Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 387193, 7 pages
doi:10.1155/2012/387193
Research Article
Fixed Point Theorems and Uniqueness of
the Periodic Solution for the Hematopoiesis Models
Jun Wu1 and Yicheng Liu2
1
College of Mathematics and Computer Science, Changsha University of Science Technology Changsha
410114, China
2
Department of Mathematics and System Sciences, College of Science, National University of Defense
Technology, Changsha 410073, China
Correspondence should be addressed to Jun Wu, junwmath@hotmail.com
Received 20 September 2011; Revised 26 December 2011; Accepted 26 December 2011
Academic Editor: Elena Braverman
Copyright q 2012 J. Wu and Y. Liu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We present some results to the existence and uniqueness of the periodic solutions for the
hematopoiesis models which are described by the functional differential equations with multiple
delays. Our methods are based on the equivalent norm techniques and a new fixed point theorem
in the continuous function space.
1. Introduction
In this paper, we aim to establish the existence and uniqueness result for the periodic
solutions to the following functional differential equations with multiple delays:
x t −atxt ft, xt − τ1 t, . . . , xt − τm t,
1.1
where a, τi ∈ CR, R , f ∈ CRm1 , R are T -periodic functions on variable t for T > 0 and m
is a positive integer.
Recently, many authors investigate the dynamics for the various hematopoiesis
models, which includ the attractivity and uniqueness of the periodic solutions. For examples,
Mackey and Glass in 1 have built the following delay differential equation:
x t −axt θn
βθn
,
xn t − τ
1.2
where a, n, β, θ, τ are positive constants, xt denotes the density of mature cells in blood
circulation, and τ is the time between the production of immature cells in the bone marrow
2
Abstract and Applied Analysis
and their maturation for release in the circulating bloodstream; Liu et al. 2,Yang 3, Saker
4, Zaghrout et al. 5, and references therein, also investigate the attractivity and uniqueness
of the periodic solutions for some hematopoiesis models.
This paper is organized as follows. In Section 2, we present two new fixed point
theorems in continuous function spaces and establish the existence and uniqueness results
for the periodic solutions of 1.1. An illustrative example to the hematopoiesis models is
exhibited in the Section 3.
2. Fixed Point Theorems and Existence Results
2.1. Fixed Point Theorems
In this subsection, we will present two new fixed point theorems in continuous function
spaces. More details about the fixed point theorems in continuous function spaces can be
found in the literature 6–8 and references therein.
Let E be a Banach space equipped with the norm · E . BCR, E which denotes
the Banach space consisting of all bounded continuous mappings from R into E with norm
uC max{utE : t ∈ R} for u ∈ BCR, E.
Theorem 2.1. Let F be a nonempty closed subset of BCR, E and A : F → F an operator. Suppose
the following:
(H1 ) there exist β ∈ 0, 1 and G : R × R → R such that for any u, v ∈ F,
Aut − AvtE ≤ βut − vtE t
t−T
Gt, sus − vsE ds for t ∈ R,
2.1
(H2 ) there exist an α ∈ 0, 1 − β and a positive bounded function y ∈ CR, R such that
t
Gt, sysds ≤ αyt
∀t ∈ R.
2.2
t−T
Then A has a unique fixed point in F.
Proof. For any given x0 ∈ F, let xn Axn−1 , n 1, 2, . . .. By H1 , we have
Axn1 t − Axn tE ≤ βxn1 t − xn tE t
t−T
Gt, sxn1 s − xn sE ds.
2.3
Set an t xn1 t − xn tE , then we get
an1 t ≤ βan t t
t−T
Gt, san sds.
2.4
Abstract and Applied Analysis
3
In order to prove that the sequence {xn } is a Cauchy sequence with respect to norm
·C , we introduce an equivalent norm and show that {xn } is a Cauchy sequence with respect
to the new one. Basing on the condition H2 , we see that there are two positive constants M
and m such that m ≤ yt ≤ M for all t ∈ R. Define the new norm · 1 by
u1 sup
1
utE : t ∈ R ,
yt
u ∈ BCR, E.
2.5
Then,
1
1
uC ≤ u1 ≤ uC .
M
m
2.6
Thus, the two norms · 1 and · C are equivalent.
Set an xn1 − xn 1 , then we have an t ≤ ytan for t ∈ R. By 2.13, we have
1
1
an1 t ≤ βan yt
yt
an
≤ βan yt
t
Gt, san sds
t−T
t
Gt, sysds ≤ β α an .
2.7
t−T
Thus,
2
n1
a0 .
an1 ≤ β α an ≤ β α an−1 ≤ · · · ≤ β α
2.8
This means {xn } is a Cauchy sequence with respect to norm · 1 . Therefore, also, {xn } is a
Cauchy sequence with respect to norm · C . Thus, we see that {xn } has a limit point in F,
say u. It is known that u is the fixed point of A in F.
Suppose both u and v u / v are the fixed points of A, then Au u, Av v. Following
the similar arguments, we prove that
u − v1 Au − Av1 ≤ β α u − v1 .
2.9
It is impossible. Thus the fixed point of A is unique. This completes the proof of Theorem 2.1.
Let P CR, E be a Banach space consisting of all T -periodic functions in BCR, E with
the norm uP max{utE : t ∈ 0, T } for u ∈ P CR, E. Then, following the similar
arguments in Theorem 2.1, we deduce Theorem 2.2 which is a useful result for achieving the
existence of periodic solutions of functional differential equations.
4
Abstract and Applied Analysis
Theorem 2.2. Let A : P CR, E → P CR, E be an operator. Suppose the following:
1 there exist β ∈ 0, 1 and G : R × R → R such that for any u, v ∈ P CR, E,
H
t
Aut − AvtE ≤ βut − vtE Gt, s
t−T
n u ηi s − v ηi s ds,
E
2.10
i1
where ηi ∈ C0, T , R with ηi s ≤ s and n is a positive integer;
2 there exist two constants α, K and a positive function y ∈ CR, R such that nKα ∈
H
0, 1 − β, yηi s ≤ Kys, and
t
Gt, sysds ≤ αyt
∀t ∈ 0, T .
2.11
t−T
Then A has a unique fixed point in P CR, E.
1 , we have
Proof. For any given x0 ∈ F, let xk Axk−1 , k 1, 2, . . .. By H
Axk1 t − Axk tE ≤ βxk1 t − xk tE t
Gt, s
t−T
n xk1 ηi s − xk ηi s ds.
E
i1
2.12
Set ak t xk1 t − xk tE , then we get
ak1 t ≤ βak t t
Gt, s
t−T
n
ak ηi s ds.
2.13
i1
2 , we see that there are two positive constants M and m
Basing on the condition H
such that m ≤ yt ≤ M for all t ∈ 0, T . Define the new norm · 2 by
u2 sup
1
utE : t ∈ 0, T ,
yt
u ∈ P CR, E.
2.14
Then,
1
1
up ≤ u2 ≤ up .
M
m
Thus, the two norms · 2 and · p are equivalent.
2.15
Abstract and Applied Analysis
5
Set ak xk1 − xk 2 , then we have ak t ≤ ytak for t ∈ 0, T . By 2.13, we have
1
1
ak1 t ≤ βak yt
yt
ak
≤ βak yt
t
Gt, s
t−T
t
t−T
n
ak ηi s ds
i1
n
Gt, s y ηi s ds ≤ β nKα ak .
2.16
i1
Thus,
2
k1
a0 .
ak1 ≤ β nKα ak ≤ β nKα ak−1 ≤ · · · ≤ β nKα
2.17
This means {xk } is a Cauchy sequence with respect to norm · 2 . Therefore, {xk } is a Cauchy
sequence with respect to norm · p . Therefore, we see that {xk } has a limit point in P CR, E,
say u. It is easy to prove that u is the fixed point of A in P CR, E. The uniqueness of the fixed
point is obvious. This completes the proof of Theorem 2.2.
2.2. Existence and Uniqueness of the Periodic Solution
In order to show the existence of periodic solutions of 1.1, we assume that the function f is
fulfilling the following conditions:
Hf there exist Li > 0 i 1, 2, . . . , m such that for any xi , yi ∈ R,
m
ft, x1 , . . . , xm − f t, y1 , . . . , ym ≤
Li xi − yi ,
2.18
i1
Hfτ for all t ∈ 0, T , t ≥ τi t ≥ 0 i 1, 2, . . . , m.
Theorem 2.3. Suppose Hf and Hfτ hold. Then the equation 1.1 has a unique T -periodic
solution in C0, T .
Proof. By direction computations, we see that ϕt is the T -periodic solution if and only if ϕt
is solution of the following integral equation:
xt eλT
λT
e −1
t
e−λt−s λ − asxs fs, xs − τ1 s, . . . , xs − τm s ds,
2.19
t−T
where λ max{|at| : t ∈ 0, T }.
Thus, we would transform the existence of periodic solution of 1.1 into a fixed point
problem. Considering the map A : P CR, R → P CR, R defined by, for t ∈ 0, T ,
Axt eλT
λT
e −1
t
t−T
e−λt−s λ − asxs fs, xs − τ1 s, . . . , xs − τm s ds.
2.20
Then, u is a T -periodic solution of 1.1 if and only if u is a fixed point of the operator A in
P CR, R.
6
Abstract and Applied Analysis
At this stage, we should check that A fulfill all conditions of Theorem 2.2. In fact, for
x, y ∈ P CR, R, by assumption Hf , we have
Axt − Ay t
eλT
≤ λT
e −1
LeλT
≤ λT
e −1
m
2λ
xs − ys
Li xs − τi s − ys − τi s
ds
t
e
−λt−s
t−T
i1
x ηi s − y ηi s ds,
t
e
−λt−s
t−T
2.21
m1
i1
where ηi s s − τi s i 1, 2, . . . , m, ηm1 s s and L max{2λ, L1 , . . . , Lm }.
1 in Theorem 2.2 holds for β 0, n m 1, and Gt, s Thus, the condition H
LeλT /eλT − 1e−λt−s .
On the other hand, we choose a constant c > 0 such that 0 < m 1LeλT /eλT −
11/c λ < 1. Take α LeλT /eλT − 11/c λ and yt ect for t ∈ 0, T , then
yηi t ≤ yt, and we have
t
Gt, sysds t
t−T
t−T
LeλT −λt−s cs
e
e ds ≤ αyt.
eλT − 1
2.22
2 in Theorem 2.2 holds for K 1.
This implies the condition H
Following Theorem 2.2, we conclude that the operator A has a unique fixed point, say
ϕ, in P CR, R. Thus, 1.1 has a unique T -periodic solution in P CR, R. This completes the
proof of Theorem 2.3.
3. Application to the Hematopoiesis Model
In this section, we consider the periodic solution of following hematopoiesis model with
delays:
x t −atxt m
bi t
,
n t − τ t
1
x
i
i1
3.1
where a, bi , τi ∈ CR, R are T -periodic functions, τi satisfies conditions Hfτ , and n ≥ 1 is a
real number i 1, . . . , m.
Theorem 3.1. The delayed hematopoiesis model 3.1 has a unique positive T -periodic solution.
Proof. Let CT {y : y ∈ CT and yt ≥ 0 for t ≥ 0}, define the operator F : CT → CT by
eλT
Axt λT
e −1
t
e
t−T
−λt−s
bi s
ds.
λ − asxs 1 xn s − τi s
i1
m
3.2
Abstract and Applied Analysis
7
It is easy to show that A is welldefined. Furthermore, since the function
gt, x1 , . . . , xm m
bi t
1
xin
i1
for xi ∈ R
3.3
with the bounded partial derivative
bi tnxin−1
∂gt, x1 , . . . , xm ≤ max
: t ∈ 0, T , 1 ≤ i ≤ m ,
∂xi
1 xin
3.4
then it is easy to prove that the condition Hf holds. Following the similar arguments of
Theorem 2.3, we claim that the operator A has a unique fixed point in CT , which is the unique
positive T -periodic solution for equation 3.1. This completes the proof of Theorem 3.1.
Remark 3.2. Theorem 3.1 exhibits that the periodic coefficients hematopoiesis model admits a
unique positive periodic solution without additional restriction. Also, Theorem 3.1 improves
Theorem 2.1 in 2 and Corollary 1 in 3.
References
1 M. C. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science, vol. 197,
no. 4300, pp. 287–289, 1977.
2 G. Liu, J. Yan, and F. Zhang, “Existence and global attractivity of unique positive periodic solution for a
model of hematopoiesis,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 157–171,
2007.
3 X.-T. Yang, “Existence and global attractivity of unique positive almost periodic solution for a model
of hematopoiesis,” Applied Mathematics B, vol. 25, no. 1, pp. 25–34, 2010.
4 S. H. Saker, “Oscillation and global attractivity in hematopoiesis model with periodic coefficients,”
Applied Mathematics and Computation, vol. 142, no. 2-3, pp. 477–494, 2003.
5 A. Zaghrout, A. Ammar, and M. M. A. El-Sheikh, “Oscillations and global attractivity in delay
differential equations of population dynamics,” Applied Mathematics and Computation, vol. 77, no. 23, pp. 195–204, 1996.
6 E. de Pascale and L. de Pascale, “Fixed points for some non-obviously contractive operators,”
Proceedings of the American Mathematical Society, vol. 130, no. 11, pp. 3249–3254, 2002.
7 Y. Liu and X. Wang, “Contraction conditions with perturbed linear operators and applications,”
Mathematical Communications, vol. 15, no. 1, pp. 25–35, 2010.
8 T. Suzuki, “Lou’s fixed point theorem in a space of continuous mappings,” Journal of the Mathematical
Society of Japan, vol. 58, no. 3, pp. 769–774, 2006.
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