# Bregmanf-Projection Operator with Applications to Variational Inequalities in Banach Spaces

Authors: Chin-Tzong Pang; Eskandar Naraghirad; Ching-Feng Wen
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Using Bregman functions, we introduce the new concept of Bregman generalizedf-projection operator Proj
C
f
,
g
:
E
*
→
C, where E is a reflexive Banach
space with dual space E
*
;
f
:
E
→
ℝ
∪
+
∞ is a proper, convex, lower semicontinuous and bounded from below function; g
:
E
→
ℝ is a strictly convex and Gâteaux differentiable function; and C is a nonempty, closed, and convex subset of E. The existence of a solution for a class of variational inequalities in Banach spaces is presented.


---

## Body

## 1. Introduction

Many nonlinear problems in functional analysis can be reduced to the search of fixed points of nonlinear operators. See, for example, [1–14] and the references therein. Let E 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 and T
:
C
→
E be a mapping. We denote by F
(
T
)
=
{
x
∈
C
:
T
x
=
x
} the set offixed points of T. Let C be a nonempty, closed, and convex subset of a smooth Banach space E; let T be a mapping from C into itself. A point p
∈
C is said to be anasymptotic fixed point [15] of T if there exists a sequence {
x
n
}
n
∈
ℕ in C which converges weakly to p and lim
⁡
n
→
∞
∥
x
n
-
T
x
n
∥
=
0. We denote the set of all asymptotic fixed points of T by F
^
(
T
). A point p
∈
C is called astrong asymptotic fixed point of T if there exists a sequence {
x
n
}
n
∈
ℕ in C which converges strongly to p and lim
⁡
n
→
∞
∥
x
n
-
T
x
n
∥
=
0. We denote the set of all strong asymptotic fixed points of T by F
~
(
T
).We recall the definition of Bregman distances. Letg
:
E
→
ℝ be a strictly convex and Gâteaux differentiable function on a Banach space E. TheBregman distance [16] (see also [17, 18]) corresponding to g is the function D
g
:
E
×
E
→
ℝ defined by
(1)
D
g
(
x
,
y
)
=
g
(
x
)
-
g
(
y
)
-
〈
x
-
y
,
∇
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
(2)
D
g
(
x
,
y
)
=
ϕ
(
x
,
y
)
:
=
∥
x
∥
2
-
2
〈
x
,
J
y
〉
+
∥
y
∥
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 C
⊆
E be nonempty. A mapping T
:
C
→
E is said to be(i)
Bregman nonexpansive if(3)
D
g
(
T
x
,
T
y
)
≤
D
g
(
x
,
y
)
,
∀
x
,
y
∈
C
.
(ii)
Bregman quasi-nonexpansive if F
(
T
)
≠
∅ and
(4)
D
g
(
p
,
T
x
)
≤
D
g
(
p
,
x
)
,
∀
x
∈
C
,
∀
p
∈
F
(
T
)
.
(iii)
Bregman relatively nonexpansive if the following conditions are satisfied:
(1)
F
(
T
) is nonempty;
(2)
D
g
(
p
,
T
v
)
≤
D
g
(
p
,
v
), ∀
p
∈
F
(
T
)
,
v
∈
C;
(3)
F
^
(
T
)
=
F
(
T
);
(iv)
Bregman weak relatively nonexpansive if the following conditions are satisfied:
(1)
F
(
T
) is nonempty;
(2)
D
g
(
p
,
T
v
)
≤
D
g
(
p
,
v
), ∀
p
∈
F
(
T
)
,
v
∈
C;
(3)
F
~
(
T
)
=
F
(
T
).It is clear that any Bregman relatively nonexpansive mapping is a Bregman quasi-nonexpansive mapping. It is also obvious that every Bregman relatively nonexpansive mapping is a Bregman weak relatively nonexpansive mapping, but the converse is not true in general; see, for example, [19]. Indeed, for any mapping T
:
C
→
C we have F
(
T
)
⊂
F
~
(
T
)
⊂
F
^
(
T
). If T is Bregman relatively nonexpansive, then F
(
T
)
=
F
~
(
T
)
=
F
^
(
T
).LetE be a reflexive Banach space, let f
:
E
→
ℝ
∪
{
+
∞
} be a proper, convex, lower semicontinuous function, let g
:
E
→
ℝ be strictly convex and Gâteaux differentiable, and let C
⊆
E be nonempty. We define a functional H
:
E
×
E
*
→
ℝ
∪
{
+
∞
} by
(5)
H
(
x
,
x
*
)
=
g
(
x
)
-
〈
x
,
x
*
〉
+
g
*
(
x
*
)
+
f
(
x
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
x
∈
E
,
x
*
∈
E
*
.
It could easily be seen that H satisfies the following properties:(1)
H
(
x
,
x
*
) is convex and continuous with respect to x
* when x is fixed;
(2)
H
(
x
,
x
*
) is convex and lower semicontinuous with respect to x when x
* is fixed.Definition 1.
LetE be a Banach space with dual space E
*, let f
:
E
→
ℝ
∪
{
+
∞
} be a proper, convex, lower semicontinuous function, let g
:
E
→
ℝ be strictly convex and Gâteaux differentiable, and let C be a nonempty, closed subset of E. We say that Proj
C
f
,
g
:
E
*
→
2
C is a Bregman generalized f-projection operator if
(6)
Proj
C
f
,
g
=
{
z
∈
C
:
H
(
z
,
x
*
)
=
inf
⁡
y
∈
C
H
(
y
,
x
*
)
}
,
∀
x
*
∈
E
*
.In this paper, using Bregman functions, we introduce the new concept of Bregman generalizedf-projection operator Proj
C
f
,
g
:
E
*
→
C, where E is a reflexive Banach space with dual space E
*, f
:
E
→
ℝ
∪
{
+
∞
} is a proper, convex, lower semicontinuous, and bounded from below function, g
:
E
→
ℝ is a strictly convex and Gâteaux differentiable function, and C is a nonempty, closed, and convex subset of E. The existence of a solution for a class of variational inequalities in Banach spaces is presented. Our results improve and generalize some known results in the current literature; see, for example, [20, 21].


## 2. Properties of Bregman Functions and Bregman Distances

LetE be a (real) Banach space, and let g
:
E
→
ℝ. For any x in E, thegradient
∇
g
(
x
) is defined to be the linear functional in E
* such that
(7)
〈
y
,
∇
g
(
x
)
〉
=
lim
⁡
t
→
0
g
(
x
+
t
y
)
-
g
(
x
)
t
,
∀
y
∈
E
.
The function g is said to beGâteaux differentiable at x if ∇
g
(
x
) is well defined, and g isGâteaux differentiable if it is Gâteaux differentiable everywhere on E. We call g Fréchet differentiable at x (see, for example, [22, page 13] or [23, page 508]) if, for all ϵ
>
0, there exists δ
>
0 such that
(8)
|
g
(
y
)
-
g
(
x
)
-
〈
y
-
x
,
∇
g
(
x
)
〉
|
≤
ϵ
∥
y
-
x
∥
whenever
∥
y
-
x
∥
≤
δ
.
The function g is said to beFréchet differentiable if it is Fréchet differentiable everywhere.For anyr
>
0, let B
r
:
=
{
z
∈
E
:
∥
z
∥
≤
r
}. A function g
:
E
→
ℝ is said to be(i)
strongly coercive if(9)
lim
⁡
∥
x
n
∥
→
+
∞
g
(
x
n
)
∥
x
n
∥
=
+
∞
;
(ii)
locally bounded if g
(
B
r
) is bounded for all r
>
0;
(iii)
locally uniformly smooth on E ([24, pages 207, 221]) if the function σ
r
:
[
0
,
+
∞
)
→
[
0
,
+
∞
], defined by
(10)
σ
r
(
t
)
=
sup
⁡
x
∈
B
r
,
y
∈
S
E
,
α
∈
(
0,1
)
(
(
α
(
1
-
α
)
)
-
1
(
α
g
(
x
+
(
1
-
α
)
t
y
)
hhhhhhhhhhhhhhhhhh
+
(
1
-
α
)
g
(
x
-
α
t
y
)
-
g
(
x
)
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
(
α
(
1
-
α
)
)
-
1
)
,
satisfies
(11)
lim
⁡
t
↓
0
σ
r
(
t
)
t
=
0
,
∀
r
>
0
;
(iv)
locally uniformly convex on E (oruniformly convex on bounded subsets of E ([24, pages 203, 221])) if the gauge ρ
r
:
[
0
,
+
∞
)
→
[
0
,
+
∞
] ofuniform convexity of g, defined by
(12)
ρ
r
(
t
)
=
inf
⁡
x
,
y
∈
B
r
,
∥
x
-
y
∥
=
t
,
α
∈
(
0,1
)
(
(
α
(
1
-
α
)
)
-
1
(
α
g
(
x
)
+
(
1
-
α
)
g
(
y
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
-
g
(
α
x
+
(
1
-
α
)
y
)
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
(
α
(
1
-
α
)
)
-
1
)
,
satisfies
(13)
ρ
r
(
t
)
>
0
,
∀
r
,
t
>
0
.For a locally uniformly convex mapg
:
E
→
ℝ, we have
(14)
g
(
α
x
+
(
1
-
α
)
y
)
≤
α
g
(
x
)
+
(
1
-
α
)
g
(
y
)
-
α
(
1
-
α
)
ρ
r
(
∥
x
-
y
∥
)
,
for all x
,
y in B
r and for all α in (
0,1
).LetE be a Banach space and g
:
E
→
ℝ a strictly convex and Gâteaux differentiable function. By (1), the Bregman distance satisfies [16]
(15)
D
g
(
x
,
z
)
=
D
g
(
x
,
y
)
+
D
g
(
y
,
z
)
+
〈
x
-
y
,
∇
g
(
y
)
-
∇
g
(
z
)
〉
,
∀
x
,
y
,
z
∈
E
.
In particular,
(16)
D
g
(
x
,
y
)
=
-
D
g
(
y
,
x
)
+
〈
y
-
x
,
∇
g
(
y
)
-
∇
g
(
x
)
〉
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
∀
x
,
y
∈
E
.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, thesubdifferential
∂
g of g is defined by
(17)
∂
g
(
x
)
=
{
x
*
∈
E
*
:
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 [25, 26]. For any lower semicontinuous convex function g
:
E
→
(
-
∞
,
+
∞
], theconjugate function
g
* of g is defined by
(18)
g
*
(
x
*
)
=
sup
⁡
x
∈
E
{
〈
x
,
x
*
〉
-
g
(
x
)
}
,
∀
x
*
∈
E
*
.
It is well known that
(19)
g
(
x
)
+
g
*
(
x
*
)
≥
〈
x
,
x
*
〉
,
∀
(
x
,
x
*
)
∈
E
×
E
*
,
(20)
(
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 isproper we mean that dom
⁡
g
:
=
{
x
∈
E
:
g
(
x
)
<
+
∞
}
≠
∅.The following definition is slightly different from that in Butnariu and Iusem [22].Definition 2 (see [23]).
LetE be a Banach space. A function g
:
E
→
ℝ is said to be aBregman 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 [22] and Za
˘linescu [24].Lemma 3.
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)
d
o
m
g
*
=
E
*
,
g
* is Gâteaux differentiable and ∇
g
*
=
(
∇
g
)
-
1.The following two results follow from [24, Proposition 3.6.4].Proposition 4.
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)
d
o
m
g
*
=
E
*
,
g
*
is locally bounded and locally uniformly smooth on
E
;
(3)
d
o
m
g
*
=
E
*
,
g
*
is Fréchet differentiable and
∇
g
*
is uniformly norm-to-norm continuous on bounded subsets of
E
*
.Proposition 5.
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)
d
o
m
g
*
=
E
*
,
g
*
is strongly coercive and locally uniformly convex on
E
.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 [27] that for x in E and x
0 in C, we have
(21)
D
g
(
x
0
,
x
)
=
min
⁡
y
∈
C
D
g
(
y
,
x
)
iff
〈
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
(22)
D
g
(
x
0
,
x
)
=
min
⁡
y
∈
C
D
g
(
y
,
x
)
.
TheBregman projection
proj
C
g from E onto C defined by proj
C
g
(
x
)
=
x
0 has the following property:
(23)
D
g
(
y
,
proj
C
g
x
)
+
D
g
(
proj
C
g
x
,
x
)
≤
D
g
(
y
,
x
)
,
hhhhhhhhhhhhhhhhhhhh
∀
y
∈
C
,
∀
x
∈
E
.
See [22] for details.Lemma 6 (see [9]).
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.Lemma 7 (see [23, 28]).
LetE be a reflexive Banach space, let g
:
E
→
ℝ be a strongly coercive Bregman function, and let V be the function defined by
(24)
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
(25)
V
(
x
,
∇
g
(
y
)
)
=
D
g
(
x
,
y
)
.Lemma 8 (see [29, Proposition 23.1]).
LetE be a real Banach space and let f
:
E
→
ℝ
∪
{
+
∞
} be a lower semicontinuous convex function. Then there exist x
*
∈
E
* and a
∈
ℝ such that
(26)
f
(
x
)
≥
x
*
(
x
)
+
a
,
∀
x
∈
E
.


## 3. Properties of Bregmanf-Projection Operator Proj
C
f
,
g

Theorem 9.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let f
:
E
→
ℝ
∪
{
+
∞
} be a proper, convex, lower semicontinuous function and let g
:
E
→
ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded, and locally uniformly convex on E. Then P
r
o
j
C
f
,
g
(
x
*
)
≠
∅ for all x
*
∈
E
*.Proof.
Letx
*
∈
E
* and λ
=
inf
⁡
y
∈
C
H
(
y
,
x
*
). Then there exists a sequence {
x
n
}
n
∈
ℕ
⊂
C such that λ
=
lim
⁡
n
→
∞
H
(
x
n
,
x
*
). We consider the following two possible cases.
Case 1. If C is bounded, then there exists a subsequence {
x
n
j
}
j
∈
ℕ of {
x
n
}
n
∈
ℕ and x
∈
C such that x
n
j
⇀
x as j
→
∞. Since H
(
z
,
x
*
) is convex and lower semicontinuous with respect to z, we deduce that H
(
z
,
x
*
) is convex and weakly lower semicontinuous with respect to z. This implies that
(27)
H
(
x
,
x
*
)
≤
lim
⁡
⁡
inf
⁡
n
→
∞
H
(
x
n
,
x
*
)
=
lim
⁡
n
→
∞
H
(
x
n
,
x
*
)
hhhlhhh
=
inf
⁡
y
∈
C
H
(
x
n
,
x
*
)
and hence x
∈
Proj
C
f
,
g
(
x
*
). This shows that Proj
C
f
,
g
≠
∅.
Case 2. Assume that C is unbounded. Since f
:
C
→
ℝ
∪
{
+
∞
} is proper, convex, and lower semicontinuous, we know that the function f
C
:
E
→
ℝ
∪
{
+
∞
}, defined by
(28)
f
C
(
x
)
=
{
f
(
x
)
,
if
x
∈
C
,
+
∞
,
if
x
∉
C
,
is proper, convex, and lower semicontinuous. In view of Lemma 8, there exist x
*
∈
E
* and a
∈
ℝ such that
(29)
f
C
(
x
)
≥
〈
x
,
x
*
〉
+
a
,
∀
x
∈
E
.
This implies that for any x
*
∈
E
* and x
∈
C
(30)
H
(
x
,
x
*
)
=
g
(
x
)
-
〈
x
,
x
*
〉
+
g
*
(
x
*
)
+
f
(
x
)
iiiiiiiiiiiiii
≥
g
(
x
)
+
g
*
(
x
*
)
+
a
.
Next, we show that {
x
n
}
n
∈
ℕ is bounded. If not, then there exists a subsequence {
x
n
j
}
j
∈
ℕ of {
x
n
}
n
∈
ℕ such that ∥
x
n
k
∥
→
+
∞ as k
→
∞. Since g is strongly coercive, we conclude that
(31)
lim
⁡
∥
x
n
k
∥
→
+
∞
H
(
x
n
k
,
x
*
)
∥
x
n
k
∥
≥
lim
⁡
∥
x
n
k
∥
→
+
∞
g
(
x
n
k
)
∥
x
n
k
∥
=
+
∞
.
This implies that
(32)
lim
⁡
∥
x
n
k
∥
→
+
∞
H
(
x
n
k
,
x
*
)
=
+
∞
.
Since f is proper in C, we obtain that λ
=
inf
⁡
y
∈
C
H
(
y
,
x
*
)
=
lim
⁡
n
→
∞
H
(
x
n
,
x
*
)
<
+
∞ which contradicts (31). By a similar argument, as in Case 1, we can prove that Proj
C
f
,
g
(
x
*
)
≠
∅ which completes the proof.Theorem 10.
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. Then the following assertions hold:(i)
for any givenx
*
∈
E
*, P
r
o
j
C
f
,
g
(
x
*
) is a nonempty, closed, and convex subset of C;
(ii)
P
r
o
j
C
f
,
g is monotone; that is, for any x
*
,
y
*
∈
E
*, x
∈
P
r
o
j
C
f
,
g
(
x
*
) and y
∈
P
r
o
j
C
f
,
g
(
y
*
),
(33)
〈
x
-
y
,
x
*
-
y
*
〉
≥
0
;
(iii)
For any givenx
*
∈
E
*, x
∈
P
r
o
j
C
f
,
g
(
x
*
) if and only if
(34)
〈
x
-
y
,
x
*
-
∇
g
(
x
)
〉
+
f
(
y
)
-
f
(
x
)
≥
0
;Proof.
(i) Letx
*
∈
E
* be fixed. In view of Theorem 9, we conclude that Proj
C
f
,
g
(
x
*
)
≠
∅. According to (20) we have g
(
x
)
+
g
*
(
x
*
)
-
〈
x
,
x
*
〉
≥
0
,
∀
(
x
,
x
*
)
∈
E
×
E
*. Let us prove that Proj
C
f
,
g
(
x
*
) is closed. Let {
x
n
}
n
∈
ℕ
⊂
Proj
C
f
,
g
(
x
*
) and x
n
→
x as n
→
∞. In view of (6), we deduce that
(35)
G
(
x
,
x
*
)
≤
liminf
⁡
n
→
∞
H
(
x
n
,
x
*
)
hhhhiihh
=
liminf
⁡
n
→
∞
H
(
x
n
,
x
*
)
=
inf
⁡
y
∈
C
H
(
y
,
x
*
)
.
This implies that x
∈
Proj
C
f
,
g
(
x
*
) and hence Proj
C
f
,
g
(
x
*
) is closed. Next, we show that Proj
C
f
,
g
(
x
*
) is convex. Let x
1
,
x
2
∈
Proj
C
f
,
g
(
x
*
) and 0
≤
t
≤
1. By the property (2) of the functional H, we obtain
(36)
H
(
t
x
1
+
(
1
-
t
)
x
2
,
x
*
)
≤
t
H
(
x
1
,
x
*
)
+
(
1
-
t
)
H
(
x
2
,
x
*
)
=
t
inf
⁡
y
∈
C
H
(
y
,
x
*
)
+
(
1
-
t
)
inf
⁡
y
∈
C
H
(
y
,
x
*
)
=
inf
⁡
y
∈
C
H
(
y
,
x
*
)
.
Thus, we have t
x
1
+
(
1
-
t
)
x
2
∈
Proj
C
f
,
g
(
x
*
) and hence Proj
C
f
,
g
(
x
*
) is convex.
(ii) Letx
1
*
,
x
2
*
∈
E
*, x
1
∈
Proj
C
f
,
g
(
x
1
*
), and x
2
∈
Proj
C
f
,
g
(
x
2
*
). Then we have
(37)
g
(
x
1
)
-
〈
x
1
,
x
1
*
〉
+
g
*
(
x
1
*
)
+
f
(
x
1
)
≤
g
(
x
2
)
-
〈
x
2
,
x
2
*
〉
+
g
*
(
x
2
*
)
+
f
(
x
2
)
,
g
(
x
2
)
-
〈
x
2
,
x
2
*
〉
+
g
*
(
x
2
*
)
+
f
(
x
2
)
≤
g
(
x
1
)
-
〈
x
1
,
x
1
*
〉
+
g
*
(
x
1
*
)
+
f
(
x
1
)
.
In view of (37), we conclude that Proj
C
f
,
g
(
x
*
) is monotone.
(iii) It is a simple matter to see thatx
∈
Proj
C
f
,
g
(
x
*
) implies that
(38)
〈
x
*
-
∇
g
(
x
)
,
x
-
y
〉
+
f
(
y
)
-
f
(
x
)
≥
0
,
∀
y
∈
C
.
To this end, let y
∈
C and t
∈
(
0,1
] be arbitrarily chosen. By the definition of Proj
C
f
,
g
(
x
*
) we see that
(39)
H
(
x
,
x
*
)
≤
H
(
x
+
t
(
y
-
x
)
,
x
*
)
.
Therefore,
(40)
g
(
x
)
+
g
*
(
x
*
)
-
〈
x
,
x
*
〉
+
f
(
x
)
≤
g
(
x
+
t
(
y
-
x
)
)
+
g
*
(
x
*
)
-
〈
x
+
t
(
y
-
x
)
,
x
*
〉
+
f
(
x
+
t
(
y
-
x
)
)
≤
g
(
x
+
t
(
y
-
x
)
)
+
g
*
(
x
*
)
-
〈
x
+
t
(
y
-
x
)
,
x
*
〉
+
t
f
(
y
)
+
(
1
-
t
)
f
(
x
)
and hence
(41)
〈
t
(
y
-
x
)
,
x
*
〉
≤
g
(
x
+
t
(
y
-
x
)
)
+
t
(
f
(
y
)
-
f
(
x
)
)
.
On the other hand, by the definition of Bregman distance, we obtain that
(42)
g
(
x
)
+
g
(
x
+
t
(
y
-
x
)
)
≥
〈
t
(
x
-
y
)
,
∇
g
(
x
+
t
(
y
-
x
)
)
〉
.
This, together with (41), implies that
(43)
〈
x
-
y
,
∇
g
(
x
+
t
(
y
-
x
)
)
〉
≥
f
(
x
)
-
f
(
y
)
+
〈
x
-
y
,
x
*
〉
.
Since ∇
g is demi-continuous, letting t
→
0 in (43), we conclude that
(44)
〈
x
-
y
,
∇
g
(
x
)
-
x
*
〉
+
f
(
y
)
-
f
(
x
)
≥
0
.
Conversely, assume that
(45)
〈
x
-
y
,
∇
g
(
x
)
-
x
*
〉
+
f
(
y
)
-
f
(
x
)
≥
0
,
∀
y
∈
K
.
This implies that
(46)
g
(
y
)
-
g
(
x
)
≥
〈
x
-
y
,
∇
g
(
x
)
〉
iiiiiiiiiiiiiiiiiii
≥
〈
x
-
y
,
x
*
〉
+
f
(
y
)
-
f
(
x
)
≥
0
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
∀
y
∈
K
.


## 4. Applications to Variational Inequalities

In this section, we investigate the existence of solution to the following variational inequality problem: find the pointx
∈
C such that
(47)
〈
y
-
x
,
A
x
〉
+
f
(
y
)
-
f
(
x
)
≥
0
,
∀
y
∈
C
,
where C is a nonempty, closed, and convex subset of the Banach space E, and A
:
C
→
E
* and f
:
C
→
ℝ
∪
{
+
∞
} are two mappings.Definition 11 (KKM mapping [30]).
LetC be a nonempty subset of a linear space X. A set-valued mapping G
:
C
→
2
X is called a KKM mapping if, for any finite subset {
y
1
,
y
2
,
…
,
y
n
} of C, we have
(48)
co
⁡
{
y
1
,
y
2
,
…
,
y
n
}
⊂
⋃
i
=
1
n
G
(
y
i
)
,
where co
⁡
{
y
1
,
y
2
,
…
,
y
n
} denotes the convex hull of {
y
1
,
y
2
,
…
,
y
n
}.Lemma 12 (Fan KKM Theorem [30]).
LetC be a nonempty convex subset of a Hausdorff topological vector X and let G
:
C
→
2
X be a KKM mapping with closed values. If there exists a point y
0
∈
C such that G
(
y
0
) is a compact subset of C, then ⋂
y
∈
C
G
(
y
)
≠
∅.Theorem 13.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E with dual space E
*. Let g
:
E
→
ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded and locally uniformly convex on E. Let A
:
C
→
E
* be a continuous mapping and f
:
E
→
ℝ
∪
{
+
∞
} be a proper, convex, lower semicontinuous function. If there exists an element y
0
∈
C such that
(49)
{
x
∈
C
:
〈
y
0
-
x
,
∇
g
(
x
)
-
A
x
〉
+
g
(
x
)
+
f
(
x
)
≤
g
(
y
0
)
+
f
(
y
0
)
}
is a compact subset of C, then the variational inequality (47) has a solution.Proof.
In view of Theorem10, we need to prove that the following inclusion has a solution:
(50)
x
∈
Proj
C
f
,
g
(
∇
g
(
x
)
-
A
x
)
.
We define a set-valued mapping V
:
C
→
2
C by
(51)
V
(
y
)
=
{
x
∈
C
:
H
(
x
,
∇
g
(
x
)
-
A
x
)
≤
H
(
y
,
∇
g
(
x
)
-
A
x
)
}
.
It is obvious that, for any y
∈
C, V
(
y
)
≠
∅. Let us prove that V
(
y
) is closed for any y
∈
C. Let {
x
n
}
n
∈
ℕ
⊂
V
(
y
) and x
n
→
x as n
→
∞. Then,
(52)
H
(
x
n
,
∇
g
(
x
n
)
-
A
x
n
)
≤
H
(
y
,
∇
g
(
x
n
)
-
A
x
n
)
.
This implies that
(53)
-
〈
x
n
,
∇
g
(
x
n
)
-
A
x
n
〉
+
g
(
x
n
)
+
f
(
x
n
)
≤
-
〈
y
,
∇
g
(
x
n
)
-
A
x
n
〉
+
g
(
y
)
+
f
(
y
)
.
Since ∇
g and A are continuous and f is lower semicontinuous, we conclude that
(54)
-
〈
x
,
∇
g
(
x
)
-
A
x
〉
+
g
(
x
)
+
f
(
x
)
≤
-
〈
y
,
∇
g
(
x
)
-
A
x
〉
+
g
(
y
)
+
f
(
y
)
.
Therefore,
(55)
H
(
x
,
∇
g
(
x
)
-
A
x
)
≤
H
(
y
,
∇
g
(
x
)
-
A
x
)
,
which implies that x
∈
V
(
y
). Now, we prove that V
:
C
→
2
C is a KKM mapping. Indeed, suppose y
1
,
y
2
,
…
,
y
n
∈
C and 0
<
a
1
,
a
2
,
…
,
a
n
≤
1 with ∑
i
=
1
n
a
i
=
1. Let z
=
∑
i
=
1
n
a
i
y
i. In view of the property (2) of H, we obtain
(56)
H
(
z
,
∇
g
(
z
)
-
A
z
)
=
H
(
∑
i
=
1
n
‍
a
i
y
i
,
∇
g
(
z
)
-
A
z
)
≤
∑
i
=
1
n
‍
a
i
H
(
y
i
,
∇
g
(
z
)
-
A
z
)
and hence
(57)
H
(
z
,
∇
g
(
z
)
-
A
z
)
≤
max
⁡
1
≤
i
≤
n
H
(
y
i
,
∇
g
(
z
)
-
A
z
)
.
Hence there exists at least one number j
=
1,2
,
…
,
n, such that
(58)
H
(
z
,
∇
g
(
z
)
-
A
z
)
≤
H
(
y
j
,
∇
g
(
z
)
-
A
z
)
.
that is, z
∈
V
(
y
). Thus, V is a KKM mapping.
Ifx
∈
V
(
y
0
), then H
(
z
,
∇
g
(
z
)
-
A
z
)
≤
H
(
y
0
,
∇
g
(
z
)
-
A
z
). By the definition of H, we obtain
(59)
-
〈
x
,
∇
g
(
x
)
-
A
x
〉
+
g
(
x
)
+
f
(
x
)
≤
-
〈
y
0
,
∇
g
(
x
)
-
A
x
〉
+
g
(
y
0
)
+
f
(
y
0
)
which is equivalent to
(60)
〈
y
0
-
x
,
∇
g
(
x
)
-
A
x
〉
+
g
(
x
)
+
f
(
x
)
≤
g
(
y
0
)
+
f
(
y
0
)
.
Therefore,
(61)
V
(
y
0
)
=
{
x
∈
C
:
〈
∇
g
(
x
)
-
A
x
,
y
0
-
x
〉
h
h
h
h
h
h
h
h
+
g
(
x
)
+
f
(
x
)
≤
g
(
y
0
)
+
f
(
y
0
)
}
.
In view of (49), we deduce that V
(
y
0
) is compact. It follows from Lemma 12 that ⋂
y
∈
C
V
(
y
)
≠
∅. Hence there exists at least one x
0
∈
⋂
y
∈
C
V
(
y
)
); that is,
(62)
H
(
x
0
,
∇
g
(
x
0
)
-
A
x
0
)
≤
H
(
y
,
∇
g
(
x
0
)
-
A
x
0
)
,
∀
y
∈
C
.
In view of the definition of Bregman f-projection operator Proj
C
f
,
g, we conclude that
(63)
x
0
∈
Proj
C
f
,
g
(
∇
g
(
x
0
)
-
A
x
0
)
.
This completes the proof.Theorem 14.
LetE be a reflexive Banach space and g
:
E
→
ℝ a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let f
:
E
→
ℝ
∪
{
+
∞
} be a proper, convex, lower semicontinuous function. Let C be a nonempty, closed, and convex subset of E and let T
:
C
→
C be a Bregman weak relatively nonexpansive mapping. Let {
α
n
}
n
∈
ℕ
∪
{
0
} be a sequence in (
0,1
) such that liminf
⁡
n
→
∞
α
n
(
1
-
α
n
)
>
0. Let {
x
n
}
n
∈
ℕ
∪
{
0
} be a sequence generated by
(64)
x
0
=
x
∈
C
c
h
o
s
e
n
a
r
b
i
t
r
a
r
i
l
y
,
C
0
=
C
,
y
n
=
∇
g
*
[
α
n
∇
g
(
x
n
)
+
(
1
-
α
n
)
∇
g
(
T
x
n
)
]
,
C
n
+
1
=
{
z
∈
C
n
:
H
(
z
,
∇
g
(
y
n
)
)
≤
H
(
z
,
∇
g
(
x
n
)
)
}
,
x
n
+
1
=
P
r
o
j
C
n
+
1
g
x
,
n
∈
ℕ
∪
{
0
}
,
where ∇
g is the gradient of g. Then {
x
n
}
n
∈
ℕ, {
T
x
n
}
n
∈
ℕ, and {
y
n
}
n
∈
ℕ converge strongly to P
r
o
j
F
g
x
0.Proof.
We divide the proof into several steps.
Step 1. We prove that C
n is closed and convex for each n
∈
ℕ
∪
{
0
}.
It is clear thatC
0
=
C is closed and convex. Let C
m be closed and convex for some m
∈
ℕ. For z
∈
C
m, we see that
(65)
H
(
z
,
∇
g
(
y
m
)
)
≤
H
(
z
,
∇
g
(
x
m
)
)
is equivalent to
(66)
〈
z
,
∇
g
(
x
m
)
-
∇
g
(
y
m
)
〉
≤
g
(
y
m
)
-
g
(
x
m
)
+
〈
x
m
,
∇
g
(
x
m
)
〉
-
〈
y
m
,
∇
g
(
y
m
)
〉
.
It could easily be seen that C
m
+
1 is closed and convex. Therefore, C
n is closed and convex for each n
∈
ℕ
∪
{
0
}.
Step 2. We claim that F
⊂
C
n for all n
∈
ℕ
∪
{
0
}.
It is obvious thatF
⊂
C
0
=
C. Assume now that F
⊂
C
m for some m
∈
ℕ. Employing Lemma 7, for any w
∈
F
⊂
C
m, we obtain
(67)
H
(
w
,
∇
g
(
y
m
)
)
=
H
(
w
,
∇
g
(
y
m
)
)
=
g
(
w
)
-
〈
w
,
∇
g
(
y
m
)
〉
+
g
*
(
∇
g
(
y
m
)
)
+
f
(
w
)
=
V
(
w
,
α
m
∇
g
(
x
m
)
+
(
1
-
α
m
)
∇
g
(
T
x
m
)
)
+
f
(
w
)
=
g
(
w
)
-
〈
w
,
α
m
∇
g
(
x
m
)
+
(
1
-
α
m
∇
g
(
T
x
m
)
)
〉
+
g
*
(
α
m
∇
g
(
x
m
)
+
(
1
-
α
m
)
∇
g
(
T
x
m
)
)
+
f
(
w
)
≤
α
m
g
(
w
)
+
(
1
-
α
m
)
g
(
w
)
+
α
m
g
*
(
∇
g
(
x
m
)
)
+
(
1
-
α
m
)
g
*
(
∇
g
(
T
x
m
)
)
+
f
(
w
)
=
α
m
V
(
w
,
∇
g
(
x
m
)
)
+
(
1
-
α
m
)
V
(
w
,
∇
g
(
T
x
m
)
)
+
f
(
w
)
=
α
m
D
g
(
w
,
x
m
)
+
(
1
-
α
m
)
D
g
(
w
,
T
x
m
)
+
f
(
w
)
≤
α
m
D
g
(
w
,
x
m
)
+
(
1
-
α
m
)
D
g
(
w
,
x
m
)
+
f
(
w
)
=
D
g
(
w
,
x
m
)
+
f
(
w
)
=
V
(
w
,
∇
g
(
x
m
)
)
+
f
(
w
)
=
H
(
w
,
∇
g
(
x
m
)
)
.
This proves that w
∈
C
m
+
1 and hence F
⊂
C
n for all n
∈
ℕ
∪
{
0
}.
Step
3. We prove that {
x
n
}
n
∈
ℕ
,
{
y
n
}
n
∈
ℕ, and {
T
x
n
}
n
∈
ℕ are bounded sequences in C.
Sincex
n
=
proj
C
n
g
x, we get that
(68)
H
(
x
n
,
∇
g
(
x
)
)
≤
H
(
w
,
∇
g
(
x
)
)
for each w
∈
F
(
T
). This implies that the sequence {
H
(
w
,
∇
g
(
x
n
)
)
}
n
∈
ℕ is bounded and hence there exists M
1
>
0 such that
(69)
H
(
x
n
,
∇
g
(
x
)
)
≤
M
1
,
∀
n
∈
ℕ
.
We claim that the sequence {
x
n
}
n
∈
ℕ is bounded. Assume on the contrary that ∥
x
n
∥
→
∞ as n
→
∞. In view of Lemma 8, there exist x
*
∈
E
* and a
∈
ℝ such that
(70)
f
(
x
)
≥
〈
x
n
,
x
*
〉
+
a
,
∀
n
∈
ℕ
.
From the definition of Bregman distance, it follows that
(71)
M
1
≥
H
(
x
n
,
∇
g
(
x
)
)
hh
=
g
(
x
n
)
-
g
(
x
)
-
〈
x
n
-
x
,
∇
g
(
x
)
〉
+
f
(
x
n
)
hh
≥
g
(
x
n
)
-
g
(
x
)
-
〈
x
n
,
∇
g
(
x
)
-
x
*
〉
+
〈
x
,
∇
g
(
x
)
〉
+
a
hh
≥
g
(
x
n
)
-
g
(
x
)
-
∥
x
n
∥
∥
∇
g
(
x
)
-
x
*
∥
iiiiii
+
〈
x
,
∇
g
(
x
)
〉
+
a
,
∀
n
∈
ℕ
.
Without loss of generality, we may assume that ∥
x
n
∥
≠
0 for each n
∈
ℕ. This implies that
(72)
M
1
∥
x
n
∥
≥
g
(
x
n
)
∥
x
n
∥
-
g
(
x
)
∥
x
n
∥
-
∥
∇
g
(
x
)
-
x
*
∥
hhhh
+
〈
x
,
∇
g
(
x
)
〉
∥
x
n
∥
+
a
∥
x
n
∥
,
∀
n
∈
ℕ
.
Since g is strongly coercive, by letting n
→
∞ in (72), we conclude that 0
≥
∞, which is a contradiction. Therefore, {
x
n
}
n
∈
ℕ is bounded. Since {
T
n
}
n
∈
ℕ is an infinite family of Bregman weak relatively nonexpansive mappings from C into itself, we have for any q
∈
F that
(73)
D
g
(
q
,
T
x
n
)
≤
D
g
(
q
,
x
n
)
,
∀
n
∈
ℕ
.
This, together with Definition 2 and the boundedness of {
x
n
}
n
∈
ℕ, implies that the sequence {
T
n
x
n
}
n
∈
ℕ is bounded.
Step 4. We show that x
n
→
v for some v
∈
F, where v
=
proj
F
g
x.
From Step 3 we know that{
x
n
}
n
∈
ℕ is bounded. By the construction of C
n, we conclude that C
m
⊂
C
n and x
m
=
proj
C
m
g
x
∈
C
m
⊂
C
n for any positive integer m
≥
n. This, together with (23), implies that
(74)
D
g
(
x
m
,
x
n
)
=
D
g
(
x
m
,
proj
C
n
g
x
)
≤
D
g
(
x
m
,
x
)
-
D
g
(
proj
C
n
g
x
,
x
)
=
D
g
(
x
m
,
x
)
-
D
g
(
x
n
,
x
)
.
In view of (21), we conclude that
(75)
D
g
(
x
n
,
x
)
=
D
g
(
proj
C
n
g
x
,
x
)
≤
D
g
(
w
,
x
)
-
D
g
(
w
,
x
n
)
hhhhihhh
≤
D
g
(
w
,
x
)
,
∀
w
∈
F
⊂
C
n
,
n
∈
ℕ
∪
{
0
}
.
It follows from (75) that the sequence {
D
g
(
x
n
,
x
)
}
n
∈
ℕ is bounded and hence there exists M
2
>
0 such that
(76)
D
g
(
x
n
,
x
)
≤
M
2
,
∀
n
∈
ℕ
.
In view of (64), we conclude that
(77)
D
g
(
x
n
,
x
)
≤
D
g
(
x
n
,
x
)
+
D
g
(
x
m
,
x
n
)
≤
D
g
(
x
m
,
x
)
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhhihhhhh
∀
m
≥
n
.
This proves that {
D
g
(
x
n
,
x
)
}
n
∈
ℕ is an increasing sequence in ℝ and hence the limit lim
⁡
n
→
∞
D
g
(
x
n
,
x
) exists. Letting m
,
n
→
∞ in (74), we deduce that D
g
(
x
m
,
x
n
)
→
0. In view of Lemma 6, we obtain that ∥
x
m
-
x
n
∥
→
0 as m
,
n
→
∞. This means that {
x
n
}
n
∈
ℕ is a Cauchy sequence. Since E is a Banach space and C is closed and convex, we conclude that there exists v
∈
C such that
(78)
lim
⁡
n
→
∞
∥
x
n
-
v
∥
=
0
.
Now, we show that v
∈
F. In view of Lemma 6 and (78), we obtain
(79)
lim
⁡
n
→
∞
D
g
(
x
n
+
1
,
x
n
)
=
0
.
Since x
n
+
1
∈
C
n
+
1, we conclude that
(80)
D
g
(
x
n
+
1
,
y
n
)
≤
D
g
(
x
n
+
1
,
x
n
)
.
This, together with (79), implies that
(81)
lim
⁡
n
→
∞
D
g
(
x
n
+
1
,
y
n
)
=
0
.
It follows from Lemma 6, (79), and (81) that
(82)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
,
lim
⁡
n
→
∞
∥
x
n
+
1
-
y
n
∥
=
0
.
In view of (78), we get
(83)
lim
⁡
n
→
∞
∥
y
n
-
u
∥
=
0
.
From (78) and (83), it follows that
(84)
lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0
.
Since ∇
g is uniformly norm-to-norm continuous on any bounded subset of E, we obtain
(85)
lim
⁡
n
→
∞
∥
∇
g
(
x
n
)
-
∇
g
(
y
n
)
∥
=
0
.
Applying Lemma 6 we derive that
(86)
lim
⁡
n
→
∞
D
g
(
y
n
,
x
n
)
=
0
.
It follows from the three-point identity (see (14)) that for any w
∈
F
(87)
|
D
g
(
w
,
x
n
)
-
D
g
(
w
,
y
n
)
|
=
|
D
g
(
w
,
y
n
)
+
D
g
(
y
n
,
x
n
)
hhhhh
+
〈
w
-
y
n
,
∇
g
(
y
n
)
-
∇
g
(
x
n
)
〉
-
D
g
(
w
,
y
n
)
|
=
|
D
g
(
y
n
,
x
n
)
-
〈
w
-
y
n
,
∇
g
(
y
n
)
-
∇
g
(
x
n
)
〉
|
≤
D
g
(
y
n
,
x
n
)
+
∥
w
-
y
n
∥
∥
∇
g
(
y
n
)
-
∇
g
(
x
n
)
∥
⟶
0
as n
→
∞.
The functiong is bounded on bounded subsets of E and, thus, ∇
g is also bounded on bounded subsets of E
* (see, e.g., [22, Proposition 1.1.11], for more details). This implies that the sequences {
∇
g
(
x
n
)
}
n
∈
ℕ, {
∇
g
(
y
n
)
}
n
∈
ℕ, and {
∇
g
(
T
x
n
)
:
n
∈
ℕ
∪
{
0
}
} are bounded in E
*.
In view of Proposition4(3), we know that dom g
*
=
E
* and g
* is strongly coercive and uniformly convex on bounded subsets of E
*. Let s
1
=
sup
⁡
{
∥
∇
g
(
x
n
)
∥
,
∥
∇
g
(
T
x
n
)
∥
:
n
∈
ℕ
∪
{
0
}
} and ρ
s
1
*
:
E
*
→
ℝ be the gauge of uniform convexity of the conjugate function g
*. We prove that for any w
∈
F
(88)
D
g
(
w
,
y
n
)
≤
D
g
(
w
,
x
n
)
-
α
n
(
1
-
α
n
)
ρ
s
1
*
iiiiiiiiiiiiiii
×
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
.
Let us show (88). For any given w
∈
F
(
T
), in view of the definition of the Bregman distance (see (2)) and Lemma 6, we obtain
(89)
D
g
(
w
,
y
n
)
=
D
g
(
w
,
∇
g
*
[
α
n
∇
g
(
x
n
)
+
(
1
-
α
n
)
∇
g
(
T
x
n
)
]
)
=
V
(
w
,
α
n
∇
g
(
x
n
)
+
(
1
-
α
n
)
∇
g
(
T
x
n
)
)
=
g
(
w
)
-
〈
w
,
α
n
∇
g
(
x
n
)
+
(
1
-
α
n
)
∇
g
(
T
x
n
)
〉
+
g
*
(
α
n
∇
g
(
x
n
)
+
(
1
-
α
n
)
∇
g
(
T
x
n
)
)
≤
α
n
g
(
w
)
+
(
1
-
α
n
)
g
(
w
)
-
α
n
〈
w
,
∇
g
(
x
n
)
〉
-
(
1
-
α
n
)
〈
w
,
∇
g
(
T
x
n
)
〉
+
α
n
g
*
(
∇
g
(
x
n
)
)
+
(
1
-
α
n
)
g
*
(
∇
g
(
T
x
n
)
)
-
α
n
(
1
-
α
n
)
ρ
s
1
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
=
α
n
V
(
w
,
∇
g
(
x
n
)
)
+
(
1
-
α
n
)
V
(
w
,
∇
g
(
T
x
n
)
)
-
α
n
(
1
-
α
n
)
ρ
s
1
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
n
x
n
)
∥
)
=
α
n
D
g
(
w
,
x
n
)
+
(
1
-
α
n
)
D
g
(
w
,
T
x
n
)
-
α
n
(
1
-
α
n
)
ρ
s
1
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
≤
α
n
D
g
(
w
,
x
n
)
+
(
1
-
α
n
)
D
g
(
w
,
x
n
)
-
α
n
(
1
-
α
n
)
ρ
s
1
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
=
D
g
(
w
,
x
n
)
-
α
n
(
1
-
α
n
)
ρ
s
1
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
.
In view of (87), we get that
(90)
D
g
(
w
,
x
n
)
-
D
g
(
w
,
y
n
)
⟶
0
as
n
⟶
∞
.
In view of (87) and (88), we conclude that
(91)
α
n
(
1
-
α
n
)
ρ
s
1
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
≤
D
g
(
w
,
x
n
)
-
D
g
(
w
,
y
n
)
⟶
0
as n
→
∞. From the assumption liminf
⁡
n
→
∞
α
n
(
1
-
α
n
)
>
0, we get
(92)
lim
⁡
n
→
∞
ρ
s
1
*
(
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
)
=
0
.
Therefore, from the property of ρ
s
1
* we deduce that
(93)
lim
⁡
n
→
∞
∥
∇
g
(
x
n
)
-
∇
g
(
T
x
n
)
∥
=
0
.
Since ∇
g
* is uniformly norm-to-norm continuous on bounded subsets of E
*, we arrive at
(94)
lim
⁡
n
→
∞
∥
x
n
-
T
x
n
∥
=
0
.
This implies that v
∈
F
(
T
).
Finally, we show thatv
=
proj
F
g
x. From x
n
=
proj
C
n
g
x, we conclude that
(95)
〈
z
-
x
n
,
∇
g
(
x
n
)
-
∇
g
(
x
)
〉
≥
0
,
∀
z
∈
C
n
.
Since F
⊂
C
n for each n
∈
ℕ, we obtain
(96)
〈
z
-
x
n
,
∇
g
(
x
n
)
-
∇
g
(
x
)
〉
≥
0
,
∀
z
∈
F
.
Letting n
→
∞ in (96), we deduce that
(97)
〈
z
-
v
,
∇
g
(
u
)
-
∇
g
(
x
)
〉
≥
0
,
∀
z
∈
F
.
In view of (21), we have v
=
proj
F
g
x, which completes the proof.Remark 15.
Theorem14 improves Theorem 4.1 of [20] in the following aspects.(1)
For the structure of Banach spaces, we extend the duality mapping to more general case, that is, a convex, continuous, and strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets.
(2)
For the mappings, we extend the mapping from a relatively nonexpansive mapping to a Bregman weak relatively nonexpansive mapping. We remove the assumptionF
^
(
T
)
=
F
(
T
) on the mapping T and extend the result to a Bregman weak relatively nonexpansive mapping, where F
^
(
T
) is the set of asymptotic fixed points of the mapping T.
(3)
Theorems9 and 10 extend and improve corresponding results of [20].



---

*Source: 594285-2014-03-20.xml*