STRONG CONVERGENCE THEOREMS FOR FIXED POINT PROBLEMS OF ASYMPTOTICALLY QUASI-ϕ-NONEXPANSIVE MAPPINGS IN THE INTERMEDIATE SENSE

Journal of Applied Mathematics & Informatics.
2014.
Sep,
32(5_6):
621-633

- Received : May 22, 2014
- Accepted : July 24, 2014
- Published : September 30, 2014

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In this paper, we introduce a general iterative algorithm for asymptotically quasi-
ϕ
-nonexpansive mappings in the intermediate sense to have the strong convergence in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding results announced by many authors.
AMS Mathematics Subject Classification : 47H09, 47H10, 47H17.
E
be a real Banach space with the dual space
E
^{*}
. Let
C
be a nonempty closed convex subset of
E
. Let
T
:
C
→
C
be a nonlinear mapping. We denote by
F
(
T
) the set of fixed points of
T
.
A mapping
T
:
C
→
C
is said to be nonexpansive if
Three classical iteration processes are often used to approximate a fixed point of nonexpansive mapping. The first one is introduced by Halpern
[3]
and is defined as follows: Take an initial point
x
_{0}
∈
C
arbitrarily and define {
x_{n}
} recursively by
where
is a sequence in the interval [0, 1]. The second iteration process is now known as Mann’s iteration process
[6]
which is defined as
where the initial point
x
_{1}
is taken in
C
arbitrarily and the sequence
is in the interval [0,1]. The third iteration process is referred to as Ishikawa’s iteration process
[5]
which is defined recursively by
where the initial point
x
_{1}
is taken in
C
arbitrarily,
are sequences in the interval [0, 1].
In general not much is known regarding the convergence of the iteration processes (1.1)-(1.3) unless the underlying space
E
has elegant properties which we briefly mention here.
Recently, Matsushita and Takahashi
[7]
proved strong convergence theorems for approximation of fixed points of relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. More precisely, they proved the following theorem.
Theorem 1.1.
Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let T be a relatively nonexpansive mapping from C into itself and let
{
α_{n}
}
be a sequence of real numbers such that
0 ≤
α_{n}
< 1
and
lim sup
_{n→∞}
α_{n}
< 1.
Suppose that
{
x_{n}
}
is given by
where J is the duality mapping on E. If F
(
T
)
is nonempty, then
{
x_{n}
}
converges strongly to
Π
_{F(T)}
x
_{0}
,
where
Π
_{F(T)}
is the generalized projection from C onto F
(
T
).
In
[4]
, Hao introduced the following iterative scheme for approximating a fixed point of asymptotically quasi-
ϕ
-nonexpansive mappings in the intermediate sense in a reflexive, strictly convex and smooth Banach space:
x
_{0}
∈
E
,
C
_{1}
=
C
,
x
_{1}
= Π
_{C1}
x
_{0}
,
where
ξ_{n}
= max{0, sup
_{p∈F(T),x∈C}
(
ϕ
(
p
,
T^{n}x
) −
ϕ
(
p
,
x
))}.
Motivated by the fact above, the purpose of this paper is to prove a strong convergence theorem for finding a fixed point of asymptotically quasi-
ϕ
-nonexpansive mappings in the intermediate sense in a reflexive, strictly convex and smooth Banach space, which has the Kadec-Klee property.
E
be a real Banach space and let
E
^{*}
be the dual space of
E
. The duality mapping
J
:
E
→ 2
^{E*}
is defined by
By Hahn-Banach theorem,
J
(
x
) is nonempty.
The modulus of convexity of
E
is the function
δ_{E}
: (0, 2] → [0, 1] defined by
E
is said to be uniformly convex if ∀
ε
∈ (0, 2], there exists a
δ
=
δ
(
ε
) > 0 such that for
x, y
∈
E
with ∥
x
∥ ≤ 1, ∥
y
∥ ≤ 1 and ∥
x
−
y
∥ ≥
ε
, then
Equivalently,
E
is uniformly convex if and only if
δ_{E}
(
ε
) > 0, ∀
ε
∈ (0, 2].
E
is strictly convex if for all
x, y
∈
E
,
x
≠
y
, ∥
x
∥ = ∥
y
∥ = 1, we have ∥
λx
+(1−
λ
)
y
∥ < 1, ∀
λ
∈ (0, 1). The space
E
is said to be smooth if the limit
exists for all
x, y
∈
S
(
E
) = {
z
∈
E
: ∥
z
∥ = 1}. It is also said to be uniformly smooth if the limit exists uniformly in
x, y
∈
S
(
E
).
It is well known that if
E
is uniformly smooth, then
J
is norm-to-norm uniformly continuous on each bounded subset of
E
. If
E
is smooth, then
J
is single-valued.
Recall that a Banach space
E
has the Kadec-Klee property if for any sequence {
x_{n}
} ⊂
E
and
x
∈
E
with
x_{n}
⇀
x
and ∥
x_{n}
∥ → ∥
x
∥, then ∥
x_{n}
−
x
∥ → 0 as
n
→ ∞. It is well known that if
E
is a uniformly convex Banach space, then
E
has the Kadec-Klee property.
In what follows, we always use
ϕ
:
E
×
E
→ ℝ to denote the Lyapunov functional defined by
It follows from the definition of
ϕ
that
and
Following Alber
[1]
, the generalized projection Π
_{C}
:
E
→
C
is defined by
The existence and uniqueness of the operator Π
_{C}
follows from the properties of the function
ϕ
(
x, y
) and strict monotonicity of mapping
J
(see
[1
,
2
,
10]
).
Lemma 2.1
(
[1]
).
Let E be a reflexive, strictly convex and smooth Banach space and C be a nonempty closed convex subset of E. Then the following conclusions hold:
Remark 2.1.
If
E
is a real Hilbert space, then
ϕ
(
x, y
) = ∥
x
−
y
∥
^{2}
and Π
_{C}
is the metric projection
P_{C}
of
E
onto
C
.
Definition 2.2
. Let
C
be a nonempty closed convex subset of
E
and let
T
be a mapping from
C
into itself. A point
p
∈
C
is said to be an asymptotic fixed point of
T
if
C
contains a sequence {
x_{n}
}, which converges weakly to
p
and lim
_{n→∞}
∥
x_{n}
−
Tx_{n}
∥ = 0.
The set of asymptotic fixed points of
T
is denoted by
Definition 2.3.
A mapping
T
:
C
→
C
is said to be
(1) relatively nonexpansive if
and
for all
x
∈
C
and
p
∈
F
(
T
);
(2) quasi-
ϕ
-nonexpansive if
F
(
T
)≠
ϕ
and
for all
x
∈
C
and
p
∈
F
(
T
);
(3) asymptotically quasi-
ϕ
-nonexpansive if
F
(
T
)≠
ϕ
and there exists a sequence {
k_{n}
} ⊂ [0,∞) with
k_{n}
→ 1 as
n
→ ∞ such that
for all
x
∈
C, p
∈
F
(
T
) and
n
≥ 1;
(4) asymptotically quasi-
ϕ
-nonexpansive in the intermediate sense if
F
(
T
)≠
ϕ
and
Put
Remark 2.2.
From the definition, it is obvious that
ξ_{n}
→ 0 as
n
→ ∞ and
Remark 2.3.
(1) It is easy to see that the class of quasi-
ϕ
-nonexpansive mappings contains the class of relatively nonexpansive mappings.
(2) The class of asymptotically quasi-
ϕ
-nonexpansive mappings is more general than the class of relatively asymptotically nonexpansive mappings.
(3) The class of asymptotically quasi-
ϕ
-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasi-nonexpansive mappings in the intermediate sense in the framework.
Recall that
T
is said to be asymptotically regular on
C
if for any bounded subset
K
of
C
,
Definition 2.4.
A mapping
T
:
C
→
C
is said to be closed if for any sequence {
x_{n}
} ⊂
C
with
x_{n}
→
x
and
Tx_{n}
→
y
,
T_{n}
=
y
.
Lemma 2.5
(
[4]
).
Let E be a reflexive, strictly convex and smooth Banach space such that both E and E
^{*}
have the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let T
:
C
→
C
be a closed and asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Then
F
(
T
)
is a closed convex subset of C.
Theorem 3.1.
Let E be a reflexive, strictly convex and smooth Banach space such that both E and E
*
have the Kadec-Klee property. Let C be a nonempty closed convex subset of E. Let T
:
C
→
C
be a closed, asymptotically regu-lar and asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Let
{
α_{n}
}
be a sequence in
[0, 1]
and
{
β_{n}
}
be a sequence in
(0, 1)
satisfying the following conditions:
Let
{
x_{n}
}
be a sequence generated by
where
ξ_{n}
= max{0, sup
_{p∈F(T);x∈C}
(
ϕ
(
p
,
T^{n}x
) −
ϕ
(
p, x
))}, Π
_{Cn+1}
is the general-ized projection of E onto
C
_{n+1}
.
If
F
(
T
)
is bounded in C, then
{
x_{n}
}
converges strongly to
Π
_{F(T)}
x
_{1}
.
Proof
. It follows from Lemma 2.2 that
F
(
T
) is a closed convex subset of
C
, so that Π
_{F(T)}
x
is well defined for any
x
∈
C
.
We split the proof into six steps.
Step 1. We first show that
C_{n}
,
n
≥ 1, is nonempty, closed and convex.
It is obvious that
C
_{1}
=
C
is closed and convex. Suppose that
C_{n}
is closed and convex for some
n
≥ 2. For
z
_{1}
,
z
_{2}
∈
C
_{n+1}
, we see that
z
_{1}
,
z
_{2}
∈
C_{n}
. It follows that
z
=
tz
_{1}
+ (1 −
t
)
z
_{2}
∈
C_{n}
, where
t
∈ (0, 1). Notice that
and
These are equivalent to
and
Multiplying
t
and 1 −
t
on both sides of (3.2) and (3.3), respectively, we obtain that
That is,
Therefore, we have
This implies that
C
_{n+1}
is closed and convex for all
n
≥ 1.
Step 2. We show that
F
(
T
) ⊂
C_{n}
, ∀
n
≥ 1.
For
n
= 1, we have
F
(
T
) ⊂
C
_{1}
=
C
. Now, assume that
F
(
T
) ⊂
C_{n}
for some
n
≥ 2. Put
w_{n}
=
J
^{−1}
(
β_{n}Jx_{n}
+ (1−
β_{n}
)
JT^{n}x_{n}
). For each
x
^{*}
∈
F
(
T
), we obtain from (2.2) and (2.3) that
and
Therefore, we have
So,
x
^{*}
∈
C
_{n+1}
. It implies that
F
(
T
) ⊂
C
_{n+1}
.
Step 3. We prove that {
x_{n}
} is bounded and lim
_{n→∞}
ϕ
(
x_{n}
,
x
_{1}
) exists.
Since
x_{n}
= Π
_{Cn}
x
_{1}
, we have from Lemma 2.1 that
Again, since
F
(
T
) ⊂
C_{n}
, we have
It follows from Lemma 2.1 that for each
u
∈
F
(
T
) and for each
n
≥ 1,
Therefore, {
ϕ
(
x_{n}
,
x
_{1}
)} is bounded. By virtue of (2.1), {
x_{n}
} is also bounded. Again, since
x_{n}
= Π
_{Cn}
x
_{1}
,
x
_{n+1}
= Π
_{Cn+1}
x
_{1}
and
x
_{n+1}
∈
C
_{n+1}
⊂
C_{n}
for all
n
≥ 1, we have
This implies that {
ϕ
(
x_{n}
,
x
_{1}
)} is nondecreasing and bounded. Hence, lim
_{n→∞}
ϕ
(
x_{n}
,
x
_{1}
) exists.
Step 4. Next, we prove that
where
is some point in
C
.
Now, since {
x_{n}
} is bounded and the space
E
is reflexive, we may assume that there exists a subsequence {
x_{ni}
} of {
x_{n}
} such that
Since
C_{n}
is closed and convex, it is easy to see that
for each
n
≥ 1. This implies that
On the other hand, it follows from the weak lower semicontinuity of the norm that
which implies that
as
n_{i}
→ ∞. Hence,
as
n_{i}
→ ∞. In view of the Kadec Klee property of
E
, we see that
as
n_{i}
→ ∞. If there exists another subsequence
such that
we have
which implies
This shows that
Step 5. Now we prove that
Since
and lim
_{n}
_{→∞}
ϕ(
x_{n}
,
x
_{1}
) exists, we see that
Hence, we have
Since
x
_{n+1}
∈
C
_{n+1}
,
x_{n}
→
and
α_{n}
→ 0, it follows from (3.1) and Remark 2.2 that
as
n
→ ∞. This implies that
Therefore we obtain
and so
This shows that {
Jy_{n}
} is bounded. Since
E
is reflexive,
E
^{*}
is reflexive. Without loss of generality, we can assume that
In view of reflexivity of
E
, we see that
J
(
E
) =
E
^{*}
. Hence, there exists
y
∈
E
such that
This implies that
J
(
y_{n}
) ⇀
Jy
. And
Taking lim inf
_{n→∞}
for both sides of (3.7), we have from (3.4) that
which shows that
and so
It follows from (3.6) and the Kadec-Klee property of
E
^{*}
that
Since
J
^{−1}
is norm-weak-continuous, we have
It follows from (3.5),(3.8) and the Kadec-Klee property of
E
that we have
On the other hand, since {
x_{n}
} is bounded and
T
is asymptotically quasi-
ϕ
-nonexapnsive in the intermediate sense, for any given
p
∈
F
(
T
), we have from (2.3) that
This implies that {
T^{n}x_{n}
} is bounded. Since
it implies that {
w_{n}
} is also bounded. From (3.1), we have
It follows from (3.9) that
as
n
→ ∞. Since
J
^{−1}
is norm-weaklycontinuous, this implies that
as
n
→ ∞. Note that
This together with (3.10) shows that
as
n
→ ∞. Since
we have
Since
By condition (ii) and (3.11), we have that
Since
J
^{−1}
is norm-weakly-continuous, this implies that
It follows from (3.12) that
This together with (3.13) and the Kadec-Klee property of
E
shows that
as
n
→ ∞. Again, by the asymptotic regularity of
T
, we have
as
n
→ ∞. That is,
It follows from the closedness of
T
that
Step 6. Finally, we prove that
Let
w
= Π
_{F(T)}
x
_{1}
. Since
w
∈
F
(
T
) ⊂
C_{n}
and
x_{n}
= Π
_{Cn}
x
_{1}
, we have
This implies that
From the definition of
and (3.14), we see that
This completes the proof.
Remark 3.1.
If we take
α_{n}
= 0 for all
n
∈ ℕ, then the iterative scheme (3.1) reduces to following scheme:
where
which is (1.2) and an improvement to (1.1).
In the framework of Hilbert spaces, Theorem 3.1 is reduced to the following.
Corollary 3.2.
Let E be a Hilbert space. Let C be a nonempty closed convex subset of H. Let T
:
C
→
C
be a closed, asymptotically regular and asymptot-ically quasi-ϕ-nonexpansive mapping in the intermediate sense. Let
{
α_{n}
}
be a sequence in
[0, 1]
and
{
β_{n}
}
be a sequence in
(0, 1)
satisfying the following condi-tions:
Let
{
x_{n}
}
be a sequence generated by
where
is the metric projection from E onto
C
_{n+1}
.
If
F
(
T
)
is bounded in
C
,
then
{
x_{n}
}
converges strongly to
P
_{F(T)}
x
_{1}
.
Proof
. . If
E
is a Hilbert space, then
J
=
I
(the identity mapping) and
ϕ
(
x, y
) = ∥
x
−
y
∥
^{2}
. We can obtain the desired conclusion easily from Theorem 3.1. This completes the proof.
If
T
is quasi-
ϕ
- nonexpansive, then Theorem 3.1 is reduced to the following without involving boundedness of
F
(
T
) and asymptotically regularity on
C
.
Corollary 3.3.
Let E be a reflexive, strictly convex and smooth Banach space such that both E and E
^{*}
have the Kadec-Klee property. Let C be a nonempty closed convex subset of E. Let T
:
C
→
C
be a closed, quasi-ϕ-nonexpansive mapping with F
(
T
) ≠
ϕ
.
Let
{
α_{n}
}
be a sequence in
[0, 1]
and
{
β_{n}
}
be a sequence in
(0, 1)
satisfying the following conditions:
Let
{
x_{n}
}
be a sequence generated by
where
is the generalized projection of
E
onto
C
_{n+1}
.
Then
{
x_{n}
}
converges strongly to
Π
_{F(T)}
x
_{1}
.
Remark 3.2.
(1) By Remark 3.1, Theorem 3.1 extends Theorem 2.1 of Hao
[4]
.
(2) Theorem 3.1 generelized Theorem 3.1 of Matsushita and Takahashi
[7]
in the following respects:
(3) Corollary 3.1 generalized and improves Corollary 2.5 of Hao
[4]
, Theorem 3.4 of Nakajo and Takahashi
[8]
and Theorem 2.1 of Su and Qin
[9]
in the following aspects:
Jae Ug Jeong received M.Sc. from Busan National University and Ph.D at Gyeongsang National University. Since 1982 he has been at Dongeui University. His research interests include fixed point theory and variational inequality problems.
Department of Mathematics, Dongeui University, Busan 614-714, South Korea.
e-mail:jujeong@deu.ac.kr

fixed point
;
nonexpansive mapping
;
asymptotically quasi-ϕ-nonexpansive mapping
;
relatively nonexpansive mapping

1. Introduction

Let
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2. Preliminaries

Let
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- (a)ϕ(x,ΠCy) +ϕ(ΠCy, y) ≤ϕ(x, y), ∀x∈C, y∈E;
- (b) If x∈E and z∈C, then z= ΠCx⇔ ⟨z−y, Jx−Jz⟩ ≥ 0, ∀y∈C;
- (c) For x, y∈E,ϕ(x, y) = 0if and only if x=y.

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3. Main results

- (i)limn→∞αn= 0;
- (ii)0 < lim infn→∞βn≤ lim supn→∞βn< 1.

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- (i)limn→∞αn= 0;
- (ii)0 < lim infn→∞βn≤ lim supn→∞βn< 1.

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- (i)limn→∞αn= 0;
- (ii)0 < lim infn→∞βn≤ lim supn→∞βn< 1.

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- (i) from the relatively nonexpansive mapping to the asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense;
- (ii) from a uniformly convex and uniformly smooth Banach space to a reflexive, strictly convex and smooth Banach space.

- (i) Algorithm of Corollary 3.1 is different from algorithms in[4,8,9].
- (ii) Corollary 3.1 includes Corollary 2.5 of Hao[4]as a special case.
- (iii) The setQnin[8,9]have been relaxed.

BIO

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Citing 'STRONG CONVERGENCE THEOREMS FOR FIXED POINT PROBLEMS OF ASYMPTOTICALLY QUASI-ϕ-NONEXPANSIVE MAPPINGS IN THE INTERMEDIATE SENSE
'

@article{ E1MCA9_2014_v32n5_6_621}
,title={STRONG CONVERGENCE THEOREMS FOR FIXED POINT PROBLEMS OF ASYMPTOTICALLY QUASI-ϕ-NONEXPANSIVE MAPPINGS IN THE INTERMEDIATE SENSE}
,volume={5_6}
, url={http://dx.doi.org/10.14317/jami.2014.621}, DOI={10.14317/jami.2014.621}
, number= {5_6}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={JEONG, JAE UG}
, year={2014}
, month={Sep}