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Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 682465, 12 pages
doi:10.1155/2012/682465
Research Article
Global Dynamical Systems Involving Generalized
f-Projection Operators and
Set-Valued Perturbation in Banach Spaces
Yun-zhi Zou,1 Xi Li,2 Nan-jing Huang,2 and Chang-yin Sun1
1
2
School of Automation, Southeast University, Jiangsu, Nanjing 210096, China
Department of Mathematics, Sichuan University, Sichuan, Chengdu 610064, China
Correspondence should be addressed to Nan-jing Huang, nanjinghuang@hotmail.com
Received 29 February 2012; Accepted 16 May 2012
Academic Editor: Zhenyu Huang
Copyright q 2012 Yun-zhi Zou et al. 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.
A new class of generalized dynamical systems involving generalized f -projection operators is
introduced and studied in Banach spaces. By using the fixed-point theorem due to Nadler, the
equilibrium points set of this class of generalized global dynamical systems is proved to be
nonempty and closed under some suitable conditions. Moreover, the solutions set of the systems
with set-valued perturbation is showed to be continuous with respect to the initial value.
1. Introduction
It is well known that dynamics system has long time been an interest of many researchers.
This is largely due to its extremely wide applications in a huge variety of scientific fields, for
instance, mechanics, optimization and control, economics, transportation, equilibrium, and
so on. For details, we refer readers to references 1–10 and the references therein.
In 1994, Friesz et al. 3 introduced a class of dynamics named global projective
dynamics based on projection operators. Recently, Xia and Wang 7 analyzed the global
asymptotic stability of the dynamical system proposed by Friesz as follows:
dx
PK x − ρNx − x,
dt
1.1
where N : Rn → Rn is a single-valued function, ρ > 0 is a constant, PK x denotes the projection of the point x on K; here K ⊂ Rn is a nonempty, closed, and convex subset.
2
Journal of Applied Mathematics
Later, in 2006, Zou et al. 9 studied a class of global set-valued projected dynamical
systems as follows:
dxt
∈ PK gxt − ρNxt − gxt ,
dt
for a.a. t ∈ 0, J,
1.2
x0 b,
where N : Rn → 2R is a set-valued function, g : Rn → Rn is a single-valued function, ρ > 0
is a constant, PK x denotes the projection of the point x on K, b is a given point in Rn .
The concept of generalized f-projection operator was first introduced by Wu and
Huang 11 in 2006. They also proved that the generalized f-projection operator is an
extension of the projection operator PK in Rn and it owns some nice properties as PK does;
see 12, 13. Some applications of generalized f-projection operator are also given in 11–13.
Very recently, Li et al. 14 studied the stability of the generalized f-projection operator with
an application in Banach spaces. We would like to point out that Cojocaru 15 introduced
and studied the projected dynamical systems on infinite Hilbert spaces in 2002.
To explore further dynamic systems in infinite dimensional spaces in more general
forms has been one of our major motivations and efforts recently, and this paper is a response
to those efforts. In this paper, we introduce and study a new class of generalized dynamical
systems involving generalized f-projection operators. By using the fixed-point theorem due
to Nadler 16, we prove that the equilibrium points set of this class of generalized global
dynamical systems is nonempty and closed. We also show that the solutions set of the
systems with set-valued perturbation is continuous with respect to the initial value. The
results presented in this paper generalize many existing results in recent literatures.
n
2. Preliminaries
Let X be a Banach space and let K ⊂ X be a closed convex set, let N : X → 2X be a set-valued
mapping, and let g : X → X be a single-valued mapping. The normalized duality mapping
J from X to X∗ is defined by
Jx x∗ ∈ X∗ : x, x∗ x2 x∗ 2 ,
2.1
for x ∈ X. For convenience, we list some properties of J· as follows.X is a smooth Banach
space, J· is single valued and hemicontinuous; that is, J is continuous from the strong
topology of X to the weak ∗ topology of X∗ .
Let CX denote the family of all nonempty compact subsets of X and let H·, · denote
the Hausdorff metric on CX defined by
HA, B max sup inf da, b, sup inf da, b ,
a∈A b∈B
b∈B a∈A
∀A, B ∈ CX.
2.2
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3
In this paper, we consider a new class of generalized set-valued dynamical system,
that is, to find those absolutely continuous functions x· from 0, h → X such that
dxt
f ∈ ΠK gxt − ρNxt − gxt,
dt
for a.a. t ∈ 0, h,
2.3
x0 b,
where b ∈ X, ρ > 0 is a constant and f : K → R ∪ {∞} is proper, convex, and lower semif
continuous and ΠK : X → 2K is a generalized f-projection operator denoted by
f
ΠK x u ∈ K : GJx, u inf GJx, ξ ,
ξ∈K
∀x ∈ X.
2.4
It is well known that many problems arising in the economics, physical equilibrium
analysis, optimization and control, transportation equilibrium, and linear and nonlinear
mathematics programming problems can be formulated as projected dynamical systems see,
e.g., 1–10, 15, 17 and the references therein. We also would like to point out that problem
2.3 includes the problems considered in Friesz et al. 3, Xia and Wang 7, and Zou et al. 9
as special cases. Therefore, it is important and interesting to study the generalized projected
dynamical system 2.3.
Definition 2.1. A point x∗ is said to be an equilibrium point of global dynamical system 2.3,
if x∗ satisfies the following inclusion:
f 0 ∈ ΠK gx − ρNx − gx.
2.5
Definition 2.2. A mapping N : X → X is said to be
i α-strongly accretive if there exists some α > 0 such that
Nx − N y , J x − y ≥ αx − y2 ,
∀x, y ∈ K;
2.6
ii ξ-Lipschitz continuous if there exists a constant ξ ≥ 0 such that
Nx − N y ≤ ξ
x − y
,
∀x, y ∈ K.
2.7
Definition 2.3. A set-valued mapping T : X → X is said to be ξ-Lipschitz continuous if there
exists a constant ξ > 0 such that
H T x, T y ≤ ξ
x − y
,
∀x, y ∈ K,
2.8
where H·, · is the Hausdorff metric on CX.
Lemma 2.4 see 14. Let X be a real reflexive and strictly convex Banach space with its dual X∗
and let K be a nonempty closed convex subset of X. If f : K → R ∪ {∞} is proper, convex, and
4
Journal of Applied Mathematics
f
f
lower semicontinuous, then ΠK is single valued. Moreover, if X has Kadec-Klee property, then ΠK is
continuous.
Lemma 2.5 see 18. Let X be a real uniformly smooth Banach space. Then X is q-uniformly
smooth if and only if there exists a constant Cq > 0 such that, for all x, y ∈ X,
x y
q ≤ xq q y, Jq x Cq y
q .
2.9
Lemma 2.6 see 19. Let X, d be a complete metric space and let T1 , T2 be two set-valued contractive mappings with same contractive constants θ ∈ 0, 1. Then
HFT1 , FT2 ≤
1
sup HT1 x, T2 x,
1 − θ x∈X
2.10
where FT1 and FT2 are fixed-point sets of T1 and T2 , respectively.
Lemma 2.7 see 19. Let X be a real strictly convex, reflexive, and smooth Banach space. For any
f
f
x1 , x2 ∈ X, let x1 ΠK x1 and x2 ΠK x2 . Then
x1 − x2 ,
Jx1 − Jx2 , x1 − x2 ≥ 2M δ
2M
2
2.11
where
M
x1 2 x2 2
.
2
2.12
We say that X is 2-uniformly convex and 2-uniformly smooth Banach space if there
exist k, c > 0 such that
δX ≥ k2 ,
2.13
ρX t ≤ ct2 ,
where
x y
: x y
1, x − y
≥ ,
δX inf 1 − 2 1 x y
x − y
− 1 : x 1, y
≤ t .
ρX t sup
2
2.14
Based on Lemma 2.7, we can obtain the following lemma.
Lemma 2.8. Let X be 2-uniformly convex and 2-uniformly smooth Banach space. Then
c
f f
ΠK x − ΠK y
≤ 64 x − y
,
k
∀x, y ∈ X.
2.15
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5
Proof. According to Lemma 2.7, we have
⎛
f ⎞
f
x
−
Π
y
Π
f
K
K f
⎠,
Jx − J y , ΠK x − ΠK y ≥ 2M12 δ⎝
2M1
2.16
where
M1 f
f
ΠK x2 ΠK y2
.
2
2.17
Since δX ≥ k2 , 2.16 yields
2
f f
ΠK x − ΠK y
≤ Jx − J y .
k
2.18
From the property of J·, we have
2M22 ρX 4
x − y
/M2
Jx − J y ≤
x − y
≤ 32c
x − y
.
2.19
It follows from 2.18 and 2.19 that
c
f f
ΠK x − ΠK y
≤ 64 x − y
.
k
2.20
This completes the proof.
3. Equilibrium Points Set
In this section, we prove that the equilibrium points set of the generalized set-valued dynamical system 2.3 is nonempty and closed.
Theorem 3.1. Let X be 2-uniformly convex and 2-uniformly smooth Banach space. Let N : X →
CX be μ-Lipschitz continuous and let g : X → X be α-Lipschitz continuous and β-strongly
accretive. If
c
1 α2 − 2βC2 64 α ρμ < 1,
k
3.1
then the equilibrium points set of the generalized set-valued dynamical system 2.3 is nonempty and
closed.
6
Journal of Applied Mathematics
Proof. Let
f T x x − gx ΠK gx − ρNx ,
∀x ∈ K.
3.2
f
Since N : X → CX and ΠK are continuous, we know that T : X → CX. From Definition
2.1, it is easy to see that x∗ is an equilibrium point of the generalized set-valued dynamical
system 2.3 if and only if x∗ is a fixed-point of T in X, that is:
f x∗ ∈ T x∗ x∗ − gx∗ ΠK gx∗ − ρNx∗ .
3.3
Thus, the equilibrium points set of 2.3 is the same as the fixed-points set of T . We first prove
that FT is nonempty. In fact, for any x, y ∈ X and a1 ∈ T x, there exists u ∈ Nx such that
f a1 x − gx ΠK gx − ρu .
3.4
Since u ∈ Nx, and N : X → CX, it follows from Nadler 16 that there exists v ∈ Ny
such that
u − v ≤ H Nx, N y .
3.5
f a2 y − g y ΠK g y − ρv .
3.6
Let
Then a2 ∈ T y. From 3.4 to 3.6, we have
f f a1 − a2 x − y − gx − g y ΠK gx − ρu − ΠK g y − ρv f f
≤ x − y − gx − g y ΠK gx − ρu − ΠK g y − ρv .
3.7
Since g is α-Lipschitz continuous and β-strongly accretive,
x − y − gx − g y 2 ≤ x − y
2 − 2 gx − g y , J x − y C2 gx − gy
2
2
≤ 1 α2 − 2βC2 x − y
.
3.8
f
From Lemma 2.8, where ΠK is Lipchitz continuous, we have
c f f
ΠK gx − ρu − ΠK g y − ρv ≤ 64 gx − g y ρu − v
k
c ≤ 64 α
x − y
ρu − v .
k
3.9
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7
From the selection of v and the Lipschitz continuity of N,
u − v ≤ H Nx, N y ≤ μ
x − y
.
3.10
In light of 3.7–3.10, we have
a1 − a2 ≤
1
α2
c
− 2βC2 64 α ρμ x − y
L
x − y
,
k
3.11
where
L
1 α2 − 2βC2 64
c
α ρμ .
k
3.12
Now 3.11 implies that
d a1 , T y inf a1 − a2 ≤ L
x − y
.
a2 ∈T y
3.13
Since a1 ∈ T x is arbitrary, we have
sup d a1 , T y ≤ L
x − y
.
a1 ∈T x
3.14
Similarly, we can prove
sup dT x, a2 ≤ L
x − y
.
a2 ∈T y 3.15
From 3.14, 3.15, and the definition of the Hausdorff metric H on CX, we have
H T x, T y ≤ L
x − y
,
∀x, y ∈ K.
3.16
Now the assumption of the theorem implies that L < 1 and so T x is a set-valued contractive
mapping. By the fixed-point theorem of Nadler 16, there exists x∗ such that x∗ ∈ T x∗ , and
thus x∗ is the equilibrium point of 2.3. This means that FT is nonempty.
Now we prove that FT is closed. Let {xn } ⊂ FT with xn → x0 n → ∞. Then
xn ∈ T xn and 3.16 imply that
HT xn , T x0 ≤ Lxn − x0 .
3.17
Thus,
dx0 , T x0 ≤ x0 − xn dxn , T xn HT xn , T x0 ≤ 1 Lxn − x0 −→ 0 as n → ∞.
It follows that x0 ∈ FT and so FT are closed. This completes the proof.
3.18
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Journal of Applied Mathematics
Remark 3.2. Theorem 3.1 is a generalization of Theorem 1 in Zou et al. 9 from Rn to Banach
space X.
4. Sensitivity of the Solutions Set
In this section, we study the sensitivity of the solutions set of the generalized dynamical system with set-valued perturbation for 2.3 as follows:
dxt
f ∈ ΠK gxt − ρNxt − gxt Fxt,
dt
for a.a.t ∈ 0, h,
4.1
x0 b,
where g and b are the same as in 2.3, F : X → 2X is a set-valued mapping, and N : X → X
is a single-valued mapping. Let Sb denote the set of all solutions of 4.1 on 0, h with
x0 b.
Now we prove the following result.
Theorem 4.1. Let X be 2-uniformly convex and 2-uniformly smooth Banach space. Let g : X → X
be α-Lipschitz continuous, let N : X → X be μ-Lipschitz continuous, and let F : X → CX be a
ω-Lipschitz continuous set-valued mapping with compact convex values. If
64
c
α ρμ α ω < 1,
k
c
h 64 α ρμ α ω < 1,
k
4.2
then Sb is nonempty and continuous.
Proof. Let
f Mx ΠK gx − ρNx − gx Fx.
4.3
Then M : X → CX is a set-valued mapping with compact convex values since F : X →
CX is a set-valued mapping with compact convex values. For any x1 , x2 ∈ X and a1 ∈
Mx1 , there exists u ∈ Fx1 such that
f a1 ΠK gx1 − ρNx1 − gx1 u.
4.4
Since u ∈ Fx1 , and F : X → CX, it follows from Nadler 16 that there exists v ∈ Fx2 such that
u − v ≤ HFx1 , Fx2 .
4.5
f a2 ΠK gx2 − ρNx2 − gx2 v.
4.6
Let
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9
Then a2 ∈ Mx2 . From 4.4 and 4.6, we have
f f
a1 − a2 ΠK gx1 − ρNx1 − ΠK gx2 − ρNx2 − gx2 − gx2 u − v
f f
≤ gx1 − gx2 ΠK gx − ρNx1 − ΠK g y − ρNx2 u − v.
4.7
Since g is α-Lipschitz continuous,
gx1 − gx2 ≤ αx2 − x2 .
4.8
f
From Lemma 2.8, ΠK is Lipschitz continuous. It follows from the continuity of N and g that
c f f
ΠK gx1 − ρNx1 −ΠK gx2 −ρNx2 ≤ 64 gx1 −gx2 ρNx1 −Nx2 k
c
≤ 64 α ρμ x1 − x2 .
k
4.9
From the selection of v and the Lipschitz continuity of F, we know
u − v ≤ HFx1 , Fx2 ≤ ωx1 − x2 .
4.10
In light of 4.7–4.10, we have
a1 − a2 ≤
α 64
c
α ρμ ω x1 − x2 θ
x − y
,
k
4.11
where
θ α 64
c
α ρμ ω.
k
4.12
Now 4.11 implies that
da1 , Mx2 inf a1 − a2 ≤ θx1 − x2 .
a2 ∈Mx2 4.13
Since a1 ∈ Mx1 is arbitrary, we obtain
sup da1 , Mx2 ≤ θx1 − x2 .
4.14
sup dMx1 , a2 ≤ θx1 − x2 .
4.15
a1 ∈Mx1 Similarly, we can prove
a2 ∈Mx2 10
Journal of Applied Mathematics
From 4.13, to 4.15, and the definition of the Hausdorff metric H on CX, we have
HMx1 , Mx2 ≤ θx1 − x2 ,
∀x1 , x2 ∈ X.
4.16
Now 4.2 implies that 0 < θ < 1, and so Mx is a set-valued contractive mapping. Let
Qx, b y ∈ C0, h, X | yt b t
zsds, zs ∈ Mxs ,
4.17
0
where
C0, h, X f : 0, h −→ X | f is continuous .
4.18
Since M : X → CX is a continuous set-valued mapping with compact convex values,
by the Michael’s selection theorem see, e.g., Theorem 16.1 in 20, we know that Qx, b is
nonempty for each x and b ∈ X. Moreover, it is easy to see that the set of fixed-points of
Qx, b coincides with Sb. It follows from 21 or 8 that Qx, b is compact and convex for
each x and b ∈ X. Suppose that bm is the initial value of 4.1; that is, x0 bm m 0, 1, 2, . . .
and bm → b0 m → ∞. Since
Qx, b0 Qx, bm − bm b0 ,
4.19
it is obvious that Qx, bm converges uniformly to Qx, b0 .
Next we prove that Qx, bm is a set-valued contractive mapping. For any given
x1 , x2 ∈ C0, h, X, since M : X → CX is a continuous set-valued mapping with compact
convex values, by the Michael’s selection theorem see, e.g., Theorem 16.1 in 20, we know
that Mx1 s has a continuous selection r1 s ∈ Mx1 s. Let
c1 t bm t
r1 sds.
4.20
0
Then c1 ∈ Qx1 , bm . Since r1 s ∈ Mx1 s is measurable and Mx2 s is a measurable
mapping with compact values, we know that there exists a measurable selection r2 s ∈
Mx2 s such that
r1 s − r2 s ≤ HMx1 s, Mx2 s.
4.21
Thus, it follows from 4.16 that
r1 s − r2 s ≤ HMx1 s, Mx2 s ≤ θx1 − x2 .
4.22
Let
c2 t bm t
0
r2 sds.
4.23
Journal of Applied Mathematics
11
Then c2 ∈ Qx2 , bm and
c1 − c2 ≤
h
r1 s − r2 sds ≤ hHMx1 s, Mx2 s
0
4.24
≤ hθx1 − x2 .
Hence, we have
dc1 , Qx2 , bm inf
c2 ∈Qx2 ,bm c1 − c2 ≤ hθx1 − x2 .
4.25
Since c1 ∈ Qx1 , bm is arbitrary, we obtain
sup
c1 ∈Qx1 ,bm dc1 , Qx2 , bm ≤ hθx1 − x2 .
4.26
dQx1 , bm , c2 ≤ hθx1 − x2 .
4.27
Similarly, we can prove that
sup
c2 ∈Qx2 ,bm From the definition of the Hausdorff metric H on CX, 4.26 and 4.27 imply that
HQx1 , bm , Qx2 , bm ≤ hθx1 − x2 ,
∀x1 , x2 ∈ X, m 0, 1, 2, . . . .
4.28
Since hθ < 1, it is easy to see that Qx, b has a fixed-point for each given b ∈ X, and so Sb
is nonempty for each given b ∈ X. Setting
Wm x Qx, bm ,
m 0, 1, 2, . . . ,
4.29
we know that Wm x are contractive mappings with the same contractive constant hθ. By
Lemma 2.6 and 4.28, we have
HFWm , FW0 ≤
1
sup HWm x, W0 x −→ 0.
1 − hθ x∈X
4.30
Thus, FWm → FW0 , which implies that Sbm → Sb; that is, the solution of 4.1 is
continuous with respect to the initial value of 4.1. This completes the proof.
Remark 4.2. Theorem 4.1 is a generalization of Theorem 2 in Zou et al. 9 from Rn to Banach
space X.
Acknowledgments
The authors appreciate greatly the editor and two anonymous referees for their useful comments and suggestions, which have helped to improve an early version of the paper. This
12
Journal of Applied Mathematics
work was supported by the National Natural Science Foundation of China 11171237 and
the Key Program of NSFC Grant no. 70831005.
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