# Split Common Fixed Point Problem for a Class of Total Asymptotic Pseudocontractions

Authors: E. E. Chima; M. O. Osilike
Journal: Journal of Applied Mathematics
 (2016)

---

## Abstract

We study the split common fixed point problem (SCFP) for a class of total asymptotically pseudocontractive mappings. We obtain some important properties of our class of mappings including the demiclosedness property and the closedness and convexity of the fixed point set. We then propose an algorithm and prove weak and strong convergence theorems for the approximation of solutions of the SCFP for certain class of these mappings.


---

## Body

## 1. Introduction

LetH
1 and H
2 be two real Hilbert spaces, K and Q nonempty closed convex subsets of H
1 and H
2, respectively, and A
:
H
1
→
H
2 a bounded linear operator. The split feasibility problem (SFP) (see, e.g., [1–9]) is (1)
f
i
n
d
x
∗
∈
H
1
s
u
c
h
t
h
a
t
x
∗
∈
K
,
A
x
∗
∈
Q
.If Q
=
{
b
}, a singleton, we have theconvexly constrained linear inverse problem (CCLIP): (2)
f
i
n
d
x
∗
∈
H
1
,
s
u
c
h
t
h
a
t
x
∗
∈
K
,
A
x
∗
=
b
.The split feasibility problem (SFP) has various important applications in several disciplines (see, e.g., [2–9]).LetS
:
H
1
→
H
1 and T
:
H
2
→
H
2 be mappings such that K
≔
F
(
S
)
=
{
x
∈
H
1
:
S
x
=
x
}
≠
∅ and Q
≔
F
(
T
)
=
{
x
∈
H
2
:
T
x
=
x
}
≠
∅. Thesplit common fixed point problem (SCFP) for S and T is to find a point x
∗
∈
H
1 such that x
∗
∈
F
(
S
) and A
x
∗
∈
F
(
T
). In sequel we use Γ to denote the set of solutions of (SCFP); that is,(3)
Γ
=
x
∈
K
=
F
S
,
A
x
∈
Q
=
F
T
.Definition 1.
LetH be a real Hilbert space, and let C be a nonempty closed convex subset of H.
A mappingT
:
C
→
C is said to be ({
μ
n
}
n
=
1
∞
,
{
ξ
n
}
n
=
1
∞
,
ϕ)-total asymptotically
k
-strictly pseudocontractive (see, e.g., [3]) if there exist a constant k
∈
[
0,1
), a continuous and strictly increasing function ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) with ϕ
(
0
)
=
0, and sequences {
μ
n
}
⊂
[
0
,
∞
) and {
ξ
n
}
⊂
[
0
,
∞
) with μ
n
→
0 and ξ
n
→
0 such that, for all x
,
y
∈
C, (4)
T
n
x
-
T
n
y
2
≤
x
-
y
2
+
k
I
-
T
n
x
-
I
-
T
n
y
2
+
μ
n
ϕ
x
-
y
+
ξ
n
.
T is said to be ({
μ
n
}
n
=
1
∞
,
{
ξ
n
}
n
=
1
∞
,
ϕ)-total asymptotically pseudocontractive if k
=
1 in (4).
Observe that ifϕ
(
t
)
=
t
2 and ξ
n
=
0
∀
n
≥
1 in (4), we obtain (5)
T
n
x
-
T
n
y
2
≤
k
n
x
-
y
2
+
k
x
-
T
n
x
-
y
-
T
n
y
2
,where k
n
=
1
+
μ
n
⊆
[
1
,
∞
) and lim
n
→
∞
⁡
k
n
=
1
. Mappings satisfying (5) are the well-known class of k
-strictly asymptotically pseudocontractive mappings, while mappings satisfying (5) for k
=
1 are called asymptotically pseudocontractive mappings. These classes of mappings are generalizations of the well-known important class ofasymptotically nonexpansive mappings introduced by Goebel and Kirk [10] (i.e., mappings T
:
C
→
C which satisfy T
n
x
-
T
n
y
≤
k
n
x
-
y, ∀
x
,
y
∈
C and for some sequence {
k
n
}
n
=
1
∞
⊆
[
1
,
∞
) with lim
n
→
∞
⁡
k
n
=
1).We consider the following examples.Example 2 (see [10]).
In the real Hilbert spacel
2, let C
=
{
x
∈
l
2
:
x
≤
1
} denote the closed unit ball, and define S
:
C
→
C by S
(
x
1
,
x
2
,
x
3
,
…
)
=
(
0
,
x
1
2
,
a
2
x
2
,
a
3
x
3
,
…
), where {
a
j
}
j
=
1
∞ is a real sequence in (
0,1
) such that ∏
j
=
1
∞
‍
a
j
=
1
/
2
. Then S is asymptotically nonexpansive and hence asymptotically strictly pseudocontractive. It follows that S is total asymptotically strictly pseudocontractive and hence total asymptotically pseudocontractive.Example 3 (see [6, 7]).
LetD be an orthogonal subspace of the Euclidean space R
n, and for each x
=
(
x
1
,
x
2
,
x
3
,
…
,
x
n
)
∈
D, define T
:
D
→
D by (6)
T
x
=
x
1
,
x
2
,
x
3
,
…
,
x
n
,
i
f
∏
j
=
1
n
x
j
<
0
,
-
x
1
,
-
x
2
,
-
x
3
,
…
,
-
x
n
,
i
f
∏
j
=
1
n
x
j
≥
0
.Then T is asymptotically nonexpansive (see, e.g., [7]) and hence total asymptotically strictly pseudocontractive (see, e.g., [6]) and hence it is total asymptotically pseudocontractive.Example 4 (see [11]).
LetR denote the reals with the usual norm, C
=
[
-
6,2
], and define T
:
C
→
C by (7)
T
x
=
x
,
i
f
x
∈
-
6,1
,
2
x
-
x
2
,
i
f
x
∈
1,2
.It is shown in [11] that |
T
x
-
T
y
|
2
≤
|
x
-
y
|
2
+
|
x
-
T
x
-
y
-
T
y
|
2 and |
T
x
-
T
y
|
≤
6
|
x
-
y
|, ∀
x
,
y
∈
[
-
6,2
]. It is easy to observe that, for all integers n
>
1, we have (8)
T
n
x
=
x
,
i
f
x
∈
-
6,1
,
2
x
-
x
2
,
i
f
x
∈
1,2
.Thus we easily obtain (9)
T
n
x
-
T
n
y
2
≤
x
-
y
2
+
x
-
T
n
x
-
y
-
T
n
y
2
,
∀
x
,
y
∈
-
6,2
,
∀
n
≥
1
.Hence T is asymptotically pseudocontractive and hence total asymptotically pseudocontractive. Furthermore, |
T
n
x
-
T
n
y
|
≤
6
|
x
-
y
|, ∀
x
,
y
∈
[
-
6,1
] so that T is uniformly L-Lipschitzian.The following is an example of a total asymptotically pseudocontractive map which is not total asymptotically strictly pseudocontractive.Example 5.
LetR denote the reals with the usual norm, C
=
[
0,1
], and define T
:
C
→
C by (10)
T
x
=
1
-
x
2
/
3
3
/
2
.Then for all integers n
≥
1 and for all x
∈
[
0,1
] we have (11)
T
n
x
=
1
-
x
2
/
3
3
/
2
,
i
f
n
i
s
o
d
d
,
x
,
i
f
n
i
s
e
v
e
n
.Thus (12)
T
n
x
-
T
n
y
2
≤
x
-
y
2
+
x
-
T
n
x
-
y
-
T
n
y
2
,
∀
x
,
y
∈
0,1
,
n
≥
1
,and hence T is total asymptotically pseudocontractive. T is not total asymptotically strictly pseudocontractive since, in every real Hilbert space H, every total asymptotically strictly pseudocontractive mapping T
:
C
⊆
H
→
C satisfies (13)
lim
sup
n
→
∞
⁡
T
n
x
-
T
n
y
≤
1
+
k
1
-
k
x
-
y
,
∀
x
,
y
∈
C
.
In [3] the authors studied the split common fixed point problem (SCFP) for a class of total asymptotically strictly pseudocontractive mappings in real Hilbert spaces. They proposed an algorithm and proved weak and strong convergence theorems for finding solutions of SCFP for the class of mappings studied.It is our purpose in this work to study the split common fixed point problem (SCFP) for a class of total asymptotically pseudocontractive mappings which is much more general than the class of mappings studied in [3]. We obtain some important properties of our class of mappings including thedemiclosedness property and then propose an algorithm and prove weak and strong convergence theorems for the approximation of solutions of the SCFP.


## 2. Preliminaries

In what follows, we will need the following.LetE be a real Banach space and C a nonempty closed convex subset of E. A mapping T
:
C
→
C is said to be s
e
m
i
c
o
m
p
a
c
t if, for any bounded sequence {
x
n
}
⊂
C with lim
n
→
∞
⁡
x
n
-
T
x
n
=
0, there exists a subsequence {
x
n
i
}
⊂
{
x
n
} such that {
x
n
i
} converges strongly to some point x
∗
∈
C.T
:
C
→
C is said to be uniformly L-Lipschitzian if there exists a constant L
≥
0, such that, for all x
,
y
∈
C, (14)
T
n
x
-
T
n
y
≤
L
x
-
y
.
E is said to have the Opial property if, for any sequence {
x
n
} with x
n
⇀
x, we have (15)
lim
inf
n
→
∞
⁡
x
n
-
x
<
lim
inf
n
→
∞
⁡
x
n
-
y
,
∀
y
∈
E
w
i
t
h
y
≠
x
.It is well known that every Hilbert space satisfies the Opial condition.Lemma 6 (see [10]).
LetH be a real Hilbert space. If {
x
n
} is a sequence in H weakly convergent to z, then (16)
lim
sup
n
→
∞
⁡
x
n
-
y
2
=
lim
sup
n
→
∞
⁡
x
n
-
z
2
+
z
-
y
2
,
∀
y
∈
H
.Lemma 7 (see [11]).
Let{
a
n
}, {
b
n
}, and {
δ
n
} be sequences of nonnegative real numbers satisfying the inequality (17)
a
n
+
1
≤
1
+
δ
n
a
n
+
b
n
,
∀
n
≥
1
.If ∑
n
=
1
∞
δ
n
<
∞ and ∑
n
=
1
∞
b
n
<
∞, then lim
n
→
∞
⁡
a
n exists. If in addition {
a
n
}
n
=
1
∞ has a subsequence which converges to zero, then lim
n
→
∞
⁡
a
n
=
0
.


## 3. Main Results

We start with the following important properties of ({
μ
n
}
n
=
1
∞
,
{
ξ
n
}
n
=
1
∞
,
ϕ)-total asymptotically pseudocontractive mappings.Proposition 8.
LetH be a real Hilbert space, C a nonempty closed convex subset of H, and T
:
C
→
C a uniformly L-Lipschitzian (
{
μ
n
}
n
=
1
∞
,
{
ξ
n
}
n
=
1
∞
,
ϕ
)-total asymptotically pseudocontractive mapping with F
(
T
)
≠
∅. Let β
∈
(
0
,
1
/
1
+
1
+
L
2
); and set G
n
=
T
n
[
(
1
-
β
)
I
+
β
T
n
]. Then for each q
∈
F
(
T
) and each x
∈
C, the following equivalent inequalities hold: (18)
x
-
G
n
x
,
x
-
q
≥
β
2
x
-
G
n
x
2
-
1
+
β
μ
n
2
ϕ
x
-
q
-
1
+
β
ξ
n
2
,
(19)
x
-
G
n
x
,
q
-
G
n
x
≤
2
-
β
2
x
-
G
n
x
2
+
1
+
β
μ
n
2
ϕ
x
-
q
+
1
+
β
ξ
n
2
.Proof.
For arbitraryx
∈
C and q
∈
F
(
T
) we have (20)
G
n
x
-
q
2
=
T
n
1
-
β
x
+
β
T
n
x
-
T
n
q
2
≤
1
-
β
x
+
β
T
n
x
-
q
2
+
1
-
β
x
+
β
T
n
x
-
G
n
x
2
+
μ
n
ϕ
x
-
q
+
ξ
n
=
1
-
β
x
-
q
+
β
T
n
x
-
q
2
+
1
-
β
x
-
G
n
x
+
β
T
n
x
-
G
n
x
2
+
μ
n
ϕ
x
-
q
+
ξ
n
=
1
-
β
x
-
q
2
+
β
T
n
x
-
q
2
-
β
1
-
β
x
-
T
n
x
2
+
1
-
β
x
-
G
n
x
2
+
β
T
n
x
-
G
n
x
2
-
β
1
-
β
x
-
T
n
x
2
+
μ
n
ϕ
x
-
q
+
ξ
n
≤
x
-
q
2
+
β
x
-
T
n
x
2
+
μ
n
ϕ
x
-
q
+
ξ
n
-
β
1
-
β
x
-
T
n
x
2
+
1
-
β
x
-
G
n
x
2
+
L
2
β
3
x
-
T
n
x
2
-
β
1
-
β
x
-
T
n
x
2
+
μ
n
ϕ
x
-
q
+
ξ
n
=
x
-
q
2
+
1
-
β
G
n
x
-
x
2
+
1
+
β
μ
n
ϕ
x
-
q
+
1
+
β
ξ
n
-
β
1
-
2
β
-
L
2
β
2
x
-
T
n
x
2
≤
x
-
q
2
+
1
-
β
G
n
x
-
x
2
+
1
+
β
μ
n
ϕ
x
-
q
+
1
+
β
ξ
n
.It follows from (20) that (21)
x
-
G
n
x
,
x
-
q
≥
β
2
x
-
G
n
x
2
-
1
+
β
μ
n
2
ϕ
x
-
q
-
1
+
β
ξ
n
2
,
x
-
G
n
x
,
q
-
G
n
x
≤
2
-
β
2
x
-
G
n
x
2
+
1
+
β
μ
n
2
ϕ
x
-
q
+
1
+
β
ξ
n
2
.Proposition 9.
LetH be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let T
:
C
→
C be a uniformly L-Lipschitzian (
{
μ
n
}
n
=
1
∞
,
{
ξ
n
}
n
=
1
∞
,
ϕ
)-total asymptotically pseudocontractive mapping with lim
n
→
∞
⁡
μ
n
=
0, lim
n
→
∞
⁡
ξ
n
=
0. Then (i)
(
I
-
T
) is demiclosed at 0;
(ii)
F
(
T
)
=
{
x
∈
C
:
T
x
=
x
} is closed and convex.Proof.
(i) Let{
x
n
}
n
=
1
∞ be a sequence in C which converges weakly to p and {
x
n
-
T
x
n
}
n
=
1
∞ converges strongly to 0. We prove that p
∈
F
(
T
). Since {
x
n
}
n
=
1
∞ converges weakly, it is bounded. For each x
∈
H, define f
:
H
→
[
0
,
∞
) by (22)
f
x
≔
limsup
n
→
∞
⁡
x
n
-
x
2
.Observe that, for arbitrary but fixed integer m
≥
1, we have (23)
x
n
-
T
m
x
n
≤
x
n
-
T
x
n
+
T
x
n
-
T
2
x
n
+
⋯
+
T
m
-
1
x
n
-
T
m
x
n
≤
m
L
x
n
-
T
x
n
⟶
0
a
s
n
⟶
∞
.Set G
m
x
≔
T
m
(
(
1
-
β
)
x
+
β
T
m
x
), where β
∈
(
0
,
1
/
1
+
L
2
+
1
)
. Then we obtain (24)
1
-
β
x
n
+
β
T
m
x
n
-
T
m
x
n
=
1
-
β
x
n
-
T
m
x
n
⟶
0
a
s
n
⟶
∞
,
T
m
x
n
-
G
m
x
n
≤
L
β
x
n
-
T
x
n
⟶
0
a
s
n
⟶
∞
.Hence(25)
1
-
β
x
n
+
β
T
m
x
n
-
G
m
x
n
≤
1
-
β
x
n
+
β
T
m
x
n
-
T
m
x
n
+
T
m
x
n
-
G
m
x
n
⟶
0
a
s
n
⟶
∞
.Furthermore, (26)
x
n
-
G
m
x
n
≤
x
n
-
T
m
x
n
+
T
m
x
n
-
G
m
x
n
⟶
0
a
s
n
⟶
∞
.Since {
x
n
-
p
} is bounded we also obtain (27)
1
-
β
x
n
+
β
T
m
x
n
-
1
-
β
p
+
β
T
m
p
≤
1
+
L
x
n
-
p
≤
D
∀
n
≥
1
,for some D
>
0.
From Lemma6, we obtain f
(
x
)
=
limsup
n
→
∞
⁡
x
n
-
p
2
+
p
-
x
2, ∀
x
∈
H. Thus f
(
x
)
=
f
(
p
)
+
p
-
x
2, ∀
x
∈
H, and hence (28)
f
G
m
p
=
f
p
+
p
-
G
m
p
2
.Observe that (29)
f
G
m
p
=
limsup
n
→
∞
⁡
x
n
-
G
m
p
2
=
limsup
n
→
∞
⁡
x
n
-
G
m
x
n
+
G
m
x
n
-
G
m
p
2
=
limsup
n
→
∞
⁡
G
m
x
n
-
G
m
p
2
=
limsup
n
→
∞
⁡
T
m
1
-
β
x
n
+
β
T
m
x
n
-
T
m
1
-
β
p
+
β
T
m
p
2
≤
limsup
n
→
∞
⁡
1
-
β
x
n
+
β
T
m
x
n
-
1
-
β
p
+
β
T
m
p
2
+
1
-
β
x
n
+
β
T
m
x
n
-
G
m
x
n
-
1
-
β
p
+
β
T
m
p
-
G
m
p
2
+
μ
m
ϕ
1
-
β
x
n
+
β
T
m
x
n
-
1
-
β
p
+
β
T
m
p
+
ξ
m
≤
limsup
n
→
∞
⁡
1
-
β
x
n
-
p
+
β
T
m
x
n
-
T
m
p
2
+
1
-
β
p
-
G
m
p
+
β
T
m
p
-
G
m
p
2
+
μ
m
ϕ
D
+
ξ
m
≤
limsup
n
→
∞
⁡
1
-
β
x
n
-
p
2
+
β
T
m
x
n
-
T
m
p
2
-
β
1
-
β
x
n
-
T
m
x
n
-
p
-
T
m
p
2
+
1
-
β
p
-
G
m
p
2
+
β
T
m
p
-
G
m
p
2
-
β
1
-
β
p
-
T
m
p
2
+
μ
m
ϕ
D
+
ξ
m
≤
limsup
n
→
∞
⁡
1
-
β
x
n
-
p
2
+
β
x
n
-
p
2
+
x
n
-
T
m
x
n
-
p
-
T
m
p
2
+
μ
m
ϕ
x
n
-
p
+
ξ
m
-
β
1
-
β
x
n
-
T
m
x
n
-
p
-
T
m
p
2
+
1
-
β
p
-
G
m
p
2
+
L
2
β
3
p
-
T
m
p
2
-
β
1
-
β
p
-
T
m
p
2
+
μ
m
ϕ
D
+
ξ
m
≤
limsup
n
→
∞
⁡
x
n
-
p
2
-
β
1
-
2
β
-
L
2
β
2
p
-
T
m
p
2
+
1
-
β
p
-
G
m
p
2
+
1
+
β
μ
m
ϕ
D
+
1
+
β
ξ
m
.It follows that(30)
f
p
+
p
-
G
m
p
2
≤
f
p
+
1
-
β
p
-
G
m
p
2
+
1
+
β
μ
m
ϕ
D
+
1
+
β
ξ
m
,and thus(31)
p
-
G
m
p
2
≤
1
β
1
+
β
μ
m
ϕ
D
+
1
+
β
ξ
m
⟶
0
⁡
a
s
m
⟶
∞
.It follows that(32)
p
-
T
m
p
≤
p
-
G
m
p
+
G
m
p
-
T
m
p
≤
p
-
G
m
p
+
L
β
p
-
T
m
p
,and hence(33)
p
-
T
m
p
≤
1
1
-
β
L
p
-
G
m
p
⟶
0
a
s
m
⟶
∞
.It now follows that T
m
p
→
p
a
s
m
→
∞. Since T is continuous, we have that T
m
+
1
p
→
T
p
a
s
m
→
∞, and hence T
p
=
p.
(ii) Let{
p
n
}
n
=
1
∞
⊆
F
(
T
) be such that p
n
→
p. We prove that p
∈
F
(
T
). Consider(34)
p
-
T
p
≤
p
-
p
n
+
p
n
-
T
p
=
p
-
p
n
+
T
p
n
-
T
p
≤
1
+
L
p
n
-
p
⟶
0
a
s
n
⟶
∞
.Hence p
∈
F
(
T
), and F
(
T
) is closed.
Letp
1
,
p
2
∈
F
(
T
) and let λ
∈
[
0,1
] be arbitrary. Set p
≔
λ
p
1
+
(
1
-
λ
)
p
2. We prove that p
∈
F
(
T
). Observe that p
-
p
1
=
(
1
-
λ
)
p
1
-
p
2 and p
-
p
2
=
λ
p
1
-
p
2. Set (35)
G
n
x
≔
T
n
1
-
β
x
+
β
T
n
x
,where β
∈
(
0
,
2
/
1
+
1
+
4
L
2
)
. Then G
n
p
1
=
p
1, and G
n
p
2
=
p
2. Observe that (36)
p
-
G
n
p
2
=
λ
p
1
-
G
n
p
+
1
-
λ
p
2
-
G
n
p
2
=
λ
p
1
-
G
n
p
2
+
1
-
λ
p
2
-
G
n
p
2
-
λ
1
-
λ
p
1
-
p
2
2
.Observe that(37)
G
n
p
-
p
1
2
=
T
n
1
-
β
p
+
β
T
n
p
-
p
1
2
=
T
n
1
-
β
p
+
β
T
n
p
-
T
n
1
-
β
p
1
+
β
T
n
p
1
2
≤
1
-
β
p
-
p
1
+
β
T
n
p
-
T
n
p
1
2
+
1
-
β
p
+
β
T
n
p
-
G
n
p
-
1
-
β
p
1
+
β
T
n
p
1
-
G
n
p
1
2
+
μ
n
ϕ
1
-
β
p
-
p
1
+
β
T
n
p
-
T
n
p
1
+
ξ
n
=
1
-
β
p
-
p
1
2
+
β
T
n
p
-
T
n
p
1
2
-
β
1
-
β
p
-
T
n
p
2
+
1
-
β
p
-
G
n
p
+
β
T
n
p
-
G
n
p
2
+
μ
n
ϕ
1
-
β
+
β
L
p
-
p
1
+
ξ
n
≤
1
-
β
p
-
p
1
2
+
β
p
-
p
1
2
+
β
p
-
T
n
p
2
+
β
μ
n
ϕ
p
-
p
1
+
β
ξ
n
-
β
1
-
β
p
-
T
n
p
2
+
1
-
β
p
-
G
n
p
2
+
β
T
n
p
-
G
n
p
2
-
β
1
-
β
p
-
T
n
p
2
+
μ
n
ϕ
1
-
β
+
β
L
p
-
p
1
+
ξ
n
≤
p
-
p
1
2
+
1
-
β
p
-
G
n
p
2
-
β
1
-
2
β
-
β
2
L
2
p
-
T
n
p
2
+
μ
n
ϕ
1
-
β
+
β
L
p
-
p
1
+
β
ϕ
p
-
p
1
+
1
+
β
ξ
n
=
p
-
p
1
2
+
1
-
β
p
-
G
n
p
2
-
β
1
-
2
β
-
β
2
L
2
p
-
T
n
p
2
+
σ
n
,where σ
n
=
μ
n
[
ϕ
(
[
1
-
β
+
β
L
]
p
-
p
1
)
+
β
ϕ
(
p
-
p
1
)
]
+
(
1
+
β
)
ξ
n.
Similarly,(38)
G
n
p
-
p
2
2
≤
p
-
p
2
2
+
1
-
β
p
-
G
n
p
2
-
β
1
-
2
β
-
β
2
L
2
p
-
T
n
p
2
+
σ
n
.Thus(39)
p
-
G
n
p
2
≤
λ
p
-
p
1
2
+
λ
1
-
β
p
-
G
n
p
2
-
λ
β
1
-
2
β
-
β
2
L
2
p
-
T
n
p
2
+
λ
σ
n
+
1
-
λ
p
-
p
2
2
+
1
-
λ
1
-
β
p
-
G
n
p
2
-
1
-
λ
β
1
-
2
β
-
β
2
L
2
p
-
T
n
p
2
+
1
-
λ
σ
n
-
λ
1
-
λ
p
1
-
p
2
2
,and it follows that (40)
β
p
-
G
n
p
2
≤
σ
n
⟶
0
a
s
n
⟶
∞
.Hence G
n
p
→
p
a
s
n
→
∞. Observe that (41)
p
-
T
n
p
2
≤
p
-
G
n
p
+
G
n
p
-
T
n
p
≤
p
-
G
n
p
+
L
β
p
-
T
n
p
.Thus (42)
1
-
L
β
p
-
T
n
p
≤
p
-
G
n
p
⟶
0
a
s
n
⟶
∞
,and hence T
n
p
→
p
a
s
n
→
∞. Since T is continuous we have (43)
T
n
+
1
p
⟶
T
p
a
s
n
⟶
∞
.Thus T
p
=
p
.We now introduce our algorithm and prove weak and strong convergence theorems for solving the split common fixed point problem for a class of total asymptotically pseudocontractive mappings in real Hilbert spaces.LetH
1 and H
2 be two real Hilbert spaces, A
:
H
1
→
H
2 a bounded linear operator, S
:
H
1
→
H
1 a uniformly L
1-Lipschitzian (
{
μ
n
1
}
,
{
ξ
n
1
}
,
ϕ
1
)-total asymptotically pseudocontractive mapping, with K
≔
F
(
S
)
≠
∅, and T
:
H
2
→
H
2 a uniformly L
2-Lipschitzian (
{
μ
n
2
}
,
{
ξ
n
2
}
,
ϕ
2
)-total asymptotically pseudocontractive mapping, with Q
≔
F
(
T
)
≠
∅. We now introduce the following iterative algorithm for approximating solutions of split common fixed point problem: x
∈
H
1 such that x
∈
F
(
S
) and A
x
∈
F
(
T
) (i.e., x
∈
Γ
=
x
∈
H
1
:
x
∈
F
(
S
) and A
x
∈
F
(
T
)
}.For arbitraryx
1
∈
H
1, the sequence {
x
n
}
n
=
1
∞ is given by(44)
u
n
=
x
n
+
γ
A
∗
T
n
1
-
β
I
+
β
T
n
-
I
A
x
n
,
n
≥
1
,
y
n
=
1
-
β
n
u
n
+
β
n
S
n
u
n
,
n
≥
1
,
x
n
+
1
=
1
-
α
n
u
n
+
α
n
S
n
y
n
,
n
≥
1
,where {
α
n
} and {
β
n
} are suitable sequences in (
0,1
) and β is a suitable parameter in (
0,1
). We prove the following.Theorem 10.
LetH
1 and H
2 be two real Hilbert spaces, A
:
H
1
→
H
2 a bounded linear operator, S
:
H
1
→
H
1 a uniformly L
1-Lipschitzian (
{
μ
n
(
1
)
}
,
{
ξ
n
(
1
)
}
,
ϕ
1
)-total asymptotically pseudocontractive mapping, with K
≔
F
(
S
)
≠
∅, and T
:
H
2
→
H
2 a uniformly L
2-Lipschitzian (
{
μ
n
(
2
)
}
,
{
ξ
n
(
2
)
}
,
ϕ
2
)-total asymptotically pseudocontractive mapping, with Q
≔
F
(
T
)
≠
∅; let μ
n
=
max
⁡
{
μ
n
(
1
)
,
μ
n
(
2
)
}, ξ
n
=
max
⁡
{
ξ
n
(
1
)
,
ξ
n
(
2
)
}, L
=
max
⁡
{
L
1
,
L
2
}, and ϕ
(
t
)
=
max
⁡
{
ϕ
1
(
t
)
,
ϕ
2
(
t
)
} such that there exist two positive real constants M and M
∗ such that ϕ
(
t
)
≤
M
∗
t
2, ∀
t
≥
M. Let ∑
n
=
1
∞
μ
n
<
∞ and ∑
n
=
1
∞
ξ
n
<
∞; let {
α
n
} and {
β
n
} be sequences of real numbers satisfying the condition:(45)
0
<
δ
≤
α
n
≤
β
n
≤
β
<
2
2
+
μ
M
∗
+
4
L
2
+
2
+
μ
M
∗
2
,where μ
=
sup
n
≥
1
⁡
μ
n and γ
∈
(
0
,
β
/
A
2
). Let Γ
=
{
x
∈
H
1
:
x
∈
F
(
S
)
a
n
d
A
x
∈
F
(
T
)
}
≠
∅. Then for arbitrary x
1
∈
H
1, the sequence {
x
n
} generated from x
1 by (44) converges weakly to a point in Γ.
If in additionS is semicompact, then {
x
n
} and {
u
n
} converge strongly to a point in Γ.Proof.
We will divide the proof into four steps.
Step 1. We prove that, for each p
∈
Γ, the following limits exist and (46)
lim
n
→
∞
⁡
x
n
-
p
=
lim
n
→
∞
⁡
u
n
-
p
.Since ϕ is a continuous and an increasing function, it follows that ϕ
(
λ
)
≤
ϕ
(
M
)
∀
λ
≤
M, and by hypothesis ϕ
(
λ
)
≤
M
∗
λ
2
∀
λ
≥
M. In either case, we can obtain that (47)
ϕ
λ
≤
ϕ
M
+
M
∗
λ
2
,
∀
λ
≥
0
.Let G
n
u
n
=
S
n
(
(
1
-
β
n
)
u
n
+
β
n
S
n
u
n
), n
≥
1
. Then, for arbitrary p
∈
F
(
S
), we obtain(48)
G
n
u
n
-
p
2
=
S
n
1
-
β
n
u
n
+
β
n
S
n
u
n
-
S
n
p
2
≤
1
-
β
n
u
n
+
β
n
S
n
u
n
-
p
2
+
1
-
β
n
u
n
+
β
n
S
n
u
n
-
G
n
u
n
2
+
μ
n
ϕ
1
-
β
n
u
n
+
β
n
S
n
u
n
-
p
+
ξ
n
≤
1
-
β
n
u
n
-
p
+
β
n
S
n
u
n
-
p
2
+
1
-
β
n
u
n
-
G
n
u
n
+
β
n
S
n
u
n
-
G
n
u
n
2
+
μ
n
ϕ
M
+
M
∗
1
-
β
n
u
n
-
p
+
β
n
S
n
u
n
-
p
2
+
ξ
n
=
1
-
β
n
u
n
-
p
2
+
β
n
S
n
u
n
-
p
2
-
β
n
1
-
β
n
u
n
-
S
n
u
n
2
+
1
-
β
n
u
n
-
G
n
u
n
2
+
β
n
S
n
u
n
-
G
n
u
n
2
-
β
n
1
-
β
n
u
n
-
S
n
u
n
2
+
μ
n
ϕ
M
+
μ
n
M
∗
1
-
β
n
u
n
-
p
2
+
β
n
S
n
u
n
-
p
2
-
β
n
1
-
β
n
u
n
-
S
n
u
n
2
+
ξ
n
≤
1
-
β
n
u
n
-
p
2
+
β
n
u
n
-
p
2
+
u
n
-
S
n
u
n
2
+
μ
n
ϕ
M
+
μ
n
M
∗
u
n
-
p
2
+
ξ
n
-
β
n
1
-
β
n
u
n
-
S
n
u
n
2
+
1
-
β
n
u
n
-
G
n
u
n
2
+
β
n
3
L
2
u
n
-
S
n
u
n
2
-
β
n
1
-
β
n
u
n
-
S
n
u
n
2
+
μ
n
ϕ
M
+
μ
n
M
∗
1
-
β
n
u
n
-
p
2
+
β
n
u
n
-
p
2
+
u
n
-
S
n
u
n
2
+
μ
n
ϕ
M
+
μ
n
M
∗
u
n
-
p
2
+
ξ
n
-
β
n
1
-
β
n
u
n
-
S
n
u
n
2
+
ξ
n
=
1
+
μ
n
M
∗
1
+
β
n
1
+
μ
n
M
∗
u
n
-
p
2
+
1
-
β
n
u
n
-
G
n
u
n
2
-
β
n
1
-
β
n
2
+
μ
M
∗
-
β
n
2
L
2
u
n
-
S
n
u
n
2
+
1
+
β
n
1
+
μ
n
M
∗
ξ
n
+
1
+
β
n
1
+
μ
n
M
∗
μ
n
ϕ
M
=
1
+
δ
n
u
n
-
p
2
+
1
-
β
n
u
n
-
G
n
u
n
2
-
β
n
1
-
β
n
2
+
μ
M
∗
-
β
n
2
L
2
u
n
-
S
n
u
n
2
+
σ
n
,where δ
n
=
μ
n
M
∗
(
1
+
β
n
(
1
+
μ
n
M
∗
)
) and (49)
σ
n
=
1
+
β
n
1
+
μ
n
M
∗
ξ
n
+
1
+
β
n
1
+
μ
n
M
∗
μ
n
ϕ
M
=
1
+
β
n
1
+
μ
n
M
∗
ξ
n
+
μ
n
ϕ
M
.
Observe that(50)
x
n
+
1
-
p
2
=
1
-
α
n
u
n
+
α
n
S
n
y
n
-
p
2
=
1
-
α
n
u
n
+
α
n
G
n
u
n
-
p
2
=
1
-
α
n
u
n
-
p
2
+
α
n
G
n
u
n
-
p
2
-
α
n
1
-
α
n
u
n
-
G
n
u
n
2
.
Using (48) in (50) we obtain (51)
x
n
+
1
-
p
2
≤
1
-
α
n
u
n
-
p
2
+
α
n
1
+
δ
n
u
n
-
p
2
+
1
-
β
n
u
n
-
G
n
u
n
2
-
β
n
1
-
β
n
2
+
μ
n
M
∗
-
β
n
2
L
2
u
n
-
S
n
u
n
2
+
σ
n
-
α
n
1
-
α
n
u
n
-
G
n
u
n
2
≤
1
+
α
n
δ
n
u
n
-
p
2
-
α
n
β
n
-
α
n
u
n
-
G
n
u
n
2
-
α
n
β
n
1
-
β
n
2
+
μ
M
∗
-
β
n
2
L
2
u
n
-
S
n
u
n
2
+
α
n
σ
n
.Observe also that if we let F
n
=
T
n
(
(
1
-
β
)
I
+
β
T
n
), then, (52)
u
n
-
p
2
=
x
n
+
γ
A
∗
T
n
1
-
β
I
+
β
T
n
-
I
A
x
n
-
p
2
=
x
n
-
p
+
γ
A
∗
F
n
-
I
A
x
n
2
=
x
n
-
p
2
+
γ
2
A
∗
F
n
-
I
A
x
n
2
+
2
γ
x
n
-
p
,
A
∗
F
n
-
I
A
x
n
.But (53)
γ
2
A
∗
F
n
-
I
A
x
n
2
=
γ
2
A
∗
F
n
-
I
A
x
n
,
A
∗
F
n
-
I
A
x
n
=
γ
2
A
A
∗
F
n
-
I
A
x
n
,
F
n
-
I
A
x
n
≤
γ
2
A
2
F
n
-
I
A
x
n
2
.Furthermore, (54)
2
γ
x
n
-
p
,
A
∗
F
n
-
I
A
x
n
=
2
γ
A
x
n
-
A
p
,
F
n
-
I
A
x
n
=
2
γ
A
x
n
-
A
p
+
F
n
-
I
A
x
n
-
F
n
-
I
A
x
n
,
F
n
-
I
A
x
n
=
2
γ
F
n
A
x
n
-
A
p
,
F
n
A
x
n
-
A
x
n
-
F
n
-
I
A
x
n
2
.Since A
p
∈
F
(
T
), we set x
=
A
x
n and q
=
A
p in (19) to obtain (55)
F
n
A
x
n
-
A
p
,
F
n
A
x
n
-
A
x
n
-
F
n
-
I
A
x
n
2
≤
2
-
β
2
F
n
-
I
A
x
n
2
+
1
+
β
2
μ
n
ϕ
A
x
n
-
A
p
+
1
+
β
2
ξ
n
-
F
n
-
I
A
x
n
2
≤
-
β
2
F
n
A
x
n
-
A
x
n
2
+
1
+
β
2
μ
n
ϕ
A
x
n
-
A
p
+
1
+
β
2
ξ
n
≤
-
β
2
F
n
A
x
n
-
A
x
n
2
+
1
+
β
2
μ
n
M
∗
A
2
x
n
-
p
2
+
ϕ
M
+
1
+
β
2
ξ
n
.Substituting (55) into (54) yields (56)
2
γ
x
n
-
p
,
A
∗
F
n
-
I
A
x
n
≤
2
γ
-
β
2
F
n
A
x
n
-
A
x
n
2
+
1
+
β
2
μ
n
M
∗
A
2
x
n
-
p
2
+
1
+
β
2
μ
n
ϕ
M
+
1
+
β
2
ξ
n
;and it follows from (52) that (57)
u
n
-
p
2
≤
x
n
-
p
2
+
γ
2
A
2
F
n
A
x
n
-
A
x
n
2
-
γ
β
F
n
A
x
n
-
A
x
n
2
+
1
+
β
γ
μ
n
M
∗
A
2
x
n
-
p
2
+
ϕ
M
+
1
+
β
γ
ξ
n
=
1
+
1
+
β
γ
μ
n
M
∗
A
2
x
n
-
p
2
-
γ
β
-
γ
A
2
F
n
A
x
n
-
A
x
n
2
+
1
+
β
γ
μ
n
ϕ
M
+
1
+
β
γ
ξ
n
=
1
+
ω
n
x
n
-
p
2
-
γ
β
-
γ
A
2
F
n
A
x
n
-
A
x
n
2
+
ν
n
,where ω
n
=
(
1
+
β
)
γ
μ
n
M
∗
A
2 and ν
n
=
(
1
+
β
)
γ
μ
n
ϕ
(
M
)
+
(
1
+
β
)
γ
ξ
n.
Substituting (57) in (51) we obtain (58)
x
n
+
1
-
p
2
≤
1
+
α
n
δ
n
1
+
ω
n
x
n
-
p
2
-
γ
β
-
γ
A
2
F
n
A
x
n
-
A
x
n
2
+
ν
n
-
α
n
β
n
-
α
n
u
n
-
G
n
u
n
2
-
α
n
β
n
1
-
β
n
2
+
μ
M
∗
-
β
n
2
L
2
u
n
-
S
n
u
n
2
+
α
n
σ
n
≤
1
+
ω
n
+
1
+
ω
n
α
n
δ
n
x
n
-
p
2
-
γ
β
-
γ
A
2
F
n
A
x
n
-
A
x
n
2
+
1
+
α
n
δ
n
α
n
ν
n
-
α
n
β
n
-
α
n
u
n
-
G
n
u
n
2
-
α
n
β
n
1
-
β
n
2
+
μ
M
∗
-
β
n
2
L
2
u
n
-
S
n
u
n
2
+
α
n
σ
n
=
1
+
θ
n
x
n
-
p
2
-
γ
β
-
γ
A
2
F
n
A
x
n
-
A
x
n
2
-
α
n
β
n
-
α
n
u
n
-
G
n
u
n
2
-
α
n
β
n
1
-
β
n
2
+
μ
M
∗
-
β
n
2
L
2
u
n
-
S
n
u
n
2
+
η
n
,where θ
n
=
ω
n
+
(
1
+
ω
n
)
α
n
δ
n and η
n
=
ν
n
+
α
n
(
δ
n
ν
n
+
σ
n
).
It follows from (58) and condition (45) that (59)
x
n
+
1
-
p
2
≤
1
+
θ
n
x
n
-
p
2
+
η
n
.The conditions ∑
n
=
1
∞
μ
n
<
∞ and ∑
n
=
1
∞
ξ
n
<
∞ imply that ∑
n
=
1
∞
θ
n
<
∞ and ∑
n
=
1
∞
η
n
<
∞. It now follows from Lemma 7 that lim
n
→
∞
⁡
x
n
-
p exists. It now follows from (58) that (60)
lim
n
→
∞
⁡
u
n
-
S
n
u
n
=
0
;
(61)
lim
n
→
∞
⁡
F
n
A
x
n
-
A
x
n
=
0
.Consequently,(62)
T
n
A
x
n
-
A
x
n
=
T
n
-
I
A
x
n
=
T
n
A
x
n
-
F
n
A
x
n
+
F
n
A
x
n
-
A
x
n
≤
T
n
A
x
n
-
F
n
A
x
n
+
F
n
A
x
n
-
A
x
n
≤
L
β
T
n
A
x
n
-
A
x
n
+
F
n
A
x
n
-
A
x
n
.Hence (63)
T
n
A
x
n
-
A
x
n
≤
1
1
-
L
β
F
n
A
x
n
-
A
x
n
⟶
0
a
s
n
⟶
∞
.This together with (61) implies that (64)
lim
n
→
∞
⁡
T
n
A
x
n
-
A
x
n
=
0
.Since lim
n
→
∞
⁡
x
n
-
p exists, it follows from (52) and (64) that lim
n
→
∞
⁡
u
n
-
p exists and(65)
lim
n
→
∞
⁡
u
n
-
p
=
lim
n
→
∞
⁡
x
n
-
p
.
Step 2. We prove that(66)
lim
n
→
∞
⁡
x
n
+
1
-
x
n
=
0
,
lim
n
→
∞
⁡
u
n
+
1
-
u
n
=
0
.From (44) we obtain(67)
x
n
+
1
-
x
n
=
1
-
α
n
u
n
+
α
n
S
n
y
n
-
x
n
=
1
-
α
n
u
n
+
α
n
G
n
u
n
-
x
n
=
1
-
α
n
x
n
+
γ
A
∗
T
n
1
-
β
I
+
β
T
n
-
I
A
x
n
+
α
n
G
n
u
n
-
x
n
=
1
-
α
n
x
n
+
γ
A
∗
F
n
-
I
A
x
n
+
α
n
G
n
u
n
-
x
n
=
1
-
α
n
γ
A
∗
F
n
-
I
A
x
n
+
α
n
G
n
u
n
-
u
n
+
α
n
u
n
-
x
n
=
1
-
α
n
γ
A
∗
F
n
-
I
A
x
n
+
α
n
G
n
u
n
-
u
n
+
α
n
x
n
+
γ
A
∗
F
n
-
I
A
x
n
-
x
n
=
γ
A
∗
F
n
-
I
A
x
n
+
α
n
G
n
u
n
-
u
n
≤
γ
A
∗
F
n
-
I
A
x
n
+
α
n
G
n
u
n
-
u
n
.Observe that (68)
u
n
-
G
n
u
n
≤
u
n
-
S
n
u
n
+
S
n
u
n
-
G
n
u
n
≤
u
n
-
S
n
u
n
+
L
β
n
u
n
-
S
n
u
n
=
1
+
L
β
n
u
n
-
S
n
u
n
.Using (68) in (67) we obtain(69)
x
n
+
1
-
x
n
≤
γ
A
∗
F
n
-
I
A
x
n
+
α
n
1
+
L
β
n
u
n
-
S
n
u
n
,and it follows from (60) and (61) that(70)
lim
n
→
∞
⁡
x
n
+
1
-
x
n
=
0
.Similarly, it follows from (44), (61), and (70) that(71)
u
n
+
1
-
u
n
=
x
n
+
1
+
γ
A
∗
T
n
+
1
1
-
β
I
+
β
T
n
+
1
-
I
A
x
n
+
1
-
x
n
-
γ
A
∗
T
n
1
-
β
I
+
β
T
n
-
I
A
x
n
=
x
n
+
1
+
γ
A
∗
F
n
+
1
-
I
A
x
n
+
1
-
x
n
+
γ
A
∗
F
n
-
I
A
x
n
≤
x
n
+
1
-
x
n
+
γ
A
∗
F
n
+
1
-
I
A
x
n
+
1
+
γ
A
∗
F
n
-
I
A
x
n
⟶
0
⁡
a
s
n
⟶
∞
.
Step 3. We prove that (72)
u
n
-
S
u
n
⟶
0
,
A
x
n
-
T
A
x
n
⟶
0
a
s
n
⟶
∞
.In fact, from (60), we have(73)
q
n
≔
u
n
-
S
n
u
n
⟶
0
,
a
s
n
⟶
∞
.Since S is uniformly L
1-Lipschitzian, it follows from (71) and (73) that (74)
u
n
-
S
u
n
≤
u
n
-
S
n
u
n
+
S
n
u
n
-
S
u
n
≤
q
n
+
L
S
n
-
1
u
n
-
u
n
≤
q
n
+
L
S
n
-
1
u
n
-
S
n
-
1
u
n
-
1
+
S
n
-
1
u
n
-
1
-
u
n
≤
q
n
+
L
2
u
n
-
u
n
-
1
+
L
S
n
-
1
u
n
-
1
-
u
n
-
1
+
u
n
-
1
-
u
n
≤
q
n
+
L
1
+
L
u
n
-
u
n
-
1
+
L
q
n
-
1
⟶
0
a
s
n
⟶
∞
.Similarly, from (64), we obtain (75)
w
n
≔
A
x
n
-
T
n
A
x
n
⟶
0
,
a
s
n
⟶
∞
.Since T is uniformly L
2-Lipschitzian, it follows from (70) and (75) that (76)
A
x
n
-
T
A
x
n
≤
A
x
n
-
T
n
A
x
n
+
T
n
A
x
n
-
T
A
x
n
≤
w
n
+
L
T
n
-
1
A
x
n
-
A
x
n
≤
w
n
+
L
T
n
-
1
A
x
n
-
T
n
-
1
A
x
n
-
1
+
T
n
-
1
A
x
n
-
1
-
A
x
n
≤
w
n
+
L
2
A
x
n
-
A
x
n
-
1
+
L
T
n
-
1
A
x
n
-
1
-
A
x
n
-
1
+
A
x
n
-
1
-
A
x
n
≤
w
n
+
L
1
+
L
A
x
n
-
A
x
n
-
1
+
L
w
n
-
1
⟶
0
a
s
n
⟶
∞
.This implies that(77)
A
x
n
-
T
A
x
n
⟶
0
,
a
s
n
⟶
∞
.
Step 4. We prove that the sequences {
x
n
} and {
u
n
} converge weakly to x
∗
∈
Γ.
Observe that since{
u
n
} is bounded, then there exists a subsequence {
u
n
i
} of {
u
n
} which converges weakly to a point x
∗
∈
H
1. Since lim
n
→
∞
⁡
u
n
-
S
u
n
=
0, we obtain (78)
lim
i
→
∞
⁡
u
n
i
-
S
u
n
i
=
0
.It follows from Proposition 9 that x
∗
∈
F
(
S
).
Furthermore, from (44) and (61), we obtain (79)
x
n
i
=
u
n
i
-
γ
A
∗
F
n
i
-
I
A
x
n
i
⇀
x
∗
.Since A is linear and bounded, we obtain A
x
n
i
⇀
A
x
∗, and it follows from (72) that (80)
lim
i
→
∞
⁡
A
x
n
i
-
T
A
x
n
i
=
0
.The demiclosedness of T at zero now yields that A
x
∗
∈
F
(
T
), and thus x
∗
∈
Γ
. Since every Hilbert space is an Opial space and {
u
n
} has a subsequence {
u
n
i
} which converges weakly to a point x
∗
∈
Γ, it follows from a standard argument that {
u
n
} converges weakly to x
∗.
IfS is semicompact, then since {
u
n
} is bounded and lim
n
→
∞
⁡
u
n
-
S
u
n
=
0, we have that there exists a subsequence {
u
n
j
} of {
u
n
} which converges strongly to a point u
∗
∈
H
1. Since {
u
n
} converges weakly to x
∗, we have u
∗
=
x
∗. Thus lim
j
→
∞
⁡
u
n
j
-
x
∗
=
0 and it follows from Lemma 7 that {
u
n
} (and hence {
x
n
}) converges strongly to x
∗
∈
Γ.Corollary 11.
LetH
1 and H
2 be two real Hilbert spaces, A
:
H
1
→
H
2 a bounded linear operator, S
:
H
1
→
H
1 a uniformly L
1-Lipschitzian asymptotically pseudocontractive mapping with sequence {
k
n
(
1
)
}
n
=
1
∞ such that ∑
n
=
1
∞
‍
(
k
n
(
1
)
-
1
)
<
∞ and K
≔
F
(
S
)
≠
∅, and T
:
H
2
→
H
2 a uniformly L
2-Lipschitzian asymptotically pseudocontractive mapping with sequence {
k
n
(
2
)
}
n
=
1
∞ such that ∑
n
=
1
∞
‍
(
k
n
(
2
)
-
1
)
<
∞ and Q
≔
F
(
T
)
≠
∅; let k
n
=
max
⁡
{
k
n
(
1
)
,
k
n
(
2
)
} and L
=
max
⁡
{
L
1
,
L
2
}. Let {
α
n
} and {
β
n
} be sequences of real numbers satisfying the condition: (81)
0
<
δ
≤
α
n
≤
β
n
≤
β
<
2
2
+
k
+
4
L
2
+
2
+
k
2
,where k
=
sup
n
≥
1
⁡
(
k
n
-
1
) and γ
∈
(
0
,
β
/
A
2
). Let Γ
=
{
x
∈
H
1
:
x
∈
F
(
S
)
a
n
d
A
x
∈
F
(
T
)
}
≠
∅. Then for arbitrary x
1
∈
H
1 the sequence {
x
n
} generated from x
1 by (44) converges weakly to a point in Γ.
If in additionS is semicompact, then {
x
n
} and {
u
n
} converge strongly to a point in Γ.Example 12.
LetC and S be as in Example 2, and let D and T be as in Example 3. Then F
(
T
)
=
{
(
0,0
,
0
,
…
,
0
)
}
∪
{
(
x
1
,
x
2
,
x
3
,
…
,
x
n
)
:
∏
j
=
1
n
‍
x
j
<
0
}, and F
(
S
)
=
{
(
0,0
,
0
,
…
)
}. Furthermore, C and D are nonempty closed convex subsets of l
2 and R
n, respectively. Define A
:
C
→
D by A
x
=
(
x
1
,
x
2
,
x
3
,
…
,
x
n
) for each x
=
(
x
1
,
x
2
,
x
3
,
…
)
∈
C. Then A is a bounded linear operator with adjoint operator A
∗
y
=
(
x
1
,
x
2
,
x
3
,
…
,
x
n
,
0,0
,
0
,
…
) for y
=
(
x
1
,
x
2
,
x
3
,
…
,
x
n
)
∈
D. Furthermore, A
=
A
∗
=
1. Thus using algorithm (44) with {
α
n
}
n
=
1
∞ and {
β
n
}
n
=
1
∞ satisfying the condition 0
<
δ
<
α
n
≤
β
n
<
β
<
1
/
1
+
5 and γ
∈
(
0
,
β
), it follows from Theorem 10 that x
n
⇀
(
0,0
,
0
,
…
)
∈
F
(
S
) and A
(
0,0
,
0
,
…
)
=
(
0,0
,
0
,
…
,
0
)
∈
F
(
T
).



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*Source: 3435078-2016-04-28.xml*