# Strong Convergence for the Split Common Fixed-Point Problem for Total Quasi-Asymptotically Nonexpansive Mappings in Hilbert Space

Authors: Lawan Bulama Mohammed; A. Kılıçman
Journal: Abstract and Applied Analysis
 (2015)

---

## Abstract

In this paper, we study and modify the algorithm of Kraikaew and Saejung for the class of total quasi-asymptotically nonexpansive case so that the strong convergence is guaranteed for the solution of split common fixed-point problems in Hilbert space. Moreover, we justify our result through an example. The results presented in this paper not only extend the result of Kraikaew and Saejung but also extend, improve, and generalize some existing results in the literature.


---

## Body

## 1. Introduction

Let·
,
· be an inner product space, · the corresponding norm, E a Banach space, H
1, H
2 two Hilbert spaces, A
:
H
1
→
H
2 a bounded linear operator, and A
*
:
H
2
→
H
1 an adjoint of A. Let {
C
i
}
i
=
1
p and {
Q
j
}
j
=
1
r be a nonempty, closed, convex subsets of H
1 and H
2, respectively.A Banach spaceE is said to satisfyOpial’s condition (see [1]) if, for any sequence {
x
n
} in E, x
n
⇀
x as n
→
∞ implies that
(1)
lim
⁡
inf
⁡
n
→
∞
x
n
-
x
<
lim
⁡
inf
⁡
n
→
∞
x
n
-
y
,
∀
y
∈
E
,
y
≠
x
.
And also, a Banach space E is said to haveKadec-Klee property (see [1]), if, for any sequence {
x
n
} in E, x
n
⇀
x and x
n
→
x as n
→
∞ implies that
(2)
x
n
⟶
x
,
as
n
⟶
∞
.Remark 1.
It is well known that each Hilbert space satisfied Opial and Kadec-Klee property.The mappingT
:
H
→
H is said to be demiclosed at zero, if any sequence {
x
n
} in H there holds the following implication:
(3)
x
n
⇀
x
,
T
x
n
⟶
0
,
as
n
⟶
∞
⟹
T
x
=
0
.
T is said to be α-strongly quasi-nonexpansive if there exists α
>
0 with the property T
x
-
z
2
≤
x
-
z
2
-
α
x
-
T
x
2,  ∀
x
∈
H and z
∈
Fix
⁡
(
T
); this is equivalent to
(4)
x
-
z
,
T
x
-
x
≤
-
1
-
α
2
x
-
T
x
2
,
∀
x
∈
H
,
z
∈
Fix
⁡
T
.
T is said to be quasi-nonexpansive, if Fix
⁡
(
T
)
≠
∅ such that p
-
T
x
≤
p
-
x,  ∀
p
∈
Fix
⁡
(
T
) and x
∈
H, and {
k
n
}-quasi-asymptotically nonexpansive mapping, if Fix
⁡
(
T
)
≠
∅ and there exists a sequence {
k
n
}
⊆
[
1
,
∞
) with k
n
→
1 such that, for each n
≥
1, p
-
T
n
x
2
≤
k
n
p
-
x
2,  ∀
p
∈
Fix
⁡
(
T
) and x
∈
H, and it is said to be (
{
v
n
}
,
{
μ
n
}
,
ξ
)-total quasi-asymptotically nonexpansive mapping if Fix
⁡
(
T
)
≠
∅; and there exist nonnegative real sequences {
v
n
}, {
μ
n
} in [
0
,
∞
) with v
n
→
0 and μ
n
→
0 and a strictly continuous function ξ
:
R
+
→
R
+ with ξ
(
0
)
=
0 such that, for each n
≥
1,
(5)
p
-
T
n
x
2
≤
p
-
x
2
+
v
n
ξ
p
-
x
+
μ
n
,
∀
p
∈
Fix
⁡
G
,
x
∈
H
.Remark 2.
It is known that, the class of quasi-nonexpansive mapping contained in the class of{
k
n
}-quasi-asymptotically nonexpansive mapping and the class of {
k
n
}-quasi-asymptotically nonexpansive mapping is contained in the class of (
{
v
n
}
,
{
μ
n
}
,
ξ
)-total quasi-asymptotically nonexpansive mapping, see [2].The mappingT is said to be uniformly L-Lipschitzian if ∃ a constant L
>
0 such that, for each n
≥
1, T
n
x
-
T
n
y
≤
L
x
-
y,  ∀
x, y
∈
H, and it is said to be semicompact, if, for any bounded sequence x
n
⊂
H with lim
⁡
n
→
∞
x
n
-
T
x
n
=
0, there exists subsequence {
x
n
i
}
⊂
{
x
n
} such that {
x
n
i
} converges strongly to some point x
*
∈
H.The convex feasibility problems (CFP) are finding a vectorx
*
∈
H
1 satisfying
(6)
x
*
∈
⋂
i
=
1
p
C
i
.
The problem of solving 6 has been intensively studied by numerous authors due to its various application in several physical problems such as approximation theorem, image recovery, signal processing, control theory, biomedical engineering, communication and geophysics (see [3–5]) and reference therein.In 2005, Censor et al. (see [6]) introduced and studied the problem of multiple set split feasibility problems (MSSFP) which is formulated as finding a vector x
*
∈
H
1 with the property
(7)
x
*
∈
⋂
i
=
1
p
C
i
,
A
x
*
∈
⋂
j
=
1
r
Q
j
.
If, in 7, we take p
=
r
=
1, we get
(8)
x
*
∈
C
,
A
x
*
∈
Q
.
Equation 8 is known as the split feasibility problems (SFP) (see [7]), where C and Q are nonempty, closed, and convex subsets of H
1 and H
2, respectively. Since every closed convex subset of Hilbert space is the fixed-point set of its associating projection, then 6 and 7 become
(9)
x
*
∈
⋂
i
=
1
p
Fix
⁡
U
i
,
(10)
x
*
∈
⋂
i
=
1
p
Fix
⁡
U
i
,
A
x
*
∈
⋂
j
=
1
r
Fix
⁡
T
j
.
Equations 9 and 10 are called the common fixed-point problems (CFPP) and split common fixed-point problems (SCFPP), respectively, where U
i
:
H
1
→
H
1  (i
=
1,2
,
3
,
…
,
p) and T
j
:
H
2
→
H
2  (j
=
1,2
,
3
,
…
,
r) are some nonlinear operators.If we takep
=
r
=
1, problem 10 is reduced to find a point x
*
∈
H
1 with property
(11)
x
*
∈
Fix
⁡
U
,
A
x
*
∈
Fix
⁡
T
.
Equation 11 is known as the two-set SCFPP.In 2009, Censor and Segal [8] introduced the concept of SCFPP 10 in finite dimensional Hilbert space, who invented an algorithm for solving 11 which generate a sequence {
x
n
} according to the following iterative procedure:
(12)
x
n
+
1
≔
U
x
n
+
γ
A
*
T
-
I
A
x
n
,
∀
n
≥
0
,
where the initial guess x
0
∈
H is choosing arbitrarily and 0
<
γ
<
2
/
A
2.In 2011, Moudafi [9] studied the convergence properties of relaxed algorithm for solving 10 for the class of quasi-nonexpansive operators T such that (
I
-
T
) is demiclosed at zero and he obtained the weak convergence results. Note that, in finite dimensional Hilbert space, weak and strong convergence are equivalent. Differently, in infinite dimensional cases, they are not the same. Moudafi’s results guarantee only weak convergence results. In 2013, Mohammed [10, 11] utilized the strongly quasi-nonexpansive operators and quasi-nonexpansive operators to solve Moudafi’s algorithm and he obtained weak and strong convergence results, respectively.In 2014, Kraikaew and Saejung [12] also modified Moudafi’s algorithm [9] and they obtained the strong convergent results as shown below.Theorem 3 (see [12]).
letU
:
H
1
→
H
1 be a strongly quasi-nonexpansive operator and let T
:
H
2
→
H
2 be a quasi-nonexpansive operator such that both (
I
-
U
) and (
I
-
T
) are demiclosed at zero. Let A
:
H
1
→
H
2 be a bounded linear operator with L
=
A
A
*. Suppose that Γ
≠
∅. Let {
x
n
}
⊂
H
1 be a sequence generated by
(13)
x
0
∈
H
,
x
n
+
1
=
α
n
x
0
+
1
-
α
n
U
x
n
+
γ
A
*
T
-
I
A
x
n
,
where the parameters γ and {
α
n
} satisfy the following conditions: (a)
γ
∈
(
0
,
1
/
L
);
(b)
{
α
n
}
⊂
(
0,1
)
,
lim
⁡
n
→
∞
α
n
=
0 and ∑
‍
α
n
=
∞. Then x
n
→
P
Γ
x
0.Motivated by these results, in this paper, we studied and modified the algorithm of Kraikaew and Saejung [12] for the class of total quasi-asymptotically nonexpansive mappings to solve the split common fixed-point problems 10 in the frame work of infinite dimensional Hilbert space. The results presented in this paper not only improve and extend some recent results of Kraikaew and Saejung [12], but also improve and extend some recent results of Censor and Segal [8], Moudafi [9], and Mohammed [10, 11] and many existing results.Throughout this paper, we adopt the following notations.(i)
I is the identity operator.
(ii)
Fix
⁡
(
T
) is the fixed-point set of T; that is, Fix
⁡
(
T
)
=
{
x
∈
H
:
T
x
=
x
}.
(iii)
“→” and “⇀” denote the strong and weak convergence, respectively.
(iv)
ω
ω
(
x
n
) denote the set of the cluster point of {
x
n
} in the weak topology, that is, {
∃
{
x
n
j
}
of
{
x
n
}
such
that
x
n
j
⇀
x
}.
(v)
Γ is the solution set of split common fixed-point problems 10; that is,
(14)
Γ
=
x
*
∈
C
=
⋂
i
=
1
p
Fix
⁡
U
i
,
A
x
*
∈
Q
=
⋂
j
=
1
r
Fix
⁡
T
j
.


## 2. Preliminaries

In the sequel, we will make use of the following lemmas in proving our main results.Lemma 4 (see [2]).
LetG
:
H
→
H be a (
{
v
n
}
,
{
μ
n
}
,
ξ
)-total quasi-asymptotically nonexpansive mapping. Then for each q
∈
Fix
⁡
(
G
) and x
∈
H, the following inequalities are equivalent: for each n
≥
1
(15)
q
-
G
n
x
2
≤
q
-
x
2
+
v
n
ξ
q
-
x
+
μ
n
;
2
x
-
G
n
x
,
x
-
q
≥
x
-
G
n
x
2
-
v
n
ξ
q
-
x
-
μ
n
;
2
x
-
G
n
x
,
q
-
G
n
≤
x
-
G
n
x
2
+
v
n
ξ
q
-
x
+
μ
n
.Lemma 5 (see [2]).
Let{
a
n
}, {
b
n
}, {
c
n
} be a sequences of nonnegative real numbers satisfying
(16)
a
n
+
1
≤
1
+
c
n
a
n
+
b
n
.
If ∑
c
n
<
∞ and ∑
b
n
<
∞, then lim
⁡
n
→
∞
a
n exists.Lemma 6.
LetH be a Hilbert space, C a nonempty closed convex subset of H, and P
C a metric projection of H onto C satisfying x
n
-
x
*
,
x
n
-
P
C
x
n
≤
0, and then P
C
x
n
-
x
n
≤
P
C
x
n
-
x
*,  ∀
n
≥
1.Proof.
Letx
*
∈
C; then
(17)
x
n
-
P
C
x
n
2
=
x
n
-
x
*
+
x
*
-
P
C
x
n
2
x
n
-
P
C
x
n
2
=
x
n
-
x
*
2
+
x
*
-
P
C
x
n
2
x
n
-
P
C
x
n
2
=
+
2
x
n
-
x
*
,
x
*
-
P
C
x
n
x
n
-
P
C
x
n
2
=
x
n
-
x
*
2
+
x
*
-
P
C
x
n
2
x
n
-
P
C
x
n
2
=
+
2
x
n
-
x
*
,
x
*
-
x
n
+
x
n
-
P
C
x
n
x
n
-
P
C
x
n
2
=
x
n
-
x
*
2
+
x
*
-
P
C
x
n
2
x
n
-
P
C
x
n
2
=
-
2
x
n
-
x
*
2
+
2
x
n
-
x
*
,
x
n
-
P
C
x
n
x
n
-
P
C
x
n
2
=
x
*
-
P
C
x
n
2
-
x
n
-
x
*
2
x
n
-
P
C
x
n
2
=
+
2
x
n
-
x
*
,
x
n
-
P
C
x
n
x
n
-
P
C
x
n
2
≤
x
*
-
P
C
x
n
2
,
⟹
x
n
-
P
C
x
n
≤
x
*
-
P
C
x
n
.Lemma 7 (see [5]).
If a sequence{
x
n
} is Fejer monotone with respect to nonempty closed convex subset C, then the following hold:(i)
x
n
⇀
x
*
∈
C if and only if ω
ω
⊂
C;
(ii)
the sequence{
P
C
x
n
} converges strongly to some point in C;
(iii)
ifx
n
⇀
x
*
∈
C, then x
*
=
lim
⁡
n
→
∞
P
C
x
n
.


## 3. Main Results

Theorem 8.
LetH
1, H
2 be two Hilbert spaces and G
:
H
1
→
H
1, T
:
H
2
→
H
2 be (
{
v
n
1
}
,
{
μ
n
1
}
,
ξ
1
), (
{
v
n
2
}
,
{
μ
n
2
}
,
ξ
2
)-total quasi-asymptotically nonexpansive mappings and uniformly L
1
,
L
2-Lipschitzian continuous mappings such that (
I
-
G
) and (
I
-
T
) are both demiclosed at zero. Let A
:
H
1
→
H
2 be a bounded linear operator and let A
*
:
H
2
→
H
1 be an adjoint of A with L
=
A
A
*. Let M and M
* be positive constants such that ξ
(
k
)
≤
ξ
(
M
)
+
M
*
k
2,  ∀
k
≥
0. Assume that the solution set of SCFPP 14 is nonempty, and let P
Γ be a metric projection of H
1 onto Γ satisfying x
n
-
x
*
,
x
n
-
P
Γ
x
n
≤
0. Define the sequence x
n
⊂
H
1 by
(18)
x
0
∈
H
1
,
u
n
=
I
+
γ
A
*
T
n
-
I
A
x
n
,
x
n
+
1
=
α
n
u
n
+
1
-
α
n
G
n
u
n
,
∀
n
≥
1
,
where the parameter γ, L, {
v
n
}, {
μ
n
}, {
ξ
n
}, and {
α
n
} satisfy the following conditions: (a)
0
<
α
n
<
1, γ
∈
(
0
,
1
/
L
), where L
=
max
⁡
⁡
{
L
1
,
L
2
};
(b)
v
n
=
max
⁡
⁡
{
v
n
1
,
v
n
2
}, μ
n
=
max
⁡
⁡
{
μ
n
1
,
μ
n
2
}, and ξ
=
max
⁡
⁡
{
ξ
1
,
ξ
2
}.Then, the sequence {
x
n
} defined by 18 converges strongly to x
*
∈
Γ.Proof.
To show thatx
n
→
x
* as n
→
∞, it suffices to show x
n
⇀
x
* and x
n
→
x
* as n
→
∞.
The proof is divided into five steps as follows.
Step 1. In this step, we show that, for each x
*
∈
Γ, the following limit exists:
(19)
lim
⁡
n
→
∞
x
n
-
x
*
=
lim
⁡
n
→
∞
u
n
-
x
*
.
Let x
*
∈
Γ; this implies that x
*
∈
C
≔
⋂
i
=
1
p
‍
Fix
⁡
(
U
i
) and A
x
*
∈
Q
=
⋂
j
=
1
r
‍
Fix
⁡
(
T
j
). From 18 and Lemma 4, we have
(20)
x
n
+
1
-
x
*
2
=
α
n
u
n
+
1
-
α
n
G
n
u
n
-
x
*
2
=
α
n
2
u
n
-
G
n
u
n
2
+
2
α
n
u
n
-
G
n
u
n
,
G
n
u
n
-
x
*
+
G
n
u
n
-
x
*
2
=
α
n
2
u
n
-
G
n
u
n
2
+
2
α
n
u
n
-
x
*
+
x
*
-
G
n
u
n
,
G
n
u
n
-
x
*
+
G
n
u
n
-
x
*
2
=
α
n
2
u
n
-
G
n
u
n
2
+
2
α
n
u
n
-
x
*
,
G
n
u
n
-
x
*
+
1
-
2
α
n
G
n
u
n
-
x
*
2
=
α
n
2
u
n
-
G
n
u
n
2
+
2
α
n
u
n
-
x
*
,
G
n
u
n
-
u
n
+
u
n
-
x
*
+
1
-
2
α
n
G
n
u
n
-
x
*
2
≤
-
α
n
1
-
α
n
u
n
-
G
n
u
n
2
+
v
n
α
n
ξ
u
n
-
x
*
+
μ
n
α
n
+
2
α
n
u
n
-
x
*
2
+
1
-
2
α
n
·
u
n
-
x
*
2
+
v
n
ξ
u
n
-
x
*
+
μ
n
=
-
α
n
1
-
α
n
u
n
-
G
n
u
n
2
+
1
-
α
n
v
n
ξ
·
u
n
-
x
*
+
u
n
-
x
*
2
+
1
-
α
n
μ
n
=
-
α
n
1
-
α
n
u
n
-
G
n
u
n
2
+
1
+
1
-
α
n
v
n
M
*
·
u
n
-
x
*
2
+
1
-
α
n
v
n
ξ
M
+
μ
n
.
On the other hand,
(21)
u
n
-
x
*
2
=
x
n
-
x
*
+
γ
A
*
T
n
-
I
A
x
n
2
=
x
n
-
x
*
2
+
γ
2
A
*
T
n
-
I
A
x
n
2
+
2
γ
x
n
-
x
*
,
A
*
T
n
-
I
A
x
n
,
(22)
γ
2
A
*
T
n
-
I
A
x
n
2
=
γ
2
A
*
T
n
-
I
A
x
n
,
A
*
T
n
-
I
A
x
n
=
γ
2
A
A
*
T
n
-
I
A
x
n
,
T
n
-
I
A
x
n
≤
γ
2
L
T
n
-
I
A
x
n
2
,
by Lemma 4, it follows that
(23)
2
γ
x
n
-
x
*
,
A
*
T
n
-
I
A
x
n
=
2
γ
A
x
n
-
A
x
*
,
T
n
-
I
A
x
n
≤
-
γ
T
n
-
I
A
x
n
2
+
γ
v
n
M
*
L
x
n
-
x
*
2
+
γ
v
n
ξ
M
+
μ
n
.
By substituting 22 and 23 into 21, we obtained
(24)
u
n
-
x
*
2
≤
1
+
γ
v
n
M
*
L
x
n
-
x
*
2
-
γ
1
-
γ
L
·
T
n
-
I
A
x
n
2
+
γ
v
n
ξ
M
+
μ
n
.
Substituting 24 into 20 and then simplifying, we have
(25)
x
n
+
1
-
x
*
2
≤
1
+
1
-
α
n
v
n
M
*
·
1
+
γ
v
n
M
*
L
x
n
-
x
*
2
·
m
-
γ
1
-
γ
L
T
n
-
I
A
x
n
2
·
m
+
γ
v
n
ξ
M
+
μ
n

1
+
γ
v
n
M
*
L
x
n
-
x
*
2
-
α
n
1
-
α
n
x
n
-
G
n
u
n
2
+
1
-
α
n
v
n
ξ
M
+
μ
n
≤
1
+
1
-
α
n
v
n
M
*
1
+
γ
v
n
M
*
L
·
x
n
-
x
*
2
-
γ
1
-
γ
L
T
n
-
I
A
x
n
2
-
α
n
1
-
α
n
x
n
-
G
n
u
n
2
+
1
+
1
-
α
n
v
n
M
*
γ
v
n
ξ
M
+
μ
n
+
1
-
α
n
v
n
ξ
M
+
μ
n
,
and from 25, we deduce that
(26)
x
n
+
1
-
x
*
2
≤
1
+
γ
v
n
M
*
L
+
1
-
α
n
v
n
M
*
1
+
γ
v
n
M
*
L
·
x
n
-
x
*
2
+
1
+
1
-
α
n
v
n
M
*
·
γ
v
n
ξ
M
+
μ
n
+
1
-
α
n
v
n
ξ
M
+
μ
n
.
Therefore, from 26, we have
(27)
x
n
+
1
-
x
*
2
≤
1
+
β
n
x
n
-
x
*
2
+
η
n
,
where
(28)
β
n
=
γ
v
n
M
*
L
+
1
-
α
n
v
n
M
*
1
+
γ
v
n
M
*
L
,
η
n
=
1
+
1
-
α
n
v
n
M
*
γ
v
n
ξ
M
+
μ
n
+
1
-
α
n
v
n
ξ
M
+
μ
n
.
Clearly, ∑
β
n
<
∞ and ∑
η
n
<
∞. Moreover, β
n
→
0 and η
n
→
0 as n
→
∞.
By Lemma5, we conclude that lim
⁡
n
→
∞
x
n
-
x
* exists.
We now prove that, for eachx
*
∈
Γ, lim
⁡
n
→
∞
u
n
-
x
* exists.
From25, we deduce that
(29)
γ
1
-
γ
L
T
n
-
I
A
x
n
2
≤
x
n
-
x
*
2
-
x
n
+
1
-
x
*
2
+
β
n
x
n
-
x
*
2
+
η
n
,
α
n
1
-
α
n
u
n
-
G
n
u
n
2
≤
x
n
-
x
*
2
-
x
n
+
1
-
x
*
2
+
β
n
x
n
-
x
*
2
+
η
n
.
From 29, we deduce that
(30)
lim
⁡
n
→
∞
u
n
-
G
n
u
n
=
0
,
lim
⁡
n
→
∞
A
x
n
-
T
n
A
x
n
=
0
.
From 24, 30, and the fact that lim
⁡
n
→
∞
x
n
-
x
* exists, then lim
⁡
n
→
∞
u
n
-
x
* exists. Moreover, from 20, we deduce that
(31)
lim
⁡
n
→
∞
u
n
-
x
*
=
lim
⁡
n
→
∞
x
n
-
x
*
.Step 2. In this step, we show that
(32)
lim
⁡
n
→
∞
x
n
+
1
-
x
n
=
0
,
lim
⁡
n
→
∞
u
n
+
1
-
u
n
=
0
.Proof.
It follows from18 that
(33)
x
n
+
1
-
x
n
=
α
n
u
n
+
1
-
α
n
G
n
u
n
-
x
n
=
1
-
α
n
G
n
u
n
-
u
n
+
u
n
-
x
n
=
1
-
α
n
G
n
u
n
-
u
n
+
A
*
T
n
-
I
A
x
n
,
and in view of 30, we obtain that
(34)
lim
⁡
n
→
∞
x
n
+
1
-
x
n
=
0
.
Similarly, it follows from 30 and 34 that
(35)
u
n
+
1
-
u
n
=
I
+
γ
A
*
T
n
+
1
-
I
A
x
n
+
1
x
x
m
+
I
+
γ
A
*
T
n
-
I
A
x
n
x
x
=
x
n
+
1
-
x
n
x
x
m
+
γ
A
*
T
n
+
1
-
I
A
x
n
+
1
-
γ
A
*
T
n
-
I
A
x
n
x
x
⟶
0
as
n
⟶
∞
⟹
lim
⁡
n
→
∞
u
n
+
1
-
u
n
=
0
.Step 3. In this step, we show that
(36)
u
n
-
G
u
n
⟶
0
,
A
x
n
-
T
x
n
⟶
0
,
as
n
⟶
∞
.Proof.
From the fact thatu
n
-
G
n
u
n
→
0 and u
n
+
1
-
u
n
→
0 and G is uniformly L-Lipschitzian continuous, it follows that
(37)
u
n
-
G
u
n
≤
u
n
-
G
n
u
n
+
G
u
n
-
G
n
u
n
≤
u
n
-
G
n
u
n
+
L
u
n
-
G
n
-
1
u
n
≤
u
n
-
G
n
u
n
+
L
G
n
-
1
u
n
-
G
n
-
1
u
n
-
1
+
L
u
n
-
G
n
-
1
u
n
-
1
≤
u
n
-
G
n
u
n
+
L
2
u
n
-
u
n
-
1
+
L
u
n
-
u
n
-
1
+
u
n
-
1
-
G
n
-
1
u
n
-
1
≤
u
n
-
G
n
u
n
+
L
L
+
1
u
n
-
u
n
-
1
+
L
u
n
-
1
-
G
n
-
1
u
n
-
1
⟶
0
⟹
u
n
-
G
u
n
⟶
0
.
Similarly, from the fact that A
x
n
-
T
n
A
x
n
→
0, x
n
+
1
-
x
n
→
0, and T is uniformly L-Lipschitzian continuous, it follows that A
x
n
-
T
A
x
n
→
0.Step 4. In this step, we show that
(38)
x
n
⇀
x
*
,
u
n
⇀
x
*
,
as
n
⟶
∞
.Proof.
Since{
u
n
} is bounded, then there exists a subsequence u
n
i
⊂
u
n such that
(39)
u
n
i
⇀
x
*
,
as
i
⟶
∞
.
From 39 and 36, we have
(40)
u
n
i
-
G
u
n
i
⟶
0
,
as
i
⟶
∞
.
From 39 and 40 and the fact that (
I
-
G
) is demiclosed at zero, we get that x
*
∈
Fix
⁡
(
G
).
Moreover, from18, 39, and the fact A
x
n
-
T
n
A
x
n
→
0, as n
→
∞, we have
(41)
x
n
i
=
α
n
u
n
i
-
γ
A
*
T
n
i
-
I
A
x
n
i
⟶
x
*
.
By the definition of A, we get
(42)
A
x
n
i
⇀
A
x
*
,
as
i
⟶
∞
.
In view of 36, we get
(43)
A
x
n
i
-
T
A
x
n
i
⟶
0
,
as
i
⟶
∞
.
From 42 and 43 and the fact that (
I
-
T
) is demiclosed at zero, we have A
x
*
∈
Fix
⁡
(
T
), and this implies that x
*
∈
Γ.
Now, we show thatx
* is unique.
Suppose to the contrary that there exists another subsequenceu
n
j
⊂
u
n such that u
n
j
⇀
y
*
∈
Γ with x
*
≠
y
* by virtue of 19 and opial property of Hilbert space; we have
(44)
lim
⁡
inf
⁡
i
→
∞
u
n
i
-
x
*
<
lim
⁡
inf
⁡
i
→
∞
u
n
i
-
y
*
=
lim
⁡
inf
⁡
n
→
∞
u
n
-
y
*
=
lim
⁡
inf
⁡
j
→
∞
u
n
j
-
y
*
<
lim
⁡
inf
⁡
j
→
∞
u
n
j
-
x
*
=
lim
⁡
inf
⁡
n
→
∞
u
n
-
x
*
=
lim
⁡
inf
⁡
i
→
∞
u
n
i
-
x
*
⟹
lim
⁡
inf
⁡
i
→
∞
u
n
i
-
x
*
<
lim
⁡
inf
⁡
i
→
∞
u
n
i
-
x
*
which is contradiction. Therefore u
n
⇀
x
*. By using 18 and 30, we have
(45)
x
n
=
u
n
-
γ
A
*
T
n
-
I
A
x
n
⇀
x
*
,
as
n
⟶
∞
.Step 5. In this step, we show that
(46)
x
n
⟶
x
*
,
as
n
⟶
∞
.
To show 46, it suffices to show that x
n
+
1
→
x
* as n
→
∞.Proof.
From Lemmas6, 7, 27, and the fact that β
n
→
0 and η
n
→
0, we have
(47)
x
n
+
1
-
x
*
2
≤
x
n
+
1
-
x
*
2
mmmmmmmm
≤
1
+
β
n
x
n
-
x
*
2
+
η
n
mmmmmmmm
=
x
n
-
x
*
2
+
β
n
x
n
-
x
*
2
+
η
n
mmmmmmmm
=
x
n
-
P
Γ
x
n
+
P
Γ
x
n
-
x
*
2
mmmmmmmm
+
β
n
x
n
-
x
*
2
+
η
n
mmmmmmmm
≤
4
P
Γ
x
n
-
x
*
2
+
β
n
x
n
-
x
*
2
+
η
n
⟹
x
n
+
1
-
x
*
2
≤
4
P
Γ
x
n
-
x
*
2
mmmmmmmnnnnnnm
+
β
n
x
n
-
x
*
2
+
η
n
⟹
lim
⁡
n
→
∞
x
n
+
1
-
x
*
2
≤
4
lim
⁡
n
→
∞
P
Γ
x
n
-
x
*
2
mmmmmmmnnnvvvnnnm
+
lim
⁡
n
→
∞
β
n
x
n
-
x
*
2
mmmmmmmnnnvvvnnnm
+
lim
⁡
n
→
∞
η
n
⟹
lim
⁡
n
→
∞
x
n
+
1
-
x
*
2
=
0
.
From38 and 46, we conclude that x
n
→
x
*, as n
→
∞.Corollary 9.
LetH
1, H
2
A, and A
*, be as in Theorem 8. Let G and T be {
k
n
1
}
,
(
{
k
n
2
}
) quasi-asymptotically nonexpansive and uniformly L
1
,
L
2-Lipschitzians continuous mappings such that (
I
-
G
) and (
I
-
T
) are both demiclosed at zero. Let L
=
A
A
*, and let M and M
* be constants such that ξ
(
k
)
≤
ξ
(
M
)
+
M
*
k
2,  ∀
k
≥
0. Assume that the solution set of SCFPP 14 is nonempty, and let P
Γ be a metric projection of H
1 onto Γ satisfying x
n
-
x
*
,
x
n
-
P
Γ
x
n
≤
0. Let the sequence {
x
n
} be defined as in Theorem 8 where the parameters α
n, γ, {
k
n
}, and L satisfy the following conditions: (a)
α
n
⊂
(
0,1
)
,
γ
∈
(
0
,
1
/
L
), where L
=
max
⁡
{
L
1
,
L
2
};
(b)
k
n
=
max
⁡
{
k
n
1
,
k
n
2
}.Then the sequence {
x
n
} defined as in Theorem 8 converges strongly to x
*
∈
Γ.Proof.
By Remark2  G and T are (
{
v
n
}
,
{
μ
n
}
,
ξ
)-total quasi-asymptotically nonexpansive mappings with {
v
n
}
=
{
k
n
-
1
}, μ
n
=
0, and ξ
(
k
)
=
k
2, ∀
k
≥
0. Therefore, all the conditions in Theorem 8 are satisfied. The conclusions of this corollary follow directly from Theorem 8.Corollary 10.
LetH
1, H
2
A, and A
* be as in Theorem 8. Let G and T be two quasi-nonexpansive and uniformly L
1
,
L
2-Lipschitzian continuous mappings such that (
I
-
G
) and (
I
-
T
) are both demiclosed at zero. Let L
=
A
A
*, and let M and M
* be positive constants such that ξ
(
k
)
≤
ξ
(
M
)
+
M
*
k
2,  ∀
k
≥
0. Assume that the solution set of SCFPP 14 is nonempty, and let P
Γ be a metric projection of H onto Γ satisfying x
n
-
x
*
,
x
n
-
P
Γ
x
n
≤
0. Let the sequence {
x
n
} be defined as in Theorem 8 where the parameters {
α
n
}, γ, and L, satisfy the following conditions:(a)
{
α
n
}
⊂
(
0,1
), γ
∈
(
0
,
1
/
L
), where L
=
max
⁡
{
L
1
,
L
2
}.Then, the sequence {
x
n
} defined as in 18 converges strongly to x
*
∈
Γ.Proof.
By Remark2  G and T are (
{
1
}
)-quasi-asymptotically nonexpansive mappings. Therefore, all the conditions in Corollary 9 are satisfied. The conclusions of this corollary follow directly from Corollary 9.Now we give an example of our theorem.Example 11.
LetB be a unit ball in a real Hilbert space l
2, and let T
:
B
→
B be a mapping define by
(48)
T
:
x
1
,
x
2
,
x
3
,
…
⟶
0
,
x
1
2
,
a
2
x
2
,
a
3
x
3
,
…
,
x
1
,
x
2
,
x
3
,
…
∈
B
,
where {
a
i
} is a sequence in (
0,1
) such that ∏
i
=
2
∞
(
a
i
)
=
1
/
2.
It is proved in Goebel and Kirk [13] that(a)
T
x
-
T
y
≤
2
x
-
y,
(b)
T
n
x
-
T
n
y
≤
2
∏
i
=
2
n
(
a
i
)
x
-
y  ∀
x
,
y
∈
B and n
≥
2.Let k
1
1
/
2
=
2 such that k
n
1
/
2
=
2
∏
i
=
2
n
(
a
i
), for n
≥
2; then
(49)
lim
⁡
n
→
∞
k
n
=
lim
⁡
n
→
∞
2
∏
i
=
2
n
a
i
=
1
.
Let v
n
=
k
n
-
1,  ∀
n
≥
1, let ξ
(
t
)
=
t
2,  ∀
t
≥
0, let and {
μ
n
} be a nonnegative real sequence such that μ
n
→
∞ as n
→
∞. From (a), (b) and  ∀
x, y
∈
B and n
≥
1, we have
(50)
T
n
x
-
T
n
y
2
≤
x
-
y
2
+
v
n
x
-
y
2
+
μ
n
.
Again, since 0
∈
B and 0
∈
Fix
⁡
(
T
), this implies that Fix
⁡
(
T
)
≠
∅. From the above equation, we have
(51)
p
-
T
n
y
2
≤
p
-
y
2
+
v
n
ξ
p
-
y
+
μ
n
.
This show that T is total quasi-asymptotically nonexpansive mapping.Example 12.
LetH
1
=
H
2
=
l
2 be a real Hilbert spaces, C
,
Q two unit balls in l
2, and T
:
C
→
C, G
:
Q
→
Q two mappings defined by
(52)
T
:
x
1
,
x
2
,
x
3
,
…
⟶
0
,
x
1
2
,
a
2
x
2
,
a
3
x
3
,
…
,
x
1
,
x
2
,
x
3
,
…
∈
C
,
G
:
y
1
,
y
2
,
y
3
,
…
⟶
0
,
y
1
2
,
b
2
y
2
,
b
3
y
3
,
…
,
y
1
,
y
2
,
y
3
,
…
∈
Q
,
such that (
I
-
T
) and (
I
-
G
) are both demiclosed at zero, where {
a
i
} and {
b
i
} are sequences in (
0,1
) such that ∏
i
=
2
∞
(
a
i
)
=
∏
i
=
2
∞
(
b
i
)
=
1
/
2. Let A, A
*, M, M
*
ξ
(
k
), and P
Γ be as in Theorem 8. And assume that conditions (a)–(b) in Theorem 8 are satisfied. Then, the sequence {
x
n
} defined in Theorem 8 converges strongly to x
*
∈
Γ.Proof.
By Example11, it follows that G and T are both (
{
v
n
}
,
{
μ
n
}
,
ξ
)-total quasi-asymptotically nonexpansive mappings; moreover from Example 11 (b) we have that G and T are both uniformly L
1
,
L
2-Lipschitzian with L
1
=
2
∏
i
=
2
n
(
a
i
) and L
2
=
2
∏
i
=
2
n
(
b
i
); also by our hypothesis (
I
-
G
) and (
I
-
T
) are both demiclosed at zero. Therefore, all the conditions in Theorem 8 are satisfied. Hence, the conclusions of this corollary follow directly from Theorem 8.



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*Source: 412318-2015-03-22.xml*