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QUADRATIC OPTIMIZATION OF FIXED POINTS
FOR A FAMILY OF NONEXPANSIVE MAPPINGS
IN HILBERT SPACE
B. E. RHO ADES
Received 10 September 2003
Given a finite family of nonexpansive self-mappings of a Hilbert space, a particular qua-
dratic functional, and a strongly positive selfadjoint bounded linear operator, Yamada
et al. defined an iteration scheme which converges to the unique minimizer of the qua-
dratic functional over the common fixed point set of the mappings. In order to obtain
their result, they needed to assume that the maps satisfy a commutative type condition.
In this paper, we establish their conclusion without the assumption of any type of com-
mutativity.
Finding an optimal point in the intersection F of the fixed point sets of a family of
nonexpansive maps is one that occurs frequently in various areas of mathematical sci-
ences and engineering. For example, the well-known convex feasibility problem reduces
to finding a point in the intersection of the fixed point sets of a family of nonexpan-
sive maps. (See, e.g., [3, 4].) The problem of finding an optimal point that minimizes a
given cost function Θ : Ᏼ
→ R over F is of wide interdisciplinary interest and practical
importance. (See, e.g., [2, 4, 5, 7, 14].) A simple algorithmic solution to the problem of
minimizing a quadratic function over F is of extreme value in many applications includ-
ing the set-theoretic signal estimation. (See, e.g., [5, 6, 10, 14].) The best approximation
problem of finding the projection P
F
(a) (in the norm induced by the inner product of
Ᏼ)fromanygivenpointa in Ᏼ is the simplest case of our problem. Some papers dealing
with this best approximation problem are [2, 9, 11].
Let Ᏼ be a Hilbert space, C a closed convex subset of Ᏼ,andT
i
,wherei = 1,2, ,N,
a finite family of nonexpansive self-maps of C,withF :=∩
n
i=1
Fix(T
i
) =∅. Define a qua-
dratic function Θ : Ᏼ → R by
Θ(u):=
1
2
Au,u−b, u∀u ∈ Ᏼ,(1)
where b ∈ Ᏼ and A is a selfadjoint strongly positive operator. We will also assume that
B := I − A satisfies B < 1, although this is not restrictive, since µA is strongly positive
Copyright © 2004 Hindawi Publishing Corporation
Fixed Point Theory and Applications 2004:2 (2004) 135–147
2000 Mathematics Subject Classification: 47H10
URL: />136 Quadratic optimization
with I − µA < 1foranyµ ∈ (0,2/A), and minimizing
Θ(u):=(1/2)µAu,u−µb, u
over F is equivalent to the original minimization problem.
Ya m ada e t a l. [13] show that there exists a unique minimizer u
∗
of Θ over C if and
only if u
∗
satisfies
Au
∗
− b,u − u
∗
≥ 0 ∀u ∈ C. (2)
In their solution of this problem, Yamada et al. [13] add the restriction that the T
i
satisfy
Fix
T
N
···T
1
=
Fix
T
1
T
N
···T
3
T
2
=
Fix
T
N−1
T
N−2
···T
1
T
N
. (3)
There are many nonexpansive maps, with a common fixed point set, that do not satisfy
(3). For example, if X = [0,1] and T
1
and T
2
are defined by T
1
x = x/2+1/4andT
2
x =
3x/4, then Fix(T
1
,T
2
) ={2/5},whereasFix(T
2
,T
1
) ={3/10}.
In our solution, we are able to remove restriction (3). We will take advantage of the
modified Wittmann iteration scheme developed by Atsushiba and Takahashi [1].
Let α
n1
,α
n2
, ,α
nN
∈ (0,1], n = 1,2, Given the mappings T
1
,T
2
, ,T
N
,onecan
define, for each n,newmappingsU
1
, ,U
N
by
U
n1
= α
n1
T
1
+
1 − α
n1
I,
U
n2
= α
n2
T
2
U
n1
+
1 − α
n2
I,
.
.
.
U
n,N−1
= α
n,N−1
T
N−1
U
n,N−2
+
1 − α
n,N−1
I,
W
n
:= U
nN
= α
nN
T
N
U
n,N−1
+
1 − α
nN
I.
(4)
From [1, Lemmas 3.1 and 3.2], if the T
i
are n onexpansive, so are the U
ni
, and both sets
of functions have the same fixed point set.
The iteration scheme we will use is the following. Let b ∈ C and choose any u
0
∈ C.
Define {u
n
} by
u
n+1
= λ
n
b +
I − λ
n
A
W
n
u
n
,(5)
where the W
n
are the self-maps of C generated by (4).
Theorem 1. Let T
i
: Ᏼ → Ᏼ (i = 1, , N) be nonexpansive with nonempty common fixed
point set F =∅. Assume that {λ
n
} and {α
ni
} satisfy
(i) 0 ≤ λ
n
≤ 1,
(ii) lim λ
n
= 0,
(iii)
n≥1
λ
n
=∞,
B. E. Rhoades 137
(iv)
n≥1
|λ
n
− λ
n−1
| < ∞,
(v)
n≥1
|α
ni
− α
n−1,i
| < ∞ for each i = 1,2, ,N.
Then, for any point u
0
∈ Ᏼ,thesequence{u
n
} generated by (5) converges strongly to the
unique minimizer u
∗
of the function Θ of (1)overF.
Proof. From [15], u
∗
exists and is unique. We will first assume that
u
0
∈ C
u
∗
:=
x ∈ Ᏼ |
x − u
∗
≤
b − Au
∗
1 −B
,(6)
where A and B are as previously defined.
For any x ∈ Ᏼ and 0 ≤ λ ≤ 1, define
T
λ
(x) = λb +(I − λA)W(x). (7)
Then, for any y ∈ Ᏼ, since W is nonexpansive,
T
λ
(x) − T
λ
(y)
=
(I − λA)
W(x) − W(y)
≤
1 − λ
1 −B
x − y. (8)
Also, since u
∗
∈ F,
T
λ
u
∗
− u
∗
= λ
b − Au
∗
. (9)
Thus,
T
λ
(x) − u
∗
≤
T
λ
(x) − T
λ
u
∗
+
T
λ
u
∗
− u
∗
≤
1 − λ
1 −B
x − u
∗
+ λ
1 −B
b − Au
∗
1 −B
≤
x − Au
∗
1 −B
.
(10)
If, in (7), we make the substitution λ = λ
n
, T
λ
(x) = u
n+1
,andW(x) = W
n
u
n
,thenit
follows from (9)and(10)thatu
n
and W
n
u
n
belong to C
u
∗
for each n.Thus,{u
n
} and
{W
n
u
n
} are bounded. Since B < 1, {BW
n
u
n
} is also bounded.
Let K denote the diameter of C
u
∗
.
We may w rite (5)intheform
u
n+1
= λ
n
b +
I − λ
n
(I − B)
W
n
u
n
= λ
n
b +
I − λ
n
W
n
u
n
+ λ
n
BW
n
u
n
.
(11)
We will first show that
lim
u
n+1
− u
n
=
0. (12)
138 Quadratic optimization
Using (11), since each W
n
is nonexpansive and B < 1,
u
n+1
− u
n
=
λ
n
b +
1 − λ
n
W
n
u
n
+ λ
n
BW
n
u
n
− λ
n−1
b
−
1 − λ
n−1
W
n−1
u
n−1
− λ
n−1
BW
n−1
u
n−1
≤
λ
n
− λ
n−1
b +
1 − λ
n
W
n
u
n
− W
n−1
u
n
+
λ
n
− λ
n−1
W
n−1
u
n−1
+
1 − λ
n
W
n−1
u
n
− W
n−1
u
n−1
+ λ
n
B
W
n
u
n
− W
n−1
u
n
+ λ
n
B
W
n−1
u
n
− W
n−1
u
n−1
+
λ
n
− λ
n−1
BW
n−1
u
n−1
≤ 3
λ
n
− λ
n−1
K +
1 − λ
n
+ λ
n
B
×
W
n
u
n
− W
n−1
u
n
+
1 − λ
n
+ λ
n
B
W
n−1
u
n
− W
n−1
u
n−1
.
(13)
From (4), since T
N
and U
n−1,N−1
are nonexpansive,
W
n
u
n
− W
n−1
u
n
=
α
nN
T
N
U
n,N−1
u
n
+
1 − α
nN
u
n
− α
n−1,N
T
N
U
n−1,N−1
u
n
−
1 − α
n−1,N
u
n
≤
α
nN
− α
n−1,N
u
n
+
α
nN
T
N
U
n,N−1
u
n
− α
n−1,N
T
N
U
n−1,N−1
u
n
≤
α
nN
− α
n−1,N
u
n
+
α
nN
T
N
U
n,N−1
u
n
− T
N
U
n−1,N−1
u
n
+
α
nN
− α
n−1,N
T
N
U
n−1,N−1
u
n
≤
α
nN
− α
n−1,N
u
n
+ α
nN
U
n,N−1
u
n
− U
n−1,N−1
u
n
+
α
nN
− α
n−1,N
K
≤ 2K
α
nN
− α
n−1,N
+ α
nN
U
n,N−1
u
n
− U
n−1,N−1
u
n
.
(14)
Again, from (4),
U
n,N−1
u
n
− U
n−1,N−1
u
n
=
α
n,N−1
T
N−1
U
n,N−2
u
n
+
1 − α
n,N−1
u
n
− α
n−1,N−1
T
N−1
U
n−1,N−2
u
n
−
1 − α
n−1,N−1
u
n
≤
α
n,N−1
− α
n−1,N−1
u
n
+
α
n,N−1
T
N−1
U
n,N−2
u
n
− α
n−1,N−1
T
N−1
U
n−1,N−2
u
n
≤
α
n,N−1
− α
n−1,N−1
u
n
+ α
n,N−1
T
N−1
U
n,N−2
u
n
− T
N−1
U
n−1,N−2
u
n
+
α
n,N−1
− α
n−1,N−1
K
≤ 2K
α
n,N−1
− α
n−1,N−1
+ α
n,N−1
U
n,N−2
u
n
− U
n−1,N−2
u
n
≤ 2K
α
n,N−1
− α
n−1,N−1
+
U
n,N−2
u
n
− U
n−1,N−2
u
n
.
(15)
B. E. Rhoades 139
Therefore,
U
n,N−1
u
n
− U
n−1,N−1
u
n
≤ 2K
α
n,N−1
− α
n−1,N−1
+2K
α
n,N−2
− α
n−1,N−2
+
U
n,N−3
u
n
− U
n−1,N−3
u
n
.
.
.
≤ 2K
N−1
i=2
α
ni
− α
n−1,i
+
U
n1
u
n
− U
n−1,1
u
n
=
α
n1
T
1
u
n
+
1 − α
n1
u
n
− α
n−1,1
T
1
u
n
−
1 − α
n−1,1
u
n
+2K
N−1
i=2
α
ni
− α
n−1,i
≤
α
n1
− α
n−1,1
u
n
+
α
n1
T
1
u
n
− α
n−1,1
T
1
u
n
+2K
N−1
i=2
α
ni
− α
n−1,i
≤ 2K
N−1
i=1
α
ni
− α
n−1,i
.
(16)
Substituting (16)into(14),
W
n
u
n
− W
n−1
u
n
≤ 2K
α
nN
− α
n−1,N
+2α
nN
K
N−1
i=1
α
ni
− α
n−1,i
≤ 2K
N
i=1
α
ni
− α
n−1,i
.
(17)
Using (17)in(13),
u
n+1
− u
n
≤
1 − λ
n
1 −B
u
n
− u
n−1
+3K
λ
n
− λ
n−1
+2
1 − λ
n
1 −B
K
N
i=1
α
ni
− α
n−1,i
.
(18)
Thus, since 0 < 1− λ
n
(1 −B) < 1foralln,
u
n+m+1
− u
n+m
≤
n+m
i=m
1 − λ
i
1 −B
u
i+1
− u
i
+3K
n+m
i=m
λ
i
− λ
i−1
+2K
n+m
i=m
N
j=1
α
ij
− α
i−1, j
.
(19)
140 Quadratic optimization
From (iii), since the product diverges to zero,
limsup
n
u
n+1
− u
n
=
limsup
n
u
n+m+1
− u
m+n
≤ 2K
∞
i=m
λ
i
− λ
i−1
+2K
∞
i=m
N
j=1
α
ij
− α
i−1, j
.
(20)
Therefore, taking the limsup
m
of both sides and using (iv) and (v),
limsup
n
u
n+1
− u
n
=
0, (21)
and (12)issatisfied.
Now, for any nonexpansive self-map T of C
u
∗
,defineG
t
: C
u
∗
→ C
u
∗
by
G
t
(x) = tb +(1− t)TG
t
(x)+tBTG
t
(x) (22)
for each t ∈ (0,1]. Using an argument similar to the proof of [8, Theorem 12.2, page
45], we will now show that if T has a fixed point, then, for each x in C
u
∗
, the strong
limit
t→0
G
t
(x) exists and is a fixed point of T.
Define y(t) = G
t
(x)andletw be a fixed point of T:
y(t) − w = t(b − w)+(1− t)
Ty(t) − w
+ tBTy(t). (23)
Since T is nonexpansive,
y(t) − w
≤ tb − w +(1− t)
Ty(t) − w
+ tB
Ty(t)
≤ tb − w +(1− t)
y(t) − w
+ tB
Ty(t)
,
t
y(t) − w
≤ tb − w +tB
Ty(t) − w
+ tBw,
(24)
or
y(t) − w
≤b − w + B
y(t) − w
+ Bw, (25)
which, since
B < 1, yields
y(t) − w
≤
1
1 −B
b − w + Bw
, (26)
and y(t) remains bounded as t → 0.
Also,
BTy(t)
<
Ty(t)
≤
Ty(t) − Tw
+ w≤
y(t) − w
+ w, (27)
and both BTy(t)andTy(t) remain bounded as t → 0.
Hence,
y(t) − Ty(t)
=
t
b − Ty(t)+BT y(t)
−→ 0ast −→ 0. (28)
B. E. Rhoades 141
Define y
n
= y(t
n
)andlett
n
→ 0. Let µ
n
be a Banach limit and f : C
u
∗
→ R
+
defined by
f (z) = µ
n
y
n
− z
2
. (29)
Since f is continuous and convex, f (z) →∞as z→∞.SinceᏴ is reflexive, f attains
it infimum over C
u
∗
.
Let M be the set of minimizers of f over C
u
∗
.Ifu ∈ C
u
∗
,then
f (Tu)
= µ
n
y
n
− Tu
2
=
µ
n
Ty
n
− Tu
2
≤
µ
n
y
n
− u
2
=
f (u). (30)
Therefore, M is invariant under T. Since it is also bounded, closed, and convex, it must
contain a fixed point of T. Denote this fixed point by v.Then,
y
n
− Ty
n
, y
n
− v
=
y
n
− v, y
n
− v
+
v − Ty
n
, y
n
− v
=
y
n
− v
2
+
Tv− Ty
n
, y
n
− v
.
(31)
But
Tv− Ty
n
, y
n
− v
≤
Tv− Ty
n
y
n
− v
≤
y
n
− v
2
, (32)
so that
y
n
− Ty
n
, y
n
− v
≥ 0. (33)
Since
y
n
= t
n
b +
1 − t
n
Ty
n
+ t
n
BTy
n
,
y
n
− b =
1 − t
n
Ty
n
− b
+ t
n
BTy
n
=
1 − t
n
Ty
n
− y
n
+
1 − t
n
y
n
− b
+ t
n
BTy
n
,
(34)
thus,
t
y
n
− b
=
1 − t
n
Ty
n
− y
n
+ t
n
BTy
n
(35)
or
y
n
− b − Bv =
1 − t
n
t
n
Ty
n
− y
n
+ BTy
n
− Bv. (36)
Therefore, from (33),
y
n
− b − Bv, y
n
− v
=
1 − t
n
t
n
Ty
n
− y
n
, y
n
− v
+
BTy
n
− Bv, y
n
− v
≤
BTy
n
− Bv, y
n
− v
.
(37)
142 Quadratic optimization
For any z ∈ C
u
∗
,
y
n
− v
2
=
y
n
−
1 − t
n
v − t
n
z + t
n
(z − v) − t
n
b − t
n
Bv + t
n
(b + Bv)
2
≥
y
n
−
1 − t
n
v − t
n
b − t
n
Bv
2
+2t
n
z − v + b+ Bv, y
n
−
1 − t
n
v − t
n
z − t
n
b − t
n
Bv
.
(38)
Let > 0begiven.SinceᏴ is uniformly smooth, there exists a t
0
> 0suchthat,forall
t
n
≤ t
0
,
z − v + b+ Bv,
y
n
− v
−
y
n
−
1 − t
n
v − t
n
z − t
n
b − t
n
Bv
< . (39)
Thus, from (38),
z − v + b+ Bv, y
n
− v
< +
z − v + b+ Bv, y
n
−
1 − t
n
v − t
n
z − t
n
b − t
n
B
< +
1
2t
y
n
− v
2
−
y
n
−
1 − t
n
v − t
n
b − t
n
Bv
2
.
(40)
Since the Gateaux derivative exists in Ᏼ,weobtain
µ
n
z − v + b+ Bv, y
n
− v
≤ 0. (41)
Setting z = θ in (41) and adding (37)and(41)yields
µ
n
y
n
− v, y
n
− v
≤ µ
n
BTy
n
− Bv, y
n
− v
(42)
or
µ
n
y
n
− v
2
≤ µ
n
BTy
n
− Bv
y
n
− v
≤ µ
n
B
Ty
n
− Tv
y
n
− v
≤
Bµ
n
y
n
− v
2
.
(43)
Therefore, µ
n
y
n
− v
2
= 0. Thus, there is a subsequence of {y
n
} converging strongly
to v. Suppose that lim
k→∞
y(t
n
k
) = v
1
and lim
k→∞
y(t
m
k
) = v
2
.From(37), we have
v
1
− b − Bv
2
,v
1
− v
2
≤
BTv
1
− Bv
2
,v
1
− v
2
,
v
2
− b − Bv
1
,v
2
− v
1
≤
BTv
2
− Bv
1
,v
2
− v
1
.
(44)
Adding these inequalitites, we obtain
v
1
− BTv
1
+ BTv
2
− v
2
,v
1
− v
2
≤ 0 (45)
or
v
1
− v
2
,v
1
− v
2
≤
BTv
1
− BTv
2
,v
1
− v
2
; (46)
B. E. Rhoades 143
that is,
v
1
− v
2
2
≤
BTv
2
− BTv
1
v
1
− v
2
≤B
Tv
2
− Tv
1
v
1
− v
2
≤B
v
2
− v
1
2
,
(47)
which, since B < 1, implies that v
1
= v
2
, and thus lim y
n
= v.
Now, setting z = θ in (41), we obtain
µ
n
b − (I − B)v, y
n
− v
≤ 0 (48)
or
µ
n
b − Av, y
n
− v
≤ 0, (49)
which, from (2), implies that v = u
∗
.
Let u
nk
denote the unique element of C
u
∗
such that
u
nk
=
1
k
b +
1 −
1
k
W
n
u
nk
+
1
k
BW
n
u
nk
. (50)
From what we have just proved, lim
k
u
nk
→ u
∗
. Using (11),
u
n+1
− W
n+1
u
n+1,k
=
λ
n
b +
1 − λ
n
+ λ
n
B
W
n
u
n
− W
n+1
u
n+1,k
≤ λ
n
b − W
n+1
u
n+1,k
+
1 − λ
n
W
n
u
n
− W
n+1
u
n+1,k
+ λ
n
B
W
n
u
n
− W
n+1
u
n+1,k
+ λ
n
BW
n+1
u
n+1,k
< 3Kλ
n
+
1 − λ
n
+ λ
n
B
W
n
u
n
− W
n
u
nk
+
W
n
u
nk
− W
n
u
n+1,k
+
W
n
u
n+1,k
− W
n+1
u
n+1,k
≤ 3Kλ
n
+
1 − λ
n
+ λ
n
B
u
n
− u
nk
+
u
nk
− u
n+1,k
+
W
n
u
n+1,k
− W
n+1
u
n+1,k
.
(51)
As in (17),
W
n
u
n+1,k
− W
n+1
u
n+1,k
≤ 2K
N
i=1
α
n+1,i
− α
ni
. (52)
From the definition of u
nk
,
u
nk
=
1
k
b +
1 −
1
k
W
n
u
nk
+
1
k
BW
n
u
nk
,
u
n+1,k
=
1
k
b +
1 −
1
k
W
n+1
u
n+1,k
+
1
k
BW
n+1
u
n+1,k
,
u
n+1,k
− u
nk
=
1 −
1
k
W
n+1
u
n+1,k
− W
n
u
nk
+
1
k
B
W
n+1
u
n+1,k
− W
n
u
nk
.
(53)
144 Quadratic optimization
Therefore, since W
n+1
is nonexpansive,
u
n+1,k
− u
nk
≤
1 −
1
k
+
1
k
B
W
n+1
u
n+1,k
− W
n
u
nk
≤
1 −
1
k
+
1
k
B
W
n+1
u
n+1,k
− W
n+1
u
nk
+
W
n+1
u
nk
− W
n
u
nk
≤
1 −
1
k
+
1
k
B
u
n+1,k
− u
nk
+
W
n+1
u
nk
− W
n
u
nk
.
(54)
Thus, using (17),
1 −B
k
u
n+1,k
− u
nk
≤
k − 1+B
k
2K
N
i=1
α
n+1,i
− α
n,i
(55)
or
u
n+1,k
− u
nk
≤
k − 1+B
1 −B
2K
N
i=1
α
n+1,i
− α
n,i
. (56)
Substituting (56)and(52)into(51)yields
u
n+1
− W
n+1
u
n+1,k
≤ 3Kλ
n
+
1 − λ
n
+ λ
n
B
u
n
− u
nk
+
k
1 −B
2K
N
i=1
α
n+1,i
− α
n,i
.
(57)
Thus, using (iii) and (v), we have
µ
n
u
n
− W
n
u
nk
2
=
µ
n
u
n+1
− W
n+1
u
n+1,k
2
≤ µ
n
u
n
− u
nk
2
. (58)
From (53),
u
nk
− u
n
=
1
k
b − u
n
+
1 −
1
k
W
n
u
nk
− u
n
+
1
k
BW
n
u
nk
. (59)
Hence,
1 −
1
k
u
n
− W
n
u
nk
=
u
n
− u
nk
−
1
k
u
n
− b
+
1
k
BW
n
u
nk
, (60)
1 −
1
k
2
u
n
− W
n
u
nk
2
≥
u
n
− u
nk
2
−
2
k
u
n
− b − BW
n
u
nk
,u
n
− u
nk
=
1 −
2
k
u
n
− u
nk
2
−
2
k
u
nk
− b − BW
n
u
nk
,u
n
− u
nk
.
(61)
B. E. Rhoades 145
Therefore, using (58)and(61),
1 −
1
k
2
µ
n
u
n
− u
nk
2
≥
1 −
1
k
2
µ
n
u
n
− W
n
u
nk
2
≥
1 −
2
k
µ
n
u
n
− u
nk
2
−
2
k
µ
n
u
nk
− b − BW
n
u
nk
,u
n
− u
nk
,
(62)
which implies that
1
2k
µ
n
u
n
− u
nk
2
≥ µ
n
b − u
nk
+ BW
n
u
nk
,u
n
− u
nk
. (63)
Since lim
k
u
nk
→ u
∗
, independent of n, it follows that
0 ≥ µ
n
b − u
∗
+ Bu
∗
,u
n
− u
∗
= µ
n
b − (I − B)u
∗
,u
n
− u
∗
=
µ
n
b − Au
∗
,u
n
− u
∗
.
(64)
From (12),
lim
b − u
∗
,u
n+1
− u
∗
−
b − u
∗
,u
n
− u
∗
=
0. (65)
We need the following result from [12]. If A is a real number and {a
1
,a
2
, }∈
∞
such
that µ
n
{a
n
}≤a for all Banach limits µ
n
and limsup
n
(a
n+1
− a
n
) ≤ 0, then limsup
n
a
n
≤ a.
Consequently,
limsup
n
b − u
∗
,u
n
− u
∗
≤ 0. (66)
Since u
∗
∈ F,
W
n
u
n
− u
∗
=
W
n
u
n
− W
n
u
∗
≤
u
n
− u
∗
. (67)
From (11),
u
n+1
− u
∗
= λ
n
b − u
∗
+
1 − λ
n
W
n
u
n
− u
∗
+ λ
n
BW
n
u
n
= λ
n
b − u
∗
+
1 − λ
n
+ λ
n
B
W
n
u
n
− u
∗
+ λ
n
Bu
∗
(68)
or
1 − λ
n
+ λ
n
B
W
n
u
n
− u
∗
=
u
n+1
− u
∗
− λ
n
b − u
∗
− λ
n
Bu
∗
. (69)
Therefore,
1 − λ
n
+ λ
n
B
2
W
n
u
n
− u
∗
2
≥
u
n+1
− u
∗
2
− 2λ
n
b − u
∗
+ Bu
∗
,u
n+1
− u
∗
,
(70)
146 Quadratic optimization
which implies that
u
n+1
− u
∗
2
≤
1 − λ
n
+ λ
n
B
2
W
n
u
n
− u
∗
2
+2λ
n
b − u
∗
,u
n+1
− u
∗
+2λ
n
Bu
∗
,u
n+1
− u
∗
.
(71)
From (ii) and the boundedness of C
u
∗
, there exists a positive integer N such that, for
all n ≥ N,
λ
n
b − u
∗
,u
n+1
− u
∗
≤
4
, λ
n
Bu
∗
,u
n+1
− u
∗
≤
4
. (72)
Therefore, for n ≥ N,
u
n+1
− u
∗
2
≤
1 − λ
n
+ λ
n
B
2
u
n
− u
∗
2
+
2
+
2
,
u
n+m
− u
∗
2
≤
n+m−1
i=m
1 − λ
i
+ λ
i
B
2
u
m
− u
∗
2
+
n+m−1
i=m
1 − λ
i
+ λ
i
B
2
.
(73)
Using (iii),
limsup
n
u
n
− u
∗
2
= limsup
n
u
n+m
− u
∗
2
≤ 0. (74)
Thus, {u
n
} converges strongly to u
∗
.
Now let u
0
∈ Ᏼ.Let{s
n
} be another sequence generated by (11)forsomes
0
∈ C
u
∗
.
Then, by what we have just proved, lims
n
= u
∗
.SinceW
n
is nonexpansive for each n,
u
n+1
− s
n+1
=
λ
n
b +
1 − λ
n
A
W
n
u
n
− λ
n
b −
1 − λ
n
A
W
n
s
n
≤
1 − λ
n
A
W
n
u
n
− W
n
s
n
≤
1 − λ
n
+ λ
n
B
W
n
u
n
− W
n
s
n
≤
1 − λ
n
+ λ
n
B
u
n
− s
n
.
(75)
By induction,
u
n
− s
n
≤
u
0
− s
0
n
k=1
1 − λ
k
1 −B
. (76)
Therefore, using (iii), limu
n
− s
n
=0andu
n
− u
∗
≤u
n
− s
n
+ s
n
− u
∗
so that
limu
n
= u
∗
.
B. E. Rhoades 147
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B. E. Rhoades: Department of Mathematics, Indiana University, Bloomington, IN 47405-7106,
USA
E-mail address:
