# Applications of Bregman-Opial Property to Bregman Nonspreading Mappings in Banach Spaces

Authors: Eskandar Naraghirad; Ngai-Ching Wong; Jen-Chih Yao
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The Opial property of Hilbert spaces and some other special Banach spaces
is a powerful tool in establishing fixed point theorems for nonexpansive and, more generally,
nonspreading mappings. Unfortunately, not every Banach space shares the Opial property.
However, every Banach space has a similar Bregman-Opial property for Bregman distances.
In this paper, using Bregman distances, we introduce the classes of Bregman nonspreading
mappings and investigate the Mann and Ishikawa iterations for these mappings. We establish
weak and strong convergence theorems for Bregman nonspreading mappings.


---

## Body

## 1. Introduction

LetE be a (real) Banach space with norm ∥
·
∥ and dual space E
*. For any x in E, we denote the value of x
* in E
* at x by 〈
x
,
x
*
〉. When {
x
n
}
n
∈
ℕ is a sequence in E, we denote the strong convergence of {
x
n
}
n
∈
ℕ to x
∈
E by x
n
→
x and the weak convergence by x
n
⇀
x. Let C be a nonempty subset of E. Let T
:
C
→
E be a map. We denote by F
(
T
)
=
{
x
∈
C
:
T
x
=
x
} the set of fixed points of T. We call the map T
(i)
nonexpansive if ∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥ for all x
,
y in C,
(ii)
quasi-nonexpansive if F
(
T
)
≠
∅ and ∥
T
x
-
y
∥
≤
∥
x
-
y
∥ for all x in C and y in F
(
T
).The nonexpansivity plays an important role in the study of theIshikawa iteration, given by
(1)
y
n
=
β
n
T
x
n
+
(
1
-
β
n
)
x
n
,
x
n
+
1
=
γ
n
T
y
n
+
(
1
-
γ
n
)
x
n
,
where the sequences {
β
n
}
n
∈
ℕ and {
γ
n
}
n
∈
ℕ satisfy some appropriate conditions. When all β
n
=
0, Ishikawa iteration (1) reduces to the classical Mann iteration. Construction of fixed points of nonexpansive mappings via Mann's and Ishikawa’s algorithms [1] has been extensively investigated in the literature (see, e.g., [2] and the references therein).A powerful tool in deriving weak or strong convergence of iterative sequences is due to Opial [3]. A Banach space E is said to satisfy the Opial property [3] if for any weakly convergent sequence {
x
n
}
n
∈
ℕ in E with weak limit x we have
(2)
limsup
⁡
n
→
∞
∥
x
n
-
x
∥
<
lim
⁡
sup
⁡
n
→
∞
∥
x
n
-
y
∥
,
for all y in E with y
≠
x. It is well known that all Hilbert spaces, all finite dimensional Banach spaces, and the Banach spaces l
p (1
≤
p
<
∞) satisfy the Opial property. However, not every Banach space satisfies the Opial property; see, for example, [4, 5].Working with the Bregman distanceD
g, the following Bregman-Opial-like inequality holds for every Banach space E:
(3)
limsup
⁡
n
→
∞
D
g
(
x
n
,
x
)
<
lim
⁡
sup
⁡
n
→
∞
D
g
(
x
n
,
y
)
,
whenever x
n
⇀
x
≠
y. See Lemma 11 for details. The Bregman-Opial property suggests introducing the notions of Bregman nonexpansive-like mappings and developing fixed point theorems and convergence results for the Ishikawa iterations for these mappings.We recall the definition of Bregman distances. Letg
:
E
→
ℝ be a strictly convex and Gâteaux differentiable function on a Banach space E. The Bregman distance [6] (see also [7, 8]) corresponding to g is the function D
g
:
E
×
E
→
ℝ defined by
(4)
D
g
(
x
,
y
)
=
g
(
x
)
-
g
(
y
)
-
〈
x
-
y
,
∇
g
(
y
)
〉
,
=
g
(
x
)
LLLLlL
-
g
(
y
)
-
∀
x
,
y
∈
E
.
It follows from the strict convexity of g that D
g
(
x
,
y
)
≥
0 for all x
,
y in E. However, D
g might not be symmetric and D
g might not satisfy the triangular inequality.WhenE is a smooth Banach space, setting g
(
x
)
=
∥
x
∥
2 for all x in E, we have that ∇
g
(
x
)
=
2
J
x for all x in E. Here J is the normalized duality mapping from E into E
*. Hence, D
g
(
·
,
·
) reduces to the usual map ϕ
(
·
,
·
) as
(5)
D
g
(
x
,
y
)
=
ϕ
(
x
,
y
)
≔
∥
x
∥
2
-
2
〈
x
,
J
y
〉
+
∥
y
∥
2
,
=
ϕ
(
x
,
y
)
≔
LLLLl
∥
x
∥
2
-
2
∀
x
,
y
∈
E
.
If E is a Hilbert space, then D
g
(
x
,
y
)
=
∥
x
-
y
∥
2.Letg
:
E
→
ℝ be strictly convex and Gâteaux differentiable, and let C
⊆
E be nonempty. A mapping T
:
C
→
E is said to be(i)
Bregman nonexpansive if
(6)
D
g
(
T
x
,
T
y
)
≤
D
g
(
x
,
y
)
,
∀
x
,
y
∈
C
;
(ii)
Bregman quasi-nonexpansive if F
(
T
)
≠
∅ and
(7)
D
g
(
p
,
T
x
)
≤
D
g
(
p
,
x
)
,
∀
x
∈
C
,
∀
p
∈
F
(
T
)
;
(iii)
Bregman skew quasi-nonexpansive if F
(
T
)
≠
∅ and
(8)
D
g
(
T
x
,
p
)
≤
D
g
(
x
,
p
)
,
∀
x
∈
C
,
∀
p
∈
F
(
T
)
,
(iv)
Bregman nonspreading if
(9)
D
g
(
T
x
,
T
y
)
+
D
g
(
T
y
,
T
x
)
≤
D
g
(
T
x
,
y
)
+
D
g
(
T
y
,
x
)
,
∀
x
,
y
∈
C
.It is obvious that every Bregman nonspreading mapT with F
(
T
)
≠
∅ is Bregman quasi-nonexpansive. Bregman nonspreading mappings include, in particular, the class of nonspreading functions studied by Takahashi and his coauthors (see, e.g., [9, 10]), which is defined with the map ϕ in (5).Let us give an example of a Bregman nonspreading mapping with nonempty fixed point set, which is not quasi-nonexpansive.Example 1.
Letg
:
ℝ
→
ℝ be defined by g
(
x
)
=
x
4. The associated Bregman distance is given by
(10)
D
g
(
x
,
y
)
=
x
4
-
y
4
-
4
(
x
-
y
)
y
3
=
x
4
+
3
y
4
-
4
x
y
3
,
∀
x
,
y
∈
R
.
Define T
:
[
0,2
]
→
[
0,2
] by
(11)
T
x
=
{
0
if
x
∈
[
0,2
)
,
1
if
x
=
2
.
We have F
(
T
)
=
{
0
}. Plainly, T is neither nonexpansive nor continuous.
However,T is Bregman nonspreading. To see this, we define f
:
[
0,2
]
×
[
0,2
]
→
ℝ by
(12)
f
(
x
,
y
)
=
D
g
(
T
x
,
T
y
)
+
D
g
(
T
y
,
T
x
)
-
D
g
(
T
x
,
y
)
-
D
g
(
T
y
,
x
)
,
∀
x
,
y
∈
[
0,2
]
.
Consider the following three possible cases.Case 1.
Ifx
=
y
=
2, then we have T
x
=
T
y
=
1 and hence
(13)
f
(
2,2
)
=
0
+
0
-
17
-
17
=
-
34
<
0
.Case 2.
Ifx
=
2 and y
∈
[
0,2
), then we have T
x
=
1, T
y
=
0, and hence
(14)
f
(
2
,
y
)
=
1
+
3
-
1
-
3
y
4
+
4
y
3
-
48
=
-
3
y
4
+
4
y
3
-
45
<
0
.Case 3.
Ifx
,
y
∈
[
0,2
), then we have T
x
=
T
y
=
0 and hence
(15)
f
(
x
,
y
)
=
-
3
(
x
4
+
y
4
)
≤
0
.
Thus we have f
(
x
,
y
)
≤
0 for all x
,
y in [
0,2
] and hence T is a Bregman nonspreading mapping.In Section2, we collect and study some basic ties of Bregman distances. In Section 3, utilizing the Bregman-Opial property, we present some fixed point theorems. In Sections 4 and 5, we investigate weak and strong convergence of the Ishikawa and Bregman-Ishikawa iterations for Bregman nonspreading mappings. Our results improve and generalize some known results in the current literature; see, for example, [11].


## 2. Bregman Functions and Bregman Distances

LetE be a (real) Banach space, and let g
:
E
→
ℝ. For any x in E, the gradient  ∇
g
(
x
) is defined to be the linear functional in E
* such that
(16)
〈
y
,
∇
g
(
x
)
〉
=
lim
⁡
t
→
0
g
(
x
+
t
y
)
-
g
(
x
)
t
,
∀
y
∈
E
.
The function g is said to be Gâteaux differentiable at x if ∇
g
(
x
) is well defined, and g is Gâteaux differentiable if it is Gâteaux differentiable everywhere on E. We call g Fréchet differentiable at x (see, e.g., [12, page 13] or [13, page 508]) if, for all ϵ
>
0, there exists δ
>
0 such that
(17)
|
g
(
y
)
-
g
(
x
)
-
〈
y
-
x
,
∇
g
(
x
)
〉
|
≤
ϵ
∥
y
-
x
∥
,
g
(
y
)
-
g
L
L
L
L
L
(
x
)
-
whenever
∥
y
-
x
∥
≤
δ
.
The function g is said to be Fréchet differentiable if it is Fréchet differentiable everywhere.LetB be the closed unit ball of a Banach space E. A function g
:
E
→
ℝ is said to be(i)
strongly coercive if
(18)
lim
⁡
∥
x
n
∥
→
+
∞
g
(
x
n
)
∥
x
n
∥
=
+
∞
;
(ii)
locally bounded if g
(
r
B
) is bounded for all r
>
0;
(iii)
locally uniformly smooth on E ([14, pp. 207, 221]) if the function σ
r
:
[
0
,
+
∞
)
→
[
0
,
+
∞
], defined by
(19)
σ
r
(
t
)
=
sup
⁡
x
∈
r
B
,
y
∈
S
E
,
α
∈
(
0,1
)
(
α
g
(
x
+
(
1
-
α
)
t
y
)
=
=
sup
⁡
x
∈
r
B
,
y
∈
S
E
,
α
∈
(
0,1
)
+
(
1
-
α
)
g
(
x
-
α
t
y
)
-
g
(
x
)
)
×
(
α
(
1
-
α
)
)
-
1
/
2
,
satisfies
(20)
lim
⁡
t
↓
0
σ
r
(
t
)
t
=
0
,
∀
r
>
0
;
(iv)
locally uniformly convex on E (or uniformly convex on bounded subsets of E ([14, pp. 203, 221])) if the gauge  ρ
r
:
[
0
,
+
∞
)
→
[
0
,
+
∞
] of uniform convexity of g, defined by
(21)
ρ
r
(
t
)
=
inf
⁡
x
,
y
∈
r
B
,
∥
x
-
y
∥
=
t
,
α
∈
(
0,1
)
(
α
g
(
x
)
+
(
1
-
α
)
g
(
y
)
=
=
inf
⁡
x
,
y
∈
r
B
,
∥
x
-
y
∥
=
t
,
α
∈
(
0,1
)
-
g
(
α
x
+
(
1
-
α
)
y
)
)
×
(
α
(
1
-
α
)
)
-
1
/
2
,
satisfies
(22)
ρ
r
(
t
)
>
0
,
∀
r
,
t
>
0
.For a locally uniformly convex mapg
:
E
→
ℝ, we have
(23)
g
(
α
x
+
(
1
-
α
)
y
)
≤
α
(
x
)
g
+
(
1
-
α
)
g
(
y
)
-
α
(
1
-
α
)
ρ
r
(
∥
x
-
y
∥
)
,
for all x
,
y in r
B and for all α in (
0,1
).LetE be a Banach space and g
:
E
→
ℝ a strictly convex and Gâteaux differentiable function. By (4), the Bregman distance satisfies that [6]
(24)
D
g
(
x
,
z
)
=
D
g
(
x
,
y
)
+
D
g
(
y
,
z
)
+
〈
x
-
y
,
∇
g
(
y
)
-
∇
g
(
z
)
〉
,
∀
x
,
y
,
z
∈
E
.
In particular,
(25)
D
g
(
x
,
y
)
=
-
D
g
(
y
,
x
)
+
〈
y
-
x
,
∇
g
(
y
)
-
∇
g
(
x
)
〉
,
∀
x
,
y
∈
E
.Lemma 2 (see [15]).
LetE be a Banach space and g
:
E
→
ℝ a Gâteaux differentiable function which is locally uniformly convex on E. Let {
x
n
}
n
∈
ℕ and {
y
n
}
n
∈
ℕ be bounded sequences in E. Then the following assertions are equivalent: (1)
lim
⁡
n
→
∞
D
g
(
x
n
,
y
n
)
=
0,
(2)
lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0.The following Bregman-Opial-like inequality has been proved in [16].Lemma 3 (see [16]).
LetE be a Banach space and let g
:
E
→
ℝ be a strictly convex and Gâteaux differentiable function. Suppose {
x
n
}
n
∈
ℕ is a sequence in E such that x
n
⇀
x for some x in E. Then
(26)
limsup
⁡
n
→
∞
D
g
(
x
n
,
x
)
<
lim
⁡
sup
⁡
n
→
∞
D
g
(
x
n
,
y
)
,
for all y in the interior of dom g with y
≠
x.We call a functiong
:
E
→
(
-
∞
,
+
∞
]  lower semicontinuous if {
x
∈
E
:
g
(
x
)
≤
r
} is closed for all r in ℝ. For a lower semicontinuous convex function g
:
E
→
ℝ, the subdifferential  ∂
g of g is defined by
(27)
∂
g
(
x
)
=
{
x
*
∈
E
*
:
g
(
x
)
∂
g
(
x
)
=
+
〈
y
-
x
,
x
*
〉
≤
g
(
y
)
,
∀
y
∈
E
}
,
for all x in E. It is well known that ∂
g
⊂
E
×
E
* is maximal monotone [17, 18]. For any lower semicontinuous convex function g
:
E
→
(
-
∞
,
+
∞
], the conjugate function  g
* of g is defined by
(28)
g
*
(
x
*
)
=
sup
⁡
x
∈
E
{
〈
x
,
x
*
〉
-
g
(
x
)
}
,
∀
x
*
∈
E
*
.
It is well known that
(29)
g
(
x
)
+
g
*
(
x
*
)
≥
〈
x
,
x
*
〉
,
∀
(
x
,
x
*
)
∈
E
×
E
*
,
(30)
(
x
,
x
*
)
∈
∂
g
is
equivalent
to
g
(
x
)
+
g
*
(
x
*
)
=
〈
x
,
x
*
〉
.
We also know that if g
:
E
→
(
-
∞
,
+
∞
] is a proper lower semicontinuous convex function, then g
*
:
E
*
→
(
-
∞
,
+
∞
] is a proper weak* lower semicontinuous convex function. Here, saying g is proper we mean that dom
⁡
g
:
=
{
x
∈
E
:
g
(
x
)
<
+
∞
}
≠
∅.The following definition is slightly different from that in Butnariu and Iusem [12].Definition 4 (see [13]).
LetE be a Banach space. A function g
:
E
→
ℝ is said to be a Bregman function if the following conditions are satisfied: (1)
g is continuous, strictly convex, and Gâteaux differentiable;
(2)
the set{
y
∈
E
:
D
g
(
x
,
y
)
≤
r
} is bounded for all x in E and r
>
0.The following lemma follows from Butnariu and Iusem [12] and Zǎlinescu [14].Lemma 5.
LetE be a reflexive Banach space and g
:
E
→
ℝ a strongly coercive Bregman function. Then(1)
∇
g
:
E
→
E
*
is one-to-one, onto, and norm-to-weak*continuous;
(2)
〈
x
-
y
,
∇
g
(
x
)
-
∇
g
(
y
)
〉
=
0
if and only if
x
=
y
;
(3)
{
x
∈
E
:
D
g
(
x
,
y
)
≤
r
}
is bounded for all  y  in  E  and
r
>
0
;
(4)
dom
⁡
g
*
=
E
*
,
g
*
is Gâteaux differentiable, and
∇
g
*
=
(
∇
g
)
-
1
.The following two results follow from [14, Proposition 3.6
.
4].Proposition 6.
LetE be a reflexive Banach space and let g
:
E
→
ℝ be a convex function which is locally bounded. The following assertions are equivalent:(1)
g is strongly coercive and locally uniformly convex on E;
(2)
dom
⁡
g
*
=
E
* and g
* is locally bounded and locally uniformly smooth on E;
(3)
dom
⁡
g
*
=
E
*
,
g
* is Fréchet differentiable, and ∇
g
* is uniformly norm-to-norm continuous on bounded subsets of E
*.Proposition 7.
LetE be a reflexive Banach space and g
:
E
→
ℝ a continuous convex function which is strongly coercive. The following assertions are equivalent: (1)
g is locally bounded and locally uniformly smooth on E;
(2)
g
* is Fréchet differentiable and ∇
g
* is uniformly norm-to-norm continuous on bounded subsets of E;
(3)
dom
⁡
g
*
=
E
*
and
g
* is strongly coercive and locally uniformly convex on E.Lemma 8 (see [13, 19]).
LetE be a reflexive Banach space, g
:
E
→
ℝ a strongly coercive Bregman function, and V the function defined by
(31)
V
(
x
,
x
*
)

=

g
(
x
)
-

〈
x
,
x
*
〉

+

g
*
(
x
*
)
,
∀
x
∈
E
,
∀
x
*
∈
E
*
.
The following assertions hold: (1)
D
g
(
x
,
∇
g
*
(
x
*
)
)
=
V
(
x
,
x
*
) for all x in E and x
* in E
*,
(2)
V
(
x
,
x
*
)
+
〈
∇
g
*
(
x
*
)
-
x
,
y
*
〉
≤
V
(
x
,
x
*
+
y
*
) for all x in E and x
*
,
y
* in E
*.It also follows from the definition thatV is convex in the second variable x
*, and
(32)
V
(
x
,
∇
g
(
y
)
)
=
D
g
(
x
,
y
)
.LetE be a Banach space and let C be a nonempty convex subset of E. Let g
:
E
→
ℝ be a strictly convex and Gâteaux differentiable function. Then, we know from [20] that, for x in E and x
0 in C, we have
(33)
D
g
(
x
0
,
x
)
=
min
⁡
y
∈
C
D
g
(
y
,
x
)
if
and
only
if
〈
y
-
x
0
,
∇
g
(
x
)
-
∇
g
(
x
0
)
〉
≤
0
,
∀
y
∈
C
.
Further, if C is a nonempty, closed, and convex subset of a reflexive Banach space E and g
:
E
→
ℝ is a strongly coercive Bregman function, then, for each x in E, there exists a unique x
0 in C such that
(34)
D
g
(
x
0
,
x
)
=
min
⁡
y
∈
C
D
g
(
y
,
x
)
.
The Bregman projection
proj
C
g from E onto C defined by proj
C
g
(
x
)
=
x
0 has the following property:
(35)
D
g
(
y
,
proj
C
g
x
)
+
D
g
(
proj
C
g
x
,
x
)
≤
D
g
(
y
,
x
)
,
∀
y
∈
C
,
∀
x
∈
E
.
See [12] for details.LetE be a reflexive Banach space and let g
:
E
→
ℝ be a lower semicontinuous, strictly convex, and Gâteaux differentiable function. Let C be a nonempty, closed, and convex subset of E and let {
x
n
}
n
∈
ℕ be a bounded sequence in E. For any x in E, we set
(36)
Br
(
x
,
{
x
n
}
)
=
lim
⁡
sup
⁡
n
→
∞
D
g
(
x
n
,
x
)
.
The Bregman asymptotic radius of {
x
n
}
n
∈
ℕ relative to C is defined by
(37)
Br
(
C
,
{
x
n
}
)
=
inf
⁡
{
Br
(
x
,
{
x
n
}
)
:
x
∈
C
}
.
The Bregman asymptotic center of {
x
n
}
n
∈
ℕ relative to C is the set
(38)
BA
(
C
,
{
x
n
}
)
=
{
x
∈
C
:
Br
(
x
,
{
x
n
}
)
=
Br
(
C
,
{
x
n
}
)
}
.Proposition 9.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E, and let g
:
E
→
ℝ be strictly convex, Gâteaux differentiable, and locally bounded on E. If {
x
n
}
n
∈
ℕ is a bounded sequence of C, then B
A
(
C
,
{
x
n
}
n
∈
ℕ
) is a singleton.Proof.
In view of the definition of Bregman asymptotic radius, we may assume that{
x
n
}
n
∈
ℕ converges weakly to z in C. By Lemma 3, we conclude that BA
(
C
,
{
x
n
}
n
∈
ℕ
)
=
{
z
}.


## 3. Fixed Point Theorems

Lemma 10 (see [21]).
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g
:
E
→
ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable, and locally bounded on E. Let T
:
C
→
E be a Bregman quasi-nonexpansive mapping. Then F
(
T
) is closed and convex.Lemma 11.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g
:
E
→
ℝ be a strictly convex and Gâteaux differentiable function. Let T
:
C
→
E be a Bregman nonspreading mapping. Then
(39)
D
g
(
x
,
T
y
)
≤
D
g
(
x
,
y
)
+
D
g
(
T
x
,
x
)
+
〈
x
-
T
x
,
∇
g
(
y
)
-
∇
g
(
T
y
)
〉
+
〈
T
x
-
T
y
,
∇
g
(
x
)
-
∇
g
(
T
x
)
〉
,
∀
x
,
y
∈
C
.Proof.
Letx
,
y
∈
C. In view of (24), we have
(40)
D
g
(
T
x
,
T
y
)
≤
D
g
(
T
x
,
y
)
+
D
g
(
T
y
,
x
)
-
D
g
(
T
y
,
T
x
)
=
D
g
(
T
x
,
x
)
+
D
g
(
x
,
y
)
+
〈
T
x
-
x
,
∇
g
(
x
)
-
∇
g
(
y
)
〉
+
D
g
(
T
y
,
T
x
)
+
D
g
(
T
x
,
x
)
+

〈
T
y

-

T
x
,
∇
g
(
T
x
)

-

∇
g
(
x
)
〉

-

D
g
(
T
y
,
T
x
)
=
D
g
(
x
,
y
)
+
2
D
g
(
T
x
,
x
)
+
〈
T
x
-
x
,
∇
g
(
x
)
-
∇
g
(
y
)
〉
+
〈
T
x
-
T
y
,
∇
g
(
x
)
-
∇
g
(
T
x
)
〉
.
This, together with (24), implies that
(41)
D
g
(
x
,
T
y
)
=
D
g
(
x
,
T
x
)
+
D
g
(
T
x
,
T
y
)
+
〈
x
-
T
x
,
∇
g
(
T
x
)
-
∇
g
(
T
y
)
〉
≤
D
g
(
x
,
T
x
)
+
D
g
(
x
,
y
)
+
2
D
g
(
T
x
,
x
)
+
〈
T
x
-
x
,
∇
g
(
x
)
-
∇
g
(
y
)
〉
+
〈
T
x
-
T
y
,
∇
g
(
x
)
-
∇
g
(
T
x
)
〉
+
〈
x
-
T
x
,
∇
g
(
T
x
)
-
∇
g
(
T
y
)
〉
=
D
g
(
x
,
y
)
+
D
g
(
T
x
,
x
)
+
〈
x
-
T
x
,
∇
g
(
x
)
-
∇
g
(
T
x
)
〉
+
〈
T
x
-
T
y
,
∇
g
(
x
)
-
∇
g
(
T
x
)
〉
+
〈
T
x
-
x
,
∇
g
(
x
)
-
∇
g
(
y
)
〉
+
〈
x
-
T
x
,
∇
g
(
T
x
)
-
∇
g
(
T
y
)
〉
=
D
g
(
x
,
y
)
+
D
g
(
T
x
,
x
)
+
〈
x
-
T
x
,
∇
g
(
y
)
-
∇
g
(
T
y
)
〉
+
〈
T
x
-
T
y
,
∇
g
(
x
)
-
∇
g
(
T
x
)
〉
.Proposition 12 (demiclosedness principle).
LetC be a nonempty subset of a reflexive Banach space E. Let g
:
E
→
ℝ be a strictly convex, Gâteaux differentiable, and locally bounded function. Let T
:
C
→
E be a Bregman nonspreading mapping. If x
n
⇀
z in C and lim
⁡
n
→
∞
∥
T
x
n
-
x
n
∥
=
0, then T
z
=
z. That is, I
-
T is demiclosed at zero, where I is the identity mapping on E.Proof.
Since{
x
n
}
n
∈
ℕ converges weakly to z and lim
⁡
n
→
∞
∥
T
x
n
-
x
n
∥
=
0, both the sequences {
x
n
}
n
∈
ℕ and {
T
x
n
}
n
∈
ℕ are bounded. Since ∇
g is uniformly norm-to-norm continuous on bounded subsets of E (see, e.g., [14]), we arrive at
(42)
lim
⁡
n
→
∞
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
=
0
.
In view of Lemma 2, we deduce that lim
⁡
n
→
∞
D
g
(
x
n
,
T
x
n
)
=
0. Set
(43)
M
1
=
sup
⁡
{
∥
T
x
n
∥
,
∥
T
z
∥
,
∥
∇
g
(
z
)
∥
,
∥
∇
g
(
T
z
)
∥
:
n
∈
ℕ
}
<
+
∞
.
By Lemma 11, for all n in ℕ,
(44)
D
g
(
x
n
,
T
z
)
≤
D
g
(
x
n
,
z
)
+
D
g
(
T
x
n
,
x
n
)
+
〈
x
n
-
T
x
n
,
∇
g
(
z
)
-
∇
g
(
T
z
)
〉
+
〈
T
x
n
-
T
z
,
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
〉
≤
D
g
(
x
n
,
z
)
+
D
g
(
T
x
n
,
x
n
)
+
∥
x
n
-
T
x
n
∥
∥
∇
g
(
z
)
-
∇
g
(
T
z
)
∥
+
∥
T
x
n
-
T
z
∥
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
≤
D
g
(
x
n
,
z
)
+
D
g
(
T
x
n
,
x
n
)
+
2
M
1
∥
x
n
-
T
x
n
∥
+
2
M
1
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
.
This implies
(45)
lim
⁡
sup
⁡
n
→
∞
D
g
(
x
n
,
T
z
)
≤
lim
⁡
sup
⁡
n
→
∞
D
g
(
x
n
,
z
)
.
From the Bregman-Opial-like property, we obtain T
z
=
z.Letℓ
∞ be the Banach lattice of bounded real sequences with the supremum norm. It is well known that there exists a bounded linear functional μ on ℓ
∞ such that the following three conditions hold:(1)
if{
t
n
}
n
∈
ℕ
∈
ℓ
∞ and t
n
≥
0 for every n in ℕ, then μ
(
{
t
n
}
)
≥
0;
(2)
ift
n
=
1 for every n in ℕ, then μ
(
{
t
n
}
)
=
1;
(3)
μ
(
{
t
n
+
1
}
)
=
μ
(
{
t
n
}
) for all {
t
n
}
n
∈
ℕ in ℓ
∞.Here,{
t
n
+
1
} denotes the sequence (
t
2
,
t
3
,
t
4
,
…
,
t
n
+
1
,
…
) in ℓ
∞. Such a functional μ is called a Banach limit and the value of μ at {
t
n
}
n
∈
ℕ in ℓ
∞ is denoted by μ
n
t
n. Therefore, condition (
3
) means μ
n
t
n
=
μ
n
t
n
+
1. If μ satisfies conditions (
1
) and (
2
), we call μ a mean on ℓ
∞. See, for example, [22].To see some examples of those mappingsT satisfying all the stated hypotheses in the following result, we refer the reader to [23].Theorem 13 (see [23]).
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g
:
E
→
ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded and locally uniformly convex on E. Let T
:
C
→
C be a mapping. Let {
x
n
}
n
∈
ℕ be a bounded sequence of C and let μ be a mean on ℓ
∞. Suppose that
(46)
μ
n
D
g
(
x
n
,
T
y
)
≤
μ
n
D
g
(
x
n
,
y
)
,
∀
y
∈
C
.
Then T has a fixed point in C.Corollary 14.
LetC be a nonempty, bounded, closed, and convex subset of a reflexive Banach space E. Let g
:
E
→
ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable function, locally bounded, and locally uniformly convex on E. Let T
:
C
→
C be a Bregman nonspreading mapping. Then T has a fixed point.Proof.
Letμ a Banach limit on ℓ
∞ and x
∈
C be such that {
T
n
x
}
n
∈
ℕ is bounded. For any n in ℕ we have
(47)
D
g
(
T
n
x
,
T
y
)
+
D
g
(
T
y
,
T
n
x
)
≤
D
g
(
T
n
x
,
y
)
+
D
g
(
T
y
,
T
n
-
1
x
)
,
∀
y
∈
C
.
This implies that
(48)
μ
n
D
g
(
T
n
x
,
T
y
)
+
μ
n
D
g
(
T
y
,
T
n
x
)
≤
μ
n
D
g
(
T
n
x
,
y
)
+
μ
n
D
g
(
T
y
,
T
n
-
1
x
)
,
∀
y
∈
C
.
Thus we have
(49)
μ
n
D
g
(
T
n
x
,
T
y
)
≤
μ
n
D
g
(
T
n
x
,
y
)
,
∀
y
∈
C
.
It follows from Theorem 13 that F
(
T
)
≠
∅.


## 4. Weak and Strong Convergence Theorems for Bregman Nonspreading Mappings

In this section, we prove weak and strong convergence theorems concerning Bregman nonspreading mappings in a reflexive Banach space.Lemma 15.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g
:
E
→
ℝ be a strictly convex and Gâteaux differentiable function. Let T
:
C
→
C be a Bregman skew quasi-nonexpansive mapping with a nonempty fixed point set F
(
T
). Let {
x
n
}
n
∈
ℕ and {
y
n
}
n
∈
ℕ be two sequences defined by (1) such that {
β
n
}
n
∈
ℕ and {
γ
n
}
n
∈
ℕ are arbitrary sequences in [
0,1
]. Then the following assertions hold:(1)
max
⁡
{
D
g
(
x
n
+
1
,
z
), D
g
(
y
n
,
z
)
}
≤
D
g
(
x
n
,
z
) for all z in F
(
T
) and n
=
1,2
,
…,
(2)
lim
⁡
n
→
∞
D
g
(
x
n
,
z
) exists for any z in F
(
T
).Proof.
Letz
∈
F
(
T
). In view of (23), we have
(50)
D
g
(
y
n
,
z
)
=
D
g
(
β
n
T
x
n
+
(
1
-
β
n
)
x
n
,
z
)
≤
β
n
D
g
(
T
x
n
,
z
)
+
(
1
-
β
n
)
D
g
(
x
n
,
z
)
≤
β
n
D
g
(
x
n
,
z
)
+
(
1
-
β
n
)
D
g
(
x
n
,
z
)
=
D
g
(
x
n
,
z
)
.
Consequently,
(51)
D
g
(
x
n
+
1
,
z
)
=
D
g
(
γ
n
T
y
n
+
(
1
-
γ
n
)
x
n
,
z
)
≤
γ
n
D
g
(
T
y
n
,
z
)
+
(
1
-
γ
n
)
D
g
(
x
n
,
z
)
≤
γ
n
D
g
(
y
n
,
z
)
+
(
1
-
γ
n
)
D
g
(
x
n
,
z
)
≤
γ
n
D
g
(
x
n
,
z
)
+
(
1
-
γ
n
)
D
g
(
x
n
,
z
)
=
D
g
(
x
n
,
z
)
.
This implies that {
D
g
(
x
n
,
z
)
}
n
∈
ℕ is a bounded and nonincreasing sequence for all z in F
(
T
). Thus we have that lim
⁡
n
→
∞
D
g
(
x
n
,
z
) exists for any z in F
(
T
).Theorem 16.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g
:
E
→
ℝ be strictly convex, Gâteaux differentiable, locally bounded, and locally uniformly convex on E. Let T
:
C
→
C be a Bregman nonspreading and Bregman skew quasi-nonexpansive mapping. Let {
β
n
}
n
∈
ℕ and {
γ
n
}
n
∈
ℕ be sequences in [
0,1
], and let {
x
n
}
n
∈
ℕ be a sequence with x
1 in C defined by (1). (a)
If{
x
n
}
n
∈
ℕ is bounded and liminf
⁡
n
→
∞
∥
T
x
n
-
x
n
∥
=
0, then the fixed point set F
(
T
)
≠
∅.
(b)
AssumeF
(
T
)
≠
∅. Then {
x
n
}
n
∈
ℕ is bounded.
(i)
lim
⁡
n
→
∞
∥
T
x
n
-
x
n
∥
=
0 when liminf
⁡
n
→
∞
γ
n
(
1
-
γ
n
)
>
0 and lim
⁡
n
→
∞
β
n
=
1.
(ii)
liminf
⁡
n
→
∞
∥
T
x
n
-
x
n
∥
=
0 when either
(1)
limsup
⁡
n
→
∞
γ
n
(
1
-
γ
n
)
>
0 and lim
⁡
n
→
∞
β
n
=
1 or
(2)
liminf
⁡
n
→
∞
γ
n
(
1
-
γ
n
)
>
0 and limsup
⁡
n
→
∞
β
n
=
1.Proof.
Assume that{
x
n
}
n
∈
ℕ is bounded and liminf
⁡
n
→
∞
∥
T
x
n
-
x
n
∥
=
0. Consequently, there is a bounded subsequence {
T
x
n
k
}
k
∈
ℕ of {
T
x
n
}
n
∈
ℕ such that lim
⁡
k
→
∞
∥
T
x
n
k
-
x
n
k
∥
=
0. Since ∇
g is uniformly norm-to-norm continuous on bounded subsets of E (see, e.g., [14]),
(52)
lim
⁡
k
→
∞
∥
∇
g
(
T
x
n
k
)
-
∇
g
(
x
n
k
)
∥
=
0
.
In view of Proposition 9, we conclude that BA
(
C
,
{
x
n
k
}
)
=
{
z
} for some z in C. Let
(53)
M
2
=
sup
⁡
{
∥
T
(
z
)
∥
,
∥
T
x
n
k
∥
,
∥
∇
g
(
z
)
∥
,
M
2
=
sup
⁡
∥
∇
g
(
T
z
)
∥
:
k
∈
ℕ
}
<
+
∞
.
It follows from Lemma 11 that
(54)
D
g
(
x
n
k
,
T
z
)
≤
D
g
(
x
n
k
,
z
)
+
D
g
(
T
x
n
k
,
x
n
k
)
+
〈
x
n
k
-
T
x
n
k
,
∇
g
(
z
)
-
∇
g
(
T
z
)
〉
+
〈
T
x
n
k
-
T
z
,
∇
g
(
x
n
k
)
-
∇
g
(
T
x
n
k
)
〉
≤
D
g
(
x
n
k
,
z
)
+
D
g
(
T
x
n
k
,
x
n
k
)
+
∥
x
n
k
-
T
x
n
k
∥
∥
∇
g
(
z
)
-
∇
g
(
T
z
)
∥
+
∥
T
x
n
k
-
T
z
∥
∥
∇
g
(
x
n
k
)
-
∇
g
(
T
x
n
k
)
∥
≤
D
g
(
x
n
k
,
z
)
+
D
g
(
T
x
n
k
,
x
n
k
)
+
2
M
2
∥
x
n
k
-
T
x
n
k
∥
+
2
M
2
∥
∇
g
(
x
n
k
)
-
∇
g
(
T
x
n
k
)
∥
,
k
=
1,2
,
…
.
This implies
(55)
limsup
⁡
k
→
∞
D
g
(
x
n
k
,
T
z
)
≤
limsup
⁡
k
→
∞
D
g
(
x
n
k
,
z
)
.
From the Bregman-Opial-like property, we obtain T
z
=
z.
LetF
(
T
)
≠
∅ and let z
∈
F
(
T
). It follows from Lemma 15 that lim
⁡
n
→
∞
∥
x
n
-
z
∥ exists and hence {
x
n
}
n
∈
ℕ is bounded. This implies that the sequence {
T
y
n
}
n
∈
ℕ is bounded too. Let s
1
=
sup
⁡
{
∥
x
n
∥
,
∥
T
y
n
∥
:
n
∈
ℕ
}
<
∞. In view of (23), we obtain a continuous, strictly increasing, and convex function ρ
s
1
:
[
0
,
+
∞
)
→
[
0
,
+
∞
) with ρ
s
1
(
0
)
=
0 such that
(56)
D
g
(
x
n
+
1
,
z
)
=
D
g
(
γ
n
T
y
n
+
(
1
-
γ
n
)
x
n
,
z
)
≤
γ
n
D
g
(
T
y
n
,
z
)
+
(
1
-
γ
n
)
D
g
(
x
n
,
z
)
-
γ
n
(
1
-
γ
n
)
ρ
s
1
(
∥
T
y
n
-
x
n
∥
)
≤
γ
n
D
g
(
y
n
,
z
)
+
(
1
-
γ
n
)
D
g
(
x
n
,
z
)
-
γ
n
(
1
-
γ
n
)
ρ
s
1
(
∥
T
y
n
-
x
n
∥
)
≤
γ
n
D
g
(
x
n
,
z
)
+
(
1
-
γ
n
)
D
g
(
x
n
,
z
)
-
γ
n
(
1
-
γ
n
)
ρ
s
1
(
∥
T
y
n
-
x
n
∥
)
=
D
g
(
x
n
,
z
)
-
γ
n
(
1
-
γ
n
)
ρ
s
1
(
∥
T
y
n
-
x
n
∥
)
.
Consequently, we conclude that
(57)
γ
n
(
1
-
γ
n
)
ρ
s
1
(
∥
T
y
n
-
x
n
∥
)
≤
D
g
(
x
n
,
z
)
-
D
g
(
x
n
+
1
,
z
)
→
0
,
as
n
→
∞
.
It follows that
(58)
liminf
⁡
n
→
∞
ρ
s
1
(
∥
T
y
n
-
x
n
∥
)
=
0
whenever
lim
⁡
sup
⁡
n
→
∞
γ
n
(
1
-
γ
n
)
>
0
.
From the property of ρ
s
1 we deduce that
(59)
lim
⁡
inf
⁡
n
→
∞
∥
T
y
n
-
x
n
∥
=
0
whenever
lim
⁡
sup
⁡
n
→
∞
γ
n
(
1
-
γ
n
)
>
0
.
In the same manner, we also obtain that
(60)
lim
⁡
n
→
∞
∥
T
y
n
-
x
n
∥
=
0
whenever
lim
⁡
inf
⁡
n
→
∞
γ
n
(
1
-
γ
n
)
>
0
.
Since ∇
g is uniformly norm-to-norm continuous on bounded subsets of E (see, e.g., [14]), we arrive at
(61)
lim
⁡
n
→
∞
∥
∇
g
(
T
y
n
)
-
∇
g
(
x
n
)
∥
=
0
.
On the other hand, from (1) we get
(62)
T
x
n
-
y
n
=
(
1
-
β
n
)
(
T
x
n
-
x
n
)
,
x
n
-
y
n
=
β
n
(
x
n
-
T
x
n
)
.
Assuming first liminf
⁡
n
→
∞
γ
n
(
1
-
γ
n
)
>
0. By (60) we see that
(63)
M
3
≔
sup
⁡
{
∥
∇
g
(
x
n
)
∥
,
∥
∇
g
(
T
x
n
)
∥
,
M
3
≔
sup
⁡
∥
∇
g
(
T
y
n
)
∥
:
n
∈
ℕ
}
<
+
∞
.
Since T is Bregman nonspreading, in view of (24), (25), and (62), we obtain
(64)
D
g
(
x
n
,
T
x
n
)
=
D
g
(
x
n
,
T
y
n
)
+
D
g
(
T
y
n
,
T
x
n
)
+
〈
x
n
-
T
y
n
,
∇
g
(
T
y
n
)
-
∇
g
(
T
x
n
)
〉
≤
D
g
(
x
n
,
T
y
n
)
+
[
D
g
(
T
y
n
,
x
n
)
+
D
g
(
T
x
n
,
y
n
)
-
D
g
(
T
x
n
,
T
y
n
)
]
+
∥
x
n
-
T
y
n
∥
∥
∇
g
(
T
y
n
)
-
∇
g
(
T
x
n
)
∥
≤
D
g
(
x
n
,
T
y
n
)
+
[
-
D
g
(
x
n
,
T
y
n
)
+
〈
x
n
-
T
y
n
,
∇
g
(
x
n
)
-
∇
g
(
T
y
n
)
〉
]
+
[
-
D
g
(
y
n
,
T
x
n
)
+
〈
y
n
-
T
x
n
,
∇
g
(
y
n
)
-
∇
g
(
T
x
n
)
〉
]
+
∥
x
n
-
T
y
n
∥
∥
∇
g
(
T
y
n
)
-
∇
g
(
T
x
n
)
∥
≤
∥
x
n
-
T
y
n
∥
∥
∇
g
(
x
n
)
-
∇
g
(
T
y
n
)
∥
+
∥
y
n
-
T
x
n
∥
∥
∇
g
(
y
n
)
-
∇
g
(
T
x
n
)
∥
+
∥
x
n
-
T
y
n
∥
∥
∇
g
(
T
y
n
)
-
∇
g
(
T
x
n
)
∥
=
(
1
-
β
n
)
∥
x
n
-
T
x
n
∥
∥
∇
g
(
y
n
)
-
∇
g
(
T
x
n
)
∥
+
∥
x
n
-
T
y
n
∥
[
∥
∇
g
(
x
n
)
-
∇
g
(
T
y
n
)
+
∥
∇
g
(
T
y
n
)
-
∇
g
(
T
x
n
)
∥
]
≤
2
(
1
-
β
n
)
M
3
∥
x
n
-
T
x
n
∥
+
4
M
3
∥
x
n
-
T
y
n
∥
.
When lim
⁡
n
→
∞
β
n
=
1, we conclude that
(65)
lim
⁡
n
→
∞
D
g
(
x
n
,
T
x
n
)
=
0
.
In view of Lemma 2, we have that
(66)
lim
⁡
n
→
∞
∥
x
n
-
T
x
n
∥
=
0
.
Finally, we assumelimsup
⁡
n
→
∞
γ
n
(
1
-
γ
n
)
>
0 and lim
⁡
n
→
∞
β
n
=
1 instead. By (59) we have subsequences {
x
n
k
}
k
∈
ℕ and {
y
n
k
}
k
∈
ℕ of {
x
n
}
n
∈
ℕ and {
y
n
}
n
∈
ℕ, respectively, such that
(67)
lim
⁡
k
→
∞
∥
T
y
n
k
-
x
n
k
∥
=
0
.
Replacing M
3 with the finite number sup
⁡
{
∥
∇
g
(
x
n
k
)
∥
,
∥
∇
g
(
T
x
n
k
)
∥
,
∥
∇
g
(
T
y
n
k
)
∥
:
k
∈
ℕ
}
<
+
∞, and dealing with the subsequences {
x
n
k
}
k
∈
ℕ and {
y
n
k
}
k
∈
ℕ in (60) and (62). Passing to a further subsequence if necessary, we will arrive at the desired conclusion with (66) that lim
⁡
k
→
∞
∥
T
x
n
k
-
x
n
k
∥
=
0. Hence, liminf
⁡
n
→
∞
∥
T
x
n
-
x
n
∥
=
0. The other case can be argued similarly.Theorem 17.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g
:
E
→
ℝ be strictly convex, Gâteaux differentiable, locally bounded, and locally uniformly convex on E. Let T
:
C
→
C be a Bregman nonspreading and Bregman skew quasi-nonexpansive mapping with F
(
T
)
≠
∅. Let {
β
n
}
n
∈
ℕ and {
γ
n
}
n
∈
ℕ be sequences in [
0,1
], and let {
x
n
}
n
∈
ℕ be a sequence with x
1 in C defined by (1). Assume that liminf
⁡
n
→
∞
γ
n
(
1
-
γ
n
)
>
0 and lim
⁡
n
→
∞
β
n
=
1. Then {
x
n
}
n
∈
ℕ converges weakly to a fixed point of T.Proof.
It follows from Theorem16 that {
x
n
}
n
∈
N is bounded and lim
⁡
n
→
∞
∥
T
x
n
-
x
n
∥
=
0. Since E is reflexive, then there exists a subsequence {
x
n
i
}
i
∈
ℕ of {
x
n
}
n
∈
ℕ such that x
n
i
⇀
p
∈
C as i
→
∞. By Proposition 12, p
∈
F
(
T
). We claim that x
n
⇀
p as n
→
∞. If not, then there exists a subsequence {
x
n
j
}
j
∈
ℕ of {
x
n
}
n
∈
N such that {
x
n
j
}
j
∈
ℕ converges weakly to some q in C with p
≠
q. In view of Proposition 12 again, we conclude that q
∈
F
(
T
). By Lemma 15, lim
⁡
n
→
∞
D
g
(
x
n
,
z
) exists for all z in F
(
T
). Thus we obtain by the Bregman-Opial-like property that
(68)
lim
⁡
n
→
∞
D
g
(
x
n
,
p
)
=
lim
⁡
i
→
∞
D
g
(
x
n
i
,
p
)
<
lim
⁡
i
→
∞
D
g
(
x
n
i
,
q
)
=
lim
⁡
n
→
∞
D
g
(
x
n
,
q
)
=
lim
⁡
j
→
∞
D
g
(
x
n
j
,
q
)
<
lim
⁡
j
→
∞
D
g
(
x
n
j
,
p
)
=
lim
⁡
n
→
∞
D
g
(
x
n
,
p
)
.
This is a contradiction. Thus we have p
=
q, and the desired assertion follows.Theorem 18.
LetC be a nonempty, compact, and convex subset of a reflexive Banach space E. Let g
:
E
→
ℝ be strictly convex, Gâteaux differentiable, locally bounded, and uniformly convex on bounded sets. Let T
:
C
→
C be a Bregman nonspreading and Bregman skew quasi-nonexpansive mapping. Let {
β
n
}
n
∈
ℕ and {
γ
n
}
n
∈
ℕ be sequences in [
0,1
]. Assume that either limsup
⁡
n
→
∞
γ
n
(
1
-
γ
n
)
>
0 and lim
⁡
n
→
∞
β
n
=
1 or liminf
⁡
n
→
∞
γ
n
(
1
-
γ
n
)
>
0 and limsup
⁡
n
→
∞
β
n
=
1. Let {
x
n
}
n
∈
ℕ be a sequence with x
1 in C defined by (1). Then {
x
n
}
n
∈
ℕ converges strongly to a fixed point z of T.Proof.
By Corollary14, we see that the fixed point set F
(
T
) of T is nonempty. In view of Theorem 16, we obtain that {
x
n
}
n
∈
ℕ is bounded and liminf
⁡
n
→
∞
∥
T
x
n
-
x
n
∥
=
0. By the compactness of C, there exists a subsequence {
x
n
k
}
k
∈
ℕ of {
x
n
}
n
∈
ℕ such that {
x
n
k
}
k
∈
ℕ converges strongly to some z in C. In view of Lemma 2 we deduce that lim
⁡
k
→
∞
D
g
(
x
n
k
,
z
)
=
0. We can even assume that lim
⁡
k
→
∞
∥
T
x
n
k
-
x
n
k
∥
=
0, and in particular, {
T
x
n
k
}
k
∈
ℕ is bounded. Since ∇
g is uniformly norm-to-norm continuous on bounded subsets of E (see, e.g., [14]),
(69)
lim
⁡
k
→
∞
∥
∇
g
(
T
x
n
k
)
-
∇
g
(
x
n
k
)
∥
=
0
.
Let M
4
=
sup
⁡
{
∥
T
z
∥
,
∥
T
x
n
k
∥
,
∥
∇
g
(
z
)
∥
,
∥
∇
g
(
T
z
)
∥
:
k
∈
ℕ
}
<
+
∞. In view of Lemma 11, we obtain
(70)
D
g
(
x
n
k
,
T
z
)
≤
D
g
(
x
n
k
,
z
)
+
D
g
(
T
x
n
k
,
x
n
k
)
+
〈
x
n
k
-
T
x
n
k
,
∇
g
(
z
)
-
∇
g
(
T
z
)
〉
+
〈
T
x
n
k
-
T
z
,
∇
g
(
x
n
k
)
-
∇
g
(
T
x
n
k
)
〉
≤
D
g
(
x
n
k
,
z
)
+
D
g
(
T
x
n
k
,
x
n
k
)

+

2
M
4
[
∥
x
n
k

-

T
x
n
k
∥

+

∥
∇
g
(
x
n
k
)


-

∇
g
(
T
x
n
k
)
∥
]
for all k in ℕ.
It follows thatlim
⁡
k
→
∞
∥
x
n
k
-
T
z
∥
=
0. Thus we have T
z
=
z. In view of Lemmas 15 and 2, we conclude that lim
⁡
n
→
∞
∥
x
n
-
z
∥
=
0. Therefore, z is the strong limit of the sequence {
x
n
}
n
∈
ℕ.


## 5. Bregman-Ishikawa’s Type Iteration for Bregman Nonspreading Mappings

We propose the following Bregman-Ishikawa's type iteration. LetE be a reflexive Banach space and let g
:
E
→
ℝ be a strictly convex and Gâteaux differentiable function. Let C be a nonempty, closed, and convex subset of E. Let T
:
C
→
C be a Bregman nonspreading mapping such that the fixed point set F
(
T
) is nonempty. Let {
x
n
}
n
∈
ℕ and {
y
n
}
n
∈
ℕ be two sequences defined by
(71)
y
n
=
∇
g
*
[
β
n
∇
g
(
T
x
n
)
+
(
1
-
β
n
)
∇
g
(
x
n
)
]
,
x
n
+
1
=
proj
C
g
(
∇
g
*
[
γ
n
∇
g
(
T
y
n
)
+
(
1
-
γ
n
)
∇
g
(
x
n
)
]
)
,
where {
β
n
}
n
∈
ℕ and {
γ
n
}
n
∈
ℕ are arbitrary sequences in [
0,1
].Lemma 19.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g
:
E
→
ℝ be a strongly coercive Bregman function. Let T
:
C
→
C be a Bregman quasi-nonexpansive mapping. Let {
x
n
}
n
∈
ℕ and {
y
n
}
n
∈
ℕ be two sequences defined by (71) such that {
β
n
}
n
∈
ℕ and {
γ
n
}
n
∈
ℕ are arbitrary sequences in [
0,1
]. Then the following assertions hold:(1)
max
⁡
{
D
g
(
z
,
x
n
+
1
)
,
D
g
(
z
,
y
n
)
}
≤
D
g
(
z
,
x
n
) for all z in F
(
T
) and n
=
1,2
,
…,
(2)
lim
⁡
n
→
∞
D
g
(
z
,
x
n
) exists for any z in F
(
T
).Proof.
Letz
∈
F
(
T
). In view of Lemma 8 and (71), we conclude that
(72)
D
g
(
z
,
y
n
)
=
D
g
(
z
,
∇
g
*
[
β
n
∇
g
(
T
x
n
)
+
(
1
-
β
n
)
∇
g
(
x
n
)
]
)
=
V
(
z
,
β
n
∇
g
(
T
x
n
)
+
(
1
-
β
n
)
∇
g
(
x
n
)
)
≤
β
n
V
(
z
,
∇
g
(
T
x
n
)
)
+
(
1
-
β
n
)
V
(
z
,
∇
g
(
x
n
)
)
=
β
n
D
g
(
z
,
T
x
n
)
+
(
1
-
β
n
)
D
g
(
z
,
x
n
)
≤
β
n
D
g
(
z
,
x
n
)
+
(
1
-
β
n
)
D
g
(
z
,
x
n
)
=
D
g
(
z
,
x
n
)
.
Consequently, using (35) we have
(73)
D
g
(
z
,
x
n
+
1
)
=
D
g
(
z
,
proj
C
g
(
∇
g
*
[
γ
n
∇
g
(
T
y
n
)
+
(
1
-
γ
n
)
∇
g
(
x
n
)
]
)
)
≤
D
g
(
z
,
∇
g
*
[
γ
n
∇
g
(
T
y
n
)
+
(
1
-
γ
n
)
∇
g
(
x
n
)
]
)
=
V
(
z
,
γ
n
∇
g
(
T
y
n
)
+
(
1
-
γ
n
)
∇
g
(
x
n
)
)
≤
γ
n
V
(
z
,
∇
g
(
T
y
n
)
)
+
(
1
-
γ
n
)
V
(
z
,
∇
g
(
x
n
)
)
=
γ
n
D
g
(
z
,
T
y
n
)
+
(
1
-
γ
n
)
D
g
(
z
,
x
n
)
≤
γ
n
D
g
(
z
,
y
n
)
+
(
1
-
γ
n
)
D
g
(
z
,
x
n
)
≤
γ
n
D
g
(
z
,
x
n
)
+
(
1
-
γ
n
)
D
g
(
z
,
x
n
)
=
D
g
(
z
,
x
n
)
.
This implies that {
D
g
(
z
,
x
n
)
}
n
∈
ℕ is a bounded and nonincreasing sequence for all z in F
(
T
). Thus we have that lim
⁡
n
→
∞
D
g
(
z
,
x
n
) exists for any z in F
(
T
).Theorem 20.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g
:
E
→
ℝ be a strongly coercive Bregman function which is locally bounded, locally uniformly convex, and locally uniformly smooth on E. Let T
:
C
→
C be a Bregman nonspreading mapping. Let {
α
n
}
n
∈
ℕ and {
β
n
}
n
∈
ℕ be two sequences in [
0,1
] satisfying the control condition:
(74)
∑
n
=
1
∞
‍
γ
n
β
n
(
1
-
β
n
)
=
+
∞
.
Let {
x
n
}
n
∈
ℕ be a sequence generated by algorithm (71). Then the following are equivalent. (1)
There exists a bounded sequence{
x
n
}
n
∈
ℕ
⊂
C such that liminf
⁡
n
→
∞
∥
T
x
n
-
x
n
∥
=
0.
(2)
The fixed point setF
(
T
)
≠
∅.Proof.
The implication(
1
)
⇒
(
2
) follows similarly as in the first part of the proof of Theorem 16.
For the implication(
2
)
⇒
(
1
), we assume F
(
T
)
≠
∅. The boundedness of the sequences {
x
n
}
n
∈
ℕ and {
y
n
}
n
∈
ℕ follows from Lemma 19 and Definition 4. Since T is a Bregman quasi-nonexpansive mapping, for any q in F
(
T
), we have
(75)
D
g
(
q
,
T
x
n
)
≤
D
g
(
q
,
x
n
)
,
∀
n
∈
ℕ
.
This, together with Definition 4 and the boundedness of {
x
n
}
n
∈
ℕ, implies that {
T
x
n
}
n
∈
ℕ is bounded.
The functiong is bounded on bounded subsets of E and therefore ∇
g is also bounded on bounded subsets of E
* (see, e.g., [12, Proposition 1.1
.
11] for more details). This implies that the sequences {
∇
g
(
x
n
)
}
n
∈
ℕ, {
∇
g
(
y
n
)
}
n
∈
ℕ, {
∇
g
(
T
y
n
)
}
n
∈
ℕ, and {
∇
g
(
T
x
n
)
}
n
∈
ℕ are bounded in E
*.
In view of Proposition7, we have that dom
⁡
g
*
=
E
* and g
* is strongly coercive and uniformly convex on bounded subsets of E
*. Let s
2
=
sup
⁡
{
∥
∇
g
(
x
n
)
∥
,
∥
∇
g
(
T
x
n
)
∥
:
n
∈
ℕ
}
<
∞ and let ρ
s
2
*
:
E
*
→
ℝ be the gauge of uniform convexity of the conjugate function g
*.
Claim. For any p in F
(
T
) and n in ℕ,
(76)
D
g
(
p
,
y
n
)
≤
D
g
(
p
,
x
n
)
-
β
n
(
1
-
β
n
)
ρ
s
2
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
.
Letp
∈
F
(
T
). For each n in ℕ, it follows from the definition of Bregman distance (4), Lemma 8, (23), and (71) that
(77)
D
g
(
p
,
y
n
)
=
g
(
p
)
-
g
(
y
n
)
-
〈
p
-
y
n
,
∇
g
(
y
n
)
〉
=
g
(
p
)
+
g
*
(
∇
g
(
y
n
)
)
-
〈
y
n
,
∇
g
(
y
n
)
〉
-
〈
p
,
∇
g
(
y
n
)
〉
+
〈
y
n
,
∇
g
(
y
n
)
〉
=
g
(
p
)
+
g
*
(
(
1
-
β
n
)
∇
g
(
x
n
)
+
β
n
∇
g
(
T
x
n
)
)
-
〈
p
,
(
1
-
β
n
)
∇
g
(
x
n
)
+
β
n
∇
g
(
T
x
n
)
〉
≤
(
1
-
β
n
)
g
(
p
)
+
β
n
g
(
p
)
+
(
1
-
β
n
)
g
*
(
∇
g
(
x
n
)
)
+
β
n
g
*
(
∇
g
(
T
x
n
)
)
-
β
n
(
1
-
β
n
)
ρ
s
2
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
-
(
1
-
β
n
)
〈
p
,
∇
g
(
x
n
)
〉
-
β
n
〈
p
,
∇
g
(
T
x
n
)
〉
=
(
1
-
β
n
)
[
g
(
p
)
+
g
*
(
∇
g
(
x
n
)
)
-
〈
p
,
∇
g
(
x
n
)
〉
]
+
β
n
[
g
(
p
)
+
g
*
(
∇
g
(
T
x
n
)
)
-
〈
p
,
∇
g
(
T
x
n
)
〉
]
-
β
n
(
1
-
β
n
)
ρ
s
2
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
=
(
1
-
β
n
)
[
g
(
p
)
-
g
(
x
n
)
+
〈
x
n
,
∇
g
(
x
n
)
〉
-
〈
p
,
∇
g
(
x
n
)
〉
]
+
β
n
[
g
(
p
)
-
g
(
T
x
n
)
+
〈
T
x
n
,
∇
g
(
T
x
n
)
〉
-
〈
p
,
∇
g
(
T
x
n
)
〉
]
-
β
n
(
1
-
β
n
)
ρ
s
2
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
=
(
1
-
β
n
)
D
(
p
,
x
n
)
+
β
n
D
(
p
,
T
x
n
)
-
β
n
(
1
-
β
n
)
ρ
s
2
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
≤
(
1
-
β
n
)
D
(
p
,
x
n
)
+
β
n
D
(
p
,
x
n
)
-
β
n
(
1
-
β
n
)
ρ
s
2
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
=
D
(
p
,
x
n
)
-
β
n
(
1
-
β
n
)
ρ
s
2
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
.
In view of Lemma 8 and (76), we obtain
(78)
D
g
(
p
,
x
n
+
1
)
=
D
g
(
p
,
∇
g
*
[
γ
n
∇
g
(
T
y
n
)
+
(
1
-
γ
n
)
∇
g
(
x
n
)
]
)
=
V
(
p
,
γ
n
∇
g
(
T
y
n
)
+
(
1
-
γ
n
)
∇
g
(
x
n
)
)
≤
γ
n
V
(
p
,
∇
g
(
T
y
n
)
)
+
(
1
-
γ
n
)
V
(
p
,
∇
g
(
x
n
)
)
=
γ
n
D
g
(
p
,
T
y
n
)
+
(
1
-
γ
n
)
D
g
(
p
,
x
n
)
≤
γ
n
D
g
(
p
,
y
n
)
+
(
1
-
γ
n
)
D
g
(
p
,
x
n
)
≤
D
g
(
p
,
x
n
)
-
γ
n
β
n
(
1
-
β
n
)
ρ
s
2
*
×
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
.
Thus we have
(79)
γ
n
β
n
(
1
-
β
n
)
ρ
s
2
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
n
x
n
)
∥
)
≤
D
g
(
p
,
x
n
)
-
D
g
(
p
,
x
n
+
1
)
.
Since {
D
g
(
p
,
x
n
)
}
n
∈
ℕ converges, together with the control condition (74), we have
(80)
liminf
⁡
n
→
∞
ρ
s
2
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
=
0
.
Therefore, from the property of ρ
s
2
* we deduce that
(81)
lim
⁡
inf
⁡
n
→
∞
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
=
0
.
Since ∇
g
* is uniformly norm-to-norm continuous on bounded subsets of E
* (see, e.g., [14]), we arrive at
(82)
lim
⁡
inf
⁡
n
→
∞
∥
x
n
-
T
x
n
∥
=
0
.Theorem 21.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g
:
E
→
ℝ be a strongly coercive Bregman function which is locally bounded, locally uniformly convex, and locally uniformly smooth on E. Let T
:
C
→
C be a Bregman nonspreading mapping with F
(
T
)
≠
∅. Let {
α
n
}
n
∈
ℕ and {
β
n
}
n
∈
ℕ be two sequences in [
0,1
] satisfying the control conditions ∑
n
=
1
∞
‍
γ
n
β
n
(
1
-
β
n
)
=
+
∞. Let {
x
n
}
n
∈
ℕ be a sequence generated by the algorithm (71). Then, there exists a subsequence {
x
n
i
}
i
∈
ℕ of {
x
n
}
n
∈
ℕ which converges weakly to a fixed point of T as i
→
∞.Proof.
It follows from Theorem20 that {
x
n
}
n
∈
ℕ is bounded and liminf
⁡
n
→
∞
∥
T
x
n
-
x
n
∥
=
0. Since E is reflexive, then there exists a subsequence {
x
n
i
}
i
∈
ℕ of {
x
n
}
n
∈
ℕ such that  x
n
i
⇀
p
∈
C as i
→
∞. In view of Proposition 12, we conclude that p
∈
F
(
T
) and the desired conclusion follows.The construction of fixed points of nonexpansive mappings via Halpern's algorithm [24] has been extensively investigated recently in the current literature (see, e.g., [2] and the references therein). Numerous results have been proved on Halpern's iterations for nonexpansive mappings in Hilbert and Banach spaces (see, e.g., [11, 25, 26]).Before dealing with the strong convergence of a Halpern-type iterative algorithm, we need the following lemmas.Lemma 22 (see [27]).
Let{
a
n
}
n
∈
ℕ be a sequence in ℝ with a subsequence {
a
n
i
}
i
∈
ℕ such that a
n
i
<
a
n
i
+
1 for all i in ℕ. Then there exists another subsequence {
a
m
k
}
k
∈
ℕ such that for all (sufficiently large) number k one have
(83)
a
m
k
≤
a
m
k
+
1
,
a
k
≤
a
m
k
+
1
.
In fact, one can set m
k
=
max
⁡
{
j
≤
k
:
a
j
<
a
j
+
1
}.Lemma 23 (see [28]).
Let{
s
n
}
n
∈
ℕ be a sequence of nonnegative real numbers satisfying
(84)
s
n
+
1
≤
(
1
-
γ
n
)
s
n
+
γ
n
δ
n
,
∀
n
≥
1
,
where {
γ
n
}
n
∈
ℕ and {
δ
n
}
n
∈
ℕ satisfy the following conditions: (i)
{
γ
n
}
n
∈
ℕ
⊂
[
0,1
] and ∑
n
=
1
∞
‍
γ
n
=
+
∞, or, equivalently, Π
n
=
1
∞
(
1
-
γ
n
)
=
0,
(ii)
lim
⁡
sup
⁡
n
→
∞
δ
n
≤
0, or
(iii)
∑
n
=
1
∞
‍
γ
n
δ
n
<
∞.Then, lim
⁡
n
→
∞
s
n
=
0.Theorem 24.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g
:
E
→
ℝ be a strongly coercive Bregman function which is locally bounded, locally uniformly convex, and locally uniformly smooth on E. Let T
:
C
→
C be a Bregman nonspreading mapping with F
(
T
)
≠
∅. Let {
α
n
}
n
∈
ℕ and {
β
n
}
n
∈
ℕ be two sequences in [
0,1
] satisfying the following control conditions: (a)
lim
⁡
n
→
∞
α
n
=
0;
(b)
∑
n
=
1
∞
‍
α
n
=
+
∞;
(c)
0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1.
Let{
x
n
}
n
∈
ℕ be a sequence generated by
(85)
u
∈
C
,
x
1
∈
C
c
h
o
s
e
n
a
r
b
i
t
r
a
r
i
l
y
,
y
n
=
∇
g
*
[
β
n
∇
g
(
x
n
)
+
(
1
-
β
n
)
∇
g
(
T
x
n
)
]
,
x
n
+
1
=
p
r
o
j
C
g
(
∇
g
*
[
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
]
)
f
o
r
n
i
n
ℕ
.
Then the sequence {
x
n
}
n
∈
ℕ defined in (85) converges strongly to p
r
o
j
F
(
T
)
g
u as n
→
∞.Proof.
We divide the proof into several steps. In view of Lemma10, we conclude that F
(
T
) is closed and convex. Set
(86)
z
=
proj
F
(
T
)
g
u
.
Step  1. We prove that {
x
n
}
n
∈
ℕ and {
y
n
}
n
∈
ℕ are bounded sequences in C.
We first show that{
x
n
}
n
∈
ℕ is bounded. Let p
∈
F
(
T
) be fixed. In view of Lemma 8 and (85), we have
(87)
D
g
(
p
,
y
n
)
=
D
g
(
p
,
∇
g
*
[
(
1
-
β
n
)
∇
g
(
x
n
)
+
β
n
∇
g
(
T
x
n
)
]
)
=
V
(
p
,
[
(
1
-
β
n
)
∇
g
(
x
n
)
+
β
n
∇
g
(
T
x
n
)
]
)
≤
(
1
-
β
n
)
V
(
p
,
∇
g
(
x
n
)
)
+
β
n
V
(
p
,
∇
g
(
T
x
n
)
)
=
(
1
-
β
n
)
D
g
(
p
,
x
n
)
+
β
n
D
g
(
p
,
T
x
n
)
≤
(
1
-
β
n
)
D
g
(
p
,
x
n
)
+
β
n
D
g
(
p
,
x
n
)
=
D
g
(
p
,
x
n
)
.
This, together with (71), implies that
(88)
D
g
(
p
,
x
n
+
1
)
=
D
g
(
p
,
proj
C
g
(
∇
g
*
[
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
]
)
)
≤
D
g
(
p
,
∇
g
*
[
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
]
)
=
V
(
p
,
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
)
≤
α
n
V
(
p
,
∇
g
(
u
)
)
+
(
1
-
α
n
)
V
(
p
,
∇
g
(
y
n
)
)
=
α
n
D
g
(
p
,
u
)
+
(
1
-
α
n
)
D
g
(
p
,
y
n
)
≤
α
n
D
g
(
p
,
u
)
+
(
1
-
α
n
)
D
g
(
p
,
y
n
)
≤
α
n
D
g
(
p
,
u
)
+
(
1
-
α
n
)
D
g
(
p
,
x
n
)
≤
max
⁡
{
D
g
(
p
,
u
)
,
D
g
(
p
,
x
n
)
}
.
By induction, we obtain
(89)
D
g
(
p
,
x
n
+
1
)
≤
max
⁡
{
D
g
(
p
,
u
)
,
D
g
(
p
,
x
1
)
}
for all n in ℕ. It follows from (89) that the sequence {
D
g
(
p
,
x
n
)
}
n
∈
ℕ is bounded and hence there exists M
7
>
0 such that
(90)
D
g
(
p
,
x
n
)
≤
M
7
,
∀
n
∈
ℕ
.
In view of Definition 4, we deduce that the sequence {
x
n
}
n
∈
ℕ is bounded. Since T is a Bregman quasi-nonexpansive mapping from C into itself, we conclude that
(91)
D
g
(
p
,
T
x
n
)
≤
D
g
(
p
,
x
n
)
,
∀
n
∈
ℕ
.
This, together with Definition 4 and the boundedness of {
x
n
}
n
∈
ℕ, implies that {
T
x
n
}
n
∈
ℕ is bounded. The function g is bounded on bounded subsets of E and therefore ∇
g is also bounded on bounded subsets of E
* (see, e.g., [12, Proposition 1.1
.
11] for more details). This, together with Step  1, implies that the sequences {
∇
g
(
x
n
)
}
n
∈
ℕ, {
∇
g
(
y
n
)
}
n
∈
ℕ, and {
∇
g
(
T
x
n
)
}
n
∈
ℕ are bounded in E
*. In view of Proposition 7, we obtain that dom
⁡
g
*
=
E
* and g
* is strongly coercive and uniformly convex on bounded subsets of E. Let s
3
=
sup
⁡
{
∥
∇
g
(
x
n
)
∥
,
∥
∇
g
(
T
x
n
)
∥
:
n
∈
ℕ
} and let ρ
s
3
*
:
E
*
→
ℝ be the gauge of uniform convexity of the conjugate function g
*.
Step  2. We prove that
(92)
D
g
(
z
,
y
n
)
≤
D
g
(
z
,
x
n
)
-
β
n
(
1
-
β
n
)
ρ
s
3
*
×
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
,
∀
n
∈
ℕ
.
For each n in ℕ, in view of the definition of Bregman distance (4), Lemma 8, and (30), we obtain
(93)
D
g
(
z
,
y
n
)
=
g
(
z
)
-
g
(
y
n
)
-
〈
z
-
y
n
,
∇
g
(
y
n
)
〉
=
g
(
z
)
+
g
*
(
∇
g
(
y
n
)
)
-
〈
y
n
,
∇
g
(
y
n
)
〉
-
〈
z
,
∇
g
(
y
n
)
〉
+
〈
y
n
,
∇
g
(
y
n
)
〉
=
g
(
z
)
+
g
*
(
(
1
-
β
n
)
∇
g
(
x
n
)
+
β
n
∇
g
(
T
x
n
)
)
-
〈
z
,
(
1
-
β
n
)
∇
g
(
x
n
)
+
β
n
∇
g
(
T
x
n
)
〉
≤
(
1
-
β
n
)
g
(
z
)
+
β
n
g
(
z
)
+
(
1
-
β
n
)
g
*
×
(
∇
g
(
x
n
)
)
+
β
n
g
*
(
∇
g
(
T
x
n
)
)
-
β
n
(
1
-
β
n
)
ρ
s
3
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
-
(
1
-
β
n
)
〈
z
,
∇
g
(
x
n
)
〉
-
β
n
〈
z
,
∇
g
(
T
x
n
)
〉
=
(
1
-
β
n
)
[
g
(
z
)
+
g
*
(
∇
g
(
x
n
)
)
-
〈
z
,
∇
g
(
x
n
)
〉
]
+
β
n
[
g
(
z
)
+
g
*
(
∇
g
(
T
x
n
)
)
-
〈
z
,
∇
g
(
T
x
n
)
〉
]
-
β
n
(
1
-
β
n
)
ρ
s
3
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
=
(
1
-
β
n
)
[
g
(
z
)
-
g
(
x
n
)
+
〈
x
n
,
∇
g
(
x
n
)
〉
-
〈
z
,
∇
g
(
x
n
)
〉
]
+
β
n
[
g
(
z
)
-
g
(
T
x
n
)
+
〈
T
x
n
,
∇
g
(
T
x
n
)
〉
-
〈
z
,
∇
g
(
T
x
n
)
〉
]
-
β
n
(
1
-
β
n
)
ρ
s
3
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
n
x
n
)
∥
)
=
(
1
-
β
n
)
D
(
z
,
x
n
)
+
β
n
D
(
z
,
T
x
n
)
-
β
n
(
1
-
β
n
)
ρ
s
3
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
≤
(
1
-
β
n
)
D
g
(
z
,
x
n
)
+
β
n
D
g
(
z
,
x
n
)
-
β
n
(
1
-
β
n
)
ρ
s
3
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
=
D
(
z
,
x
n
)
-
β
n
(
1
-
β
n
)
ρ
s
3
*
×
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
.
In view of Lemma 8 and (92), we obtain
(94)
D
g
(
z
,
x
n
+
1
)
=
D
g
(
z
,
proj
C
g
(
∇
g
*
[
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
]
)
)
≤
D
g
(
z
,
∇
g
*
[
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
]
)
=
V
(
z
,
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
)
≤
α
n
V
(
z
,
∇
g
(
u
)
)
+
(
1
-
α
n
)
V
(
z
,
∇
g
(
y
n
)
)
=
α
n
D
g
(
z
,
u
)
+
(
1
-
α
n
)
D
g
(
z
,
y
n
)
≤
α
n
D
g
(
z
,
u
)
+
(
1
-
α
n
)
[
D
g
(
z
,
x
n
)
-
β
n
(
1
-
β
n
)
ρ
s
3
*
×

(
∥
∇
g
(
x
n
)
-

∇
g
(
T
x
n
)
∥
)
D
g
(
z
,
x
n
)
]
.
Let
(95)
M
8
≔
sup
⁡
{
|
D
g
(
z
,
u
)
-
D
g
(
z
,
x
n
)
|
+
β
n
(
1
-
β
n
)
ρ
s
3
*
×
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
:
n
∈
ℕ
|
D
g
(
z
,
u
)
-
D
g
(
z
,
x
n
)
|
}
.
It follows from (94) that
(96)
β
n
(
1
-
β
n
)
ρ
s
3
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
≤
D
g
(
z
,
x
n
)
-
D
g
(
z
,
x
n
+
1
)
+
α
n
M
8
.
Let
(97)
z
n
=
∇
g
*
[
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
]
.
Then x
n
+
1
=
proj
C
g
(
z
n
) for all n in ℕ. In view of Lemma 8 and (92) we obtain
(98)
D
g
(
z
,
x
n
+
1
)
=
D
g
(
z
,
proj
C
g
(
∇
g
*
[
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
]
)
)
≤
D
g
(
z
,
∇
g
*
[
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
]
)
=
V
(
z
,
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
)
≤
V
(
z
,
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
-
α
n
(
∇
g
(
u
)
-
∇
g
(
z
)
)
)
-
〈
∇
g
*
[
α
n
∇
g
(
u
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
]
-
z
,
-
α
n
(
∇
g
(
u
)
-
∇
g
(
z
)
)
〉
=
V
(
z
,
α
n
∇
g
(
z
)
+
(
1
-
α
n
)
∇
g
(
y
n
)
)
+
α
n
〈
z
n
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
≤
α
n
V
(
z
,
∇
g
(
z
)
)
+
(
1
-
α
n
)
V
(
z
,
∇
g
(
y
n
)
)
+
α
n
〈
z
n
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
=
α
n
D
g
(
z
,
z
)
+
(
1
-
α
n
)
D
g
(
z
,
y
n
)
+
α
n
〈
z
n
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
=
(
1
-
α
n
)
D
g
(
z
,
x
n
)
+
α
n
〈
z
n
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
.
Step  3. We show that x
n
→
z as n
→
∞.
Case  1. If there exists n
0 in ℕ such that {
D
g
(
z
,
x
n
)
}
n
=
n
0
∞ is non-increasing, then {
D
g
(
z
,
x
n
)
}
n
∈
ℕ is convergent. Thus, we have D
g
(
z
,
x
n
)
-
D
g
(
z
,
x
n
+
1
)
→
0 as n
→
∞. This, together with (96) and conditions (a) and (c), implies that
(99)
lim
⁡
n
→
∞
ρ
s
3
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
=
0
.
Therefore, from the property of ρ
s
3
* we deduce that
(100)
lim
⁡
n
→
∞
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
=
0
.
Since ∇
g
*
=
(
∇
g
)
-
1 (Lemma 5) is uniformly norm-to-norm continuous on bounded subsets of E
* (see, e.g., [14]), we arrive at
(101)
lim
⁡
n
→
∞
∥
x
n
-
T
x
n
∥
=
0
.
On the other hand, we have
(102)
D
g
(
T
x
n
,
y
n
)
=
D
g
(
T
x
n
,
∇
g
*
[
β
n
∇
g
(
x
n
)
+
(
1
-
β
n
)
∇
g
(
T
x
n
)
]
)
=
V
(
T
x
n
,
β
n
∇
g
(
x
n
)
+
(
1
-
β
n
)
∇
g
(
T
x
n
)
)
≤
β
n
V
(
T
x
n
,
∇
g
(
x
n
)
)
+
(
1
-
β
n
)
V
(
T
x
n
,
∇
g
(
T
x
n
)
)
=
β
n
D
g
(
T
x
n
,
x
n
)
+
(
1
-
β
n
)
D
g
(
T
x
n
,
T
x
n
)
=
β
n
D
g
(
T
x
n
,
x
n
)
.
This, together with Lemma 2 and (101), implies that
(103)
lim
⁡
n
→
∞
D
g
(
T
x
n
,
y
n
)
=
0
.
Similarly, we have
(104)
D
g
(
y
n
,
z
n
)
≤
α
n
D
g
(
y
n
,
u
)
+
(
1
-
α
n
)
D
g
(
y
n
,
y
n
)
=
α
n
D
g
(
y
n
,
u
)
→
0
as
n
→
∞
.
In view of Lemma 2 and (101), we conclude that
(105)
lim
⁡
n
→
∞
∥
y
n
-
T
x
n
∥
=
0
,
lim
⁡
n
→
∞
∥
z
n
-
x
n
∥
=
0
.
Since {
x
n
}
n
∈
ℕ is bounded, together with (33) we can assume that there exists a subsequence {
x
n
i
}
i
∈
ℕ of {
x
n
}
n
∈
ℕ such that x
n
i
⇀
y
∈
F
(
T
) (Proposition 12) and
(106)
lim
⁡
sup
⁡
n
→
∞
〈
x
n
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
=
lim
⁡
i
→
∞
〈
x
n
i
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
=
〈
y
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
≤
0
.
We thus conclude
(107)
lim
⁡
sup
⁡
n
→
∞
〈
z
n
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
=
lim
⁡
sup
⁡
n
→
∞
〈
x
n
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
≤
0
.
The desired result follows from Lemmas 2 and 23 and (98).
Case  2. Suppose there exists a subsequence {
n
i
}
i
∈
ℕ of {
n
}
n
∈
ℕ such that
(108)
D
g
(
z
,
x
n
i
)
<
D
g
(
z
,
x
n
i
+
1
)
for all i in ℕ. By Lemma 22, there exists a nondecreasing sequence {
m
k
}
k
∈
ℕ of positive integers such that m
k
→
∞,
(109)
D
g
(
z
,
x
m
k
)
<
D
g
(
z
,
x
m
k
+
1
)
,
D
g
(
z
,
x
k
)
≤
D
g
(
z
,
x
m
k
+
1
)
,
∀
k
∈
ℕ
.
This, together with (96), implies that
(110)
β
m
k
(
1
-
β
m
k
)
ρ
s
3
*
(
∥
∇
g
(
x
m
k
)
-
∇
g
(
T
x
m
k
)
∥
)
≤
D
g
(
z
,
x
m
k
)
-
D
g
(
z
,
x
m
k
+
1
)
+
α
m
k
M
8
≤
α
m
k
M
8
,
∀
k
∈
ℕ
.
Then, by conditions (a) and (c), we get
(111)
lim
⁡
k
→
∞
ρ
s
3
*
(
∥
∇
g
(
x
m
k
)
-
∇
g
(
T
x
m
k
)
∥
)
=
0
.
By the same argument, as in Case  1, we arrive at
(112)
lim
⁡
sup
⁡
k
→
∞
〈
z
m
k
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
=
lim
⁡
sup
⁡
k
→
∞
〈
x
m
k
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
≤
0
.
It follows from (98) that
(113)
D
g
(
z
,
x
m
k
+
1
)
≤
(
1
-
α
m
k
)
D
g
(
z
,
x
m
k
)
+
α
m
k
〈
z
m
k
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
.
Since D
g
(
z
,
x
m
k
)
≤
D
g
(
z
,
x
m
k
+
1
), we have that
(114)
α
m
k
D
g
(
z
,
x
m
k
)
≤
D
g
(
z
,
x
m
k
)
-
D
g
(
z
,
x
m
k
+
1
)
+
α
m
k
〈
z
m
k
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
≤
α
m
k
〈
z
m
k
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
.
In particular, since α
m
k
>
0, we obtain
(115)
D
g
(
z
,
x
m
k
)
≤
〈
z
m
k
-
z
,
∇
g
(
u
)
-
∇
g
(
z
)
〉
.
In view of (112), we deduce that
(116)
lim
⁡
k
→
∞
D
g
(
z
,
x
m
k
)
=
0
.
This, together with (113), implies that
(117)
lim
⁡
k
→
∞
D
g
(
z
,
x
m
k
+
1
)
=
0
.
On the other hand, we have D
g
(
z
,
x
k
)
≤
D
g
(
z
,
x
m
k
+
1
) for all k in ℕ. This ensures that x
k
→
z as k
→
∞ by Lemma 2.



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*Source: 272867-2014-01-16.xml*