# Convergence Theorems for Fixed Points of Multivalued Mappings in Hilbert Spaces

Authors: N. Djitte; M. Sene
Journal: International Journal of Analysis
 (2014)

---

## Abstract

LetH be a real Hilbert space and K a nonempty closed convex subset of H. Suppose T
:
K
→
CB
(
K
) is a multivalued Lipschitz pseudocontractive mapping such that F
(
T
)
≠
∅. An Ishikawa-type iterative algorithm is constructed and it is shown that, for the corresponding sequence {
x
n
}, under appropriate conditions on the iteration parameters, lim inf
n
→
∞
⁡
d
(
x
n
,
T
x
n
)
=
0 holds. Finally, convergence theorems are proved under approximate additional conditions. Our theorems are  significant improvement on important recent results of Panyanak (2007) and Sastry and Babu (2005).


---

## Body

## 1. Introduction

LetK be a nonempty subset of a normed space E. The set K is calledproximinal (see, e.g., [1–4]) if for each x
∈
E there exists u
∈
K such that
(1)
d
(
x
,
u
)
=
inf
⁡
{
∥
x
-
y
∥
:
y
∈
K
}
=
d
(
x
,
K
)
,
where d
(
x
,
y
)
=
∥
x
-
y
∥ for all x
,
y
∈
E. It is known that every nonempty closed convex subset of a real Hilbert space is proximinal. Let CB
(
K
) and P
(
K
) denote the families of nonempty closed bounded subsets and nonempty proximinal bounded subsets of K, respectively. TheHausdorff metric on CB
(
K
) is defined by
(2)
H
(
A
,
B
)
=
max
⁡
{
sup
⁡
a
∈
A
d
(
a
,
B
)
,
sup
⁡
b
∈
B
d
(
b
,
A
)
}
for all A
,
B
∈
CB
(
K
). Let T
:
D
(
T
)
⊆
E
→
CB
(
E
) be amultivalued mapping on E. A point x
∈
D
(
T
) is called afixed point of
T if and only if x
∈
T
x. The set F
(
T
)
:
=
{
x
∈
D
(
T
)
:
x
∈
T
x
} is called thefixed point set of
T.A multivalued mappingT
:
D
(
T
)
⊆
E
→
2
E is calledLipschitzian if there exists L
>
0 such that
(3)
H
(
T
x
,
T
y
)
≤
L
∥
x
-
y
∥
∀
x
,
y
∈
D
(
T
)
.In (3), if L
∈
(
0,1
), T is said to be acontraction, and T is callednonexpansive if L
=
1.Existence theorem for fixed point ofmultivalued contractions and nonexpansive mappings using the Hausdorff metric have been proved by several authors (see, e.g., Nadler Jr. [5], Markin [6], and Lim [7]). Later, an interesting and rich fixed point theory for such maps and more general maps was developed which has applications in control theory, convex optimization, differential inclusion, and economics (see Gorniewicz [8] and references cited therein).Several theorems have been proved on the approximation of fixed points ofmultivalued nonexpansive mappings (see, e.g., [1–4, 9, 10] and the references therein) and their generalizations (see, e.g., [11, 12]).Sastry and Babu [2] introduced the following iterative scheme. Let T
:
E
→
P
(
E
) be a multivalued mapping and let x
* be a fixed point of T. The sequence of iterates is given for x
0
∈
E by
(4)
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
y
n
∀
n
≥
0
,
y
n
∈
T
x
n
,
∥
y
n
-
x
*
∥
=
d
(
T
x
n
,
x
*
)
,
where α
n is a real sequence in (0,1) satisfying the following conditions:(i)
∑
n
=
1
∞
‍
α
n
=
∞;
(ii)
lim
⁡
α
n
=
0.They also introduced the following sequence:(5)
y
n
=
(
1
-
β
n
)
x
n
+
β
n
z
n
,
z
n
∈
T
x
n
,
∥
z
n
-
x
*
∥
=
d
(
x
*
,
T
x
n
)
,
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
u
n
,
u
n
∈
T
y
n
,
∥
u
n
-
x
*
∥
=
d
(
T
y
n
,
x
*
)
,
where {
α
n
}, {
β
n
} are real sequences satisfying the following conditions:(i)
0
≤
α
n
,
β
n
<
1;
(ii)
lim
⁡
n
→
∞
⁡
β
n
=
0;
(iii)
∑
n
=
1
∞
α
n
β
n
=
∞.Sastry and Babu called the process defined by (4) a Mann iteration process and the process defined by (5) where the iteration parameters α
n
,
β
n satisfy conditions (i), (ii), and (iii) an Ishikawa iteration process. They proved in [2] that the Mann and Ishikawa iteration schemes for a multivalued map T with fixed point p converge to a fixed point of T under certain conditions. More precisely, they proved the following result for a multivalued nonexpansive map with compact domain.Theorem  SB (Sastry and Babu [2]).Let
H
be real Hilbert space,
K
a nonempty compact convex subset of
H
, and
T
:
K
→
P
(
K
)
a multivalued nonexpansive map with a fixed point
p
. Assume that (i)  0
≤
α
n
,
β
n
<
1
; (ii)  β
n
→
0
; and (iii)  ∑
‍
α
n
β
n
=
∞
. Then, the sequence
{
x
n
}
defined by (5) converges strongly to a fixed point of
T
.Panyanak [1] extended the above result of Sastry and Babu [2] to uniformly convex real Banach spaces. He proved the following result.Theorem  P1 (Panyanak [1]).Let
E
be a uniformly convex real Banach space,
K
a nonempty compact convex subset of
E
, and
T
:
K
→
P
(
K
)
a multivalued nonexpansive map with a fixed point
p
. Assume that (i)  0
≤
α
n
,
β
n
<
1
; (ii)  β
n
→
0
; and (iii)  ∑
α
n
β
n
=
∞
. Then, the sequence
{
x
n
}
defined by (5) converges strongly to a fixed point of
T
.Panyanak [1] also modified the iteration schemes of Sastry and Babu [2]. Let K be a nonempty closed convex subset of a real Banach space and let T
:
K
→
P
(
K
) be a multivalued map with F
(
T
) a nonempty proximinal subset of K.The sequence of Mann iterates is defined byx
0
∈
K,
(6)
h
h
h
h
i
h
h
h
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
y
n
,
α
n
∈
[
a
,
b
]
,
0
<
a
<
b
<
1
,
n
≥
0
,
where y
n
∈
T
x
n is such that ∥
y
n
-
u
n
∥
=
d
(
u
n
,
T
x
n
) and u
n
∈
F
(
T
) is such that ∥
x
n
-
u
n
∥
=
d
(
x
n
,
F
(
T
)
).The sequence of Ishikawa iterates is defined byx
0
∈
K,
(7)
h
h
h
h
i
h
h
h
h
h
y
n
=
(
1
-
β
n
)
x
n
+
β
n
z
n
,
β
n
∈
[
a
,
b
]
,
0
<
a
<
b
<
1
,
n
≥
0
,
where z
n
∈
T
x
n is such that ∥
z
n
-
u
n
∥
=
d
(
u
n
,
T
x
n
) and u
n
∈
F
(
T
) is such that ∥
x
n
-
u
n
∥
=
d
(
x
n
,
F
(
T
)
). Consider
(8)
h
h
h
h
i
h
h
h
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
z
n
′
,
α
n
∈
[
a
,
b
]
,
0
<
a
<
b
<
1
,
n
≥
0
,
where z
n
′
∈
T
y
n is such that ∥
z
n
′
-
v
n
∥
=
d
(
v
n
,
T
y
n
) and v
n
∈
F
(
T
) is such that ∥
y
n
-
v
n
∥
=
d
(
y
n
,
F
(
T
)
). Before we state his theorem, we need the following definition.A mappingT
:
K
→
CB
(
K
) is said to satisfycondition (I) if there exists a strictly increasing function f
:
[
0
,
∞
)
→
[
0
,
∞
) with f
(
0
)
=
0
,
f
(
r
)
>
0 for all r
∈
(
0
,
∞
) such that
(9)
d
(
x
,
T
(
x
)
)
≥
f
(
d
(
x
,
F
(
T
)
)
)
∀
x
∈
D
(
T
)
.Theorem  P2 (Panyanak [1]).Let
E
be a uniformly convex real Banach space,
K
a nonempty closed bounded convex subset of
E
, and
T
:
K
→
P
(
K
)
a multivalued nonexpansive map that satisfies condition (I). Assume that (i)  0
≤
α
n
<
1
and (ii)  ∑
α
n
=
∞
. Suppose that
F
(
T
)
is a nonempty proximinal subset of
K
. Then, the sequence
{
x
n
}
defined by (6) converges strongly to a fixed point of
T
.Panyanak [1] then asked the following question.Question (P).Is Theorem  P2 true for the Ishikawa iterates defined by (7) and (8)?As remarked by Nadler Jr. [5], the definition of the Hausdorff metric on CB
(
E
) gives the following useful result.Lemma 1.
LetA
,
B
∈
CB
(
X
) and a
∈
A. For every γ
>
0, there exists b
∈
B such that
(10)
d
(
a
,
b
)
≤
H
(
A
,
B
)
+
γ
.Song and Wang [3, 4] modified the iteration process by Panyanak [1] and improved the results therein. They gave their iteration scheme as follows.LetK be a nonempty closed convex subset of a real Banach space and let T
:
K
→
CB
(
K
) be a multivalued map. Let α
n, β
n
∈
[
0,1
] and γ
n
∈
(
0
,
∞
) be such that lim
⁡
n
→
∞
⁡
γ
n
=
0. Choosing x
0
∈
K,
(11)
y
n
=
(
1
-
β
n
)
x
n
+
β
n
z
n
,
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
u
n
,
where z
n
∈
T
x
n, u
n
∈
T
y
n are such that
(12)
∥
z
n
-
u
n
∥
≤
H
(
T
x
n
,
T
y
n
)
+
γ
n
,
∥
z
n
+
1
-
u
n
∥
≤
H
(
T
x
n
+
1
,
T
y
n
)
+
γ
n
.They then proved the following result.Theorem  SW (Song and Wang [3, 4]).Let
K
be a nonempty compact convex subset of a uniformly convex real Banach space
E
. Let
T
:
K
→
C
B
(
K
)
be a multivalued nonexpansive mapping with
F
(
T
)
≠
∅
satisfying
T
(
p
)
=
{
p
}
for all
p
∈
F
(
T
)
. Assume that (i)  0
≤
α
n
,
β
n
<
1
; (ii)  β
n
→
0
; and (iii)  ∑
‍
α
n
β
n
=
∞
. Then, the Ishikawa sequence defined by (11) converges strongly to a fixed point of
T
.Shahzad and Zegeye [13] extended and improved the results of Sastry and Babu [2], Panyanak [1], and Son and Wang [3, 4] to multivalued quasi-nonexpansive maps. Also, in an attempt to remove the restriction T
p
=
{
p
}
∀
p
∈
F
(
T
) in Theorem  SW, they introduced a new iteration scheme as follows.LetK be a nonempty closed convex subset of a real Banach space, T
:
K
→
P
(
K
) a multivalued map, and P
T
x
:
=
{
y
∈
T
x
:
∥
x
-
y
∥
=
d
(
x
,
T
x
)
}. Let α
n
,
β
n
∈
[
0,1
]. Choose x
0
∈
K, and define {
x
n
} as follows:
(13)
y
n
=
(
1
-
β
n
)
x
n
+
β
n
z
n
,
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
u
n
,
where z
n
∈
P
T
x
n, u
n
∈
P
T
y
n. They then proved the following result.Theorem  SZ (Shahzad and Zegeye [13]).Let
X
be a uniformly convex real Banach space,
K
a nonempty convex subset of
X
, and
T
:
K
→
P
(
K
)
a multivalued map with
F
(
T
)
≠
∅
such that
P
T
is nonexpansive. Let
{
x
n
}
be the Ishikawa iterates defined by (13). Assume that
T
satisfies condition (I) and
α
n
,
β
n
∈
[
a
,
b
]
⊂
(
0,1
)
. Then,
{
x
n
}
converges strongly to a fixed point of
T
.Remark 2.
In recursion formula (4), the authors take y
n
∈
T
(
x
n
) such that ∥
y
n
-
x
*
∥
=
d
(
x
*
,
T
x
n
). Theexistence of y
n satisfying this condition is guaranteed by the assumption that T
x
n is proximinal. In general such a y
n is extremely difficult to pick. If T
x
n is proximinal, it is not difficult to prove that it is closed. If, in addition, it is a convex subset of a real Hilbert space, then y
n isunique and is characterized by
(14)
〈
x
*
-
y
n
,
y
n
-
u
n
〉
≥
0
∀
u
n
∈
T
x
n
.
One can see from this inequality that it is not easy to picky
n
∈
T
x
n satisfying
(15)
∥
y
n
-
x
*
∥
=
d
(
x
*
,
T
x
n
)
at every step of the iteration process. So, recursion formula (4) is not convenient to use in any possible application. Also, the recursion formulas defined in (7) and (8) are not convenient to use in any possible application. The sequences {
z
n
} and {
z
n
′
} are not known precisely. The restrictions z
n
∈
T
x
n, ∥
z
n
-
u
n
∥
=
d
(
u
n
,
T
x
n
), u
n
∈
F
(
T
), and z
n
′
∈
T
y
n
,
∥
z
n
′
-
v
n
∥
=
d
(
v
n
,
T
y
n
), v
n
∈
F
(
T
), make them difficult to use. These restrictions on z
n and z
n
′ depend on F
(
T
), the fixed points set. So, the recursions formulas (7) and (8) are not easily useable.Definition 3.
LetK be a nonempty subset of a real Hilbert space H. A map T
:
K
→
H is called k
-strictly pseudocontractive if there exists k
∈
(
0,1
) such that
(16)
∥
T
x
-
T
y
∥
2
≤
∥
x
-
y
∥
2
+
k
∥
x
-
y
-
(
T
x
-
T
y
)
∥
2
hhhhhhhihhhhhhhhhhhihhhhhhhh
∀
x
,
y
∈
K
.
Ifk
=
1 in (16), the map T is said to bepseudocontractive.Browder and Petryshyn [14] introduced and studied the class of strictly pseudocontractive maps as an important generalization of the class of nonexpansive maps (mappings T
:
K
→
K satisfying ∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥
∀
x
,
y
∈
K). It is trivial to see that every nonexpansive map is strictly pseudocontractive.Motivated by this, Chidume et al. [15] introduced the class ofmultivalued strictly pseudocontractive maps defined on a real Hilbert space H as follows.Definition 4.
A multivalued mapT
:
D
(
T
)
⊂
H
→
2
H is called k
-strictly pseudocontractive if there exists k
∈
(
0,1
) such that, for all x
,
y
∈
D
(
T
),
(17)
(
H
(
T
x
,
T
y
)
)
2
≤
∥
x
-
y
∥
2
+
k
∥
x
-
y
-
(
u
-
v
)
∥
2
hhhhhhhhhhhhhhhhhhhhh
∀
u
∈
T
x
,
v
∈
T
y
.
Ifk
=
1 in (17), the map T is said to bepseudocontractive.We observe from (17) that every nonexpansive mapping is strict pseudocontractive and hence the class of pseudocontractive mappings is a more general class of mappings.Then, they proved strong convergence theorems for this class of mappings. The recursion formula used is of theKrasnoselskii-type [16].Theorem  CA1 (Chidume et al. [15]).Let
K
be a nonempty closed convex subset of a real Hilbert space
H
. Suppose that
T
:
K
→
C
B
(
K
)
is a multivalued
k
-strictly pseudocontractive mapping such that
F
(
T
)
≠
∅
. Assume that
T
p
=
{
p
}
∀
p
∈
F
(
T
)
. Let
{
x
n
}
be a sequence defined by
x
0
∈
K
,
(18)
x
n
+
1
=
(
1
-
λ
)
x
n
+
λ
y
n
,
n
≥
0
,
where
y
n
∈
T
x
n
and
λ
∈
(
0,1
-
k
)
. Then,
lim
⁡
n
→
∞
d
(
x
n
,
T
x
n
)
=
0
.Theorem  CA2 (Chidume et al. [15]).Let
K
be a nonempty compact convex subset of a real Hilbert space
H
and let
T
:
K
→
C
B
(
K
)
be a multivalued
k
-strictly pseudocontractive mapping with
F
(
T
)
≠
∅
such that
T
p
=
{
p
}
∀
p
∈
F
(
T
)
. Suppose that
T
is continuous. Let
{
x
n
}
be a sequence defined by
x
0
∈
K
,
(19)
x
n
+
1
=
(
1
-
λ
)
x
n
+
λ
y
n
,
where
y
n
∈
T
x
n
and
λ
∈
(
0,1
-
k
)
. Then, the sequence
{
x
n
}
converges strongly to a fixed point of
T
.Remark 5.
We note that, for the more general situation of approximating a fixed point of amultivalued Lipschitz pseudocontractive map in a real Hilbert space, an example of Chidume and Mutangadura [17] shows that, even in the single-valued case, the Mann iteration method does not always converge in the setting of Theorem  CA2.We now give an example of multivalued pseudocontractive map (Definition4) which is not nonexpansive.Example 6.
LetT
:
[
0,1
]
)
→
2
R be the multivalued map defined by
(20)
T
x
=
{
{
2
}
if
x
=
0
,
{
0
,
x
}
if
x
≠
0
.
(i)
T satisfies, for all x
,
y
∈
D
(
T
),
(21)
(
H
(
T
x
,
T
y
)
)
2
≤
|
x
-
y
|
2
+
|
(
x
-
u
)
-
(
y
-
v
)
|
2
,
hhhhhhhhhhhhhhhhhhhhh
∀
u
∈
T
x
,
v
∈
T
y
.
(ii)
T is not nonexpansive.Proof of (i).
Inequality (21) is obvious for x
=
y
=
0. Now for (
x
,
y
)
≠
(
0,0
), we proceed as follows.
Case  1. Assume that x
,
y
∈
(
0,1
]. In this case, T
x
=
{
0
,
x
} and T
y
=
{
0
,
y
}. Therefore, we have
(22)
H
(
T
x
,
T
y
)
=
max
⁡
{
sup
⁡
a
∈
T
y
d
(
a
,
{
0
,
x
}
)
,
sup
⁡
b
∈
T
x
d
(
b
,
{
0
,
y
}
)
}
=
max
⁡
{
min
⁡
{
x
,
|
x
-
y
|
}
,
min
⁡
{
y
,
|
x
-
y
|
}
}
=
{
max
⁡
{
|
x
-
y
|
,
min
⁡
{
y
,
|
x
-
y
|
}
}
if
x
≥
y
max
⁡
{
min
⁡
{
x
,
|
x
-
y
|
}
,
|
x
-
y
|
}
if
x
≤
y
=
|
x
-
y
|
.
Hence, for allx
,
y
∈
(
0,1
], we have
(23)
(
H
(
T
x
,
T
y
)
)
2
≤
|
x
-
y
|
2
+
|
(
x
-
u
)
-
(
y
-
v
)
|
2
,
hhhhhhhhhhhhhhhhhhhhh
∀
u
∈
T
x
,
v
∈
T
y
.
Case  2. Assume that x
∈
(
0,1
] and y
=
0. In this case, we have T
x
=
{
0
,
x
} and T
y
=
T
0
=
{
2
}.
Therefore,(24)
H
(
T
x
,
T
y
)
=
max
⁡
{
sup
⁡
a
∈
{
2
}
d
(
a
,
{
0
,
x
}
)
,
sup
⁡
b
∈
{
0
,
x
}
|
2
-
b
|
}
=
sup
⁡
b
∈
{
0
,
x
}
|
2
-
b
|
=
sup
⁡
{
2
,
-
x
+
2
}
=
2
.
On the other hand, letu
∈
T
x
=
{
0
,
x
} and v
∈
T
y
=
{
2
}.(i)
Ifu
=
0, then
(25)
|
(
x
-
u
)
-
(
y
-
v
)
|
2
=
(
x
+
2
)
2
≥
(
H
(
T
x
,
T
y
)
)
2
.
(ii)
Ifu
=
x, we have
(26)
|
(
x
-
u
)
-
(
y
-
v
)
|
2
=
4
=
(
H
(
T
x
,
T
y
)
)
2
.
Therefore,(27)
(
H
(
T
x
,
T
y
)
)
2
≤
|
x
-
y
|
2
+
|
(
x
-
u
)
-
(
y
-
v
)
|
2
,
hhhhhhhhhhhhhhhhhhhhh
∀
u
∈
T
x
,
v
∈
T
y
.
This completes the proof of (i).Proof of (ii).
Ifx
∈
(
0,1
] and y
=
0, we have that H
(
T
x
,
T
y
)
=
2 and |
x
-
y
|
=
x. So,
(28)
H
(
T
x
,
T
y
)
>
|
x
-
y
|
.
This proves that T is not nonexpansive.It is our purpose in this paper to prove strong convergence theorems for the class ofmultivalued Lipschitz pseudocontractive maps in real Hilbert spaces. We use the recursion formula (11),dispensing with thesecond restriction on the sequences {
z
n
} and {
u
n
}: ∥
z
n
+
1
-
u
n
-
1
∥
≤
H
(
T
x
n
+
1
,
T
y
n
)
+
γ
n, n
≥
1. This class of maps is much more larger than that of multivalued nonexpansive maps used in Theorem  SW. So, in the setting of real Hilbert spaces, our theorem improves and extends the result of Song and Wang [3, 4].


## 2. Preliminaries

In the sequel, we will need the following results.Lemma 7 (Daffer and Kaneko [12]).
Let{
a
n
} and {
γ
n
} be sequences of nonnegative real numbers satisfying the following relation:
(29)
a
n
+
1
≤
a
n
+
γ
n
∀
n
≥
n
0
,
where n
0 is a nonnegative integer. If ∑
γ
n
<
∞, then lim
⁡
n
→
∞
⁡
a
n exists.Lemma 8.
LetH be a real Hilbert space. Then
(30)
∥
λ
x
+
(
1
-
λ
)
y
∥
2
=
λ
∥
x
∥
2
+
(
1
-
λ
)
∥
y
∥
2
-
λ
(
1
-
λ
)
∥
x
-
y
∥
2
for all x
,
y
∈
H, and λ
∈
[
0,1
].


## 3. Main Results

We use the following iteration scheme.LetK be a nonempty closed convex subset of a real Hilbert space H and α
n, β
n, and γ
n real sequences in (
0,1
]. Let {
x
n
} be the sequence defined from arbitrary x
1
∈
K by
(31)
y
n
=
(
1
-
β
n
)
x
n
+
β
n
u
n
,
n
≥
1
,
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
w
n
,
n
≥
1
,
where u
n
∈
T
x
n, w
n
∈
T
y
n are such that
(32)
∥
u
n
-
w
n
∥
≤
H
(
T
x
n
,
T
y
n
)
+
γ
n
.
We first prove the following theorem.Theorem 9.
LetK be a nonempty closed convex subset of a real Hilbert space H and T
:
K
→
C
B
(
K
) a multivalued L-Lipschitz pseudocontractive mapping with F
(
T
)
≠
∅ and T
p
=
{
p
}
∀
p
∈
F
(
T
). Let {
x
n
} be the sequence defined by (31) and (32). Assume that (i)  0
≤
α
n
≤
β
n
<
1
∀
n
≥
0; (ii)   lim
⁡
β
n
=
0; (iii)  ∑
‍
α
n
β
n
=
∞ and ∑
γ
n
2
<
∞. Then, liminf
⁡
n
→
∞
d
(
x
n
,
T
x
n
)
=
0.Proof.
Letp
∈
F
(
T
). Using Lemma 8, the fact that T is pseudocontractive, and the assumption T
p
=
{
p
}
∀
p
∈
F
(
T
), we have
(33)
∥
x
n
+
1
-
p
∥
2
=
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
w
n
-
p
∥
2
-
α
n
(
1
-
α
n
)
∥
x
n
-
w
n
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
(
H
(
T
y
n
,
T
p
)
)
2
-
α
n
(
1
-
α
n
)
∥
x
n
-
w
n
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
(
∥
y
n
-
p
∥
2
+
∥
y
n
-
w
n
∥
2
)
-
α
n
(
1
-
α
n
)
∥
x
n
-
w
n
∥
2
=
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
y
n
-
p
∥
2
+
α
n
∥
y
n
-
w
n
∥
2
-
α
n
(
1
-
α
n
)
∥
x
n
-
w
n
∥
2
.
Observing that(34)
∥
y
n
-
w
n
∥
2
=
(
1
-
β
n
)
∥
x
n
-
w
n
∥
2
+
β
n
∥
u
n
-
w
n
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
u
n
∥
2
,
then, from inequality (33) and identity (34), we have that
(35)
∥
x
n
+
1
-
p
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
y
n
-
p
∥
2
+
α
n
(
(
1
-
β
n
)
∥
x
n
-
w
n
∥
2
+
β
n
∥
u
n
-
w
n
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
u
n
∥
2
)
-
α
n
(
1
-
α
n
)
∥
x
n
-
w
n
∥
2
.
Using again Lemma8, the fact that T is pseudocontractive, and the assumption T
p
=
{
p
}
∀
p
∈
F
(
T
), we obtain the following estimates:
(36)
∥
y
n
-
p
∥
2
=
(
1
-
β
n
)
∥
x
n
-
p
∥
2
+
β
n
∥
u
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
u
n
∥
2
≤
(
1
-
β
n
)
∥
x
n
-
p
∥
2
+
β
n
(
H
(
T
x
n
,
T
p
)
)
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
u
n
∥
2
≤
(
1
-
β
n
)
∥
x
n
-
p
∥
2
+
β
n
(
∥
x
n
-
p
∥
2
+
∥
u
n
-
x
n
∥
2
)
-
β
n
(
1
-
β
n
)
∥
x
n
-
u
n
∥
2
=
∥
x
n
-
p
∥
2
+
β
n
2
∥
x
n
-
u
n
∥
2
.
Therefore, inequalities (35) and (36) and condition (i) imply that
(37)
∥
x
n
+
1
-
p
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
(
∥
x
n
-
p
∥
2
+
β
n
2
∥
x
n
-
u
n
∥
2
)
+
α
n
(
(
1
-
β
n
)
∥
x
n
-
w
n
∥
2
+
β
n
∥
u
n
-
w
n
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
u
n
∥
2
)
-
α
n
(
1
-
α
n
)
∥
x
n
-
w
n
∥
2
≤
∥
x
n
-
p
∥
2
+
α
n
β
n
2
∥
x
n
-
u
n
∥
2
+
α
n
(
1
-
β
n
)
∥
x
n
-
w
n
∥
2
+
α
n
β
n
∥
u
n
-
w
n
∥
2
-
α
n
β
n
(
1
-
β
n
)
∥
x
n
-
u
n
∥
2
-
α
n
(
1
-
α
n
)
∥
x
n
-
w
n
∥
2
=
∥
x
n
-
p
∥
2
-
α
n
β
n
(
1
-
2
β
n
)
∥
x
n
-
u
n
∥
2
-
α
n
(
β
n
-
α
n
)
∥
x
n
-
w
n
∥
2
+
α
n
β
n
∥
u
n
-
w
n
∥
2
≤
∥
x
n
-
p
∥
2
-
α
n
β
n
(
1
-
2
β
n
)
∥
x
n
-
u
n
∥
2
+
α
n
β
n
∥
u
n
-
w
n
∥
2
.
Using inequality (32), the fact that T is L-Lipschitzian, and the recursion formula (31), we have
(38)
∥
u
n
-
w
n
∥
2
≤
2
β
n
2
L
2
∥
x
n
-
u
n
∥
2
+
2
γ
n
2
.
Therefore, from inequalities (37) and (38), we obtain
(39)
∥
x
n
+
1
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
α
n
β
n
(
1
-
2
β
n
-
2
L
2
β
n
2
)
×
∥
x
n
-
u
n
∥
2
+
2
γ
n
2
.
Observing that condition (ii) yields thatβ
n
+
L
2
β
n
2
≤
1
/
4, for all n
≥
N for some N, it then follows that
(40)
1
2
∑
n
=
N
∞
‍
α
n
β
n
∥
x
n
-
u
n
∥
2
≤
∥
x
N
-
p
∥
2
+
2
∑
n
=
N
∞
‍
γ
n
2
<
∞
,
which implies, by condition (iii), that liminf
⁡
n
→
∞
∥
x
n
-
u
n
∥
=
0. Since u
n
∈
T
x
n, it follows that d
(
x
n
,
T
x
n
)
≤
∥
x
n
-
u
n
∥. Therefore, liminf
⁡
n
→
∞
d
(
x
n
,
T
x
n
)
=
0.We now prove the following corollaries of Theorem9.Corollary 10.
LetK be a nonempty closed convex subset of a real Hilbert space H and T
:
K
→
C
B
(
K
) a multivalued Lipschitz pseudocontractive mapping with F
(
T
)
≠
∅ and T
p
=
{
p
}
∀
p
∈
F
(
T
). Let {
x
n
} be the sequence defined by (31) and (32). Assume that T is hemicompact, and (i)  0
≤
α
n
≤
β
n
<
1
∀
n
≥
0; (ii)  lim
⁡
β
n
=
0; (iii)  ∑
α
n
β
n
=
∞; and (iv)  ∑
γ
n
2
<
∞. Then, {
x
n
} converges strongly to a fixed point of T.Proof.
From Theorem9, we have that liminf
⁡
n
→
∞
d
(
x
n
,
T
x
n
)
=
0. So there exists a subsequence {
x
n
k
} of {
x
n
} such that lim
⁡
k
→
∞
d
(
x
n
k
,
T
x
n
k
)
=
0. Using the fact that T is hemicompact, the sequence {
x
n
k
} has a subsequence denoted again by {
x
n
k
} that converges strongly to some q
∈
K. Since T is continuous, we have d
(
x
n
k
,
T
x
n
k
)
→
d
(
q
,
T
q
). Therefore, d
(
q
,
T
q
)
=
0 and so q
∈
F
(
T
). Now setting p
=
q in inequality (39) and using condition (ii) we have that
(41)
∥
x
n
+
1
-
q
∥
2
≤
∥
x
n
-
q
∥
2
+
2
γ
n
2
,
for all n
≥
N for some N
≥
1. Therefore, Lemma 7 implies that lim
⁡
n
→
∞
⁡
∥
x
n
-
q
∥ exists. Since lim
⁡
k
→
∞
∥
x
n
k
-
q
∥
=
0, it then follows that {
x
n
} converges strongly to q
∈
F
(
T
), completing the proof.We can easily observe that ifT is nonexpansive, then it is Lipschitzian and pseudocontractive. Therefore, the following corollary generalizes Theorem  SW of Song and Wang [3, 4] in the setting of Hilbert spaces.Corollary 11.
LetK be a nonempty compact convex subset of a real Hilbert space H and T
:
K
→
C
B
(
K
) a multivalued Lipschitz pseudocontractive mapping with F
(
T
)
≠
∅ and T
p
=
{
p
}
∀
p
∈
F
(
T
). Let {
x
n
} be the sequence defined by (31) and (32). Assume that (i)  0
≤
α
n
≤
β
n
<
1
∀
n
≥
0; (ii)  lim
⁡
β
n
=
0; (iii)  ∑
α
n
β
n
=
∞ and ∑
γ
n
2
<
∞. Then, {
x
n
} converges strongly to a fixed point of T.Proof.
SinceK is compact, it follows that T is hemicompact. So, the proof follows from Corollary 10.Corollary 12.
LetK be a nonempty closed convex subset of a real Hilbert space H, and let T
:
K
→
C
B
(
K
) be a multivalued Lipschitz pseudocontractive mapping with F
(
T
)
≠
∅ and T
p
=
{
p
}
∀
p
∈
F
(
T
). Let {
x
n
} be the sequence defined by (31) and (32). Assume that T satisfies condition (I) and (i)  0
≤
α
n
≤
β
n
<
1
∀
n
≥
0; (ii)  lim
⁡
β
n
=
0; (iii)  ∑
α
n
β
n
=
∞; and (iv)  ∑
γ
n
2
<
∞. Then, {
x
n
} converges strongly to a fixed point of T.Proof.
From Theorem9, we have that liminf
⁡
n
→
∞
d
(
x
n
,
T
x
n
)
=
0. So there exists a subsequence {
x
n
k
} of {
x
n
} such that lim
⁡
k
→
∞
d
(
x
n
k
,
T
x
n
k
)
=
0. Since T satisfies condition (I), we have lim
⁡
k
→
∞
d
(
x
n
k
,
F
(
T
)
)
=
0. Thus there exist a subsequence of {
x
n
k
} denoted again by {
x
n
k
} and a sequence {
p
k
}
⊂
F
(
T
) such that
(42)
∥
x
n
k
-
p
k
∥
≤
1
2
k
∀
k
.
By settingp
=
p
k in inequality (39) and using the condition (ii), we have
(43)
∥
x
n
k
+
1
-
p
k
∥
2
≤
∥
x
n
k
-
p
k
∥
2
+
2
γ
n
k
2
≤
1
2
k
+
2
γ
n
k
2
for all k
≥
k
0, for some k
0
≥
1.
We now show that{
p
k
} is a Cauchy sequence in K. Notice that, for k
≥
k
0,
(44)
∥
p
k
+
1
-
p
k
∥
2
≤
2
∥
p
k
+
1
-
x
n
k
+
1
∥
2
+
2
∥
x
n
k
+
1
-
p
k
∥
2
≤
2
(
1
2
2
k
+
2
+
1
2
k
+
2
γ
n
k
2
)
≤
2
(
1
2
k
-
1
+
2
γ
n
k
2
)
.
From condition (iv), it follows that{
p
k
} is a Cauchy sequence in K and thus converges strongly to some q
∈
K. Using the fact that T is L-Lipschitzian and p
k
→
q, we have
(45)
d
(
p
k
,
T
q
)
≤
H
(
T
p
k
,
T
q
)
≤
L
∥
p
k
-
q
∥
,
so that d
(
q
,
T
q
)
=
0 and thus q
∈
T
q. Therefore, q
∈
F
(
T
) and {
x
n
k
} converges strongly to q. Now setting p
=
q in inequality (39) and using condition (ii) we have that
(46)
∥
x
n
+
1
-
q
∥
2
≤
∥
x
n
-
q
∥
2
+
2
γ
n
2
,
for all n
≥
N for some N
≥
1. Therefore, Lemma 7 implies that lim
⁡
n
→
∞
⁡
∥
x
n
-
q
∥ exists. Since lim
⁡
k
→
∞
∥
x
n
k
-
q
∥
=
0, it then follows that {
x
n
} converges strongly to q
∈
F
(
T
), completing the proof.Remark 13.
Our theorem and corollaries improve convergence theorems for multivalued nonexpansive mappings in [1–4, 12, 13] in the following sense.
(i) In our algorithm,u
n
∈
T
x
n
,
w
n
∈
T
y
n do not have to satisfy the restrictive conditions ∥
u
n
-
x
*
∥
=
d
(
x
*
,
T
x
n
) and ∥
w
n
-
x
*
∥
=
d
(
x
*
,
T
y
n
) in the recursion formula (5) and similar restrictions in the recursion formulas (7) and (8). These restrictions on u
n and w
n depend on x
*, a fixed point that is being approximated or the fixed points set F
(
T
). Also, in our algorithm, the second restriction on the sequences {
z
n
} and {
u
n
}
:
∥
z
n
+
1
-
u
n
-
1
∥
≤
H
(
T
x
n
+
1
,
T
y
n
)
+
γ
n, n
≥
1, in the recursion formula (11), is removed.
(ii) Our theorem and corollaries are proved for the class of multivalued Lipschitz pseudocontractive mappings which is much more general than that of multivalued nonexpansive mappingsRemark 14.
Corollary11 is an extension of the Theorem of Ishikawa [18] from single-valued to multivalued Lipschitz pseudocontractive mappings.Remark 15.
Real sequences that satisfy the hypotheses of Theorems9 are α
n
=
(
n
+
1
)
-
1
/
2, β
n
=
(
n
+
1
)
-
1
/
2, and γ
n
=
(
n
+
1
)
-
1
,
n
≥
0.



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*Source: 269786-2014-09-24.xml*