An. Şt. Univ. Ovidius Constanţa
Vol. 19(1), 2011, 313–330
Iterative methods for k-strict
pseudo-contractive mappings in Hilbert spaces
Yonghong Yao, Yeong-Cheng Liou, Shin Min Kang
Abstract
In this paper, we investigate two iterative methods for k-strict pseudocontractive mappings in a real Hilbert space. We prove that the proposed iterative algorithms converge strongly to some fixed point of a
strict pseudo-contractive mapping.
1. Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H. Let
T : C → C be a mapping. We use F ix(T ) to denote the set of fixed points
of T . Recall that T is said to be a strict pseudo-contractive mapping if there
exists a constant 0 ≤ k < 1 such that
(1.1) kT x − T yk2 ≤ kx − yk2 + kk(I − T )x − (I − T )yk2
for all x, y ∈ C.
For such case, we also say that T is a k-strict pseudo-contractive mapping.
When k = 0, T is said to be nonexpansive, and it is said to be pseudocontractive if k = 1. T is said to be strongly pseudo-contractive if there exists
a constant α ∈ (0, 1) such that hT x − T y, x − yi ≤ αkx − yk2 for all x, y ∈ C.
Clearly, the class of k-strict pseudo-contractive mappings falls into the one
between classes of nonexpansive mappings and pseudo-contractive mappings.
Key Words: Fixed point, strict pseudo-contractive mapping, iterative method, strong
convergence, Hilbert space.
Mathematics Subject Classification: 47H09, 65J15
Received: January, 2010
Accepted: December, 2010
313
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Yonghong Yao, Yeong-Cheng Liou, Shin Min Kang
We remark also that the class of strongly pseudo-contractive mappings is independent of the class of k-strict pseudo-contractive mappings (see, e.g., [3-5]).
It is clear that, in a real Hilbert space H, (1.1) is equivalent to
(1.2)
1−k
k(I − T )x − (I − T )yk2 for all x, y ∈ C.
hT x − T y, x − yi ≤ kx − yk2 −
2
Recall also that a mapping f : C → C is called contractive if there exists a
constant α ∈ [0, 1) such that
kf (x) − f (y)k ≤ αkx − yk
for all x, y ∈ C.
Iterative methods for nonexpansive mappings have been extensively investigated; see [1, 2, 6-15, 17-18, 20-22, 24-36] and the references therein. However
iterative methods for strict pseudo-contractive mappings are far less developed than those for nonexpansive mappings though Browder and Petryshyn
[5] initiated their work in 1967; the reason is probably that the second term
appearing in the right-hand side of (1.1) impedes the convergence analysis for
iterative methods used to find a fixed point of the strict pseudo-contractive
mapping T . However, on the other hand, strict pseudo-contractive mappings
have more powerful applications than nonexpansive mappings do in solving
inverse problems; see Scherzer [23]. Therefore it is interesting to develop the
iterative methods for strict pseudo-contractive mappings. As a matter of fact,
Browder and Petryshyn [5] show that if a k-strict pseudo-contractive mappings
T has a fixed point in C, then starting with an initial x0 ∈ C, the sequence
{xn } generated by the recursive formula:
xn+1 = λxn + (1 − λ)T xn ,
n ≥ 0,
where λ is a constant such that k < λ < 1, converges weakly to a fixed point
of T .
Recently, Marino and Xu [16] have extended Browder and Petryshyn’s
result by proving that the sequence {xn } generated by the following Mann’s
method:
xn+1 = λn xn + (1 − λn )T xn ,
n≥0
converges weakly to a fixed point of T , provided the
sequence {λn }
Pcontrol
∞
satisfies the conditions that k < λn < 1 for all n and n=0 (λn − k)(1 − λn ) =
∞. However, this convergence is in general not strong. So in order to get
strong convergence for strict pseudo-contractive mappings, Marino and Xu
[16] further suggested the following modified Mann’s method based on the CQ
Iterative methods for k-strict pseudo-contractive mappings ...
315
method:
(1.3)

x ∈ C chosen arbitrarily,


 0


yn = αn xn + (1 − αn )T xn ,




 C = {z ∈ C : ky − zk2 ≤ kx − zk2
n
n
n

+ (1 − αn )(k − αn )kxn − T xn k2 },





Qn = {z ∈ C : hxn − z, x0 − xn i ≥ 0},




xn+1 = PCn ∩Qn x0 .
They proved that the sequence {xn } generated by (1.3) is strongly convergent to a fixed point of T for any choice of the control sequence {αn } such
that αn < 1 for all n.
We observe that the CQ method (1.3) generates a sequence {xn } by projecting the initial guess x0 onto the intersection of two appropriately constructed closed convex subsets Cn and Qn . Hence, how to construct closed
convex subsets Cn and Qn is very crucial for the CQ method.
It is the purpose of this paper to suggest and analyze some iterative methods for strict pseudo-contractive mappings in the sense of Browder-Petryshyn
in a real Hilbert space. First, we consider a modified Mann’s method which
is different from (1.3). Secondly, we study another modified method for strict
pseudo-contractive mappings. We prove that the proposed iterative methods
converge strongly to some fixed point of a strict pseudo-contractive mapping.
2. Preliminaries
In this section, we collect some facts in a real Hilbert space H which are
listed as below.
Let H be a real Hilbert space with inner product h·, ·i and norm k · k and
let C be a closed convex subset of H. For every point x ∈ H there exists a
unique nearest point in C, denoted by PC x, such that
kx − PC xk ≤ kx − yk
for all y ∈ C,
where PC is called the metric projection of H onto C. We know that PC is
a nonexpansive mapping. For given sequence {xn } ⊂ C, let ωw (xn ) = {x :
∃xnj → x weakly} denote the weak ω-limit set of {xn }.
We note the following Lemmas 2.1 and 2.2 are well-known.
Lemma 2.1. Let H be a real Hilbert space. Then there hold the following
well-known identities:
(a) kx − yk2 = kxk2 − 2hx, yi + kyk2 for all x, y ∈ H;
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Yonghong Yao, Yeong-Cheng Liou, Shin Min Kang
(b) ktx + (1 − t)yk2 = tkxk2 + (1 − t)kyk2 − t(1 − t)kx − yk2 for all t ∈ [0, 1],
x, y ∈ H.
Lemma 2.2. Let C be a closed convex subset of a real Hilbert space H.
Given x ∈ H and z ∈ C. Then z = PC x if and only if there holds the relation
hx − z, y − zi ≤ 0
for all y ∈ C.
Lemma 2.3. ([16]) Assume C is a closed convex subset of a real Hilbert
space H, let T : C → C be a k-strict pseudo-contractive mapping. Then, the
mapping I − T is demiclosed at zero. That is, if {xn } is a sequence in C such
that xn → x∗ weakly and (I − T )xn → 0 strongly, then (I − T )x∗ = 0.
Lemma 2.4. ([17]) Let C be a closed convex subset of a real Hilbert space
H. Let {xn } be a sequence in H and u ∈ H. Let q = PC u. If {xn } is such
that ωw (xn ) ⊂ C and satisfies the condition
kxn − uk ≤ ku − qk
for all n.
Then xn → q.
Lemma 2.5. ([25]) Let {xn } and {yn } be bounded sequences in a Banach space E such that xn+1 = σn xn + (1 − σn )yn , n ≥ 0 where {σn } is a
sequence in [0, 1] such that 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1. Assume
lim supn→∞ (kyn+1 − yn k − kxn+1 − xn k) ≤ 0. Then limn→∞ kyn − xn k = 0.
Lemma 2.6. Let C be a nonempty closed convex subset of a real Hilbert
space H. Let T : C → C be a k-strict pseudo-contractive mapping with
F ix(T ) 6= ∅. Then, F ix(T ) is convex and closed.
Proof. First, we prove that F ix(T ) is convex. For all p, q ∈ F ix(T ) and
ξ ∈ [0, 1], we set R = ξp + (1 − ξ)q. From Lemma 2.1(b) and (1.1), we have
kR − T Rk2
= kξ(p − T R) + (1 − ξ)(q − T R)k2
= ξkp − T Rk2 + (1 − ξ)kq − T Rk2 − ξ(1 − ξ)kp − qk2
= ξkT p − T Rk2 + (1 − ξ)kT q − T Rk2 − ξ(1 − ξ)kp − qk2
≤ ξ[kp − Rk2 + kkR − T Rk2 ] + (1 − ξ)[kq − Rk2 + kkR − T Rk2 ]
− ξ(1 − ξ)kp − qk2
= ξkp − Rk2 + (1 − ξ)kq − Rk2 − ξ(1 − ξ)kp − qk2 + kkR − T Rk2
= kξ(p − R) + (1 − ξ)(q − R)k2 + kkR − T Rk2
= kkR − T Rk2 ,
Iterative methods for k-strict pseudo-contractive mappings ...
which implies that
317
(1 − k)kR − T Rk2 ≤ 0,
that is
R = T R.
Therefore, R ∈ F ix(T ) and F ix(T ) is convex.
Next, we prove that F ix(T ) is closed. First, we note that T is Lipschitz
1+k
with Lipschitzian constant L = 1−k
. If we take pn ∈ F ix(T ) satisfying pn → p
as n → ∞, then
p = lim pn = lim T pn = T p.
n→∞
n→∞
Therefore, p ∈ F ix(T ). Hence, F ix(T ) is closed. This completes the proof.
Lemma 2.7. Let C be a nonempty closed convex subset of a real Hilbert
space H. Let T : C → C be a k-strict pseudo-contractive mapping with
F ix(T ) 6= ∅. Let f : C → C be a continuous strongly pseudo-contractive
mapping. For any t ∈ (0, 1), let xt be the unique fixed point of tf + (1 − t)T .
Then as t → 0+, the path {xt } converges strongly to p = PF ix(T ) f (p) which is
the unique solution of the variational inequality: hf (p) − p, z − pi ≤ 0 for all
z ∈ F ix(T ).
Proof. From Lemma 2.6, we know that F ix(T ) is closed and convex.
Hence, PF ix(T ) is well defined. For fixed u ∈ C arbitrarily, let the path
{yt : t ∈ (0, 1)} be defined by yt = tu + (1 − t)T xt for all t ∈ (0, 1). Then, from
[19], it is clear that the path {yt } converges strongly to p = PF ix(T ) u. Taking
u = f (p). Then, the path {yt } defined by yt = tf (p) + (1 − t)T yt converges
strongly to p = PF ix(T ) f (p). Note that
kxt − yt k2 = thf (xt ) − f (p), xt − yt i + (1 − t)hT xt − T yt , xt − yt i
= thf (xt ) − f (yt ), xt − yt i + thf (yt ) − f (p), xt − yt i
+ (1 − t)hT xt − T yt , xt − yt i
≤ tαkxt − yt k2 + tkf (yt ) − f (p)kkxt − yt k + (1 − t)kxt − yt k2 ,
that is,
1
kf (yt ) − f (p)k.
1−α
Since yt → p = PF ix(T ) f (p) and f is continuous, then xt → p = PF ix(T ) f (p).
From Lemma 2.2, it is clear that p is the unique solution of the variational
inequality: hf (p) − p, z − pi ≤ 0 for all z ∈ F ix(T ). This completes the
proof.
kxt − yt k ≤
Lemma 2.8. ([21]) Assume {an } is a sequence of nonnegative real numbers
such that an+1 ≤ (1 − bn )an + cn , n ≥ 0, where {bn } is a sequence in (0, 1)
and {cn } is a sequence in R such that
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Yonghong Yao, Yeong-Cheng Liou, Shin Min Kang
P∞
(a) n=0 bn = ∞;
P∞
(b) lim supn→∞ cn /bn ≤ 0 or n=0 |cn | < ∞.
Then limn→∞ an = 0.
3. Main Results
3.1. Projection methods for strict pseudo-contractive mappings
First, we introduce the following modified Mann’s method based on the
projection methods which is different from (1.3).
Algorithm 3.1.Let C be a nonempty closed convex subset of a real Hilbert
space H. Let T : C → C be a k-strict pseudo-contractive mapping for some
0 ≤ k < 1. Let {αn } be a real sequence in [0, 1). For C1 = C and x1 = PC1 x0 ,
let the sequence {xn } be generated by the following method:
(3.1)


 yn = αn xn + (1 − αn )T xn ,
Cn+1 = {z ∈ Cn : kyn − zk ≤ kxn − zk},


xn+1 = PCn+1 x0 .
Now we prove the following strong convergence theorem concerning the
above projection method (3.1).
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert
space H. Let T : C → C be a k-strict pseudo-contractive mapping for some
0 ≤ k < 1 with F ix(T ) 6= ∅. Let the sequence {xn } be generated by the method
(3.1). Assume that the sequence {αn } is chosen so that αn ∈ [a, b] ⊂ [k, 1) for
all n ≥ 0. Then {xn } converges strongly to PF ix(T ) x0 .
Proof. First, we note that Cn is convex and closed. As a matter of fact, we
observe that kyn −zk ≤ kxn −zk is equivalent to kyn −xn k2 +2hyn −xn , xn −zi ≥
0. So Cn is closed and convex. Hence, {xn } is well-defined.
Next we show that F ix(T ) ⊂ Cn for all n. From (1.1) and (1.2), we note
that for all p ∈ F ix(T ),
hT xn − p, xn − pi ≤ kxn − pk2 −
1−k
kxn − T xn k2
2
and
kT xn − pk2 ≤ kxn − pk2 + kkxn − T xn k2 .
Iterative methods for k-strict pseudo-contractive mappings ...
319
Then, we have
kyn − pk2 = kαn (xn − p) + (1 − αn )(T xn − p)k2
= αn2 kxn − pk2 + 2αn (1 − αn )hT xn − p, xn − pi
+ (1 − αn )2 kT xn − pk2
¸
·
1−k
≤ αn2 kxn − pk2 + 2αn (1 − αn ) kxn − pk2 −
kxn − T xn k2
2
+ (1 − αn )2 [kxn − pk2 + kkxn − T xn k2 ]
= kxn − pk2 + (1 − αn )(k − αn )kxn − T xn k2
≤ kxn − pk2 ,
that is, kyn − pk ≤ kxn − pk. So p ∈ Cn+1 ⊂ Cn for all n. This implies that
F ix(T ) ⊂ Cn ,
n ≥ 0.
From xn = PCn x0 , we have
hx0 − xn , xn − yi ≥ 0 for all y ∈ Cn .
Using F ix(T ) ⊂ Cn , we also have
hx0 − xn , xn − pi ≥ 0 for all p ∈ F ix(T ).
So, for p ∈ F ix(T ), we have
0 ≤ hx0 − xn , xn − pi
= hx0 − xn , xn − x0 + x0 − pi
= −kx0 − xn k2 + hx0 − xn , x0 − pi
≤ −kx0 − xn k2 + kx0 − xn kkx0 − pk.
Hence,
kx0 − xn k ≤ kx0 − pk
for all p ∈ F ix(T ).
In particular, {xn } is bounded and
(3.2)
kx0 − xn k ≤ kx0 − qk,
where q = PF ix(T ) x0 .
From xn = PCn x0 and xn+1 = PCn+1 x0 ∈ Cn+1 ⊂ Cn , we have
(3.3)
hx0 − xn , xn − xn+1 i ≥ 0.
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Yonghong Yao, Yeong-Cheng Liou, Shin Min Kang
Hence,
0 ≤ hx0 − xn , xn − xn+1 i
= hx0 − xn , xn − x0 + x0 − xn+1 i
= −kx0 − xn k2 + hx0 − xn , x0 − xn+1 i
≤ −kx0 − xn k2 + kx0 − xn kkx0 − xn+1 k,
and therefore
kx0 − xn k ≤ kx0 − xn+1 k,
which implies that limn→∞ kxn − x0 k exists.
From Lemma 2.1(a) and (3.3), we obtain
kxn+1 − xn k2 = k(xn+1 − x0 ) − (xn − x0 )k2
= kxn+1 − x0 k2 − kxn − x0 k2 − 2hxn+1 − xn , xn − x0 i
≤ kxn+1 − x0 k2 − kxn − x0 k2
→ 0.
By the fact xn+1 ∈ Cn+1 ⊂ Cn , we get
kyn − xn+1 k ≤ kxn − xn+1 k,
which implies that kyn − xn+1 k → 0. At the same time, we note that
kyn − xn+1 k2 = kαn (xn − xn+1 ) + (1 − αn )(T xn − xn+1 )k2
(3.4)
= αn kxn − xn+1 k2 + (1 − αn )kT xn − xn+1 k2
− αn (1 − αn )kxn − T xn k2
and
kT xn − xn+1 k2 = kT xn − xn + xn − xn+1 k2
(3.5)
= kT xn − xn k2 + 2hT xn − xn , xn − xn+1 i
+ kxn − xn+1 k2 .
Substitute (3.5) into (3.4) to get
kyn − xn+1 k2 = αn kxn − xn+1 k2 + (1 − αn )kT xn − xn k2
+ 2(1 − αn )hT xn − xn , xn − xn+1 i + (1 − αn )kxn − xn+1 k2
− αn (1 − αn )kxn − T xn k2
= kxn − xn+1 k2 + 2(1 − αn )hT xn − xn , xn − xn+1 i
+ (1 − αn )2 kxn − T xn k2 .
Iterative methods for k-strict pseudo-contractive mappings ...
321
This together with xn − xn+1 → 0 and yn − xn+1 → 0 imply that
(3.6)
kxn − T xn k → 0.
Now (3.6) and Lemma 2.3 guarantee that every weak limit point of {xn } is a
fixed point of T . That is, ωw (xn ) ⊂ F (T ). This fact, the inequality (3.2) and
Lemma 2.4 ensure the strong convergence of {xn } to PF (T ) x0 . This completes
the proof.
Corollary 3.1. Let C be a nonempty closed convex subset of a real Hilbert
space H. Let T : C → C be a nonexpansive mapping with F ix(T ) 6= ∅. Let
the sequence {xn } be generated by the method (3.1). Assume that the sequence
{αn } is chosen so that αn ∈ [a, b] ⊂ [0, 1) for all n ≥ 0. Then the sequence
{xn } defined by (3.1) converges strongly to PF ix(T ) x0 .
3.2. Modified methods for strict pseudo-contractive mappings
Below is another modified method for strict pseudo-contractive mappings.
Algorithm 3.2. Let C be a nonempty closed convex subset of a real Hilbert
space H. Let T : C → C be a k-strict pseudo-contractive mapping with 0 ≤
k < 1. Let f : C → C be a contractive mapping. Let {αn }, {βn } and {γn } be
real sequences in [0, 1] such that αn + βn + γn = 1, n ≥ 0. Let the sequence
{xn } be generated from an arbitrary x0 ∈ C by the following iterative method:
(3.7)
xn+1 = αn f (xn ) + βn xn + γn T xn ,
n ≥ 0.
In particular, if we take f ≡ u, then (3.7) reduces to
(3.8)
xn+1 = αn u + βn xn + γn T xn ,
n ≥ 0.
Now we state and prove the following strong convergence theorem.
Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert
space H, and T : C → C be a k-strict pseudo-contractive mapping with
F ix(T ) 6= ∅. Let f : C → C be a contractive mapping with contractive coefficient α. Let {αn }, {βn } and {γn } be three real sequences in [0, 1] satisfying
the following conditions:
(C1) lim
P∞n→∞ αn = 0;
(C2) n=0 αn = ∞;
(C3) βn ∈ [a, b] ⊂ (k, 1).
For initial guess x0 ∈ C, then the sequence {xn } defined by (3.7) converges
strongly to p ∈ F ix(T ) which solves the variational inequality:
hf (p) − p, z − pi ≤ 0
for all z ∈ F ix(T ).
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Yonghong Yao, Yeong-Cheng Liou, Shin Min Kang
Proof. We first show that {xn } is bounded. Indeed, take a point p ∈
F ix(T ) to get
(3.9)
kxn+1 − pk = kαn (f (xn ) − p) + βn (xn − p) + γn (T xn − p)]k
≤ αn kf (xn ) − f (p)k + αn kf (p) − pk
+ kβn (xn − p) + γn (T xn − p)k.
From (1.1) and (1.2), we obtain
kβn (xn − p) + γn (T xn − p)k2
= βn2 kxn − pk2 + γn2 kT xn − pk2 + 2βn γn hT xn − p, xn − pi
≤ βn2 kxn − pk2 + γn2 [kxn − pk2 + kkxn − T xn k2 ]
1−k
+ 2βn γn [kxn − pk2 −
kxn − T xn k2 ]
2
= (βn + γn )2 kxn − pk2 + [γn2 k − (1 − k)βn γn ]kxn − T xn k2
= (βn + γn )2 kxn − pk2 + γn [(βn + γn )k − βn ]kxn − T xn k2
≤ (βn + γn )2 kxn − pk2 ,
which implies that
(3.10)
kβn (xn − p) + γn (T xn − p)k ≤ (βn + γn )kxn − pk.
It follows from (3.9) and (3.10) that
kxn+1 − pk ≤ αn αkxn − pk + αn kf (p) − pk + (βn + γn )kxn − pk
= [1 − (1 − α)αn ]kxn − pk + (1 − α)αn
kf (p) − pk
.
1−α
By induction, we obtain, for all n ≥ 0,
½
¾
kf (p) − pk
kxn − pk ≤ max kx0 − pk,
.
1−α
Hence, {xn } is bounded.
We note that (3.7) can be rewritten as
µ
¶
kγn
γn
xn +
[kxn + (1 − k)T xn ].
xn+1 = αn f (xn ) + βn −
1−k
1−k
γn
kγn
kγn
1
)+ 1−k
= 1 and (βn − 1−k
) ∈ ( a−k
It is clear that αn +(βn − 1−k
1−k , 1−k ) ⊂ (0, 1).
Iterative methods for k-strict pseudo-contractive mappings ...
323
kγn
kγn
Now we define xn+1 = (βn − 1−k
)xn + (1 − βn + 1−k
)yn , n ≥ 0. It follows that
yn+1 − yn
kγn+1
kγn
xn+1 − (βn − 1−k
)xn
1−k )xn+1
=
−
kγn+1
kγn
1 − βn + 1−k
1 − βn+1 + 1−k
γn+1
αn+1 f (xn+1 ) + 1−k [kxn+1 + (1 − k)T xn+1 ]
=
n+1
1 − βn+1 + kγ1−k
γn
αn f (xn ) + 1−k
[kxn + (1 − k)T xn ]
−
kγn
1 − βn + 1−k
xn+2 − (βn+1 −
(3.11)
=
αn+1 f (xn+1 )
1 − βn+1 +
+
+
kγn+1
1−k
−
γn+1
1−k
αn f (xn )
kγn
1 − βn + 1−k
£
¤
k(xn+1 − xn ) + (1 − k)(T xn+1 − T xn )
kγn+1
1−k
1 − βn+1 +
Ã
γn+1
1−k
1 − βn+1 +
kγn+1
1−k
−
!
γn
1−k
1 − βn +
kγn
1−k
(kxn + (1 − k)T xn ).
It follows from (3.11) that
kyn+1 − yn k
αn+1
≤
1 − βn+1 +
(3.12)
+
kγn+1
1−k
kf (xn+1 )k +
γn+1
1−k
1 − βn+1 +
¯
γn+1
¯
¯
1−k
+¯
¯ 1 − βn+1 +
kγn+1
1−k
αn
1 − βn +
kγn
1−k
kf (xn )k
°
°
°k(xn+1 − xn ) + (1 − k)(T xn+1 − T xn )°
kγn+1
1−k
−
γn
1−k
1 − βn +
kγn
1−k
¯
¯
¯
¯kkxn + (1 − k)T xn k.
¯
Again from Lemma 2.1(b) and (1.1), we have
kk(xn+1 − xn ) + (1 − k)(T xn+1 − T xn )k2
= kkxn+1 − xn k2 + (1 − k)kT xn+1 − T xn k2
− k(1 − k)k(I − T )xn+1 − (I − T )xn k2
≤ kkxn+1 − xn k2 + (1 − k)[kxn+1 − xn k2 + kk(I − T )xn+1 − (I − T )xn k2 ]
− k(1 − k)k(I − T )xn+1 − (I − T )xn k2
= kxn+1 − xn k2 ,
324
Yonghong Yao, Yeong-Cheng Liou, Shin Min Kang
that is,
(3.13)
kk(xn+1 − xn ) + (1 − k)(T xn+1 − T xn )k ≤ kxn+1 − xn k.
At the same time, we observe that
γn+1
1−k
1 − βn+1 +
=
kγn+1
1−k
γn+1
1 − k + (1 − αn+1 )k − βn+1
1 − αn+1 k − βn+1 + (k − 1)αn+1
1 − αn+1 k − βn+1
(1 − k)αn+1
=1−
.
1 − αn+1 k − βn+1
=
Hence,
γn+1
1−k
kγn+1
1−βn+1 + 1−k
∈ [0, 1] and
γn+1
1−k
kγn+1
1−βn+1 + 1−k
γn+1
1−k
1 − βn+1 +
kγn+1
1−k
−
→ 1 as n → ∞. Therefore,
γn
1−k
kγn
1−k
1 − βn +
→ 0.
Combining (3.12) and (3.13) yields
kyn+1 − yn k ≤
αn+1
1 − βn+1 +
kγn+1
1−k
kf (xn+1 )k +
¯
γn+1
¯
¯
1−k
+ kxn+1 − xn k + ¯
¯ 1 − βn+1 +
αn
1 − βn +
kγn+1
1−k
−
kγn
1−k
kf (xn )k
γn
1−k
1 − βn +
kγn
1−k
¯
¯
¯
¯M,
¯
where M is a constant such that supn {kkxn + (1 − k)T xn k} ≤ M . Since
αn → 0, the last inequality implies
lim sup(kyn+1 − yn k − kxn+1 − xn k) ≤ 0.
n→∞
Apply Lemma 2.5 to get
lim kyn − xn k = 0.
n→∞
Consequently,
µ
¶
kγn
lim kxn+1 − xn k = lim 1 − βn +
kyn − xn k = 0.
n→∞
n→∞
1−k
Note that
kxn − T xn k ≤ kxn+1 − xn k + kxn+1 − T xn k
≤ kxn+1 − xn k + αn kf (xn ) − T xn k + βn kxn − T xn k,
Iterative methods for k-strict pseudo-contractive mappings ...
325
that is,
kxn − T xn k ≤
1
(kxn+1 − xn k + αn kf (xn ) − T xn k) → 0.
1 − βn
Next we prove that
lim suphf (p) − p, xn − pi ≤ 0
n→∞
for all p ∈ F ix(T ).
Let zt be the unique solution to the equation
zt = tf (zt ) + (1 − t)T zt .
Since f is contractive, it must be strict pseudo-contractive. From Lemma 2.7,
we know that zt → p ∈ F ix(T ) which solves the variational inequality
hf (p) − p, z − pi ≤ 0 for all z ∈ F ix(T ).
We can write
zt − xn = (1 − t)(T zt − xn ) + t(f (zt ) − xn ).
It follows that
kzt − xn k2 = (1 − t)hT zt − xn , zt − xn i + thf (zt ) − xn , zt − xn i
= (1 − t)(hT zt − T xn , zt − xn i + hT xn − xn , zt − xn i)
+ thf (zt ) − zt , zt − xn i + thzt − xn , zt − xn i
≤ (1 − t)kzt − xn k2 + (1 − t)kT xn − xn kkzt − xn k
+ thf (zt ) − zt , zt − xn i + tkzt − xn k2
= kzt − xn k2 + (1 − t)kT xn − xn kkzt − xn k
+ thf (zt ) − zt , zt − xn i,
and hence
hf (zt ) − zt , xn − zt i ≤
1−t
kT xn − xn kkzt − xn k.
t
Note that kxn − T xn k → 0, zt → p and {zt } and {xn } are all bounded. By
using the standard proof, it is easy to obtain
(3.14)
lim suphf (p) − p, xn − pi ≤ 0
n→∞
for all p ∈ F ix(T ).
326
Yonghong Yao, Yeong-Cheng Liou, Shin Min Kang
Finally we claim that xn → p in norm. Indeed, we have
kxn+1 − pk2
= αn hf (xn ) − p, xn+1 − pi + hβn (xn − p) + γn (T xn − p), xn+1 − pi
≤ αn hf (xn ) − p, xn+1 − pi + kβn (xn − p) + γn (T xn − p)kkxn+1 − pk
≤ αn hf (xn ) − f (p), xn+1 − pi + αn hf (p) − p, xn+1 − pi
+ (βn + γn )kxn − pkkxn+1 − pk
≤ αn αkxn − pkkxn+1 − pk + αn hf (p) − p, xn+1 − pi
+ (1 − αn )kxn − pkkxn+1 − pk
1
≤ [1 − (1 − α)αn ] (kxn − pk2 + kxn+1 − pk2 ) + αn hf (p) − p, xn+1 − pi,
2
that is,
kxn+1 − pk2 ≤ [1 − (1 − α)αn ]kxn − pk2
+ 2αn hf (p) − p, xn+1 − pi,
which implies that
(3.15)
kxn+1 − pk2 ≤ [1 − (1 − α)αn ]kxn − pk2
½
¾
2
+ (1 − α)αn
hf (p) − p, xn+1 − pi .
1−α
So combining Lemma 2.8 with (3.14) and (3.15) we conclude that kxn −pk → 0.
This completes the proof.
As a direct corollary of Theorem 3.2, we obtain the following.
Corollary 3.2. Let C be a nonempty closed convex subset of a real
Hilbert space H, and T : C → C be a k-strict pseudo-contractive mapping
with F ix(T ) 6= ∅. Let {αn }, {βn } and {γn } be three real sequences in [0, 1]
satisfying the following conditions:
(C1) lim
P∞n→∞ αn = 0;
(C2) n=0 αn = ∞;
(C3) βn ∈ [a, b] ⊂ (k, 1).
For initial guess x0 ∈ C and fixed u ∈ C, then the sequence {xn } defined by
(3.8) converges strongly to p = PF ix(T ) u.
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.
Iterative methods for k-strict pseudo-contractive mappings ...
327
References
[1] K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation
of common fixed points of a countable family of nonexpansive mappings
in a Banach space, Nonlinear Anal., 67 (2007), 2350–2360.
[2] H. Bauschke, The approximation of fixed points of compositions of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 202 (1996),
150–159.
[3] F. E. Browder, Fixed point theorems for noncompact mappings in Hilbert
spaces, Proc. Natl. Acad. Sci. USA, 53 (1965), 1272–1276.
[4] F. E. Browder, Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. Ration. Mech. Anal., 24
(1967), 82–90.
[5] F. E. Browder and W. V. Petryshyn, Construction of fixed points of
nonlinear mappings in Hiblert space, J. Math. Anal. Appl., 20 (1967),
197–228.
[6] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103–120.
[7] S. S. Chang, Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 323 (2006),
1402–1416.
[8] C. E. Chidume and C. O. Chidume, Iterative approximation of fixed
points of nonexpansive mappings, J. Math. Anal. Appl., 318 (2006), 288–
295.
[9] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc.,
73 (1967), 957–961.
[10] J. S. Jung, Iterative approaches to common fixed points of nonexpansive
mappings in Banach spaces, J. Math. Anal. Appl., 302 (2005), 509–520.
[11] T. H. Kim and H. K. Xu, Strong convergence of modified Mann iterations,
Nonlinear Anal., 61 (2005), 51–60.
[12] T. H. Kim and H. K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal., 64 (2006), 1140–1152.
328
Yonghong Yao, Yeong-Cheng Liou, Shin Min Kang
[13] P. L. Lions, Approximation de points fixes de contractions, C. R. Acad.
Sci. Paris Ser. A-B 284 (1977), 1357–1359.
[14] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc.,
4 (1953), 506–510.
[15] G. Marino and H. K. Xu, Convergence of generalized proximal point
algorithms, Commun. Pure Appl. Anal., 3 (2004), 791–808.
[16] G. Marino and H. K. Xu, Weak and strong convergence theorems for strict
pseudocontractions in Hilbert spaces, J. Math. Anal. Appl., 329 (2007),
336–349.
[17] C. Matinez-Yanes and H. K. Xu, Strong convergence of the CQ method
for fixed point processes, Nonlinear Anal., 64 (2006), 2400–2411.
[18] S. Y. Matsushita and W. Takahashi, A strong convergence theorem for
relatively nonexpansive mappings in a Banach space, J. Approx. Theory
134 (2005), 257–266.
[19] C. H. Morales and J. S. Jung, Convergence of paths for pseudocontractive
mappings in Banach spaces, Proc. Amer. Math. Soc., 128 (2000), 3411–
3419.
[20] K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl.,
279 (2003), 372–379.
[21] J. G. O’Hara, P. Pillay and H. K. Xu, Iterative approaches to finding
nearest common fixed points of nonexpansive mappings in Hilbert spaces,
Nonlinear Anal., 54 (2003), 1417–1426.
[22] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274–276.
[23] O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, J. Math. Anal. Appl., 194
(1991), 911–933.
[24] N. Shioji and W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math.
Soc., 125 (1997), 3641–3645.
[25] T. Suzuki, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc., 135 (2007),
99–106.
Iterative methods for k-strict pseudo-contractive mappings ...
329
[26] R. Wittmann, Approximation of fixed points of nonexpansive mappings,
Arch. Math., 58 (1992), 486–491.
[27] D. P. Wu, S. S. Chang and G. X. Yuan, Approximation of common fixed
points for a family of finite nonexpansive mappings in Banach space,
Nonlinear Anal., 63 (2005), 987–999.
[28] H. K. Xu, Strong convergence of an iterative method for nonexpansive
mappings and accretive operators, J. Math. Anal. Appl., 314 (2006), 631–
643.
[29] H. K. Xu, Strong convergence of approximating fixed point sequences for
nonexpansive mappings, Bull. Austral. Math. Soc., 74 (2006), 143–151.
[30] Y. Yao, A general iterative method for a finite family of nonexpansive
mappings, Nonlinear Anal., 66 (2007), 2676–2687.
[31] Y. Yao and M. A. Noor, On viscosity iterative methods for variational
inequalities, J. Math. Anal. Appl., 325 (2007), 776–787.
[32] Y. Yao and J. C. Yao, On modified iterative method for nonexpansive
mappings and monotone mappings, Appl. Math. Comput., 186 (2007),
1551–1558.
[33] H. Zegeye and N. Shahzad, Viscosity approximation methods for a common fixed point of a family of quasi-nonexpansive mappings, Nonlinear
Anal., 68 (2008), 2005–2012.
[34] L. C. Zeng, N. C. Wong and J. C. Yao, Strong convergence theorems
for strictly pseudo-contractive mappings of Browder-Petryshyn type, Taiwanese J. Math., 10 (2006), 837–849.
[35] L.C. Zeng and J. C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings,
Nonlinear Anal., 64 (2006), 2507–2515.
[36] L. C. Zeng and J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,
Taiwanese J. Math., 10 (2006), 1293–1303.
330
Yonghong Yao, Yeong-Cheng Liou, Shin Min Kang
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
Department of Mathematics and the RINS
Gyeongsang National University
Jinju 660-701, Korea
E-mail: smkang@gnu.ac.kr