Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 658905, 11 pages
doi:10.1155/2012/658905
Research Article
Two General Algorithms for Computing
Fixed Points of Nonexpansive Mappings in
Banach Spaces
Shuang Wang
School of Mathematical Sciences, Yancheng Teachers University, Jiangsu, Yancheng 224051, China
Correspondence should be addressed to Shuang Wang, wangshuang19841119@163.com
Received 16 January 2012; Accepted 23 February 2012
Academic Editor: Yonghong Yao
Copyright q 2012 Shuang Wang. 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.
Recently, Yao et al. 2011 introduced two algorithms for solving a system of nonlinear variational
inequalities. In this paper, we consider two general algorithms and obtain the extension results for
computing fixed points of nonexpansive mappings in Banach spaces. Moreover, the fixed points
solve the same system of nonlinear variational inequalities.
1. Introduction
Let X be a real Banach space and let C be a nonempty closed convex subset of X. Recall that
a mapping T : C → C is said to be nonexpansive if T x − T y ≤ x − y, for all x, y ∈ C. We
denote by FixT the set of fixed points of T .
Recently, Yao et al. 1 considered the following algorithms:
xt ΠC I − tFΠC I − λAΠC I − μB xt ,
1.1
and for an arbitrary point x0 ∈ C,
xn1 βn xn 1 − βn ΠC I − αn FΠC I − λAΠC I − μB xn ,
n ≥ 0,
1.2
where ΠC : X → C is a sunny nonexpansive retraction, F : C → X is a strongly positive
bounded linear operator and A, B : C → X are α-inverse-strongly accretive and β-inversestrongly accretive operators, respectively. They proved that the {xt } defined by 1.1 and
{xn } defined by 1.2 converge strongly to a unique solution x of the variational inequality
2
Journal of Applied Mathematics
Fx, jx − z ≤ 0. Furthermore, they proved that the above algorithms converge strongly to
some solutions of a system of nonlinear inequalities, which involves finding x∗ , y∗ ∈ C × C
such that
λAy∗ x∗ − y∗ , jx − x∗ ≥ 0,
μBx∗ y∗ − x∗ , j x − y∗ ≥ 0,
∀x ∈ C,
∀x ∈ C.
1.3
For related works, please see 2–5 and the references therein.
In this paper, we introduce two general algorithms 3.3 and 3.22 defined below
and prove that the proposed algorithms strongly converge to x∗ ∈ FixT which solves the
variational inequality Fx∗ , jx∗ − u ≤ 0, u ∈ FixT , where F : C → X is a β-Lipschitzian
and η-strongly accretive operator. It is worth pointing out that our proofs contain some new
techniques.
2. Preliminaries
Let X be a real Banach space with norm · and let X ∗ be its dual space. The value of f ∈ X ∗
and x ∈ X will be denoted by x, f. For the sequence {xn } in X, we write xn x to indicate
that the sequence {xn } converges weakly to x. xn → x means that {xn } converges strongly
to x.
Let η > 0, a mapping F of C into X is said to be η-strongly accretive if there exists
jx − y ∈ Jx − y such that
2
Fx − Fy, j x − y ≥ ηx − y ,
2.1
for all x, y ∈ C. A mapping F from C into X is said to be β-Lipschitzian if, for β > 0,
Fx − Fy ≤ βx − y,
2.2
for all x, y ∈ C. From the definition of F see 1, we note that a strongly positive bounded
linear operator F is a F-Lipschitzian and γ-strongly accretive operator.
Let U {x ∈ X : x 1}. A Banach space X is said to be uniformly convex if for each
∈ 0, 2, there exists δ > 0 such that for any x, y ∈ U,
x − y ≥ x y
implies 2 ≤ 1 − δ.
2.3
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach
space X is said to be smooth if the limit
x ty − x
lim
,
t→0
t
2.4
Journal of Applied Mathematics
3
exists for all x, y ∈ U. It is said to be uniformly smooth if the limit 2.4 is attained uniformly
for x, y ∈ U. Also, we define a function ρ : 0, ∞ → 0, ∞ called the modulus of smoothness
of X as follows:
ρτ sup
1 x y x − y − 1 : x, y ∈ X, x 1, y τ .
2
2.5
It is known that X is uniformly smooth if and only if limτ → 0 ρτ/τ 0. Let q be a fixed real
number with 1 < q ≤ 2. Then a Banach space X is said to be q-uniformly smooth if there exists
a constant c > 0 such that ρτ ≤ cτ q for all τ > 0.
In order to prove our main results, we need the following lemmas.
Lemma 2.1 see 6. Let q be a given real number with 1 < q ≤ 2 and let X be a q-uniformly smooth
Banach space. Then
x yq ≤ xq q y, Jq x 2Kyq ,
2.6
for all x, y ∈ X, where K is the q-uniformly smooth constant of X and Jq is the generalized duality
∗
mapping from X into 2X defined by
Jq x f ∈ X ∗ : x, f xq , f xq−1 ,
2.7
for all x ∈ X.
Lemma 2.2 see 7. Let C be a closed convex subset of a smooth Banach space X, let D be a
nonempty subset of C and Π be a retraction from C onto D. Then Π is sunny and nonexpansive
if and only if
u − Πu, j y − Πu ≤ 0,
2.8
for all u ∈ C and y ∈ D.
Lemma 2.3 see 8. Let C be a nonempty bounded closed convex subset of a uniformly convex
Banach space X and let T be a nonexpansive mapping of C into itself. If {xn } is a sequence of C such
that xn x and xn − T xn → 0, then x is a fixed point of T .
Lemma 2.4 see 9, 10. Let {sn } be a sequence of nonnegative real numbers satisfying
sn1 ≤ 1 − λn sn λn δn γn ,
n ≥ 0,
2.9
where {λn }, {δn } and {γn } satisfy the following conditions: (i) {λn } ⊂ 0, 1 and ∞
n0 λn ∞, (ii)
∞
λ
δ
<
∞,
and
(iii)
γ
≥
0
n
≥
0,
γ
<
∞.
Then
limn → ∞ sn 0.
lim supn → ∞ δn ≤ 0 or ∞
n
n
n
n
n0
n0
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Journal of Applied Mathematics
Lemma 2.5 see 11. Let {xn } and {zn } be bounded sequences in Banach space E and {γn } be a
sequence in 0, 1 which satisfies the following condition:
0 < lim inf γ n ≤ lim sup γn < 1.
n→∞
n→∞
2.10
Suppose that xn1 γn xn 1 − γn zn , n ≥ 0, and lim supn → ∞ zn1 − zn − xn1 − xn ≤ 0. Then
limn → ∞ zn − xn 0.
In addition, we need the following extension of Lemma 2.5 in Wang and Hu 2 in a
2-uniformly smooth Banach space.
Lemma 2.6. Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space
√
X. Let F : C → X be a β-Lipschitzian and η-strongly accretive operator with 0 < η ≤ 2βK
and 0< t < η/2β2 K 2 . Then S I − tF : C → X is a contraction with contraction coefficient
τt 1 − 2tη − tβ2 K 2 .
Proof. By Lemma 2.1, we have
2
Sx − Sy2 x − y − t Fx − Fy 2
2
x − y − 2t Fx − Fy, j x − y 2t2 K 2 Fx − Fy
2
2
2
≤ x − y − 2tηx − y 2t2 β2 K 2 x − y
2
1 − 2t η − tβ2 K 2 x − y ,
for all x, y ∈ C. From 0 < η ≤
√
2βK and 0 < t < η/2β2 K 2 , we have
Sx − Sy ≤ τt x − y,
where τt τt .
2.11
2.12
1 − 2tη − tβ2 K 2 ∈ 0, 1. Hence S is a contraction with contraction coefficient
3. Main Results
Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth
Banach space X. Let T : C → C be a nonexpansive mapping with√FixT /
∅. Let F : C → X
2βK.
Let
t ∈ 0, η/2β2 K 2 be a β-Lipschitzian
and
η-strongly
accretive
operator
with
0
<
η
≤
and τt 1 − 2tη − tβ2 K 2 , consider a mapping St on C defined by
St x ΠC I − tF T x,
x ∈ C,
3.1
Journal of Applied Mathematics
5
where ΠC is a sunny nonexpansive retraction from X onto C. It is easy to see that St is a
contraction. Indeed, from Lemma 2.6, we have
St x − St y ≤ ΠC I − tF T x − ΠC I − tF T y
≤ I − tF T x − I − tF T y
≤ τt T x − T y
≤ τt x − y,
3.2
for all x, y ∈ C. Therefore, the following implicit method is well defined:
xt ΠC I − tF T xt ,
xt ∈ C.
3.3
Theorem 3.1. The net {xt } generated by the implicit method 3.3 converges in norm, as t → 0 to
the unique solution x∗ ∈ FixT of the variational inequality:
Fx∗ , jx∗ − u ≤ 0,
u ∈ FixT .
3.4
Proof. We first show that the solution set of 3.4 is singleton. As a matter of fact, we assume
that x∗ ∈ FixT and x ∈ FixT both are solutions to 3.4, then
Fx∗ , jx∗ − x
≤ 0,
F x,
jx − x∗ ≤ 0.
3.5
3.6
Adding 3.5 to 3.6, we get
Fx∗ − F x,
jx∗ − x
≤ 0.
3.7
The strong accretive of F implies that x∗ x,
and the uniqueness is proved. Below we use
x∗ ∈ FixT to denote the unique solution of 3.4.
Next, we prove that {xt } is bounded. Taking u ∈ FixT , from 3.3 and using
Lemma 2.6, we have
xt − u ΠC I − tF T xt − ΠC u
≤ I − tF T xt − I − tF T u − tFT u
≤ I − tF T xt − I − tF T u tFu
≤ τt xt − u tFu,
3.8
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Journal of Applied Mathematics
that is,
xt − u ≤
t Fu.
1 − τt
3.9
Observe that
lim
t→0
t
1
.
1 − τt η
3.10
From t → 0 , we may assume, without loss of generality, that t ≤ η/2β2 K 2 − , where is an arbitrarily small positive number. Thus, we have t/1 − τt to be continuous, for all
t ∈ 0, η/2β2 K 2 − . Therefore, we obtain
t
M1 sup
:t∈
1 − τt
0,
η
−
2β2 K 2
< ∞.
3.11
From 3.9 and 3.11, we have {xt } bounded and so is {FT xt }.
On the other hand, from 3.3, we obtain
xt − T xt ΠC I − tF T xt − ΠC T xt ≤ I − tF T xt − T xt tFT xt −→ 0
t −→ 0 .
3.12
Next, we show that {xt } is relatively norm-compact as t → 0 . Assume that {tn } ∈
0, η/2β2 K 2 such that tn → 0 as n → ∞. Put xn : xtn . It follows from 3.12 that
xn − T xn −→ 0 n −→ ∞.
3.13
For a given u ∈ FixT , by 3.3 and using Lemma 2.2, we have
xt − I − tF T xt , jxt − u ≤ 0.
3.14
By 3.14 and using Lemma 2.6, we have
xt − u2 xt − u, jxt − u
xt − I − tF T xt , jxt − u I − tF T xt − u, jxt − u
≤
I − tF T xt − u, jxt − u
≤
I − tF T xt − I − tF T u, jxt − u t Fu, ju − xt ≤ τt xt − u2 t Fu, ju − xt ,
3.15
Journal of Applied Mathematics
7
that is,
xt − u2 ≤
t Fu, ju − xt ≤ M1 Fu, ju − xt .
1 − τt
3.16
In particular,
xn − u2 ≤ M1 Fu, ju − xn .
3.17
Since {xt } is bounded, without loss of generality, we may assume that {xn } converges
weakly to a point x.
Noticing 3.13 we can use Lemma 2.3 to get x ∈ FixT . Therefore we
can substitute x for u in 3.17 to get
≤ M1 F x,
jx − xn .
xn − x
3.18
Consequently, the weak convergence of {xn } to x actually implies that xn → x.
This has
proved the relative norm compactness of the net {xt } as t → 0 .
We next show that x solves the variational inequality 3.4. Observe that
xt ΠC I − tF T xt − I − tF T xt − I − tF xt I − tF T xt xt − tFxt 1 ⇒ Fxt ΠC I − tF T xt − I − tF T xt − I − tF xt I − tF T xt .
t
3.19
For any u ∈ FixT , we have
1 ΠC I − tF T xt − I − tF T xt , jxt − u
t
1 −
I − tF xt − I − tF T xt , jxt − u
t
1
≤ − xt − T xt , jxt − u Fxt − FT xt , jxt − u
t
1
≤ − I − T xt − I − T u, jxt − u βxt − T xt xt − u
t
≤ βM2 xt − T xt ,
Fxt , jxt − u 3.20
where M2 sup{xt − u, t ∈ 0, η/2β2 K 2 }.
Now replacing t in 3.20 with tn and letting n → ∞, we have
F x,
jx − u ≤ 0.
3.21
That is, x ∈ FixT is a solution of 3.4, hence x x∗ by uniqueness. In summary, we have
shown that each cluster point of {xt } at t → 0 equals x∗ . Therefore, xt → x∗ as t → 0.
8
Journal of Applied Mathematics
Theorem 3.2. Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly
smooth Banach space X with a weakly sequentially continuous duality
mapping j. Let F : C → X be
√
a β-Lipschitzian and η-strongly accretive operator with 0 < η ≤ 2βK. Suppose that T : C → C is
a nonexpansive mapping with FixT /
∅. Let ΠC be a sunny nonexpansive retraction from X onto C.
Let {αn } and {βn } be two real sequences in 0, 1 and satisfy the conditions:
A1 limn → ∞ αn 0 and
∞
n0
αn ∞,
A2 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1.
For given x1 ∈ C arbitrarily, let the sequence {xn } be generated by
xn1
yn ΠC I − αn F T xn ,
βn xn 1 − βn yn , n ≥ 0.
3.22
Then the sequence {xn } strongly converges to x∗ ∈ FixT which solves the variational inequality
3.4.
Proof. We proceed with the following steps.
Step 1. We claim that {xn } is bounded. From limn → ∞ αn 0, we may assume, without loss of
generality, that 0 < αn ≤ η/2β2 K 2 − for all n. In fact, let u ∈ FixT , from 3.22 and using
Lemma 2.6, we have
yn − u ΠC I − αn F T xn − ΠC u
≤ I − αn F T xn − I − αn F T u − αn Fu
≤ ταn xn − u αn Fu,
where ταn 3.23
1 − 2αn η − αn β2 K 2 ∈ 0, 1. Then from 3.22 and 3.23, we obtain
xn1 − u ≤ βn xn − u 1 − βn yn − u
≤ βn xn − u 1 − βn ταn xn − u αn Fu
≤ 1 − 1 − βn 1 − ταn xn − u 1 − βn αn Fu
⎫
⎧
⎨
αn Fu ⎬
.
≤ max xn − u,
⎩
1 − ταn ⎭
3.24
By induction, we have
xn − u ≤ max x1 − u, M3 Fu ,
3.25
where M3 sup{αn /1 − ταn : 0 < αn ≤ η/2β2 K 2 − } < ∞. Therefore, {xn } is bounded. We
also obtain that {yn } and {FT xn } are bounded.
Journal of Applied Mathematics
9
Step 2. We claim that limn → ∞ xn − yn 0. Observe that
yn1 − yn ΠC I − αn1 F T xn1 − ΠC I − αn F T xn ≤ I − αn1 F T xn1 − I − αn F T xn ≤ T xn1 − T xn αn1 FT xn1 αn FT xn ≤ xn1 − xn αn1 FT xn1 αn FT xn .
3.26
Therefore, we have
lim sup yn1 − yn − xn1 − xn ≤ 0.
n→∞
3.27
From 3.22, 3.27, and using Lemma 2.5, we have limn → ∞ xn − yn 0.
Step 3. We claim that limn → ∞ yn − T yn 0. Observe that
yn − T yn ΠC I − αn F T xn − ΠC T yn ≤ T xn − T yn αn FT xn ≤ xn − yn αn FT xn .
3.28
Hence, from Step 2 and limn → ∞ αn 0, we have
lim yn − T yn 0.
n→∞
3.29
Step 4. We claim that lim supn → ∞ Fx∗ , jx∗ − yn ≤ 0, where x∗ limt → 0 xt and xt is defined
by 3.3. Since yn is bounded, there exists a subsequence {ynk } of {yn } which converges
weakly to ω. From Step 3, we obtain T ynk ω. From Lemma 2.3, we have ω ∈ FixT .
Hence, using Theorem 3.1, we have x∗ ∈ FixT and
∗
Fx , jx∗ − ω ≤ 0.
lim sup Fx∗ , j x∗ − yn lim Fx∗ , j x∗ − ynk
k→∞
n→∞
3.30
Step 5. We claim that {xn } converges strongly to x∗ ∈ FixT . From 3.22 and using
Lemma 2.2, we have
ΠC I − αn F T xn − I − αn F T xn , j yn − x∗ ≤ 0.
3.31
10
Journal of Applied Mathematics
Observe that
yn − x∗ 2 ΠC I − αn F T xn − x∗ , j yn − x∗
ΠC I − αn F T xn − I − αn F T xn , j yn − x∗
I − αn F T xn − x∗ , j yn − x∗
≤
I − αn F T xn − x∗ , j yn − x∗
≤
I − αn F T xn − I − αn F T x∗ , j yn − x∗ αn Fx∗ , j x∗ − yn
≤ I − αn F T xn − I − αn F T x∗ yn − x∗ αn Fx∗ , j x∗ − yn
≤ ταn xn − x∗ yn − x∗ αn Fx∗ , j x∗ − yn
≤
3.32
2
τα2n
1
xn − x∗ 2 yn − x∗ αn Fx∗ , j x∗ − yn ,
2
2
that is,
yn − x∗ 2 ≤ τα xn − x∗ 2 2αn Fx∗ , j x∗ − yn .
n
3.33
By 3.22 and 3.33, we have
2
xn1 − x∗ 2 ≤ βn xn − x∗ 2 1 − βn yn − x∗ ≤ βn xn − x∗ 2 1 − βn ταn xn − x∗ 2 2αn Fx∗ , j x∗ − yn
≤ 1 − 1 − βn 1 − ταn xn − x∗ 2 2M3 1 − βn 1 − ταn Fx∗ , j x∗ − yn
1 − λn xn − x∗ 2 λn δn ,
3.34
where λn 1 − βn 1 − ταn , δn 2M3 Fx∗ , jx∗ − yn . It is easy to see that λn → 0,
∞
n1 λn ∞ and lim supn → ∞ δn ≤ 0. Hence, by Lemma 2.4, the sequence {xn } converges
strongly to x∗ ∈ FixT . From x∗ limt → 0 xt and Theorem 3.1, we have x∗ to be the unique
solution of the variational inequality 3.4.
Taking T ΠC I − λAΠC I − μB and F F, where 0 < λ ≤ α/K 2 and 0 < μ ≤ η/K 2 ,
we obtain the following theorems immediately.
Corollary 3.3 see 1, Theorem 3.5. The net {xt } generated by the implicit method 1.1 converges
in norm, as t → 0 , to the unique solution x of variational inequality
x ∈ Ω,
Fx,
jx − z ≤ 0,
z ∈ Ω.
3.35
Journal of Applied Mathematics
11
Corollary 3.4 see 1, Theorem 3.7. Let C be a nonempty closed convex subset of a uniformly
convex and 2-uniformly smooth Banach space X and let ΠC be a sunny nonexpansive retraction from
X onto C. Let the mappings A, B : C → X be α-inverse-strongly accretive and β-inverse-strongly
accretive operators, respectively. Let F : C → H be a strongly positive linear bounded operator with
coefficient γ > 0. For given x0 ∈ C, let the sequence {xn } be generated iteratively by 1.2. Suppose
that the sequences {αn } and {βn } satisfy the conditions (A1) and (A2), then {xn } converges strongly
to x ∈ Ω which solves the variational inequality 3.35.
Acknowledgments
Supported by the Natural Science Foundation of Yancheng Teachers University under Grant
11YCKL009 and Professor and Doctor Foundation of Yancheng Teachers University under
Grant 11YSYJB0202.
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