Theoretical Mathematics & Applications, vol.3, no.3, 2013, 49-61
ISSN: 1792-9687 (print), 1792-9709 (online)
Scienpress Ltd, 2013
Convergence theorems for a finite family
of nonspreading and nonexpansive
multivalued mappings and equilibrium
problems with application
Hong Bo Liu1
Abstract
We introduce a finite family of nonspreading multivalued mappings
and a finite family of nonexpansive multivalued mappings in Hilbert
space. We establish some weak and strong convergence theorems of
the sequences generated by our iterative process. Some new iterative
sequences for finding a common element of the set of solutions of an
equilibrium problem and the set of common fixed points it was introduced. The results improve and extend the corresponding results of
Mohammad Eslamian (Optim. Lett., 7(3):547-557, 2012).
Mathematics Subject Classification: 47J05; 47H09; 49J25
Keywords: Equilibrium problem; onspreading mapping multivalued; nonexpansive multivalued mapping; common fixed point
1
School of Science, Southwest University of Science and Technology, Mianyang,
Sichuan 621010, P. R. China.
Article Info: Received : May 17, 2013. Revised : July 10, 2013
Published online : September 1, 2013
50
1
Convergence theorems for a finite family of nonspreading
Introduction
Throughout this paper, we denote by N and R the sets of positive integers
and real numbers, respectively. Let E be a nonempty closed subset of a real
Hilbert space H. Let N (E) and CB(E) denote the family of nonempty subsets
and nonempty closed bounded subsets of E, respectively. The Hausdorff metric
on CB(E) is defined by
H(A1 , A2 ) = max{ sup d(x, A2 ), sup d(y, A1 )}
x∈A1
y∈A2
for A1 , A2 ∈ CB(E), where d(x, A1 ) = inf{||x − y||, y ∈ A1 }. An element
p ∈ E is called a fixed point of T : E → N (E) if p ∈ T (p). The set of fixed
points of T is represented by F (T ).
A subset E ⊂ H is called proximal if for each x ∈ H, there exists an
element y ∈ H such that
kx − yk = d(x.E) = inf{kx − zk : z ∈ E}.
We denote by K(E) and P (E) nonempty compact subsets and nonempty proximal bounded subsets of E, respectively.
The multi-valued mapping T : E → CB(E) is called nonexpansive if
H(T x, T y) ≤ ||x − y||,
f or
∀x, y ∈ E.
The multi-valued mapping T : E → CB(E) is called quasi-nonexpansive if
F (T ) 6= ∅ and
H(T x, T p) ≤ ||x − p||,
f or
∀x ∈ E
p ∈ F (T ).
Iterative process for approximating fixed points (and common fixed points)
of nonexpansive multivalued mappings have been investigated by various authors (see[1], [2], [3]).
Recently, Kohsaka and Takahashi (see[4], [5])introduced an important class
of mappings which they called the class of nonspreading mappings. Let E be a
subset of Hilbert space H, then they called a mapping T : E → E nonspreading
if
2kT x − T yk2 ≤ kT x − yk2 + kT y − xk2 ,
∀x, y ∈ E.
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H. Liu
Lemoto and Takahashi [6] proved that T : E → E is nonspreading if and only
if
kT x − T yk2 ≤ kx − yk2 + 2hx − T y, y − T yi,
∀x, y ∈ E.
Now, inspired by [4] and [5], we propose a definition as follows,
Definition 1.1. The multivalued mapping T : E → CB(E) is called nonspreading if
2kux − uy k2 ≤ kux − yk2 + kuy − xk2 ,
(1)
for ux ∈ T x, uy ∈ T y, ∀x, y ∈ E.
By Takahashi [6], We get also the multivalued mapping T : E → CB(E)
is nonspreading if and only if
kux − uy k2 ≤ kx − yk2 + 2hx − ux , y − uy i,
for
(2)
ux ∈ T x, uy ∈ T y, ∀x, y ∈ E. In fact,
2kux − uy k2 ≤ kux − yk2 + kuy − xk2
⇔ 2kux − uy k2 ≤ kux − xk2 + 2hux − x, x − yi + kx − yk2
+kux − xk2 + 2hx − ux , ux − uy i + kux − uy k2
⇔ 2kux − uy k2 ≤ 2kux − xk2 + kx − yk2 + kux − uy k2
+2hux − x, x − ux − (y − uy )i
⇔ kux − uy k2 ≤ kx − yk2 + 2hx − ux , y − uy i.
The equilibrium problem for φ : E × E → R is to find x ∈ E such that
φ(x, y) ≥ 0, ∀y ∈ E. The set of such solution is denoted by EP (φ). Given a
mapping T : E → CB(E), let φ(x, y) = hx, yi for all y ∈ E. The x ∈ EP (φ)
if and only if x ∈ E is a solution of the variational inequality hT x, yi ≥ 0 for
all y ∈ E.
Numerous problems in physics, optimization, and economics can be reduced to find a solution of the equilibrium problem. Some methods have been
proposed to solve the equilibrium problem. For instance, see Blum and Oettli
[7], Combettes and Hirstoaga [8], Li and Li [9], Giannessi, Maugeri, and Pardalos [10], Moudafi and Thera [11] and Pardalos,Rassias and Khan [12]. In the
52
Convergence theorems for a finite family of nonspreading
recent years, the problem of finding a common element of the set of solutions of
equilibrium problems and the set of fixed points of singlevalued nonexpansive
mappings in the framework of Hilbert spaces has been intensively studied by
many authors.
In this paper, inspired by [13], we propose an iterative process for finding
a common element of the set of solutions of equilibrium problem and the
set of common fixed points of a countable family of nonexpansive multivalued
mappings and a countable family of nonspreading multivalued mappings in the
setting of real Hilbert spaces. We also prove the strong and weak convergence
of the sequences generated by our iterative process. The results presented in
the paper improve and extend the corresponding results in [15].
2
Preliminaries
In the sequel, we begin by recalling some preliminaries and lemmas which
will be used in the proof. Let H be a real Hilbert spaces with the norm k · k.
It is also known that H satisfies Opial, s condition, i.e., for any sequence xn
with xn * x, the inequality
lim inf kxn − xk < lim inf kxn − yk
n→∞
n→∞
holds for every y ∈ H with y 6= x.
Lemma 2.1. (see[16]) Let H be a real Hilbert space, then the equality
kx − yk2 = kxk2 − kyk2 − 2hx − y, yi
holds for all x, y ∈ H
Lemma 2.2. (see[17]) Let H be a real Hilbert space. Then for all x, y, z ∈ H
and α, β, γ ∈ [0, 1] with α + β + γ = 1, we have
kαx + βy + γzk2 = αkxk2 + βkyk2 + γkzk2 − αβkx − yk2
−βγky − zk2 − γαkz − xk2 .
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H. Liu
For solving the equilibrium problem, we assume that the bifunction φ :
E × E → R satisfies the following conditions for all x, y, z ∈ E:
(A1 ) φ(x, x) = 0,
(A2 ) φ is monotone, i.e., φ(x, y) + φ(y, x) ≤ 0,
(A3 ) φ is upper-hemicontinuous, i.e.lim supt→0+ φ(tz + (1 − t)x, y) ≤ φ(x, y),
(A4 ) φ(x, .) = 0 is convex and lower semicontinuous.
Lemma 2.3. (see[7]) Let E be a nonempty closed convex subset of H and
let φ be a bifunction satisfying (A1 ) − (A4 ). Let r > 0 and x ∈ H. Then, there
existsz ∈ E such that
1
φ(z, y) + hy − z, z − xi ≥ 0,
r
∀y ∈ E.
The following Lemma is established in [8].
Lemma 2.4. Let φ be a bifunction satisfying (A1 ) − (A4 ). For r > 0 and
x ∈ H, define a mapping Tr : H → E as follows
1
Tr x = {z ∈ E : φ(z, y) + hy − z, z − xi ≥ 0,
r
∀y ∈ E}.
Then, the following results hold
(a) Tr is single valued;
(b) Tr is firmly nonexpansive,i.e.,for any x, y ∈ H,kTr x − Tr yk2 ≤ hTr x −
Tr y, x − yi;
(c) F (Tr ) = EP (φ);
(d) EP (φ) is closed and convex.
The following Lemma is established in [3].
Lemma 2.5. Let E be a nonempty closed convex subset of a real Hilbert
space H. Let T : E → K(E) be a nonexpansive multivalued mapping. If
xn * p and limn→∞ d(xn , T xn ) = 0, then p ∈ T p.
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Convergence theorems for a finite family of nonspreading
Lemma 2.6. Let E be a nonempty closed convex subset of a real Hilbert
space H. Let T : E → E be a nonspreading multivalued mapping such that
F (T ) 6= ∅. Then S is demiclosed, i.e., xn * p and limn→∞ d(xn , T xn ) = 0, we
have p ∈ T p.
Proof. Let {xn } ⊂ E be a sequence such that xn * p and
lim d(xn , T xn ) = 0.
n→∞
¯ T p. From
Then the sequences {xn } and T xn are bounded. Suppose that p∈
,
Definition 1.1 and Opial s condition, we have
lim inf kxn − pk2
n→∞
< lim inf kxn − up k2
n→∞
= lim inf (kxn − uxn k2 + kuxn − up k2 + 2hxn − uxn , uxn − up i)
n→∞
≤ lim inf (kxn − uxn k2 + kxn − pk2 + 2hxn − uxn , p − up i
n→∞
+2hxn − uxn , uxn − up i)
= lim inf kxn − pk2 ,
n→∞
where up ∈ T p and uxn ∈ T xn . This is a contradiction. Hence we get the
conclusion.
3
Main results
Theorem 3.1. Let E be a nonempty closed convex subset of a real Hilbert
space H, φ be a bifunction of E × E into R satisfying (A1 ) − (A2 ). Let for
i = 1, 2, 3, · · · N , fi : E → E be a finite family of nonspreading multivalued
mappings and Ti : E → K(E) be a finite family of nonexpansive multivalued
T
T
T
F (fi ) EP (φ) 6= ∅ and Ti p = {p}
mappings. Assume that F = N
i=1 F (Ti )
for each p ∈ F. Let {xn } and {wn } be sequences generated initially by an
arbitrary element x1 ∈ E and

φ(w , y) + 1 hy − w , w − x i ≥ 0, ∀y ∈ E,
n
n
n
n
rn
n
xn+1 = αn wn + βn u + γn zn , ∀n ≥ 1,
n
where uin ∈ fi wn ,zn ∈ Tn wn , Tn = TN (mod(N )) and
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H. Liu
(i) {αn }, {βn }, {γn } ⊂ [a, 1] ⊂ (0, 1),
(ii) αn + βn + γn = 1,∀n ≥ 1,
(iii) rn > 0,and lim inf n→∞ rn > 0.
Then, the sequences {xn } and {wn } converge weakly to an element of F.
Proof. (I) First, We claim that limn→∞ kxn − qk exist for all q ∈ F.
Indeed, let q ∈ F, then from the definition of Tr in lemma 2.4,we have wn =
Trn xn and therefore
kwn − qk = kTrn xn − Trn qk ≤ kxn − qk ∀n ≥ 1.
By Lemma 2.2 and Since for each i,fi is nonspearding multivalued mappings,
we have
kxn+1 − qk2
= kαn wn + βn unn + γn zn − qk2
≤ αn kwn − qk2 + βn kunn − qk2 + γn kzn − qk2 − αn βn kwn − unn k2
−αn γn kwn − zn k2
= αn kwn − qk2 + βn kunn − qk2 + γn d2 (zn , Tn q) − αn βn kwn − unn k2
−αn γn kwn − zn k2
≤ αn kwn − qk2 + βn kunn − qk2 + γn H 2 (Tn wn , Tn q) − αn βn kwn − unn k2
−αn γn kwn − zn k2
≤ αn kwn − qk2 + βn kwn − qk2 + γn kwn − qk2 − αn βn kwn − unn k2
−αn γn kwn − zn k2
= kwn − qk2 − αn βn kwn − unn k2 − αn γn kwn − zn k2
≤ kxn − qk2 − αn βn kwn − unn k2 − αn γn kwn − zn k2 .
(3)
This implies that kxn+1 − qk ≤ kxn − qk ,hence we get limn→∞ kxn − qk exist
for all q ∈ F.
(II) Next, we prove that limn→∞ kwn − wn+i k = 0, for all i ≥ 1.
From (3), we have
αn βn kwn − unn k2 ≤ kxn − qk2 − kxn+1 − qk2 .
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Convergence theorems for a finite family of nonspreading
It follows from our assumption that
a2 kwn − unn k2 ≤ kxn − qk2 − kxn+1 − qk2 .
Taking the limit as n → ∞, we have limn→∞ kwn − unn k = 0. Observing (3)
again and our assumption, we have
a2 kwn − zn k2 ≤ kxn − qk2 − kxn+1 − qk2 .
Taking the limit as n → ∞, we get limn→∞ kwn − zn k = 0. Thus
lim d(wn , Tn wn ) ≤ lim kwn − zn k = 0.
(4)
lim kxn+1 − wn k ≤ lim βn kwn − unn k + lim γn kwn − zn k = 0.
(5)
n→∞
n→∞
Also we have
n→∞
n→∞
n→∞
Since wn = Trn xn , we have
kwn − qk2 = kTrn xn − Trn qk2
≤ hTrn xn − Trn q, xn − qi
= hwn − q, xn − qi
1
=
(kwn − qk2 + kxn − qk2 − kxn − wn k2 ),
2
and hence
kwn − qk2 ≤ kxn − qk2 − kxn − wn k2 .
(6)
Using (3) and the last inequality (6), we have
kxn+1 − qk2 ≤ kxn − qk2 − kxn − wn k2
and hence
kxn − wn k2 ≤ kxn − qk2 − kxn+1 − qk2 .
This imply that
lim kxn − wn k = 0.
(7)
lim kwn − wn+1 k ≤ lim (kxn+1 − wn+1 k + kxn+1 − wn k) = 0.
(8)
n→∞
Using (5) and (7), we get
n→∞
n→∞
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H. Liu
It follows that
lim kwn − wn+i k = 0,
∀i ∈ {1, 2, 3, · · · N }.
n→∞
(9)
Applying (7)-(9), we obtain that
lim kxn+1 − xn k = 0.
(10)
n→∞
This also implies that
lim kxn+i − xn k = 0,
∀i ∈ {1, 2, 3, · · · N }.
n→∞
(11)
(III) We claim that for i ∈ {1, 2, · · · N },
lim d(wn , Ti wn ) = lim d(wn , fi wn ) = 0
n→∞
n→∞
lim d(xn , Ti xn ) = lim d(xn , fi xn ) = 0.
n→∞
n→∞
Observing that for i ∈ N ,
lim d(wn , Tn+i wn )
n→∞
≤ lim [kwn − wn+i k + d(wn+i , Tn+i wn+i ) + H(Tn+i wn+i , Tn+i wn )]
n→∞
≤ lim [2kwn − wn+i k + d(wn+i , Tn+i wn+i )] = 0,
n→∞
which implies that the sequence
N
[
{d(wn , Tn+i wn )} → 0,
i=1
we have
lim d(wn , Ti wn ) = lim d(wn , Tn+(i−n) wn ) = 0.
n→∞
n→∞
Also we have
lim d(wn , fn+i wn )
n→∞
≤ lim [kwn − wn+i k + d(wn+i , fn+i wn+i ) + H(fn+i wn+i , fn+i wn )]
n→∞
n+i
n+i
≤ lim [kwn − wn+i k + kwn+i − un+i
n+i k + kun+i − un k]
n→∞
≤ lim [kwn − wn+i k + kwn+i − un+i
n+i k
n→∞
1
n+i
+(kwn − wn+i k2 + 2|hwn+i − un+i
, wn − unn+i i|) 2 = 0
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Convergence theorems for a finite family of nonspreading
which imply that
lim d(wn , fi wn ) = 0, ∀i ∈ {1, 2, · · · N }.
n→∞
Hence for ∀i ∈ {1, 2, · · · N }, we have
lim d(xn , Ti xn ) ≤
n→∞
≤
lim [kxn − wn k + d(wn , Ti wn ) + H(Ti wn , Ti xn )]
n→∞
lim [2kxn − wn k + d(wn , Ti xn )] = 0,
n→∞
and
lim d(xn , fi xn ) ≤
n→∞
≤
lim [kxn − wn k + d(wn , fi wn ) + H(fi wn , fi xn )]
n→∞
lim [kxn − wn k + d(wn , fi wn )
n→∞
1
+(kwn − xn k2 + 2|hwn − uin , xn − uin i|) 2 = 0.
(IV) Now we prove that ωw (xn ) ⊂ F where
ωw (xn ) = {x ∈ H : xni * x, {xni } ⊂ {xn }}.
Since {xn } is bounded and H is reflexive,ωw (xn ) is nonempty. Let µ ∈
ωw (xn ) is be an arbitrary element. Then there exists a subsequence {xni ⊂
{xn } converging weakly to µ. From (3.5) we get that wni * µ as i → ∞. We
show that µ ∈ EP (φ). Since wn = Trn xn , we have
φ(wn , y) +
1
hy − wn , wn − xn i ≥ 0,
rn
∀y ∈ E.
By (A4 ), we have
1
hy − wn , wn − xn i ≥ φ(y, wn ),
rn
and hence
hy − wni ,
wni − xni
i ≥ φ(y, wni ).
rni
Therefore, we have φ(y, µ) ≤ 0 for all y ∈ E.
For t ∈ (0, 1], let yt = ty + (1 − t)µ.Since y, µ ∈ E and E is convex, we
have yt ∈ E and hence φ(yt , µ) ≤ 0. Form (A1 ), (A4 ) we have
φ(yt , yt ) = φ(yt , ty+ (1 − t)µ) ≤ tφ(yt , y) + (1 − t)φ(yt , µ) ≤ tφ(yt , y),
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H. Liu
which gives φ(yt , y) ≥ 0. From (A3 ) we have φ(µ, y) ≥ 0, and hence µ ∈ EP (φ).
T
T
F (fi ). Therefore ωw (xn ) ⊂
By Lemma 2.5 and 2.6, we have µ ∈ N
i=1 F (Ti )
F.
(V) We show that {xn } and {wn } converge weakly to an element of F.
To verify that the aberration is valid, it is sufficient to show that ωw (xn ) is
a single point set. Let µ1 , µ2 ∈ ωw (xn ), and {xnk }, {xnm } ⊂ {xn }, such that
xnk → µ1 , xnm → µ2 . Since limn→∞ kxn −qk exists for all q ∈ F, and µ1 , µ2 ∈ F,
we have limn→∞ kxn − µ1 k and limn→∞ kxn − µ2 k. Now let µ1 6= µ2 , then by
Opial, s property,
lim kxn − µ1 k =
n→∞
<
=
=
<
=
lim kxnk − µ1 k
k→∞
lim kxnk − µ2 k
k→∞
lim kxn − µ2 k
n→∞
lim kxnm − µ2 k
m→∞
lim kxnm − µ1 k
m→∞
lim kxn − µ1 k,
n→∞
which is a contradiction. Therefore µ1 = µ2 , This show that ωw (xn ) is a single
point set. This complete the proof of Theorem 3.1.
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