An. Şt. Univ. Ovidius Constanţa
Vol. 19(1), 2011, 331–346
Some unified algorithms for finding minimum
norm fixed point of nonexpansive semigroups
in Hilbert spaces
Yonghong Yao, Yeong-Cheng Liou
Abstract
In this paper, we introduce two general algorithms (one implicit and
one explicit) for finding a common fixed point of a nonexpansive semigroup {T (s)}s≥0 in Hilbert spaces. We prove that both approaches
converge strongly to a common fixed point of {T (s)}s≥0 . Such common fixed point x∗ is the unique solution of some variational inequality,
which is the optimality condition for some minimization problem. As
special cases of the above two algorithms, we obtain two schemes which
both converge strongly to the minimum norm common fixed point of
{T (s)}s≥0 .
1
Introduction
Let H be a real Hilbert space with inner product h·, ·i and norm k · k, respectively. Let C be a nonempty closed convex subset of H. Recall a mapping
f : C → H is called to a contraction if, for all x, y ∈ C, there exists ρ ∈ [0, 1)
such that kf (x) − f (y)k ≤ ρkx − yk. A mapping T : C → C is said to be
nonexpansive if kT x − T yk ≤ kx − yk, ∀x, y ∈ C. Denote the set of fixed points
of T by F ix(T ). Let A be a strongly positive bounded linear operator on H,
i.e., there exists a constant γ̄ > 0 such that hAx, xi ≥ γ̄kxk2 for all x ∈ H.
Key Words: Common fixed point; Variational inequality; Nonexpansive Semigroup;
Algorithms; Minimum norm.
Mathematics Subject Classification: 47H05; 47H10; 47H17.
Received: January, 2010
Accepted: December, 2010
331
332
Yonghong Yao, Yeong-Cheng Liou
Iterative methods for nonexpansive mappings are widely used to solve convex minimization problems, see, for instance, [1],[2], [4]-[6], [9],[11]-[14],[16][30], [32], [33]. A typical problem is to minimize a function over the set of
fixed points of a nonexpansive mapping T ,
min
x∈F ix(T )
1
hAx, xi − hx, bi.
2
(1.1)
In [27], Xu proved that the sequence {xn } defined by xn+1 = αn b + (1 −
αn A)T xn , n ≥ 0 strongly converges to the unique solution of (1.1) under
certain conditions. Recently, Marino and Xu [17] introduced the viscosity approximation method xn+1 = αn γf (xn )+(1−αn A)T xn , n ≥ 0 and proved that
the sequence {xn } converges strongly to the unique solution of the variational
inequality
h(A − γf )z, x − zi ≥ 0, ∀x ∈ F ix(T )
which is the optimality condition for the minimization problem
min
x∈F ix(T )
1
hAx, xi − h(x)
2
0
where h is a potential function for γf (i.e., h = γf on H).
In this paper, we focus on nonexpansive semigroup {T (s)}s≥0 . Recall that
a family S := {T (s)}s≥0 of mappings of C into itself is called a nonexpansive
semigroup if it satisfies the following conditions:
(S1) T (0)x = x for all x ∈ C;
(S2) T (s + t) = T (s)T (t) for all s, t ≥ 0;
(S3) kT (s)x − T (s)yk ≤ kx − yk for all x, y ∈ C and s ≥ 0;
(S4) for all x ∈ H, s → T (s)x is continuous.
We denote by F ix(T (s)) the set of fixed points ofTT (s) and by F ix(S) the set
of all common fixed points of S, i.e. F ix(S) = s≥0 F ix(T (s)). It is known
that F ix(S) is closed and convex (Lemma 1 in [1]).
Algorithms for nonexpansive semigroups have been considered by some authors, please consult [3], [7],[8],[10],[15], [31]. The following interesting problem
arises: Can one construct some more general algorithms which unify the above
algorithms?
On the other hand, we also notice that it is quite often to seek a particular
solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a closed convex subset C of a Hilbert space H1 and a
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Some unified algorithms ...
bounded linear operator R : H1 → H2 , where H2 is another Hilbert space. The
†
C-constrained pseudoinverse of R, RC
, is then defined as the minimum-norm
solution of the constrained minimization problem
†
RC
(b) := arg min kRx − bk
x∈C
which is equivalent to the fixed point problem
x = PC (x − λR∗ (Rx − b))
where PC is the metric projection from H1 onto C, R∗ is the adjoint of R,
λ > 0 is a constant, and b ∈ H2 is such that PR(C) (b) ∈ R(C).
It is therefore another interesting problem to invent some algorithms that
can generate schemes which converge strongly to the minimum-norm solution
of a given problem.
In this paper, we introduce two general algorithms (one implicit and one explicit) for finding a common fixed point of a nonexpansive semigroup {T (s)}s≥0
in Hilbert spaces. We prove that both approaches converge strongly to a common fixed point of {T (s)}s≥0 . Such common fixed point x∗ is the unique
solution of some variational inequality, which is the optimality condition for
some minimization problem. As special cases of the above two algorithms,
We obtain two schemes which both converge strongly to the minimum norm
common fixed point of {T (s)}s≥0 .
2
Preliminaries
Let C be a nonempty closed convex subset of a real Hilbert space H. The
metric (or nearest point) projection from H onto C is the mapping PC : H →
C which assigns to each point x ∈ C the unique point PC x ∈ C satisfying the
property
kx − PC xk = inf kx − yk =: d(x, C).
y∈C
It is well known that PC is a nonexpansive mapping and satisfies
hx − y, PC x − PC yi ≥ kPC x − PC yk2 , ∀x, y ∈ H.
Moreover, PC is characterized by the following properties:
hx − PC x, y − PC xi ≤ 0,
(2.1)
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Yonghong Yao, Yeong-Cheng Liou
and
kx − yk2 ≥ kx − PC xk2 + ky − PC xk2 ,
for all x ∈ H and y ∈ C.
We need the following lemmas for proving our main results.
Lemma 2.1. ([23]) Let C be a nonempty bounded closed convex subset of a
Hilbert space H and let {T (s)}s≥0 be a nonexpansive semigroup on C. Then,
for every h ≥ 0,
Z
°1 Z t
°
1 t
°
°
lim supx∈C °
T (s)xds − T (h)
T (s)xds° = 0.
t→∞
t 0
t 0
Lemma 2.2. ([12]) Let C be a closed convex subset of a real Hilbert space H
and let S : C → C be a nonexpansive mapping. Then, the mapping I − S is
demiclosed. That is, if {xn } is a sequence in C such that xn → x∗ weakly and
(I − S)xn → y strongly, then (I − S)x∗ = y.
Lemma 2.3. ([18]) Let {xn } and {yn } be bounded sequences in a Banach
space X and let {γn } be a sequence in [0, 1] with 0 < lim inf n→∞ βn ≤
lim supn→∞ βn < 1. Suppose that xn+1 = (1 − γn )xn + γn yn for all n ≥ 0 and
lim supn→∞ (kyn − yn−1 k − kxn − xn−1 k) ≤ 0. Then, limn→∞ kyn − xn k = 0.
Lemma 2.4. ([26]) Assume {an } is a sequence of nonnegative real numbers
such that
an+1 ≤ (1 − γn )an + δn γn ,
where {γn } is a sequence in (0, 1) and {δn } is a sequence such that
P∞
(1)
n=1 γn = ∞;
P∞
(2) lim supn→∞ δn ≤ 0 or n=1 |δn γn | < ∞.
Then limn→∞ an = 0.
3
Main results
In this section we will show our main results.
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Some unified algorithms ...
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert
space H. Let S = {T (s)}s≥0 : C → C be a nonexpansive semigroup with
F ix(S) 6= ∅. Let f : C → H be a ρ-contraction (possibly non-self). Let A be
a strongly positive linear bounded self-adjoint operator on H with coefficient
γ̄ > 0. Let {λt }0<t<1 be a continuous net of positive real numbers such that
limt→0 λt = +∞. Let γ and β be two real numbers such that 0 < γ < γ̄/ρ and
β ∈ [0, 1). Let the net {xt } be defined by the following implicit scheme:
1
xt = PC [tγf (xt ) + βxt + ((1 − β)I − tA)
λt
Z
λt
T (s)xt ds], t ∈ (0, 1). (3.1)
0
Then, as t → 0+, the net {xt } strongly converges to x∗ ∈ F ix(S) which is the
unique solution of the following variational inequality:
x∗ ∈ F ix(S),
h(γf − A)x∗ , x − x∗ i ≤ 0,
(3.2)
∀x ∈ F ix(S).
In particular, if we take f = 0 and A = I, then the net {xt } defined by (3.1)
reduces to
Z λt
1
xt = PC [βxt + (1 − β − t)
T (s)xt ds], t ∈ (0, 1).
(3.3)
λt 0
In this case, the net {xt } defined by (3.3) converges in norm to the minimum
norm fixed point x∗ of F ix(S), namely, the point x∗ is the unique solution to
the minimization problem:
x∗ = arg
min
x∈F ix(S)
kxk.
(3.4)
Proof. First, we note that the net {xt } defined by (3.1) is well-defined. We
define a mapping
Gx := PC [tγf (x) + βx + ((1 − β)I − tA)
1
λt
Z
λt
T (s)xds], t ∈ (0, 1).
0
It follows that
kGx − Gyk ≤
1
ktγ(f (x) − f (y)) + β(x − y) + ((1 − β)I − tA)
λt
≤ tγkf (x) − f (y)k + βkx − yk + k((1 − β)I − tA)
≤ tγρkx − yk + βkx − yk + (1 − β − tγ̄)kx − yk
= (1 − (γ̄ − γρ)t)kx − yk.
Z
1
λt
λt
(T (s)x − T (s)y)dsk
0
Z
λt
(T (s)x − T (s)y)dsk
0
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Yonghong Yao, Yeong-Cheng Liou
This implies that the mapping G is a contraction and so it has a unique fixed
point. Therefore, the net {xt } defined by (3.1) is well-defined.
Take p ∈ F ix(S). By (3.1), we have
kxt − pk
= kPC [tγf (xt ) + βxt + ((1 − β)I − tA)
1
λt
Z
λt
T (s)xt ds] − pk
0
≤ kt(γf (xt ) − Ap) + β(xt − p) + ((1 − β)I − tA)(
1
λt
Z
λt
T (s)xt ds − p)k
0
Z λt
1
kT (s)xt − T (s)pkds
λt 0
≤ tγkf (xt ) − f (p)k + tkγf (p) − Apk + βkxt − pk + (1 − β − γ̄t)kxt − pk
≤ tγρkxt − pk + tkγf (p) − Apk + βkxt − pk + (1 − β − γ̄t)kxt − pk.
≤ tkγf (xt ) − Apk + βkxt − pk + (1 − β − γ̄t)
It follows that
kxt − pk ≤
1
kγf (p) − Apk
γ̄ − γρ
which implies that the net {xt } is bounded.
Set R :=
1
γ̄−γρ kγf (p)
1
k
λt
Z
− Apk. It is clear that {xt } ⊂ B(p, R). Notice that
λt
T (s)xt ds − pk ≤ kxt − pk ≤ R.
0
Moreover, we observe that if x ∈ B(p, R) then
kT (s)x − pk ≤ kT (s)x − T (s)pk ≤ kx − pk ≤ R,
i.e., B(p, R) is T (s)-invariant for all s.
Set yt = tγf (xt ) + βxt + ((1 − β)I − tA) λ1t
R λt
0
T (s)xt ds. From (3.1), we
Some unified algorithms ...
337
deduce
kT (τ )xt − xt k
= PC [T (τ )xt ] − PC [yt ]k
≤ kT (τ )xt − yt k
Z λt
1
≤ kT (τ )xt − T (τ )
T (s)xt dsk
λt 0
Z λt
Z λt
1
1
+kT (τ )
T (s)xt ds −
T (s)xt dsk
λt 0
λt 0
Z λt
1
T (s)xt ds − yt k
+k
λt 0
Z λt
Z λt
1
1
≤ kxt −
T (s)xt dsk + kT (τ )
T (s)xt ds −
λt 0
λt 0
Z λt
Z λt
1
1
T (s)xt dsk + k
T (s)xt ds − yt k
−
λt 0
λt 0
Z
2t
A λt
≤
kγf (xt ) −
T (s)xt dsk
1−β
λt 0
Z λt
Z λt
1
1
+kT (τ )
T (s)xt ds −
T (s)xt dsk.
λt 0
λt 0
By Lemma 2.1, we deduce for all 0 ≤ τ < ∞
lim kT (τ )xt − xt k = 0.
t→0
(3.5)
Note that xt = PC [yt ]. By using the property of the metric projection (2.1),
we have
kxt − pk2
= hxt − yt , xt − pi + hyt − p, xt − pi
≤ hyt − p, xt − pi
= thγf (xt ) − Ap, xt − pi + βkxt − pk2
Z λt
1
+h((1 − β)I − tA)(
T (s)xt ds − p), xt − pi
λt 0
≤ βkxt − pk2 + (1 − β − γ̄t)kxt − pk2
+tγhf (xt ) − f (p), xt − pi + thγf (p) − Ap, xt − pi
≤ [1 − (γ̄ − γρ)t]kxt − pk2 + thγf (p) − Ap, xt − pi.
Therefore,
kxt − pk2 ≤
1
hγf (p) − Ap, xt − pi, ∀p ∈ F ix(S).
γ̄ − γρ
338
Yonghong Yao, Yeong-Cheng Liou
From this inequality, we have immediately that ωw (xt ) = ωs (xt ), where ωw (xt )
and ωs (xt ) denote the set of weak and strong cluster points of {xt }, respectively.
Let {tn } ⊂ (0, 1) be a sequence such that tn → 0 as n → ∞. Put xn := xtn ,
yn := ytn and λn := λtn . Since {xn } is bounded, without loss of generality,
we may assume that {xn } converges weakly to a point x∗ ∈ C. Also yn → x∗
weakly. Noticing (3.5) we can use Lemma 2.2 to get x∗ ∈ F ix(S).
We can rewrite (3.1) as
Z
1
1
(A − γf )xt = − ((1 − β)I − tA)[xt −
t
λt
λt
0
1
T (s)xt ds] + (xt − yt ).
t
Therefore,
Z λt
1−β 1
[
h(I − T (s))xt − (I − T (s))p, xt − pids]
t
λt 0
Z λt
1
1
+ hA
[xt − T (s)xt ]ds, xt − pi + hxt − yt , xt − pi.
λt
t
0
h(A − γf )xt , xt − pi
= −
Noting that I − T (s) is monotone and hxt − yt , xt − pi ≤ 0, so
h(A − γf )xt , xt − pi
≤
1
hA
λt
Z
λt
[xt − T (s)xt ]ds, xt − pi
0
= hAxt −
A
λt
≤ kAkkxt −
≤
Z
λt
T (s)xt ds, xt − pi
0
1
λt
Z
λt
T (s)xt dskkxt − pk
0
t
1
kAkkγf (xt ) − A
1−β
λt
Z
λt
T (s)xt dskkxt − pk.
0
Taking the limit through t := tni → 0, we have
h(A − γf )x∗ , x∗ − pi = lim h(A − γf )xni , xni − pi ≤ 0.
i→∞
Since the solution of the variational inequality (3.2) is unique. Hence ωw (xt ) =
ωs (xt ) is singleton. Therefore, xt → x∗ .
In particular, if we take f = 0 and A = I, then it follows that xt → x∗ =
PF ix(S) (0), which implies that x∗ is the minimum norm fixed point of S. As
a matter of fact, by (3.2), we deduce
hx∗ , x∗ − xi ≤ 0,
∀x ∈ F ix(S),
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Some unified algorithms ...
that is,
kx∗ k2 ≤ hx∗ , xi ≤ kx∗ kkxk,
∀x ∈ F ix(S).
Therefore, the point x∗ is the unique solution to the minimization problem
x∗ = arg
min
x∈F ix(S)
kxk.
This completes the proof.
Next we introduce an explicit algorithm for finding a solution of minimization problem (3.4). This scheme is obtained by discretizing the implicit scheme
(3.1). We will show the strong convergence of this algorithm.
Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert
space H. Let S = {T (s)}s≥0 : C → C be a nonexpansive semigroup with
F ix(S) 6= ∅. Let f : C → H be a ρ-contraction (possibly non-self) with
ρ ∈ [0, 1). Let A be a strongly positive linear bounded self-adjoint operator
on H with coefficient γ̄ > 0. Let γ and β be two real numbers such that
0 < γ < γ̄/ρ and β ∈ [0, 1). Let the sequence {xn } be generated iteratively by
the following explicit algorithm:
xn+1 = (1−γn )xn +γn PC [αn γf (xn )+βxn +((1−β)I−αn A)
1
λn
Z
λn
T (s)xn ds], n ≥ 0,
0
(3.6)
where {αn } and {γn } are real number sequence in [0, 1] and {λn } is a positive
real number. Suppose that the following conditions are satisfied:
P∞
(i) limn→∞ αn = 0 and n=1 αn = ∞;
(ii) limn→∞ λn = ∞ and limn→∞
λn −λn−1
λn
= 0;
(iii) 0 < lim inf n→∞ γn ≤ lim supn→∞ γn < 1.
Then the sequence {xn } strongly converges to x∗ ∈ F ix(S) which is the unique
solution of the variational inequality (3.2).
In particular, if we take f = 0 and A = I, then the sequence {xn } generated
by (3.6) reduces to
xn+1 = (1 − γn )xn + γn PC [βxn + (1 − αn − β)
1
λn
Z
λn
T (s)xn ds], n ≥ 0. (3.7)
0
In this case, the sequence {xn } converges in norm to the minimum norm fixed
point x∗ of F ix(S).
340
Yonghong Yao, Yeong-Cheng Liou
Proof. Take p ∈ F ix(S). From (3.6), we have
³
(1 − γn )kxn − pk + γn αn kγf (xn ) − Apk + βkxn − pk
Z λn
´
1
+(1 − β − γ̄αn )
kT (s)xn − T (s)pkds
λn 0
³
≤ (1 − γn )kxn − pk + γn αn γρkxn − pk + αn kγf (p) − Apk + βkxn − pk
´
+(1 − β − αn γ̄)kxn − pk
kxn+1 − pk ≤
= [1 − (γ̄ − ργ)αn γn ]kxn − pk + αn γn kγf (p) − Apk.
It follows that by induction
kxn − pk ≤ max{kx0 − pk,
kγf (p) − Apk
}.
γ̄ − γρ
Set yn = PC [αn γf (xn ) + βxn + ((1 − β)I − αn A)zn ] for all n ≥ 0, where
Rλ
zn = λ1n 0 n T (s)xn ds. Hence, we have
kyn − yn−1 k
≤
kαn γf (xn ) + βxn + ((1 − β)I − αn A)zn
−αn−1 γf (xn−1 ) − βxn−1 − ((1 − β)I − αn−1 A)zn−1 k
= kγαn (f (xn ) − f (xn−1 )) + γ(αn − αn−1 )f (xn−1 ) + β(xn − xn−1 )
+((1 − β)I − αn A)(zn − zn−1 ) + (αn−1 − αn )Azn−1 k
≤ γαn kf (xn ) − f (xn−1 )k + |αn − αn−1 |(kγf (xn−1 )k + kAzn−1 k)
+βkxn − xn−1 k + (1 − β − αn γ̄)kzn − zn−1 k
and
kzn − zn−1 k
= k
1
λn
Z
λn
[T (s)xn − T (s)xn−1 ]ds + (
0
1
1
−
)
λn
λn−1
Z
λn−1
T (s)xn−1 ds
0
Z λn
1
+
T (s)xn−1 dsk
λn λn−1
Z λn
Z λn
1
1
≤
kT (s)xn − T (s)xn−1 kds +
k
[T (s)xn−1 − T (s)p]dsk
λn 0
λn λn−1
Z λn−1
1
1
kT (s)xn−1 − T (s)pkds
+|
−
|
λn
λn−1 0
2|λn − λn−1 |
≤ kxn − xn−1 k +
kxn−1 − pk.
λn
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Some unified algorithms ...
Therefore,
kyn − yn−1 k ≤
γαn ρkxn − xn−1 k + |αn − αn−1 |(kγf (xn−1 )k + kAzn−1 k)
+βkxn − xn−1 k + (1 − β − αn γ̄)kxn − xn−1 k
2|λn − λn−1 |
+
kxn−1 − pk
λn
|λn − λn−1 |
),
≤ [1 − (γ̄ − γρ)αn ]kxn − xn−1 k + M (|αn − αn−1 | +
λn
where M > 0 is a constant such that
sup{kγf (xn−1 )k + kAzn−1 k, 2kxn−1 − pk} ≤ M.
n
Hence, we get
lim sup(kyn − yn−1 k − kxn − xn−1 k) ≤ 0.
n→∞
This together with Lemma 2.3 imply that
lim kyn − xn k = 0.
n→∞
Therefore,
lim kxn+1 − xn k = lim γn kyn − xn k = 0.
n→∞
n→∞
Note that
Z λn
1
kT (τ )xn − xn k ≤ kT (τ )xn − T (τ )
T (s)xn dsk
λn 0
Z λn
Z λn
1
1
T (s)xn ds −
T (s)xn dsk
+kT (τ )
λn 0
λn 0
Z λn
1
+k
T (s)xn ds − xn k
λn 0
Z λn
Z λn
1
1
T (s)xn ds −
T (s)xn dsk
≤ kT (τ )
λn 0
λn 0
Z λn
1
+2kxn −
T (s)xn dsk.
(3.8)
λn 0
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Yonghong Yao, Yeong-Cheng Liou
From (3.6), we have
1
kxn −
λn
Z
λn
T (s)xn dsk
1
kxn − xn+1 k + kxn+1 −
λn
≤
0
≤
Z
T (s)xn dsk
0
kxn − xn+1 k + (1 − γn )kxn −
+γn αn γkf (xn )k + γn βkxn −
+γn αn kA
1
λn
Z
λn
1
λn
1
λn
Z
λn
T (s)xn dsk
Z
0
λn
T (s)xn dsk
0
λn
T (s)xn dsk.
0
It follows that
kxn −
1
λn
Z
λn
T (s)xn dsk
≤
0
→
n
1
kxn − xn+1 k + γn αn γkf (xn )k
(1 − β)γn
Z λn
o
1
+γn αn kA
T (s)xn dsk
λn 0
0.
(3.9)
From (3.8), (3.9) and Lemma 2.1, we have
lim kT (τ )xn − xn k = 0 for every τ ≥ 0.
n→∞
(3.10)
Notice that {xn } is a bounded sequence. Let x̃ be a weak limit of {xn }.
Putting x∗ = PF ix(S) (I − A + γf ). Then there exists R such that B(x∗ , R)
contains {xn }. Moreover, B(x∗ , R) is T (s)-invariant for every s ≥ 0; therefore,
without loss of generality, we can assume that {T (s)}s≥0 is a nonexpansive
semigroup on B(x∗ , R). By the demiclosedness principle (Lemma 2.2) and
(3.10), we have x̃ ∈ F ix(S). Therefore,
lim suphγf (x∗ ) − Ax∗ , xn+1 − x∗ i = lim hγf (x∗ ) − Ax∗ , x̃ − x∗ i ≤ 0.
n→∞
n→∞
Rλ
Finally, we prove xn → x∗ . Set un = tγf (xn )+βxn +((1−β)I−tA) λ1n 0 n T (s)xn ds.
It follows that yn = PC [un ] for all n ≥ 0. By using the property of the metric
projection (2.1), we have
hyn − un , yn − x∗ i ≤ 0.
Some unified algorithms ...
343
So,
kyn − x∗ k2
hyn − x∗ , yn − x∗ i
hyn − un , yn − x∗ i + hun − x∗ , yn − x∗ i
hun − x∗ , yn − x∗ i
αn γhf (xn ) − f (x∗ ), yn − x∗ i + αn hγf (x∗ ) − Ax∗ , yn − x∗ i
+βhxn − x∗ , yn − x∗ i + h((1 − β)I − αn A)(zn − x∗ ), yn − x∗ i
≤ αn γkf (xn ) − f (x∗ )kkyn − x∗ k + αn hγf (x∗ ) − Ax∗ , yn − x∗ i
+βkxn − x∗ kkyn − x∗ k + (1 − β − γ̄αn )kzn − x∗ kkyn − x∗ k
≤ [1 − (γ̄ − γρ)αn ]kxn − x∗ kkyn − x∗ k + αn hγf (x∗ ) − Ax∗ , yn − x∗ i
1 − (γ̄ − γρ)αn
1
≤
kxn − x∗ k2 + kyn − x∗ k + αn hγf (x∗ ) − Ax∗ , yn − x∗ i,
2
2
=
=
≤
=
that is,
kyn − x∗ k2
≤
[1 − (γ̄ − γρ)αn ]kxn − x∗ k2 + 2αn hγf (x∗ ) − Ax∗ , yn − x∗ i.
By the convexity of the norm, we have
kxn+1 − x∗ k2
≤
≤
(1 − γn )kxn − x∗ k2 + γn kyn − x∗ k2
[1 − (γ̄ − γρ)αn γn ]kxn − x∗ k2 + 2αn γn hγf (x∗ ) − Ax∗ , yn − x∗ i.
Hence, all conditions of Lemma 2.4 are satisfied. Therefore, we immediately
deduce that xn → x∗ .
In particular, if we take f = 0 and A = I, then it is clear that x∗ = PF ix(S) (0)
is the unique solution to the minimization problem x∗ = arg minx∈F ix(S) kxk.
This completes the proof.
Acknowledgment
The first author was supported in part by Colleges and Universities Science
and Technology Development Foundation (20091003) of Tianjin and NSFC
11071279. The second author was supported in part by NSC 99-2221-E-230006.
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Department of Mathematics, Tianjin Polytechnic University,
Tianjin 300160, People’s Republic of China
e-mail: yaoyonghong@yahoo.cn
Department of Information Management,
Cheng Shiu University, Kaohsiung 833, Taiwan,
e-mail: simplex liou@hotmail.com