STRONG CONVERGENCE THEOREMS FOR STRICTLY ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN HILBERT SPACES | Saluja | Journal of the Indonesian Mathematical Society 29 101 1 PB
J. Indones. Math. Soc.
Vol. 16, No. 1 (2010), pp. 25–38.
STRONG CONVERGENCE THEOREMS FOR STRICTLY
ASYMPTOTICALLY PSEUDOCONTRACTIVE
MAPPINGS IN HILBERT SPACES
Gurucharan Singh Saluja1 and Hemant Kumar Nashine2
1
Department of Mathematics & Information Technology, Govt. Nagarjuna
P.G. College of Science,Raipur(Chhattisgarh),
India saluja 1963@rediffmail.com
2
Department of Mathematics, Disha Institute of Management and
Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud,
Raipur-492101(Chhattisgarh), India hnashine@rediffmail.com,
nashine 09@rediffmail.com
Abstract. We propose a new (CQ) algorithm for strictly asymptotically pseudocontractive mappings and obtain a strong convergence theorem for this class of
mappings in the framework of Hilbert spaces.
Key words and Phrases: CQ-iteration, strictly asymptotically pseudo-contractive
mapping, common fixed point, implicit iteration, strong convergence, Hilbert space.
Abstrak. Penulis mengajukan suatu algoritma baru (CQ) untuk pemetaan pseudocontractive yang asimtotik kuat dan memperoleh sebuah teorema konvergesi kuat
untuk kelas pemetaan ini pada kerangka kerja ruang Hilbert.
Key words and Phrases: Iterasi CQ, pemetaan pseudo-contractive yang asimtotik
kuat, titik tetap umum, iterasi implicit, konvergensi kuat, ruang Hilbert
1. Introduction and Preliminaries
Throughout this paper, let H be a real Hilbert space with the scalar product
and norm denoted by the symbols h., .i and k . k respectively. Let C be a closed
convex subset of H, we denote by PC (.) the metric projection from H onto C. It is
known that z = PC (x) is equivalent to hz − y, x − zi ≥ 0 for every y ∈ C. A point
x ∈ C is a fixed point of T provided that T x = x. Denote by F (T ) the set of fixed
point of T , that is, F (T ) = {x ∈ C : T x = x}. It is known that F (T ) is closed and
convex. Let T be a (possibly) nonlinear mapping from C into C. We now consider
the following classes:
2000 Mathematics Subject Classification: 47H09, 47H10.
Received: 20-02-2010, accepted: 30-07-2010.
25
G. S. Saluja and H. K. Nashine
26
(I) T is contractive, i.e., there exists a constant k < 1 such that
kT x − T yk ≤
k kx − yk ,
(1)
for all x, y ∈ C.
(II) T is nonexpansive, i.e.,
kT x − T yk ≤ kx − yk ,
(2)
for all x, y ∈ C.
(III) T is uniformly L-Lipschitzian, i.e., if there exists a constant L > 0 such
that
kT n x − T n yk ≤
L kx − yk ,
(3)
for all x, y ∈ C and n ∈ N.
(IV) T is pseudo-contractive, i.e.,
2
hT x − T y, j(x − y)i ≤ kx − yk ,
(4)
for all x, y ∈ C.
(V) T is strictly pseudo-contractive, i.e., there exists a constant k ∈ [0, 1) such
that
2
kT x − T yk
2
2
≤ kx − yk + k k(x − T x) − (y − T y)k ,
(5)
for all x, y ∈ C.
(VI) T is asymptotically nonexpansive [4], i.e., if there exists a sequence {rn } ⊂
[0, ∞) with limn→∞ rn = 0 such that
kT n x − T n yk ≤
(1 + rn ) kx − yk ,
(6)
for all x, y ∈ C and n ∈ N.
(VII) T is k-strictly asymptotically pseudo-contractive [14], i.e., if there exists a
sequence {rn } ⊂ [0, ∞) with limn→∞ rn = 0 such that
2
kT n x − T n yk
2
≤ (1 + rn )2 kx − yk
+k k(x − T n x) − (y − T n y)k
2
(7)
for some k ∈ [0, 1) for all x, y ∈ C and n ∈ N.
Remark 1.1. [14] . If T is k-strictly asymptotically pseudo-contractive mapping,
then it is uniformly L-Lipschitzian, but the converse does not hold. It can be shown
as:
Observe that if T is k-strictly asymptotically pseudo-contractive, taking 1 +
rn = an in (7) with limn→∞ an = 1 since limn→∞ rn = 0, then for all x, y ∈ C, we
have from (7) that
kT n x − T n yk
2
2
2
≤ a2n kx − yk + k k(x − T n x) − (y − T n y)k
√
= (an kx − yk)2 + ( k k(x − T n x) − (y − T n y)k)2
√
≤ (an kx − yk + k k(x − T n x) − (y − T n y)k)2 .
Strong Convergence Theorems for Strictly
27
Thus
kT n x − T n yk ≤
≤
an kx − yk +
an kx − yk +
√
√
k k(x − T n x) − (y − T n y)k
√
k kx − yk − k kT n x − T n yk ,
so that
√
an + k
√
kx − yk .
kT x − T yk ≤
1− k
Since {an } is bounded, then an ≤ a for all n ≥ 0 and for some a > 0. Hence
√
an + k
√ kx − yk = L kx − yk ,
kT n x − T n yk ≤
1− k
n
n
√
√k . This implies that a k-strictly asymptotically pseudo-contractive
where L = a+
1− k
mapping is uniformly L-Lipschitzian.
The class of strictly pseudo-contractive mappings have been studied by several
authors (see, for example [2, 5, 11, 16] and references therein.).
In case of contractive mapping, the Banach Contraction Principle guarantee
not only the existence of unique fixed point, but also to obtain the fixed point
by successive approximation (or Picard iteration). But for outside the class of
contractive mapping, the classical iteration scheme no longer applies. So some
other iteration scheme is required.
Two iteration processes are often used to approximate fixed point of nonexpansive and pseudo-contractive mappings. The first iteration process is known as
Mann’s iteration [12], where {xn } is defined as
xn+1
= αn xn + (1 − αn )T xn , n ≥ 0
(8)
αn xn + (1 − αn )T yn ,
βn xn + (1 − βn )T xn ; n ≥ 0
(9)
where the initial guess x0 is taken in C arbitrary and the sequence {αn } is in the
interval [0, 1].
The second iteration process is known as Ishikawa iteration process [6] which
is defined by
xn+1
yn
=
=
where the initial guess x0 is taken in C arbitrary and {αn } and {βn } are sequences
in the interval [0, 1].
Process (9) is indeed more general than the process (8). But research has been
concentrated on the later, probably due to the reason that process (8) is simpler
and that a convergence theorem for process (8) may possibly lead to a convergence
theorem for process (9), provided that the sequence {βn } satisfy certain appropriate
conditions.
If T is a nonexpansive
P∞ mapping with a fixed point and the control sequence
{αn } is chosen so that n=0 αn (1 − αn ) = ∞, then the sequence {xn } generated
by Mann’s iteration process (8) converges weakly to a fixed point of T (this is
G. S. Saluja and H. K. Nashine
28
also valid in a uniformly convex Banach space with the Fréchet differentiable norm
[18]). However we note that Mann’s iterations have only weak convergence even in
Hilbert space [3].
Attempts to modify the Mann iteration method (8) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [13] proposed
the following modification of Mann iteration method (8) for a single nonexpansive
mapping T in a Hilbert space H:
x0 ∈ C chosen arbitrary
,
yn
Cn
Qn
xn+1
=
=
=
=
αn xn + (1 − αn )T xn ,
{z ∈ C : kyn − zk ≤ kxn − zk},
{z ∈ C : hxn − z, x0 − xn i ≥ 0},
PCn ∩Qn (x0 ).
(10)
They proved that if the sequence {αn } is bounded above from one, then the sequence {xn } generated by (10) converges strongly to PF (T ) (x0 ).
Algorithm (10) is called a (CQ) algorithm for the Mann iteration method
because at each step the Mann iterate (denoted by yn in (10)) is used to construct
the sets Cn and Qn which are in turn used to construct the next iterate xn+1 .
In algorithm (10) the initial guess x0 is projected onto the intersection of two
suitably constructed closed convex subsets Cn and Qn .
In 2006, Kim and Xu [7] adapted the iteration (10) to asymptotically nonexpansive mappings. They introduced the following iteration process for asymptotically nonexpansive mappings in Hilbert space H:
x0 ∈ C chosen arbitrary,
yn
=
Cn
Qn
=
=
=
xn+1
αn xn + (1 − αn )T xn ,
2
2
{z ∈ C : kyn − zk ≤ kxn − zk + θn },
{z ∈ C : hxn − z, x0 − xn i ≥ 0},
PCn ∩Qn (x0 ),
(11)
where
θn = (1 − αn )((1 + rn )2 − 1)(diam C)2 → 0 as n → ∞.
Recently, Marino and Xu [11] extended (CQ) algorithm from nonexpansive
mappings to strict pseudo-contractive mappings.
It is important note that the set Cn in the (CQ) algorithm differs among
distinct classes of mappings.
In recent years, the implicit iteration scheme for approximating fixed points
of nonlinear mappings has been introduced and studied by several authors.
In 2001, Xu and Ori [20] have introduced the following implicit iteration
process for common fixed points of a finite family of nonexpansive mappings {Ti }N
i=1
Strong Convergence Theorems for Strictly
29
in Hilbert spaces:
tn xn−1 + (1 − tn )Tn xn , n ≥ 1
=
xn
(12)
where Tn = Tn mod N . (Here the mod N function takes values in {1, 2, . . . , N }).
And they proved the weak convergence of the process (12).
Very recently, Acedo and Xu [1] still in the framework of Hilbert spaces
introduced the following cyclic algorithm.
N −1
Let C be a closed convex subset of a Hilbert space H and let {Ti }i=0
be
TN −1
N k-strict pseudo-contractions on C such that F = i=0 F (Ti ) 6= ∅. Let x0 ∈ C
and let {αn } be a sequence in (0, 1). The cyclic algorithm generates a sequence
{xn }∞
n=1 in the following way:
x1
x2
xN
xN +1
=
=
..
.
=
=
..
.
α0 x0 + (1 − α0 )T0 x0 ,
α1 x1 + (1 − α1 )T1 x1 ,
αN −1 xN −1 + (1 − αN −1 )TN −1 xN −1 ,
αN xN + (1 − αN )T0 xN ,
In general, {xn+1 } is defined by
xn+1
=
αn xn + (1 − αn )T[n] xn ,
(13)
where T[n] = Ti with i = n (mod N ), 0 ≤ i ≤ N − 1. They also proved a weak
convergence theorem for k-strict pseudo-contractions in Hilbert spaces by cyclic
algorithm (13).
We note that it is the same as Mann’s iterations that have only weak convergence theorems with implicit iteration scheme (12) and (13). In this paper, we
introduce the following implicit iteration scheme and modify it by hybrid method,
so strong convergence theorems are obtained:
−1
Let C be a closed convex subset of a Hilbert space H and let {Ti }N
be N kTN −1i=0
strictly asymptotically pseudo-contractions on C such that F = i=0 F (Ti ) 6= ∅.
Let x0 ∈ C and let {αn } be a sequence in (0, 1). The implicit iteration scheme
generates a sequence {xn }∞
n=0 in the following way:
xn+1
=
s
αn xn + (1 − αn )T[n]
xn ,
(14)
s
where T[n]
= Tns (mod N ) = Tis with n = sN + i and i ∈ I = {0, 1, . . . , N − 1}.
Observe that if C is a nonempty closed convex subset of a real Hilbert space H
N −1
and {Ti }i=0
: C → C be N k-strictly asymptotically pseudo-contractive mappings.
If (1 − αn )L < 1, where L = max{Li : i = 0, 1, . . . , N − 1}, then for given xn ∈ C,
the mapping Wn : C → C defined by
Wn (x)
s
= αn xn + (1 − αn )T[n]
x, ∀n ≥ 1,
(15)
G. S. Saluja and H. K. Nashine
30
is a contraction mapping. In fact, the following are observed
°
°
°
°
s
s
x − (αn xn + (1 − αn )T[n]
y)°
kWn x − Wn yk = °αn xn + (1 − αn )T[n]
°
°
° s
s °
x − T[n]
y°
= (1 − αn ) °T[n]
≤
(1 − αn )L kx − yk , ∀x, y ∈ C.
(16)
Since (1 − αn )L < 1 for all n ≥ 1, hence Wn : C → C is a contraction mapping.
By Banach contraction mapping principle, there exists a unique fixed point xn ∈ C
such that
xn
s
= αn xn + (1 − αn )T[n]
x, ∀n ≥ 1.
(17)
Therefore, if (1 − αn )L < 1 for all n ≥ 1, then the iterative sequence (14) can
be employed for the approximation of common fixed points for a finite family of
k-strictly asymptotically pseudo-contractive mappings.
It is the purpose of this paper to modify iteration process (14) by hybrid
method as follows:
x0 ∈ C chosen arbitrary,
yn
=
s
αn xn + (1 − αn )T[n]
xn ,
Cn
=
{z ∈ C : kyn − zk ≤ kxn − zk
Qn
xn+1
=
=
2
2
°
°2
°
°
s
+(k − αn ) °xn − T[n]
xn ° + λn },
{z ∈ C : hxn − z, x0 − xn i ≥ 0},
PCn ∩Qn (x0 ),
s
where T[n]
= Tns (mod
N)
(18)
= Tis with n = sN + i and i ∈ I = {0, 1, . . . , N − 1} and
λn = (1 − αn )((1 + rn )2 − 1)(diam C)2 → 0 as n → ∞.
The purpose of this paper is to establish strong convergence theorems of
newly proposed (CQ) algorithm (18) for finite family of k-strictly asymptotically
pseudo-contractive mappings in Hilbert spaces. Our results extend the corresponding results of Liu [9], Kim and Xu [8], Osilike and Akuchu [15], Thakur [19] and
some others.
In the sequel, we will need the following lemmas.
Lemma 1.2. Let H be a real Hilbert space. There holds the following identities:
2
2
2
(i) kx − yk = kxk − kyk − 2hx − y, yi ∀ x, y ∈ H.
2
2
2
2
(ii) ktx + (1 − t)yk = t kxk + (1 − t) kyk − t(1 − t) kx − yk ,
∀ t ∈ [0, 1], ∀ x, y ∈ H.
(iii) If {xn } be a sequence in H weakly converges to z, then
2
2
2
lim sup kxn − yk = lim sup kxn − zk + kz − yk
n→∞
n→∞
∀y ∈ H.
Strong Convergence Theorems for Strictly
31
Lemma 1.3. Let H be a real Hilbert space. Given a closed convex subset C ⊂ H
and points x, y, z ∈ H. Given also a real number a ∈ R. The set
2
2
{v ∈ C : ky − vk ≤ kx − vk + hz, vi + a}
is convex (and closed).
Lemma 1.4. Let K be a closed convex subset of a real Hilbert space H. Given
x ∈ H and y ∈ K. Then z = PK x if and only if there holds the relation
hx − z, y − zi ≤ 0 ∀y ∈ K,
where PK is the nearest point projection from H onto K, that is, PK x is the unique
point in K with the property
kx − PK xk ≤ kx − yk ∀x ∈ K.
We use following notation:
1. ⇀ for weak convergence and → for strong convergence.
2. ωw (xn ) = {x : ∃ xnj ⇀ x} denotes the weak ω-limit set of {xn }.
Lemma 1.5. [10]. Let K be a closed convex subset of H. Let {xn } be a sequence
in H and u ∈ H. Let q = PK u. If {xn } is such that ωw (xn ) ⊂ K and satisfies the
condition
kxn − uk =
ku − qk ∀n.
(19)
Then xn → q.
∞
∞
Lemma 1.6. [17]. Let {an }∞
n=1 , {βn }n=1 and {rn }n=1 be sequences of nonnegative
real numbers satisfying the inequality
an+1 ≤ (1 + rn )an + βn , n ≥ 1.
P∞
If n=1 rn < ∞ and n=1 βn < ∞, then limn→∞ an exists. If in addition
{an }∞
n=1 has a subsequence which converges strongly to zero, then limn→∞ an = 0.
P∞
Lemma 1.7. Let H be a real Hilbert space, let C be a nonempty closed convex subset
of H, and let Ti : C → C be a ki -strictly asymptotically pseudocontractive
P∞ mapping
for i = 0, 1, . . . , N − 1 with a sequence {rni } ⊂ [0, ∞) such that n=1 rni < ∞
and for some 0 ≤ ki < 1, then there exist constants L > 0 and k ∈ [0, 1) and a
sequence {rn } ⊂ [0, ∞) with limn→∞ rn = 0 such that for any x, y ∈ C and for
each i = 0, 1, . . . , N − 1 and each n ≥ 1, the following hold:
kTin x − Tin yk ≤
2
(1 + rn )2 kx − yk
2
+k k(x − Tin x) − (y − Tin y)k ,
(20)
and
kTin x − Tin yk ≤
L kx − yk .
(21)
Proof. Since for each i = 0, 1, . . . , N − 1, Ti is ki -strictly asymptotically pseudocontractive, where ki ∈ [0, 1) and {rni } ⊂ [0, ∞) with limn→∞ rni = 0. By
G. S. Saluja and H. K. Nashine
32
Remark 1.1, Ti is Li -Lipschitzian. Taking rn = max{rni , i = 0, 1, . . . , N − 1} and
k = max{ki , i = 0, 1, . . . , N − 1}, hence, for each i = 0, 1, . . . , N − 1, we have
kTin x − Tin yk ≤
(1 + rni )2 kx − yk
2
2
+ki k(x − Tin x) − (y − Tin y)k ,
≤ (1 + rn )2 kx − yk
2
2
+k k(x − Tin x) − (y − Tin y)k .
(22)
The conclusion (20) is proved. Again taking L = max{Li : i = 0, 1, . . . , N − 1} for
any x, y ∈ C, we have
kTin x − Tin yk ≤
Li kx − yk ≤ L kx − yk .
(23)
This completes the proof of lemma.
¤
2. Main Results
Theorem 2.1. Let C be a closed convex subset of a Hilbert space H. Let N ≥ 1 be
an integer. Let for each 0 ≤ i ≤ N − 1, Ti : C → C be N P
ki -strictly asymptotically
∞
pseudo-contraction mappings for some 0 ≤ ki < 1 and n=1 rn < ∞. Let k =
max{ki : 0 ≤ i ≤ N − 1} and rn = max{rni : 0 ≤ i ≤ N − 1}. Assume that
TN −1
F = i=0 F (Ti ) 6= ∅. Given x0 ∈ C, let {xn }∞
n=0 be the sequence generated
by an implicit iteration scheme (14). Assume that the control sequence {αn } is
chosen so
° that k + ǫ° < αn < 1 − ǫ for all n and for some ǫ ∈ (0, 1). Then
limn→∞ °xn − T[l] xn ° = 0 for all l ∈ I = {0, 1, . . . , N − 1}.
Proof. Let p ∈ F =
2
kxn+1 − pk
TN −1
i=0
=
=
=
≤
F (Ti ). It follows from (14) and Lemma 1.2 (ii) that
°
°2
°
°
s
x n − p°
°αn xn + (1 − αn )T[n]
°
°2
°
°
s
xn − p)°
°αn (xn − p) + (1 − αn )(T[n]
°2
°
°
° s
2
αn kxn − pk + (1 − αn ) °T[n]
xn − p°
°
°2
°
°
s
−αn (1 − αn ) °xn − T[n]
xn °
h
2
2
αn kxn − pk + (1 − αn ) (1 + rn )2 kxn − pk
°
°2
°
°2 i
°
°
°
°
s
s
xn °
+k °xn − T[n]
xn ° − αn (1 − αn ) °xn − T[n]
(24)
(25)
Strong Convergence Theorems for Strictly
33
h
i
2
αn (1 + rn )2 + (1 − αn )(1 + rn )2 kxn − pk
°
°2
°
°
s
−(αn − k)(1 − αn ) °xn − T[n]
xn °
°
°2
°
°
2
s
xn °
= (1 + rn )2 kxn − pk − (αn − k)(1 − αn ) °xn − T[n]
°
°2
°
°
2
s
xn °
= (1 + dn ) kxn − pk − (αn − k)(1 − αn ) °xn − T[n]
≤
where dn = rn2 + 2rn . Since k + ǫ < αn < 1 − ǫ for all n, from (24) we have
°
°2
°
°
2
2
s
xn ° .
kxn+1 − pk ≤ (1 + dn ) kxn − pk − ǫ2 °xn − T[n]
(26)
Now (26) implies that
2
2
(27)
kxn+1 − pk ≤ (1 + dn ) kxn − pk .
P∞
P∞
Since n=1 rn < ∞ thus n=1 dn < ∞, it follows by Lemma 1.2, we know that
limn→∞ kxn − pk exists and so {xn } is bounded. Consider (26) again yields that
°
°
dn
1
°
°2
2
2
2
s
[kxn − pk − kxn+1 − pk ] + 2 kxn − pk .
(28)
xn ° ≤
°xn − T[n]
ǫ2
ǫ
Since {xn } is bounded and dn → 0 as n → ∞. So, we get
°
°
°
°
s
xn ° → 0 as n → ∞.
°xn − T[n]
(29)
From the definition of {xn }, we have
°
°
°
°
s
kxn+1 − xn k = (1 − αn ) °xn − T[n]
xn ° → 0, as n → ∞.
(30)
So, kxn − xn+l k → 0 as n → ∞ and for all l < N . Now for n ≥ N , and since T is
uniformly Lipschitzian (by Remark 1.1) with Lipschitz constant L > 0, so we have
°
°
° °
°
°
°
° ° s
s
°xn − T[n] xn ° ≤ °
xn − T[n] xn °
xn ° + °T[n]
°xn − T[n]
°
°
°
°
°
°
°
° s−1
s
≤ °xn − T[n]
xn ° + L °T[n]
xn − xn °
°
°
°
h°
°
°
° s−1
°
s−1
s
≤ °xn − T[n]
xn ° + L °T[n]
xn − T[n−N
x
°
n−N
]
°
°
i
°
° s−1
(31)
+ °T[n−N
] xn−N − xn−N ° + kxn−N − xn k .
Since for each n ≥ N , n ≡
(31), we have
°
°
°xn − T[n] xn ° ≤
(n − N ) (mod N ). Thus T[n] = T[n−N ] , therefore from
°
°
°
°
s
xn ° + L2 kxn − xn−N k
°xn − T[n]
°
°
° s−1
°
+L °T[n−N
] xn−N − xn−N ° + L kxn−N − xn k .
(32)
G. S. Saluja and H. K. Nashine
34
From (29) and (32), we obtain
°
°
°xn − T[n] xn ° → 0 as n → ∞.
Consequently, for any l ∈ I = {0, 1, . . . , N − 1},
°
°
°
°
°xn − T[n+l] xn ° ≤ kxn − xn+l k + °xn+l − T[n+l] xn+l °
°
°
+ °T[n+l] xn+l − T[n+l] xn °
°
°
≤ (1 + L) kxn − xn+l k + °xn+l − T[n+l] xn+l °
(33)
→ 0 as n → ∞.
(34)
°
°
lim °xn − T[l] xn ° = 0, ∀ l ∈ I = {0, 1, . . . , N − 1}.
(35)
This implies that
n→∞
This completes the proof.
¤
Theorem 2.2. Let C be a closed convex compact subset of a Hilbert space H. Let
N ≥ 1 be an integer. Let for each 0 ≤ i ≤ N − 1, Ti : C → C be NPki -strictly
∞
asymptotically pseudo-contraction mappings for some 0 ≤ ki < 1 and n=1 rn <
∞. Let k = max{ki : 0 ≤ i ≤ N − 1} and rn = max{rni : 0 ≤ i ≤ N − 1}. Assume
TN −1
that F = i=0 F (Ti ) 6= ∅. Given x0 ∈ C, let {xn }∞
n=0 be the sequence generated by
an implicit iteration scheme (14). Assume that the control sequence {αn } is chosen
so that k + ǫ < αn < 1 − ǫ for all n and for some ǫ ∈ (0, 1). Then {xn } converges
N −1
strongly to a common fixed point of the family {Ti }i=0
.
Proof. We only conclude the difference. By compactness of C this immediately
implies that there is a subsequence {xnj } of {xn } which converges to a com−1
mon fixed point of {Ti }N
i=0 , say, p. Combining (27) with Lemma 1.6, we have
limn→∞ kxn − pk = 0. This completes the proof.
¤
Remark 2.3. Theorem 2.2 extends and improves the corresponding result of Liu [9]
in the following ways:
(i) We removed the uniformly L-Lipschitzian condition.
(ii) The modified Mann iteration process is replaced by implicit iteration process for a finite family of mappings.
For our next result, we shall need the following definition:
Definition 2.4. A mapping T : C → C is said to be semi-compact, if for any
bounded sequence {xn } in C such that limn→∞ kxn − T xn k = 0 there exists a subsequence {xni } ⊂ {xn } such that limi→∞ xni = x ∈ C.
Theorem 2.5. Let C be a closed convex subset of a Hilbert space H. Let N ≥ 1 be
an integer. Let for each 0 ≤ i ≤ N − 1, Ti : C → C be N P
ki -strictly asymptotically
∞
pseudo-contraction mappings for some 0 ≤ ki < 1 and n=1 rn < ∞. Let k =
max{ki : 0 ≤ i ≤ N − 1} and rn = max{rni : 0 ≤ i ≤ N − 1}. Assume that
TN −1
F = i=0 F (Ti ) 6= ∅. Given x0 ∈ C, let {xn }∞
n=0 be the sequence generated by an
implicit iteration scheme (14). Assume that the control sequence {αn } is chosen
Strong Convergence Theorems for Strictly
35
so that k + ǫ < αn < 1 − ǫ for all n and for some ǫ ∈ (0, 1). Assume that one
N −1
member of the family {Ti }i=0
be semi-compact. Then {xn } converges strongly to
N −1
a common fixed point of the family {Ti }i=0
.
Proof. Without loss of generality, we can assume that T1 is semi-compact. It follows
from (35) that
°
°
(36)
lim °xn − T[1] xn ° = 0.
n→∞
By the semi-compactness of T1 , there exists a subsequence {xnk } of {xn } such that
xnk → u ∈ C strongly. Since C is closed, u ∈ C, and furthermore,
°
°
°
°
(37)
lim °xnk − T[l] xnk ° = °u − T[l] u° = 0,
nk →∞
for all l ∈ I = {0, 1, . . . , N − 1}. Thus u ∈ F . Since {xnk } converges strongly
to u and limn→∞ kxn − uk exists, it follows from Lemma 1.2 that {xn } converges
strongly to u. This completes the proof.
¤
Remark 2.6. Theorem 2.5 extends and improves the corresponding result of Kim
and Xu [8].
Remark 2.7. Theorem 2.5 also extends and improves Theorem 1.6 of Osilike and
Akuchu [15] from asymptotically pseudocontractive mappings to strictly asymptotically pseudocontractive mappings.
We now prove strong convergence of k-strictly asymptotically pseudo-contractive
N −1
mappings {Ti }i=0
using algorithm (18):
Theorem 2.8. Let C be a closed convex subset of a Hilbert space H. Let N ≥ 1 be
an integer. Let for each 0 ≤ i ≤ N − 1, Ti : C → C be
N ki -strictly asymptotically
P∞
pseudo-contraction mappings for some 0 ≤ ki < 1, n=1 rn < ∞ and I − T[n] is
demiclosed at zero. Let k = max{ki : 0 ≤ i ≤ N − 1} and rn = max{rni : 0 ≤
TN −1
i ≤ N − 1}. Assume that F = i=0 F (Ti ) 6= ∅. Let {xn }∞
n=0 be the sequence
generated by an the algorithm (18). Assume that the sequence {αn } is chosen so
that supn≥0 αn < 1. Then {xn } converges strongly to PF (x0 ).
Proof. By Lemma 1.3, we observe that Cn is convex.
Now, for all p ∈ F , using Lemma 1.2(ii), we have
2
kyn − pk
=
=
=
°
°2
°
°
s
x n − p°
°αn xn + (1 − αn )T[n]
°2
°
°
°
s
xn − p)°
°αn (xn − p) + (1 − αn )(T[n]
°
°2
°
° s
2
αn kxn − pk + (1 − αn ) °T[n]
xn − p°
°
°2
°
°
s
−αn (1 − αn ) °xn − T[n]
xn °
(38)
G. S. Saluja and H. K. Nashine
36
≤
≤
=
≤
≤
h
2
2
αn kxn − pk + (1 − αn ) (1 + rn )2 kxn − pk
°
°
°2 i
°2
°
°
°
°
s
s
+k °xn − T[n]
xn ° − αn (1 − αn ) °xn − T[n]
xn °
h
i
2
αn (1 + rn )2 + (1 − αn )(1 + rn )2 kxn − pk
°
°2
°
°
s
xn °
−(αn − k)(1 − αn ) °xn − T[n]
°
°2
°
°
2
s
(1 + rn )2 kxn − pk − (αn − k)(1 − αn ) °xn − T[n]
xn °
°
°2
°
°
2
s
(1 + rn )2 kxn − pk + (k − αn )(1 − αn ) °xn − T[n]
xn °
°
°2
°
°
2
s
xn ° + λn ,
kxn − pk + (k − αn ) °xn − T[n]
(39)
so p ∈ Cn for all n. Thus F ⊂ Cn for all n.
Next we show that F ⊂ Qn for all n ≥ 0, for this we use induction.
For n = 0, we have F ⊂ C = Q0 . Assume that F ⊂ Qn .
Since xn+1 is the projection of x0 onto Cn ∩ Qn , by Lemma 1.4, we have
hxn+1 − z, x0 − xn+1 i ≥ 0 ∀z ∈ Cn ∩ Qn .
As F ⊂ Cn ∩ Qn by the induction assumption, the last inequality holds, in particular, for all z ∈ F . This together with the definition of Qn+1 implies that F ⊂ Qn+1 .
Hence F ⊂ Qn for all n ≥ 0.
Now, since xn = PQn (x0 ) (by the definition of Qn ), and since F ⊂ Qn , we
have
kxn − x0 k ≤ kp − x0 k ∀p ∈ F.
In particular, {xn } is bounded and
kxn − x0 k ≤ kq − x0 k , where q = PF (x0 ).
(40)
The fact xn+1 ∈ Qn asserts that hxn+1 − xn , xn − x0 i ≥ 0. This together with
Lemma 1.2(i), implies that
2
kxn+1 − xn k
=
=
2
k(xn+1 − x0 ) − (xn − x0 )k
2
2
2
2
kxn+1 − x0 k − kxn − x0 k − 2hxn+1 − xn , xn − x0 i
≤ kxn+1 − x0 k − kxn − x0 k .
It follows that,
kxn+1 − xn k → 0.
By the fact xn+1 ∈ Cn we get
kxn+1 − yn k
2
(41)
2
≤ kxn+1 − xn k
°
°2
°
°
s
+(k − αn ) °xn − T[n]
xn ° + λn .
(42)
Strong Convergence Theorems for Strictly
37
s
Moreover, since yn = αn xn + (1 − αn )T[n]
xn , we deduce that
kxn+1 − yn k
2
=
Substituting (43) into (42) to get
2
αn kxn+1 − xn k
°
°2
°
°
s
+(1 − αn ) °xn+1 − T[n]
xn °
°
°2
°
°
s
−αn (1 − αn ) °xn − T[n]
xn ° .
°
°2
°
°
s
(1 − αn ) °xn+1 − T[n]
xn °
(43)
2
≤ (1 − αn ) kxn+1 − xn k
°
°2
°
°
s
xn ° + λn .
+k °xn − T[n]
Since αn < 1 for all n, the last inequality becomes,
°
°
°2
°2
°
°
°
°
2
s
s
xn °
xn ° ≤ kxn+1 − xn k + k °xn − T[n]
°xn+1 − T[n]
λn
,
ρ
for some positive number ρ > 0, such that αn < ρ < 1.
But on the other hand, we compute
°
°2
°
°
2
s
s
xn ° = kxn+1 − xn k + 2hxn+1 − xn , xn − T[n]
xn i
°xn+1 − T[n]
°
°2
°
°
s
xn ° .
+ °xn − T[n]
+
By (44) and (45), we get
°
°2
°
°
s
(1 − k) °xn − T[n]
xn °
Therefore
°
°2
°
°
s
xn °
°xn − T[n]
Now,
°
°
°xn − T[n] xn °
≤
≤
λn
s
− 2hxn+1 − xn , xn − T[n]
xn i.
ρ
2
λn
s
−
hxn+1 − xn , xn − T[n]
xn i
ρ(1 − k) 1 − k
→ 0 as n → ∞.
° °
°
°
°
° ° s
°
s
xn ° + °T[n]
≤ °xn − T[n]
xn − T[n] xn °
°
°
°
°
°
°
° s−1
°
s
≤ °xn − T[n]
xn ° + (1 + r1 ) °T[n]
xn − xn °
→ 0 as n → ∞.
(44)
(45)
(46)
(47)
(48)
Now, since I − T[n] is demiclosed at zero, (48) imply that xn ⇀ x, where x is a
weak limit of {xn } and hence ωw (xn ) ⊂ F (Ti ) for any i = 0, 1, . . . , N − 1. So,
TN −1
ωw (xn ) ⊂ F = i=0 . This fact, the inequality (40) and Lemma 1.5 implies that
{xn } → q = PF (x0 ), that is, {xn } converges strongly to PF (x0 ). This completes
the proof.
¤
38
G. S. Saluja and H. K. Nashine
Remark 2.9. Theorem 2.8 extends Theorem 3.1 of Thakur [19] to the case of finite
family of mappings and implicit iteration process considered in this paper.
Acknowledgement. Our sincere thanks to the referee for his valuable suggestions
and comments on the manuscript and also thanks to Prof. A. Banerjee for providing
a manuscript of Osilike [14].
References
[1] Acedo, G. L. and Xu, H. K., ”Iterative methods for strict pseudo-contractions in Hilbert
spaces”, Nonlinear Anal. 67 (2007), 2258 - 2271.
[2] Browder, F. E. and Petryshyn, W. V., ”Construction of fixed points of nonlinear mappings
in Hilbert spaces”, J. Math. Anal. Appl. 20 (1967), 197 - 228.
[3] Genel, A. and Lindenstrauss, J., ”An example concerning fixed points”, Israel J. Math. 22(1)
(1975), 81 - 86.
[4] Goebel, K. and Kirk, W. A., ”A fixed point theorem for asymptotically nonexpansive mappings”, Proc. Amer. Math. Soc. 35 (1972), 171 - 174.
[5] Hicks, T. L. and Kubicek, J. R., ”On the Mann iterative process in Hilbert space”, J. Math.
Anal. Appl. 59 (1977), 498 - 504.
[6] Ishikawa, S., ”Fixed points by a new iteration method”, Proc. Amer. Math. Soc. 44 (1974),
147 - 150.
[7] Kim, T. H. and Xu, H. K., ”Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups”, Nonlinear Anal. 64 (2006), 1140 - 1152.
[8] Kim, T. H. and Xu, H. K., ”Convergence of the modified Mann’s iteration
method for asymptotically strictly pseudocontractive mapping”, Nonlinear Anal. (2007),
doi:10.1016/j.na.2007.02.029.
[9] Liu, Q., ”Convergence theorems of the sequence of iterates for asymptotically demicontractive
and hemicontractive mappings”, Nonlinear Anal. 26 (1996), 1835 - 1842.
[10] Martinez-Yanex, C. and Xu, H. K., ”Strong convergence of the CQ method for fixed point
processes”, Nonlinear Anal. 64 (2006), 2400 - 2411.
[11] Marino, G. and Xu, H. K., ”Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces”, J. Math. Anal. Appl. 329 (2007), 336 - 346.
[12] Mann, W. R., ”Mean value methods in iteration”, Proc. Amer. Math. Soc. 4 (1953), 506 510.
[13] Nakajo, K. and Takahashi, W., ”Strong convergence theorems for nonexpansive mappings
and nonexpansive semigroups”, J. Math. Anal. Appl. 279 (2003), 372 - 379.
[14] Osilike, M. O., ”Iterative approximation of fixed points of asymptotically demicontractive
mappings”, Indian J. pure appl. Math. 29(12) (1998), 1291 - 1300.
[15] Osilike, M. O. and Akuchu, B. G., ”Common fixed points of a finite family of asymptotically
pseudocontractive maps”, Fixed Point Theory and Appl. 2(2004), 81 - 88.
[16] Osilike, M. O. and Udomene, A., ”Demiclosedness principle and convergence results for
strictly pseudocontractive mappings of Browder-Petryshyn type”, J. Math. Anal. Appl. 256
(2001), 431 - 445.
[17] Osilike, M. O., Aniagbosor, S. C. and Akuchu, B. G., ”Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces”, PanAm. Math. J. 12 (2002), 77 - 78.
[18] Reich, S., ”Weak convergence theorems for nonexpansive mappings in Banach spaces”, J.
Math. Anal. Appl. 67 (1979), 274 - 276.
[19] Thakur, B. S., ”Convergence of strictly asymptotically pseudo-contractions”, Thai J. Math.
5(1) (2007), 41 - 52.
[20] Xu, H. K. and Ori, R. G., ”An implicit iteration process for nonexpansive mappings”, Numer.
Funct. Anal. Optim. 22 (2001), 767 - 773.
Vol. 16, No. 1 (2010), pp. 25–38.
STRONG CONVERGENCE THEOREMS FOR STRICTLY
ASYMPTOTICALLY PSEUDOCONTRACTIVE
MAPPINGS IN HILBERT SPACES
Gurucharan Singh Saluja1 and Hemant Kumar Nashine2
1
Department of Mathematics & Information Technology, Govt. Nagarjuna
P.G. College of Science,Raipur(Chhattisgarh),
India saluja 1963@rediffmail.com
2
Department of Mathematics, Disha Institute of Management and
Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud,
Raipur-492101(Chhattisgarh), India hnashine@rediffmail.com,
nashine 09@rediffmail.com
Abstract. We propose a new (CQ) algorithm for strictly asymptotically pseudocontractive mappings and obtain a strong convergence theorem for this class of
mappings in the framework of Hilbert spaces.
Key words and Phrases: CQ-iteration, strictly asymptotically pseudo-contractive
mapping, common fixed point, implicit iteration, strong convergence, Hilbert space.
Abstrak. Penulis mengajukan suatu algoritma baru (CQ) untuk pemetaan pseudocontractive yang asimtotik kuat dan memperoleh sebuah teorema konvergesi kuat
untuk kelas pemetaan ini pada kerangka kerja ruang Hilbert.
Key words and Phrases: Iterasi CQ, pemetaan pseudo-contractive yang asimtotik
kuat, titik tetap umum, iterasi implicit, konvergensi kuat, ruang Hilbert
1. Introduction and Preliminaries
Throughout this paper, let H be a real Hilbert space with the scalar product
and norm denoted by the symbols h., .i and k . k respectively. Let C be a closed
convex subset of H, we denote by PC (.) the metric projection from H onto C. It is
known that z = PC (x) is equivalent to hz − y, x − zi ≥ 0 for every y ∈ C. A point
x ∈ C is a fixed point of T provided that T x = x. Denote by F (T ) the set of fixed
point of T , that is, F (T ) = {x ∈ C : T x = x}. It is known that F (T ) is closed and
convex. Let T be a (possibly) nonlinear mapping from C into C. We now consider
the following classes:
2000 Mathematics Subject Classification: 47H09, 47H10.
Received: 20-02-2010, accepted: 30-07-2010.
25
G. S. Saluja and H. K. Nashine
26
(I) T is contractive, i.e., there exists a constant k < 1 such that
kT x − T yk ≤
k kx − yk ,
(1)
for all x, y ∈ C.
(II) T is nonexpansive, i.e.,
kT x − T yk ≤ kx − yk ,
(2)
for all x, y ∈ C.
(III) T is uniformly L-Lipschitzian, i.e., if there exists a constant L > 0 such
that
kT n x − T n yk ≤
L kx − yk ,
(3)
for all x, y ∈ C and n ∈ N.
(IV) T is pseudo-contractive, i.e.,
2
hT x − T y, j(x − y)i ≤ kx − yk ,
(4)
for all x, y ∈ C.
(V) T is strictly pseudo-contractive, i.e., there exists a constant k ∈ [0, 1) such
that
2
kT x − T yk
2
2
≤ kx − yk + k k(x − T x) − (y − T y)k ,
(5)
for all x, y ∈ C.
(VI) T is asymptotically nonexpansive [4], i.e., if there exists a sequence {rn } ⊂
[0, ∞) with limn→∞ rn = 0 such that
kT n x − T n yk ≤
(1 + rn ) kx − yk ,
(6)
for all x, y ∈ C and n ∈ N.
(VII) T is k-strictly asymptotically pseudo-contractive [14], i.e., if there exists a
sequence {rn } ⊂ [0, ∞) with limn→∞ rn = 0 such that
2
kT n x − T n yk
2
≤ (1 + rn )2 kx − yk
+k k(x − T n x) − (y − T n y)k
2
(7)
for some k ∈ [0, 1) for all x, y ∈ C and n ∈ N.
Remark 1.1. [14] . If T is k-strictly asymptotically pseudo-contractive mapping,
then it is uniformly L-Lipschitzian, but the converse does not hold. It can be shown
as:
Observe that if T is k-strictly asymptotically pseudo-contractive, taking 1 +
rn = an in (7) with limn→∞ an = 1 since limn→∞ rn = 0, then for all x, y ∈ C, we
have from (7) that
kT n x − T n yk
2
2
2
≤ a2n kx − yk + k k(x − T n x) − (y − T n y)k
√
= (an kx − yk)2 + ( k k(x − T n x) − (y − T n y)k)2
√
≤ (an kx − yk + k k(x − T n x) − (y − T n y)k)2 .
Strong Convergence Theorems for Strictly
27
Thus
kT n x − T n yk ≤
≤
an kx − yk +
an kx − yk +
√
√
k k(x − T n x) − (y − T n y)k
√
k kx − yk − k kT n x − T n yk ,
so that
√
an + k
√
kx − yk .
kT x − T yk ≤
1− k
Since {an } is bounded, then an ≤ a for all n ≥ 0 and for some a > 0. Hence
√
an + k
√ kx − yk = L kx − yk ,
kT n x − T n yk ≤
1− k
n
n
√
√k . This implies that a k-strictly asymptotically pseudo-contractive
where L = a+
1− k
mapping is uniformly L-Lipschitzian.
The class of strictly pseudo-contractive mappings have been studied by several
authors (see, for example [2, 5, 11, 16] and references therein.).
In case of contractive mapping, the Banach Contraction Principle guarantee
not only the existence of unique fixed point, but also to obtain the fixed point
by successive approximation (or Picard iteration). But for outside the class of
contractive mapping, the classical iteration scheme no longer applies. So some
other iteration scheme is required.
Two iteration processes are often used to approximate fixed point of nonexpansive and pseudo-contractive mappings. The first iteration process is known as
Mann’s iteration [12], where {xn } is defined as
xn+1
= αn xn + (1 − αn )T xn , n ≥ 0
(8)
αn xn + (1 − αn )T yn ,
βn xn + (1 − βn )T xn ; n ≥ 0
(9)
where the initial guess x0 is taken in C arbitrary and the sequence {αn } is in the
interval [0, 1].
The second iteration process is known as Ishikawa iteration process [6] which
is defined by
xn+1
yn
=
=
where the initial guess x0 is taken in C arbitrary and {αn } and {βn } are sequences
in the interval [0, 1].
Process (9) is indeed more general than the process (8). But research has been
concentrated on the later, probably due to the reason that process (8) is simpler
and that a convergence theorem for process (8) may possibly lead to a convergence
theorem for process (9), provided that the sequence {βn } satisfy certain appropriate
conditions.
If T is a nonexpansive
P∞ mapping with a fixed point and the control sequence
{αn } is chosen so that n=0 αn (1 − αn ) = ∞, then the sequence {xn } generated
by Mann’s iteration process (8) converges weakly to a fixed point of T (this is
G. S. Saluja and H. K. Nashine
28
also valid in a uniformly convex Banach space with the Fréchet differentiable norm
[18]). However we note that Mann’s iterations have only weak convergence even in
Hilbert space [3].
Attempts to modify the Mann iteration method (8) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [13] proposed
the following modification of Mann iteration method (8) for a single nonexpansive
mapping T in a Hilbert space H:
x0 ∈ C chosen arbitrary
,
yn
Cn
Qn
xn+1
=
=
=
=
αn xn + (1 − αn )T xn ,
{z ∈ C : kyn − zk ≤ kxn − zk},
{z ∈ C : hxn − z, x0 − xn i ≥ 0},
PCn ∩Qn (x0 ).
(10)
They proved that if the sequence {αn } is bounded above from one, then the sequence {xn } generated by (10) converges strongly to PF (T ) (x0 ).
Algorithm (10) is called a (CQ) algorithm for the Mann iteration method
because at each step the Mann iterate (denoted by yn in (10)) is used to construct
the sets Cn and Qn which are in turn used to construct the next iterate xn+1 .
In algorithm (10) the initial guess x0 is projected onto the intersection of two
suitably constructed closed convex subsets Cn and Qn .
In 2006, Kim and Xu [7] adapted the iteration (10) to asymptotically nonexpansive mappings. They introduced the following iteration process for asymptotically nonexpansive mappings in Hilbert space H:
x0 ∈ C chosen arbitrary,
yn
=
Cn
Qn
=
=
=
xn+1
αn xn + (1 − αn )T xn ,
2
2
{z ∈ C : kyn − zk ≤ kxn − zk + θn },
{z ∈ C : hxn − z, x0 − xn i ≥ 0},
PCn ∩Qn (x0 ),
(11)
where
θn = (1 − αn )((1 + rn )2 − 1)(diam C)2 → 0 as n → ∞.
Recently, Marino and Xu [11] extended (CQ) algorithm from nonexpansive
mappings to strict pseudo-contractive mappings.
It is important note that the set Cn in the (CQ) algorithm differs among
distinct classes of mappings.
In recent years, the implicit iteration scheme for approximating fixed points
of nonlinear mappings has been introduced and studied by several authors.
In 2001, Xu and Ori [20] have introduced the following implicit iteration
process for common fixed points of a finite family of nonexpansive mappings {Ti }N
i=1
Strong Convergence Theorems for Strictly
29
in Hilbert spaces:
tn xn−1 + (1 − tn )Tn xn , n ≥ 1
=
xn
(12)
where Tn = Tn mod N . (Here the mod N function takes values in {1, 2, . . . , N }).
And they proved the weak convergence of the process (12).
Very recently, Acedo and Xu [1] still in the framework of Hilbert spaces
introduced the following cyclic algorithm.
N −1
Let C be a closed convex subset of a Hilbert space H and let {Ti }i=0
be
TN −1
N k-strict pseudo-contractions on C such that F = i=0 F (Ti ) 6= ∅. Let x0 ∈ C
and let {αn } be a sequence in (0, 1). The cyclic algorithm generates a sequence
{xn }∞
n=1 in the following way:
x1
x2
xN
xN +1
=
=
..
.
=
=
..
.
α0 x0 + (1 − α0 )T0 x0 ,
α1 x1 + (1 − α1 )T1 x1 ,
αN −1 xN −1 + (1 − αN −1 )TN −1 xN −1 ,
αN xN + (1 − αN )T0 xN ,
In general, {xn+1 } is defined by
xn+1
=
αn xn + (1 − αn )T[n] xn ,
(13)
where T[n] = Ti with i = n (mod N ), 0 ≤ i ≤ N − 1. They also proved a weak
convergence theorem for k-strict pseudo-contractions in Hilbert spaces by cyclic
algorithm (13).
We note that it is the same as Mann’s iterations that have only weak convergence theorems with implicit iteration scheme (12) and (13). In this paper, we
introduce the following implicit iteration scheme and modify it by hybrid method,
so strong convergence theorems are obtained:
−1
Let C be a closed convex subset of a Hilbert space H and let {Ti }N
be N kTN −1i=0
strictly asymptotically pseudo-contractions on C such that F = i=0 F (Ti ) 6= ∅.
Let x0 ∈ C and let {αn } be a sequence in (0, 1). The implicit iteration scheme
generates a sequence {xn }∞
n=0 in the following way:
xn+1
=
s
αn xn + (1 − αn )T[n]
xn ,
(14)
s
where T[n]
= Tns (mod N ) = Tis with n = sN + i and i ∈ I = {0, 1, . . . , N − 1}.
Observe that if C is a nonempty closed convex subset of a real Hilbert space H
N −1
and {Ti }i=0
: C → C be N k-strictly asymptotically pseudo-contractive mappings.
If (1 − αn )L < 1, where L = max{Li : i = 0, 1, . . . , N − 1}, then for given xn ∈ C,
the mapping Wn : C → C defined by
Wn (x)
s
= αn xn + (1 − αn )T[n]
x, ∀n ≥ 1,
(15)
G. S. Saluja and H. K. Nashine
30
is a contraction mapping. In fact, the following are observed
°
°
°
°
s
s
x − (αn xn + (1 − αn )T[n]
y)°
kWn x − Wn yk = °αn xn + (1 − αn )T[n]
°
°
° s
s °
x − T[n]
y°
= (1 − αn ) °T[n]
≤
(1 − αn )L kx − yk , ∀x, y ∈ C.
(16)
Since (1 − αn )L < 1 for all n ≥ 1, hence Wn : C → C is a contraction mapping.
By Banach contraction mapping principle, there exists a unique fixed point xn ∈ C
such that
xn
s
= αn xn + (1 − αn )T[n]
x, ∀n ≥ 1.
(17)
Therefore, if (1 − αn )L < 1 for all n ≥ 1, then the iterative sequence (14) can
be employed for the approximation of common fixed points for a finite family of
k-strictly asymptotically pseudo-contractive mappings.
It is the purpose of this paper to modify iteration process (14) by hybrid
method as follows:
x0 ∈ C chosen arbitrary,
yn
=
s
αn xn + (1 − αn )T[n]
xn ,
Cn
=
{z ∈ C : kyn − zk ≤ kxn − zk
Qn
xn+1
=
=
2
2
°
°2
°
°
s
+(k − αn ) °xn − T[n]
xn ° + λn },
{z ∈ C : hxn − z, x0 − xn i ≥ 0},
PCn ∩Qn (x0 ),
s
where T[n]
= Tns (mod
N)
(18)
= Tis with n = sN + i and i ∈ I = {0, 1, . . . , N − 1} and
λn = (1 − αn )((1 + rn )2 − 1)(diam C)2 → 0 as n → ∞.
The purpose of this paper is to establish strong convergence theorems of
newly proposed (CQ) algorithm (18) for finite family of k-strictly asymptotically
pseudo-contractive mappings in Hilbert spaces. Our results extend the corresponding results of Liu [9], Kim and Xu [8], Osilike and Akuchu [15], Thakur [19] and
some others.
In the sequel, we will need the following lemmas.
Lemma 1.2. Let H be a real Hilbert space. There holds the following identities:
2
2
2
(i) kx − yk = kxk − kyk − 2hx − y, yi ∀ x, y ∈ H.
2
2
2
2
(ii) ktx + (1 − t)yk = t kxk + (1 − t) kyk − t(1 − t) kx − yk ,
∀ t ∈ [0, 1], ∀ x, y ∈ H.
(iii) If {xn } be a sequence in H weakly converges to z, then
2
2
2
lim sup kxn − yk = lim sup kxn − zk + kz − yk
n→∞
n→∞
∀y ∈ H.
Strong Convergence Theorems for Strictly
31
Lemma 1.3. Let H be a real Hilbert space. Given a closed convex subset C ⊂ H
and points x, y, z ∈ H. Given also a real number a ∈ R. The set
2
2
{v ∈ C : ky − vk ≤ kx − vk + hz, vi + a}
is convex (and closed).
Lemma 1.4. Let K be a closed convex subset of a real Hilbert space H. Given
x ∈ H and y ∈ K. Then z = PK x if and only if there holds the relation
hx − z, y − zi ≤ 0 ∀y ∈ K,
where PK is the nearest point projection from H onto K, that is, PK x is the unique
point in K with the property
kx − PK xk ≤ kx − yk ∀x ∈ K.
We use following notation:
1. ⇀ for weak convergence and → for strong convergence.
2. ωw (xn ) = {x : ∃ xnj ⇀ x} denotes the weak ω-limit set of {xn }.
Lemma 1.5. [10]. Let K be a closed convex subset of H. Let {xn } be a sequence
in H and u ∈ H. Let q = PK u. If {xn } is such that ωw (xn ) ⊂ K and satisfies the
condition
kxn − uk =
ku − qk ∀n.
(19)
Then xn → q.
∞
∞
Lemma 1.6. [17]. Let {an }∞
n=1 , {βn }n=1 and {rn }n=1 be sequences of nonnegative
real numbers satisfying the inequality
an+1 ≤ (1 + rn )an + βn , n ≥ 1.
P∞
If n=1 rn < ∞ and n=1 βn < ∞, then limn→∞ an exists. If in addition
{an }∞
n=1 has a subsequence which converges strongly to zero, then limn→∞ an = 0.
P∞
Lemma 1.7. Let H be a real Hilbert space, let C be a nonempty closed convex subset
of H, and let Ti : C → C be a ki -strictly asymptotically pseudocontractive
P∞ mapping
for i = 0, 1, . . . , N − 1 with a sequence {rni } ⊂ [0, ∞) such that n=1 rni < ∞
and for some 0 ≤ ki < 1, then there exist constants L > 0 and k ∈ [0, 1) and a
sequence {rn } ⊂ [0, ∞) with limn→∞ rn = 0 such that for any x, y ∈ C and for
each i = 0, 1, . . . , N − 1 and each n ≥ 1, the following hold:
kTin x − Tin yk ≤
2
(1 + rn )2 kx − yk
2
+k k(x − Tin x) − (y − Tin y)k ,
(20)
and
kTin x − Tin yk ≤
L kx − yk .
(21)
Proof. Since for each i = 0, 1, . . . , N − 1, Ti is ki -strictly asymptotically pseudocontractive, where ki ∈ [0, 1) and {rni } ⊂ [0, ∞) with limn→∞ rni = 0. By
G. S. Saluja and H. K. Nashine
32
Remark 1.1, Ti is Li -Lipschitzian. Taking rn = max{rni , i = 0, 1, . . . , N − 1} and
k = max{ki , i = 0, 1, . . . , N − 1}, hence, for each i = 0, 1, . . . , N − 1, we have
kTin x − Tin yk ≤
(1 + rni )2 kx − yk
2
2
+ki k(x − Tin x) − (y − Tin y)k ,
≤ (1 + rn )2 kx − yk
2
2
+k k(x − Tin x) − (y − Tin y)k .
(22)
The conclusion (20) is proved. Again taking L = max{Li : i = 0, 1, . . . , N − 1} for
any x, y ∈ C, we have
kTin x − Tin yk ≤
Li kx − yk ≤ L kx − yk .
(23)
This completes the proof of lemma.
¤
2. Main Results
Theorem 2.1. Let C be a closed convex subset of a Hilbert space H. Let N ≥ 1 be
an integer. Let for each 0 ≤ i ≤ N − 1, Ti : C → C be N P
ki -strictly asymptotically
∞
pseudo-contraction mappings for some 0 ≤ ki < 1 and n=1 rn < ∞. Let k =
max{ki : 0 ≤ i ≤ N − 1} and rn = max{rni : 0 ≤ i ≤ N − 1}. Assume that
TN −1
F = i=0 F (Ti ) 6= ∅. Given x0 ∈ C, let {xn }∞
n=0 be the sequence generated
by an implicit iteration scheme (14). Assume that the control sequence {αn } is
chosen so
° that k + ǫ° < αn < 1 − ǫ for all n and for some ǫ ∈ (0, 1). Then
limn→∞ °xn − T[l] xn ° = 0 for all l ∈ I = {0, 1, . . . , N − 1}.
Proof. Let p ∈ F =
2
kxn+1 − pk
TN −1
i=0
=
=
=
≤
F (Ti ). It follows from (14) and Lemma 1.2 (ii) that
°
°2
°
°
s
x n − p°
°αn xn + (1 − αn )T[n]
°
°2
°
°
s
xn − p)°
°αn (xn − p) + (1 − αn )(T[n]
°2
°
°
° s
2
αn kxn − pk + (1 − αn ) °T[n]
xn − p°
°
°2
°
°
s
−αn (1 − αn ) °xn − T[n]
xn °
h
2
2
αn kxn − pk + (1 − αn ) (1 + rn )2 kxn − pk
°
°2
°
°2 i
°
°
°
°
s
s
xn °
+k °xn − T[n]
xn ° − αn (1 − αn ) °xn − T[n]
(24)
(25)
Strong Convergence Theorems for Strictly
33
h
i
2
αn (1 + rn )2 + (1 − αn )(1 + rn )2 kxn − pk
°
°2
°
°
s
−(αn − k)(1 − αn ) °xn − T[n]
xn °
°
°2
°
°
2
s
xn °
= (1 + rn )2 kxn − pk − (αn − k)(1 − αn ) °xn − T[n]
°
°2
°
°
2
s
xn °
= (1 + dn ) kxn − pk − (αn − k)(1 − αn ) °xn − T[n]
≤
where dn = rn2 + 2rn . Since k + ǫ < αn < 1 − ǫ for all n, from (24) we have
°
°2
°
°
2
2
s
xn ° .
kxn+1 − pk ≤ (1 + dn ) kxn − pk − ǫ2 °xn − T[n]
(26)
Now (26) implies that
2
2
(27)
kxn+1 − pk ≤ (1 + dn ) kxn − pk .
P∞
P∞
Since n=1 rn < ∞ thus n=1 dn < ∞, it follows by Lemma 1.2, we know that
limn→∞ kxn − pk exists and so {xn } is bounded. Consider (26) again yields that
°
°
dn
1
°
°2
2
2
2
s
[kxn − pk − kxn+1 − pk ] + 2 kxn − pk .
(28)
xn ° ≤
°xn − T[n]
ǫ2
ǫ
Since {xn } is bounded and dn → 0 as n → ∞. So, we get
°
°
°
°
s
xn ° → 0 as n → ∞.
°xn − T[n]
(29)
From the definition of {xn }, we have
°
°
°
°
s
kxn+1 − xn k = (1 − αn ) °xn − T[n]
xn ° → 0, as n → ∞.
(30)
So, kxn − xn+l k → 0 as n → ∞ and for all l < N . Now for n ≥ N , and since T is
uniformly Lipschitzian (by Remark 1.1) with Lipschitz constant L > 0, so we have
°
°
° °
°
°
°
° ° s
s
°xn − T[n] xn ° ≤ °
xn − T[n] xn °
xn ° + °T[n]
°xn − T[n]
°
°
°
°
°
°
°
° s−1
s
≤ °xn − T[n]
xn ° + L °T[n]
xn − xn °
°
°
°
h°
°
°
° s−1
°
s−1
s
≤ °xn − T[n]
xn ° + L °T[n]
xn − T[n−N
x
°
n−N
]
°
°
i
°
° s−1
(31)
+ °T[n−N
] xn−N − xn−N ° + kxn−N − xn k .
Since for each n ≥ N , n ≡
(31), we have
°
°
°xn − T[n] xn ° ≤
(n − N ) (mod N ). Thus T[n] = T[n−N ] , therefore from
°
°
°
°
s
xn ° + L2 kxn − xn−N k
°xn − T[n]
°
°
° s−1
°
+L °T[n−N
] xn−N − xn−N ° + L kxn−N − xn k .
(32)
G. S. Saluja and H. K. Nashine
34
From (29) and (32), we obtain
°
°
°xn − T[n] xn ° → 0 as n → ∞.
Consequently, for any l ∈ I = {0, 1, . . . , N − 1},
°
°
°
°
°xn − T[n+l] xn ° ≤ kxn − xn+l k + °xn+l − T[n+l] xn+l °
°
°
+ °T[n+l] xn+l − T[n+l] xn °
°
°
≤ (1 + L) kxn − xn+l k + °xn+l − T[n+l] xn+l °
(33)
→ 0 as n → ∞.
(34)
°
°
lim °xn − T[l] xn ° = 0, ∀ l ∈ I = {0, 1, . . . , N − 1}.
(35)
This implies that
n→∞
This completes the proof.
¤
Theorem 2.2. Let C be a closed convex compact subset of a Hilbert space H. Let
N ≥ 1 be an integer. Let for each 0 ≤ i ≤ N − 1, Ti : C → C be NPki -strictly
∞
asymptotically pseudo-contraction mappings for some 0 ≤ ki < 1 and n=1 rn <
∞. Let k = max{ki : 0 ≤ i ≤ N − 1} and rn = max{rni : 0 ≤ i ≤ N − 1}. Assume
TN −1
that F = i=0 F (Ti ) 6= ∅. Given x0 ∈ C, let {xn }∞
n=0 be the sequence generated by
an implicit iteration scheme (14). Assume that the control sequence {αn } is chosen
so that k + ǫ < αn < 1 − ǫ for all n and for some ǫ ∈ (0, 1). Then {xn } converges
N −1
strongly to a common fixed point of the family {Ti }i=0
.
Proof. We only conclude the difference. By compactness of C this immediately
implies that there is a subsequence {xnj } of {xn } which converges to a com−1
mon fixed point of {Ti }N
i=0 , say, p. Combining (27) with Lemma 1.6, we have
limn→∞ kxn − pk = 0. This completes the proof.
¤
Remark 2.3. Theorem 2.2 extends and improves the corresponding result of Liu [9]
in the following ways:
(i) We removed the uniformly L-Lipschitzian condition.
(ii) The modified Mann iteration process is replaced by implicit iteration process for a finite family of mappings.
For our next result, we shall need the following definition:
Definition 2.4. A mapping T : C → C is said to be semi-compact, if for any
bounded sequence {xn } in C such that limn→∞ kxn − T xn k = 0 there exists a subsequence {xni } ⊂ {xn } such that limi→∞ xni = x ∈ C.
Theorem 2.5. Let C be a closed convex subset of a Hilbert space H. Let N ≥ 1 be
an integer. Let for each 0 ≤ i ≤ N − 1, Ti : C → C be N P
ki -strictly asymptotically
∞
pseudo-contraction mappings for some 0 ≤ ki < 1 and n=1 rn < ∞. Let k =
max{ki : 0 ≤ i ≤ N − 1} and rn = max{rni : 0 ≤ i ≤ N − 1}. Assume that
TN −1
F = i=0 F (Ti ) 6= ∅. Given x0 ∈ C, let {xn }∞
n=0 be the sequence generated by an
implicit iteration scheme (14). Assume that the control sequence {αn } is chosen
Strong Convergence Theorems for Strictly
35
so that k + ǫ < αn < 1 − ǫ for all n and for some ǫ ∈ (0, 1). Assume that one
N −1
member of the family {Ti }i=0
be semi-compact. Then {xn } converges strongly to
N −1
a common fixed point of the family {Ti }i=0
.
Proof. Without loss of generality, we can assume that T1 is semi-compact. It follows
from (35) that
°
°
(36)
lim °xn − T[1] xn ° = 0.
n→∞
By the semi-compactness of T1 , there exists a subsequence {xnk } of {xn } such that
xnk → u ∈ C strongly. Since C is closed, u ∈ C, and furthermore,
°
°
°
°
(37)
lim °xnk − T[l] xnk ° = °u − T[l] u° = 0,
nk →∞
for all l ∈ I = {0, 1, . . . , N − 1}. Thus u ∈ F . Since {xnk } converges strongly
to u and limn→∞ kxn − uk exists, it follows from Lemma 1.2 that {xn } converges
strongly to u. This completes the proof.
¤
Remark 2.6. Theorem 2.5 extends and improves the corresponding result of Kim
and Xu [8].
Remark 2.7. Theorem 2.5 also extends and improves Theorem 1.6 of Osilike and
Akuchu [15] from asymptotically pseudocontractive mappings to strictly asymptotically pseudocontractive mappings.
We now prove strong convergence of k-strictly asymptotically pseudo-contractive
N −1
mappings {Ti }i=0
using algorithm (18):
Theorem 2.8. Let C be a closed convex subset of a Hilbert space H. Let N ≥ 1 be
an integer. Let for each 0 ≤ i ≤ N − 1, Ti : C → C be
N ki -strictly asymptotically
P∞
pseudo-contraction mappings for some 0 ≤ ki < 1, n=1 rn < ∞ and I − T[n] is
demiclosed at zero. Let k = max{ki : 0 ≤ i ≤ N − 1} and rn = max{rni : 0 ≤
TN −1
i ≤ N − 1}. Assume that F = i=0 F (Ti ) 6= ∅. Let {xn }∞
n=0 be the sequence
generated by an the algorithm (18). Assume that the sequence {αn } is chosen so
that supn≥0 αn < 1. Then {xn } converges strongly to PF (x0 ).
Proof. By Lemma 1.3, we observe that Cn is convex.
Now, for all p ∈ F , using Lemma 1.2(ii), we have
2
kyn − pk
=
=
=
°
°2
°
°
s
x n − p°
°αn xn + (1 − αn )T[n]
°2
°
°
°
s
xn − p)°
°αn (xn − p) + (1 − αn )(T[n]
°
°2
°
° s
2
αn kxn − pk + (1 − αn ) °T[n]
xn − p°
°
°2
°
°
s
−αn (1 − αn ) °xn − T[n]
xn °
(38)
G. S. Saluja and H. K. Nashine
36
≤
≤
=
≤
≤
h
2
2
αn kxn − pk + (1 − αn ) (1 + rn )2 kxn − pk
°
°
°2 i
°2
°
°
°
°
s
s
+k °xn − T[n]
xn ° − αn (1 − αn ) °xn − T[n]
xn °
h
i
2
αn (1 + rn )2 + (1 − αn )(1 + rn )2 kxn − pk
°
°2
°
°
s
xn °
−(αn − k)(1 − αn ) °xn − T[n]
°
°2
°
°
2
s
(1 + rn )2 kxn − pk − (αn − k)(1 − αn ) °xn − T[n]
xn °
°
°2
°
°
2
s
(1 + rn )2 kxn − pk + (k − αn )(1 − αn ) °xn − T[n]
xn °
°
°2
°
°
2
s
xn ° + λn ,
kxn − pk + (k − αn ) °xn − T[n]
(39)
so p ∈ Cn for all n. Thus F ⊂ Cn for all n.
Next we show that F ⊂ Qn for all n ≥ 0, for this we use induction.
For n = 0, we have F ⊂ C = Q0 . Assume that F ⊂ Qn .
Since xn+1 is the projection of x0 onto Cn ∩ Qn , by Lemma 1.4, we have
hxn+1 − z, x0 − xn+1 i ≥ 0 ∀z ∈ Cn ∩ Qn .
As F ⊂ Cn ∩ Qn by the induction assumption, the last inequality holds, in particular, for all z ∈ F . This together with the definition of Qn+1 implies that F ⊂ Qn+1 .
Hence F ⊂ Qn for all n ≥ 0.
Now, since xn = PQn (x0 ) (by the definition of Qn ), and since F ⊂ Qn , we
have
kxn − x0 k ≤ kp − x0 k ∀p ∈ F.
In particular, {xn } is bounded and
kxn − x0 k ≤ kq − x0 k , where q = PF (x0 ).
(40)
The fact xn+1 ∈ Qn asserts that hxn+1 − xn , xn − x0 i ≥ 0. This together with
Lemma 1.2(i), implies that
2
kxn+1 − xn k
=
=
2
k(xn+1 − x0 ) − (xn − x0 )k
2
2
2
2
kxn+1 − x0 k − kxn − x0 k − 2hxn+1 − xn , xn − x0 i
≤ kxn+1 − x0 k − kxn − x0 k .
It follows that,
kxn+1 − xn k → 0.
By the fact xn+1 ∈ Cn we get
kxn+1 − yn k
2
(41)
2
≤ kxn+1 − xn k
°
°2
°
°
s
+(k − αn ) °xn − T[n]
xn ° + λn .
(42)
Strong Convergence Theorems for Strictly
37
s
Moreover, since yn = αn xn + (1 − αn )T[n]
xn , we deduce that
kxn+1 − yn k
2
=
Substituting (43) into (42) to get
2
αn kxn+1 − xn k
°
°2
°
°
s
+(1 − αn ) °xn+1 − T[n]
xn °
°
°2
°
°
s
−αn (1 − αn ) °xn − T[n]
xn ° .
°
°2
°
°
s
(1 − αn ) °xn+1 − T[n]
xn °
(43)
2
≤ (1 − αn ) kxn+1 − xn k
°
°2
°
°
s
xn ° + λn .
+k °xn − T[n]
Since αn < 1 for all n, the last inequality becomes,
°
°
°2
°2
°
°
°
°
2
s
s
xn °
xn ° ≤ kxn+1 − xn k + k °xn − T[n]
°xn+1 − T[n]
λn
,
ρ
for some positive number ρ > 0, such that αn < ρ < 1.
But on the other hand, we compute
°
°2
°
°
2
s
s
xn ° = kxn+1 − xn k + 2hxn+1 − xn , xn − T[n]
xn i
°xn+1 − T[n]
°
°2
°
°
s
xn ° .
+ °xn − T[n]
+
By (44) and (45), we get
°
°2
°
°
s
(1 − k) °xn − T[n]
xn °
Therefore
°
°2
°
°
s
xn °
°xn − T[n]
Now,
°
°
°xn − T[n] xn °
≤
≤
λn
s
− 2hxn+1 − xn , xn − T[n]
xn i.
ρ
2
λn
s
−
hxn+1 − xn , xn − T[n]
xn i
ρ(1 − k) 1 − k
→ 0 as n → ∞.
° °
°
°
°
° ° s
°
s
xn ° + °T[n]
≤ °xn − T[n]
xn − T[n] xn °
°
°
°
°
°
°
° s−1
°
s
≤ °xn − T[n]
xn ° + (1 + r1 ) °T[n]
xn − xn °
→ 0 as n → ∞.
(44)
(45)
(46)
(47)
(48)
Now, since I − T[n] is demiclosed at zero, (48) imply that xn ⇀ x, where x is a
weak limit of {xn } and hence ωw (xn ) ⊂ F (Ti ) for any i = 0, 1, . . . , N − 1. So,
TN −1
ωw (xn ) ⊂ F = i=0 . This fact, the inequality (40) and Lemma 1.5 implies that
{xn } → q = PF (x0 ), that is, {xn } converges strongly to PF (x0 ). This completes
the proof.
¤
38
G. S. Saluja and H. K. Nashine
Remark 2.9. Theorem 2.8 extends Theorem 3.1 of Thakur [19] to the case of finite
family of mappings and implicit iteration process considered in this paper.
Acknowledgement. Our sincere thanks to the referee for his valuable suggestions
and comments on the manuscript and also thanks to Prof. A. Banerjee for providing
a manuscript of Osilike [14].
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