An. Şt. Univ. Ovidius Constanţa
Vol. 18(1), 2010, 163–180
CONVERGENCE THEOREMS FOR TOTAL
ASYMPTOTICALLY NONEXPANSIVE
MAPPINGS
Yan Hao
Abstract
In this paper, a demiclosed principle for total asymptotically nonexpansive mappings is established. An implicit iterative method for
the class of total asymptotically nonexpansive mappings is considered.
Weak and strong convergence theorems are established in a real Hilbert
space. As applications of main results, an equilibrium problem is considered based on implicit iterative process.
1. Introduction and Preliminaries
Throughout this paper, we assume that H is a real Hilbert space, whose
inner product and norm are denoted by h·, ·i and k · k. We also assume that
ψ : [0, ∞) → [0, ∞) is strictly increasing continuous function with ψ(0) = 0. →
and ⇀ are denoted by strong convergence and weak convergence, respectively.
Let C be a nonempty closed and convex subset of H and T : C → C a
mapping. In this paper, we use F (T ) to denote the fixed point set of the
mapping T . Recall that T is said to be nonexpansive if
kT x − T yk ≤ kx − yk,
∀x, y ∈ C.
(1.1)
Key Words: nonexpansive mapping; asymptotically nonexpanvie mapping; total asymptotically nonexpansive mapping; fixed point; equilibrium
Mathematics Subject Classification: 47H09, 47H10, 47J25.
Received: June, 2009
Accepted: January, 2010
163
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Yan Hao
T is said to be asymptotically nonexpansive if there exists a positive sequence hn ⊂ [1, ∞) with limn→∞ hn = 1 such that
kT n x − T n yk ≤ hn kx − yk,
∀x, y ∈ C, n ≥ 1.
(1.2)
The class of asymptotically nonexpansive mappings was introduced by Goebel
and Kirk [14] as a generalization of the class of nonexpansive mappings. They
proved that if C is a nonempty closed convex and bounded subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive mapping
on C, then T has a fixed point.
T is said to be asymptotically nonexpansive in the intermediate sense if it
is continuous and the following inequality holds:
lim sup sup (kT n x − T n yk − kx − yk) ≤ 0.
(1.3)
n→∞ x,y∈C
Observe that if we define
σn = sup (kT n x − T n yk − kx − yk)
and νn = max{0, σn },
x,y∈C
then νn → 0 as n → ∞. It follows that (1.3) is reduced to
kT n x − T n yk ≤ kx − yk + νn ,
∀x, y ∈ C, n ≥ 1.
(1.4)
The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck, Kuczumow and Reich [2]. It is known
[16] that if C is a nonempty closed convex bounded subset of a uniformly
convex Banach space E and T is asymptotically nonexpansive in the intermediate sense, then T has a fixed point. It is worth mentioning that the class
of mappings which are asymptotically nonexpansive in the intermediate sense
contains properly the class of asymptotically nonexpansive mappings.
Recently, Alber, Chidume and Zegeye [1] introduced the concept of total asymptotically nonexpansive mappings. Recall that T is said to be total
asymptotically nonexpansive if
kT n x − T n yk ≤ kx − yk + µn ψ(kx − yk) + νn ,
∀x, y ∈ C,
(1.5)
where {µn } and {νn } are nonnegative real sequences such that µn → 0 and
νn → 0 as n → ∞. From the definition, we see that the class of total asymptotically nonexpansive mappings include the class of asymptotically nonexpansive
mappings as a special case; see also [13] for more details.
Recently, weak convergence problems of implicit (or non-implicit) iterative
processes to a common fixed point for a finite family of nonexpansive mappings
CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY
NONEXPANSIVE MAPPINGS
165
and asymptotically nonexpansive mappings have been considered by a number
of authors (see, for example, [1],[8],[9],[12],[15],[19],[21],[23],[24],[27]-[32],[34][36],[38]).
In 2001, Xu and Ori [34] introduced the following implicit iteration process
for a finite family of nonexpansive mappings {T1 , T2 , . . . , TN }, with {αn } a real
sequence in (0, 1), and an initial point x0 ∈ C:
x1 = α1 x0 + (1 − α1 )T1 x1 ,
x2 = α2 x1 + (1 − α2 )T2 x2 ,
..
.
xN = αN xN −1 + (1 − αN )TN xN ,
xN +1 = αN +1 xN + (1 − αN +1 )T1 xN +1 ,
..
.
which can be re-written in the following compact form:
xn = αn xn−1 + (1 − αn )Tn xn ,
∀n ≥ 1,
(1.6)
where Tn = Tn(modN ) (here the mod N function takes values in {1, 2, . . . , N }).
They obtained the following results in a real Hilbert space.
Theorem XO. Let H be a real Hilbert space, C be a nonempty closed convex
subset of H, and T : C → C be a finite family of nonexpansive self-mappings
on C such that F = ∩N
i=1 F (Ti ) 6= ∅. Let {xn } be defined by (1.6). If {αn } is
chosen so that αn → 0, as n → ∞, then {xn } converges weakly to a common
fixed point of the family of {Ti }N
i=1 .
In 2006, Chang et al. [9] improved the results of Xu and Ori [34] from the
class of nonexpansive mappings to the class of asymptotically nonexpansive
mappings which is defined on a nonempty closed and convex subset C of H
such that C + C ⊂ C. To be more precise, they considered the following
implicit iterative process for a finite family of asymptotically nonexpansive
mappings {T1 , T2 , . . . , TN }, with {αn } a real sequence in (0, 1), {un } a bounded
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Yan Hao
sequence in C, and an initial point x0 ∈ C:
x1 = α1 x0 + (1 − α1 )T1 x1 + u1 ,
x2 = α2 x1 + (1 − α2 )T2 x2 + u2 ,
..
.
xN = αN xN −1 + (1 − αN )TN xN + uN ,
xN +1 = αN +1 xN + (1 − αN +1 )T1 xN +1 + uN +1 ,
..
.
x2N = α2N x2N −1 + (1 − α2N )TN2 x2N + u2N ,
x2N +1 = α2N +1 x2N + (1 − α2N +1 )T13 x2N +1 + u2N +1 ,
..
.
Since for each n ≥ 1, it can be written as n = (k − 1)N + i, where i = i(n) ∈
{1, 2, . . . , N }, k = k(n) ≥ 1 is a positive integer and k(n) → ∞ as n → ∞.
Hence the above table can be rewritten in the following compact form:
k(n)
xn = αn xn−1 + (1 − αn )Ti(n) xn + un ,
∀n ≥ 1.
(1.7)
They obtained weak convergence theorems of the implicit iterative scheme
(1.7) for a finite family of asymptotically nonexpansive mappings {T1 , T2 , . . . , TN };
see [9] for more details.
The purpose of this paper is to establish weak and strong convergence
theorems of the implicit iteration process (1.7) for a finite family of uniformly
Lipschitz total asymptotically nonexpansive mappings in a real Hilbert space.
Next, we recall some well-known concepts.
Recall that a space X is said to satisfy Opial’s condition [18] if for each
sequence {xn } in X, the condition that the sequence xn → x weakly implies
that
lim inf kxn − xk < lim inf kxn − yk
n→∞
n→∞
for all y ∈ E and y 6= x.
Recall that a mapping T : C → C is semicompact if any sequence {xn } in
C satisfying limn→∞ kxn − T xn k = 0 has a convergent subsequence.
Recall that the mapping T is said to be demiclosed at the origin if for each
sequence {xn } in C, the condition xn → x0 weakly and T xn → 0 strongly
implies T x0 = 0.
CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY
NONEXPANSIVE MAPPINGS
167
Next, we show that the process (1.7) is well-defined if the control sequence
satisfies 0 < L−1
< αn < 1, where L = max1≤i≤N {Li }. Indeed, define a
L
mapping
k(n)
Wn x = αn xn−1 + (1 − αn )Ti(n) x + un ,
∀n ≥ 1,
∀x ∈ C.
It follows that
kWn x − Wn yk ≤ (1 − αn )Lkx − yk,
∀x, y ∈ C.
Since (1 − αn )L < 1, it follows that Wn is a contractive mapping and hence
has a unique fixed point xn in C. This is, the process (1.7) is well-defined.
In order to prove our main results, we also need the following lemmas.
Lemma 1.1 ([30]). Let {an }, {bn } and {cn } be three nonnegative sequences
satisfying the following condition:
an+1 ≤ (1 + bn )an + cn , ∀n ≥ n0 ,
P∞
P∞
where n0 is some nonnegative integer, n=0 cn < ∞ and n=0 bn < ∞. Then
limn→∞ an exists.
Lemma 1.2 ([27]). Let H be a real Hilbert space and 0 < p ≤ tn ≤ q < 1 for
all n ≥ 0. Suppose that {xn } and {yn } are sequences of H such that
lim sup kxn k ≤ r,
n→∞
lim sup kyn k ≤ r
n→∞
and
lim ktn xn + (1 − tn )yn k = r
n→∞
hold for some r ≥ 0. Then limn→∞ kxn − yn k = 0.
Lemma 1.3. Let C be a nonempty closed convex and bounded subset of a real
Hilbert space H and T be a L-Lipschitz continuous and total asymptotically
nonexpansive mapping with the function ψ and sequences {µn } and {νn } such
that µn → 0 and νn → 0 as n → ∞. Then I − T is demiclosed at zero.
Proof. Let {xn } be a sequence in C such that xn ⇀ x∗ and xn − T xn → 0
as n → ∞. Next, we show that x∗ ∈ C and x∗ = T x∗ . Since C is closed and
convex, we see that x∗ ∈ C. It is sufficient to show that x∗ = T x∗ . Choose
1
) and define yα,m = (1 − α)x∗ + αT m x∗ for arbitrary but fixed
α ∈ (0, 1+L
m ≥ 1. Note that
kxn − T m xn k ≤ kxn − T xn k + kT xn − T 2 xn k + · · · + kT m−1 xn − T m xn k
≤ mLkxn − T xn k.
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Yan Hao
It follows that
lim kxn − T m xn k = 0.
(1.8)
n→∞
Note that
hx∗ − yα,m , yα,m − T m yα,m i
= hx∗ − xn , yα,m − T m yα,m i + hxn − yα,m , yα,m − T m yα,m i =
= hx∗ − xn , yα,m − T m yα,m i + hxn − yα,m , T m xn − T m yα,m i−
− hxn − yα,m , xn − yα,m i + hxn − yα,m , xn − T m xn i
≤ hx∗ − xn , yα,m − T m yα,m i+
(1.9)
+ kxn − yα,m k(kxn − yα,m k + µn ψ(kxn − yα,m k) + νm )−
− kxn − yα,m k2 + kxn − yα,m kkxn − T m xn k ≤
≤ hx∗ − xn , yα,m − T m yα,m i + µm M ψ(M ) + νm M +
+ kxn − yα,m kkxn − T m xn k,
where M = supn≥0 {kxn − yα,m k}. Since xn ⇀ x∗ and (1.8), we arrive at
hx∗ − yα,m , yα,m − T m yα,m i ≤ µm M ψ(M ) + νm M.
(1.10)
On the other hand, we have
hx∗ − yα,m , (x∗ − T m x∗ ) − (yα,m − T m yα,m )i ≤ (1 + L)kx∗ − yα,m k2
= (1 + L)α2 kx∗ − T m x∗ k2 .
(1.11)
Note that
kx∗ − T m x∗ k2 = hx∗ − T m x∗ , x∗ − T m x∗ i =
1
= hx∗ − yα,m , x∗ − T m x∗ i =
α
1
= hx∗ − yα,m , (x∗ − T m x∗ ) − (yα,m − T m yα,m )i+
α
1
+ hx∗ − yα,m , yα,m − T m yα,m i.
α
(1.12)
Substituting (1.10) and (1.11) into (1.12), we arrive at
kx∗ − T m x∗ k2 ≤ (1 + L)αkx∗ − T m x∗ k2 +
µm M ψ(M ) + νm M
.
α
This implies that
α[1 − (1 + L)α]kx∗ − T m x∗ k2 ≤ µm M ψ(M ) + νm M,
∀m ≥ 1.
(1.13)
CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY
NONEXPANSIVE MAPPINGS
169
Letting m → ∞ in (1.13), we see that T m x∗ → x∗ . Since T is uniformly
L-Lipschitz, we obtain that x∗ = T x∗ . This completes the proof.
2. Main results
Theorem 2.1. Let H be a real Hilbert space and C be a nonempty closed
convex and bounded subset of H such that C + C ⊂ C. Let Ti : C → C be
a uniformly Li -Lipschitz total asymptotically nonexpansive mapping with the
(i)
(i)
function ψi and sequences {µn }, {νn } for each i ∈ {1, 2, . . . , N }. Assume
P∞
P∞ (i)
(i)
that n=1 µn < ∞ and n=1 νn < ∞
for each i ∈ {1, 2, . . . , N }. Let {un }
P∞
be a bounded sequence in C such that n=1 kun k < ∞ and {αn } a sequence
in [ L−1
L , a], where L = max1≤i≤N {Li } > 1 and a is some constant in (0, 1).
Assume that F = ∩N
i=1 F (Ti ) 6= ∅. Let {xn } be a sequence generated by (1.7).
Then the sequence {xn } converges weakly to some point x∗ ∈ F.
Proof. Define the following sequences
(2)
(N )
µn = max{µ(1)
n , µn , . . . , µn }
and
νn = max{νn(1) , νn(2) , . . . , νn(N ) }.
P∞
P∞
It is easy to see that n=1 µn < ∞ and n=1 νn < ∞. Fixing p ∈ F, we
have
k(n)
kxn − pk ≤ αn kxn−1 − pk + (1 − αn )kTi(n) xn − pk + kun k ≤
≤ αn kxn−1 − pk + (1 − αn ) kxn − pk + µk(n) ψi(n) (kxn − pk) + νk(n) + kun k ≤
≤ kxn−1 − pk + µk(n) ψi(n) ((diam C)) + νk(n) + kun kleq
≤ kxn−1 − pk + µk(n) ψr ((diam C)) + νk(n) + kun k,
(2.1)
where
ψr ((diam C)) = max{ψ1 ((diam C)), ψ2 ((diam C)), . . . , ψN ((diam C))}.
In view of Lemma 1.1, we obtain that the limit of the sequence {kxn − pk}
exits. Next, we assume that
d = lim kxn − pk,
(2.2)
n→∞
where d > 0 is some constant. It follows that
lim sup kxn−1 − p − un k ≤ lim sup kxn−1 − pk + lim sup kun k = d.
n→∞
n→∞
n→∞
(2.3)
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Yan Hao
Note that
k(n)
k(n)
kTi(n) xn − p + un k ≤ kTi(n) xn − pk + kun k ≤
≤ kxn − pk + µk(n) ψi(n) (kxn − pk) + νk(n) + kun k ≤
≤ kxn − pk + µk(n) ψr ((diam C)) + νk(n) + kun k.
It follows that
k(n)
lim sup kTi(n) xn − p + un k ≤ d.
(2.4)
n→∞
On the other hand, we have
k(n)
d = lim kxn − pk = lim kαn (xn−1 − p − un ) + (1 − αn )(Ti(n) xn − p + un )k.
n→∞
n→∞
(2.5)
Combining (2.3), (2.4) with (2.4) and from Lemma 1.2, we obtain that
k(n)
lim kxn−1 − Ti(n) xn k = 0.
(2.6)
n→∞
Note that
k(n)
kxn − xn−1 k ≤ (1 − αn )kxn−1 − Ti(n) xn k + kun k.
From (2.6), we arrive at
lim kxn − xn−1 k = 0.
(2.7)
n→∞
On the other hand, we have
k(n)
k(n)
kTi(n) xn − xn k ≤ kTi(n) xn − xn−1 k + kxn−1 − xn k,
which combined with (2.6) gives that
k(n)
lim kTi(n) xn − xn k = 0.
(2.8)
n→∞
From (2.7), we also have
lim kxn − xn+l k = 0,
n→∞
∀l = 1, 2, . . . , N.
(2.9)
For any positive n > N , it can be rewritten as n = (k(n) − 1)N + i(n),
i(n) ∈ {1, 2, . . . , N }. Note that
k(n)
k(n)
kxn−1 − Tn xn k ≤ kxn−1 − Ti(n) xn k + kTi(n) xn − Tn xn k ≤
k(n)
k(n)−1
≤ kxn−1 − Ti(n) xn k + LkTi(n)
≤ kxn−1 −
k(n)
Ti(n) xn k
k(n)−1
+
xn − xn k ≤
k(n)−1
L{kTi(n) xn
k(n)−1
− Ti(n−N ) xn−N k+
+ kTi(n−N ) xn−N − x(n−N )−1 k + kx(n−N )−1 − xn k}.
(2.10)
CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY
NONEXPANSIVE MAPPINGS
171
On the other hand, we have n − N = ((k(n) − 1) − 1)N + i(n) = ((k(n) − 1) −
1)N + i(n − N ), i.e.,
k(n − N ) = k(n) − 1
and
i(n − N ) = i(n).
It follows that
k(n)−1
kTi(n)
k(n)−1
xn − Ti(n−N ) xn−N k ≤ Lkxn − xn−N k
(2.11)
k(n−N )
(2.12)
and
k(n)−1
kTi(n−N ) xn−N − x(n−N )−1 k = kTi(n−N ) xn−N − x(n−N )−1 k.
Combining (2.6), (2.9) with (2.10), we see that
lim kxn−1 − Tn xn k = 0.
n→∞
(2.13)
On the other hand, we have that
kxn − Tn xn k ≤ kxn − xn−1 k + kxn−1 − Tn xn k.
From (2.7) and (2.13), we obtain that
lim kxn − Tn xn k = 0.
n→∞
(2.14)
Consequently, for any j = 1, 2, . . . , N , we see that
kxn − Tn+j xn k ≤ kxn − xn+j k + kxn+j − Tn+j xn+j k + kTn+j xn+j − Tn+j xn k ≤
≤ (1 + L)kxn − xn+j k + kxn+j − Tn+j xn+j k.
From (2.9) and (2.14), we arrive at
lim kxn − Tn+j xn k = 0.
n→∞
Therefore, for ∀i ∈ {1, 2, . . . , N }, there exists some e ∈ {1, 2, . . . , N } such that
n + e = i(mod N ). It follows that
lim kxn − Ti xn k = lim kxn − Tn+e xn k = 0.
n→∞
n→∞
(2.15)
Since {xn } is bounded, we see that there exists a subsequence {xni } ⊂ {xn }
such that xni ⇀ x∗ . From Lemma 1.3, we can obtain that x∗ ∈ F. Next we
prove that {xn } converges weakly to x∗ . Suppose the contrary. Then we see
that there exists some subsequence {xnj } ⊂ {xn } such that {xnj } converges
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Yan Hao
weakly to x̄ ∈ C and x̄ 6= x∗ . From Lemma 1.3, we also have x̄ ∈ F. Put
w = limn→∞ kxn − x∗ k. Since H is an Opial’s space, we see that
w = lim kxni − x∗ k < lim kxni − x̄k =
ni →∞
ni →∞
= lim kxnj − x̄k < lim kxnj − x∗ k =
nj →∞
nj →∞
∗
= lim kxni − x k = w.
ni →∞
This derives a contradiction. It follows that x̄ = x∗ . This completes the proof.
Remark 2.2. Theorem 2.1 improves Theorem 1 of Chang et al. [9] from
asymptotically nonexpansive mappings to total asymptotically nonexpansive
mappings.
Corollary 2.3. Let H be a real Hilbert space and C be a nonempty closed
convex and bounded subset of H. Let Ti : C → C be a uniformly Li -Lipschitz
total asymptotically nonexpansive mapping with the function ψi and sequences
P∞
(i)
(i)
(i)
{µn }, {νn } for each i ∈ {1, 2, . . . , N }. Assume that n=1 µn < ∞ and
P∞ (i)
L−1
n=1 νn < ∞ for each i ∈ {1, 2, . . . , N }. Let {αn } be a sequence in [ L , a],
where L = max1≤i≤N {Li } > 1 and a is some constant in (0, 1). Assume
that F = ∩N
i=1 F (Ti ) 6= ∅. Let {xn } be a sequence generated by the following
manner:
x0 ∈ C,
k(n)
xn = αn xn−1 + (1 − αn )Ti(n) xn ,
∀n ≥ 1.
(2.16)
Then the sequence {xn } converges weakly to some point x∗ ∈ F.
Next, we prove a strong convergence theorem under the condition of semicompactness.
Theorem 2.4. Let H be a real Hilbert space and C a nonempty closed convex and bounded subset of H such that C + C ⊂ C. Let Ti : C → C be
a uniformly Li -Lipschitz total asymptotically nonexpansive mapping with the
(i)
(i)
function ψi and sequences {µn }, {νn } for each i ∈ {1, 2, . . . , N }. Assume
P∞
P∞ (i)
(i)
that n=1 µn < ∞ and n=1 νn < ∞
for each i ∈ {1, 2, . . . , N }. Let {un }
P∞
be a bounded sequence in C such that n=1 kun k < ∞ and {αn } a sequence
in [ L−1
L , a], where L = max1≤i≤N {Li } > 1 and a is some constant in (0, 1).
Assume that F = ∩N
i=1 F (Ti ) 6= ∅ and at least there exists a mapping Tr which
is semicompact. Let {xn } be a sequence generated by (1.7). Then the sequence
{xn } converges weakly to some point x∗ ∈ F.
Proof. Without loss of generality, we may assume that T1 is semicompact. It
follows from (2.15) that
lim kxn − T1 xn k = 0.
n→∞
CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY
NONEXPANSIVE MAPPINGS
173
By the semicompactness of T1 , we have there exists a subsequence {xni } of
{xn } such that xni → x ∈ C strongly. From (2.15), we have
lim kxni − Tl xni k = kx − Tl xk = 0,
ni →∞
for all l = 1, 2, · · · , N. This implies that x ∈ F. From Theorem 2.1, we know
that limn→∞ kxn − pk exists for each p ∈ F. This shows that limn→∞ kxn − xk
exists. From xni → x, we have
lim kxn − x∗ k = 0.
n→∞
This completes the proof of Theorem 2.4.
Corollary 2.5. Let H be a real Hilbert space and C be a nonempty closed
convex bounded subset of H. Let Ti : C → C be a uniformly Li -Lipschitz
total asymptotically nonexpansive mapping with the function ψi and sequences
P∞
(i)
(i)
(i)
{µn }, {νn } for each i ∈ {1, 2, . . . , N }. Assume that
n=1 µn < ∞ and
P∞ (i)
L−1
n=1 νn < ∞ for each i ∈ {1, 2, . . . , N }. Let {αn } be a sequence in [ L , a],
where L = max1≤i≤N {Li } > 1 and a is some constant in (0, 1). Assume
that F = ∩N
i=1 F (Ti ) 6= ∅ and at least there exists a mapping Tr which is
semicompact. Let {xn } be a sequence generated by (2.16). Then the sequence
{xn } converges weakly to some point x∗ ∈ F.
3. Applications
Let A : C → H be a mapping. Recall that the classical variational inequality problem is to find x ∈ C such that
hAx, y − xi ≥ 0,
∀y ∈ C.
(3.1)
It is known that x ∈ C is a solution to the variational inequality problem
(3.1) if and only if x is a fixed point of the mapping PC (I − ρA), where
PC is the metric projection from H onto C, ρ > 0 is a constant and I is
the identity mapping. This implies that the variational inequality problem
(3.1) is equivalent to a fixed point problem. This alternative formula is very
important form the numerical analysis point of view. Recently, many authors
studied the variational inequality (3.1) by iterative methods.
Let F be a bifunction of C × C into R, where R is the set of real numbers.
We consider the following equilibrium problem:
Find x ∈ C such that F (x, y) ≥ 0,
∀y ∈ C.
(3.2)
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Yan Hao
In this paper, the set of such x ∈ C is denoted by EP (F ), i.e.,
EP (F ) = {x ∈ C : F (x, y) ≥ 0,
∀y ∈ C}.
Numerous problems in physics, optimization and economics reduce to find a
solution of (1.1); see ([3],[5],[10],[11]). Approximating solutions of the problem
(3.2) based on iterative methods was studied by many authors, see, for example, ([4]-[7],[17],[20],[22],[25],[26],[33],[37]) and the reference therein. Putting
F (x, y) = hAx, y − xi, ∀x, y ∈ C, we see that z ∈ F P (F ) if and only if
hAz, y − zi ≥ 0, ∀y ∈ C. That is, z is a solution to the variational inequality
(3.1). Numerous problems in physics, optimization, and economics reduce to
find a solution of the problem (3.2). To study the problem (3.2), we may
assume that the bifunction F : C × C → R satisfies the following conditions:
(A1) F (x, x) = 0 for all x ∈ C;
(A2) F is monotone, i.e., F (x, y) + F (y, x) ≤ 0 for all x, y ∈ C;
(A3) for each x, y, z ∈ C, limt↓0 F (tz + (1 − t)x, y) ≤ F (x, y);
(A4) for each x ∈ C, y 7→ F (x, y) is convex and weakly lower semi-continuous.
Next, we consider the convergence of implicit iterative process (1.7) for the
equilibrium problem (3.2). To prove the main results in this section, we need
the following lemma which can be found in [3] and [4].
Lemma 3.1. Let C be a nonempty closed convex subset of H ad let F :
C × C → R be a bifunction satisfying (A1)-(A4). Then, for any r > 0 and
x ∈ H, there exists z ∈ C such that F (z, y) + 1r hy − z, z − xi ≥ 0, ∀y ∈ C.
Further, define a mapping
1
Tr x = {z ∈ C : F (z, y) + hy − z, z − xi ≥ 0,
r
∀y ∈ C}
for all r > 0 and x ∈ H. Then, the following hold:
(1) Tr is single-valued;
(2) Tr is firmly nonexpansive, i.e., for any x, y ∈ H,
kTr x − Tr yk2 ≤ hTr x − Tr y, x − yi;
(3) F (Tr ) = EP (F );
(4) EP (F ) is closed and convex.
CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY
NONEXPANSIVE MAPPINGS
175
Theorem 3.2. Let C be a nonempty closed and convex subset of a real Hilbert
space H. Let F1 be a bifunction from C × C to R satisfying (A1)-(A4) such
that EP (F1 ) is nonempty. Let {xn } be a sequence generated in the following
manner:


x0 ∈ H, chosen arbitrily
(3.3)
F1 (yn , y) + r1n hy − yn , yn − xn i ≥ 0, ∀y ∈ C,


xn = αn xn−1 + (1 − αn )yn + un , ∀n ≥ 1.
where {rn } ⊂ (0, ∞), {αn } ⊂ (0, 1) and {un } is a bounded sequence. Assume
that the following conditions are satisfied
(1) a ≤ αn ≤ b, where 0 < a < b < 1;
(2) lim inf n→∞ rn > 0;
P∞
(3)
n=1 kun k < ∞.
Then the sequence {xn } generated in (3.3) converges weakly to some point in
EP (F ).
Proof. Putting yn = Trn xn for each n ≥ 1, we from Lemma 3.1 see that
Trn is firmly nonexpansive. Whenever needed, we shall equivalently write the
implicit iteration (3.3) as
x0 ∈ H,
xn = αn xn−1 + (1 − αn )Trn xn + un ,
∀n ≥ 1,
(3.4)
Fixing p ∈ EP (F ), we see that
kxn − pk ≤ αn kxn−1 − pk + (1 − αn )kTrn xn − pk + kun k
≤ αn kxn−1 − pk + (1 − αn )kxn − pk + kun k.
This in turn implies that
kxn − pk ≤ kxn−1 − pk +
kun k
.
a
From Lemma 1.1, we obtain that limn→∞ kxn − pk exits. Next, we assume
that limn→∞ kxn − pk = r > 0. Note that
lim sup kxn−1 − p + un k ≤ r
(3.5)
n→∞
and
lim sup kTrn xn − p + un k ≤ lim sup(kxn − pk + kun k) ≤ r.
n→∞
n→∞
(3.6)
176
Yan Hao
On the other hand, we have
lim kxn −pk = lim kαn (xn−1 −p+un )+(1−αn )(Trn xn −p+un )k = r. (3.7)
n→∞
n→∞
Combining (3.5), (3.6) with (3.7), from Lemma 1.2, we obtain that
lim kxn−1 − Trn xn k = 0.
(3.8)
n→∞
From (3.1), we arrive at
xn − xn−1 = (1 − αn )(Trn xn − xn−1 ) + un .
From the conditions (3) and (3.8), we see that
lim kxn−1 − xn k = 0.
(3.9)
n→∞
Note that
kxn − Trn xn k ≤ kxn − xn−1 k + kxn−1 − Trn xn k.
In view of (3.8) and (3.9), we obtain that
lim kxn − Trn xn k = 0.
(3.9)
n→∞
Since the sequence {xn } is a bounded, we see that there exists a subsequence
{xni } of {xn } such that xni ⇀ q.
Next, we show that q ∈ EP (T ). Since yn = Trn xn , we have
F (yn , y) +
1
hy − yn , yn − xn i ≥ 0,
rn
∀y ∈ C.
It follows from (A2) that
hy − yn ,
and hence
hy − yni ,
y
yn − x n
i ≥ F (y, yn )
rn
yni − xni
i ≥ F (y, yni ).
rni
−x
Since nirn ni → 0, yni ⇀ q and (A4), we have F (y, q) ≤ 0 for all y ∈ C. For
i
t with 0 < t ≤ 1 and y ∈ C, let yt = ty + (1 − t)q. Since y ∈ C and q ∈ C, we
have yt ∈ C and hence F (yt , q) ≤ 0. So, from (A1) and (A4), we have
0 = F (yt , yt ) ≤ tF (yt , y) + (1 − t)F (yt , q) ≤ tF (yt , y).
CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY
NONEXPANSIVE MAPPINGS
177
That is, F (yt , y) ≥ 0. It follows from (A3) that F (q, y) ≥ 0 for all y ∈ C and
hence q ∈ EP (F ). This proves that q ∈ EP (T ).
Finally, we show that the sequence {xn } converges weakly to q. Suppose
the contrary holds. It follows that there exists some subsequence {xnj } of
{xn } such that xnj ⇀ q̄ and q 6= q̄. By the same method as given above, we
can prove that q̄ ∈ EP (T ). Put
lim kxn − qk = d1
n→∞
and
lim kxn − q̄k = d2 ,
n→∞
where d1 and d2 are two nonnegative numbers. In view of Opial’s condition,
we see that
d1 = lim inf kxni −qk < lim inf kxni −q̄k = lim inf kxnj −q̄k < lim inf kxnj −qk = d1 .
i→∞
j→∞
j→∞
j→∞
This is a contradiction. Hence q̄ = q. This shows that the sequence {xn }
converges weakly to q. The proof is completed.
Acknowledgments
The author are extremely grateful to the referee for useful suggestions that
improved the contents of the paper.
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Zhejiang Ocean University
School of Mathematics, Physics and Information Science
Zhoushan 316004, China
Email: zjhaoyan@yahoo.cn