Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 417234, 13 pages
doi:10.1155/2012/417234
Research Article
Approximation of Common Fixed Points of
Nonexpansive Semigroups in Hilbert Spaces
Dan Zhang, Xiaolong Qin, and Feng Gu
Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University, Hangzhou,
Zhejiang 310036, China
Correspondence should be addressed to Feng Gu, gufeng99@sohu.com
Received 14 October 2011; Accepted 11 December 2011
Academic Editor: Rudong Chen
Copyright q 2012 Dan Zhang 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 real Hilbert space. Consider on H a nonexpansive semigroup S {T s : 0 ≤ s < ∞}
with a common fixed point, a contraction f with the coefficient 0 < α < 1, and a strongly positive
linear bounded self-adjoint operator A with the coefficient γ > 0. Let 0 < γ < γ/α. It is proved
that the sequence {x
sn } generated by the iterative method x0 ∈ H, xn1 αn γfxn βn xn 1 −
βn I − αn A1/sn 0 n T sxn ds, n ≥ 0 converges strongly to a common fixed point x∗ ∈ FS,
where FS denotes the common fixed point of the nonexpansive semigroup. The point x∗ solves
the variational inequality γf − Ax∗ , x − x∗ ≤ 0 for all x ∈ FS.
1. Introduction and Preliminaries
Let H be a real Hilbert space and T be a nonlinear mapping with the domain DT . A point
x ∈ DT is a fixed point of T provided T x x. Denote by FT the set of fixed points of T ;
that is, FT {x ∈ DT : T x x}. Recall that T is said to be nonexpansive if
T x − T y ≤ x − y,
∀x, y ∈ DA.
1.1
Recall that a family S {T s | s ≥ 0} of mappings from H into itself is called a
one-parameter nonexpansive semigroup if it satisfies the following conditions:
i T 0x x, for all x ∈ H;
ii T s tx T sT tx, for all s, t ≥ 0 and for all x ∈ H;
iii T sx − T sy
≤ x − y
, for all s ≥ 0 and for all x, y ∈ H;
iv for all x ∈ C, s → T sx is continuous.
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We denote by FS the set of common fixed points of S, that is, FS 0≤s<∞ FT s.
It is known that FS is closed and convex; see 1. Let C be a nonempty closed and convex
subset of H. One classical way to study nonexpansive mappings is to use contractions to
approximate a nonexpansive mapping; see 2, 3. More precisely, take t ∈ 0, 1 and define a
contraction Tt : C → C by
Tt x tu 1 − tT x,
x ∈ C,
1.2
where u ∈ C is a fixed point. Banach’s contraction mapping principle guarantees that Tt has
a unique fixed point xt in C. If T enjoys a nonempty fixed point set, Browder 2 proved the
following well-known strong convergence theorem.
Theorem B. Let C be a bounded closed convex subset of a Hilbert space H and let T be a nonexpansive
mapping on C. Fix u ∈ C and define zt ∈ C as zt tu 1 − tT zt for t ∈ 0, 1. Then as t → 0,
{zt } converges strongly to a element of FT nearest to u.
As motivated by Theorem B, Halpern 4 considered the following explicit iteration:
x0 ∈ C,
xn1 αn u 1 − αn T xn ,
n ≥ 0,
1.3
and proved the following theorem.
Theorem H. Let C be a bounded closed convex subset of a Hilbert space H and let T be a nonexpansive mapping on C. Define a real sequence {αn } in 0, 1 by αn n−θ , 0 < θ < 1. Define a
sequence {xn } by 1.3. Then {xn } converges strongly to the element of FT nearest to u.
In 1977, Lions 5 improved the result of Halpern 4, still in Hilbert spaces, by proving
the strong convergence of {xn } to a fixed point of T where the real sequence {αn } satisfies the
following conditions:
C1 limn → ∞ αn 0;
C2 ∞
n1 αn ∞;
C3 limn → ∞ αn1 − αn /α2n1 0.
It was observed that both Halpern’s and Lions’s conditions on the real sequence {αn }
excluded the canonical choice αn 1/n 1. This was overcome in 1992 by Wittmann 6,
who proved, still in Hilbert spaces, the strong convergence of {xn } to a fixed point of T if {αn }
satisfies the following conditions:
C1 limn → ∞ αn 0;
C2 ∞
n1 αn ∞;
∞
C3 n1 |αn1 − αn | < ∞.
Recall that a mapping f : H → H is an α-contraction if there exists a constant α ∈ 0, 1
such that
fx − f y ≤ αx − y,
∀x, y ∈ H.
1.4
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3
Recall that an operator A is strongly positive on H if there exists a constant γ > 0 such
that
Ax, x ≥ γ
x
2 ,
∀x ∈ H.
1.5
Iterative methods for nonexpansive mappings have recently been applied to solve
convex minimization problems; see, for example, 7–13 and the references therein. A typical
problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive
mapping T on a real Hilbert space H:
min
1
x∈FT 2
Ax, x − x, b,
1.6
where A is a linear bounded operator on H and b is a given point in H. In 11, it is proved
that the sequence {xn } defined by the iterative method below, with the initial guess x0 ∈ H
chosen arbitrarily,
xn1 I − αn AT xn αn b,
n ≥ 0,
1.7
strongly converges to the unique solution of the minimization problem 1.6 provided that
the sequence {αn } satisfies certain conditions.
Recently, Marino and Xu 9 studied the following continuous scheme:
xt tγfxt I − tAT xt ,
1.8
where f is an α-contraction on a real Hilbert space H, A is a bounded linear strongly positive
operator and γ > 0 is a constant. They showed that {xt } strongly converges to a fixed
point x of T . Also in 9, they introduced a general explicit iterative scheme by the viscosity
approximation method:
xn ∈ H,
xn1 αn γfxn I − αn AT xn ,
n≥0
1.9
and proved that the sequence {xn } generated by 1.9 converges strongly to a unique solution
of the variational inequality
A − γf x∗ , x − x∗ ≥ 0,
∀x ∈ FT ,
1.10
which is the optimality condition for the minimization problem
min
1
x∈FT 2
Ax, x − hx,
where h is a potential function for γf i.e., h x γfx for x ∈ H.
1.11
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In this paper, motivated by Li et al. 8, Marino and Xu 9, Plubtieng and Punpaeng
14, Shioji and Takahashi 15, and Shimizu and Takahashi 16, we consider the mapping Tt
defined as follows:
Tt x tγfx I − tA
1
λt
λt
1.12
T sx ds,
0
where γ > 0 is a constant, f is an α-contraction, A is a bounded linear strongly positive selfadjoint operator and {λt } is a positive real divergent net. If γα < γ for each 0 < t < A
−1 , one
can see that Tt is a 1 − tγ − γα-contraction. So, by Banach’s contraction mapping principle,
there exists an unique solution xt of the fixed point equation
1
xt tγfxt I − tA
λt
λt
1.13
T sxt ds.
0
We show that the sequence {xt } generated by above continuous scheme strongly converges
to a common fixed point x∗ ∈ FS, which is the unique point in FS solving the variational
inequality γf − Ax∗ , x − x∗ ≤ 0 for all x ∈ FS. Furthermore, we also study the following
explicit iterative scheme:
x0 ∈ H,
xn1 αn γfxn βn xn 1
1 − βn I − αn A
sn
sn
T sxn ds,
n ≥ 0.
1.14
0
We prove that the sequence {xn } generated by 1.14 converges strongly to the same x∗ .
The results presented in this paper improve and extend the corresponding results
announced by Marino and Xu 9, Plubtieng and Punpaeng 14, Shioji and Takahashi 15,
and Shimizu and Takahashi 16.
In order to prove our main result, we need the following lemmas.
Lemma 1.1 see 16. Let D be a nonempty bounded closed convex subset of a Hilbert space H and
let S {T t : 0 ≤ t < ∞} be a nonexpansive semigroup on D. Then, for any 0 ≤ h < ∞,
1 t
1 t
T sx ds − T h
T sx ds 0.
lim sup
t → ∞ x∈D t 0
t 0
1.15
Lemma 1.2 see 17. Let H be a Hilbert space, C a closed convex subset of H, and T : C → C
a nonexpansive mapping with FT /
∅. Then I − T is demiclosed, that is, if {xn } is a sequence in C
weakly converging to x and if {I − T xn } strongly converges to y, then I − T x y.
Lemma 1.3 see 18. Let C be a nonempty closed convex subset of a real Hilbert space H and let
PC be the metric projection from H onto C( i.e., for x ∈ H, PC x is the only point in C such that
x − PC x
inf{
x − z
: z ∈ C}). Given x ∈ H and z ∈ C. Then z PC x if and only if there holds
the relations
x − z, y − z ≤ 0,
∀y ∈ C.
1.16
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5
Lemma 1.4. Let H be a Hilbert space, f a α-contraction, and A a strongly positive linear bounded
self-adjoint operator with the coefficient γ > 0. Then, for 0 < γ < γ/α,
2
x − y, A − γf x − A − γf y ≥ γ − γα x − y ,
x, y ∈ H.
1.17
That is, A − γf is strongly monotone with coefficient γ − αγ.
Proof. From the definition of strongly positive linear bounded operator, we have
2
x − y, A x − y ≥ γ x − y .
1.18
On the other hand, it is easy to see
2
x − y, γfx − γfy ≤ γαx − y .
1.19
Therefore, we have
x − y, A − γf x − A − γf y x − y, A x − y − x − y, γfx − γfy
2
≥ γ − γα x − y
1.20
for all x, y ∈ H. This completes the proof.
Remark 1.5. Taking γ 1 and A I, the identity mapping, we have the following inequality:
2
x − y, I − f x − I − f y ≥ 1 − αx − y ,
x, y ∈ H.
1.21
Furthermore, if f is a nonexpansive mapping in Remark 1.5, we have
x − y, I − f x − I − f y ≥ 0,
x, y ∈ H.
1.22
Lemma 1.6 see 9. Assume A is a strongly positive linear bounded self-adjoint operator on a Hilbert space H with coefficient γ > 0 and 0 < ρ ≤ A
−1 . Then I − ρA
≤ 1 − ργ.
Lemma 1.7 see 12. Let {αn } be a sequence of nonnegative real numbers satisfying the following
condition:
αn1 ≤ 1 − γn αn γn σn ,
∀n ≥ 0,
where {γn } is a sequence in 0, 1 and {σn } is a sequence of real numbers such that
i limn → ∞ γn 0 and ∞
n0 γn ∞,
ii either lim supn → ∞ σn ≤ 0 or ∞
n0 |γn σn | < ∞.
Then {αn }∞
n0 converges to zero.
1.23
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2. Main Results
Lemma 2.1. Let H a real Hilbert space and S {T s : 0 ≤ s < ∞} a nonexpansive semigroup on H
such that FS /
∅. Let {λt }0<t<1 be a continuous net of positive real numbers such that limt → 0 λt ∞.
Let f : H → H be an α-contraction, A a strongly positive linear bounded self-adjoint operator of H
into itself with coefficient γ > 0. Assume that 0 < γ < γ/α. Let {xt } be a sequence defined by 1.13.
Then
i {xt } is bounded for all t ∈ 0, A
−1 ;
ii limt → 0 T τxt − xt 0 for all 0 ≤ τ < ∞;
iii xt defines a continuous curve from 0, A
−1 into H.
Proof. i Taking p ∈ FS, we have
1 λt
xt − p ≤ T sxt ds − p
tγfxt I − tA
λt 0
λt
1
T sxt − pds
≤ tγfxt − Ap 1 − tγ
λt 0
≤ tγfxt − Ap 1 − tγ xt − p
≤ tγ fxt − f p tγf p − Ap 1 − tγ xt − p
≤ 1 − t γ − γα xt − p tγf p − Ap.
2.1
It follows that
xt − p ≤
1 γf p − Ap.
γ − αγ
2.2
This implies that {xt } is not only bounded, but also that {xt } is contained in Bp, 1/γ −
γα
γfp − Ap
of center p and radius 1/γ − γα
γfp − Ap
, for all fixed p ∈ FS.
Moreover for p ∈ FS and t ∈ 0, A
−1 ,
1 λt 1 λt
T sxt ds − p T sxt − T sp ds
λt 0
λt 0
≤ xt − p
1 γf p − Ap.
≤
γ − γα
ii Observe that
λ
t
1
T sxt ds T τxt − xt ≤ T τxt − T τ
λt 0
λ
t
1
1 λt
T τ
T sxt ds −
T sxt ds
λt 0
λt 0
2.3
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1 λt
T sxt ds − xt λt 0
1 λt
≤ 2xt −
T sxt ds
λt 0
λ
t
1
1 λt
T τ
T sxt ds −
T sxt ds
λt 0
λt 0
1 λt
2tγfxt − A
T sxt ds
λt 0
λ
t
1
1 λt
T τ
T sxt ds −
T sxt ds.
λt 0
λt 0
2.4
Taking Bp, 1/γ − γα
γfp − Ap
as D in Lemma 1.1 and passing to limt → 0 in 2.4, we
can obtain ii immediately.
iii Taking t1 , t2 ∈ 0, A
−1 and fixing p ∈ FS, we see that
xt1 − xt2 1 λ t1
≤ t1 − t2 γfxt1 t2 γ fxt1 − fxt2 − t1 − t2 A
T sxt1 ds
λt1 0
1 λ t1
1 λ t2
I − t2 A
T sxt1 ds −
T sxt2 ds λt1 0
λt2 0
1 λ t1
≤ |t1 − t2 |γ fxt1 t2 γα
xt1 − xt2 |t1 − t2 |
A
T sxt1 ds
λt1 0
λ t1
1 λ t1
1 λ t2
1
1 − t2 γ T sxt1 ds −
T sxt2 ds −
T sxt2 ds
λt1 0
λt2 0
λt2 λt
1
1 λ t1
≤ |t1 − t2 |γ fxt1 t2 γα
xt1 − xt2 |t1 − t2 |
A
T sxt1 ds
λt1 0
λt
λ
t2
1
1
1
1
1 − t2 γ xt1 − xt2 −
T
ds
T
ds
sx
sx
.
t2
t2
λt2 λt
λt1 λt2 0
2.5
1
Thus applying 2.3, we arrive at
xt1 − xt2 1 γf p − Ap p
≤ |t1 − t2 |γ fxt1 t2 γα
xt1 − xt2 |t1 − t2 |
A
γ − γα
1 2
1 − t2 γ xt1 − xt2 p
γf
p
−
Ap
−
λ
|λt
t1 |
λt2 2
γ − γα
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1 γf p − Ap p
≤ |t1 − t2 | γ fxt1 A
γ − γα
1 2
γf p − Ap p .
1 − t2 γ − γα xt1 − xt2 |λt2 − λt1 |
λt2
γ − γα
2.6
It follows that
xt1 − xt2 ≤ M1 |t1 − t2 | M2 |λt2 − λt1 |,
2.7
γ γ − γα fxt1 A
γf p − Ap γ − γα A
p
M1 2
t2 γ − γα
2.8
2 γf p − Ap γ − γα p
M2 .
2
λt2 t2 γ − γα
2.9
where
and
This inequality, together with the continuity of the net {λt }, gives the continuity of the curve
{xt }.
Theorem 2.2. Let H be a real Hilbert space H and S {T s : 0 ≤ s < ∞} a nonexpansive
semigroup such that FS /
∅. Let {λt }0<t<1 be a net of positive real numbers such that limt → 0 λt ∞.
Let f be an α-contraction and let A be a strongly positive linear bounded self-adjoint operator on H
with the coefficient γ > 0. Assume that 0 < γ < γ/α. Then sequence {xt } defined by 1.13 strongly
converges as t → 0 to x∗ ∈ FS, which solves the following variational inequality:
γf − A x∗ , p − x∗ ≤ 0,
∀p ∈ FS.
2.10
Equivalently, one has
PFS I − A γf x∗ x∗ .
2.11
Proof. The uniqueness of the solution of the variational inequality 2.10 is a consequence of
the strong monotonicity of A − γf Lemma 1.4 and it was proved in 9. Next, we will use
x∗ ∈ FS to denote the unique solution of 2.10. To prove that xt → x∗ t → 0, we write,
for a given p ∈ FS,
xt − p t γfxt − Ap I − tA
1
λt
λt
0
T sxt ds − p .
2.12
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9
Using xt − p to make inner product, we obtain that
xt − p2 I − tA
1
λt
λt
T sxt ds − p , xt − p
t γfxt − Ap, xt − p
0
2 ≤ 1 − tγ xt − p t γfxt − Ap, xt − p .
2.13
It follows that
xt − p2 ≤ 1 γ fxt − f p , xt − p γf p − Ap, xt − p
γ
γα xt − p2 1 γf p − Ap, xt − p ,
≤
γ
γ
2.14
which yields that
xt − p2 ≤
1 γf p − Ap, xt − p .
γ − αγ
2.15
Since H is a Hilbert space and {xt } is bounded as t → 0, we have that if {tn } is a sequence in
0, 1 such that tn → 0 and xtn x. By 2.15, we see xtn → x. Moreover, by ii of Lemma 2.1
we have x ∈ FS. We next prove that x solves the variational inequality 2.10. From 1.13,
we arrive at
1
1 λt
T sxt ds .
A − γf xt − I − tA xt −
t
λt 0
For p ∈ FS, it follows from 1.22 that
1 λt
T sxt ds , xt − p
I − tA xt −
λt 0
1 1 λt −
I − T sxt − I − T sp ds, xt − p
t λt 0
1 λt
A
I − T sxt ds, xt − p
λt 0
1 λt −
I − T sxt − I − T sp, xt − p ds
tλt 0
1 λt
A
I − T sxt ds, xt − p
λt 0
1 λt
≤ A
I − T sxt ds, xt − p
λt 0
1
A − γf xt , xt − p −
t
2.16
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Journal of Applied Mathematics
1 λt
A tγfxt − tA
T sxt ds , xt − p
λt 0
1 λt
T sxt ds , xt − p .
t A γfxt − A
λt 0
2.17
Passing to limt → 0 , since {xt } is a bounded sequence, we obtain
A − γf x, x − p ≤ 0,
2.18
that is, x satisfies the variational inequality 2.10. By the uniqueness it follows x x∗ . In
a summary, we have shown that each cluster point of {xt } as t → 0 equals x∗ . Therefore,
xt → x∗ as t → 0. The variational inequality 2.10 can be rewritten as
I − A γf x∗ − x∗ , x∗ − p ,
p ∈ FS.
2.19
This, by Lemma 1.3, is equivalent to
PFS I − A γf x∗ x∗ .
2.20
This completes the proof.
Remark 2.3. Theorem 2.2 which include the corresponding results of Shioji and Takahashi 15
as a special case is reduced to Theorem 3.1 of Plubtieng and Punpaeng 14 when A I, the
identity mapping and γ 1.
Theorem 2.4. Let H be a real Hilbert space H and S {T s : 0 ≤ s < ∞} a nonexpansive
{β } be
semigroup such that FS / ∅. Let {sn } be a positive real divergent sequence and let {αn } and
n
sequences in 0, 1 satisfying the following conditions limn → ∞ αn limn → ∞ βn 0 and ∞
n0 αn ∞. Let f be an α-contraction and let A be a strongly positive linear bounded self-adjoint operator
with the coefficient γ > 0. Assume that 0 < γ < γ/α. Then sequence {xn } defined by 1.14 strongly
converges to x∗ ∈ FS, which solves the variational inequality 2.10.
Proof. We divide the proof into three parts.
Step 1. Show the sequence {xn } is bounded.
Noticing that limn → ∞ αn limn → ∞ βn 0, we may assume, with no loss of generality,
that αn /1 − βn < A
−1 for all n ≥ 0. From Lemma 1.6, we know that 1 − βn I − αn A
≤
1 − βn − αn γ. Picking p ∈ FS, we have
xn1 − p
1 sn
αn γfxn − Ap βn xn − p 1 − βn I − αn A
T sxn ds − p sn 0
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1 sn
T
ds
−
p
≤ αn γfxn − Ap βn xn − p 1 − βn − αn γ sx
n
s
n 0
≤ αn γ fxn − f p αn γf p − Ap βn xn − p 1 − βn − αn γ xn − p
≤ 1 − αn γ − γα xn − p αn γf p − Ap.
2.21
By simple inductions, we see that
Ap − γf p xn − p ≤ max x0 − p,
,
γ − γα
2.22
which yields that the sequence {xn } is bounded.
Step 2. Show that
lim sup γf − A x∗ , yn − x∗ ≤ 0,
n→∞
2.23
s
where x∗ is obtained in Theorem 2.2 and yn 1/sn 0 n T sxn ds.
Putting z0 PFS x0 , from 2.22 we see that the closed ball M of center z0 and radius
max{
z0 − p
, Az0 − γfz0 /γ − γα} is T s-invariant for each s ∈ 0, ∞ and contain {xn }.
Therefore, we assume, without loss of generality, S {T s : 0 ≤ s < ∞} is a nonexpansive
semigroup on M. It follows from Lemma 1.1 that
lim yn − T hyn 0
n→∞
2.24
for all 0 ≤ h < ∞. Taking a suitable subsequence {yni } of {yn }, we see that
lim sup γf − A x∗ , yn − x∗ lim γf − A x∗ , yni − x∗ .
i→∞
n→∞
2.25
Since the sequence {yn } is also bounded, we may assume that yni x. From the
demiclosedness principle, we have x ∈ FS. Therefore, we have
lim sup γf − A x∗ , yn − x∗ γf − A x∗ , x − x∗ ≤ 0.
n→∞
2.26
On the other hand, we have
xn1 − yn ≤ αn γfxn − Axn βn xn − yn .
2.27
From the assumption limn → ∞ αn limn → ∞ βn 0, we see that
lim xn1 − yn 0,
n→∞
2.28
12
Journal of Applied Mathematics
which combines with 2.26 gives that
lim sup γf − A x∗ , xn1 − x∗ ≤ 0.
2.29
n→∞
Step 3. Show xn → x∗ as n → ∞.
Note that
xn1 − x∗ 2
αn γfxn − Ax∗ βn xn − x∗ 1 − βn I − αn A yn − x∗ , xn1 − x∗
αn γfxn − Ax∗ , xn1 − x∗ βn xn − x∗ , xn1 − x∗ 1 − βn I − αn A yn − x∗ , xn1 − x∗
≤ αn γ fxn − fx∗ , xn1 − x∗ γfx∗ − Ax∗ , xn1 − x∗
βn xn − x∗ xn1 − x∗ 1 − βn I − αn Ayn − x∗ xn1 − x∗ ≤ αn αγ
xn − x∗ xn1 − x∗ αn γfx∗ − Ax∗ , xn1 − x∗
βn xn − x∗ xn1 − x∗ 1 − βn − αn γ xn − x∗ xn1 − x∗ 1 − αn γ − γα xn − x∗ xn1 − x∗ αn γfx∗ − Ax∗ , xn1 − x∗
1 − αn γ − γα ≤
xn − x∗ 2 xn1 − x∗ 2 αn γfx∗ − Ax∗ , xn1 − x∗ .
2
1 − αn γ − γα
1
≤
xn − x∗ 2 xn1 − x∗ 2 αn γfx∗ − Ax∗ , xn1 − x∗ .
2
2
2.30
It follows that
xn1 − x∗ 2 ≤ 1 − αn γ − γα xn − x∗ 2 2αn γfx∗ − Ax∗ , xn1 − x∗ .
2.31
By using Lemma 1.7, we can obtain the desired conclusion easily.
Remark 2.5. If γ 1 and A I, the identity mapping, then Theorem 2.4 is reduced to Theorem
3.3 of Plubtieng and Punpaeng 14.
If the sequence {βn } ≡ 0, then Theorem 2.4 is reduced to the following.
Corollary 2.6. Let H be a real Hilbert space H and S {T s : 0 ≤ s < ∞} a nonexpansive
semigroup such that FS /
∅. Let {sn } be a positive real divergent sequence and let {αn } be a sequence
in 0, 1 satisfying the following conditions limn → ∞ αn 0 and ∞
n0 αn ∞. Let f be a α-contraction
and let A be a strongly positive linear bounded self-adjoint operator with the coefficient γ > 0. Assume
that 0 < γ < γ/α. Let {xn } be a sequence generated by the following manner:
x0 ∈ H,
xn1 αn γfxn I − αn A
1
sn
sn
T sxn ds,
n ≥ 0.
2.32
0
Then the sequence {xn } defined by above iterative algorithm converges strongly to x∗ ∈ FS, which
solves the variational inequality 2.10.
Remark 2.7. Corollary 2.6 includes Theorem 2 of Shioji and Takahashi 15 as a special case.
Journal of Applied Mathematics
13
Remark 2.8. Theorem 2.2 and Corollary 2.6 improve Theorem 3.2 and Theorem 3.4 of Marino
and Xu 9 from a single nonexpansive mapping to a nonexpansive semigroup, respectively.
Acknowledgment
The present studies were supported by the National Natural Science Foundation of China
11071169, 11126334 and the Natural Science Foundation of Zhejiang Province Y6110287.
References
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