Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 547828, 9 pages
doi:10.1155/2010/547828
Research Article
Convergence of Paths for Perturbed Maximal
Monotone Mappings in Hilbert Spaces
Yuan Qing,
1
Xiaolong Qin,
1
Haiyun Zhou,
2
and Shin Min Kang
3
1
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
2
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China
3
Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea
Correspondence should be addressed to Shin Min Kang, [email protected]
Received 16 July 2010; Revised 30 November 2010; Accepted 20 December 2010
Academic Editor: Ljubomir B. Ciric
Copyright q 2010 Yuan Qing 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.
Let H be a Hilbert space and C a nonempty closed convex subset of H.LetA : C →
H be a maximal monotone mapping and f : C → C a bounded demicontinuous strong
pseudocontraction. Let {x
t
2 Fixed Point Theory and Applications
3 A : C → H is said to be pseudomonotone if for any sequence {x
n
} in C which
converges weakly to an element x in C with lim sup
n →∞
Ax
n
,x
n
− x≤0 we have
lim inf
n →∞
Ax
n
,x
n
− y
≥
Ax, x − y
, ∀y ∈ C.
1.2
4 A : C → H is said to be bounded if it carries bounded sets into bounded sets; it is
coercive if Ax, x/x→∞as x→∞.
5 Let X, Y be linear normed spaces. T : DT ⊂ X → Y is said to be demicontinuous if,
for any {x
≤
x − y
2
, ∀x, y ∈ D
f
.
1.3
8 The mapping f with the domain Df and the range Rf in H is said to be strongly
pseudocontractive if there exists a constant β ∈ 0, 1 such that
f
x
− f
y
,x− y
≤ β
x − y
In 1968, Browder 4 proved the existence results of zero points for maximal monotone
mappings in reflexive Banach spaces. To be more precise, he proved the following theorem.
Fixed Point Theory and Applications 3
Theorem B2. Let X be a reflexive Banach space, T
1
: DT
1
⊆ X → 2
X
∗
a maximal monotone
mapping and T
2
a bounded, pseudomonotone and coercive mapping. Then, for any h ∈ X
∗
,thereexists
u ∈ X such that h ∈ T
1
T
2
u,orRT
1
T
2
is all of X
∗
.
For the existence of continuous paths for continuous pseudocontractions in Banach
spaces, Morales and Jung 15 proved the following theorem.
Theorem MJ. Let E be a Banach space. Suppose that C is a nonempty closed convex subset of E and
demicontinuous monotone mapping. Then T is pseudomonotone.
Proof. For any sequence {x
n
}⊂C which converges weakly to an element x in C such that
lim sup
n →∞
Tx
n
,x
n
− x
≤ 0,
2.1
we see from the monotonicity of T that
Tx,x
n
− x
≤
Tx
n
,x
n
− x
. 2.2
,x− z
, 2.4
4 Fixed Point Theory and Applications
which yields that
lim inf
n →∞
Tx
n
,x
n
− z
lim inf
n →∞
Tx
n
,x− z
.
2.5
Noticing that
g,x
n
− z
≤
g
t
,x− y
≤ lim inf
n →∞
Tx
n
,x− y
.
2.8
Noting that z
t
→ x, t → 0, g
t
Tz
t
,andT : C → H is demicontinuous, we have g
t
Tz
t
Tx as t → 0, and hence
lim inf
n →∞
Tx
n
ii If {x
t
} is bounded, then Ax
t
→0 as t →∞.
iii If A
−1
0
/
∅,then{x
t
} is bounded and satisfies
x
t
− f
x
t
,x
t
− p
≤ 0, ∀p ∈ A
−1
0
,
2.12
From the boundedness of f and {x
t
}, one has lim
t →∞
Ax
t
0.
iii For p ∈ A
−1
0, one obtains that
x
t
− p
2
x
t
− p, x
t
− p
f
x
Ax
t
,x
t
− p
≤ β
x
t
− p
2
f
p
− p, x
t
− p
.
2.13
It follows that
−tAx
t
, one arrives at
x
t
− f
x
t
,x
t
− p
−t
Ax
t
,x
t
− p
≤ 0, ∀t>0. 2.15
This completes the proof.
Lemma 2.4. Let C be a nonempty closed convex subset of a Hilbert space H and A a maximal
monotone mapping. Then C ⊆ I AC. If one defines g : C → C by gxI A
−1
x,for
all x ∈ C,theng : C → C is a nonexpansive mapping with FgA
y
≤
x − y
, 2.16
which yields that g is nonexpansive mapping. Notice that
x g
x
⇐⇒
I A
x x ⇐⇒ Ax 0. 2.17
6 Fixed Point Theory and Applications
That is, FgA
−1
0. On the other hand, for any x ∈ C, we have
x − g
x
I A
x − x
Ax
.
2.18
This completes the proof.
Set S 0, 1.LetBS denote the Banach space of all bounded real value functions
on S with the supremum norm, X a subspace of BS,andμ an element in X
∗
, where X
∗
denotes the dual space of X. Denote by μf the value of μ at f ∈ X.Ifes1, for all x ∈ S,
sometimes μe will be denoted by μ1. When X contains constants, a linear functional μ on
X is called a mean on X if μ1μ 1. We also know that if X contains constants, then the
following are equivalent.
1 μ1μ 1.
2 inf
s∈S
fs ≤ μf ≤ sup
s∈S
μ
t
y − z, x
t
− z
≤ 0,y∈ C. 2.20
Now, we are in a position to prove the main results of this work.
Theorem 2.6. Let H be a Hilbert space and C a nonempty closed convex subset of H.LetA :
C → H be a maximal monotone mapping and f : C → C a bounded demicontinuous strong
pseudocontraction. Let {x
t
} be as in Lemma 2.3.Then{x
t
} is bounded if and only if {x
t
} converges
strongly to a zero point p of A as t →∞which is the unique solution in A
−1
0 to the following
variational inequality:
f
p
− p, y − p
≤ 0, ∀y ∈ A
y∈C
h
y
.
2.22
From the convexity and continuity of h, we can get the convexity and continuity of the set
K. Since h is continuous and H is a Hilbert space, we see that h attains its infimum over
K;see20 for more details. Then K is nonempty bounded and closed convex subset of C.
Indeed, K contains one point only. Set gxI A
−1
x, where g : K → K.Noticethat
g is nonexpansive. Since every nonempty bounded and closed convex subset has the fixed
point property for nonexpansive self-mapping in the framework of Hilbert spaces, then g has
a fixed point p in K,thatis,gpp. It follows from Lemma 2.4 that Ap0. On the other
hand, one has μ
t
x
t
− p min
y∈C
hy.InviewofLemma 2.5,weobtainthat
μ
t
y − p, x
t
− p
}→p.Fromiii of Lemma 2.3, one has
x
t
α
− f
x
t
α
,x
t
α
− y
≤ 0, ∀y ∈ A
−1
0
.
2.25
Taking limit in 2.25, one gets that
p − f
p
,p− y≤0, ∀y ∈ A
−1
x
t
β
,x
t
β
− p
≤ 0. 2.28
Taking limit in 2.28,weobtainthat
q − f
q
,q− p
≤ 0. 2.29
8 Fixed Point Theory and Applications
Adding 2.27 and 2.29, we have
p − q f
q
− f
p
2.31
It follows that p q.Thatis,{x
t
} converges strongly to p ∈ A
−1
0, which is the unique
solution to the f ollowing variational inequality:
f
p
− p, y − p
≤ 0, ∀y ∈ A
−1
0
.
2.32
Remark 2.7. From Theorem 2.6, we can obtain the following interesting fixed point theorem.
The composition of bounded, demicontinuous, and strong pseudocontractions with the
metric projection has a unique fixed point. That is, p Pfp.
Acknowledgment
The third author was supported by the National Natural Science Foundation of China Grant
no. 10771050.
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