Puy chosuantai2

Weak and strong convergence to common
fixed points of a countable family of
multivalued mappings in Banach spaces
∗
W. Cholamjiak1,2 , P. Cholamjiak3 , Y. J. Cho4 , S. Suantai1,2
1
Department of Mathematics, Faculty of Science, Chiang Mai University,
Chiang Mai 50200, Thailand
2
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand
3
Department of Mathematics, Faculty of Science, University of Phayao,
Phayao 56000, Thailand
4
Department of Mathematics Education and the RINS, Gyeongsang National University,
Chinju 660-701, Republic of Korea

Abstract

In this paper, we introduce a modified Mann iteration for a countable family
of multivalued mappings. We use the best approximation operator to obtain
weak and strong convergence theorems in a Banach space. We apply the main
results to the problem of finding a common fixed point of a countable family of
nonexpansive multivalued mappings.

Keywords: Nonexpansive multivalued mapping; fixed point; weak convergence; strong con-
vergence.

AMS Subject Classification: 47H10, 47H09.

∗
Corresponding author.
Email addresses: c-wchp007@hotmail.com (W. Cholamjiak), prasitch2008@yahoo.com(P.
Cholamjiak), yjcho@gnu.ac.kr (Y. J. Cho), scmti005@chiangmai.ac.th (S. Suantai)
1

2 W. Cholamjiak, P. Cholamjiak, Y. J. Cho, S. Suantai

1 Introduction

Let D be a nonempty convex subset of a Banach spaces E. The set D is called
proximinal if for each x ∈ E, there exists an element y ∈ D such that x − y =
d(x, D), where d(x, D) = inf{ x − z : z ∈ D}. Let CB(D), CCB(D), K(D) and
P (D) denote the families of nonempty closed bounded subsets, nonempty closed
convex bounded subsets, nonempty compact subsets, and nonempty proximinal
bounded subsets of D, respectively. The Hausdorff metric on CB(D) is defined
by
H(A, B) = max sup d(x, B), sup d(y, A)
x∈A y∈B
for A, B ∈ CB(D). A single-valued map T : D → D is called nonexpansive if
T x − T y ≤ x − y for all x, y ∈ D. A multivalued mapping T : D → CB(D)
is said to be nonexpansive if H(T x, T y) ≤ x − y for all x, y ∈ D. An element
p ∈ D is called a fixed point of T : D → D (respectively, T : D → CB(D)) if
p = T p (respectively, p ∈ T p). The set of fixed points of T is denoted by F (T ).
The mapping T : D → CB(D) is called quasi-nonexpansive[23] if F (T ) = ∅ and
H(T x, T p) ≤ x − p for all x ∈ D and all p ∈ F (T ). It is clear that every
nonexpansive multivalued mapping T with F (T ) = ∅ is quasi-nonexpansive. But
there exist quasi-nonexpansive mappings that are not nonexpansive, see [22]. It is
known that if T is a quasi-nonexpansive multivalued mapping, then F (T ) is closed.
Throughout this paper, we denote the weak convergence and strong convergence
are and →, respectively. The mapping T : D → CB(D) is called hemicompact
if, for any sequence {xn } in D such that d(xn , T xn ) → 0 as n → ∞, there exists a
subsequence {xnk } of {xn } such that xnk → p ∈ D. We note that if D is compact,
then every multivalued mapping T : D → CB(D) is hemicompact.
A Banach space E is said to satisfy Opial’s property; see [16] if for each x ∈ E
and each sequence {xn } weakly convergent to x, the following condition holds for
all x = y :
lim inf xn − x < lim inf xn − y .
n→∞ n→∞
The mapping T : D → CB(D) is said to be dimi-closed if for every sequence
{xn } ⊂ D and any yn ∈ T xn such that xn x and yn → y, we have x ∈ D and
y ∈ T x.
Remark 1.1. [11] If the space satisfies Opial’s property, then I − T is demi-closed at
0, where T : D → K(D) is a nonexpansive multivalued mapping.

For single valued nonexpansive mappings, in 1953, Mann [14] introduced the fol-
lowing iterative procedure to approximate a fixed point of a nonexpansive mapping

Weak and strong convergence to common fixed points... 3

T in a Hilbert space H:

xn+1 = αn xn + (1 − αn )T xn , ∀n ∈ N, (1.1)

where the initial point x1 is taken in D arbitrarily and {αn } is a sequence in [0,1].
However, we note that Mann’s iteration process (1.1) has only weak convergence,
in general; for instance, see [2, 7, 18].
Subsequently, Mann iteration has extensively been studied by many authors, for
more details, see in [3, 4, 6, 9, 10, 12, 24, 25, 27, 30] and many other results which
cannot be mentioned here. However, the studying of multivalued nonexpansive
mappings is harder than that of single valued nonexpansive mappings in both Hilbert
spaces and Banach spaces. The result of fixed points for multivalued contractions
and nonexpansive mappings by using the Hausdorff metric was initiated by Markin
[13]. Later, different iterative processes have been used to approximate fixed points
of multivalued nonexpansive mappings (see [1, 17, 19, 20, 29, 23, 26, 28]).

In 2009, Song and Wang [28] proved strong and weak convergence for Mann
iteration of a multivalued nonexpansive mapping T in a Banach space. They gave
a strong convergence of the modified Mann iteration which is independent of the
implicit anchor-like continuous path zt ∈ tu + (1 − t)T zt .
Let D be a nonempty closed subset of a Banach space E, βn ∈ [0, 1], αn ∈ [0, 1]
and γn ∈ (0, +∞) such that limn→∞ γn = 0.
(A) Choose x0 ∈ D,

xn+1 = (1 − αn )xn + αn yn , ∀n ≥ 0,

where yn ∈ T xn such that yn+1 − yn ≤ H(T xn+1 , T xn ) + γn .
(B) For fixed u ∈ D, the sequence of modified Mann iteration is defined by x0 ∈ D,

xn+1 = βn u + αn xn + (1 − αn − βn )yn , ∀n ≥ 0,

where yn ∈ T xn such that yn+1 − yn ≤ H(T xn+1 , T xn ) + γn .

Very recently, Shahzad and Zegeye [22] obtained the strong convergence theo-
rems for a quasi-nonexpansive multivalued mapping. They relaxed compactness of
the domain of T and constructed an iteration scheme which removes the restriction
of T namely T p = {p} for any p ∈ F (T ). The results provided an affirmative answer
to some questions in [17]. They introduced new iterations as follows:
Let D be a nonempty convex subset of a Banach space E and T : D → CB(D)
and let αn , αn ∈ [0, 1],


(C) The sequence of Ishikawa iterates is defined by x0 ∈ D,

yn = αn zn + (1 − αn )xn ,
xn+1 = αn zn + (1 − αn )xn , ∀n ≥ 0,

where zn ∈ T xn and zn ∈ T yn .
(D) Let T : D → P (D) and PT x = {y ∈ T x : x − y = d(x, T x)}, we denote PT
is the best approximation operator. The sequence of Ishikawa iterates is defined by
x0 ∈ D,

yn = αn zn + (1 − αn )xn ,
xn+1 = αn zn + (1 − αn )xn , ∀n ≥ 0,

where zn ∈ PT xn and zn ∈ PT yn .

It is remarked that Hussain and Khan [8], in 2003, employed the best approx-
imation operator PT to study fixed points of *-nonexpansive multivalued mapping
T and strong convergence of its iterates to a fixed point of T defined on a closed
and convex subset of a Hilbert space.

Let D be a nonempty closed convex subset of a Banach space E. Let {Tn }∞
n=1
be a family of multivalued mappings from D into 2D and PTn x = {yn ∈ Tn x :
x − yn = d(x, Tn x)}, n ≥ 1. Let {αn } be a sequence in (0, 1).
(E) The sequence of the modified Mann iteration is defined by x1 ∈ D and

xn+1 ∈ αn xn + (1 − αn )PTn xn , ∀n ≥ 1. (1.2)

In this paper, we modify Mann iteration by using the best approximation oper-
ator PTn , n ≥ 1 to find a common fixed point of a countable family of nonexpansive
multivalued mappings {Tn }∞ , n ≥ 1. Then we obtain weak and strong conver-
n=1
gence theorems for a countable family of multivalued mappings in Banach spaces.
Moreover, we apply our results to the problem of finding a common fixed point of a
family of nonexpansive multivalued mappings.

2 Preliminaries

In this section, we give some characterizations and properties of the metric
projection in a Hilbert space.


Let H be a real Hilbert space with inner product ·, · and norm · . Let D
be a closed and convex subset of H. If, for any point x ∈ H, there exists a unique
nearest point in D, denoted by PD x, such that

x − PD x ≤ x − y , ∀y ∈ D,

then PD is called the metric projection of H onto D. We know that PD is a nonex-
pansive mapping of H onto D.

Lemma 2.1. [31] Let D be a closed and convex subset of a real Hilbert space H
and PD be the metric projection from H onto D. Then, for any x ∈ H and z ∈ D,
z = PD x if and only if the following holds:

x − z, y − z ≤ 0, ∀y ∈ D.

Using the proof line in Lemma 3.1.3 of [31], we obtain the following result.

Proposition 2.2. Let D be a closed and convex subset of a real Hilbert space H.
Let T : D → CCB(D) be a multivalued mapping and PT the best approximation
operator. Then, for any x ∈ D, z ∈ PT x if and only if the following holds:

x − z, y − z ≤ 0, ∀y ∈ T x.

Lemma 2.3. [31] Let H be a real Hilbert space. Then the following equations hold:
(1) x − y 2 = x 2 − y 2 − 2 x − y, y for all x, y ∈ H;
(2) tx + (1 − t)y 2 = t x 2 + (1 − t) y 2 − t(1 − t) x − y 2 for all t ∈ [0, 1]
and x, y ∈ H.

Next, we show that PT is nonexpansive under some suitable conditions imposed
on T .
Remark 2.4. Let D be a closed and convex subset of a real Hilbert space H. Let
T : D → CCB(D) be a multivalued mapping. If T x = T y, ∀x, y ∈ D, then PT is a
nonexpansive multivalued mapping.

Proof. Let x, y ∈ D. For each a ∈ PT x, we have

d(a, PT y) ≤ a − b , ∀b ∈ PT y. (2.1)

By Proposition 2.2, we have

x − y − (a − b), a − b = x − a, a − b + y − b, b − a ≥ 0.


It follows that

2
a−b = x − y, a − b + a − b − (x − y), a − b
≤ x − y, a − b
≤ x−y a−b . (2.2)

This implies that
a−b ≤ x−y . (2.3)

From (2.1) and (2.3), we obtain

d(a, PT y) ≤ x − y

for every a ∈ PT x. Hence supa∈PT x d(a, PT y) ≤ x − y . Similarly, we can show
that supb∈PT y d(PT x, b) ≤ x − y . This shows that H(PT x, PT y) ≤ x − y .

It is clear that if a nonexpansive multivalued mapping T satisfies the condition
that T x = T y, ∀x, y ∈ D, then PT is nonexpansive. The following example shows
that if T is a nonexpansive multivalued mapping satisfying the property that T x =
T y, ∀x, y ∈ D, then T x is not a singleton for all x ∈ D.
Example 1. Consider D = [0, 1] with the usual norm. Define T : D → K(D)
by
T x = [0, a], a ∈ (0, 1].

For x, y ∈ D, we have H(T x, T y) = 0 ≤ x − y . Hence T is nonexpansive and
F (T ) = [0, a].
Next, we show that there exists a nonexpansive multivalued mapping T which
PT is nonexpansive but T x = T y, ∀x, y ∈ D, x = y.
Example 2. Consider D = [0, 1] with the usual norm. Define T : D → K(D)
by
T x = [0, x].

For x, y ∈ D, x = y, we have PT x = {x} and PT y = {y}. Then H(T x, T y) =
x − y = H(PT x, PT y). Hence T and PT are nonexpansive.
Open Question: Can we remove the assumption that PT is nonexpansive ?
Now, we give an example which T is not nonexpansive, but PT is nonexpansive.
Example 3. Consider D = [0, 1] with the usual norm. Define T : D → K(D) by

[0, x] , x ∈ [0, 1 ],
2
Tx =
{ 1 } , x ∈ ( 1 , 1].
2 2


Since H(T ( 1 ), T ( 3 )) = H([0, 5 ], { 2 }) = 1 > 2 = 1 − 3 , T is not nonexpansive.
5 5
1 1
2 5 5 5
However, PT is nonexpansive. In fact,
Case 1, if x, y ∈ [0, 1 ] then H(PT x, PT y) = x − y .
2
Case 2, if x ∈ [0, 2 ] and y ∈ ( 1 , 1] then H(PT x, PT y) = x − 1 ≤ x − y .
1
2 2
1
Case 3, if x, y ∈ ( 2 , 1] then H(PT x, PT y) = 0.
It would be interesting to study the convergence of a multivalued mapping T by
using the best approximation operator PT .
In order to deal with a family of nonlinear mappings, we consider the following
conditions. Let E be a Banach space and D be a subset of E.
(1) Let {Tn } and τ be two families of nonlinear mappings of D into itself with
∅ = F (τ ) = ∞ F (Tn ), where F (Tn ) is the set of all fixed points of Tn and F (τ )
n=1
is the set of all common fixed points of τ . The family {Tn } is said to satisfy the
NST-condition [15] with respect to τ if, for each bounded sequence {zn } in C,

lim zn − Tn zn = 0 =⇒ lim zn − T zn = 0, ∀T ∈ τ.
n→∞ n→∞

(2) Let {Tn } and τ be two families of multivalued mappings of D into 2D with
∅ = F (τ ) = ∞ F (Tn ), where F (Tn ) is the set of all fixed points of Tn and F (τ ) is
n=1
the set of all common fixed points of τ . The family {Tn } is said to satisfy the SC-
condition with respect to τ if, for each bounded sequence {zn } in D and sn ∈ Tn zn ,

lim zn − sn = 0 =⇒ lim zn − cn = 0, ∃cn ∈ T zn , ∀T ∈ τ.
n→∞ n→∞

It is easy to see that, if the family {Tn } of nonexpansive mappings satisfies the
N ST −condition, then {Tn } satisfies the SC −condition for single valued mappings.
(3) Let T be a multivalued mapping from D into 2D with F (T ) = ∅. The
mapping T is said to satisfy Condition I if there is a nondecreasing function f :
[0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0 for r ∈ (0, ∞) such that d(x, T x) ≥
f (d(x, F (T ))) for all x ∈ D.
The following result can be found in [21].

Lemma 2.5. [21] Let D be a bounded closed subset of a Banach space E. Suppose
that a nonexpansive multivalued mapping T : D → P (D) has a nonempty fixed
point set. If I − T is closed, then T satisfies Condition I on D.

(4) Let {Tn } and τ be two families of multivalued mappings of D into 2D with
∅ = F (τ ) = ∞ F (Tn ), where F (Tn ) is the set of all fixed points of Tn and F (τ ) is
n=1
the set of all common fixed points of τ . The family {Tn } is said to satisfy Condition


(A) if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0
for r ∈ (0, ∞) such that there exists T ∈ τ , d(x, T x) ≥ f (d(x, F (τ ))) for all x ∈ D.

We will give the example of {Tn } which satisfies the SC-condition and Condition
(A) in the last section. Now, we need the following lemmas to prove our main results.

Lemma 2.6. [32] Let X be uniformly convex Banach space and Br (0) be a closed
ball of X. Then there exists a continuous strictly increasing convex function g :
[0, ∞) → [0, ∞) with g(0) = 0 such that
2 2 2
λx + (1 − λ)y ≤λ x + (1 − λ) y − λ(1 − λ)g( x − y )

for all x, y ∈ Br (0) and λ ∈ [0, 1].

Lemma 2.7. [5] Let E be a uniformly convex Banach space and Br (0) = {x ∈ E :
x ≤ r} be a closed ball of E. Then there exists a continuous strictly increasing con-
vex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that, for any j ∈ {1, 2, · · · , m},
m m m
2
2 αj
αi xi ≤ αi xi − αi g( xj − xi )
m−1
i=1 i=1 i=1
m
for all m ∈ N, xi ∈ Br (0) and αi ∈ [0, 1] for all i = 1, 2, · · · , m with i=1 αi = 1.

3 Strong and weak convergence of the modified Mann
iteration in Banach spaces

In this section, we prove strong convergence theorems of the modified Mann
iteration for a countable family of multivalued mappings under the SC-condition
and Condition (A) and obtain weak convergence theorem under the SC-condition
in Banach spaces. Moreover, we apply the main result to obtain weak and strong
convergence theorems for a countable family of nonexpansive multivalued mappings.

Theorem 3.1. Let D be a closed and convex subset of a uniformly convex Banach
space E which satisfies Opial’s property. Let {Tn } and τ be two families of multi-
valued mappings from D into P (D) with F (τ ) = ∩∞ F (Tn ) = ∅. Let {αn } be a
n=1
sequence in (0, 1) such that 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1. Let {xn } be
generated by (1.2). Assume that
(A1) for each n ∈ N, H(PTn x, PTn p) ≤ x − p , ∀x ∈ D, p ∈ F (τ );
(A2) I − T is demi-closed at 0 for all T ∈ τ .
If {Tn } satisfies the SC-condition, then the sequence {xn } converges weakly to an
element in F (τ ).


Proof. Since xn+1 ∈ αn xn + (1 − αn )PTn xn , there exists zn ∈ PTn xn such that
xn+1 = αn xn + (1 − αn )zn . We note that PTn p = {p} for all p ∈ F (τ ) and n ∈ N. It
follows from (A1) that

xn+1 − p ≤ αn xn − p + (1 − αn ) zn − p
= αn xn − p + (1 − αn )d(zn , PTn p)
≤ αn xn − p + (1 − αn )H(PTn xn , PTn p)
≤ xn − p (3.1)

for every p ∈ F (τ ). Then { xn −p } is a decreasing sequence and hence limn→∞ xn −
p exists for every p ∈ F (τ ). Next, for p ∈ F (τ ), since {xn } and {zn } are
bounded, by Lemma 2.6, there exists a continuous strictly increasing convex function
g : [0, ∞) → [0, ∞) with g(0) = 0 such that

2 2
xn+1 − p = αn (xn − p) + (1 − αn )(zn − p)
2 2
≤ αn xn − p + (1 − αn ) zn − p − αn (1 − αn )g( xn − zn )
2
= αn xn − p + (1 − αn )d(zn , PTn p)2 − αn (1 − αn )g( xn − zn )
2
≤ αn xn − p + (1 − αn )H(PTn xn , PTn p)2 − αn (1 − αn )g( xn − zn )
2
≤ xn − p − αn (1 − αn )g( xn − zn ).

It follows that

2
αn (1 − αn )g( xn − zn ) ≤ xn − p − xn+1 − p 2 .

Since limn→∞ xn − p exists and 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1,

lim g( xn − zn ) = 0.
n→∞

By the properties of g, we can conclude that

lim xn − zn = 0.
n→∞

Since {Tn } satisﬁes the SC-condition, there exists cn ∈ T xn such that

lim xn − cn = 0 (3.2)
n→∞

for every T ∈ τ . Since {xn } is bounded, there exists a subsequence {xnk } of {xn }
converges weakly to some q1 ∈ D. It follows from (A2) and (3.2) that q1 ∈ T q1
for every T ∈ τ . Next, we show that {xn } converges weakly to q1 , take another
subsequence {xmk } of {xn } converging weakly to some q2 ∈ D. Again, as above, we


can conclude that q2 ∈ T q2 for every T ∈ τ . Finally, we show that q1 = q2 . Assume
q1 = q2 . Then by the Opial’s property of E, we have

lim xn − q1 = lim xnk − q1
n→∞ k→∞
< lim xnk − q2
k→∞
= lim xn − q2
n→∞
= lim xmk − q2
k→∞
< lim xmk − q1
k→∞
= lim xn − q1 ,
n→∞

which is a contradiction. Therefore q1 = q2 . This shows that {xn } converges weakly
to a fixed point of T for every T ∈ τ . This completes the proof.

Using the above results and Remark 1.1, we obtain the following:

Corollary 3.2. Let D be a closed and convex subset of a uniformly convex Banach
space E which satisfies Opial’s property. Let {Tn } and τ be two families of nonex-
pansive multivalued mappings from D into K(D) with F (τ ) = ∩∞ F (Tn ) = ∅. Let
n=1
{αn } be a sequence in (0, 1) such that 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1. Let
{xn } be generated by (1.2). Assume that for each n ∈ N, H(PTn x, PTn p) ≤ x − p ,
∀x ∈ D, p ∈ F (τ ). If {Tn } satisfies the SC-condition, then the sequence {xn }
converges weakly to an element in F (τ ).

Theorem 3.3. Let D be a closed and convex subset of a uniformly convex Banach
space E. Let {Tn } and τ be two families of multivalued mappings from D into
P (D) with F (τ ) = ∩∞ F (Tn ) = ∅. Let {αn } be a sequence in (0, 1) such that
n=1
0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1. Let {xn } be generated by (1.2). Assume
that
(B1) for each n ∈ N, H(PTn x, PTn p) ≤ x − p , ∀x ∈ D, p ∈ F (τ );
(B2) the best approximation operator PT is nonexpansive for every T ∈ τ ;
(B3) F (τ ) is closed.
If {Tn } satisfies the SC-condition and Condition (A), then the sequence {xn } con-
verges strongly to an element in F (τ ).

Proof. It follows from the proof of Theorem 3.1 that limn→∞ xn − p exists for
every p ∈ F (τ ) and limn→∞ xn − zn = 0 where zn ∈ PTn xn . Since {Tn } satisfies
the SC-condition, there exists cn ∈ T xn such that

lim xn − cn = 0
n→∞


for every T ∈ τ . This implies that

lim d(xn , T xn ) ≤ lim d(xn , PT xn ) ≤ lim xn − cn = 0
n→∞ n→∞ n→∞

for every T ∈ τ . Since that {Tn } satisﬁes Condition (A), we have limn→∞ d(xn , F (τ )) =
0. It follows from (B3), there is subsequence {xnk } of {xn } and a sequence {pk } ⊂
F (τ ) such that
1
xnk − pk < k (3.3)
2
for all k. From 3.1, we obtain

xnk+1 − p ≤ xnk+1 −1 − p
≤ xnk+1 −2 − p
.
.
.
≤ xnk − p

for all p ∈ F (τ ). This implies that
1
xnk+1 − pk ≤ xnk − pk < . (3.4)
2k
Next, we show that {pk } is a Cauchy sequence in D. From (3.3) and (3.4), we have

pk+1 − pk ≤ pk+1 − xnk+1 + xnk+1 − pk
1
< k−1
. (3.5)
2
This implies that {pk } is a Cauchy sequence in D and thus converges to q ∈ D.
Since PT is nonexpansive for every T ∈ τ ,

d(pk , T q) ≤ d(pk , PT q) ≤ H(PT pk , PT q) ≤ pk − q (3.6)

for every T ∈ τ . It follows that d(q, T q) = 0 for every T ∈ τ and thus q ∈ F (τ ). It
implies by (3.3) that {xnk } converges strongly to q. Since limn→∞ xn − q exists,
it follows that {xn } converges strongly to q. This completes the proof.

We known that if T is a quasi-nonexpansive multivalued mapping, then F (T ) is
closed. So we have the following result.

space E. Let {Tn } and τ be two families of nonexpansive multivalued mappings
from D into P (D) with F (τ ) = ∩∞ F (Tn ) = ∅. Let {αn } be a sequence in (0, 1)
n=1
such that 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1. Let {xn } be generated by (1.2).


Assume that for each n ∈ N, H(PTn x, PTn p) ≤ x − p , ∀x ∈ D, p ∈ F (τ ) and the
best approximation operator PT is nonexpansive for every T ∈ τ . If {Tn } satisfies
the SC-condition and Condition (A), then the sequence {xn } converges strongly to
an element in F (τ ).

4 Application

Let E be a real Banach space and D be a nonempty closed convex subset of
E. Let {Ti }N be a finite family of nonexpansive multivalued mappings of D into
i=0
CB(D) and {βi,n } ⊂ [0, 1] be such that N βi,n = 1 for all n ∈ N. We define a
i=0
mapping Sn : D → 2 D as follows:

N
Sn = βi,n PTi , (4.1)
i=0

where T0 = I, I denotes the identity mapping. We also show that the mapping Sn
defined by (4.1) satisfies the condition imposed on our main theorem.

Lemma 4.1. Let D be a closed and convex subset of a uniformly convex Banach
space E. Let {Ti }N be a finite family of nonexpansive multivalued mappings of D
i=1
into P (D) and, let {βi,n }N be sequences in (0, 1) such that 0 < lim inf n→∞ βi,n ≤
i=0
lim supn→∞ βi,n < 1 for all i ∈ {0, 1, · · · , N } and N βi,n = 1 for all n ∈ N. For
i=0
all n ∈ N, let Sn be the mapping defined by (4.1). Assume that the best approximation
operator PTi is nonexpansive for all i ∈ {1, 2, ..., N }. Then the followings hold:
(1) ∩∞ F (Sn ) = ∩N F (Ti );
n=1 i=0

(2) {Sn } satisfies the SC-condition;
(3) for each n ∈ N, H(PSn x, PSn p) ≤ x − p for all x ∈ D and p ∈ ∩N F (Ti ).
i=0

Proof. (1) It is easy to see that ∩N F (Ti ) ⊂ ∩∞ F (Sn ). Next, we show that
i=0 n=1
∩∞ F (Sn ) ⊂ ∩N F (Ti ). Let p ∈ ∩∞ F (Sn ) and x∗ ∈ ∩N F (Ti ). Then there
n=1 i=1 n=1 i=0
exists zi ∈ PTi p such that p = β0,n p + N βi,n zi for all n ∈ N. By Lemma 2.7,
i=1
there exists a continuous strictly increasing convex function g : [0, ∞) → [0, ∞) with


g(0) = 0 such that
N
∗ 2 ∗
p−x = β0,n (p − x ) + βi,n (zi − x∗ ) 2

i=1
N N
β0,n
≤ β0,n p − x∗ 2
+ βi,n zi − x∗ 2
− βi,n g( zi − p )
N
i=1 i=1
N N
β0,n
= β0,n p − x∗ 2
+ βi,n d(zi , PTi x∗ )2 − βi,n g( zi − p )
N
i=1 i=1
N N
∗ 2 β0,n
∗ 2
≤ β0,n p − x + βi,n H(PTi p, PTi x ) − βi,n g( zi − p )
N
i=1 i=1

N
∗ 2 β0,n
≤ p−x − βi,n g( zi − p ) .
N
i=1

By the properties of g, we can conclude that

zi = p, ∀i = 1, 2, · · · , N.

Hence p ∈ ∩N F (Ti ). This completes the proof.
i=1

(2) Let {vn } ⊂ D be a bounded sequence and an ∈ Sn vn be such that limn→∞ an −
vn = 0. Then there exists zi,n ∈ PTi vn , i = 1, 2, · · · , N such that an = β0,n vn +
N
i=1 βi,n zi,n . Since {vn } and {zi,n } are bounded, by Lemma 2.7, there exists a
continuous strictly increasing convex function g : [0, ∞) → [0, ∞) with g(0) = 0
such that
N
2 2
an − p = β0,n (vn − p) + βi,n (zi,n − p)
i=1
N N
2 2 β0,n
≤ β0,n vn − p + βi,n zi,n − p − βi,n g( zi,n − vn )
N
i=1 i=1
N N
2 β0,n
= β0,n vn − p + βi,n d(zi,n , PTi p)2 − βi,n g( zi,n − vn )
N
i=1 i=1
N N
2 β0,n
≤ β0,n vn − p + βi,n H(PTi vn , PTi p)2 − βi,n g( zi,n − vn )
N
i=1 i=1
N
2 β0,n
≤ vn − p − βi,n g( zi,n − vn ) , ∀p ∈ ∩N F (Ti ).
i=1
N
i=1

This implies that
N
β0,n
βi,n g( zi,n − vn ) ≤ vn − an vn − p + an − p , ∀p ∈ ∩N F (Ti ).
i=1
N
i=1


By our assumptions, we get limn→∞ g( zi,n − vn ) = 0 for all i = 1, 2, · · · , N . By
the properties of g, we can conclude that

lim zi,n − vn = 0, ∀i = 1, 2, · · · , N.
n→∞

Hence {Sn } satisfies the SC-condition.
(3) For p ∈ ∩N F (Ti ), we know that PTi p = {p} for every i ∈ {1, 2, ..., N }. Let
i=1
n ∈ N. Then for each x ∈ D and an ∈ PSn x, there exists zi,n ∈ PTi x such that
an = β0,n x + N βi,n zi,n . It follows that
i=1

N
an − p ≤ β0,n x − p + βi,n zi,n − p
i=1
N
= β0,n x − p + βi,n d(zi,n , PTi p)
i=1
N
≤ β0,n x − p + βi,n H(PTi x, PTi p)
i=1
≤ x−p . (4.2)

This implies that d(PSn x, p) ≤ an − p ≤ x − p . It follows from 4.2 that
supan ∈PSn x d(an , PSn p) ≤ x−p . Since PSn p = {p}, supp∈PSn p d(PSn x, p) ≤ x−p .
This shows that H(PSn x, PSn p) ≤ x − p for all x ∈ D and p ∈ ∩N F (Ti ). This
i=1
completes the proof.

Lemma 4.2. Let D be a closed and convex subset of a Banach space E. Let {Ti }N i=1
be a finite family of nonexpansive multivalued mappings of D into P (D) and, let
{βi,n }N be sequences in (0, 1) such that 0 < lim inf n→∞ βi,n ≤ lim supn→∞ βi,n < 1
i=0
for all i ∈ {0, 1, · · · , N } and N βi,n = 1 for all n ∈ N. For each n ∈ N, let Sn be
i=0
the mapping defined by (4.1). Assume that there exists i0 ∈ {1, 2, ..., N } such that
F (Ti0 ) = ∩N F (Ti ) = ∅ and I − Ti0 is closed. Then {Sn } satisfies Condition (A).
i=1

Proof. Since I − Ti0 is closed, it follows by Lemma 2.5 that Ti0 satisfies Condition I.
Then there is a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0
for r ∈ (0, ∞) such that

d(x, Ti0 x) ≥ f (d(x, F (Ti0 ))) = f (d(x, ∩N F (Ti ))
i=1

for all x ∈ D. This completes the proof.

Using Lemma 4.1, we have the following:


space E which satisfies Opial’s property. Let {Ti }N be a finite family of nonexpan-
i=1
sive multivalued mappings of D into P (D) with ∩N F (Ti ) = ∅ and, let {βi,n }N
i=1 i=0
be sequences in (0, 1) such that 0 < lim inf n→∞ βi,n ≤ lim supn→∞ βi,n < 1 for all
N
i ∈ {0, 1, · · · , N } and i=0 βi,n = 1 for all n ∈ N. For each n ∈ N, let Sn be
the mapping defined by (4.1). Assume that the best approximation operator PTi is
nonexpansive for all i ∈ {1, 2, ..., N }. Let {xn } be generated by

xn+1 ∈ αn xn + (1 − αn )PSn xn , n ≥ 1. (4.3)

If 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1, then the sequence {xn } converges
weakly to an element in ∩N F (Ti ).
i=1

Proof. Putting Tn = Sn for all n ≥ 1 in the Theorem 3.1, we obtain the desired
result.

Now, using Lemma 4.1 and Lemma 4.2, we have the following:

space E. Let {Ti }N be a finite family of nonexpansive multivalued mappings of D
i=1
N F (T ) = ∅ and, let {β }N be sequences in (0, 1) such that 0 <
into P (D) with ∩i=1 i i,n i=0
lim inf n→∞ βi,n ≤ lim supn→∞ βi,n < 1 for all i ∈ {0, 1, · · · , N } and N βi,n = 1
i=0
for all n ∈ N. For each n ∈ N, let Sn be the mapping defined by (4.1). Assume
that the best approximation operator PTi is nonexpansive for all i ∈ {1, 2, ..., N } and
there exists i0 ∈ {1, 2, ..., N } such that F (Ti0 ) = ∩N F (Ti ) with I − Ti0 is closed.
i=1
Let {xn } be generated by

xn+1 ∈ αn xn + (1 − αn )PSn xn , n ≥ 1. (4.4)

If 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1, then the sequence {xn } converges
strongly to an element in ∩N F (Ti ).
i=1

Proof. Putting Tn = Sn for all n ≥ 1 in the Theorem 3.3, we obtain the desired
result.

Acknowledgement. The authors would like to thank the referees for valuable sug-
gestion on the manuscript. The first and fourth authors supported by the Center of
Excellence in Mathematics, the Commission on Higher Education and the Graduate
School of Chiang Mai University for the financial support. The second author was


supported by the Royal Golden Jubilee Grant PHD/0261/2551 and the third au-
thor was supported by the Korea Research Foundation Grant funded by the Korean
Government (KRF-2008-313-C00050).

References

[1] M. Abbas, S. H. Khan, R. P. Agarwal, Common fixed points of two multival-
ued nonexpansive mappings by one-step iterative scheme, Appl. Math. Lett. (2010),
doi:10.1016/j.aml.2010.08.025.

[2] H. H. Bauschke, E. Matouˇkov´ and S. Reich, Projection and proximal point methods:
s a
convergence results and counterexamples, Nonlinear Anal. 56 (2004) 715-738.

[3] D. Boonchari, S. Saejung, Construction of common fixed points of a countable family
of λ-demicontractive mapping in arbitrary Banach spaces, Appl. Math. Comput. 216
(2010) 173-178.

[4] P. Cholamjiak, S. Suantai, Weak convergence theorems for a countable family of strict
pseudocontractions in Banach spaces, Fixed Point Theory Appl. Volume 2010, Article
ID 632137, 16 pages.

[5] W. Cholamjiak, S. Suantai, Weak and strong convergence theorems for a finite family
of generalized asymptotically quasi-nonexpansive mappings, Comput. Math. Appl. 60
(2010) 1917-1923.

[6] Y. J. Cho, S. M. Kang, H. Zhou, Some Conditions on Iterative Methods, Communica-
tions on Applied Nonlinear Analysis 12 (2005) 27-34.

[7] A. Genal, J. Lindenstrass, An example concerning fixed points, Israel J. Math. 22 (1975)
81-86.

[8] N. Hussain and A. R. Khan, Applications of the best approximation operator to *-
nonexpansive maps in Hilbert spaces, Numer. Funct. Anal. Optim., 24 (2003) 327-338.

[9] S. Itoh, W. Takahashi, Singlevalued mappings, multivalued mappings and fixed point
theorems, J. Math. Anal. Appl. 59 (1977) 514-521.

[10] W. A. Kirk, Transfinte methods in metric fixed point theory, Abstr. Appl. Anal. 2003
(5)(2003) 311-324.

[11] E. Lami Dozo, Multivalued nonexpansive mappings and Opial’s condition, Proc. Amer.
Math. Soc. 38 (1973) 286-292.


[12] T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976)
179-182.

[13] J. T. Markin, Continuous dependence of fixed point sets, Proc. Amer. Math. Soc. 38
(1973) 545-547.

[14] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506-
510.

[15] K. Nakajo, K. Shimoji, W. Takahashi, Strong convergence to common fixed points of
families of nonexpansive mappings in Banach spaces, J. Nonlinear and Convex Anal. 8
(2007) 11-34.

[16] Z. Opial, Weak convergence of the sequence of successive approximations for nonex-
pansive mappings, Bull. Amer. Math. Soc. 73 (1967) 591-597.

[17] B. Panyanak, Mann and Ishikawa iterative processes for multivalued mappings in Ba-
nach spaces, Comput. Math. Appl. 54 (2007) 872-877.

[18] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J.
Math. Anal. Appl. 67 (1979) 274-276.

[19] D. R. Sahu, Strong convergence theorems for nonexpansive type and non-self multi-
valued mappings, Nonlinear Anal. 37 (1999) 401-407.

[20] K. P. R. Sastry, G. V. R. Babu, Convergence of Ishikawa iterates for a multivalued
mappings with a fixed point, Czechoslovak Math. J. 55 (2005) 817-826.

[21] H. F. Senter, W. G. Dotson, Approximating fixed points of nonexpansive mappings,
Proc. Amer. Math. Soc. 44 (1974) 375380.

[22] N. Shahzad, H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued
maps in Banach spaces, Nonlinear Analysis 71 (2009) 838-844.

[23] C. Shiau, K. K. Tan, C. S. Wong, Quasi-nonexpansive multi-valued maps and selection,
Fund. Math. 87 (1975) 109-119.

[24] N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpan-
sive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997) 3641-3645.

[25] Y. Song, On a Mann type implicit iteration processes for continuous pseudo-contractive
mappings, Nonlinear Anal. 67 (2007) 3058-3063.

[26] Y. Song, Y. J. Cho, Iterative approximations for multivalued nonexpansive mappings
in reflexive Banach spaces, Math. Inequal. Appl. In press.


[27] Y. Song, S. Hu, Strong convergence theorems for nonexpansive semigroup in Banach
spaces, J. Math. Anal. Appl. 338 (2008) 152-161.

[28] Y. Song, H. Wang, Covergence of iterative algorithms for multivalued mappings in
Banach spaces, Nonlinear Anal. 70 (2009) 1547-1556].

[29] Y. Song, H. Wang, Erratum to ”Mann and Ishikawa iterative processes for multivalued
mappings in Banach spaces”[Comput. Math. Appl. 54 (2007) 872-877]. Comput. Math.
Appl. 55 (2008) 2999-3002.

[30] T. Suzuki, Strong convergence theorems for inﬁnite families of nonexpansive map-
pings in general Banach space, Fixed Point Theory Appl. 2005 (1)(2005) 103-123.
doi:10.1155/FPTA.2005.103.

[31] W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and its Applications,
Yokohama Publishers inc, Yokohama, 2000 (in Eiglish).

[32] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991)
1127-1138.

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