# Strong Convergence Algorithms of the Split Common Fixed Point Problem for Total Quasi-Asymptotically Pseudocontractive Operators

Authors: Peiyuan Wang; Hy Zhou
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We present a new algorithm for solving the two-set split common fixed point problem with total quasi-asymptotically pseudocontractive operators and consider the case of quasi-pseudocontractive operators. Under some appropriate conditions, we prove that the proposed algorithms have strong convergence. The results presented in this paper improve and extend the previous algorithms and results of Censor and Segal (2009), Moudafi (2011 and 2010), Mohammed (2013), Yang et al. (2011), Chang et al. (2012), and others.


---

## Body

## 1. Introduction

LetC and Q be nonempty closed convex subsets of real Hilbert spaces H
1 and H
2, respectively. Let A
:
H
1
→
H
2 be a bounded linear operator. To allow for constraints both in the domain and range of A, Censor and Elfving [1] originally formulated the split feasibility problem (SFP), which is to find a member of set
(1)
Ω
=
{
x
∈
C
:
A
x
∈
Q
}
≠
∅
.
A recent generalization, due to Censor and Segal in [2], is called the split common fixed point problem (SCFPP), which is to find a point x
* satisfying
(2)
x
*
∈
C
∶
=
⋂
i
=
1
t
‍
Fix
⁡
(
U
i
)
,
A
x
*
∈
Q
∶
=
⋂
j
=
1
r
‍
Fix
⁡
(
T
j
)
,
where U
i
:
H
1
→
H
1 (i
=
1,2
,
…
t) and T
j
:
H
2
→
H
2 (j
=
1,2
,
…
,
r
) are some nonlinear operators and A
:
H
1
→
H
2 is also a bounded linear operator. Denote the solution set of SCFPP by
(3)
Γ
=
{
x
*
∈
C
∣
A
x
*
∈
Q
}
.In particular, ift
=
r
=
1, problem (2) is reduced to the two-set SCFPP, where C
∶
=
Fix
⁡
(
U
) and Q
∶
=
Fix
⁡
(
T
), and the SFP can be retrieved by picking as operators U and T orthogonal projections.Censor and Segal [2] invented the following CQ-algorithm with directed operators to solve the two-set SCFPP:
(4)
∀
x
0
∈
H
1
,
x
n
+
1
=
U
(
x
n
-
γ
A
*
(
I
-
T
)
A
x
n
)
,
n
≥
0
,
where x
0
∈
H and γ
∈
(
0,2
/
L
); L is the largest eigenvalue of the matrix A
*
A.Inspired by the work of Censor and Segal, forα
n
∈
(
0,1
), Moudafi presented the following iteration with the demicontractive mappings and quasi-nonexpansive operators in papers [3] and [4], respectively:
(5)
u
n
=
x
n
-
γ
A
*
(
I
-
T
)
A
x
n
,
x
n
+
1
=
(
1
-
α
n
)
u
n
+
α
n
U
(
u
n
)
,
∀
x
0
∈
H
1
,
n
≥
0
.
Moudafi's results are weak convergence. In [5, 6], Mohammed utilized the strongly quasi-nonexpansive operators and quasi-nonexpansive operators to solve recursion (5) and obtain weak and strong convergence, respectively. Strong convergence of (5) with pseudo-demicontractive and firmly pseudo-demicontractive mappings can be found in [7, 8]. Furthermore, for several different strong convergence recursions with nonexpansive operators for solving the SCFPP see [9, 10]. For the purpose of generalization, papers [11–13] discussed the total asymptotically strictly pseudocontractive mappings and asymptotically strict pseudocontractive mappings for solving (2) and multiple-set fixed point problem (MSSFP) by the following iteration:
(6)
u
n
=
x
n
-
γ
A
*
(
I
-
T
n
)
A
x
n
,
x
n
+
1
=
(
1
-
α
n
)
u
n
+
α
n
U
n
(
u
n
)
,
∀
x
1
∈
H
1
,
n
≥
1
,
which is of weak convergence; when U is semicompact, strong convergence of (6) can be deduced. Obviously, (5) is the particular case of (6). On the other hand, papers [14, 15] presented cyclic algorithms of the SCFPP for directed operators and demicontractive mappings, and the results converge weakly.However, we found that the strong convergence of (6) needs the condition of U to be semicompact. In order to obtain strong algorithm for the two-set SCFPP without more constraints on U or T and continue to generalize the operators, in this paper, we propose a different iteration, which can ensure the strong convergence with more general case when the operators are total quasi-asymptotically pseudocontractive, demiclosed at the origin. We can choose an initial data x
1
∈
H
1 arbitrarily and define the sequence {
x
n
} by the recursion:
(7)
u
n
=
x
n
-
γ
A
(
I
-
T
n
)
A
x
n
,
y
n
=
(
1
-
β
)
u
n
+
β
U
n
(
u
n
)
,
x
n
+
1
=
α
n
ψ
(
y
n
)
+
(
1
-
α
n
)
y
n
,
n
≥
1
,
where ψ
:
H
1
→
H
1 is a δ-contraction with δ
∈
(
0,1
),  T and U are total quasi-asymptotically pseudocontractive mappings, and {
α
n
}, {
β
n
}, and {
γ
n
} are three real sequences satisfying appropriate conditions. Under some mild conditions, we prove that the sequence {
x
n
} generated by (7) converges strongly to the solution of the two-set SCFPP.


## 2. Preliminaries

In order to reach the main results, we first recall the following facts.LetC be a nonempty closed and convex subset of a real Hilbert space H with the inner product 〈
·
,
·
〉 and norm ∥
·
∥. Denote by Fix
⁡
(
T
) the set of fixed points of a mapping T; that is, Fix
⁡
(
T
)
=
{
x
∈
C
:
T
x
=
x
}.Definition 1 (see [2, 3, 16, 17]).
(i) Recalled thatT
:
C
→
C is said to be a directed or firmly quasi-nonexpansive operator; if p
∈
Fix
⁡
(
T
), then
(8)
∥
T
x
-
p
∥
2
≤
∥
x
-
p
∥
2
-
∥
x
-
T
x
∥
2
,
∀
x
∈
C
.
(ii) LetD be a closed convex nonempty set of C; T
:
C
→
C is nonexpansive; we say that T is attracting with respect to D, if, for every x
∈
C
∖
D, p
∈
D,
(9)
∥
T
x
-
p
∥
<
∥
x
-
p
∥
.
(iii) A mappingT
:
C
→
C is said to be paracontracting or quasi-nonexpansive; if p
∈
Fix
⁡
(
T
), then
(10)
∥
T
x
-
p
∥
≤
∥
x
-
p
∥
.
(iv) A mappingT
:
C
→
C is said to be demicontractive or strictly quasi-pseudocontractive; for p
∈
Fix
⁡
(
T
), there exists a constant β
∈
[
0,1
) such that
(11)
∥
T
x
-
p
∥
2
≤
∥
x
-
p
∥
2
+
β
∥
x
-
T
x
∥
2
,
∀
x
∈
C
.Definition 2 (see [11, 18]).
(i) LetT
:
C
→
C be a total quasi-asymptotically strictly pseudocontractive if F
(
T
)
≠
∅, and there exist a constant β
∈
[
0,1
], sequences {
μ
n
}
⊂
[
0
,
∞
), and {
ξ
n
}
⊂
[
0
,
∞
) with μ
n
→
0 and ξ
n
→
0 as n
→
∞ such that
(12)
∥
T
n
x
-
p
∥
2
≤
∥
x
-
p
∥
2
+
β
∥
x
-
T
n
x
∥
2
+
μ
n
ϕ
(
∥
x
-
p
∥
)
+
ξ
n
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∀
n
≥
1
,
x
∈
C
,
p
∈
Fix
⁡
(
T
)
,
where ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) is a continuous and strictly increasing function with ϕ
(
0
)
=
0.
(ii) A mappingT
:
C
→
C is said to be total quasi-asymptotically pseudocontractive if F
(
T
)
≠
∅, and there exist sequences {
μ
n
}
⊂
[
0
,
∞
) and {
ξ
n
}
⊂
[
0
,
∞
) with μ
n
→
0 and ξ
n
→
0 as n
→
∞ such that
(13)
∥
T
n
x
-
p
∥
2
≤
∥
x
-
p
∥
2
+
∥
x
-
T
n
x
∥
2
+
μ
n
ϕ
(
∥
x
-
p
∥
)
+
ξ
n
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∀
n
≥
1
,
x
∈
C
,
p
∈
Fix
⁡
(
T
)
,
where ϕ
:
[
0
,
∞
)
→
[
0
,
∞
) is a continuous and strictly increasing function with ϕ
(
0
)
=
0.
(iii) A mappingT
:
C
→
C is said to be quasi-pseudocontractive if Fix
⁡
(
T
)
≠
∅, such that
(14)
∥
T
x
-
p
∥
2

≤

∥
x

-

p
∥
2

+

∥
x
-
T
x
∥
2
,
x
∈
C
,
p
∈
Fix
⁡
(
T
)
.
(iv) A mappingT
:
C
→
C is said to be uniformly k-Lipschitzian if there is a constant k
>
0, such that
(15)
∥
T
n
x
-
T
n
y
∥
≤
k
∥
x
-
y
∥
,
∀
n
≥
1
,
∀
x
,
y
∈
C
.Remark 3.
Note that the classes of directed operators and attracting operators belong to the class of paracontracting operators. The class of paracontracting operators belongs to the class of demicontractive operators, while the class of quasi-pseudocontractive operators includes the class of demicontractive operators. Further, the class of total quasi-asymptotically pseudocontractive operators, with quasi-pseudocontractive operators as a special case, includes the class of total quasi-asymptotically strictly pseudocontractive operators.Remark 4.
LetT
:
C
→
C be a total quasi-asymptotically pseudocontractive, if F
(
T
)
≠
∅, for each x
∈
C and q
∈
Fix
⁡
(
T
); from (13) we can easily obtain the following equivalent inequalities:
(16)
〈
x
-
T
n
x
,
x
-
p
〉
≥
-
μ
n
2
ϕ
(
∥
x
-
p
∥
)
-
ξ
n
2
;
(17)
〈
x
-
T
n
x
,
p
-
T
n
x
〉
≤
∥
x
-
T
n
x
∥
2
+
μ
n
2
ϕ
(
∥
x
-
p
∥
)
+
ξ
n
2
;
(18)
〈
x
-
p
,
T
n
x
-
p
〉
≤
∥
x
-
p
∥
2
+
μ
n
2
ϕ
(
∥
x
-
p
∥
)
+
ξ
n
2
.Lemma 5 (see [19]).
Consider
(i)∥
x
±
y
∥
2
=
∥
x
∥
2
±
2
〈
x
,
y
〉
+
∥
y
∥
2, for all x
,
y
∈
H;
(ii)∥
(
1
-
t
)
x
+
t
y
∥
2
=
(
1
-
t
)
∥
x
∥
2
+
t
∥
y
∥
2
-
t
(
1
-
t
)
∥
x
-
y
∥
2, for all x
,
y
∈
H and t
∈
ℝ.Lemma 6 (see [18]).
LetC be a bounded and closed convex subset of a real Hilbert space H. Let T
:
C
→
C be a uniformly L-Lipschitz and total quasi-asymptotically pseudocontractive mapping with Fix
⁡
(
T
)
≠
∅. Suppose there exist positive constants M and M
*, for the function ϕ in (13), ϕ
(
ζ
)
≤
M
*
ζ
2 for all ζ
≥
M such that
(19)
ϕ
(
ζ
)
≤
ϕ
(
M
)
+
M
*
ζ
2
.
Then Fix
⁡
(
T
) is a closed convex subset of C.Lemma 7 (see [20]).
A mappingI
-
T
:
C
→
C is said to be demiclosed at zero, if for any sequence {
x
n
}
∈
C, such that x
n
⇀
x
*
∈
C and (
I
-
T
)
x
n
→
0 as n
→
∞; then (
I
-
T
)
x
*
=
0.Lemma 8 (see [21]).
Let{
r
n
}, {
s
n
}, and {
t
n
} be sequences of nonnegative real numbers satisfying
(20)
r
n
+
1
≤
(
1
+
t
n
)
r
n
+
s
n
,
n
≥
1
.
If ∑
n
=
1
∞
‍
t
n
<
∞ and ∑
n
=
1
∞
‍
s
n
<
∞, then the limit lim
⁡
n
→
∞
r
n exists.Lemma 9 (see [22]).
Let a sequence{
t
n
}
∈
[
0,1
) satisfy lim
⁡
n
→
∞
t
n
=
0 and ∑
n
=
1
∞
‍
t
n
=
∞. Let {
a
n
} be a sequence of nonnegative real numbers that satisfies any of the following conditions.(i)
For allɛ
>
0, there exists an integer N
≥
1 such that, for all n
≥
N,
(21)
a
n
+
1
≤
(
1
-
t
n
)
a
n
+
t
n
ɛ
;
(ii)
a
n
+
1
≤
(
1
-
t
n
)
a
n
+
o
n,  n
≥
0, where o
n
≥
0 satisfies lim
⁡
n
→
∞
o
n
/
t
n
=
0;
(iii)
a
n
+
1
≤
(
1
-
t
n
)
a
n
+
t
n
c
n, where lim
⁡
-
n
→
∞
c
n
≤
0.Then lim
⁡
n
→
∞
a
n
=
0.


## 3. Main Results

In this section, we will prove the strong convergence of (7) to solve the two-set SCFPP.Theorem 10.
LetC and Q be nonempty closed convex subsets of real Hilbert spaces H
1 and H
2, respectively. Let U
:
H
1
→
H
1 be a uniformly k
1-Lipschitz and (
{
μ
n
(
1
)
}
,
{
ξ
n
(
1
)
}
,
ϕ
1
)-total quasi-asymptotically pseudocontractive mapping, T
:
H
2
→
H
2 a uniformly k
2-Lipschitz, and (
{
μ
n
(
2
)
}
,
{
ξ
n
(
2
)
}
,
ϕ
2
)-total quasi-asymptotically pseudocontractive mappings satisfying the following conditions: (
C
1
)
C
∶
=
Fix
⁡
(
U
)
≠
∅, Q
∶
=
Fix
⁡
(
T
)
=
∅;
(
C
2
)
μ
n
=
max
⁡
{
μ
n
(
1
)
,
μ
n
(
2
)
}, ξ
n
=
max
⁡
{
ξ
n
(
1
)
,
ξ
n
(
2
)
}, n
≥
1, and ∑
n
=
1
∞
‍
μ
n
<
∞, ∑
n
=
1
∞
‍
ξ
n
<
∞;
(
C
3
)
ϕ
=
max
⁡
{
ϕ
1
,
ϕ
2
} and ∃
M
,
M
*
>
0.
Letψ
:
H
1
→
H
1 be a δ-contraction with δ
∈
(
0,1
). Let A
:
H
1
→
H
2 be a bounded linear operator. For ∀
x
1
∈
H
1, sequence {
x
n
} can be generated by the iteration (7), where the sequence {
α
n
}
⊂
(
0,1
) satisfies (i) lim
⁡
n
→
∞
α
n
=
0 and (ii) ∑
n
=
0
∞
‍
α
n
=
∞, {
β
}
⊂
[
a
,
b
] with a
,
b
∈
(
0
,
1
/
(
1
+
k
1
)
), and {
γ
}
⊂
(
0,2
/
L
) with L being the largest eigenvalue of the matrix A
T
A. Assume that I
-
U and I
-
T are demiclosed at zero. If Γ
≠
∅, then {
x
n
} generated by (7) converges strongly to a solution of the two-set SCFPP.Proof.
(1) First of all, we show that, for∀
p
∈
Γ,  {
x
n
} generated by (7) is bounded.
From (7), (16), and Lemma 6, we have
(22)
∥
u
n
-
p
∥
2
=
∥
x
n
-
p
+
γ
A
*
(
T
n
-
I
)
A
x
n
∥
2
=
∥
x
n
-
p
∥
2
+
γ
2
∥
A
*
(
I
-
T
n
)
A
x
n
∥
2
+
2
γ
〈
A
x
n
-
A
p
,
T
n
A
x
n
-
A
x
n
〉
(23)
≤
∥
x
n
-
p
∥
2
+
γ
2
L
2
∥
(
I
-
T
n
)
A
x
n
∥
2
+
γ
μ
n
ϕ
(
∥
A
x
n
-
A
p
∥
)
+
γ
ξ
n
(24)
≤
(
1
+
γ
μ
n
M
*
L
)
∥
x
n
-
p
∥
2
+
γ
2
L
2
∥
(
I

-

T
n
)
A
x
n
∥
2

+

γ
μ
n
ϕ
(
M
)

+

γ
ξ
n
.
Since(25)
∥
(
I
-
T
n
)
A
x
n
∥
≤
∥
A
x
n
-
A
p
∥
+
∥
T
n
A
x
n
-
A
p
∥
≤
(
∥
A
∥
+
k
2
∥
A
∥
)
∥
x
n
-
p
∥
,
substituting (25) into (24), we have
(26)
∥
u
n
-
p
∥
2
≤
[
1
+
γ
μ
n
M
*
L
+
γ
2
L
2
(
∥
A
∥
+
k
2
∥
A
∥
)
2
]
×
∥
x
n
-
p
∥
2
+
γ
μ
n
ϕ
(
M
)
+
γ
ξ
n
=
(
1
+
a
n
)
∥
x
n
-
p
∥
2
+
γ
μ
n
ϕ
(
M
)
+
γ
ξ
n
,
where a
n
=
γ
μ
n
M
*
L
+
γ
2
L
2
(
∥
A
∥
+
k
2
∥
A
∥
)
2; by condition (C
2), we know
(27)
∑
n
=
1
∞
‍
a
n
<
∞
.
Next, from (7), (13), and Lemma 5, we can get
(28)
∥
y
n
-
p
∥
2
=
∥
(
1
-
β
)
(
u
n
-
p
)
+
β
(
U
n
(
u
n
)
-
p
)
∥
2
=
(
1
-
β
)
∥
u
n
-
p
∥
2
+
β
∥
U
n
(
u
n
)
-
p
∥
2
-
β
(
1
-
β
)
∥
u
n
-
U
n
(
u
n
)
∥
2
≤
(
1
-
β
)
∥
u
n
-
p
∥
2

+

β
[
∥
u
n

-

p
∥
2

+

∥
u
n

-

U
n
(
u
n
)
∥
2
+
μ
n
ϕ
(
∥
u
n
-
p
∥
)
+
ξ
n
∥
u
n
-
p
∥
2
]
-
β
(
1
-
β
)
∥
u
n
-
U
n
(
u
n
)
∥
2
=
∥
u
n
-
p
∥
2
+
β
2
∥
u
n
-
U
n
(
u
n
)
∥
2
+
β
μ
n
ϕ
(
∥
u
n
-
p
∥
)
+
β
ξ
n
;
we also can see that
(29)
∥
u
n
-
U
n
(
u
n
)
∥
≤
∥
u
n
-
p
∥
+
∥
U
n
(
u
n
)
-
p
∥
≤
(
1
+
k
1
)
∥
u
n
-
p
∥
;
then substituting (29) into (28) and from (26), we have
(30)
∥
y
n
-
p
∥
2
≤
[
1
+
β
2
(
1
+
k
1
)
2
]
∥
u
n
-
p
∥
2
+
β
μ
n
ϕ
(
∥
u
n
-
p
∥
)
+
β
ξ
n
≤
[
1
+
β
2
(
1
+
k
1
)
2
+
β
μ
n
M
*
]
∥
u
n
-
p
∥
2
+
β
μ
n
ϕ
(
M
)
+
β
ξ
n
≤
(
1
+
b
n
)
(
1
+
a
n
)
∥
x
n
-
p
∥
2
+
μ
n
ϕ
(
M
)
×
[
(
1
+
b
n
)
γ
+
β
]
+
ξ
n
[
(
1
+
b
n
)
γ
+
β
]
,
where b
n
=
β
2
(
1
+
k
1
)
2
+
β
μ
n
M
*, and we also know that
(31)
∑
n
=
1
∞
‍
b
n
<
∞
.
From (7) and Lemma 5, we also have
(32)
∥
x
n
+
1
-
p
∥
2
=
∥
α
n
[
ψ
(
y
n
)
-
p
]
+
(
1
-
α
n
)
(
y
n
-
p
)
∥
2
≤
α
n
∥
ψ
(
y
n
)
-
p
∥
2
+
(
1
-
α
n
)
∥
y
n
-
p
∥
2
≤
(
1
+
2
α
n
δ
)
∥
y
n
-
p
∥
2
+
2
α
n
∥
ψ
(
p
)
-
p
∥
2
-
α
n
∥
y
n
-
p
∥
2
≤
(
1
+
2
α
n
δ
)
∥
y
n
-
p
∥
2
+
2
α
n
∥
ψ
(
p
)
-
p
∥
2
.
Substituting (30) into (32) and simplifying it we have
(33)
∥
x
n
+
1
-
p
∥
2
≤
(
1
+
2
α
n
δ
)
(
1
+
b
n
)
(
1
+
a
n
)
×
∥
x
n
-
p
∥
2
+
2
α
n
∥
ψ
(
p
)
-
p
∥
2
+
(
1
+
2
α
n
δ
)
{
μ
n
ϕ
(
M
)
[
(
1
+
b
n
)
γ
+
β
]
+
ξ
n
[
(
1
+
b
n
)
γ
+
β
]
}
=
[
1
+
a
n
+
(
1
+
a
n
)
(
b
n
+
2
a
n
δ
+
2
a
n
b
n
δ
)
]
×
∥
x
n
-
p
∥
2
+
2
α
n
∥
ψ
(
p
)
-
p
∥
2
+
(
1
+
2
α
n
δ
)
{
μ
n
ϕ
(
M
)
[
(
1
+
b
n
)
γ
+
β
]
+
ξ
n
[
(
1
+
b
n
)
γ
+
β
]
}
;
set
(34)
t
n
=
a
n
+
(
1
+
a
n
)
(
b
n
+
2
a
n
δ
+
2
a
n
b
n
δ
)
,
s
n
=
2
α
n
∥
ψ
(
p
)
-
p
∥
2
+
(
1
+
2
α
n
δ
)
×
{
μ
n
ϕ
(
M
)
[
(
1
+
b
n
)
γ
+
β
]
+
ξ
n
[
(
1
+
b
n
)
γ
+
β
]
}
;
(33) can be rewritten as
(35)
∥
x
n
+
1
-
p
∥
2
≤
(
1
+
t
n
)
∥
x
n
-
p
∥
2
+
s
n
;
by condition (C
2), (27), and (31), we know that ∑
n
=
1
∞
‍
t
n
<
∞ and ∑
n
=
1
∞
‍
s
n
<
∞. Thus it follows from Lemma 8 that the following limit exists:
(36)
lim
⁡
n
→
∞
∥
x
n
-
p
∥
.
Therefore, we obtain that {
x
n
} is bounded, so is {
u
n
}. Set z
n
=
U
n
(
u
n
). Then {
z
n
} is also bounded.
(2) Next we proveli
m
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0, li
m
n
→
∞
∥
u
n
+
1
-
u
n
∥=0.
For eachn
≥
1, ∀
u
n
∈
H
1, assume there exists v
i
(
n
)
∈
C
(
i
=
1,2
) such that u
n
=
w
v
1
(
n
)
+
(
1
-
w
)
v
2
(
n
) for w
∈
(
0,1
). Then for all q
∈
C, and by virtue of (16), we have
(37)
∥
u
n
-
U
n
(
u
n
)
∥
2
=
〈
u
n
-
U
n
(
u
n
)
,
u
n
-
U
n
(
u
n
)
〉
=
1
β
〈
u
n
-
y
n
,
u
n
-
U
n
(
u
n
)
〉
=

1
β
〈
u
n

-

U
n
(
u
n
)

-

(
y
n

-

U
n
(
y
n
)
)
,
u
n

-

y
n
〉
+
1
β
〈
y
n
-
U
n
(
y
n
)
,
u
n
-
y
n
〉
≤
1
β
(
∥
u
n
-
y
n
∥
+
∥
U
n
(
u
n
)
-
U
n
(
y
n
)
∥
)
×
∥
u
n

-

y
n
∥

+

1
β
〈
u
n

-

q
,
y
n

-

U
n
(
y
n
)
〉
+
1
β
〈
q
-
y
n
,
y
n
-
U
n
(
y
n
)
〉
≤
1

+

k
1
β
∥
u
n
-
y
n
∥
2

+

1
β
〈
u
n

-

q
,
y
n

-

U
n
(
y
n
)
〉
+
1
β
[
μ
n
2
ϕ
(
∥
y
n
-
q
∥
)
+
ξ
n
2
]
≤
(
1
+
k
1
)
β
∥
u
n
-
U
n
(
u
n
)
∥
2
+
1
β
〈
u
n
-
q
,
y
n
-
U
n
(
y
n
)
〉
+
1
β
[
μ
n
2
[
M
*
∥
y
n
-
q
∥
2
+
ϕ
(
M
)
]
+
ξ
n
2
]
,
which implies that(38)
β
[
1
-
(
1
+
k
1
)
β
]
∥
u
n
-
U
n
(
u
n
)
∥
2
≤
〈
u
n
-
z
,
y
n
-
U
n
(
y
n
)
〉
+
[
μ
n
2
[
M
*
∥
y
n
-
z
∥
2
+
ϕ
(
M
)
]
+
ξ
n
2
]
.
Now we take q
=
v
i
(
n
)
(
i
=
1,2
) in (38); multiplying w and (
1
-
w
) on the two side of (38), respectively, and then adding up, we can obtain
(39)
β
[
1
-
(
1
+
k
1
)
β
]
∥
u
n
-
U
n
(
u
n
)
∥
2
≤
[
μ
n
2
[
M
*
∥
y
n
-
z
∥
2
+
ϕ
(
M
)
]
+
ξ
n
2
]
.
Letting n
→
∞ in (39), we have
(40)
lim
⁡
n
→
∞
∥
u
n
-
U
n
(
u
n
)
∥
=
0
.
From (7), we know that
(41)
∥
x
n
+
1
-
p
∥
2
=
∥
y
n
-
p
+
α
n
(
ψ
(
y
n
)
-
y
n
)
∥
2
=
∥
y
n
-
p
∥
2
+
2
α
n
〈
y
n
-
p
,
ψ
(
y
n
)
-
y
n
〉
+
α
n
2
∥
ψ
(
y
n
)
-
y
n
∥
2
.
Letting n
→
∞ in (41) and by condition (i) in Theorem 10, we know
(42)
lim
⁡
n
→
∞
∥
x
n
-
p
∥
=
lim
⁡
n
→
∞
∥
y
n
-
p
∥
.
Similarly,
(43)
∥
y
n
-
p
∥
2
=
∥
u
n
-
p
+
β
(
u
n
-
U
n
(
u
n
)
)
∥
2
=
∥
u
n
-
p
∥
2
+
2
β
〈
u
n
-
p
,
u
n
-
U
n
(
u
n
)
〉
+
β
2
∥
u
n
-
U
n
(
u
n
)
∥
2
;from (40) the limit of ∥
y
n
-
p
∥ exists and
(44)
lim
⁡
n
→
∞
∥
x
n
-
p
∥
=
lim
⁡
n
→
∞
∥
y
n
-
p
∥
=
lim
⁡
n
→
∞
∥
u
n
-
p
∥
.
Therefore, when we take limit on both sides of (22), we can deduce that
(45)
lim
⁡
n
→
∞
∥
A
x
n
-
T
n
A
x
n
∥
=
0
.
Then,(46)
∥
x
n
+
1
-
x
n
∥
≤
∥
y
n
-
x
n
∥
+
α
n
∥
ψ
(
y
n
)
-
y
n
∥
≤
∥
u
n
-
x
n
∥
+
β
∥
u
n
-
U
n
(
u
n
)
∥
+
α
n
∥
ψ
(
y
n
)
-
y
n
∥
≤
γ
∥
A
∥
∥
A
x
n
-
T
n
A
x
n
∥
+
β
∥
u
n
-
U
n
(
u
n
)
∥
+
α
n
∥
ψ
(
y
n
)
-
y
n
∥
.
In view of (40) and (45) we have that
(47)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
Similarly, it follows from (7), (45), and (47) that
(48)
∥
u
n
+
1
-
u
n
∥
=
∥
x
n
+
1

-

x
n

-

γ
A
*
(
I

-

T
n
+
1
)
A
x
n
+
1

-

γ
A
*
(
I

-

T
n
)
A
x
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
γ
∥
A
*
∥
∥
A
x
n
+
1
T
n
+
1
-
A
x
n
+
1
∥
+
r
∥
A
*
∥
∥
A
x
n
T
n
-
A
x
n
∥
⟶
0
(
n
⟶
∞
)
.
(3) Next we prove that∥
x
n
-
U
(
u
n
)
∥
→
0, as n
→
∞.
From (40) and (48), we have
(49)
∥
u
n
-
U
(
u
n
)
∥
≤
∥
u
n
-
U
n
(
u
n
)
∥
+
∥
U
n
(
u
n
)
-
U
(
u
n
)
∥
≤
∥
u
n
-
U
n
(
u
n
)
∥
+
k
1
∥
U
n
-
1
(
u
n
)
-
u
n
∥
≤
∥
u
n
-
U
n
(
u
n
)
∥
+
k
1
[
∥
U
n
-
1
(
u
n
)
-
U
n
-
1
(
u
n
-
1
)
∥

+

∥
U
n
-
1
(
u
n
-
1
)

-

u
n
∥
]
≤
∥
u
n
-
U
n
(
u
n
)
∥
+
k
1
2
∥
u
n
-
u
n
-
1
∥
+
k
1
[
∥
U
n
-
1
(
u
n
-
1
)
-
u
n
-
1
∥
+
∥
u
n
-
1
-
u
n
∥
]
⟶
∞
(
n
→
∞
)
.
By the same way, from (45) and (47) we can also prove that
(50)
∥
A
x
n
-
T
A
x
n
∥
⟶
0
,
n
⟶
∞
.
Therefore, from (44) and (49), we know
(51)
∥
x
n
-
U
(
u
n
)
∥
≤
∥
x
n
-
u
n
∥
+
∥
u
n
-
U
(
u
n
)
∥
⟶
0
(
n
⟶
∞
)
.
Since {
x
n
} is bounded, there exists a subsequence {
x
n
i
} of {
x
n
} which converges weakly to a point x
*. Without loss of generality, we may assume that {
x
n
} converges weakly to x
*. Therefore, from (49)–(51) and Lemma 7, we have x
*
∈
Fix
⁡
(
U
).
(4) Finally, we prove thatx
n
→
x
* in norm. To do this, we calculate
(52)
∥
x
n
+
1
-
x
*
∥
2
=
〈
α
n
ψ
(
y
n
)
+
(
1
-
α
n
)
y
n
-
x
*
,
x
n
+
1
-
x
*
〉
=
〈
α
n
ψ
(
y
n
)
+
(
1
-
α
n
)
y
n
-
x
*
,
x
n
+
1
-
x
*
〉
=
α
n
〈
ψ
(
y
n
)
-
x
*
,
x
n
+
1
-
x
*
〉
+
(
1
-
α
n
)
〈
y
n
-
x
*
,
x
n
+
1
-
x
*
〉
≤
α
n
〈
ψ
(
y
n
)
-
ψ
(
x
*
)
,
x
n
+
1
-
x
*
〉
+
α
n
〈
ψ
(
x
*
)
-
x
*
,
x
n
+
1
-
x
*
〉
+
1
-
α
n
2
∥
y
n
-
x
*
∥
2
+
1
-
α
n
2
∥
x
n
+
1
-
x
*
∥
2
≤
α
n
δ
2
∥
y
n
-
x
*
∥
2
+
α
n
2
∥
x
n
+
1
-
x
*
∥
2
+
α
n
〈
ψ
(
x
*
)
-
x
*
,
x
n
+
1
-
x
*
〉
+
1
-
α
n
2
∥
y
n
-
x
*
∥
2
+
1
-
α
n
2
∥
x
n
+
1
-
x
*
∥
2
=
1
-
(
1
-
δ
)
α
n
2
∥
y
n
-
x
*
∥
2
+
1
2
∥
x
n
+
1
-
x
*
∥
2
+
α
n
〈
ψ
(
x
*
)
-
x
*
,
x
n
+
1
-
x
*
〉
.
Therefore, we have
(53)
∥
x
n
+
1
-
x
*
∥
2
≤
(
1
-
(
1
-
δ
)
α
n
)
∥
y
n
-
x
*
∥
2
+
2
α
n
〈
ψ
(
x
*
)
-
x
*
,
x
n
+
1
-
x
*
〉
.
Substituting (23) into (28), we have
(54)
∥
y
n
-
x
*
∥
2
≤
∥
x
n
-
p
∥
2
+
γ
2
L
2
∥
(
I
-
T
n
)
A
x
n
∥
2
+
β
2
∥
u
n
-
U
n
(
u
n
)
∥
2
+
μ
n
[
γ
ϕ
(
∥
A
x
n
-
A
p
∥
)
+
β
ϕ
(
∥
u
n
-
p
∥
)
]
+
ξ
n
(
γ
+
β
)
.
Since (
1
-
δ
)
α
n
∈
(
0,1
) and substituting (53) into (51), we get
(55)
∥
x
n
+
1
-
x
*
∥
2
≤
(
1
-
(
1
-
δ
)
α
n
)
∥
x
n
-
p
∥
2
+
(
1
-
(
1
-
δ
)
α
n
)
×
{
γ
2
L
2
∥
(
I
-
T
n
)
A
x
n
∥
2
+
β
2
∥
u
n
-
U
n
(
u
n
)
∥
2
+
μ
n
[
γ
ϕ
(
∥
A
x
n
-
A
p
∥
)
+
β
ϕ
(
∥
u
n
-
p
∥
)
]
+
ξ
n
(
γ
+
β
)
}
.
Let
(56)
o
n
=
(
1
-
(
1
-
δ
)
α
n
)
×
{
γ
2
L
2
∥
(
I
-
T
n
)
A
x
n
∥
2
+
β
2
∥
u
n
-
U
n
(
u
n
)
∥
2
+
μ
n
[
γ
ϕ
(
∥
A
x
n

-

A
p
∥
)

+

β
ϕ
(
∥
u
n

-

p
∥
)
]

+

ξ
n
(
γ

+

β
)
}
.
Equation (55) can be rewritten as
(57)
∥
x
n
+
1
-
x
*
∥
2
≤
(
1
-
(
1
-
δ
)
α
n
)
∥
x
n
-
p
∥
2
+
o
n
.
Evidently, from (40), (45), and Lemma 9 (ii), we can conclude that x
n
+
1
-
x
*
→
0
(
n
→
∞
).
This completes the proof.The following theorem can be concluded from Theorem10 immediately.Theorem 11.
LetC and Q be nonempty closed convex subsets of real Hilbert spaces H
1 and H
2, respectively. Let U
:
H
1
→
H
1 be a uniformly k
1-Lipschitz and quasi-pseudocontractive mapping with C
∶
=
Fix
⁡
(
U
)
≠
∅. Let T
:
H
2
→
H
2 be a uniformly k
2-Lipschitz and quasi-pseudocontractive mapping with Q
∶
=
Fix
⁡
(
T
)
=
∅. Let ψ
:
H
1
→
H
1 be a δ-contraction with δ
∈
(
0,1
). Let A
:
H
1
→
H
2 be a bounded linear operator. For ∀
x
1
∈
H
1, sequence {
x
n
} can be generated by the iteration:
(58)
u
n
=
x
n
-
γ
A
(
I
-
T
)
A
x
n
,
y
n
=
(
1
-
β
)
u
n
+
β
U
(
u
n
)
,
x
n
+
1
=
α
n
ψ
(
y
n
)
+
(
1
-
α
n
)
y
n
,
n
≥
1
,
where the sequence {
α
n
}
⊂
(
0,1
) satisfies (i) lim
⁡
n
→
∞
α
n
=
0 and (ii) ∑
n
=
0
∞
‍
α
n
=
∞, {
β
}
⊂
[
a
,
b
] with a
,
b
∈
(
0
,
1
/
(
1
+
k
1
)
), and {
γ
}
⊂
(
0,2
/
L
) with L being the largest eigenvalue of the matrix A
T
A. Assume that I
-
U and I
-
T are demiclosed at zero. If Γ
≠
∅, then {
x
n
} generated by (58) converges strongly to a solution of the two-set SCFPP.Proof.
For eachp
∈
Γ, if we take T
=
T
n, U
=
U
n, μ
n
→
0 and ξ
n
→
0, and follow the proof of Theorem 10, we can also prove that {
x
n
} converges strongly to x
*
∈
Γ by the same way.Remark 12.
Algorithm (7) and Theorems 10 and 11 improve and extend the corresponding results of Censor and Segal [2], Moudafi [3, 4], Mohammed [5, 6], Chang et al. [11, 13], Yang et al. [12], and others.


## 4. Concluding Remarks

In this work, we develop the split common fixed point problem with more general classes of total quasi-asymptotically pseudocontractive and quasi-pseudocontractive operators; corresponding algorithms are improved based on the viscosity iteration; thus we can obtain strong convergence without more constraints on operators.



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*Source: 309368-2014-04-03.xml*