
t∈R ij∈J
It is not difficult to verify that X is a Banach space under the norm k · kX . For
every ij ∈ J, we denote
a+
ij := sup aij (t),
t∈R
kl
Cij
:=
a−
ij := inf aij (t),
L+
ij := sup |Lij (t)|,
t∈R
kl
sup |Cij
(t)|,
t∈R
kl
Dij
:=
t∈R
kl
sup |Dij
(t)|.
t∈R
We will use the following assumptions:
kl
kl
(H1) For every ij ∈ J, aij , Cij
, Dij
and Lij are both almost periodic functions,
−
and aij > 0.
(H2) There exist four functions f1 , f2 , g1 , g2 : R → R and four positive constants
Lf1 , Lf2 , Lg1 , Lg2 such that f = f1 f2 , g = g1 g2 and for all u, v ∈ R, there
holds
|fi (u) − fi (v)| ≤ Lfi |u − v|,
|gi (u) − gi (v)| ≤ Lgi |u − v|,
(H3) There exists a constant λ0 > 0 such that
Z ∞
Z ∞
|Kij (u)|eλ0 u du < +∞,
|Jij (u)|eλ0 u du < +∞,
0
i = 1, 2.
ij ∈ J.
0
(H4) There exists a constant d > 0 such that
n
Aij + L+
o
Aij + L+
ij
ij −
,
max
(a+
≤ d,
max max
ij + aij )
−
−
ij∈J
ij∈J
aij
aij
n
Bij (0) +
Bij (0) o
−
,
max
(a
+
a
)
< 1,
max max
ij
ij
ij∈J
ij∈J
a−
a−
ij
ij
where Mfi = sup|x|≤d |fi (x)|, Mgi = sup|x|≤d |gi (x)|, i = 1, 2,
Z ∞
X
kl
Cij (dMf1 Mf2 )
Aij =
|Kij (u)|du
0
Ckl ∈Nr (i,j)
+
X
kl (dM M )
Dij
g1
g2
Z
∞
|Jij (u)|du,
0
Dkl ∈Ns (i,j)
and
Bij (0) =
X
Ckl ∈Nr (i,j)
Z
h
kl (M M )
Cij
f1
f2
0
∞
|Kij (u)|du
4
Q.-L. LIU, H.-S. DING
EJDE-2014/23
Z
∞
Z
∞
i
|Kij (u)|du
0
Z ∞
h
kl
|Jij (u)|du
Dij (Mg1 Mg2 )
+ d(Mf1 Lf2 + Mf2 Lf1 )
X
+
0
Dkl ∈Ns (i,j)
+ d(Mg1 Lg2 + Mg2 Lg1 )
i
|Jij (u)|du ,
0
Theorem 2.1. Under assumptions H1)–(H4), there exists a unique continuously
differentiable almost periodic solution of (1.1) in the region Ω = {ϕ ∈ X : kϕkX ≤
d}.
Proof. For ω ∈ (0, λ0 ], we denote
Z
h
X
kl (M M )
Bij (ω) =
Cij
f1
f2
∞
|Kij (u)|du
0
Ckl ∈Nr (i,j)
Z
∞
Z
∞
i
|Kij (u)|eωu du
0
Z ∞
h
kl
Dij (Mg1 Mg2 )
|Jij (u)|du
+ d(Mf1 Lf2 + Mf2 Lf1 )
X
+
0
Dkl ∈Ns (i,j)
+ d(Mg1 Lg2 + Mg2 Lg1 )
i
|Jij (u)|eωu du .
0
For each ϕ ∈ X, we consider the almost periodic differential equations
Z ∞
X
kl
x0ij (t) = −aij (t)xij (t) −
Cij
(t)
Kij (u)f (ϕkl (t − u))duϕij (t)
−
0
Ckl ∈Nr (i,j)
Z ∞
kl
Dij
(t)
Jij (u)g(ϕ0kl (t
0
Dkl ∈Ns (i,j)
X
− u))duϕij (t) + Lij (t),
ij ∈ J.
(2.1)
Combining (H1) and Lemma 1.2, we know that the inhomogeneous part of equation
(2.1) is an almost periodic function. Noting that a−
ij > 0, by [14, Theorem 7.7], we
conclude that (2.1) has a unique almost periodic solution xϕ satisfying
Z ∞
nZ t
h
R
X
ϕ
− st aij (u)du
kl
x (t) =
e
−
Cij (s)
Kij (u)f (ϕkl (s − u))duϕij (s)
−∞
X
−
kl
Dij
(s)
Ckl ∈Nr (i,j)
∞
Z
0
i o
Jij (u)g(ϕ0kl (s − u))duϕij (s) + Lij (s) ds
.
ij∈J
0
Dkl ∈Ns (i,j)
Now, define a mapping T on Ω = {ϕ ∈ X : kϕkX ≤ d} by
(T ϕ)(t) = xϕ (t),
∀ϕ ∈ Ω.
It is easy to show that T (Ω) ⊂ X.
Next, let us check that T (Ω) ⊂ Ω. It suffices to prove that kT ϕkX ≤ d for all
ϕ ∈ Ω. By (H2) and (H3), we have
kT ϕk
n Z
= sup max t∈R ij∈J
t
−∞
e−
Rt
s
aij (u)du
EJDE-2014/23
EXISTENCE AND STABILITY FOR SICNNS
h
× −
X
kl
Cij
(s)
Kij (u)f (ϕkl (s − u))duϕij (s)
0
Ckl ∈Nr (i,j)
X
−
kl
Dij
(s)
∞
Z
0
Dkl ∈Ns (i,j)
t
nZ
∞
Z
i o
Jij (u)g(ϕ0kl (s − u))duϕij (s) + Lij (s) ds
Rt
e− s aij (u)du
t∈R ij∈J
−∞
Z ∞
h
X
kl
×
|Kij (u)||f (ϕkl (s − u))|du|ϕij (s)|
Cij
≤ sup max
0
Ckl ∈Nr (i,j)
X
+
kl
Dij
t
nZ
t∈R ij∈J
e
a−
ij (s−t)
h
−∞
X
+
i o
|Jij (u)||g(ϕ0kl (s − u))|du|ϕij (s)| + |Lij (s)| ds
0
Dkl ∈Ns (i,j)
≤ sup max
∞
Z
X
nh
ij∈J
+
X
kl (dM M )
Cij
f1
f2
Z
∞
|Kij (u)|du
0
kl (dM M )
Dij
g1
g2
∞
Z
i
o
−
|Jij (u)|du + L+
ij /aij
0
Dkl ∈Ns (i,j)
= max
|Kij (u)|du
0
Ckl ∈Nr (i,j)
∞
i o
|Jij (u)|du + L+
ij ds
kl (dM M )
Dij
g1
g2
X
Z
0
Ckl ∈Nr (i,j)
Z ∞
Dkl ∈Ns (i,j)
≤ max
kl (dM M )
Cij
f1
f2
+
Aij + Lij
a−
ij
ij∈J
,
and
Z
n
k(T ϕ)0 k = sup max − aij (t)
t∈R ij∈J
h
× −
t
e−
X
X
kl
Cij
(t)
kl
Dij
(t)
Z
Kij (u)f (ϕkl (s − u))duϕij (s)
Z
∞
kl
Cij
(t)
X
Dkl ∈Ns (i,j)
i
Jij (u)g(ϕ0kl (s − u))duϕij (s) + Lij (s) ds
Z
kl
Dij
(t)
∞
Kij (u)f (ϕkl (t − u))duϕij (t)
0
Ckl ∈Nr (i,j)
−
∞
0
X
+ −
aij (u)du
0
Dkl ∈Ns (i,j)
h
s
−∞
Ckl ∈Nr (i,j)
−
Rt
Z
0
∞
io
Jij (u)g(ϕ0kl (t − u))duϕij (t) + Lij (t) n
o
Aij + L+
ij
+
≤ sup max a+
·
+
A
+
L
ij
ij
ij
a−
t∈R ij∈J
ij
n Aij + L+
o
ij
+
−
= max
(a
+
a
)
.
ij
ij
ij∈J
a−
ij
5
6
Q.-L. LIU, H.-S. DING
EJDE-2014/23
Then, from (H4) it follows that
n
Aij + L+
o
Aij + L+
ij ij
−
k(T ϕ)kX ≤ max max
≤ d,
,
max
(a+
ij + aij )
−
−
ij∈J
ij∈J
aij
aij
which implies that T (Ω) ⊂ Ω.
Let ϕ, ψ ∈ Ω, and for ij ∈ J denote
Z
X
kl
αij (s) =
Cij
(s)
∞
Kij (u)f (ϕkl (s − u))duϕij (s)
0
Ckl ∈Nr (i,j)
Z ∞
Kij (u)f (ψkl (s − u))duψij (s) ,
−
0
and
X
βij (s) =
Z
∞
kl
Dij
(s)
Dkl ∈Ns (i,j)
Z ∞
Jij (u)g(ϕ0kl (s − u))duϕij (s)
0
0
Jij (u)g(ψkl
(s − u))duψij (s) .
−
0
By (H2), for each ij ∈ J, we obtain
|αij (s)|
X
≤
kl
Cij
n Z
Ckl ∈Nr (i,j)
Z ∞
∞
Kij (u)f (ϕkl (s − u))duϕij (s)
0
Kij (u)f (ϕkl (s − u))duψij (s)
−
0
∞
o
∞
Kij (u)f (ϕkl (s − u))duψij (s) −
Kij (u)f (ψkl (s − u))duψij (s)
0
0
h Z ∞
X
kl
Cij
Kij (u)f1 (ϕkl (s − u))f2 (ϕkl (s − u))duϕij (s)
Z
+
≤
Z
Ckl ∈Nr (i,j)
Z ∞
0
i
Kij (u)f1 (ϕkl (s − u))f2 (ϕkl (s − u))duψij (s)
−
0
+
hZ
Z
∞
Kij (u)f1 (ϕkl (s − u))f2 (ϕkl (s − u))duψij (s)
0
∞
−
Kij (u)f1 (ϕkl (s − u))f2 (ψkl (s − u))duψij (s)
i
0
+
hZ
Z
−
≤
∞
Kij (u)f1 (ϕkl (s − u))f2 (ψkl (s − u))duψij (s)
0
∞
i
Kij (u)f1 (ψkl (s − u))f2 (ψkl (s − u))duψij (s) 0
Z ∞
X
kl
Cij [Mf1 Mf2 + d(Mf1 Lf2 + Mf2 Lf1 )]
|Kij (u)|dukϕ − ψkX .
Ckl ∈Nr (i,j)
0
Similarly, for each ij ∈ J, we have
Z
X
kl [M M + d(M L + M L )]
Dij
|βij (s)| ≤
g1
g2
g1 g2
g2 g1
Dkl ∈Ns (i,j)
0
∞
|Jij (u)|dukϕ − ψkX .
EJDE-2014/23
EXISTENCE AND STABILITY FOR SICNNS
7
Thus,
|αij (s)| + |βij (s)| ≤ Bij (0)||ϕ − ψ||X .
It follows that
n Z
kT ϕ − T ψk = sup max t∈R ij∈J
t
e−
Rt
s
aij (u)du
−∞
Z
t
e−
≤ sup max
t∈R ij∈J
≤ sup max
−∞
nZ t
t∈R ij∈J
Rt
s
e−
aij (u)du
Rt
s
o
[αij (s) + βij (s)]ds
(|αij (s)| + |βij (s)|)ds
aij (u)du
o
ds Bij (0)kϕ − ψkX
−∞
n B (0) o
ij
≤ max
kϕ − ψkX ,
ij∈J
a−
ij
and
k(T ϕ − T ψ)0 k
Z
n
= sup max − aij (t)
t∈R ij∈J
e−
Rt
s
aij (u)ds
−∞
n Z
≤ sup max a+
ij t∈R ij∈J
t
t
e−
Rt
s
aij (u)du
−∞
o
(αij (s) + βij (s))ds + (αij (t) + βij (t))
o
(αij (s) + βij (s))ds + (|αij (t)| + |βij (t)|)
o
n B (0)
ij
kϕ − ψkX + Bij (0)kϕ − ψkX
≤ sup max a+
ij
−
aij
t∈R ij∈J
o
n B (0)
ij
+
−
(a
+
a
)
kϕ − ψkX .
≤ max
ij
ij
ij∈J
a−
ij
Combining the above two inequalities, we obtain
n
Bij (0) Bij (0) +
o
kT ϕ − T ψkX ≤ max max
, max
(aij + a−
kϕ − ψkX .
ij )
−
−
ij∈J
ij∈J
aij
aij
Noticing that
n
Bij (0) +
o
Bij (0) , max
(aij + a−
< 1,
max max
ij )
−
−
ij∈J
ij∈J
aij
aij
By the Banach contraction principle, T has a unique fixed point x in Ω, which is
just a continuously differentiable almost periodic solution of Equation (1.1).
3. Stability of almost periodic solutions
In this section, we will establish some results about the locally exponential stability of the almost periodic solution for Equation (1.1).
Theorem 3.1. Assume (H1)–(H4) hold. Let x(t) = {xij (t)} be the unique continuously differentiable almost periodic solution of (1.1) in Ω, and y(t) = {yij (t)} be
an arbitrary continuously differentiable solution of Equation (1.1) in the region Ω.
Then, there exist two constants λ, M > 0 such that
kx(t) − y(t)k1 ≤ M e−λt ,
∀t ∈ R,
where
0
kx(t) − y(t)k1 := max{max |xij (t) − yij (t)|, max |x0ij (t) − yij
(t)|}.
ij∈J
ij∈J
8
Q.-L. LIU, H.-S. DING
EJDE-2014/23
Proof. For ω ∈ [0, λ0 ], we denote
Tij (ω) = a−
ij − ω − Bij (ω),
+
−
Sij (ω) = a−
ij − ω − (aij + aij )Bij (ω).
By (H4), we have Tij (0) > 0 and Sij (0) > 0 for all ij ∈ J. Then, due to the
continuity
ij (ω), there exists a sufficiently small positive constant
of Tij (ω) and S
λ < min minij∈J {a−
ij }, λ0 such that
Tij (λ) > 0,
ij ∈ J,
Sij (λ) > 0,
which means that
Bij (λ)
< 1,
a−
ij − λ
Bij (λ) +
(aij + a−
ij ∈ J.
(3.1)
ij ) < 1,
a−
−
λ
ij
a− for all ij ∈ J. Setting M0 = maxij∈J Bijij(0) , the following three inequalities hold:
M0 > 1,
Bij (λ)
1
≤ 0,
− −
M0
aij − λ
Bij (λ)(
a+
ij
a−
ij − λ
+ 1) < 1,
ij ∈ J.
(3.2)
Now, we denote
z(t) = zij (t) : zij (t) = xij (t) − yij (t) ,
hZ ∞
X
kl
Rij (s) =
Cij (s)
Kij (u)f (ykl (s − u))duyij (s)
0
Ckl ∈Nr (i,j)
Z ∞
−
Qij (s) =
i
Kij (u)f (xkl (s − u))duxij (s) ,
0
hZ ∞
X
kl
0
Dij (s)
Jij (u)g(ykl
(s − u))duyij (s)
0
Dkl ∈Ns (i,j)
Z ∞
i
Jij (u)g(x0kl (s − u))duxij (s) .
−
0
Since x(t) and y(t) are both solutions to equation (1.1), we have
0
zij
(s) + aij (s)zij (s) = Rij (s) + Qij (s).
Multiplying by e
Rs
aij (u)du
and integrating on [0, t], we obtain
Z t R
Rt
t
zij (t) = zij (0)e− 0 aij (u)du +
e− s aij (u)du (Rij (s) + Qij (s))ds
0
(3.3)
(3.4)
0
Let
0
M := M0 · max sup max |xij (t) − yij (t)|, sup max |x0ij (t) − yij
(t)| .
t≤0 ij∈J
t≤0 ij∈J
Without loss for generality, we can assume that M > 0. Then, for all t ≤ 0, noting
that M0 > 1, we have
0
kz(t)k1 = max max |xij (t) − yij (t)|, max |x0ij (t) − yij
(t)| < M e−λt .
ij∈J
ij∈J
Next, we prove the inequality
kz(t)k1 ≤ M e−λt ,
t > 0,
by contradiction. If the above inequality is not true, then
V := {t > 0 : kz(t)k1 > M e−λt } =
6 ∅.
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EXISTENCE AND STABILITY FOR SICNNS
9
Letting t1 = inf V , then t1 > 0 and
kz(t)k1 ≤ M e−λt ,
kz(t1 )k1 = M e−λt1 .
∀t ∈ (−∞, t1 ),
(3.5)
For s ∈ [0, t1 ] and ij ∈ J, by the assumptions and (3.5), we have
|Rij (s)|
X
=
Z
kl
|Cij
(s)|
≤
Kij (u)f (ykl (s − u))duyij (s)
0
Ckl ∈Nr (i,j)
Z ∞
−
∞
Kij (u)f (xkl (s − u))duxij (s)
0
n Z ∞
X
kl Kij (u)f (ykl (s − u))duyij (s)
Cij
Ckl ∈Nr (i,j)
Z ∞
0
Kij (u)f (ykl (s − u))duxij (s)
−
0
∞
Z ∞
o
Kij (u)f (ykl (s − u))duxij (s) −
Kij (u)f (xkl (s − u))duxij (s)
0
0
n Z ∞
X
kl
Kij (u)f1 (ykl (s − u))f2 (ykl (s − u))duyij (s)
Cij Z
+
≤
Ckl ∈Nr (i,j)
Z ∞
−
0
0
Kij (u)f1 (ykl (s − u))f2 (ykl (s − u))duxij (s)
Z ∞
+
Kij (u)f1 (ykl (s − u))f2 (ykl (s − u))duxij (s)
0
Z ∞
−
Kij (u)f1 (xkl (s − u))f2 (ykl (s − u))duxij (s)
0
Z ∞
Kij (u)f1 (xkl (s − u))f2 (ykl (s − u))duxij (s)
+
0
Z ∞
o
−
Kij (u)f1 (xkl (s − u))f2 (xkl (s − u))duxij (s)
0
Z ∞
n
X
kl (M M )
≤
Cij
|Kij (u)|du|zij (s)|
f1
f2
0
Ckl ∈Nr (i,j)
Z
∞
+ d(Mf1 Lf2 + Mf2 Lf1 )
o
|Kij (u)||zkl (s − u)|du
0
≤
X
Z
n
kl (M M )
Cij
f1
f2
∞
|Kij (u)|du
0
Ckl ∈Nr (i,j)
Z
+ d(Mf1 Lf2 + Mf2 Lf1 )
∞
o
|Kij (u)|eλu du M e−λs .
0
Similarly, for s ∈ [0, t1 ] and ij ∈ J, we have
Z
n
X
kl (M M )
|Qij (s)| ≤
Dij
g1
g2
∞
|Jij (u)|du|zij (s)|
0
Dkl ∈Ns (i,j)
Z
+ d(Mg1 Lg2 + Mg2 Lg1 )
0
∞
o
0
|Jij (u)||zkl
(s − u)|du
10
Q.-L. LIU, H.-S. DING
≤
EJDE-2014/23
∞
Z
n
kl (M M )
Dij
g1
g2
X
|Jij (u)|du
0
Dkl ∈Ns (i,j)
Z
∞
+ d(Mg1 Lg2 + Mg2 Lg1 )
o
|Jij (u)|eλu du M e−λs .
0
Then we have
|Rij (s)| + |Qij (s)| ≤ M e−λs Bij (λ),
s ∈ [0, t1 ], ij ∈ J.
(3.6)
Combining (3.4) and (3.6), we have
Z t1 R
R t1
t1
|zij (t1 )| = zij (0)e− 0 aij (u)du +
e− s aij (u)du (Rij (s) + Qij (s))ds
0
Z t1
−
−
−
M −aij t1
≤
e
+
e(aij −λ)s−aij t1 ds · M Bij (λ)
M0
0
−
M −a−
(e−λt1 − e−aij t1 )Bij (λ)
=
e ij t1 +
·M
M0
a−
ij − λ
−
n e(λ−a−
ij )t1
[1 − e(λ−aij )t1 ]Bij (λ)) o
≤ M e−λt1
+
M0
a−
ij − λ
n 1
Bij (λ) (λ−a−
Bij (λ) o
ij )t1 +
− −
= M e−λt1
e
.
M0
aij − λ
a−
ij − λ
(3.7)
Then, by (3.1) and (3.2), we deduce that
|zij (t1 )| < M e−λt1 ,
ij ∈ J.
(3.8)
Recalling that
0
zij
(t) = −aij (t)zij (t) + Rij (t) + Qij (t),
by (3.6) and (3.7), we have
0
|zij
(t1 )| = | − aij (t1 )zij (t1 ) + Rij (t1 ) + Qij (t1 )|
≤ a+
ij |zij (t1 )| + |Rij (t1 )| + |Qij (t1 )|
n 1
Bij (λ) (λ−a−
Bij (λ) o
−λt1
ij )t1 +
< a+
M
e
−
e
+ M e−λt1 Bij (λ)
ij
−
M0
a−
−
λ
a
−
λ
ij
ij
o
n 1
a+
−
Bij (λ) (λ−aij )t1
ij
= M e−λt1 a+
−
e
+
B
(λ)(
+ 1)
ij
ij
−
−
M0
aij − λ
aij − λ
Then, from (3.2) it follows that
0
|zij
(t1 )| < M e−λt1 ,
ij ∈ J.
(3.9)
Combining (3.8) and (3.9), we obtain
kz(t1 )k1 < M e−λt1 ,
which contradicts with (3.5). Thus, we obtain
kz(t)k1 ≤ M e−λt ,
for all t ∈ R. This completes the proof.
Remark 3.2. Compared with the results in [17], our Lipschitz conditions are
weaker, and thus our results may have a wider range of applications.
EJDE-2014/23
EXISTENCE AND STABILITY FOR SICNNS
11
Next, we give an example to illustrate our main results.
Example 3.3. Consider the following SICNNs with neutral delays:
x0ij (t)
= −aij (t)xij (t) −
−
X
kl
Cij
(t)
Z
∞
Kij (u)f (xkl (t − u))duxij (t)
0
Ckl ∈N1 (i,j)
Z
∞
X
kl
Dij
(t)
Jij (u)g(x0kl (t
0
Dkl ∈N1 (i,j)
(3.10)
− u))duxij (t) + Lij (t),
where i = 1, 2, 3, j = 1, 2, 3. For i, j = 1, 2, 3 and t ∈ R, let
f (t) =
1
|t| cos t,
4
√
1 2
1
(t − 1), Lij (t) = | sin 2t + sin t|,
16
8
Kij (t) = e−6t , Jij (t) = e−4t .
g(t) =
Moreover,
√
 
5 + | sin t|
5 + | sin 2t|
a11 (t) a12 (t) a13 (t)
√
2t|
a21 (t) a22 (t) a23 (t) = 
6 + | cos t| 6 + | cos t+cos
2√
a31 (t) a32 (t) a33 (t)
8 + | sin t|
8 + | sin 2t|
√
 

0.1| sin t| 0.3| sin √2t|
C11 (t) C12 (t) C13 (t)
C21 (t) C22 (t) C23 (t) = 0.2| cos t| 0.1| cos 2t|
√
C31 (t) C32 (t) C33 (t)
0.1| sin t| 0.2| sin 2t|
√
 

0.1| cos t| 0.3| cos √ 2t|
D11 (t) D12 (t) D13 (t)
D21 (t) D22 (t) D23 (t) =  0.2| sin t| 0.1| sin 2t|
√
D31 (t) D32 (t) D33 (t)
0.1| cos t| 0.2| cos 2t|

√ 
9 + | sin 3t|
√ 
7 + | cos √ 2t| ,
5 + | sin 3t|
√ 
0.5| sin √3t|
0.2| cos √3t|
0.1| cos 3t|
√ 
0.5| cos √ 3t|
0.2| sin √3t|  .
0.1| cos 3t|
Let d = 1, λ0 = 1, f1 (t) = 21 |t|, f2 (x) = 21 cos t, g1 (t) = 41 (t − 1), g2 (t) = 41 (t + 1).
By a direct calculation, we obtain Lf1 = Lf2 = 12 , Lg1 = Lg2 = 41 , Mf1 = Mf2 = 12 ,
1
Mg1 = Mg2 = 12 , L+
ij = 4 . Then, it is easy to see that (H1)–(H3) hold.
Next, let us verify (H4). We have
a−
11
a−
21
a−
31
P

a−
12
a−
22
a−
32
 
a−
5
13
−
a23 = 6
8
a−
33
P
kl
5
6
8

9
7 ,
5
kl
Ckl ∈N1 (1,1) C11
Ckl ∈N1 (1,1) C12
P
P
kl
kl
 Ckl ∈N1 (1,1) C21
Ckl ∈N1 (1,1) C22
P
P
kl
kl
Ckl ∈N1 (1,1) C31
Ckl ∈N1 (1,1) C32
P
P
kl
kl
Dkl ∈N1 (1,1) D11
Dkl ∈N1 (1,1) D12
P
P

kl
kl
 Dkl ∈N1 (1,1) D21
Dkl ∈N1 (1,1) D22
P
P
kl
kl
Dkl ∈N1 (1,1) D31
Dkl ∈N1 (1,1) D32
 +
a11
a+
21
a+
31
P
a+
12
a+
22
a+
32
 

a+
6 6 10
13
+
a23 = 7 7 8  ,
9 9 6
a+
33



C kl
0.7 1.4 1.1
PCkl ∈N1 (1,1) 13

C kl  =  1 1.8 1.4 ,
PCkl ∈N1 (1,1) 23
kl
0.6 0.9 0.6
Ckl ∈N1 (1,1) C33

P


Dkl
0.7 1.4 1.1
PDkl ∈N1 (1,1) 13

Dkl  =  1 1.8 1.4 ,
PDkl ∈N1 (1,1) 23
0.6 0.9 0.6
Dkl
Dkl ∈N1 (1,1)
33
12
Q.-L. LIU, H.-S. DING
X
Aij =
EJDE-2014/23
kl (dM M )
Cij
f1
f2
|Kij (u)|du
0
Ckl ∈N1 (i,j)
X
+
∞
Z
kl (dM M )
Dij
g1
g2
|Jij (u)|du
0
Dkl ∈N1 (i,j)
=
5
48
X
kl ,
Cij
Ckl ∈N1 (i,j)
X
Bij (0) =
∞
Z
kl
Cij
Z
h
(Mf1 Mf2 )
∞
|Kij (u)|du
0
Ckl ∈N1 (i,j)
Z
∞
Z
∞
i
|Kij (u)|du
0
Z ∞
h
kl
Dij (Mg1 Mg2 )
|Jij (u)|du
+ d(Mf1 Lf2 + Mf2 Lf1 )
X
+
0
Dkl ∈N1 (i,j)
+ d(Mg1 Lg2 + Mg2 Lg1 )
i
|Jij (u)|du
0
1
=
4
X
kl .
Cij
Ckl ∈N1 (i,j)
Then, we obtain
  3.5
7
A13
48
48
5
9
A23  =  48 48
3
4.5
A33
48

 48  0.7
B11 (0) B12 (0) B13 (0)
4
B21 (0) B22 (0) B23 (0) =  1
4
0.6
B31 (0) B32 (0) B33 (0)
4

A11
A21
A31
A12
A22
A32
5.5
48
7 
,
48
3
48

1.4
1.1
4
4
1.8
1.4 
4
4
0.9
0.6
4
4

It follows that
Aij + L+
Aij + L+
o
ij ij
+
−
,
max
(a
+
a
)
ij
ij
ij∈J
ij∈J
a−
a−
ij
ij
o
n Aij + L+
ij
+
−
(a
= max
+
a
)
ij
ij
ij∈J
a−
ij
max
n
max
−
a+
ij + aij +
≤ max A+
+
max
L
max
ij
ij
ij∈J
ij∈J
ij∈J
a−
ij
−
1 a+
11 + a11
4
a−
11
9
1 11
=
+
·
< 1 = d,
48 4
5
= A22 +
and
Bij (0) Bij (0) +
o
−
,
max
(a
+
a
)
ij
ij
ij∈J
ij∈J
a−
a−
ij
ij
Bij (0) +
= max
(a + a−
ij )
ij∈J
aij − ij
max
n
max
EJDE-2014/23
EXISTENCE AND STABILITY FOR SICNNS
≤ max{Bij (0)} max
ij∈J
ij∈J
13
−
a+
ij + aij a−
ij
−
a+
11 + a11
−
a11
1.8 11
=
·
< 1.
4
5
Thus, condition (H4) holds. By Theorem 2.1 and Theorem 3.1, Equation (3.10)
admits a unique differentiable almost periodic solution x∗ (t) in the region Ω = {ϕ ∈
X : kϕkX ≤ 1}, and x∗ (t) is locally exponentially stable in Ω.
= B22 (0)
Remark 3.4. In Example 3.3, let the iterative sequences xn (t) = {xnij (t)} be
∀t ∈ R, i = 1, 2, 3, j = 1, 2, 3,
x0ij (t) = 0,
and
Z
t
e−
xnij (t) =
Rt
s
aij (u)du
−∞
h
X
× −
kl
Cij
(s)
X
Dkl ∈N1 (i,j)
kl
Dij
(s)
Kij (u)f (xn−1kl (s − u))du · xn−1ij (s)
0
Ckl ∈N1 (i,j)
−
∞
Z
Z
0
∞
i
Jij (u)g(x0n−1kl (s − u))du · xn−1ij (s) + Lij (s) ds,
for all t ∈ R, i = 1, 2, 3, j = 1, 2, 3, and n = 1, 2, 3, . . .. From the proof of Theorem
2.1, it follows
kxn − x∗ kX → 0, n → ∞,
∗
where x is the unique almost periodic solution of (3.10) in the region Ω = {ϕ ∈
X : kϕkX ≤ 1}. So one can use this method to compute numerically the almost
periodic solution x∗ .
Remark 3.5. In the above example, f and g do not satisfy the global Lipschitz
condition. So the results in [17] can not be applied to this example.
Acknowledgements. Qing-Long Liu acknowledges support from the Graduate
Innovation Fund of Jiangxi Province (YC2013-S100). Hui-Sheng Ding acknowledges
support from the NSF of China (11101192), the Program for Cultivating Young
Scientist of Jiangxi Province (20133BCB23009), and the NSF of Jiangxi Province.
The authors are grateful to the anonymous reviewers for their valuable suggestions,
which improve the quality of this article.
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Qing-Long Liu
College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
E-mail address: 758155543@qq.com
Hui-Sheng Ding (corresponding author)
College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
E-mail address: dinghs@mail.ustc.edu.cn