Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 368137, 15 pages
doi:10.1155/2011/368137
Research Article
Convergence of Iterative Sequences for Fixed Point
and Variational Inclusion Problems
Li Yu and Ma Liang
School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China
Correspondence should be addressed to Li Yu, [email protected]
Received 14 November 2010; Accepted 8 February 2011
Academic Editor: Yeol J. Cho
Copyright q 2011 L. Yu and M. Liang. 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.
An iterative process is considered for finding a common element in the fixed point set of a strict
pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal
monotone operator and an inverse strongly monotone mapping. Strong convergence theorems of
common elements are established in real Hilbert spaces.
1. Introduction and Preliminaries
Throughout this paper, we always assume that H is a real Hilbert space with the inner
product ·, · and the norm ·.
Let C be a nonempty closed convex subset of H and S : C → C a nonlinear mapping.
In this paper, we use FS to denote the fixed point set of S. Recall that the mapping S is said
to be nonexpansive if
Sx − Sy≤x − y, ∀x, y ∈ C. 1.1
S is said to be κ-strictly pseudocontractive if there exists a constant κ ∈ 0, 1 such that
Sx − Sy
Ax − Ay, x − y
≥ 0, ∀x, y ∈ C. 1.3
A is said to be inverse strongly monotone if there exists a constant α>0 such that
Ax − Ay, x − y≥α
Ax − Ay
2
, ∀x, y ∈ C.
1.4
For such a case, A is also said to be α-inverse strongly monotone.
Let M : H → 2
H
be a set-valued mapping. The set DM defined by DM{x ∈ H :
Mx
/
∅} is said to be the domain of M.ThesetRM defined by RM
x∈H
Mx is said to
be the range of M.ThesetGM defined by GM{x, y ∈ H × H : x ∈ DM,y ∈ RM}
is said to be the graph of M.
Recall that M is said to be monotone if
x − y, f − g
> 0, ∀
≥ 0, ∀y ∈ C. 1.6
Denote by VIC, A of the solution set of 1.6. It is known that x ∈ C is a solution to 1.6 if
and only if x is a fixed point of the mapping P
C
I − λA, where λ>0 is a constant and I is the
identity mapping.
Recently, many authors considered the convergence of iterative sequences for the
variational inequality 1.6 and fixed point problems of nonlinear mappings see, for example,
1–32.
In 2005, Iiduka and Takahashi 7 proved the following theorem.
Theorem IT. Let C be a closed convex subset of a real Hilbert space H.LetA be an α-inverse-
strongly monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such
that FS ∩ VIC, A
/
∅. Suppose that x
1
x ∈ C and {x
n
} is given by
x
n1
α
n
x
1 − α
n
SP
C
n
∞,
∞
n1
|
α
n1
− α
n
|
< ∞,
∞
n1
|
λ
n1
− λ
n
|
< ∞,
1.8
then {x
n
} converges strongly to P
FS∩VIC,A
x.
Fixed Point Theory and Applications 3
In 2007, Y. Yao and J C. Yao 31 further obtained the following theorem.
α
n
u β
n
x
n
γ
n
SP
C
I − λ
n
A
y
n
,n≥ 1,
1.9
where {α
n
}, {β
n
}, and {γ
n
} are three sequences in 0, 1 and {λ
n
} is a sequence in 0, 2a.If{α
n
},
n
≤ lim sup
n →∞
β
n
< 1,
d lim
n →∞
λ
n1
− λ
n
0,
then {x
n
} converges strongly to P
FS∩Ω
u.
In this work, motivated by the above results, we consider the problem of finding a
common element in the fixed point set of a strict pseudocontraction and in the zero set of a
nonlinear mapping which is the sum of a maximal monotone operator and a inverse strongly
monotone mapping. Strong convergence theorems of common elements are established in
real Hilbert spaces. The results presented in this paper improve and extend the corresponding
results announced by Iiduka and Takahashi 7 and Y. Yao and J C. Yao 31.
In order to prove our main results, we also need the following lemmas.
Lemma 1.1 see 22. Let C be a nonempty closed convex subset of a Hilbert space H, A : C → H
a mapping, and M : H → 2
H
a maximal monotone mapping. Then,
F
} is a sequence in C with x
n
xand x
n
− Sx
n
→ 0,
then x ∈ FS.
4 Fixed Point Theory and Applications
Lemma 1.4 see 28. Let {x
n
} and {y
n
} be bounded sequences in a Hilbert space H, and let {β
n
}
be a sequence in 0, 1 with
0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1.
1.11
Suppose that x
n1
1 − β
Lemma 1.5 see 29. Assume that {α
n
} is a sequence of nonnegative real numbers such that
α
n1
≤
1 − γ
n
α
n
δ
n
, 1.13
where {γ
n
} is a sequence in 0, 1 and {δ
n
} is a sequence such that
a
∞
n1
γ
n
∞,
b lim sup
n →∞
δ
r
x − J
s
x, J
r
x − x
, ∀s, t > 0,x∈ H,
1.14
where J
r
I rM
−1
and J
s
I sM
−1
.
2. Main Results
Theorem 2.1. Let H be a real Hilbert space H and C a nonempty close and convex subset of H.Let
M : H → 2
H
and W : H → 2
H
two maximal monotone operators such that DM ⊂ C and
DW ⊂ C, respectively. Let S : C → C be a κ-strict pseudocontraction, A : C → H an α-inverse
strongly monotone mapping, and B : C → H a β-inverse strongly monotone mapping. Assume that
F : FS ∩ A M
−1
0 ∩ B W
x
n
γ
n
δ
n
J
r
n
y
n
− r
n
Ay
n
1 − δ
n
SJ
r
n
y
n
− r
n
}, and {δ
n
} are sequences in 0, 1.
Fixed Point Theory and Applications 5
Assume that the following restrictions are satisfied:
a 0 <a≤ r
n
≤ b<2α, lim
n →∞
r
n
− r
n1
0,
b 0 <c≤ s
n
≤ d<2β
n
, lim
n →∞
s
n
− s
n1
0,
c 0 ≤ κ ≤ δ
n
<e<1, lim
n →∞
Proof. The proof is split into five steps.
Step 1. Show that {x
n
} is bounded.
Note that I − r
n
A and I − s
n
B are nonexpansive for each fixed n ≥ 1. Indeed, we see
from the restriction a that
I − r
n
Ax −
I − r
n
A
y
2
x − y
Ax − Ay
2
≤
x − y
2
, ∀x, y ∈ C.
2.2
This shows that I − r
n
A is nonexpansive f or each fixed n ≥ 1, so is I − s
n
B.Put
S
n
x δ
n
x
1 − δ
n
Sx, ∀x ∈ C. 2.3
In view of the restriction c,weobtainfromLemma 1.2 that S
n
Ay
n
− p
≤ α
n
u − p β
n
x
n
− p γ
n
J
r
n
y
n
− r
n
Ay
n
− p
≤ α
n
u − p β
n
x
n
n1
− y
n
≤
x
n1
− s
n1
Bx
n1
−
x
n
− s
n
Bx
n
J
s
n1
x
n
− s
n
|
s
n1
− s
n
|
s
n1
J
s
n1
x
n
− s
n
Bx
n
−
x
n
− s
n
Bx
n
≤x
J
s
n1
x
n
− s
n
Bx
n
−
x
n
− s
n
Bx
n
s
n1
. 2.6
Put
z
n
J
n
Ay
n
J
r
n1
y
n
− r
n
Ay
n
− J
r
n
y
n
− r
n
Ay
n
≤
r
n1
y
n
− r
n
Ay
n
−
y
n
− r
n
Ay
n
≤y
n1
− y
n
2M
2
|
r
n1
− r
n
−
y
n
− r
n
Ay
n
r
n1
. 2.9
Substituting 2.5 into 2.8 yields that
z
n1
− z
n
≤x
n1
− x
n
M
3
|
s
z
n1
− S
n
z
n
≤z
n1
− z
n
z
n
− Sz
n
|
δ
n
− δ
n1
|
≤x
n1
− x
n
M
4
|
s
{
z
n
− Sz
n
}
,M
3
. 2.13
Put
l
n
x
n1
− β
n
x
n
1 − β
n
, ∀n ≥ 1.
2.14
Note that
l
n1
− l
n
n
1 − β
n
u
γ
n1
1 − β
n1
S
n1
z
n1
− S
n
z
n
γ
n1
1 − β
n1
−
γ
n
1 − β
n
n1
z
n1
− S
n
z
n
.
2.15
It follows from 2.12 that
l
n1
− l
n
≤
α
n1
1 − β
n1
−
α
n
1 − β
α
n1
1 − β
n1
−
α
n
1 − β
n
u − S
n
z
n
x
n1
− x
n
M
4
|
n
−
x
n1
− x
n
≤ 0.
2.17
8 Fixed Point Theory and Applications
From Lemma 1.4,weobtainthat
lim
n →∞
l
n
− x
n
0.
2.18
Notice that
x
n1
− x
n
1 − β
are nonexpansive, we see that
z
n
− p
2
≤
x
n
− p
2
− r
n
2α − r
n
Ay
n
− Ap
n
− Bp
2
.
2.22
It follows from 2.21 that
x
n1
− p
2
≤ α
n
u − p
2
β
n
x
n
− p
2
γ
n
z
n
− p
2
≤ α
n
u − p
2
x
n
− p
Ay
n
− Ap
2
≤ α
n
u − p
2
x
n
− x
n1
x
n
− p x
n1
− p
.
2.24
In view of 2.20, we see from the restrictions a, d,ande that
lim
2
γ
n
S
n
z
n
− p
2
≤ α
n
u − p
2
β
n
x
n
− p
x
n
− p
2
γ
n
y
n
− p
2
≤ α
n
u − p
2
x
n
Bx
n
− Bp
2
≤ α
n
u − p
2
x
n
− x
n1
x
n
− p x
n1
− p
n
Ay
n
− J
r
n
p − r
n
Ap
2
≤
z
n
− p,
y
n
− r
n
Ay
n
−
p − r
n
Ap
z
n
− p
−
y
n
− r
n
Ay
n
−
p − r
n
Ap
2
≤
1
2
2
1
2
z
n
− p
2
y
n
− p
2
−
z
n
2
z
n
− p
2
y
n
− p
2
−
z
n
− y
n
2
2r
x
n
− p
2
−
z
n
− y
n
2
2r
n
z
n
− y
n
Ay
n
− Ap
.
2.29
z
n
− y
n
Ay
n
− Ap.
2.30
10 Fixed Point Theory and Applications
In a similar way, we can obtain that
y
n
− p
2
≤
x
n
− p
2
−
u − p
2
β
n
x
n
− p
2
γ
n
S
n
z
n
− p
2
≤ α
n
2
x
n
− p
2
− γ
n
z
n
− y
n
2
2r
n
z
n
− y
n
Ay
n
− p x
n1
− p
2r
n
z
n
− y
n
Ay
n
− Ap.
2.33
In view of 2.25, we obtain from the restrictions d and e that
lim
n →∞
z
n
− y
n
0.
2.34
Notice from 2.31,weseethat
x
n1
− p
2
≤ α
n
u − p
2
β
n
x
n
− p
2
γ
n
z
n
− p
2
u − p
2
x
n
− p
2
− γ
n
y
n
− x
n
2
2s
n
y
n
x
n
− p x
n1
− p
2s
n
y
n
− x
n
Bx
n
− Bp.
2.36
In view of 2.28, we obtain from the restrictions d and e that
lim
n →∞
y
n
− x
n
0.
n
− x
n
. 2.39
In view of 2.20, we see from the restriction d that
lim
n →∞
S
n
z
n
− x
n
0.
2.40
Note that
Sz
n
− x
n
S
n
z
n
− x
n
1 − δ
n
≤
Sx
n
− Sz
n
Sz
n
− x
n
. 2.43
In view of 2.38 and 2.42,weseefromLemma 1.3 that
lim
n →∞
Sx
n
− x
n
0.
2.44
This completes Step 3.
Step 4. Show that lim sup
n →∞
i
} is bounded, we can choose a subsequence {x
n
i
j
} of {x
n
i
} converging weakly to
x. We may, without loss of generality, assume that x
n
i
x, where denotes the weak
convergence. Next, we prove that x ∈F.Inviewof2.44, we can conclude from Lemma 1.3
that x ∈ FS easily. Notice that
y
n
− r
n
Ay
n
∈ z
n
r
n
Mz
n
. 2.46
12 Fixed Point Theory and Applications
Let μ ∈ Mν. Since M is monotone, we have
u − q, x
n
− q
≤ 0.
2.49
This completes Step 4.
Step 5. Show that x
n
→ q as n →∞.
Notice that
x
n1
− q
2
α
n
u − q, x
n1
− q β
n
x
n
− q, x
n1
x
n
− q
2
x
n1
− q
2
γ
n
2
S
n
J
r
β
n
2
x
n
− q
2
x
n1
− q
2
γ
n
2
x
n
− q
2
x
n1
− q
2
.
2.50
This in turn implies that
x
n1
− q
2
≤
1 − α
2.52
This completes Step 5. This whole proof is completed.
Fixed Point Theory and Applications 13
If S is a nonexpansive mapping and δ
n
0, then Theorem 2.1 is reduced to the
following.
Corollary 2.2. Let H be a real Hilbert space H and C a nonempty close and convex subset of H.Let
M : H → 2
H
and W : H → 2
H
be two maximal monotone operators such that DM ⊂ C and
DW ⊂ C, respectively. Let S : C → C be a nonexpansive mapping, A : C → H an α-inverse
strongly monotone mapping and B : C → H a β-inverse strongly monotone mapping. Assume that
F : FS ∩ A M
−1
0 ∩ B W
−1
0
/
∅.Let{x
n
} be a sequence generated in the following
manner:
x
1
∈ C,
y
n
Ay
n
, ∀n ≥ 1,
2.53
where u ∈ C is a fixed element, J
r
n
I r
n
M
−1
and J
s
n
I s
n
W
−1
, {r
n
} is a sequence in
0, 2α, {s
n
} is a sequence in 0, 2β and {α
n
}, {β
n
} and {γ
n
α
n
∞,
d 0 < lim inf
n →∞
β
n
≤ lim inf
n →∞
β
n
< 1.
Then, the sequence {x
n
} converges strongly to q P
F
u.
Next, we consider the problem of finding common fixed points of three strict
pseudocontractions.
Theorem 2.3. Let C be a nonempty closed convex subset of a real Hilbert space H and P
C
the metric
projection from H onto C.LetS : C → C be a κ-strict pseudocontraction, T
A
: C → H an α-strict
pseudocontraction, and B : C → H a β-strict pseudocontraction. Assume that F : FS ∩ FT
A
∩
FT
B
z
n
r
n
T
A
z
n
x
n1
α
n
u β
n
x
n
γ
n
δ
n
y
n
1 − δ
n
Sy
n
, lim
n →∞
s
n
− s
n1
0,
c 0 ≤ κ ≤ δ
n
<e<1, lim
n →∞
δ
n
− δ
n1
0,
14 Fixed Point Theory and Applications
d lim
n →∞
α
n
0,
∞
n1
α
n
∞,
e 0 < lim inf
r
n
Tx
n
. Putting B I −T
B
,weseethat
B is 1−β/2-inverse strongly monotone. We also have FT
B
VIC, B and P
C
x
n
−s
n
Bx
n
1−s
n
x
n
s
n
Ru
n
.InviewofTheorem 2.1, we can obtain the desired results immediately.
Acknowledgments
The authors are extremely grateful to the referees for useful suggestions that improved the
contents of the paper. This work was supported by the National Natural Science Foundation
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