Ceng et al. Journal of Inequalities and Applications 2014, 2014:462
http://www.journalofinequalitiesandapplications.com/content/2014/1/462
RESEARCH
Open Access
On a ﬁnite family of variational inclusions
with the constraints of generalized mixed
equilibrium and ﬁxed point problems
Lu-Chuan Ceng1,2 , Chi-Ming Chen3 and Chin-Tzong Pang4,5*
*
Correspondence:
[email protected]
4
Department of Information
Management, Yuan Ze University,
Chung-Li, 32003, Taiwan
5
Innovation Center for Big Data and
Digital Convergence, Yuan Ze
University, Chung-Li, 32003, Taiwan
Full list of author information is
available at the end of the article
Abstract
In this paper, we introduce two iterative algorithms for ﬁnding common solutions of a
ﬁnite family of variational inclusions for maximal monotone and inverse-strongly
monotone mappings with the constraints of two problems: a generalized mixed
equilibrium problem and a common ﬁxed point problem of an inﬁnite family of
nonexpansive mappings and an asymptotically strict pseudocontractive mapping in
the intermediate sense in a real Hilbert space. We prove some strong and weak
convergence theorems for the proposed iterative algorithms under suitable
conditions.
MSC: Primary 49J30; 47H09; secondary 47J20; 49M05
Keywords: generalized mixed equilibrium; variational inclusion; nonexpansive
mapping; asymptotically strict pseudocontractive mapping in the intermediate sense;
maximal monotone mapping; inverse-strongly monotone mapping
1 Introduction
Let H be a real Hilbert space with inner product ·, · and norm · , C be a nonempty
closed convex subset of H and PC be the metric projection of H onto C. Let S : C → H
be a nonlinear mapping on C. We denote by Fix(S) the set of ﬁxed points of S and by R
the set of all real numbers. A mapping V is called strongly positive on H if there exists a
constant γ¯ ∈ (, ] such that
Vx, x ≥ γ¯ x ,
∀x ∈ H.
A mapping S : C → H is called L-Lipschitz-continuous if there exists a constant L >  such
that
Sx – Sy ≤ Lx – y,
∀x, y ∈ C.
In particular, if L =  then S is called a nonexpansive mapping; if L ∈ (, ) then A is called
a contraction.
Let ϕ : C → R be a real-valued function, A : H → H be a nonlinear mapping and Θ : C ×
C → R be a bifunction. We consider the generalized mixed equilibrium problem (GMEP)
©2014 Ceng et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
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Ceng et al. Journal of Inequalities and Applications 2014, 2014:462
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[] of ﬁnding x ∈ C such that
Θ(x, y) + ϕ(y) – ϕ(x) + Ax, y – x ≥ ,
∀y ∈ C.
(.)
We denote the set of solutions of GMEP (.) by GMEP(Θ, ϕ, A). The GMEP (.) is very
general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and
others. The GMEP is further considered and studied in, e.g., [–].
Throughout this paper, it is assumed as in [] that Θ : C × C → R is a bifunction satisfying conditions (H)-(H) and ϕ : C → R is a lower semicontinuous and convex function
with restriction (H), where
(H) Θ(x, x) =  for all x ∈ C;
(H) Θ is monotone, i.e., Θ(x, y) + Θ(y, x) ≤  for any x, y ∈ C;
(H) Θ is upper-hemicontinuous, i.e., for each x, y, z ∈ C,
lim sup Θ tz + ( – t)x, y ≤ Θ(x, y);
t→+
(H) Θ(x, ·) is convex and lower semicontinuous for each x ∈ C;
(H) for each x ∈ H and r > , there exist a bounded subset Dx ⊂ C and yx ∈ C such
that for any z ∈ C \ Dx ,

Θ(z, yx ) + ϕ(yx ) – ϕ(z) + yx – z, z – x < .
r
Let Θ , Θ : C × C → R be two bifunctions, and B , B : C → H be two nonlinear mappings. Consider the system of generalized equilibrium problems (SGEP): ﬁnd (x∗ , y∗ ) ∈
C × C such that
Θ (x∗ , x) + B y∗ , x – x∗ +
∗
∗
∗
Θ (y , y) + B x , y – y +

x∗
μ

y∗
μ
– y∗ , x – x∗ ≥ ,
∗
∗
– x , y – y ≥ ,
∀x ∈ C,
∀y ∈ C,
(.)
where μ and μ are two positive constants.
∞
Let {Tn }∞
n= be an inﬁnite family of nonexpansive self-mappings on C and {λn }n= be a
sequence of nonnegative numbers in [, ]. For any n ≥ , deﬁne a self-mapping Wn on H
as follows:
⎧
Un,n+ = I,
⎪
⎪
⎪
⎪
⎪
Un,n = λn Tn Un,n+ + ( – λn )I,
⎪
⎪
⎪
⎪
⎪
U
n,n– = λn– Tn– Un,n + ( – λn– )I,
⎪
⎪
⎪
⎪
.
⎪
.
⎪
⎪
⎨.
Un,k = λk Tk Un,k+ + ( – λk )I,
⎪
⎪
⎪Un,k– = λk– Tk– Un,k + ( – λk– )I,
⎪
⎪
⎪
⎪
..
⎪
⎪
⎪
.
⎪
⎪
⎪
⎪
⎪
Un, = λ T Un, + ( – λ )I,
⎪
⎪
⎩
Wn = Un, = λ T Un, + ( – λ )I.
(.)
Such a mapping Wn is called the W -mapping generated by Tn , Tn– , . . . , T and λn , λn– ,
. . . , λ .
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Let f : H → H be a contraction and V be a strongly positive bounded linear operator on H. Assume that ϕ : H → R is a lower semicontinuous and convex functional,
that Θ, Θ , Θ : H × H → R satisfy conditions (H)-(H), and that A, B , B : H → H are
inverse-strongly monotone. Very recently, motivated by Yao et al. [], Cai and Bu [] introduced the following hybrid extragradient-like iterative algorithm:
⎧
(Θ,ϕ)
⎪
⎨zn = Srn (xn – rn Axn ),
yn = TμΘ (I – μ B )TμΘ (I – μ B )zn ,
⎪
⎩
xn+ = αn (u + γ f (xn )) + βn xn + (( – βn )I – αn (I + μV ))Wn yn ,
(.)
∀n ≥ ,
for ﬁnding a common solution of GMEP (.), SGEP (.), and the ﬁxed point problem of
an inﬁnite family of nonexpansive mappings {Ti }∞
i= on H, where {rn } ⊂ (, ∞), {αn }, {βn } ⊂
(, ), and x , u ∈ H are given. The authors proved the strong convergence of the se
quence generated by the hybrid iterative algorithm (.) to a point x∗ ∈ ( ∞
i= Fix(Ti )) ∩
GMEP(Θ, ϕ, A) ∩ SGEP(G) under some suitable conditions, where SGEP(G) is the ﬁxed
point set of the mapping G := TμΘ (I – μ B )TμΘ (I – μ B ). This point x∗ also solves the
following optimization problem:
x∈(

μ
Vx, x + x – u – h(x),

n= Fix(Tn ))∩GMEP(Θ,ϕ,A)∩SGEP(G) 
∞
min
(OP)
where h : H → R is the potential function of γ f .
Let B be a single-valued mapping of C into H and R be a set-valued mapping with
D(R) = C. Consider the following variational inclusion: ﬁnd a point x ∈ C such that
 ∈ Bx + Rx.
(.)
We denote by I(B, R) the solution set of the variational inclusion (.). In particular, if
B = R = , then I(B, R) = C. If B = , then problem (.) becomes the inclusion problem
introduced by Rockafellar []. It is known that problem (.) provides a convenient framework for the uniﬁed study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, etc. Let a set-valued
mapping R : D(R) ⊂ H → H be maximal monotone. We deﬁne the resolvent operator
JR,λ : H → D(R) associated with R and λ as follows:
JR,λ = (I + λR)– ,
∀x ∈ H,
where λ is a positive number.
In , Huang [] studied problem (.) in the case where R is maximal monotone and
B is strongly monotone and Lipschitz-continuous with D(R) = C = H. Subsequently, Zeng
et al. [] further studied problem (.) in the case which is more general than Huang’s [].
Moreover, the authors [] obtained the same strong convergence conclusion as in Huang’s
result []. In addition, the authors also gave the geometric convergence rate estimate for
approximate solutions. Also, various types of iterative algorithms for solving variational
inclusions have been further studied and developed; for more details, refer to [, –]
and the references therein.
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In , for the case where C = H, Yao et al. [] introduced and analyzed an iterative
algorithms for ﬁnding a common element of the set of solutions of the GMEP (.), the set
of solutions of the variational inclusion (.) for maximal monotone and inverse-strongly
monotone mappings and the set of ﬁxed points of a countable family of nonexpansive
mappings on H.
Recently, Kim and Xu [] introduced the concept of asymptotically κ-strict pseudocontractive mappings in a Hilbert space.
Deﬁnition . Let C be a nonempty subset of a Hilbert space H. A mapping S : C → C
is said to be an asymptotically κ-strict pseudocontractive mapping with sequence {γn } if
there exist a constant κ ∈ [, ) and a sequence {γn } in [, ∞) with limn→∞ γn =  such that
n
S x – Sn y
 ≤ ( + γn )x – y + κ x – Sn x – y – Sn y  ,
∀n ≥ , ∀x, y ∈ C.
Subsequently, Sahu et al. [] considered the concept of asymptotically κ-strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.
Deﬁnition . Let C be a nonempty subset of a Hilbert space H. A mapping S : C → C is
said to be an asymptotically κ-strict pseudocontractive mapping in the intermediate sense
with sequence {γn } if there exist a constant κ ∈ [, ) and a sequence {γn } in [, ∞) with
limn→∞ γn =  such that

 lim sup sup Sn x – Sn y
– ( + γn )x – y – κ x – Sn x – y – Sn y ≤ .
(.)
n→∞ x,y∈C
Put cn := max{, supx,y∈C (Sn x – Sn y – ( + γn )x – y – κx – Sn x – (y – Sn y) )}. Then
cn ≥  (∀n ≥ ), cn →  (n → ∞), and (.) reduce to the relation
n
S x – Sn y
 ≤ ( + γn )x – y + κ x – Sn x – y – Sn y  + cn ,
∀n ≥ , ∀x, y ∈ C.
(.)
Whenever cn =  for all n ≥  in (.), then S is an asymptotically κ-strict pseudocontractive mapping with sequence {γn }. The authors [] derived the weak and strong convergence of the modiﬁed Mann iteration processes for an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γn }. More precisely, they
ﬁrst established one weak convergence theorem for the following iterative scheme:
x = x ∈ C chosen arbitrarily,
xn+ = ( – αn )xn + αn Sn xn , ∀n ≥ ,
∞
where  < δ ≤ αn ≤  – κ – δ, ∞
n= αn cn < ∞, and
n= γn < ∞; and then obtained another
strong convergence theorem for the following iterative scheme:
⎧
⎪
x = x ∈ C chosen arbitrary,
⎪
⎪ 
⎪
n
⎪
⎪
⎨yn = ( – αn )xn + αn S xn ,
Cn = {z ∈ C : yn – z ≤ xn – z + θn },
⎪
⎪
⎪
Qn = {z ∈ C : xn – z, x – xn ≥ },
⎪
⎪
⎪
⎩x = P
∀n ≥ ,
n+
Cn ∩Qn x,
where  < δ ≤ αn ≤  – κ, θn = cn + γn Δn , and Δn = sup{xn – z : z ∈ Fix(S)} < ∞.
Ceng et al. Journal of Inequalities and Applications 2014, 2014:462
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Inspired by the above facts, we in this paper introduce two iterative algorithms for ﬁnding common solutions of a ﬁnite family of variational inclusions for maximal monotone
and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common ﬁxed point problem of an inﬁnite family
of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the
intermediate sense in a real Hilbert space. We prove some strong and weak convergence
theorems for the proposed iterative algorithms under suitable conditions. The results presented in this paper are the supplement, extension, improvement, and generalization of
the previously known results in this area.
2 Preliminaries
Throughout this paper, we assume that H is a real Hilbert space whose inner product
and norm are denoted by ·, · and · , respectively. Let C be a nonempty closed convex
subset of H. We write xn x to indicate that the sequence {xn } converges weakly to x
and xn → x to indicate that the sequence {xn } converges strongly to x. Moreover, we use
ωw (xn ) to denote the weak ω-limit set of the sequence {xn }, i.e.,
ωw (xn ) := x ∈ H : xni x for some subsequence {xni } of {xn } .
Deﬁnition . A mapping A : C → H is called
(i) monotone if
Ax – Ay, x – y ≥ ,
∀x, y ∈ C;
(ii) η-strongly monotone if there exists a constant η >  such that
Ax – Ay, x – y ≥ ηx – y ,
∀x, y ∈ C;
(iii) ζ -inverse-strongly monotone if there exists a constant ζ >  such that
Ax – Ay, x – y ≥ ζ Ax – Ay ,
∀x, y ∈ C.
It is easy to see that the projection PC is -inverse-strongly monotone (in short, -ism).
Inverse-strongly monotone (also referred to as co-coercive) operators have been applied
widely in solving practical problems in various ﬁelds.
Deﬁnition . A diﬀerentiable function K : H → R is called:
(i) convex, if
K(y) – K(x) ≥ K (x), y – x ,
∀x, y ∈ H,
where K (x) is the Frechet derivative of K at x;
(ii) strongly convex, if there exists a constant σ >  such that
σ
K(y) – K(x) – K (x), y – x ≥ x – y ,

∀x, y ∈ H.
It is easy to see that if K : H → R is a diﬀerentiable strongly convex function with constant σ >  then K : H → H is strongly monotone with constant σ > .
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The metric (or nearest point) projection from H onto C is the mapping PC : H → C
which assigns to each point x ∈ H the unique point PC x ∈ C satisfying the property
x – PC x = inf x – y =: d(x, C).
y∈C
Some important properties of projections are gathered in the following proposition.
Proposition . For given x ∈ H and z ∈ C:
(i) z = PC x ⇔ x – z, y – z ≤ , ∀y ∈ C;
(ii) z = PC x ⇔ x – z ≤ x – y – y – z , ∀y ∈ C;
(iii) PC x – PC y, x – y ≥ PC x – PC y , ∀y ∈ H. (This implies that PC is nonexpansive
and monotone.)
By using the technique of [], we can readily obtain the following elementary result.
Proposition . (see [, Lemma  and Proposition ]) Let C be a nonempty closed convex
subset of a real Hilbert space H and let ϕ : C → R be a lower semicontinuous and convex
function. Let Θ : C × C → R be a bifunction satisfying the conditions (H)-(H). Assume
that
(i) K : H → R is strongly convex with constant σ >  and the function x → y – x, K (x)
is weakly upper semicontinuous for each y ∈ H;
(ii) for each x ∈ H and r > , there exist a bounded subset Dx ⊂ C and yx ∈ C such that
for any z ∈ C \ Dx ,
Θ(z, yx ) + ϕ(yx ) – ϕ(z) +
 K (z) – K (x), yx – z < .
r
Then the following hold:
(a) for each x ∈ H, Sr(Θ,ϕ) (x) = ∅;
(Θ,ϕ)
is single-valued;
(b) Sr
(Θ,ϕ)
is nonexpansive if K is Lipschitz-continuous with constant ν >  and
(c) Sr
K (x ) – K (x ), u – u ≥ K (u ) – K (u ), u – u ,
∀(x , x ) ∈ H × H,
(Θ,ϕ)
where ui = Sr (xi ) for i = , ;
(d) for all s, t >  and x ∈ H
(Θ,ϕ) (Θ,ϕ) (Θ,ϕ) K Ss x – K St x , Ss(Θ,ϕ) x – St x
≤
s – t (Θ,ϕ) (Θ,ϕ) K Ss x – K (x), Ss(Θ,ϕ) x – St x ;
s
(e) Fix(Sr(Θ,ϕ) ) = MEP(Θ, ϕ);
(f ) MEP(Θ, ϕ) is closed and convex.
In particular, whenever Θ : C × C → R is a bifunction satisfying the conditions (H)-(H)
and K(x) =  x , ∀x ∈ H, then, for any x, y ∈ H,
 (Θ,ϕ)
S
x – Sr(Θ,ϕ) y
≤ Sr(Θ,ϕ) x – Sr(Θ,ϕ) y, x – y
r
Ceng et al. Journal of Inequalities and Applications 2014, 2014:462
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(Θ,ϕ)
(Sr
Page 7 of 38
is ﬁrmly nonexpansive) and
(Θ,ϕ)
|s – t| (Θ,ϕ) S(Θ,ϕ) x – x
,
S
x – St x ≤
s
s
s
∀s, t > , x ∈ H.
In this case, Sr(Θ,ϕ) is rewritten as Tr(Θ,ϕ) . If, in addition, ϕ ≡ , then Tr(Θ,ϕ) is rewritten as
TrΘ (see [, Lemma .] for more details).
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
Lemma . Let X be a real inner product space. Then we have the following inequality:
x + y ≤ x + y, x + y,
∀x, y ∈ X.
Lemma . Let H be a real Hilbert space. Then the following hold:
(a) x – y = x – y – x – y, y for all x, y ∈ H;
(b) λx + μy = λx + μy – λμx – y for all x, y ∈ H and λ, μ ∈ [, ] with
λ + μ = ;
(c) If {xn } is a sequence in H such that xn x, it follows that
lim sup xn – y = lim sup xn – x + x – y ,
n→∞
n→∞
∀y ∈ H.
Lemma . ([, Lemma .]) Let H be a real Hilbert space. Given a nonempty closed
convex subset of H and points x, y, z ∈ H and given also a real number a ∈ R, the set
v ∈ C : y – v ≤ x – v + z, v + a
is convex (and closed).
Lemma . ([, Lemma .]) Let C be a nonempty subset of a Hilbert space H and S :
C → C be an asymptotically κ-strict pseudocontractive mapping in the intermediate sense
with sequence {γn }. Then
n
S x – Sn y
≤
 κx – y +
–κ
 + ( – κ)γn x – y + ( – κ)cn
for all x, y ∈ C and n ≥ .
Lemma . ([, Lemma .]) Let C be a nonempty subset of a Hilbert space H and S :
C → C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in
the intermediate sense with sequence {γn }. Let {xn } be a sequence in C such that xn –
xn+ →  and xn – Sn xn →  as n → ∞. Then xn – Sxn →  as n → ∞.
Lemma . (Demiclosedness principle [, Proposition .]) Let C be a nonempty closed
convex subset of a Hilbert space H and S : C → C be a continuous asymptotically κ-strict
pseudocontractive mapping in the intermediate sense with sequence {γn }. Then I – S is
demiclosed at zero in the sense that if {xn } is a sequence in C such that xn x ∈ C and
lim supm→∞ lim supn→∞ xn – Sm xn = , then (I – S)x = .
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Lemma . ([, Proposition .]) Let C be a nonempty closed convex subset of a Hilbert
space H and S : C → C be a continuous asymptotically κ-strict pseudocontractive mapping
in the intermediate sense with sequence {γn } such that Fix(S) = ∅. Then Fix(S) is closed and
convex.
Remark . Lemmas . and . give some basic properties of an asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γn }. Moreover,
Lemma . extends the demiclosedness principles studied for certain classes of nonlinear
mappings; see [] for more details.
∞
∞
Lemma . ([, p.]) Let {an }∞
n= , {bn }n= , and {δn }n= be sequences of nonnegative real
numbers satisfying the inequality
an+ ≤ ( + δn )an + bn ,
∀n ≥ .
∞
∞
If ∞
n= δn < ∞ and
n= bn < ∞, then limn→∞ an exists. If, in addition, {an }n= has a subsequence which converges to zero, then limn→∞ an = .
Recall that a Banach space X is said to satisfy the Opial condition [] if, for any given
sequence {xn } ⊂ X which converges weakly to an element x ∈ X, we have the inequality
lim sup xn – x < lim sup xn – y,
n→∞
n→∞
∀y ∈ X, y = x.
It is well known in [] that every Hilbert space H satisﬁes the Opial condition.
Lemma . (see [, Proposition .]) Let C be a nonempty closed convex subset of a real
Hilbert space H and let {xn } be a sequence in H. Suppose that
xn+ – p ≤ ( + λn )xn – p + δn ,
∀p ∈ C, n ≥ ,
where {λn } and {δn } are sequences of nonnegative real numbers such that
∞
n= δn < ∞. Then {PC xn } converges strongly in C.
∞
n= λn
< ∞ and
Lemma . (see []) Let C be a closed convex subset of a real Hilbert space H. Let {xn }
be a sequence in H and u ∈ H. Let q = PC u. If {xn } is such that ωw (xn ) ⊂ C and satisﬁes the
condition
xn – u ≤ u – q,
for all n,
then xn → q as n → ∞.
Lemma . (see [, Lemma .]) Let C be a nonempty closed convex subset of a real
Hilbert space H. Let {Tn }∞
n= be a sequence of nonexpansive self-mappings on C such that
∞
n= Fix(Tn ) = ∅ and let {λn } be a sequence in (, b] for some b ∈ (, ). Then, for every x ∈ C
and k ≥  the limit limn→∞ Un,k x exists.
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Remark . (see [, Remark .]) It can be known from Lemma . that if D is a
nonempty bounded subset of C, then for >  there exists n ≥ k such that for all n > n
sup Un,k x – Uk x ≤ .
x∈D
Remark . (see [, Remark .]) Utilizing Lemma ., we deﬁne a mapping W : C → C
as follows:
Wx = lim Wn x = lim Un, x,
n→∞
n→∞
∀x ∈ C.
Such a W is called the W -mapping generated by T , T , . . . and λ , λ , . . . . Since Wn is nonexpansive, W : C → C is also nonexpansive. Indeed, observe that for each x, y ∈ C
Wx – Wy = lim Wn x – Wn y ≤ x – y.
n→∞
If {xn } is a bounded sequence in C, then we put D = {xn : n ≥ }. Hence, it is clear from
Remark . that for an arbitrary >  there exists N ≥  such that for all n > N
Wn xn – Wxn = Un, xn – U xn ≤ sup Un, x – U x ≤ .
x∈D
This implies that
lim Wn xn – Wxn = .
n→∞
Lemma . (see [, Lemma .]) Let C be a nonempty closed convex subset of a real
Hilbert space H. Let {Tn }∞
n= be a sequence of nonexpansive self-mappings on C such that
∞
Fix(T
)
=
∅,
and
let
{λn } be a sequence in (, b] for some b ∈ (, ). Then Fix(W ) =
n
n=
∞
n= Fix(Tn ).
Lemma . (see [, Theorem . (Demiclosedness Principle)]) Let C be a nonempty
closed convex subset of a real Hilbert space H. Let T : C → C be nonexpansive. Then I – T
is demiclosed on C. That is, whenever {xn } is a sequence in C weakly converging to some
x ∈ C and the sequence {(I – T)xn } strongly converges to some y, it follows that (I – T)x = y.
Here I is the identity operator of H.
Recall that a set-valued mapping R : D(R) ⊂ H → H is called monotone if, for all x, y ∈
D(R), f ∈ R(x), and g ∈ R(y) imply
f – g, x – y ≥ .
A set-valued mapping R is called maximal monotone if R is monotone and (I + λR)D(R) =
H for each λ > , where I is the identity mapping of H. We denote by G(R) the graph
of R. It is known that a monotone mapping R is maximal if and only if, for (x, f ) ∈ H × H,
f – g, x – y ≥  for every (y, g) ∈ G(R), we have f ∈ R(x). We illustrate the concept of
maximal monotone mapping with the following example.
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Let A : C → H be a monotone, k-Lipschitz-continuous mapping and let NC v be the
normal cone to C at v ∈ C, i.e.,
NC v = w ∈ H : v – u, w ≥ , ∀u ∈ C .
Deﬁne
Tv =
Av + NC v if v ∈ C,
∅
if v ∈/ C.
Then T is maximal monotone and  ∈ Tv if and only if Av, y – v ≥  for all y ∈ C (see []).
Assume that R : D(R) ⊂ H → H is a maximal monotone mapping. Let λ > . In terms
of Huang [] (see also []), we have the following property for the resolvent operator
JR,λ : H → D(R).
Lemma . JR,λ is single-valued and ﬁrmly nonexpansive, i.e.,
JR,λ x – JR,λ y, x – y ≥ JR,λ x – JR,λ y ,
∀x, y ∈ H.
Consequently, JR,λ is nonexpansive and monotone.
Lemma . (see []) Let R be a maximal monotone mapping with D(R) = C. Then for
any given λ > , u ∈ C is a solution of problem (.) if and only if u ∈ C satisﬁes
u = JR,λ (u – λBu).
Lemma . (see []) Let R be a maximal monotone mapping with D(R) = C and let
B : C → H be a strongly monotone, continuous, and single-valued mapping. Then for each
z ∈ H, the equation z ∈ (B + λR)x has a unique solution xλ for λ > .
Lemma . (see []) Let R be a maximal monotone mapping with D(R) = C and B : C →
H be a monotone, continuous and single-valued mapping. Then (I + λ(R + B))C = H for each
λ > . In this case, R + B is maximal monotone.
Lemma . (see []) Let C be a nonempty closed convex subset of a real Hilbert space H,
and g : C → R ∪ +∞ be a proper lower semicontinuous diﬀerentiable convex function. If x∗
is a solution the minimization problem
g x∗ = inf g(x),
x∈C
then
g (x), x – x∗ ≥ ,
∀x ∈ C.
In particular, if x∗ solves (OP), then
u + γ f – (I + μV ) x∗ , x – x∗ ≤ .
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3 Strong convergence theorems
In this section, we introduce and analyze an iterative algorithm for ﬁnding common solutions of a ﬁnite family of variational inclusions for maximal monotone and inversestrongly monotone mappings with the constraints of two problems: a generalized mixed
equilibrium problem and a common ﬁxed point problem of an inﬁnite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Under appropriate conditions imposed on the parameter
sequences we will prove strong convergence of the proposed algorithm.
Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H. Let N
be an integer. Let Θ be a bifunction from C × C to R satisfying (H)-(H) and ϕ : C → R
be a lower semicontinuous and convex functional. Let Ri : C → H be a maximal monotone mapping and let A : H → H and Bi : C → H be ζ -inverse-strongly monotone and ηi inverse-strongly monotone, respectively, where i ∈ {, , . . . , N}. Let S : C → C be a uniformly
continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense
for some  ≤ κ <  with sequence {γn } ⊂ [, ∞) such that limn→∞ γn =  and {cn } ⊂ [, ∞)
such that limn→∞ cn = . Let {Tn }∞
n= be a sequence of nonexpansive self-mappings on C and
{λn } be a sequence in (, b] for some b ∈ (, ). Let V be a γ¯ -strongly positive bounded linear operator and f : H → H be an l-Lipschitzian mapping with γ l < ( + μ)γ¯ . Assume that
N
Ω := ( ∞
n= Fix(Tn )) ∩ GMEP(Θ, ϕ, A) ∩ ( i= I(Bi , Ri )) ∩ Fix(S) is nonempty and bounded.
Let Wn be the W -mapping deﬁned by (.) and {αn }, {βn } and {δn } be three sequences in
(, ) such that limn→∞ αn =  and κ ≤ δn ≤ d < . Assume that:
(i) K : H → R is strongly convex with constant σ >  and its derivative K is
Lipschitz-continuous with constant ν >  such that the function x → y – x, K (x) is
weakly upper semicontinuous for each y ∈ H;
(ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any
y ∈/ Dx ,
Θ(y, zx ) + ϕ(zx ) – ϕ(y) +
 K (y) – K (x), zx – y < ;
r
(iii)  < lim infn→∞ βn ≤ lim supn→∞ βn < ;
(iv) {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ), ∀i ∈ {, , . . . , N}, and {rn } ⊂ [, ζ ] satisﬁes
 < lim inf rn ≤ lim sup rn < ζ .
n→∞
n→∞
Pick any x ∈ H and set C = C, x = PC x . Let {xn } be a sequence generated by the following
algorithm:
⎧
un = Sr(Θ,ϕ)
(I – rn A)xn ,
⎪
n
⎪
⎪
⎪
⎪
⎪
⎪zn = JRN ,λN,n (I – λN,n BN )JRN– ,λN–,n (I – λN–,n BN– ) · · · JR ,λ,n (I – λ,n B )un ,
⎪
⎨k = δ z + ( – δ )Sn z ,
n
n n
n
n
⎪yn = αn (u + γ f (xn )) + βn kn + (( – βn )I – αn (I + μV ))Wn zn ,
⎪
⎪
⎪
⎪
⎪
Cn+ = {z ∈ Cn : yn – z ≤ xn – z + θn },
⎪
⎪
⎩
xn+ = PCn+ x , ∀n ≥ ,
(.)
where θn = (αn + γn )Δn + cn , Δn = sup{xn – p + u + (γ f – I – μV )p : p ∈ Ω} < ∞,
and = –sup αn < ∞. If Sr(Θ,ϕ) is ﬁrmly nonexpansive, then the following statements hold:
n≥
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(I) {xn } converges strongly to PΩ x ;
(II) {xn } converges strongly to PΩ x , which solves the optimization problem
min
x∈Ω

μ
Vx, x + x – u – h(x),


(OP)
provided γn + cn + xn – yn = o(αn ) additionally, where h : H → R is the potential
function of γ f .
Proof Since limn→∞ αn =  and  < lim infn→∞ βn ≤ lim supn→∞ βn < , we may assume,
without loss of generality, that αn ≤ ( – βn )( + μV )– . Since V is a γ¯ -strongly positive
bounded linear operator on H, we know that
V = sup Vu, u : u ∈ H, u =  .
Observe that
( – βn )I – αn (I + μV ) u, u =  – βn – αn – αn μVu, u
≥  – βn – αn – αn μV ≥ ,
that is, ( – βn )I – αn (I + μV ) is positive. It follows that
( – βn )I – αn (I + μV )
= sup ( – βn )I – αn (I + μV ) u, u : u ∈ H, u = 
= sup  – βn – αn – αn μVu, u : u ∈ H, u = 
≤  – βn – αn – αn μγ¯ .
Put
Λin = JRi ,λi,n (I – λi,n Bi )JRi– ,λi–,n (I – λi–,n Bi– ) · · · JR ,λ,n (I – λ,n B )
for all i ∈ {, , . . . , N} and n ≥ , and Λn = I, where I is the identity mapping on H. Then
we have that zn = ΛN
n un . We divide the rest of the proof into several steps.
Step . We show that {xn } is well deﬁned. It is obvious that Cn is closed and convex. As
the deﬁning inequality in Cn is equivalent to the inequality
(xn – zn ), z ≤ xn  – zn  + θn ,
by Lemma . we know that Cn is convex and closed for every n ≥ .
First of all, we show that Ω ⊂ Cn for all n ≥ . Suppose that Ω ⊂ Cn for some n ≥ .
Take p ∈ Ω arbitrarily. Since p = Sr(Θ,ϕ)
(p – rn Ap), A is ζ -inverse strongly monotone and
n
 ≤ rn ≤ ζ , we have

(I – rn A)xn – Sr(Θ,ϕ)
(I – rn A)p
un – p = Sr(Θ,ϕ)
n
n

≤ (I – rn A)xn – (I – rn A)p
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
= (xn – p) – rn (Axn – Ap)
= xn – p – rn xn – p, Axn – Ap + rn Axn – Ap
≤ xn – p – rn ζ Axn – Ap + rn Axn – Ap
= xn – p + rn (rn – ζ )Axn – Ap
≤ xn – p .
(.)
Since p = JRi ,λi,n (I – λi,n Bi )p, Λin p = p, and Bi is ηi -inverse-strongly monotone, where ηi ∈
(, ηi ), i ∈ {, , . . . , N}, by Lemma . we deduce that
N– 
zn – p = JRN ,λN,n (I – λN,n BN )ΛN–
n un – JRN ,λN,n (I – λN,n BN )Λn p
N– 
≤ (I – λN,n BN )ΛN–
n un – (I – λN,n BN )Λn p

= ΛN– un – ΛN– p – λN,n BN ΛN– un – BN ΛN– p n
n
n
n
N– 
N– 
≤ ΛN–
+ λN,n (λN,n – ηN )
BN ΛN–
n un – Λn p
n un – BN Λn p
N– 
≤ ΛN–
n un – Λn p
..
.

≤ Λn un – Λn p
= un – p .
(.)
Combining (.) and (.), we have
zn – p ≤ xn – p.
(.)
By Lemma .(b), we deduce from (.) and (.) that

kn – p = δn (zn – p) + ( – δn ) Sn zn – p 

= δn zn – p + ( – δn )
Sn zn – p
– δn ( – δn )
zn – Sn zn ≤ δn zn – p + ( – δn ) ( + γn )zn – p


+ κ zn – Sn zn + cn – δn ( – δn )
zn – Sn zn 
=  + γn ( – δn ) zn – p + ( – δn )(κ – δn )
zn – Sn zn + ( – δn )cn

≤ ( + γn )zn – p + ( – δn )(κ – δn )
zn – Sn zn + cn
≤ ( + γn )zn – p + cn .
(.)
Set V¯ = I + μV . Then, for γ l ≤ ( + μ)γ¯ , by Lemma . we obtain from (.), (.), and
(.)
yn – p

= αn u + γ f (xn ) + βn kn + ( – βn )I – αn V¯ Wn zn – p
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
= αn u + γ f (xn ) – V¯ p + βn (kn – p) + ( – βn )I – αn V¯ (Wn zn – p)
= αn u + γ f (p) – V¯ p + αn γ f (xn ) – f (p)

+ βn (kn – p) + ( – βn )I – αn V¯ (Wn zn – p)

≤ αn γ f (xn ) – f (p) + βn (kn – p) + ( – βn )I – αn V¯ (Wn zn – p)
+ αn u + γ f (p) – V¯ p , yn – p

≤ αn γ f (xn ) – f (p)
+ βn kn – p + ( – βn )I – αn V¯ (Wn zn – p)
+ αn u + γ f (p) – V¯ p , yn – p

≤ αn γ lxn – p + βn kn – p + ( – βn – αn – αn μγ¯ )Wn zn – p
+ αn u + γ f (p) – V¯ p
yn – p

≤ αn ( + μ)γ¯ xn – p + βn kn – p +  – βn – αn ( + μ)γ¯ zn – p

+ αn u + γ f (p) – V¯ p
+ yn – p
≤ αn ( + μ)γ¯ xn – p + βn kn – p +  – βn – αn ( + μ)γ¯ zn – p

+ αn u + γ f (p) – V¯ p
+ yn – p
≤ αn ( + μ)γ¯ xn – p + βn ( + γn )zn – p + cn

+  – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p
+ yn – p
≤ αn ( + μ)γ¯ xn – p + βn ( + γn )zn – p + cn
+  – βn – αn ( + μ)γ¯ ( + γn )zn – p + cn

+ αn u + γ f (p) – V¯ p
+ yn – p
= αn ( + μ)γ¯ xn – p +  – αn ( + μ)γ¯ ( + γn )zn – p + cn

+ αn u + γ f (p) – V¯ p
+ yn – p
≤ αn ( + μ)γ¯ ( + γn )xn – p + cn
+  – αn ( + μ)γ¯ ( + γn )xn – p + cn

+ αn u + γ f (p) – V¯ p
+ yn – p

= ( + γn )xn – p + cn + αn u + γ f (p) – V¯ p
+ yn – p ,
which hence yields
 + γn
αn u + γ f (p) – V¯ p
 +  cn
xn – p +
 – αn
 – αn
 – αn

α n + γn
αn 
= +
cn
u + γ f (p) – V¯ p
+
xn – p +
 – αn
 – αn
 – αn
α n + γn
α n + γn u + γ f (p) – V¯ p
 +  cn
≤ +
xn – p +
 – αn
 – αn
 – αn
αn + γn 

= xn – p +
cn
xn – p + u + γ f (p) – V¯ p
+
 – αn
 – αn
yn – p ≤
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 ≤ xn – p + (αn + γn ) xn – p + u + γ f (p) – V¯ p
+ cn
≤ xn – p + (αn + γn )Δn + cn = xn – p + θn ,
(.)
where θn = (αn + γn )Δn + cn , Δn = sup{xn – p + u + γ f (p) – V¯ p : p ∈ Ω} < ∞, and
= –sup αn < ∞ (due to {αn } ⊂ (, ) and limn→∞ αn = ). Hence p ∈ Cn+ . This implies
n≥
that Ω ⊂ Cn for all n ≥ . Therefore, {xn } is well deﬁned.
Step . We prove that xn – kn →  as n → ∞.
Indeed, let v = PΩ x . From xn = PCn x and v ∈ Ω ⊂ Cn , we obtain
xn – x ≤ v – x .
(.)
This implies that {xn } is bounded and hence {un }, {zn }, {kn }, and {yn } are also bounded.
Since xn+ ∈ Cn+ ⊂ Cn and xn = PCn x , we have
xn – x ≤ xn+ – x ,
∀n ≥ .
Therefore limn→∞ xn – x exists. From xn = PCn x , xn+ ∈ Cn+ ⊂ Cn , by Proposition .(ii) we obtain
xn+ – xn  ≤ x – xn+  – x – xn  ,
which implies
lim xn+ – xn = .
n→∞
(.)
It follows from xn+ ∈ Cn+ that yn – xn+  ≤ xn – xn+  + θn and hence
xn – yn  ≤  xn – xn+  + xn+ – yn 
≤  xn – xn+  + xn – xn+  + θn
=  xn – xn+  + θn .
From (.) and limn→∞ θn = , we have
lim xn – yn = .
n→∞
Also, utilizing Lemmas . and .(b) we obtain from (.), (.), and (.)
yn – p

= αn u + γ f (xn ) – V¯ Wn zn + βn (kn – p) + ( – βn )(Wn zn – p)

≤ βn (kn – p) + ( – βn )(Wn zn – p)
+ αn u + γ f (xn ) – V¯ Wn zn , yn – p
= βn kn – p + ( – βn )Wn zn – p – βn ( – βn )kn – Wn zn 
+ αn u + γ f (xn ) – V¯ Wn zn yn – p
(.)
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≤ βn kn – p + ( – βn )zn – p – βn ( – βn )kn – Wn zn 
+ αn u + γ f (xn ) – V¯ Wn zn yn – p
≤ βn ( + γn )zn – p + cn + ( – βn )zn – p – βn ( – βn )kn – Wn zn 
+ αn u + γ f (xn ) – V¯ Wn zn yn – p
≤ βn ( + γn )zn – p + cn + ( – βn ) ( + γn )zn – p + cn
– βn ( – βn )kn – Wn zn  + αn u + γ f (xn ) – V¯ Wn zn yn – p
= ( + γn )zn – p + cn – βn ( – βn )kn – Wn zn 
+ αn u + γ f (xn ) – V¯ Wn zn yn – p
≤ ( + γn )xn – p + cn – βn ( – βn )kn – Wn zn 
+ αn u + γ f (xn ) – V¯ Wn zn yn – p,
which leads to
βn ( – βn )kn – Wn zn 
≤ xn – p – yn – p + γn xn – p + cn
+ αn u + γ f (xn ) – V¯ Wn zn yn – p
≤ xn – yn xn – p + yn – p + γn xn – p + cn
+ αn u + γ f (xn ) – V¯ Wn zn yn – p.
Since limn→∞ αn = , limn→∞ γn = , and limn→∞ cn = , it follows from (.) and condition
(iii) that
lim kn – Wn zn = .
n→∞
(.)
Note that
yn – kn = αn u + γ f (xn ) – V¯ Wn zn + ( – βn )(Wn zn – kn ),
which yields
xn – kn ≤ xn – yn + yn – kn ≤ xn – yn + αn u + γ f (xn ) – V¯ Wn zn + ( – βn )(Wn zn – kn )
≤ xn – yn + αn u + γ f (xn ) – V¯ Wn zn + ( – βn )Wn zn – kn ≤ xn – yn + αn u + γ f (xn ) – V¯ Wn zn + Wn zn – kn .
So, from (.), (.), and limn→∞ αn = , we get
lim xn – kn = .
n→∞
(.)
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Step . We prove that xn – un → , un – zn → , zn – Wzn → , and zn – Sn zn →
 as n → ∞.
Indeed, taking into consideration that  < lim infn→∞ rn ≤ lim supn→∞ rn < ζ , we may
assume, without loss of generality, that {rn } ⊂ [c, d] ⊂ (, ζ ). From (.) and (.) it follows
that

kn – p ≤  + γn ( – δn ) zn – p + ( – δn )(k – δn )
zn – Sn zn + ( – δn )cn
≤ zn – p + γn zn – p + cn
≤ zn – p + γn xn – p + cn .
(.)
Next we prove that
lim xn – un = .
(.)
n→∞
For p ∈ Ω, we ﬁnd that

(I – rn A)xn – Sr(Θ,ϕ)
(I – rn A)p
un – p = Sr(Θ,ϕ)
n
n

≤ (I – rn A)xn – (I – rn A)p

= xn – p – rn (Axn – Ap)
≤ xn – p + rn (rn – ζ )Axn – Ap .
(.)
By (.), (.), and (.), we obtain
kn – p ≤ zn – p + γn xn – p + cn
≤ un – p + γn xn – p + cn
≤ xn – p + rn (rn – ζ )Axn – Ap + γn xn – p + cn ,
which implies that
c(ζ – d)Axn – Ap ≤ rn (ζ – rn )Axn – Ap
≤ xn – p – kn – p + γn xn – p + cn
≤ xn – kn xn – p + kn – p + γn xn – p + cn .
From limn→∞ γn = , limn→∞ cn = , and (.), we have
lim Axn – Ap = .
n→∞
and Lemma .(a), we have
By the ﬁrm nonexpansivity of Sr(,ϕ)
n
un – p

= Sr(Θ,ϕ)
(I – rn A)xn – Sr(Θ,ϕ)
(I – rn A)p
n
n
(.)
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≤ (I – rn A)xn – (I – rn A)p, un – p
=
 (I – rn A)xn – (I – rn A)p
 + un – p

 – (I – rn A)xn – (I – rn A)p – (un – p)
 
xn – p + un – p – xn – un – rn (Axn – Ap)


= xn – p + un – p – xn – un  + rn Axn – Ap, xn – un 
– rn Axn – Ap ,
≤
which implies that
un – p ≤ xn – p – xn – un  + rn Axn – Apxn – un .
(.)
Combining (.) and (.), we have
kn – p ≤ zn – p + γn xn – p + cn
≤ un – p + γn xn – p + cn
≤ xn – p – xn – un  + rn Axn – Apxn – un + γn xn – p + cn ,
which implies
xn – un 
≤ xn – p – kn – p + rn Axn – Apxn – un + γn xn – p + cn
≤ xn – kn xn – p + kn – p + rn Axn – Apxn – un + γn xn – p + cn .
From limn→∞ γn = , limn→∞ cn = , (.), and (.), we know that (.) holds.
Next we show that limn→∞ Bi Λin un –Bi p = , i = , , . . . , N . It follows from Lemma .
that
i
Λ un – p
 = JR ,λ (I – λi,n Bi )Λi– un – JR ,λ (I – λi,n Bi )p

n
n
i i,n
i i,n

≤ (I – λi,n Bi )Λi–
n un – (I – λi,n Bi )p

+ λi,n (λi,n – ηi )
Bi Λi– un – Bi p

≤ Λi–
n un – p
n

≤ un – p + λi,n (λi,n – ηi )
Bi Λi– un – Bi p
n

≤ xn – p + λi,n (λi,n – ηi )
Bi Λi–
n un – Bi p .
Combining (.) and (.), we have
kn – p ≤ zn – p + γn xn – p + cn

≤ Λin un – p
+ γn xn – p + cn
(.)
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
≤ xn – p + λi,n (λi,n – ηi )
Bi Λi–
n un – Bi p
+ γn xn – p + cn ,
together with {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ), i ∈ {, , . . . , N}, implies

≤ λi,n (ηi – λi,n )
Bi Λi– un – Bi p

ai (ηi – bi )
Bi Λi–
n un – Bi p
n
≤ xn – p – kn – p + γn xn – p + cn
≤ xn – kn xn – p + kn – p
+ γn xn – p + cn .
From limn→∞ γn = , limn→∞ cn = , and (.), we obtain
lim Bi Λi–
n un – Bi p = ,
n→∞
i = , , . . . , N.
(.)
By Lemma . and Lemma .(a), we obtain
i
Λ un – p
 = JR ,λ (I – λi,n Bi )Λi– un – JR ,λ (I – λi,n Bi )p

n
n
i i,n
i i,n
i
≤ (I – λi,n Bi )Λi–
n un – (I – λi,n Bi )p, Λn un – p
=
 (I – λi,n Bi )Λi– un – (I – λi,n Bi )p
 + Λi un – p

n
n

 – (I – λi,n Bi )Λi– un – (I – λi,n Bi )p – Λi un – p n
n
 i
 + Λ un – p

≤ Λi–
n un – p
n

 – Λi– un – Λi un – λi,n Bi Λi– un – Bi p n
n
n


≤ un – p + Λin un – p

 – Λi– un – Λi un – λi,n Bi Λi– un – Bi p n
n
n


≤ xn – p + Λin un – p

 – Λi– un – Λi un – λi,n Bi Λi– un – Bi p ,
n
n
n
which implies
i
Λ un – p

n

i
i–
≤ xn – p – Λi–
n un – Λn un – λi,n Bi Λn un – Bi p

i
– λ Bi Λi– un – Bi p

= xn – p – Λi–
n un – Λn un
i,n
n
i
i–
+ λi,n Λi–
n un – Λn un , Bi Λn un – Bi p

i
≤ xn – p – Λi–
n un – Λn un
i
Bi Λi– un – Bi p
.
+ λi,n Λi–
n un – Λn un
n
(.)
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Combining (.) and (.) we get
kn – p ≤ zn – p + γn xn – p + cn

≤ Λin un – p
+ γn xn – p + cn

i
≤ xn – p – Λi–
n un – Λn un
i–
+ λi,n Λn un – Λin un Bi Λi–
n un – Bi p
+ γn xn – p + cn ,
which implies
i–
Λ un – Λi un 
n
n
i
Bi Λi– un – Bi p
≤ xn – p – kn – p + λi,n Λi–
n un – Λn un
n
+ γn xn – p + cn
i
Bi Λi– un – Bi p
≤ xn – kn xn – p + kn – p + λi,n Λi–
n un – Λn un
n
+ γn xn – p + cn .
From (.), (.), limn→∞ γn = , and limn→∞ cn = , we have
i
lim Λi–
n un – Λn un = ,
n→∞
i = , , . . . , N.
(.)
From (.) we get
un – zn = Λn un – ΛN
n un

N
≤ Λn un – Λn un + Λn un – Λn un + · · · + ΛN–
n un – Λn un
→  as n → ∞.
(.)
By (.) and (.), we have
xn – zn ≤ xn – un + un – zn →  as n → ∞.
(.)
From (.) and (.), we have
zn+ – zn ≤ zn+ – xn+ + xn+ – xn + xn – zn →  as n → ∞.
(.)
By (.), (.), and (.), we get
kn – zn ≤ kn – xn + xn – un + un – zn →
as n → ∞.
(.)
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We observe that
kn – zn = ( – δn ) Sn zn – zn .
From δn ≤ d <  and (.), we have
lim Sn zn – zn = .
n→∞
(.)
We note that
n
S zn – Sn+ zn ≤ Sn zn – zn + zn – zn+ + zn+ – Sn+ zn+ + Sn+ zn+ – Sn+ zn .
From (.), (.), and Lemma ., we obtain
lim Sn zn – Sn+ zn = .
n→∞
(.)
On the other hand, we note that
zn – Szn ≤ zn – Sn zn + Sn zn – Sn+ zn + Sn+ zn – Szn .
From (.), (.), and the uniform continuity of S, we have
lim zn – Szn = .
n→∞
(.)
In addition, note that
zn – Wzn ≤ zn – kn + kn – Wn zn + Wn zn – Wzn .
So, from (.), (.), and Remark . it follows that
lim zn – Wzn = .
n→∞
(.)
Step . we prove that xn → v = PΩ x as n → ∞.
Indeed, since {xn } is bounded, there exists a subsequence {xni } which converges weakly
to some w. From (.) and (.)-(.), we see that uni w, Λm
ni uni w, and zni w,
where m ∈ {, , . . . , N}. Since S is uniformly continuous, by (.) we get limn→∞ zn –
Sm zn =  for any m ≥ . Hence from Lemma ., we obtain w ∈ Fix(S). In the meantime,
utilizing Lemma ., we deduce from (.) and zni w that w ∈ Fix(W ) = ∞
n= Fix(Tn )
N
(due to Lemma .). Next, we prove that w ∈ m= I(Bm , Rm ). As a matter of fact, since Bm
is ηm -inverse-strongly monotone, Bm is a monotone and Lipschitz-continuous mapping. It
follows from Lemma . that Rm + Bm is maximal monotone. Let (v, g) ∈ G(Rm + Bm ), i.e.,
m–
g – Bm v ∈ Rm v. Again, since Λm
n un = JRm ,λm,n (I – λm,n Bm )Λn un , n ≥ , m ∈ {, , . . . , N},
we have
m–
m
Λm–
n un – λm,n Bm Λn un ∈ (I + λm,n Rm )Λn un ,
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that is,
 m–
m–
m
Λ un – Λm
n un – λm,n Bm Λn un ∈ Rm Λn un .
λm,n n
In terms of the monotonicity of Rm , we get
v – Λm
n un , g
 m–
m
m–
– Bm v –
Λ un – Λn un – λm,n Bm Λn un ≥ 
λm,n n
and hence
v – Λm
n un , g
 m–
m
m–
≥ v – Λm
u
,
B
v
+
u
–
Λ
u
–
λ
B
Λ
u
Λ
n
m
n
n
m,n
m
n
n
n
n
λm,n n
 m–
m
m
m
m–
m
Λ un – Λn un
= v – Λn un , Bm v – Bm Λn un + Bm Λn un – Bm Λn un +
λm,n n
 m–
m
m–
m
m
u
,
B
Λ
u
–
B
u
u
,
u
–
Λ
u
+
v
–
Λ
Λ
≥ v – Λm
m n
n
n
n n m n n
n n
n n .
λm,n n
In particular,
m
m
m–
v – Λm
ni uni , g ≥ v – Λni uni , Bm Λni uni – Bm Λni uni
 m–
m
m
+ v – Λni uni ,
Λ uni – Λni uni .
λm,ni ni
m–
m
m–
Since Λm
n un – Λn un →  (due to (.)) and Bm Λn un – Bm Λn un →  (due to the
m
Lipschitz-continuity of Bm ), we conclude from Λni uni w and {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ),
i ∈ {, , . . . , N} that
lim v – Λm
ni uni , g = v – w, g ≥ .
i→∞
It follows from the maximal monotonicity of Bm + Rm that  ∈ (Rm + Bm )w, i.e., w ∈
I(Bm , Rm ). Therefore, w ∈ N
m= I(Bm , Rm ).
(Θ,ϕ)
Next, we show that w ∈ GMEP(Θ, ϕ, A). In fact, from zn = Srn (I – rn A)xn , we know
that
Θ(un , y) + ϕ(y) – ϕ(un ) + Axn , y – un +
 K (un ) – K (xn ), y – un ≥ ,
rn
∀y ∈ C.
From (H) it follows that
ϕ(y) – ϕ(un ) + Axn , y – un +
 K (un ) – K (xn ), y – un ≥ Θ(y, un ),
rn
∀y ∈ C.
Replacing n by ni , we have
ϕ(y) – ϕ(uni ) + Axni , y – uni +
∀y ∈ C.
K (uni ) – K (xni )
, y – uni ≥ Θ(y, uni ),
rni
(.)
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Put ut = ty + ( – t)w for all t ∈ (, ] and y ∈ C. Then, from (.), we have
ut – uni , Aut ≥ ut – uni , Aut – ϕ(ut ) + ϕ(uni ) – ut – uni , Axni K (uni ) – K (xni )
, ut – uni + Θ(ut , uni )
–
rni
≥ ut – uni , Aut – Auni + ut – uni , Auni – Axni – ϕ(ut ) + ϕ(uni )
K (uni ) – K (xni )
, ut – uni + Θ(ut , uni ).
–
rni
Since uni – xni →  as i → ∞, we deduce from the Lipschitz-continuity of A and K that Auni – Axni →  and K (uni ) – K (xni ) →  as i → ∞. Further, from the monotonicity of A, we have ut – uni , Aut – Auni ≥ . So, from (H), we have the weakly lower
semicontinuity of ϕ,
K (uni )–K (xni )
rni
→  and uni w, then we have
ut – w, Aut ≥ –ϕ(ut ) + ϕ(w) + Θ(ut , w),
as i → ∞.
(.)
From (H), (H), and (.) we also have
 = Θ(ut , ut ) + ϕ(ut ) – ϕ(ut )
≤ tΘ(ut , y) + ( – t)Θ(ut , w) + tϕ(y) + ( – t)ϕ(w) – ϕ(ut )
= t Θ(ut , y) + ϕ(y) – ϕ(ut ) + ( – t) Θ(ut , w) + ϕ(w) – ϕ(w) – ϕ(ut )
≤ t Θ(ut , y) + ϕ(y) – ϕ(ut ) + ( – t)ut – w, Aut = t Θ(ut , y) + ϕ(y) – ϕ(ut ) + ( – t)ty – w, Aut ,
and hence
 ≤ Θ(ut , y) + ϕ(y) – ϕ(ut ) + ( – t)y – w, Aut .
Letting t → , we have, for each y ∈ C,
 ≤ Θ(w, y) + ϕ(y) – ϕ(w) + Aw, y – w.
This implies that w ∈ GMEP(Θ, ϕ, A). Therefore,
w∈
∞
n=
Fix(Tn ) ∩ GMEP(Θ, ϕ, A) ∩
N
I(Bi , Ri ) ∩ Fix(S) := Ω.
i=
This shows that ωw (xn ) ⊂ Ω. From (.) and Lemma . we infer that xn → v = PΩ x as
n → ∞.
Finally, assume additionally that γn + cn + xn – yn = o(αn ). Note that V is a γ¯ -strongly
positive bounded linear operator and f : H → H is an l-Lipschitzian mapping with γ l <
( + μ)γ¯ . It is clear that
V¯ x – u + γ f (x) – V¯ y – u + γ f (y) , x – y ≥ ( + μ)γ¯ – γ l x – y ,
∀x, y ∈ H.
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Hence we deduce that V¯ x – (u + γ f (x)) is (( + μ)γ¯ – γ l)-strongly monotone. In the
meantime, it is easy to see that V¯ x – (u + γ f (x)) is (V¯ + γ l)-Lipschitzian with constant
V¯ + γ l > . Thus, there exists a unique solution p in Ω to the VIP
V¯ p – u + γ f (p) , u – p ≥ ,
∀u ∈ Ω.
Equivalently, p ∈ Ω solves (OP) (due to Lemma .). Consequently, we deduce from
(.) and xn → v = PΩ x (n → ∞) that
lim sup u + γ f (p) – V¯ p, yn – p
n→∞
= lim sup
u + γ f (p) – V¯ p, xn – p + u + γ f (p) – V¯ p, yn – xn
n→∞
= lim sup u + γ f (p) – V¯ p, xn – p
n→∞
= u + γ f (p) – V¯ p, v – p ≤ .
Furthermore, by Lemma . we conclude from (.), (.), and (.) that
yn – p
= αn u + γ f (p) – V¯ p + αn γ f (xn ) – f (p) + βn (kn – p)

+ ( – βn )I – αn V¯ (Wn zn – p)

≤ αn γ f (xn ) – f (p) + βn (kn – p) + ( – βn )I – αn V¯ (Wn zn – p)
+ αn u + γ f (p) – V¯ p , yn – p

≤ αn γ f (xn ) – f (p)
+ βn kn – p + ( – βn )I – αn V¯ (Wn zn – p)
+ αn u + γ f (p) – V¯ p , yn – p

≤ αn γ lxn – p + βn kn – p + ( – βn – αn – αn μγ¯ )Wn zn – p
+ αn u + γ f (p) – V¯ p , yn – p

≤ αn γ lxn – p + βn kn – p +  – βn – αn ( + μ)γ¯ zn – p
+ αn u + γ f (p) – V¯ p , yn – p
γl
xn – p + βn kn – p
= αn ( + μ)γ¯ ·
( + μ)γ¯

+  – βn – αn ( + μ)γ¯ zn – p
+ αn u + γ f (p) – V¯ p , yn – p
(γ l)
xn – p + βn kn – p
( + μ) γ¯ 
+  – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p , yn – p
≤ αn ( + μ)γ¯ ·
≤ αn ( + μ)γ¯ ·
(γ l)
xn – p + βn ( + γn )zn – p + cn


( + μ) γ¯
(.)
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+  – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p , yn – p
(γ l)
xn – p + βn ( + γn )zn – p + cn
( + μ)γ¯
+  – βn – αn ( + μ)γ¯ ( + γn )zn – p + cn
+ αn u + γ f (p) – V¯ p , yn – p
≤ αn
(γ l)
xn – p +  – αn ( + μ)γ¯ ( + γn )zn – p + cn
( + μ)γ¯
+ αn u + γ f (p) – V¯ p , yn – p
= αn
(γ l) ( + γn )xn – p + cn
( + μ)γ¯
+  – αn ( + μ)γ¯ ( + γn )xn – p + cn + αn u + γ f (p) – V¯ p , yn – p
( + μ) γ¯  – (γ l) ( + γn )xn – p + cn
=  – αn
( + μ)γ¯
+ αn u + γ f (p) – V¯ p , yn – p
( + μ) γ¯  – (γ l)
xn – p + γn xn – p + cn
≤  – αn
( + μ)γ¯
+ αn u + γ f (p) – V¯ p , yn – p ,
≤ αn
which hence yields
( + μ) γ¯  – (γ l)
xn – p
( + μ)γ¯
xn – p – yn – p γn xn – p + cn
+
+  u + γ f (p) – V¯ p , yn – p
αn
αn
γn + cn xn – yn xn – p + yn – p +
xn – p + 
≤
αn
αn
+  u + γ f (p) – V¯ p , yn – p .
≤
Since γn + cn = o(αn ), xn – yn = o(αn ), and xn → v = PΩ x , we infer from (.) and  ≤
γ l < ( + μ)γ¯ that as n → ∞
( + μ) γ¯  – (γ l)
v – p ≤ .
( + μ)γ¯
That is, p = v = PΩ x . This completes the proof.
Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H. Let
Θ be a bifunction from C × C to R satisfying (H)-(H) and ϕ : C → R be a lower
semicontinuous and convex functional. Let Ri : C → H be a maximal monotone mapping and let A : H → H and Bi : C → H be ζ -inverse strongly monotone and ηi -inversestrongly monotone, respectively, for i = , . Let S : C → C be a uniformly continuous
asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some
 ≤ κ <  with sequence {γn } ⊂ [, ∞) such that limn→∞ γn =  and {cn } ⊂ [, ∞) such
Ceng et al. Journal of Inequalities and Applications 2014, 2014:462
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that limn→∞ cn = . Let {Tn }∞
n= be a sequence of nonexpansive self-mappings on C and
{λn } be a sequence in (, b] for some b ∈ (, ). Let V be a γ¯ -strongly positive bounded linear operator and f : H → H be an l-Lipschitzian mapping with γ l < ( + μ)γ¯ . Assume
that Ω := ( ∞
n= Fix(Tn )) ∩ GMEP(Θ, ϕ, A) ∩ I(B , R ) ∩ I(B , R ) ∩ Fix(S) is nonempty and
bounded. Let Wn be the W -mapping deﬁned by (.) and {αn }, {βn }, and {δn } be three sequences in (, ) such that limn→∞ αn =  and κ ≤ δn ≤ d < . Assume that:
(i) K : H → R is strongly convex with constant σ >  and its derivative K is
Lipschitz-continuous with constant ν >  such that the function x → y – x, K (x) is
weakly upper semicontinuous for each y ∈ H;
(ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any
y ∈/ Dx ,
Θ(y, zx ) + ϕ(zx ) – ϕ(y) +
 K (y) – K (x), zx – y < ;
r
(iii)  < lim infn→∞ βn ≤ lim supn→∞ βn < ;
(iv) {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ) for i = , , and {rn } ⊂ [, ζ ] satisﬁes
 < lim inf rn ≤ lim sup rn < ζ .
n→∞
n→∞
Pick any x ∈ H and set C = C, x = PC x . Let {xn } be a sequence generated by the following
algorithm:
⎧
u = Sr(Θ,ϕ)
(I – rn A)xn ,
⎪
n
⎪
⎪ n
⎪
⎪
zn = JR ,λ,n (I – λ,n B )JR ,λ,n (I – λ,n B )un ,
⎪
⎪
⎪
⎨k = δ z + ( – δ )Sn z ,
n
n n
n
n
⎪
yn = αn (u + γ f (xn )) + βn kn + (( – βn )I – αn (I + μV ))Wn zn ,
⎪
⎪
⎪
⎪


⎪
⎪Cn+ = {z ∈ Cn : yn – z ≤ xn – z + θn },
⎪
⎩
xn+ = PCn+ x , ∀n ≥ ,
(.)
where θn = (αn + γn )Δn + cn , Δn = sup{xn – p + u + (γ f – I – μV )p : p ∈ Ω} < ∞,
(Θ,ϕ)
and = –sup αn < ∞. If Sr
is ﬁrmly nonexpansive, then the following statements hold:
n≥
(I) {xn } converges strongly to PΩ x ;
(II) {xn } converges strongly to PΩ x , which solves the optimization problem
min
x∈Ω
μ

Vx, x + x – u – h(x),


(OP)
provided γn + cn + xn – yn = o(αn ) additionally, where h : H → R is the potential
function of γ f .
Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be
a bifunction from C × C to R satisfying (H)-(H) and ϕ : C → R be a lower semicontinuous
and convex functional. Let R : C → H be a maximal monotone mapping and let A : H → H
and B : C → H be ζ -inverse strongly monotone and ξ -inverse-strongly monotone, respectively. Let S : C → C be a uniformly continuous asymptotically κ-strict pseudocontractive
mapping in the intermediate sense for some  ≤ κ <  with sequence {γn } ⊂ [, ∞) such that
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limn→∞ γn =  and {cn } ⊂ [, ∞) such that limn→∞ cn = . Let {Tn }∞
n= be a sequence of nonexpansive self-mappings on C and {λn } be a sequence in (, b] for some b ∈ (, ). Let V be a
γ¯ -strongly positive bounded linear operator and f : H → H be an l-Lipschitzian mapping
with γ l < ( + μ)γ¯ . Assume that Ω := ( ∞
n= Fix(Tn )) ∩ GMEP(Θ, ϕ, A) ∩ I(B, R) ∩ Fix(S) is
nonempty and bounded. Let Wn be the W -mapping deﬁned by (.) and {αn }, {βn }, and
{δn } be three sequences in (, ) such that limn→∞ αn =  and κ ≤ δn ≤ d < . Assume that:
(i) K : H → R is strongly convex with constant σ >  and its derivative K is
Lipschitz-continuous with constant ν >  such that the function x → y – x, K (x) is
weakly upper semicontinuous for each y ∈ H;
(ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any
y ∈/ Dx ,
Θ(y, zx ) + ϕ(zx ) – ϕ(y) +
 K (y) – K (x), zx – y < ;
r
(iii)  < lim infn→∞ βn ≤ lim supn→∞ βn < ;
(iv) {ρn } ⊂ [a, b] ⊂ (, ξ ), and {rn } ⊂ [, ζ ] satisﬁes
 < lim inf rn ≤ lim sup rn < ζ .
n→∞
n→∞
Pick any x ∈ H and set C = C, x = PC x . Let {xn } be a sequence generated by the following
algorithm:
⎧
⎪
un = Sr(Θ,ϕ)
(I – rn A)xn ,
⎪
n
⎪
⎪
n
⎪
⎪
k
=
δ
J
⎨ n n R,ρn (I – ρn B)un + ( – δn )S JR,ρn (I – ρn B)un ,
yn = αn (u + γ f (xn )) + βn kn + (( – βn )I – αn (I + μV ))Wn JR,ρn (I – ρn B)un ,
⎪
⎪


⎪
C
⎪
n+ = {z ∈ Cn : yn – z ≤ xn – z + θn },
⎪
⎪
⎩x = P
∀n ≥ ,
n+
Cn+ x ,
(.)
where θn = (αn + γn )Δn + cn , Δn = sup{xn – p + u + (γ f – I – μV )p : p ∈ Ω} < ∞,
(Θ,ϕ)
is ﬁrmly nonexpansive, then the following statements hold:
and = –sup αn < ∞. If Sr
n≥
(I) {xn } converges strongly to PΩ x ;
(II) {xn } converges strongly to PΩ x , which solves the optimization problem
min
x∈Ω
μ

Vx, x + x – u – h(x),


(OP)
provided γn + cn + xn – yn = o(αn ) additionally, where h : H → R is the potential
function of γ f .
4 Weak convergence theorems
In this section, we introduce and analyze another iterative algorithm for ﬁnding common
solutions of a ﬁnite family of variational inclusions for maximal monotone and inversestrongly monotone mappings with the constraints of two problems: a generalized mixed
equilibrium problem and a common ﬁxed point problem of an inﬁnite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Under mild conditions imposed on the parameter
sequences we will prove weak convergence of the proposed algorithm.
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Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H. Let N
be an integer. Let Θ be a bifunction from C × C to R satisfying (H)-(H) and ϕ : C → R
be a lower semicontinuous and convex functional. Let Ri : C → H be a maximal monotone mapping and let A : H → H and Bi : C → H be ζ -inverse-strongly monotone and
ηi -inverse-strongly monotone, respectively, where i ∈ {, , . . . , N}. Let S : C → C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some  ≤ κ <  with sequences {γn } ⊂ [, ∞) and {cn } ⊂ [, ∞). Let {Tn }∞
n=
be a sequence of nonexpansive self-mappings on C and {λn } be a sequence in (, b] for
some b ∈ (, ). Let V be a γ¯ -strongly positive bounded linear operator and f : H → H
be an l-Lipschitzian mapping with γ l < ( + μ)γ¯ . Assume that Ω := ( ∞
n= Fix(Tn )) ∩
N
GMEP(Θ, ϕ, A) ∩ ( i= I(Bi , Ri )) ∩ Fix(S) is nonempty. Let Wn be the W -mapping deﬁned
by (.) and {αn }, {βn } and {δn } be three sequences in (, ) such that  < κ + ε ≤ δn ≤ d < .
Assume that:
(i) K : H → R is strongly convex with constant σ >  and its derivative K is
Lipschitz-continuous with constant ν >  such that the function x → y – x, K (x) is
weakly upper semicontinuous for each y ∈ H;
(ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any
y ∈/ Dx ,
Θ(y, zx ) + ϕ(zx ) – ϕ(y) +
 K (y) – K (x), zx – y < ;
r
∞
(iii)
n= (αn + γn + cn ) < ∞ and  < lim infn→∞ βn ≤ lim supn→∞ βn < ;
(iv) {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ), ∀i ∈ {, , . . . , N}, and {rn } ⊂ [, ζ ] satisﬁes
 < lim inf rn ≤ lim sup rn < ζ .
n→∞
n→∞
Pick any x ∈ H and let {xn } be a sequence generated by the following algorithm:
⎧
un = Sr(Θ,ϕ)
(I – rn A)xn ,
⎪
n
⎪
⎪
⎨
zn = JRN ,λN,n (I – λN,n BN )JRN– ,λN–,n (I – λN–,n BN– ) · · · JR ,λ,n (I – λ,n B )un ,
⎪kn = δn zn + ( – δn )Sn zn ,
⎪
⎪
⎩
xn+ = αn (u + γ f (xn )) + βn kn + (( – βn )I – αn (I + μV ))Wn zn , ∀n ≥ .
(Θ,ϕ)
Then {xn } converges weakly to w = limn→∞ PΩ xn provided Sr
(.)
is ﬁrmly nonexpansive.
Proof First, let us show that limn→∞ xn – p exists for any p ∈ Ω. Put
Λin = JRi ,λi,n (I – λi,n Bi )JRi– ,λi–,n (I – λi–,n Bi– ) · · · JR ,λ,n (I – λ,n B )
for all i ∈ {, , . . . , N}, n ≥ , and Λn = I, where I is the identity mapping on H. Then we see
that zn = ΛN
n un . Take p ∈ Ω arbitrarily. Similarly to the proof of Theorem ., we obtain
un – p ≤ xn – p,
(.)
zn – p ≤ un – p,
(.)
un – p ≤ xn – p + rn (rn – ζ )Axn – Ap ,
(.)
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un – p ≤ xn – p – xn – un  + rn Axn – Apxn – un ,
i
Λ un – p
 ≤ xn – p + λi,n (λi,n – ηi )
Bi Λi– un – Bi p
 ,
n
n
(.)
i ∈ {, , . . . , N},
i
Λ un – p
 ≤ xn – p – Λi– un – Λi un 
n
n
n
i
Bi Λi– un – Bi p
,
+ λi,n Λi–
n un – Λn un
n
(.)
i ∈ {, , . . . , N}.
(.)
By Lemma .(b) we get
kn – p

= δn (zn – p) + ( – δn ) Sn zn – p 

= δn zn – p + ( – δn )
Sn zn – p
– δn ( – δn )
zn – Sn zn 
≤ δn zn – p + ( – δn ) ( + γn )zn – p + κ zn – Sn zn + cn

– δn ( – δn )
zn – Sn zn 
=  + γn ( – δn ) zn – p + ( – δn )(κ – δn )
zn – Sn zn + ( – δn )cn

≤ ( + γn )zn – p + ( – δn )(κ – δn )
zn – Sn zn + cn
≤ ( + γn )zn – p + cn .
Repeating the same arguments as in the proof of Theorem . we have
( – βn )I – αn (I + μV )
≤  – βn – αn – αn μγ¯ .
Then by Lemma . we deduce from (.), (.), (.), and  ≤ γ l ≤ ( + μ)γ¯ that
xn+ – p
= αn u + γ f (p) – V¯ p + αn γ f (xn ) – f (p) + βn (kn – p)

+ ( – βn )I – αn V¯ (Wn zn – p)

≤ αn γ f (xn ) – f (p) + βn (kn – p) + ( – βn )I – αn V¯ (Wn zn – p)
+ αn u + γ f (p) – V¯ p , xn+ – p

≤ αn γ f (xn ) – f (p)
+ βn kn – p + ( – βn )I – αn V¯ (Wn zn – p)
+ αn u + γ f (p) – V¯ p
xn+ – p

≤ αn γ lxn – p + βn kn – p + ( – βn – αn – αn μγ¯ )Wn zn – p
+ αn u + γ f (p) – V¯ p
xn+ – p

≤ αn ( + μ)γ¯ xn – p + βn kn – p +  – βn – αn ( + μ)γ¯ zn – p
(.)
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+ αn u + γ f (p) – V¯ p
xn+ – p
≤ αn ( + μ)γ¯ xn – p + βn kn – p +  – βn – αn ( + μ)γ¯ zn – p
+ αn u + γ f (p) – V¯ p
xn+ – p
≤ αn ( + μ)γ¯ xn – p + βn ( + γn )zn – p + cn
+  – βn – αn ( + μ)γ¯ zn – p + αn u + γ f (p) – V¯ p
xn+ – p
≤ αn ( + μ)γ¯ xn – p + βn ( + γn )xn – p + cn
+  – βn – αn ( + μ)γ¯ xn – p + αn u + γ f (p) – V¯ p
xn+ – p
≤ αn ( + μ)γ¯ ( + γn )xn – p + cn + βn ( + γn )xn – p + cn
+  – βn – αn ( + μ)γ¯ ( + γn )xn – p + cn
+ αn u + γ f (p) – V¯ p
xn+ – p
= ( + γn )xn – p + cn + αn u + γ f (p) – V¯ p
xn+ – p

≤ ( + γn )xn – p + cn + αn u + γ f (p) – V¯ p
+ xn+ – p ,
which hence yields
xn+ – p
 + γn
αn u + γ f (p) – V¯ p
 +  cn
xn – p +
 – αn
 – αn
 – αn

αn + γn
αn 
xn – p +
= +
cn
u + γ f (p) – V¯ p
+
 – αn
 – αn
 – αn

≤  + (αn + γn ) xn – p + αn u + γ f (p) – V¯ p
+ cn ,
≤
(.)
where = –sup αn < ∞ (due to {αn } ⊂ (, ) and limn→∞ αn = ). By Lemma ., we see
n≥
from ∞
n= (αn + γn + cn ) < ∞ that limn→∞ xn – p exists. Thus {xn } is bounded and so are
the sequences {un }, {zn }, and {kn }.
Also, utilizing Lemmas . and .(b) we obtain from (.), (.), and (.)
xn+ – p

= αn u + γ f (xn ) – V¯ Wn zn + βn (kn – p) + ( – βn )(Wn zn – p)

≤ βn (kn – p) + ( – βn )(Wn zn – p)
+ αn u + γ f (xn ) – V¯ Wn zn , xn+ – p
= βn kn – p + ( – βn )Wn zn – p – βn ( – βn )kn – Wn zn 
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p
≤ βn kn – p + ( – βn )zn – p – βn ( – βn )kn – Wn zn 
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p
≤ βn ( + γn )zn – p + cn + ( – βn )zn – p
– βn ( – βn )kn – Wn zn 
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+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p
≤ βn ( + γn )zn – p + cn + ( – βn ) ( + γn )zn – p + cn
– βn ( – βn )kn – Wn zn 
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p
≤ ( + γn )zn – p + cn – βn ( – βn )kn – Wn zn 
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p
≤ ( + γn )xn – p + cn – βn ( – βn )kn – Wn zn 
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p,
(.)
which leads to
βn ( – βn )kn – Wn zn 
≤ xn – p – xn+ – p + γn xn – p + cn
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p.
Since limn→∞ αn = , limn→∞ γn = , and limn→∞ cn = , it follows from the existence of
limn→∞ xn – p and condition (iii) that
lim kn – Wn zn = .
n→∞
(.)
Note that
xn+ – kn = αn u + γ f (xn ) – V¯ Wn zn + ( – βn )(Wn zn – kn ),
which yields
xn+ – kn ≤ αn u + γ f (xn ) – V¯ Wn zn + ( – βn )Wn zn – kn ≤ αn u + γ f (xn ) – V¯ Wn zn + Wn zn – kn .
So, from (.) and limn→∞ αn = , we get
lim xn+ – kn = .
n→∞
In the meantime, we conclude from (.), (.), (.), and (.) that
xn+ – p
≤ βn kn – p + ( – βn )zn – p – βn ( – βn )kn – Wn zn 
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p
≤ βn kn – p + ( – βn )zn – p + αn u + γ f (xn ) – V¯ Wn zn xn+ – p

≤ βn ( + γn )zn – p + ( – δn )(κ – δn )
zn – Sn zn + cn
(.)
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+ ( – βn )zn – p + αn u + γ f (xn ) – V¯ Wn zn xn+ – p

≤ βn ( + γn )zn – p + ( – δn )(κ – δn )
zn – Sn zn + cn
+ ( – βn ) ( + γn )zn – p + cn
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p

≤ ( + γn )zn – p + βn ( – δn )(κ – δn )
zn – Sn zn + cn
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p

≤ ( + γn )xn – p + βn ( – δn )(κ – δn )
zn – Sn zn + cn
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p,
which, together with  < κ + ≤ δn ≤ d < , implies that


( – d)βn zn – Sn zn ≤ βn ( – δn )(δn – κ)
zn – Sn zn ≤ xn – p – xn+ – p + γn xn – p + cn
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p.
Consequently, from limn→∞ αn = , limn→∞ γn = , limn→∞ cn = , condition (iii), and the
existence of limn→∞ xn – p, we get
lim zn – Sn zn = .
n→∞
(.)
Since kn – zn = ( – δn )(Sn zn – zn ), from (.) we have
lim kn – zn = .
n→∞
(.)
Combining (.), (.), and (.), we have
xn+ – p ≤ βn ( + γn )zn – p + cn + ( – βn )zn – p – βn ( – βn )kn – Wn zn 
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p
≤ βn ( + γn )zn – p + cn + ( – βn ) ( + γn )zn – p + cn
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p
= ( + γn )zn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p
≤ un – p + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p
≤ xn – p + rn (rn – ζ )Axn – Ap + γn xn – p + cn
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p,
which implies
rn (ζ – rn )Axn – Ap
≤ xn – p – xn+ – p + γn xn – p + cn
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p.
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From condition (iv), limn→∞ αn = , limn→∞ γn = , limn→∞ cn = , and the existence of
limn→∞ xn – p, we get
lim Axn – Ap = .
(.)
n→∞
Combining (.), (.), and (.), we have
xn+ – p ≤ zn – p + γn zn – p + cn
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p
≤ un – p + γn xn – p + cn
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p
≤ xn – p – xn – un  + rn Axn – Apxn – un + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p,
which implies
xn – un  ≤ xn – p – xn+ – p + rn Axn – Apxn – un + γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p.
From (.), limn→∞ αn = , limn→∞ γn = , limn→∞ cn = , and the existence of
limn→∞ xn – p, we obtain
lim xn – un = .
(.)
n→∞
Combining (.), (.), and (.), we have
xn+ – p
≤ zn – p + γn zn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p

≤ Λin un – p
+ γn zn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p

+ γn xn – p + cn
≤ xn – p + λi,n (λi,n – ηi )
Bi Λi–
n un – Bi p
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p,
which implies

λi,n (λi,n – ηi )
Bi Λi–
n un – Bi p
≤ xn – p – xn+ – p + γn xn – p + cn
+ αn u + γ f (xn ) – V¯ Wn zn xn+ – p.
From {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ), i ∈ {, , . . . , N}, limn→∞ αn = , limn→∞ γn = , limn→∞ cn =
, and the existence of limn→∞ xn – p, we obtain
lim Bi Λi–
n un – Bi p = ,
n→∞
i ∈ {, , . . . , N}.
(.)
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Combining (.), (.), and (.), we get
xn+ – p
≤ zn – p + γn zn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p

≤ Λin un – p
+ γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p

i
+ λi,n Λi– un – Λi un Bi Λi– un – Bi p
≤ xn – p – Λi–
n un – Λn un
n
n
n
+ γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p,
which implies
i–
Λ un – Λi un 
n
n
i
Bi Λi– un – Bi p
≤ xn – p – xn+ – p + λi,n Λi–
n un – Λn un
n
+ γn xn – p + cn + αn u + γ f (xn ) – V¯ Wn zn xn+ – p.
From (.), limn→∞ αn = , limn→∞ γn = , limn→∞ cn = , and the existence of
limn→∞ xn – p, we obtain
i
lim Λi–
n un – Λn un = ,
n→∞
i ∈ {, , . . . , N}.
(.)
By (.), we have
un – zn = Λn un – ΛN
n un

N
≤ Λn un – Λn un + Λn un – Λn un + · · · + ΛN–
n un – Λn un
→  as n → ∞.
(.)
From (.) and (.), we have
xn – zn ≤ xn – un + un – zn →  as n → ∞.
(.)
By (.) and (.), we obtain
kn – xn ≤ kn – zn + zn – xn →  as n → ∞,
(.)
which, together with (.) and (.), implies that
xn+ – xn ≤ xn+ – kn + kn – xn →
as n → ∞.
On the other hand, we observe that
zn+ – zn ≤ zn+ – xn+ + xn+ – xn + xn – zn .
(.)
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By (.) and (.), we have
lim zn+ – zn = .
n→∞
(.)
We note that
zn – Szn ≤ zn – zn+ + zn+ – Sn+ zn+ + Sn+ zn+ – Sn+ zn + Sn+ zn – Szn .
From (.), (.), Lemma ., and the uniform continuity of S, we obtain
lim zn – Szn = .
n→∞
(.)
In addition, note that
zn – Wzn ≤ zn – kn + kn – Wn zn + Wn zn – Wzn So, from (.), (.), and Remark . it follows that
lim zn – Wzn = .
n→∞
(.)
Since {xn } is bounded, there exists a subsequence {xni } of {xn } which converges weakly to w.
From (.) and (.), we have zni w and kni w. From (.) and the uniform continuity of S, we have limn→∞ zn – Sm zn =  for any m ≥ . So, from Lemma ., we have
w ∈ Fix(S). In the meantime, by (.) and Lemma ., we get w ∈ Fix(W ) = ∞
n= Fix(Tn )
(due to Lemma .). Utilizing similar arguments to those in the proof of Theorem ., we
can derive w ∈ GMEP(Θ, ϕ, A) ∩ ( N
i= I(Bi , Ri )). Consequently, w ∈ Ω. This shows that
ωw (xn ) ⊂ Ω.
Next let us show that ωw (xn ) is a single-point set. As a matter of fact, let {xnj } be another
subsequence of {xn } such that xnj w . Then we get w ∈ Ω. If w = w , from the Opial
condition, we have
lim xn – w = lim xni – w
n→∞
i→∞
< lim xni – w i→∞
= lim xn – w n→∞
= lim xnj – w j→∞
< lim xnj – w
j→∞
= lim xn – w.
n→∞
This attains a contradiction. So we have w = w . Put vn = PΩ xn . Since w ∈ Ω, we have
xn – vn , vn – w ≥ . By Lemma ., we see that {vn } converges strongly to some w ∈ Ω.
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Since {xn } converges weakly to w, we have
w – w , w – w ≥ .
Therefore we obtain w = w = limn→∞ PΩ xn . This completes the proof.
Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ
be a bifunction from C × C to R satisfying (H)-(H) and ϕ : C → R be a lower semicontinuous and convex functional. Let Ri : C → H be a maximal monotone mapping and
let A : H → H and Bi : C → H be ζ -inverse-strongly monotone and ηi -inverse-strongly
monotone, respectively, for i = , . Let S : C → C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some  ≤ κ <  with
sequences {γn } ⊂ [, ∞) and {cn } ⊂ [, ∞). Let {Tn }∞
n= be a sequence of nonexpansive selfmappings on C and {λn } be a sequence in (, b] for some b ∈ (, ). Let V be a γ¯ -strongly
positive bounded linear operator and f : H → H be an l-Lipschitzian mapping with γ l <
( + μ)γ¯ . Assume that Ω := ( ∞
n= Fix(Tn )) ∩ GMEP(Θ, ϕ, A) ∩ I(B , R ) ∩ I(B , R ) ∩ Fix(S)
is nonempty. Let Wn be the W -mapping deﬁned by (.) and {αn }, {βn }, and {δn } be three
sequences in (, ) such that  < κ + ε ≤ δn ≤ d < . Assume that:
(i) K : H → R is strongly convex with constant σ >  and its derivative K is
Lipschitz-continuous with constant ν >  such that the function x → y – x, K (x) is
weakly upper semicontinuous for each y ∈ H;
(ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any
y ∈/ Dx ,
Θ(y, zx ) + ϕ(zx ) – ϕ(y) +
 K (y) – K (x), zx – y < ;
r
∞
(iii)
n= (αn + γn + cn ) < ∞ and  < lim infn→∞ βn ≤ lim supn→∞ βn < ;
(iv) {λi,n } ⊂ [ai , bi ] ⊂ (, ηi ) for i = , , and {rn } ⊂ [, ζ ] satisﬁes
 < lim inf rn ≤ lim sup rn < ζ .
n→∞
n→∞
Pick any x ∈ H and let {xn } be a sequence generated by the following algorithm:
⎧
un = Sr(Θ,ϕ)
(I – rn A)xn ,
⎪
n
⎪
⎪
⎨
zn = JR ,λ,n (I – λ,n B )JR ,λ,n (I – λ,n B )un ,
⎪kn = δn zn + ( – δn )Sn zn ,
⎪
⎪
⎩
xn+ = αn (u + γ f (xn )) + βn kn + (( – βn )I – αn (I + μV ))Wn zn ,
(.)
∀n ≥ .
Then {xn } converges weakly to w = limn→∞ PΩ xn provided Sr(Θ,ϕ) is ﬁrmly nonexpansive.
Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be
a bifunction from C × C to R satisfying (H)-(H) and ϕ : C → R be a lower semicontinuous
and convex functional. Let R : C → H be a maximal monotone mapping and let A : H → H
and B : C → H be ζ -inverse-strongly monotone and ξ -inverse-strongly monotone, respectively. Let S : C → C be a uniformly continuous asymptotically κ-strict pseudocontractive
mapping in the intermediate sense for some  ≤ κ <  with sequences {γn } ⊂ [, ∞) and
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{cn } ⊂ [, ∞). Let {Tn }∞
n= be a sequence of nonexpansive self-mappings on C and {λn }
be a sequence in (, b] for some b ∈ (, ). Let V be a γ¯ -strongly positive bounded linear
operator and f : H → H be an l-Lipschitzian mapping with γ l < ( + μ)γ¯ . Assume that
Ω := ( ∞
n= Fix(Tn )) ∩ GMEP(Θ, ϕ, A) ∩ I(B, R) ∩ Fix(S) is nonempty. Let Wn be the W mapping deﬁned by (.) and {αn }, {βn }, and {δn } be three sequences in (, ) such that
 < κ + ε ≤ δn ≤ d < . Assume that:
(i) K : H → R is strongly convex with constant σ >  and its derivative K is
Lipschitz-continuous with constant ν >  such that the function x → y – x, K (x) is
weakly upper semicontinuous for each y ∈ H;
(ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any
y ∈/ Dx ,
Θ(y, zx ) + ϕ(zx ) – ϕ(y) +
 K (y) – K (x), zx – y < ;
r
∞
(iii)
n= (αn + γn + cn ) < ∞ and  < lim infn→∞ βn ≤ lim supn→∞ βn < ;
(iv) {ρn } ⊂ [a, b] ⊂ (, ξ ), and {rn } ⊂ [, ζ ] satisﬁes
 < lim inf rn ≤ lim sup rn < ζ .
n→∞
n→∞
Pick any x ∈ H and let {xn } be a sequence generated by the following algorithm:
⎧
un = Sr(Θ,ϕ)
(I – rn A)xn ,
⎪
n
⎪
⎪
⎨
kn = δn JR,ρn (I – ρn B)un + ( – δn )Sn JR,ρn (I – ρn B)un ,
⎪xn+ = αn (u + γ f (xn )) + βn kn + (( – βn )I
⎪
⎪
⎩
– αn (I + μV ))Wn JR,ρn (I – ρn B)un , ∀n ≥ .
(.)
Then {xn } converges weakly to w = limn→∞ PΩ xn provided Sr(Θ,ϕ) is ﬁrmly nonexpansive.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript.
Author details
1
Department of Mathematics, Shanghai Normal University, Shanghai, 200234, P.R. China. 2 Scientiﬁc Computing Key
Laboratory of Shanghai Universities, Shanghai, 200234, P.R. China. 3 Department of Applied Mathematics, National
Hsinchu University of Education, Hsinchu, 30033, Taiwan. 4 Department of Information Management, Yuan Ze University,
Chung-Li, 32003, Taiwan. 5 Innovation Center for Big Data and Digital Convergence, Yuan Ze University, Chung-Li, 32003,
Taiwan.
Acknowledgements
In this research, ﬁrst author was partially supported by the National Science Foundation of China (11071169), Innovation
Program of Shanghai Municipal Education Commission (09ZZ133) and Ph.D. Program Foundation of Ministry of
Education of China (20123127110002). The second author and third author were supported partly by the National
Science Council of the Republic of China.
Received: 14 August 2014 Accepted: 14 October 2014 Published: 21 Nov 2014
References
1. Peng, JW, Yao, JC: A new hybrid-extragradient method for generalized mixed equilibrium problems, ﬁxed point
problems and variational inequality problems. Taiwan. J. Math. 12, 1401-1432 (2008)
Ceng et al. Journal of Inequalities and Applications 2014, 2014:462
http://www.journalofinequalitiesandapplications.com/content/2014/1/462
2. Ceng, LC, Hu, HY, Wong, MM: Strong and weak convergence theorems for generalized mixed equilibrium problem
with perturbation and ﬁxed point problem of inﬁnitely many nonexpansive mappings. Taiwan. J. Math. 15(3),
1341-1367 (2011)
3. Yao, Y, Liou, YC, Yao, JC: New relaxed hybrid-extragradient method for ﬁxed point problems, a general system of
variational inequality problems and generalized mixed equilibrium problems. Optimization 60(3), 395-412 (2011)
4. Cai, G, Bu, SQ: Strong and weak convergence theorems for general mixed equilibrium problems and variational
inequality problems and ﬁxed point problems in Hilbert spaces. J. Comput. Appl. Math. 247, 34-52 (2013)
5. Yao, Y, Cho, YJ, Liou, YC: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and
ﬁxed point problems. Eur. J. Oper. Res. 212, 242-250 (2011)
6. Ceng, LC, Ansari, QH, Schaible, S: Hybrid extragradient-like methods for generalized mixed equilibrium problems,
system of generalized equilibrium problems and optimization problems. J. Glob. Optim. 53, 69-96 (2012)
7. Ceng, LC, Guu, SM, Yao, JC: Hybrid iterative method for ﬁnding common solutions of generalized mixed equilibrium
and ﬁxed point problems. Fixed Point Theory Appl. 2012, Article ID 92 (2012)
8. Ceng, LC, Petrusel, A: Relaxed extragradient-like method for general system of generalized mixed equilibria and ﬁxed
point problem. Taiwan. J. Math. 16(2), 445-478 (2012)
9. Rockafellar, RT: Monotone operators and the proximal point algorithms. SIAM J. Control Optim. 14, 877-898 (1976)
10. Huang, NJ: A new completely general class of variational inclusions with noncompact valued mappings. Comput.
Math. Appl. 35(10), 9-14 (1998)
11. Zeng, LC, Guu, SM, Yao, JC: Characterization of H-monotone operators with applications to variational inclusions.
Comput. Math. Appl. 50(3-4), 329-337 (2005)
12. Ceng, LC, Guu, SM, Yao, JC: Iterative approximation of solutions for a class of completely generalized set-valued
quasi-variational inclusions. Comput. Math. Appl. 56, 978-987 (2008)
13. Ceng, LC, Guu, SM, Yao, JC: Iterative algorithm for ﬁnding approximate solutions of mixed quasi-variational-like
inclusions. Comput. Math. Appl. 56, 942-952 (2008)
14. Ceng, LC, Ansari, QH, Wong, MM, Yao, JC: Mann type hybrid extragradient method for variational inequalities,
variational inclusions and ﬁxed point problems. Comput. Math. Appl. 13(2), 403-422 (2012)
15. Ahmad, R, Ansari, QH, Irfan, SS: Generalized variational inclusions and generalized resolvent equations in Banach
spaces. Comput. Math. Appl. 29, 1825-1835 (2005)
16. Ahmad, R, Ansari, QH: Generalized variational inclusions and H-resolvent equations with H-accretive operators.
Taiwan. J. Math. 111, 703-716 (2007)
17. Ahmad, R, Ansari, QH: System of vector quasi-variational inclusions with some applications. Nonlinear Anal. 69,
2812-2824 (2008)
18. Kim, TH, Xu, HK: Convergence of the modiﬁed Mann’s iteration method for asymptotically strict pseudocontractions.
Nonlinear Anal. 68, 2828-2836 (2008)
19. Sahu, DR, Xu, HK, Yao, JC: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear
Anal. 70, 3502-3511 (2009)
20. Ceng, LC, Yao, JC: A hybrid iterative scheme for mixed equilibrium problems and ﬁxed point problems. J. Comput.
Appl. Math. 214, 186-201 (2008)
21. Ceng, LC, Yao, JC: A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system
of generalized equilibria and a ﬁxed point problem. Nonlinear Anal. 72, 1922-1937 (2010)
22. Osilike, MO, Aniagbosor, SC, Akuchu, BG: Fixed points of asymptotically demicontractive mappings in arbitrary
Banach space. Panam. Math. J. 12, 77-88 (2002)
23. Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am.
Math. Soc. 73, 591-597 (1967)
24. Huang, S: Hybrid extragradient methods for asymptotically strict pseudo-contractions in the intermediate sense and
variational inequality problems. Optimization 60, 739-754 (2011)
25. Martinez-Yanes, C, Xu, HK: Strong convergence of the CQ method for ﬁxed point iteration process. Nonlinear Anal. 64,
2400-2411 (2006)
26. O’Hara, JG, Pillay, P, Xu, HK: Iterative approaches to convex feasibility problems in Banach spaces. Nonlinear Anal.
64(9), 2022-2042 (2006)
27. Yao, Y, Liou, YC, Yao, JC: Convergence theorem for equilibrium problems and ﬁxed point problems of inﬁnite family of
nonexpansive mappings. Fixed Point Theory Appl. 2007, Article ID 64363 (2007)
28. Goebel, K, Kirk, WA: Topics on Metric Fixed-Point Theory. Cambridge University Press, Cambridge (1990)
29. Yao, Y, Noor, MA, Zainab, S, Liou, YC: Mixed equilibrium problems and optimization problems. J. Math. Anal. Appl. 354,
319-329 (2009)
10.1186/1029-242X-2014-462
Cite this article as: Ceng et al.: On a ﬁnite family of variational inclusions with the constraints of generalized mixed
equilibrium and ﬁxed point problems. Journal of Inequalities and Applications 2014, 2014:462
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