# Weak and strong convergence of hybrid subgradient method for pseudomonotone

```Wen Fixed Point Theory and Applications 2014, 2014:232
http://www.fixedpointtheoryandapplications.com/content/2014/1/232
RESEARCH
Open Access
Weak and strong convergence of hybrid
equilibrium problem and multivalued
nonexpansive mappings
Dao-Jun Wen*
*
Correspondence:
[email protected]
College of Mathematics and
Statistics, Chongqing Technology
400067, China
Abstract
In this paper, we introduce a hybrid subgradient method for ﬁnding a common
element of the set of solutions of a class of pseudomonotone equilibrium problems
and the set of ﬁxed points of a ﬁnite family of multivalued nonexpansive mappings in
Hilbert space. The proposed method involves only one projection rather than two as
in the existing extragradient method and the inexact subgradient method for an
equilibrium problem. We establish some weak and strong convergence theorems of
the sequences generated by our iterative method under some suitable conditions.
Moreover, a numerical example is given to illustrate our algorithm and our results.
MSC: 47H05; 47H09; 47H10
Keywords: pseudomonotone equilibrium problem; multivalued nonexpansive
mapping; hybrid subgradient method; ﬁxed point; weak and strong convergence
1 Introduction
Let H be a real Hilbert space with inner product ·, · and norm · , respectively. Let
K be a nonempty closed convex subset of H. Let F : K × K → R be a bifunction, where
R denotes the set of real numbers. We consider the following equilibrium problem: Find
x ∈ K such that
F(x, y) ≥ ,
∀y ∈ K.
(.)
The set of solution of equilibrium problem is denoted by EP(F, K). It is well known that
some important problems such as convex programs, variational inequalities, ﬁxed point
problems, minimax problems, and Nash equilibrium problem in noncooperative games
and others can be reduced to ﬁnding a solution of the equilibrium problem (.); see [–]
and the references therein.
Recall that a mapping T : K → K is said to be nonexpansive if
Tx – Ty ≤ x – y,
∀x, y ∈ K.
A subset K ⊂ H is called proximal if for each x ∈ H, there exists an element y ∈ K such that
dist(x, K) := x – y = inf x – z : z ∈ K .
medium, provided the original work is properly cited.
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We denote by B(K), C(K), and P(K) the collection of all nonempty closed bounded subsets,
nonempty compact subsets and nonempty proximal bounded subsets of K , respectively.
The Hausdorﬀ metric H on B(H) is deﬁned by
H(K , K ) := max sup dist(x, K ), sup dist(y, K ) ,
x∈K
∀K , K ∈ B(H).
y∈K
Let T : H → H be a multivalued mapping, of which the set of ﬁxed points is denoted
by Fix(T), i.e., Fix(T) := {x ∈ Tx : x ∈ K}. A multivalued mapping T : K → B(K) is said to
be nonexpansive if
H(Tx, Ty) ≤ x – y,
∀x, y ∈ K.
(.)
T is said to be quasi-nonexpansive if, for all p ∈ Fix(T),
H(Tx, p) ≤ x – p,
∀x ∈ K.
(.)
Recently, the problem of ﬁnding a common element of the set of solutions of equilibrium problems and the set of ﬁxed points of nonlinear mappings has become an attractive
subject, and various methods have been extensively investigated by many authors. It is
worth mentioning that almost all the existing algorithms for this problem are based on
the proximal point method applied to the equilibrium problem combining with a Mann
iteration to ﬁxed point problems of nonexpansive mappings, of which the convergence
analysis has been considered if the bifunction F is monotone. This is because the proximal
point method is not valid when the underlying operator F is pseudomonotone. Another
basic idea for solving equilibrium problems is the projection method. However, Facchinei
and Pang [] show that the projection method is not convergent for monotone inequality, which is a special case of monotone equilibrium problems. In order to obtain convergence of the projection method for equilibrium problems, Tran et al. [] introduced
an extragradient method for pseudomonotone equilibrium problems, which is computationally expensive because of the two projections deﬁned onto the constrained set. Eﬀorts
for deducing the computational costs in computing the projection have been made by using penalty function methods or relaxing the constrained convex set by polyhedral convex
ones; see, e.g., [–].
In , Santos and Scheimberg [] further proposed an inexact subgradient algorithm
for solving a wide class of equilibrium problems that requires only one projection rather
than two as in the extragradient method, and of which computational results show the
eﬃciency of this algorithm in ﬁnite dimensional Euclidean spaces. On the other hand, iterative schemes for multivalued nonexpansive mappings are far less developed than those
for nonexpansive mappings though they have more powerful applications in solving optimization problems; see, e.g., [–] and the references therein.
In , Eslamian [] considered a proximal point method for nonspreading mappings
and multivalued nonexpansive mappings and equilibrium problems. To be more precise,
they proposed the following iterative method:
F(un , z) + rn y – un , un – xn ≥ , ∀y ∈ K,
xn+ = αn un + βn fn un + γn zn , n ≥ ,
(.)
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where Tn = Tn(mod N) , zn ∈ Tn un , αn + βn + γn =  for all n ≥  and fi , Ti are ﬁnite families
of nonspreading mappings and multivalued nonexpansive mappings for i = , , . . . , N , respectively. Moreover, he further proved the weak and strong convergence theorems of the
iterative sequences under the condition of monotone deﬁned on a bifunction F.
In this paper, inspired and motivated by research going on in this area, we introduce
a hybrid subgradient method for the pseudomonotone equilibrium problem and a ﬁnite
family of multivalued nonexpansive mappings, which is deﬁned in the following way:
⎧
⎪
⎨ wn ∈ ∂n F(xn , ·)xn ,
un = PK (xn – γn wn ), γn = max{σβnn,wn } ,
⎪
⎩
xn+ = αn xn + ( – αn )zn , n ≥ ,
(.)
where Tn = Tn(mod N) , zn ∈ Tn un , and {αn }, {βn }, {n }, and {σn } are nonnegative real sequences.
Our purpose is not only to modify the proximal point iterative schemes (.) for the
equilibrium problem to a hybrid subgradient method for a class of pseudomonotone equilibrium problems and a ﬁnite family of multivalued nonexpansive mappings, but also to
establish weak and strong convergence theorems involving only one projection rather than
two as in the extragradient method [] and the inexact subgradient method [] for the
equilibrium problem. Our theorems presented in this paper improve and extend the corresponding results of [, , , ].
2 Preliminaries
Let K be a nonempty closed convex subset of a real Hilbert space H with inner product
·, · and norm · , respectively. For every point x ∈ H, there exists a unique nearest point
in K , denoted by PK (x), such that
x – PK (x) ≤ x – y,
∀y ∈ K.
Then PK is called the metric projection of H onto K . It is well known that PK is nonexpansive and satisﬁes the following properties:
x – PK (x), PK (x) – y ≥ , ∀x ∈ H, y ∈ K,
 
x – y ≥ x – PK (x) + y – PK (x) , ∀x ∈ H, y ∈ K.
(.)
(.)
Recall also that a bifunction F : K × K → R is said to be
(i) r-strongly monotone if there exists a number r >  such that
F(x, y) + F(y, x) ≤ –rx – y ,
∀x, y ∈ K;
(ii) monotone on K if
F(x, y) + F(y, x) ≤ ,
∀x, y ∈ K;
(iii) pseudomonotone on K with respect to x ∈ K if
F(x, y) ≥ 
⇒
F(y, x) ≤ ,
∀y ∈ K.
(.)
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It is clear that (i) ⇒ (ii) ⇒ (iii), for every x ∈ K . Moreover, F is said to be pseudomonotone
on K with respect to A ⊆ K , if it is pseudomonotone on K with respect to every x ∈ A.
When A ≡ K , F is called pseudomonotone on K .
The following example, taken from [], shows that a bifunction may not be pseudomonotone on K , but yet is pseudomonotone on K with respect to the solution set of
the equilibrium problem deﬁned by F and K :
F(x, y) := y|x|(y – x) + xy|y – x|,
∀x, y ∈ R,
K := [–, ].
Clearly, EP(F, K) = {}. Since F(y, ) =  for every y ∈ K , this bifunction is pseudomonotone on K with respect to the solution x∗ = . However, F is not pseudomonotone on K .
In fact, both F(–., .) = . >  and F(., –.) = . > .
To study the equilibrium problem (.), we may assume that is an open convex set
containing K and the bifunction F : × → R satisfy the following assumptions:
(C) F(x, x) =  for each x ∈ K and F(x, ·) is convex and lower semicontinuous on K ;
(C) F(·, y) is weakly upper semicontinuous for each y ∈ K on the open set ;
(C) F is pseudomonotone on K with respect to EP(F, K) and satisﬁes the strict
paramonotonicity property, i.e., F(y, x) =  for x ∈ EP(F, K) and y ∈ K implies
y ∈ EP(F, K);
(C) if {xn } ⊆ K is bounded and n →  as n → ∞, then the sequence {wn } with
wn ∈ ∂n F(xn , ·)xn is bounded, where ∂ F(x, ·)x stands for the -subdiﬀerential of
the convex function F(x, ·) at x.
Throughout this paper, weak and strong convergence of a sequence {xn } in H to x are
denoted by xn x and xn → x, respectively. In order to prove our main results, we need
the following lemmas.
Lemma . [] Let H be a real Hilbert space. For all x, y ∈ H, we have the following identity:
x – y = x – y – x – y, y.
Lemma . [] Let H be a real Hilbert space and α, β, γ ∈ [, ] with α + β + γ = . For
all x, y, z ∈ H, we have the following identity:
αx + βy + γ z = αx + βy + γ z – αβx – y
– αγ x – z – βγ y – z .
Lemma . [] Let {an } and {bn } be two sequences of nonnegative real numbers such that
an+ ≤ an + bn ,
where
∞
n= bn
n ≥ ,
< ∞. Then the sequence {an } is convergent.
Lemma . [] Let K be a nonempty closed convex subset of a real Hilbert space H. Let T :
K → C(K) be a multivalued nonexpansive mapping. If xn q and limn→∞ dist(xn , Txn ) =
, then q ∈ Tq.
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3 Weak convergence
Theorem . Let K be a nonempty closed convex subset of a Hilbert space H and F : K ×
: K → C(K) be a ﬁnite family of
K → R be a bifunction satisfying (C)-(C). Let {Ti }N
i=
Fix(T
multivalued nonexpansive mappings such that = N
i ) ∩ EP(F, K) = φ and Ti (q) =
i=
{q} for i = , , . . . , N and q ∈ . For a given point x ∈ K ,  < c < σn < σ , {αn }, {βn }, and {n }
are nonnegative sequences satisfying the following conditions:
(i) αn ∈ [a, b] ⊂ (, );
∞ 
∞
∞
(ii)
n= βn = ∞,
n= βn < ∞, and
n= βn n < ∞.
Then the sequence {xn } generated by (.) converges weakly to x ∈ .
Proof First, we show the existence of limn→∞ xn – p for every p ∈ . It follows from (.)
and Lemmas . and . that

xn+ – p = αn (xn – p) + ( – αn )(zn – p)
= αn xn – p + ( – αn )zn – p – αn ( – αn )xn – zn 
= αn xn – p + ( – αn ) dist(zn , Tn p) – αn ( – αn )xn – zn 
≤ αn xn – p + ( – αn )H(Tn un , Tn p) – αn ( – αn )xn – zn 
≤ αn xn – p + ( – αn )un – p – αn ( – αn )xn – zn 
= αn xn – p + ( – αn ) xn – p – un – xn  + xn – un , p – un – αn ( – αn )xn – zn 
≤ xn – p + ( – αn )xn – un , p – un – αn ( – αn )xn – zn  .
(.)
By un = PK (xn – γn wn ) and (.), we have
xn – un , p – un ≤ γn wn , p – un .
Using un = PK (xn – γn wn ) and xn ∈ K again, we obtain (note that γn =
(.)
βn
)
max{σn ,wn }
xn – un  = xn – un , xn – un ≤ γn wn , xn – un ≤ γn wn xn – un ≤ βn xn – un ,
which implies that xn – un ≤ βn . Substituting (.) into (.) yields
xn+ – p ≤ xn – p + ( – αn )γn wn , p – un – αn ( – αn )xn – zn 
= xn – p + ( – αn )γn wn , p – xn + ( – αn )γn wn , xn – un – αn ( – αn )xn – zn 
≤ xn – p + ( – αn )γn wn , p – xn + ( – αn )γn wn xn – un – αn ( – αn )xn – zn 
(.)
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≤ xn – p + ( – αn )γn wn , p – xn + ( – αn )βn
– αn ( – αn )xn – zn  .
(.)
Since wn ∈ ∂n F(xn , ·)xn and F(x, x) =  for all x ∈ K , we have
wn , p – xn ≤ F(xn , p) – F(xn , xn ) + n
≤ F(xn , p) + n .
(.)
On the other hand, since p ∈ EP(F, K), i.e., F(p, x) ≥  for all x ∈ K , by the pseudomonotonicity of F with respect to p, we have F(x, p) ≤  for all x ∈ K . Replacing x by xn ∈ K , we
get F(xn , p) ≤ . Then from (.) and (.), it follows that
xn+ – p ≤ xn – p + ( – αn )γn F(xn , p) + ( – αn )γn n + ( – αn )βn
– αn ( – αn )xn – zn 
≤ xn – p + ( – αn )γn n + ( – αn )βn – αn ( – αn )xn – zn 
≤ xn – p + ( – αn )γn n + ( – αn )βn .
(.)
Applying Lemma . to (.), by condition (ii), we obtain the existence of limn→∞ xn –
p = d.
Now, we claim that lim supn→∞ F(xn , p) =  for every p ∈ . Indeed, since F is pseudomonotone on K and F(p, xn ) ≥ , we have –F(xn , p) ≥ . From (.), we have
( – αn )γn –F(xn , p) ≤ xn – p – xn+ – p
+ ( – αn )γn n + ( – αn )βn .
(.)
Summing up (.) for every n, we obtain
≤
∞
( – αn )γn –F(xn , p)
n=
≤ x – p + 
∞
γn n + 
∞
n=
βn < +∞.
(.)
n=
By the assumption (C), we can ﬁnd a real number w such that wn ≤ w for every n.
Setting L := max{σ , w}, where σ is a real number such that  < σn < σ for every n, it follows
from (i) that
∞
≤
( – b) βn –F(xn , p)
L
n=
≤
∞
( – αn )γn –F(xn , p) < +∞,
n=
which implies that
∞
n=
βn –F(xn , p) < +∞.
(.)
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Combining with –F(xn , p) ≥  and ∞
n= βn = ∞, we can deduced that lim supn→∞ F(xn ,
p) =  as desired.
Next, we show that any weak subsequential limit of the sequence of {xn } is an element
of = N
i= Fix(Ti ) ∩ EP(F, K). To do this, suppose that {xni } is a subsequence of {xn }. For
simplicity of notation, without loss of generality, we may assume that xni x as i → ∞.
By convexity, K is weakly closed and hence x ∈ K . Since F(·, p) is weakly upper semicontinuous for p ∈ , we have
F(x, p) ≥ lim sup F(xni , p) = lim F(xni , p)
i→∞
i→∞
= lim sup F(xn , p) = .
(.)
n→∞
By the pseudomonotonicity of F with respect to p and F(p, x) ≥ , we obtain F(x, p) ≤ .
Thus F(x, p) = . Moreover, by the assumption (C), we can deduce that x is a solution of
EP(F, K). On the other hand, it follows from (.) and condition (ii) that
lim xn – un = .
(.)
n→∞
From (.) and conditions (i)-(ii), we have
αn ( – αn )xn – zn  ≤ xn – p – xn+ – p + ( – αn )γn n + ( – αn )βn ,
taking the limit as n → ∞ yields
lim xn – zn = ,
(.)
n→∞
and thus
lim dist(xn , Tn un ) ≤ lim xn – zn = .
n→∞
n→∞
(.)
Using (.) again, we have
lim xn+ – xn = lim ( – αn )xn – zn = .
n→∞
n→∞
(.)
It follows that
lim xn+i – xn = ,
n→∞
i = , , . . . , N.
(.)
Note that
un+ – un ≤ un+ – xn+ + xn+ – xn + xn – un .
Combining (.) and (.), we obtain
lim un+ – un = .
n→∞
(.)
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This also implies that
lim un+i – un = ,
i = , , . . . , N.
n→∞
(.)
Observe that
dist(un , Tn+i un ) ≤ un – xn + xn – xn+i + dist(xn+i , Tn+i un+i ) + H(Tn+i un+i , Tn+i un )
≤ un – xn + xn – xn+i + dist(xn+i , Tn+i un+i ) + un+i – un .
Together with (.), (.), (.), and (.), we have
lim dist(un , Tn+i un ) = ,
i = , , . . . , N,
n→∞
(.)
which implies that the sequence
N
dist(un , Tn+i un )
n≥
→  as n → ∞.
(.)
i=
For i = , , . . . , N , we note also that
dist(un , Ti un )
n≥
= dist(un , Tn+(i–n) un ) n≥
= dist(un , Tn+in un ) n≥
⊂
N
dist(un , Tn+i un )
n≥
,
i=
where i – n = in (mod N) and in ∈ {, , . . . , N}. Therefore, we have
lim dist(un , Ti un ) = ,
n→∞
i = , , . . . , N.
(.)
Similarly, for i = , , . . . , N , we obtain
dist(xn , Ti xn ) ≤ xn – un + dist(un , Ti un ) + H(Ti un , Ti xn )
≤ xn – un + dist(un , Ti un ).
It follows from (.) and (.) that
lim dist(xn , Ti xn ) = ,
n→∞
i = , , . . . , N.
(.)
Applying Lemma . to (.), we can deduce that x ∈ Fix(Ti ) for i = , , . . . , N and hence
x ∈ .
Finally, we prove that {xn } converges weakly to an element of . Indeed to verify that
the claim is valid it is suﬃcient to show that ωw (xn ) is a single point set, where ωw (xn ) =
{x ∈ H : xni x} for some subsequence {xni } of {xn }. Indeed since {xn } is bounded and H
is reﬂexive, ωw (xn ) is nonempty. Taking w , w ∈ ωw (xn ) arbitrarily, let {xnk } and {xnj } be
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subsequences of {xn } such that xnk w and xnj w , respectively. Since limn→∞ xn – p
exists for all p ∈ and w , w ∈ , we see that limn→∞ xn – w and limn→∞ xn – w exist. Now let w = w , then by Opial’s property,
lim xn – w = lim xnk – w n→∞
k→∞
< lim xnk – w = lim xn – w k→∞
n→∞
= lim xnj – w < lim xnj – w j→∞
j→∞
= lim xn – w ,
n→∞
which is a contradiction. Therefore, w = w . This shows that ωw (xn ) is a single point set,
i.e., xn x. This completes the proof.
Theorem . Let K be a nonempty closed convex subset of a Hilbert space H and F :
K × K → R be a bifunction satisfying (C)-(C). Let T : K → C(K) be a multivalued nonexpansive mapping such that = Fix(T) ∩ EP(F, K) = φ and T(q) = {q} for all q ∈ . For a
given point x ∈ K ,  < c < σn < σ , let {xn } be deﬁned by
⎧
⎪
⎨ wn ∈ ∂n F(xn , ·)xn ,
un = PK (xn – γn wn ), γn = max{σβnn,wn } ,
⎪
⎩
xn+ = αn xn + ( – αn )zn , n ≥ ,
where zn ∈ Tun , {αn }, {βn }, and {n } are nonnegative sequences satisfying the following conditions:
(i) αn ∈ [a, b] ⊂ (, );
∞
∞ 
∞
(ii)
n= βn = ∞,
n= βn < ∞, and
n= βn n < ∞.
Then the sequence {xn } converges weakly to x ∈ .
Proof Putting N = , then Ti = T, a single multivalued nonexpansive mapping, and the
conclusion follows immediately from Theorem .. This completes the proof.
4 Strong convergence
To obtain strong convergence results, we either add the control condition limn→∞ αn =  ,
or we remove the condition T(q) = {q} for all q ∈ and adjust the nonempty compact
subset C(K) to a proximal bounded subset P(K) of K as follows.
Theorem . Let K be a nonempty closed convex subset of a Hilbert space H and F : K ×
K → R be a bifunction satisfying (C)-(C). Let {Ti }N
: K → C(K) be a ﬁnite family of
i=
multivalued nonexpansive mappings such that = N
Fix(T
i ) ∩ EP(F, K) = φ and Ti (q) =
i=
{q} for i = , , . . . , N and q ∈ . For a given point x ∈ K ,  < c < σn < σ , let {xn } be deﬁned
by
⎧
⎪
⎨ wn ∈ ∂n F(xn , ·)xn ,
un = PK (xn – γn wn ), γn = max{σβnn,wn } ,
⎪
⎩
xn+ = αn xn + ( – αn )zn , n ≥ ,
(.)
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where Tn = Tn(mod N) , zn ∈ Tn un , {αn }, {βn }, and {n } are nonnegative sequences satisfying
the following conditions:
(i) αn ∈ [a, b] ⊂ (, ) and limn→∞ αn =  ;
∞ 
∞
∞
(ii)
n= βn = ∞,
n= βn < ∞, and
n= βn n < ∞.
Then the sequence {xn } generated by (.) converges strongly to x∗ ∈ .
Proof By a similar argument to the proof of Theorem . and (.), we have
zn – P
(xn ) ≤ zn – xn  – xn – P
(xn ) .
(.)
It follows from (.) that
xn+ – P
(xn+ ) ≤ αn xn – P
(xn ) + ( – αn ) zn – P
(xn ) 


≤ αn xn – P
(xn ) + ( – αn )zn – P
(xn )

≤ (αn – )xn – P
(xn ) + ( – αn )zn – xn  .
(.)
Combining (.), limn→∞ αn =  , and the boundedness of the sequence {xn – P
(xn )}, we
obtain
lim xn+ – P
(xn+ ) = .
(.)
n→∞
By the assumptions (C) and (C), the set is convex (see the proof of Theorem  in []).
For all m > n, we have  (P
(xm ) + P
(xn )) ∈ , and therefore
P
(xm ) – P
(xn ) = xm – P
(xm ) + xm – P
(xn )


– xm – P
(xm ) + P
(xn ) 



≤ xm – P
(xm ) + xm – P
(xn ) – xm – P
(xm )


= xm – P
(xn ) – xm – P
(xm ) .
(.)
Using (.) with p = P
(xn ), we have
xm – P
(xn ) ≤ xm– – P
(xn ) + ( – αm– )γm– m– + ( – αm– )β 
m–

≤ xm– – P
(xn ) + ηm– + ηm–
≤ ···
m–
 ≤ xn – P
(xn ) +
ηj ,
(.)
j=n
where ηj = ( – αj )γj j + ( – αj )βj . It follows from (.) and (.) that
m–

P
(xm ) – P
(xn ) ≤ xn – P
(xn ) + 
ηj – xm – P
(xm ) .
j=n
(.)
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Together with (.) and m–
j=n ηj < ∞, this implies that {P
(xn )} is a Cauchy sequence.
Hence {P
(xn )} strongly converges to some point x∗ ∈ . Moreover, we obtain
x∗ = lim P
(xni ) = P
(x) = x,
i→∞
(.)
which implies that P
(xn ) → x∗ = x ∈ . Then, from (.), (.), and (.), we can con
clude that xn → x∗ . This completes the proof.
Theorem . Let K be a nonempty closed convex subset of a Hilbert space H and F : K ×
K → R be a bifunction satisfying (C)-(C). Let {Ti }N
i= : K → P(K) be a ﬁnite family of
multivalued mappings such that PTi is nonexpansive, where PTi := {y ∈ Ti x : dist(x, Ti x) =
x – y} and = N
i= Fix(Ti ) ∩ EP(F, K) = φ. For a given point x ∈ K ,  < c < σn < σ , let
{xn } be deﬁned by
⎧
⎪
⎨ wn ∈ ∂n F(xn , ·)xn ,
un = PK (xn – γn wn ), γn = max{σβnn,wn } ,
⎪
⎩
xn+ = αn xn + ( – αn )zn , n ≥ ,
(.)
where Tn = Tn(mod N) , zn ∈ PTn un , {αn }, {βn }, and {n } are nonnegative sequences satisfying
the following conditions:
(i) αn ∈ [a, b] ⊂ (, );
∞ 
∞
∞
(ii)
n= βn = ∞,
n= βn < ∞, and
n= βn n < ∞.
Then the sequence {xn } converges strongly to x∗ ∈ .
Proof Taking p ∈ , then PTn (p) = {p}. By substituting PT instead of T and similar argument as (.) in the proof of Theorem . we obtain
lim dist xn , Ti (xn ) ≤ lim dist xn , PTi (xn ) = .
n→∞
n→∞
(.)
By compactness of K , there exists a subsequence {xnk } of {xn } such that limk→∞ xnk = x∗ ,
for some x∗ ∈ K . Since PTi is nonexpansive for i = , , . . . , N , we have
dist x∗ , Ti x∗ ≤ dist x∗ , PTi x∗
≤ x∗ – xnk + dist xnk , PTi (xnk ) + H PTi (xnk ), PTi x∗
≤ x∗ – xnk + dist xnk , PTi (xnk ) .
(.)
It follows from (.) and (.) that
lim dist x∗ , Ti x∗ = ,
k→∞
(.)
∗
which implies that x∗ ∈ N
i= Fix(Ti ). Since {xnk } converges strongly to x and limn→∞ xn –
∗
x exists (as in the proof of Theorem .), we ﬁnd that {xn } converges strongly to x∗ . This
completes the proof.
In addition, we supply an example and numerical results to illustrate our method and
the main results of this paper.
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Example . Let H = R and K := [, ] with usual metric. Consider the nonsmooth equilibrium problem deﬁned by the bifunction
F(x, y) = xy(y – x) + xy|y – x|,
∀x, y ∈ K.
Clearly, F is pseudomonotone on K . Note that F(x, ·) is convex for x ∈ K and ∂F(x, ·)x =
[x , x ] by taking n =  for all n ∈ N.
(i) Let Tx := [ x , x ] deﬁned on K := [, ]. Note that T is a multivalued nonexpansive
mapping and Fix(T) ∩ EP(F, K) = {}. Setting N = , σn = , αn =  , βn = n , and xn – x∗ ≤
– as stop criteria, we obtain the results of algorithm (.) with diﬀerent initial points in
Table .
(ii) Let Tx := [–x, –x] deﬁned on [, ∞) → R . Note that T is not nonexpansive but
PT x = {–x} is nonexpansive for all x ∈ [, ∞). Indeed, for each u ∈ Tx, u = –ax,  ≤ a ≤ ,
choose v = –ay. Then
|u – v| = –ax – (–ay) = a|x – y| ≤ |x – y| = H(Tx, Ty).
On the other hand, for any x, we have  ∈ [, ∞) and T = {}. It follows that Fix(T) ∩
EP(F, K) = {}. Setting N = , σn = , αn =  , βn = n , and xn – x∗ ≤ – as stop criteria,
we obtain the results of algorithm (.) with diﬀerent initial points to be found in Table .
The computations are performed by Matlab Ra running on a PC Desktop Intel(R)
Core(TM) i-M, CPU @. GHz,  MHz, . GB,  GB RAM.
Remark . Our hybrid subgradient method improves the extragradient method of Tran
et al. [] and the inexact subgradient algorithm of Santos and Scheimberg [] for an equilibrium problem in deducing the computational costs of an iterative process.
Table 1 Numerical results for an initial point x0 = 0.2, 0.5, 0.8
Iter. (n)
xn(1)
xn(2)
xn(3)
0
1
2
3
4
5
6
0.2000
0.1459
0.0816
0.0314
0.0027
0.0000
0.0000
0.5000
0.3618
0.2157
0.1031
0.0351
0.0059
0.0000
0.8000
0.4782
0.2391
0.1206
0.0524
0.0093
0.0001
Table 2 Numerical results for an initial point x0 = 0.2, 0.5, 0.8
xn(1)
xn(2)
xn(3)
0
1
2
3
4
5
6
..
.
0.2000
0.1683
0.1247
0.0925
0.0621
0.0319
0.0042
..
.
0.5000
0.4136
0.3914
0.2518
0.1492
0.0991
0.0427
..
.
0.8000
0.6839
0.5284
0.3855
0.2679
0.1732
0.1043
..
.
11
12
0.0000
0.0000
0.0035
0.0001
0.0086
0.0001
Iter. (n)
Wen Fixed Point Theory and Applications 2014, 2014:232
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Remark . Our results generalize the results of Eslamian [], a proximal point method
for an equilibrium problem, to a hybrid subgradient method for a pseudomonotone equilibrium problem.
Competing interests
The author declares to have no competing interests.
Acknowledgements
The author is grateful to the anonymous referees for valuable remarks suggestions which helped him very much in
improving this manuscript. This work was supported by the National Science Foundation of China (11471059, 11271388),
Basic and Advanced Research Project of Chongqing (cstc2014jcyjA00037) and Science and Technology Research Project
of Chongqing Municipal Education Commission (KJ1400618).
Received: 23 April 2014 Accepted: 30 October 2014 Published: 17 Nov 2014
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