Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 7 (2014), 218–228
Research Article
Positive periodic solution for a nonlinear neutral
delay population equation with feedback control
Payam Nasertayoob, S. Mansour Vaezpour∗
Department of Mathematics, Amirkabir University of Technology (Polytechnic), Hafez Ave., P. O. Box 15914, Tehran, Iran.
Communicated by C. Park
Abstract
In this paper, sufficient conditions are investigated for the existence of positive periodic solution for a
nonlinear neutral delay population system with feedback control. The proof is based on the fixed-point
theorem of strict-set-contraction operators. We also present an example of nonlinear neutral delay population
c
system with feedback control to show the validity of conditions and efficiency of our results.2014
All rights
reserved.
Keywords: Fixed point theory, neutral nonlinear equation, feedback control, strict-set-contraction.
2010 MSC: 34K40, 92D25, 47H10.
1. Introduction and Preliminaries
The preliminary mathematical model of the population growth is given by the following logistic equation:
dx
= ρx(t)[1 − ax(t)].
dt
(1.1)
In the real problems, conditions for the population of species are more complicated and the simple logistic
model (1.1) could be generalized in many ways. For some kinds of population systems, when density of
species depends not only on the population at time, but also on the population unit earlier, the equation
(1.1) may be recovered as follows [15]:
dx
= ρx(t)[1 − ax(t) − bx(t − τ )].
dt
∗
Corresponding author
Email addresses: nasertayoob@aut.ac.ir (Payam Nasertayoob), vaez@aut.ac.ir (S. Mansour Vaezpour )
Received 2013-1-9
(1.2)
P. Nasertayoob, S. M. Vaezpour, J. Nonlinear Sci. Appl. 7 (2014), 218–228
219
One can also consider the non-autonomous version of (1.2) [18], i.e.,
dx
= ρ(t)x(t)[1 − a(t)x(t) − b(t)x(t − τ )].
dt
(1.3)
In the more realistic situation, the biological systems or ecosystems are continuously perturbed via unpredictable forces. These perturbations are generally results of the change in the system’s parameters. In the
language of the control theory, these perturbation functions may be regarded as control variables and, consequently, one should ask a question that whether or not an ecosystem can withstand those unpredictable
perturbations which persist for a finite periodic time. In the mathematical biology, the following population
model of systems of differential equations is a famous feedback control model with delays [6],
dx
= ρx(t)[1 − ax(t − τ ) − bu(t)],
dt
du
= −ηu(t) + gx(t − τ ),
dt
where, a, b, g, µ, ρ ∈ (0, ∞) and u represent an indirect feedback control mechanism. In the recent years,
study of the feedback control models has been further developed and the literature in mathematical biology
has been riched with study of such models [6, 10, 16]. Besides, many scholars have worked on neutral
systems. Some results can be found in [1, 9].
In [14], existence of positive periodic solutions for neutral population model was studied. Subsequently,
Lu et. al., [13] investigated the positive periodic solutions for such a neutral differential system with feedback
control. Besides, several scholars have paid their attention to the nonlinear population dynamics [3, 4, 5, 11],
because such kinds of systems could simulate the real world more accurately. Recently, the almost periodic
solution of the following nonlinear population dynamics with feedback control has been studied [17],
n
X
dx
= x(t)[ρ(x) − a(x)xα (t) −
bi (t)xβi (t − σi ) − c(t)u(t)],
dt
i=1
du
= −η(t)u(t) +
dt
n
X
g(t)xβi (t − σi ).
(1.4)
i=1
Investigation of the nonlinear system above is based on the properties of almost periodic systems and
Lyapunov-Razumikhin technique.
In this paper, we consider the following nonlinear neutral non-autonomous delay population model
n
X
dx
= x(t)[ρ(x) − a(x)xα (t) −
bi (t)xβi (t − σi ) − γ(t)x0 (t − τ ) − c(t)u(t)],
dt
i=1
du
= −η(t)u(t) +
dt
n
X
g(t)xβi (t − σi ),
(1.5)
i=1
which is a generalization of neutral system dynamics in [13], where, a, bi , g, µ, ρ, η, γ are positive ω-periodic
functions and α, βi are belong to (0, ∞).
Our investigation in this paper is based on a fixed point theorem of strict-set-contractive operator which
goes back to Cac-Gatica [2] and which was used for similar purpose in [13].
We recall some preliminaries that will be used in the further sections.
Definition 1.1. Let X be a Banach space. For the bounded set Ω ⊂ X, Kuratowski measure of noncompactness defines by:
µX (Ω) = inf{d > 0 : there is a finite number of subsets Ωi ⊂ Ω
[
such that Ω =
Ωi and diamΩi < d}.
i
P. Nasertayoob, S. M. Vaezpour, J. Nonlinear Sci. Appl. 7 (2014), 218–228
220
Definition 1.2. ([7, 8]) Let X and Y are Banach spaces. A continuous and bounded map
F : E ⊂ X → Y is called a k-set contraction if for any bounded set Ω ⊂ E
µY (F (Ω)) ≤ kµX (Ω).
The operator F is called a strict-set-contraction, where 0 ≤ k < 1.
Theorem 1.3. ([2, 12]) Let Π be a semi-ordered cone in a Banach space X and
Πr,R = {x ∈ Π : 0 < r ≤ kxk ≤ R}.
Let F : Πr,R → Π be a strict-set-contraction, satisfying
Fx x
f or any x ∈ Πr,R and kxk = r,
(1.6)
Fx x
f or any x ∈ Πr,R and kxk = R.
(1.7)
and
Then, F : Πr,R → Π has at least one fixed point in Πr,R .
One may easily shows that the following function is a solution of the second equation in (1.5),
Z
t+ω
(Φx)(t) =
n
X
G(t, s){
g(s)xβi (s − σi )}ds,
t
where
(1.8)
i=1
Rs
exp( t η(θ)dθ)
Rω
G(t, s) =
,
exp( 0 η(θ)dθ) − 1
s ∈ [t, t + ω], t ∈ R.
Therefore, existence of the ω-periodic solution of the system (1.5) is equivalent to the existence of the
solution of the following equation:
n
X
dx
= x(t)[ρ(t) − a(t)xα (t) −
bi (t)xβi (t − σi ) − γ(t)x0 (t − τ ) − c(t)(Φx)(t)].
dt
(1.9)
i=1
On the other hand, each ω-periodic solution of the following integral equation
Z
x(t) =
t+ω
e s)x(s)[a(s)xα (s) +
G(t,
t
n
X
bi (s)xβi (s − σi ) + γ(s)x0 (s − τ ) + c(s)(Φx)(s)]ds,
i=1
is a solution of equation (1.9), where
e s) =
G(t,
Rs
exp(− t ρ(θ)dθ)
Rω
,
1 − exp(− 0 ρ(θ)dθ)
s ∈ [t, t + ω], t ∈ R.
Also, in what follows, we employ the following notations:
f=
sup f (t),
t∈[t,t+ω]
Z
λ = exp(−
ω
ρ(θ)dθ) < 1,
0
f=
inf
f (t),
t∈[t,t+ω]
Z
κ = exp(
ω
η(θ)dθ) > 1,
0
Cω = {x ∈ C(R, (0, +∞)) : x(t + ω) = x(t)},
(1.10)
P. Nasertayoob, S. M. Vaezpour, J. Nonlinear Sci. Appl. 7 (2014), 218–228
221
Cω1 = {x ∈ C 1 (R, (0, +∞)) : x(t + ω) = x(t)},
A = min{1, α, β1 , ..., βn },
B = max{1, α, β1 , ..., βn },
Z t+ω
G(t, s)g(s)ds,
Ψ(t) =
t
M=
sup [a(t) +
n
X
t∈[t,t+ω]
Z
N=
bi (t) + γ(t) + nc(t)Ψ(t)],
i=1
ω
[a(s) +
0
n
X
bi (s) + γ(s) + nc(s)Ψ(s)]ds.
i=1
Clearly, two spaces (Cω , kk) and (Cω1 , kk1 ) are Banach spaces. Where
kxk = max |x(t)|,
t∈[t,t+ω]
and
kxk1 = max{kx(t)k, kx0 (t)k}.
We define the following integral operator F : Π 7→ Cω1
Z
t+ω
e s)x(s)[a(s)xα (s) +
G(t,
(F x)(t) =
t
n
X
bi (s)xβi (s − σi ) + γ(s)x0 (s − τ ) + c(s)(Φx)(s)]ds,
(1.11)
i=1
where, Π = {x ∈ Cω1 : x(t) ≥ λkxk1 } is a semi-ordered cone in Cω1 .
2. Main results
Lemma 2.1. Let R ≤ 1, ρ ≤ 1 and there exists a positive real number Z such that
Z ≤ λB rB−A ,
and
Zλ2
N,
(1 − λ)
(2.2)
bi (t) + nc(t)Ψ(t)},
(2.3)
M ≤ (ρ + 1)
and
γ(t) ≤ Z{a(t) +
n
X
(2.1)
i=1
then, the integral operator F maps Πr,R into Π.
Proof : For x ∈ Πr,R with R ≤ 1, we have
Z ≤ λB rB−A ≤ λB kxkB−A
≤ λξ kxkξ−A
,
1
1
f or any ξ ∈ {1, α, β1 , ..., βn },
therefore
ξ
ξ
ξ
ZkxkA
1 ≤ λ kxk1 ≤ x (t),
f or any ξ ∈ {1, α, β1 , ..., βn }.
(2.4)
On the other hand, since kxk1 ≤ 1, we obtain
kxkξ1 ≤ kxkA
1,
f or any ξ ∈ {1, α, β1 , ..., βn },
(2.5)
thus
A
nZkxkA
1 Ψ(t) ≤ (Φx)(t) ≤ nkxk1 Ψ(t).
(2.6)
P. Nasertayoob, S. M. Vaezpour, J. Nonlinear Sci. Appl. 7 (2014), 218–228
222
Applying (2.3), (2.4) and (2.6), we have
A
A
kxkA
1 γ(t) ≤ Zkxk1 a(t) + Zkxk1
n
X
bi (t) + nZkxkA
1 c(t)Ψ(t).
i=1
Thus
±x0 (t)γ(t) ≤ kxk1 γ(t) ≤ a(t)xα (t) +
n
X
bi (t)xβi (t − σi ) + c(t)(Φx)(t).
i=1
Consequently,
0 ≤ a(t)xα (t) +
n
X
bi (t)xβi (t − σi ) ± x0 (t)γ(t) + c(t)(Φx)(t).
(2.7)
i=1
On the other hand, since ρ is a positive and ω-periodic function we obtain
λ
e s) ≤ 1 .
≤ G(t,
1−λ
1−λ
(2.8)
Step 1. We show (F x)(t) ≥ λkF xk.
kF xk =
sup |(F x)(t)|
t∈[t,t+ω]
Z
=
t+ω
e s)x(s)[a(s)xα (s) +
G(t,
sup
t∈[t,t+ω] t
n
X
bi (s)xβi (s − σi )
i=1
+γ(s)x0 (s − τ ) + c(s)(Φx)(s)]ds
Z t+ω
n
X
1
α
x(s)[a(s)x (s) +
bi (s)xβi (s − σi )
≤
1−λ t
i=1
+γ(s)x0 (s − τ ) + c(s)(Φx)(s)]ds
Z t+ω
n
X
1
λ
α
=
{
x(s)[a(s)x (s) +
bi (s)xβi (s − σi )
λ t
1−λ
i=1
+γ(s)x0 (s − τ ) + c(s)(Φx)(s)]ds}
Z
n
X
1 t+ω e
α
=
G(t, s)x(s)[a(s)x (s) +
bi (s)xβi (s − σi )
λ t
i=1
+γ(s)x0 (s − τ ) + c(s)(Φx)(s)]ds
1
=
(F x)(t).
(2.9)
λ
Step 2. We show (F x)0 (t) ≤ F x. Based on the Leibniz integral rule, relations (2.2) and (2.8), we obtain
e + ω, t)x(t + ω)
(F x)0 (t) = G(t
α
× [a(t + ω)x (t + ω) +
n
X
bi (t + ω)xβi (t + ω − σi )
i=1
0
+ γ(t + ω)x (t + ω − τ ) + c(t + ω)(Φx)(t + ω)]
n
X
α
e
− G(t)x(t)[a(t)x (t) +
bi (t)xβi (t − σi )
i=1
0
+ γ(t)x (t − τ ) + c(t)(Φx)(t)]
n
X
λ
1
= (
−
)x(t)[a(t)xα (t) +
bi (t)xβi (t − σi )
1−λ 1−λ
i=1
0
+ γ(t)x (t − τ ) + c(t)(Φx)(t)] − ρ(t)(F x)(t)
≤ ρ(t)(F x)(t) ≤ (F x)(t).
(2.10)
P. Nasertayoob, S. M. Vaezpour, J. Nonlinear Sci. Appl. 7 (2014), 218–228
223
Step 3. We show −(F x)0 (t) ≤ (F x)(t). Applying (2.2), (2.10), (2.5), (2.8) and (2.4), we obtain
−(F x)0 (t) = x(t)[a(t)xα (t) +
n
X
bi (t)xβi (t − σi )
i=1
0
+ γ(t)x (t − τ ) + c(t)(Φx)(t)] − ρ(t)(F x)(t)
n
X
≤ kxkA+1
sup
[a(t)
+
bi (t) + γ(t) + nc(t)Ψ(t)] − ρ(t)(F x)(t)
1
t∈[t,t+ω]
≤
kxkA+1
M
1
− ρ(F x)(t)
≤ kxkA+1
{ρ + 1}
1
Z
i=1
ω
= {ρ + 1}
0
Zλ2
N − ρ(F x)(t)
(1 − λ)
n
X
λ
(λkxk1 )[a(s)(ZkxkA
bi (s)(ZkxkA
1)+
1)
1−λ
i=1
A
+ γ(s)(ZkxkA
1 ) + c(s)n(Zkxk1 )Ψ(t)]ds − ρ(F x)(t)
Z t+ω
n
X
e s)x(s)[a(s)xα (s) +
≤ {ρ + 1}
G(t,
bi (s)xβi (s − σi )
t
i=1
0
+ γ(s)x (s − τ ) + c(s)(Φx)(s)]ds − ρ(F x)(t)
= {ρ + 1}(F x)(t) − ρ(F x)(t) = (F x)(t).
Steps 1, 2 and 3 result that (F x)(t) ≥ λkF xk1 . Thus, F x ∈ Π and the proof is completed.
Lemma 2.2. Let the relation (2.2) satisfies and Rγ ≤ 1, then F : Πr,R 7→ Π is a strict-set-contraction.
Proof : Clearly, one may indicate that F is continuous and bounded operator. Let Ω ⊂ Πr,R be any
bounded set and
S µCω1 (Ω) = d, then, for any positive real number ε ≤ Rγd there exists a finite family {Ωi }
such that Ω = i Ωi and diamΩi ≤ d + ε. Thus,
kx − yk1 ≤ d + ε,
f or any x, y ∈ Ωi .
(2.11)
S
On the other hand, Ωi is precompact in Cω thus there is a finite family of subsets Ωij such that Ωi = j Ωij
and
(2.12)
max{kx − yk, kxα − y α k, kxβ1 − y β1 k, ..., kxβn − y βn k} ≤ ε,
for any x, y ∈ Ωij . Also, F (Ω) is precompact in Cω . To see this, note that
Z ω
n
X
1
|(F x)(t)| ≤
{a(t)
+
bi (t) + γ(t) + nc(t)Ψ(t)}dt
kxkA+1
1
1−λ
0
i=1
≤
N
.
γ A+1 (1 − λ)
This inequality together with (2.6) give
|(F x)0 (t)| = |x(t)[a(t)xα (t) +
n
X
bi (t)xβi (t − σi )
i=1
+ γ(t)x0 (t − τ ) + c(t + ω)(Φx)(t)] − ρ(t)(F x)(t)|
n
X
1
≤ kxkA+1
|a(t)
+
bi (t) + γ(t) + nc(t)Ψ(t)| + |(F x)(t)|
1
λ
i=1
≤
1
γ A+1
{M +
N
} = %.
λ(1 − λ)
(2.13)
P. Nasertayoob, S. M. Vaezpour, J. Nonlinear Sci. Appl. 7 (2014), 218–228
224
Suppose {ξm } is an arbitrary sequences on Ω. Clearly, {ξm } is bounded. Based on the definition of integral
operator F in (1.10) the function (F ξm )(t) is differentiable for all m ∈ N and t ∈ [0, ω]. For given ε > 0, if
we consider δ = %ε , then for all m ∈ N and t, t0 ∈ [0, ω] with |t − t0 | < δ we obtain
|(F ξm )(t) − (F ξm )(t0 )| ≤ %|t − t0 | ≤ ε.
Thus, {(F ξm )(t)} as a sequence of functions on [0, ω] is equicontinuous. Therefore, based on Arzela-Ascoli
theorem there exists a subsequent of {(F ξm )(t)}, say {(F ξmi )(t)}, which is uniformly convergence on [0, ω].
Consequently, F is a compact boundedSoperator and F (Ω) is precompact in Cω . As a result, there exists a
family of subsets Ωijk such that Ωij = k Ωijk and
kF x − F yk ≤ ε,
f or any x, y ∈ Ωijk .
(2.14)
On the other hand, applying (2.10), (2.11), (2.12) and (2.14), for any x, y ∈ Ωijk , we obtain
k(F x)0 − (F y)0 k =
sup |(F x)0 (t) − (F y)0 (t)|
t∈[t,t+ω]
≤
sup |ρ(t)(F x)0 (t) − ρ(t)(F y)0 (t)|
t∈[t,t+ω]
+
sup |x(t)[a(t)xα (t) +
t∈[t,t+ω]
n
X
bi (t)xβi (t − σi )
i=1
0
+ γ(t)x (t − τ ) + c(t)(Φx)(t)] − y(t)[a(t)y α (t)
n
X
+
bi (t)y βi (t − σi ) + γ(t)y 0 (t − τ ) + c(t)(Φy)(t)]|
i=1
≤ kρkk(F x)0 − (F y)0 k
+
sup |(x(t) − y(t))[a(t)y α (t) +
t∈[t,t+ω]
n
X
bi (t)y βi (t − σi )
i=1
0
+ γ(t)y (t − τ ) + c(t)(Φy)(t)]|
n
n
X
κ X βi
1
bi Rβi + γR + cg
ε + {aRα +
R ω}ε
≤
λ
κ−1
i=1
+ R{aε +
n
X
bi ε + γ(d + ε) + cg
i=1
i=1
κ
nεω}
κ−1
≤ Rγd + Jε,
(2.15)
where
n
J
=
X
1
+ a{Rα + R} +
bi {Rβi + R}
λ
+ 2γR + cg
κ
ω{
κ−1
i=1
n
X
Rβi + R}.
i=1
Therefore, from (2.14) and (2.15) and the condition ε ≤ Rγd, we obtain
kF x − F yk1 ≤ Rγd + Jε,
f or any x, y ∈ Ωijk .
Since ε is arbitrary small, we have
µCω1 (F (Ω)) ≤ RγµCω1 (Ω),
and the proof of lemma is completed. P. Nasertayoob, S. M. Vaezpour, J. Nonlinear Sci. Appl. 7 (2014), 218–228
225
Theorem 2.3. Let conditions of the Lemma 2.1 hold, also
r<{
1−λ 1
}A ,
N
(2.16)
and
1−λ 1
} A < R,
ZλN
then, the integral operator (1.11) has at least one periodic solution in Πr,R .
{
(2.17)
Proof : step 1. Let x ∈ Πr,R and kxk = r. If F x = x, then the operator F has fixed point.
Let F x > x. This means that F x − x ∈ Π − {0}, which implies that (F x)(t) − x(t) ≥ λkF x − xk1 ,
consequently,
kxk ≤ kF xk.
(2.18)
Applying (2.8), (2.5) and inequality (2.16), we have
Z t+ω
n
X
e s)x(s)[a(s)xα (s) +
(F x)(t) =
G(t,
bi (s)xβi (s − σi )
t
i=1
+γ(s)x0 (s − τ ) + c(s)(Φx)(s)]ds
Z t+ω
n
X
1
x(s)[a(s)xα (s) +
bi (s)xβi (s − σi )
≤
1−λ t
i=1
+γ(s)x0 (s − τ ) + c(s)(Φx)(s)]ds
Z t+ω
n
X
1
≤
kxk[a(s)kxkα1 +
bi (s)kxkβ1 i
1−λ t
i=1
+γ(s)kxk1 + c(s)
n
X
kxk1βi Ψ(s)]ds
i=1
≤
=
kxkkxkA
1
1−λ
Z
ω
[a(s) +
0
n
X
bi (s) + γ(s) + nc(s)Ψ(s)]ds
i=1
kxkrA N
< kxk,
1−λ
(2.19)
therefore,
kxk ≤ kF xk < kxk.
which is a contradiction.
step 2. Let x ∈ Πr,R and kxk = R. If F x = x, then the operator F has fixed point.
Let F x > x. This means that x − F x ∈ Π − {0}, which implies that x(t) − (F x)(t) ≥ λkF x − xk1 ,
consequently, for any t ∈ [0, ω] we have
(F x)(t) ≤ x(t).
(2.20)
Applying (2.20), (2.8), (2.4) and (2.17), we have
x(t) ≥ (F x)(t)
Z t+ω
n
X
α
e
=
G(t, s)x(s)[a(s)x (s) +
bi (s)xβi (s − σi )
t
i=1
0
+γ(s)x (s − τ ) + c(s)(Φx)(s)]ds
Z
n
X
λkxk1 t+ω
α
[a(s)x (s) +
bi (s)xβi (s − σi )
≥
1−λ t
i=1
0
+γ(s)x (s − τ ) + c(s)(Φx)(s)]ds
P. Nasertayoob, S. M. Vaezpour, J. Nonlinear Sci. Appl. 7 (2014), 218–228
≥
λkxk1
1−λ
Z
t+ω
α
[a(s)(λkxk1 ) +
t
n
X
226
bi (s)(λkxk1 )βi
i=1
n
X
+γ(s)(λkxk1 ) + c(s)
(λkxk1 )βi Ψ(s)]ds
i=1
≤
=
Zλkxk1 kxkA
1
Z
1−λ
ω
[a(s) +
0
n
X
bi (s) + γ(s) + nc(s)Ψ(s)]ds
i=1
Zλkxk1 RA
N > kxk1 .
1−λ
That is a contradiction. Therefore, (1.6) and (1.7) hold. By Theorem 1.3 we see that the integral operator
F has at least one fixed point in Πr,R under appropriate conditions. Remark 2.4. Note that
Z ω
n
X
N =
[a(s) +
bi (s) + γ(s) + nc(s)Ψ(s)]ds
0
Z
i=1
ω
≤
sup [a(t) +
0
t∈[t,t+ω]
n
X
bi (t) + γ(t) + nc(t)Ψ(t)]ds
i=1
≤ ωM.
Thus, for any arbitrary positive ω-periodic functions a, bi , γ, c and g the real number ω is bounded below
N
.
by M
On the other hand, inequality (2.2) yields
N
1−λ
≤
≤ ω.
λ2 Z{ρ + 1}
M
(2.21)
This shows that ω is also bounded below by (1 − λ)(λ2 Z{ρ + 1})−1 . However, λ depends on the both select
of the function ρ and period number ω. This means that ρ is a ω-periodic function with the following
property:
Z ω
Z ω
exp(2
ρ(θ)dθ) − exp(
ρ(θ)dθ) ≤ ωZ(ρ + 1),
0
that is valid for ρ(t) =
1+sin(2πt)
,
32
0
ω = 1 and Z = 0.9.
Remark 2.5. According to the Lemma 2.2, F : Πr,R 7→ Π is a strict-set-contraction operator so long as
Rγ ≤ 1. Or
1
.
R
Thus, with due attention to inequality (2.17), we obtain
γ(t) ≤
γ(t) ≤ {
ZλN 1
}A .
1−λ
1
1
On the other hand, inequality (2.16) yields N A < {1 − λ} A r−1 , consequently,
1
γ(t) <
{Zλ} A
.
r
(2.22)
P. Nasertayoob, S. M. Vaezpour, J. Nonlinear Sci. Appl. 7 (2014), 218–228
227
1
Thus, γ is bounded above by {Zλ} A r−1 .
Remark 2.6. Combining inequalities (2.16) and (2.17), we obtain
1−λ
1−λ
<N < A ,
ZλRA
r
(2.23)
that indicate that N is bounded above and below.
3. Illustrative Example
Consider the following system of neutral population dynamics with delay and feedback control
31
dN
= N (t)[1 + sin(2πt) − w(1 − cos(2πt))N 32 (t)
dt
32
33
1
− w(1 + cos(2πt))(N 33 (t − σ1 ) + N 34 (t − σ2 ))
2
10
− w(2 + cos(2πt))N 0 (t − τ ) − w(1 +
cos(2πt))u(t)]
18
32
33
du
1 − sin(2πt)
= (−1 − cos(2πt))u(t) +
(N 33 (t − σ1 ) + N 34 (t − σ2 )),
dt
0.582
which is an example of nonlinear population system (1.5), with
n = 2,
ω = 1,
10
cos(2πt)),
18
1 + sin(2πt)
ρ(t) =
,
32
η(t) = 1 + cos(2πt),
a(t) = w(1 − cos(2πt)),
c(t) = w(1 +
1 − sin(2πt)
,
0.582
γ(t) = w(2 + cos(2πt)),
g(t) =
1
cos(2πt)),
2
31
32
33
α=
, β1 = , β2 = ,
32
33
34
Z 1
Z 1
1 + sin(2πθ)
1
√ = 0.9692 < 1,
λ = exp(−
ρ(θ)dθ) = exp(−
(
)dθ) = 32
32
e
0
0
Z 1
Z 1
κ = exp(
η(θ)dθ) = exp(
1 + cos(2πθ)dθ)) = e > 1,
b1 (t) = b2 (t) = w(1 +
0
0
R = 0.99, Z = 0.9, r = 0.5.
Taking into consideration aforesaid data, we have
and
1 1
Z = 0.9 < 0.9692 × ( ) 32 = 0.948 = λB rB−A ,
2
(3.1)
10.31w
Zλ2
M
=
= 1.97 < (ρ + 1)
= 27.44.
N
5.27w
(1 − λ)
(3.2)
Besides, according to the inequality
1
≤ G(s, t),
e−1
P. Nasertayoob, S. M. Vaezpour, J. Nonlinear Sci. Appl. 7 (2014), 218–228
228
we obtain,
2 + cos(2πt) ≤ 4.5 + cos(2πt)
1
1
10
cos(2πt) + 1 + cos(2πt) + 2(1 +
cos(2πt))}
2
2
18
1
1
0.9{1 − cos(2πt) + 1 + cos(2πt) + 1 + cos(2πt)
2
2
Z 1
10
1 − sin(2πt)
1
2(1 +
cos(2πt))
}
18
e−1 0
0.582
1
1
0.9{1 − cos(2πt) + 1 + cos(2πt) + 1 + cos(2πt)
2
2
Z 1
10
1 − sin(2πt)
2(1 +
cos(2πt))
G(s, t)
}.
18
0.582
0
= 0.9{1 − cos(2πt) + 1 +
=
+
≤
+
(3.3)
These three expressions, (3.1), (3.2) and (3.3) show that conditions (2.1), (2.2) and (2.3) in Lemma 2.1 are
valid for our example. At the end, for the inequality (2.23) we obtain 0.0439 < N < 0.0745, that is valid by
proper choosing of w.
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