R E S E A R C H
Open Access
On weak convergence of an iterative
algorithm for common solutions of inclusion
problems and fixed point problems in Hilbert
spaces
Yuan Hecai
**Correspondence:
[email protected] School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China
Abstract
In this paper, a monotone inclusion problem and a fixed point problem of
nonexpansive mappings are investigated based on a Mann-type iterative algorithm with mixed errors. Strong convergence theorems of common elements are established in the framework of Hilbert spaces.
MSC: 47H05; 47H09; 47J25
Keywords: maximal monotone operator; nonexpansive mapping; Hilbert space; fixed point
1 Introduction
Variational inclusion has become rich of inspiration in pure and applied mathematics. In recent years, classical variational inclusion problems have been extended and generalized to study a large variety of problems arising in image recovery, economics, and signal pro-cessing; for more details, see [–]. Based on the projection technique, it has been shown that the variational inclusion problems are equivalent to the fixed point problems. This al-ternative formulation has played a fundamental and significant part in developing several numerical methods for solving variational inclusion problems and related optimization problems.
The purposes of this paper is to study the zero point problem of the sum of a maximal monotone mapping and an inverse-strongly monotone mapping, and the fixed point prob-lem of a nonexpansive mapping. The organization of this paper is as follows. In Section , we provide some necessary preliminaries. In Section , a Mann-type iterative algorithm with mixed errors is investigated. A weak convergence theorem is established. Applica-tions of the main results are also discussed in this section.
2 Preliminaries
Throughout this paper, we always assume thatH is a real Hilbert space with the inner product·,·and the norm · , respectively. LetCbe a nonempty closed convex subset ofHand letPCbe the metric projection fromHontoC.
LetS:C→Cbe a mapping.F(S) stands for the fixed point set ofS; that is,F(S) :={x∈ C:x=Sx}.
Recall thatSis said to be nonexpansive iff
Sx–Sy ≤ x–y, ∀x,y∈C.
If Cis a bounded, closed, and convex subset ofH, thenF(S) is not empty, closed, and convex; see [].
LetA:C→Hbe a mapping. Recall thatAis said to be monotone iff
Ax–Ay,x–y ≥, ∀x,y∈C.
Ais said to be strongly monotone iff there exists a constantα> such that
Ax–Ay,x–y ≥αx–y, ∀x,y∈C.
For such a case,Ais also said to beα-strongly monotone.Ais said to be inverse-strongly monotone iff there exists a constantα> such that
Ax–Ay,x–y ≥αAx–Ay, ∀x,y∈C.
For such a case,Ais also said to beα-inverse-strongly monotone. It is not hard to see that inverse-strongly monotone mappings are monotone and Lipschitz continuous.
Recall that the classical variational inequality is to find anx∈Csuch that
Ax,y–x ≥, ∀y∈C. (.)
In this paper, we useVI(C,A) to denote the solution set of (.). It is known thatω∈Cis a solution to (.) iffωis a fixed point of the mappingPC(I–λA), whereλ> is a constant, andIstands for the identity mapping. IfAisα-inverse-strongly monotone andλ∈(, α], then the mappingPC(I–λA) is nonexpansive. Indeed, we have
(I–λA)x– (I–λA)y
=(x–y) –λ(Ax–Ay)
=x–y– λx–y,Ax–Ay+λAx–Ay
≤ x–y–λ(α–λ)Ax–Ay.
This shows thatPC(I–λA) is nonexpansive. It follows thatVI(C,A) is closed and convex. A multivalued operatorT :H→H with the domainD(T) ={x∈H:Tx=∅}and the
rangeR(T) ={Tx:x∈D(T)}is said to be monotone if forx∈D(T),x∈D(T),y∈Tx, andy∈Tx, we havex–x,y–y ≥. A monotone operatorTis said to be maximal if its graph G(T) ={(x,y) :y∈Tx}is not properly contained in the graph of any other monotone operator. Let I denote the identity operator onH and let T :H→H be a
maximal monotone operator. Then we can define, for eachλ> , a nonexpansive single-valued mappingJλ:H→HbyJλ= (I+λT)–. It is called the resolvent ofT. We know that
T– =F(J
λ) for allλ> andJλis firmly nonexpansive, that is,
Jλx–Jλy≤ Jλx–Jλy,x–y, ∀x,y∈H;
In [], Kamimura and Takahashi investigated the problem of finding zero points of a maximal monotone operator based on the following algorithm:
x∈H, xn+=αnxn+ ( –αn)Jλnxn, n= , , , . . . ,
where {αn} is a sequence in (, ),{λn}is a positive sequence,T :H→H is maximal
monotone andJλn= (I+λnT)–. They showed that the sequence{xn}converges weakly to somez∈T–() provided that the control sequence satisfies some restrictions. Further, using this result, they also investigated the case thatT=∂f, wheref :H→(–∞,∞] is a proper lower semicontinuous convex function. Convergence theorems are established in the framework of real Hilbert spaces.
In [], Takahashi an Toyoda investigated the problem of finding a common solutions of the variational inequality problem (.) and a fixed point problem of nonexpansive map-pings based on the following algorithm:
x∈C, xn+=αnxn+ ( –αn)SPC(xn–λnAxn), ∀n≥,
where{αn}is a sequence in (, ),{λn}is a positive sequence,S:C→Cis a nonexpansive
mapping and A:C→H is an inverse-strongly monotone mapping. They showed that the sequence{xn}converges weakly to somez∈VI(C,A)∩F(S) provided that the control sequence satisfies some restrictions.
In [], Tada and Takahashi investigated the problem of finding common solutions of an equilibrium problem and a fixed point problem of nonexpansive mappings based on the following algorithm:x∈Hand
⎧ ⎨ ⎩
un∈Csuch thatF(un,u) +r
nu–un,un–xn ≥, ∀u∈C,
xn+=αnxn+ ( –αn)Sun, ∀n≥,
where{αn}is a sequence in (, ),{rn}is a positive sequence,S:C→Cis a nonexpansive
mapping andF:C×C→Ris a bifunction. They showed that the sequence{xn}converges weakly to somez∈VI(C,A)∩F(S) provided that the control sequence satisfies some re-strictions.
Recently, fixed point and zero point problems have been studied by many authors based on iterative methods; see, for example, [–] and the references therein. In this paper, motivated by the above results, we consider the problem of finding a common solution to the zero point problem and the fixed point problem based on Mann-type iterative meth-ods with errors. Weak convergence theorems are established in the framework of Hilbert spaces.
To obtain our main results in this paper, we need the following lemmas.
Recall that a space is said to satisfy Opial’s condition [] if, for any sequence{xn} ⊂H
withxnx, wheredenotes the weak convergence, the inequality
lim inf
n→∞ xn–x<lim infn→∞ xn–y
holds for everyy∈Hwithy=x. Indeed, the above inequality is equivalent to the following:
lim sup
Lemma .[] Let C be a nonempty,closed,and convex subset of H,let A:C→H be a mapping,and let B:H⇒H be a maximal monotone operator.Then F(Jλ(I–λA)) =
(A+B)–(),where J
λ(I–λA)is the resolvent of B forλ> .
Lemma .[] Let{an},{bn},and{cn}be three nonnegative sequences satisfying the fol-lowing condition:
an+≤( +bn)an+cn, ∀n≥n,
where n is some nonnegative integer,
∞
n=bn<∞ and
∞
n=cn<∞. Then the limit
limn→∞anexists.
Lemma .[] Suppose that H is a real Hilbert space and <p≤tn≤q< for all n≥. Suppose further that{xn}and{yn}are sequences of H such that
lim sup n→∞
xn ≤r, lim sup n→∞
yn ≤r
and
lim
n→∞tnxn+ ( –tn)yn=r
hold for some r≥.Thenlimn→∞xn–yn= .
Lemma .[] Let C be a nonempty,closed,and convex subset of H.Let S:C→C be a nonexpansive mapping.Then the mapping I–S is demiclosed at zero,that is,if{xn}is a sequence in C such that xnx and xn¯ –Sxn→,thenx¯∈F(S).
3 Main results
Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H, let A:C→H be anα-inverse-strongly monotone mapping,let S:C→C be a nonexpan-sive mapping and let B be a maximal monotone operator on H such that the domain of B is included in C.Assume thatF=F(S)∩(A+B)–()=∅.Let{α
n},{βn},and{γn}be real number sequences in(, )such thatαn+βn+γn= .Let{λn}be a positive real number se-quence and let{en}be a bounded error sequence in C.Let{xn}be a sequence in C generated in the following iterative process:
x∈C, xn+=αnSxn+βnJλn(xn–λnAxn) +γnen (.)
for all n∈N,where Jλn= (I+λnB)–.Assume that the sequences{αn},{βn},{γn},and{λn}
satisfy the following restrictions:
(a) <a≤βn≤b< , (b) <c≤λn≤d< α,
(c) ∞n=γn<∞,
Proof Notice thatI–λnAis nonexpansive. Indeed, we have
(I–λnA)x– (I–λnA)y
=(x–y) –λn(Ax–Ay)
=x–y– λnx–y,Ax–Ay+λnAx–Ay
≤ x–y–λn(α–λn)Ax–Ay.
In view of the restriction (b), we find thatI–λnAis nonexpansive. Fixingp∈F, we find from Lemma . that
p=Sp=Jλn(p–λnAp).
Putyn=Jλn(xn–λnAxn). SinceJλnandI–λnAare nonexpansive, we have
yn–p ≤(xn–λnAxn) – (p–λnAp)
≤ xn–p. (.)
On the other hand, we have
xn+–p ≤αnxn–p+βnyn–p+γnen–p
≤ xn–p+γnen–p. (.)
We find thatlimn→∞xn–pexists with the aid of Lemma .. This in turn implies that
{xn}and{yn}are bounded. Putlimn→∞xn–p=L> . Notice that
Sxn–p+γn(en–Sxn)≤ Sxn–p+γnen–Sxn
≤ xn–p+γnen–Sxn.
This implies from the restriction (c) that
lim sup n→∞
Sxn–p+γn(en–Sxn)≤L.
We also have
yn–p+γn(en–Sxn)≤ yn–p+γnen–Sxn
≤ xn–p+γnen–Sxn.
This implies from the restriction (c) that
lim sup n→∞
yn–p+γn(en–Sxn)≤L.
On the other hand, we have
xn+–p=( –βn)
Sxn–p+γn(en–Sxn)
+βn
It follows from Lemma . that
lim
n→∞Sxn–yn= . (.)
Notice that
yn–p≤(xn–λnAxn) – (p–λnAp)
≤ x–p– αλnAx–Ap+λnAx–Ap
=x–p–λn(α–λn)Ax–Ap. (.)
This implies that
xn+–p≤αnxn–p+βnyn–p+γnen–p
≤αnxn–p+βnxn–p–βnλn(α–λn)Axn–Ap+γnen–p
≤ xn–p–βnλn(α–λn)Axn–Ap+γnen–p. (.)
It follows that
βnλn(α–λn)Axn–Ap≤ xn–p–xn+–p+γnen–p.
In view of the restrictions (a), (b), and (c), we obtain that
lim
n→∞Axn–Ap= . (.)
Notice that
yn–p=Jλn(xn–λnAxn) –Jλn(p–λnAp)
≤ (xn–λnAxn) – (p–λnAp),yn–p
=
(xn–λnAxn) – (p–λnAp)
+yn–p
–(xn–λnAxn) – (p–λnAp) – (yn–p)
≤
xn–p+yn–p–xn–yn–λn(Axn–Ap)
≤
xn–p+yn–p–xn–yn–λnAxn–Ap
+ λnxn–ynAxn–Ap
≤
xn–p+yn–p–xn–yn+ λnxn–ynAxn–Ap
.
It follows that
On the other hand, we have
xn+–p≤αnxn–p+βnyn–p+γnen–p. (.)
Substituting (.) into (.), we arrive at
xn+–p≤αnxn–p+βnxn–p–βnxn–yn
+ βnλnxn–ynAxn–Ap+γnen–p
≤ xn–p–βnxn–yn+ βnλnxn–ynAxn–Ap+γnen–p.
It derives that
βnxn–yn≤ xn–p–xn+–p+ βnλnxn–ynAxn–Ap+γnen–p.
In view of the restrictions (a) and (c), we find from (.) that
lim
n→∞xn–yn= . (.)
Notice that
Sxn–xn ≤ Sxn–yn+yn–xn.
It follows from (.) and (.) that
lim
n→∞Sxn–xn= . (.)
Since{xn} is bounded, there exists a subsequence{xni} of{xn}such thatxni ω∈C, wheredenotes the weak convergence. From Lemma ., we find thatω∈F(S). In view of (.), we can choose a subsequence{yni}of{yn}such thatyni ω. Notice that
yn=Jλn(xn–λnAxn).
This implies that
xn–λnAxn∈(I+λnB)yn.
That is,
xn–yn
λn
–Axn∈Byn.
SinceBis monotone, we get for any (u,v)∈Bthat
yn–u, xn–yn
λn
–Axn–v
Replacingnbyniand lettingi→ ∞, we obtain from (.) that
ω–u, –Aω–v ≥.
This means –Aω∈Bω, that is, ∈(A+B)(ω). Hence, we getω∈(A+B)–(). This com-pletes the proof thatω∈F.
Suppose that there is another subsequence{xnj}of{xn}such thatxnj ω∗. Then we can show thatω∗∈Fin exactly the same way. Assume thatω=ω∗sincelimn→∞xn–pexits
for anyp∈F. Putlimn→∞xn–ω=d. Since the space satisfies Opial’s condition, we see that
d=lim inf
i→∞ xni–ω
<lim inf i→∞ xni–ω
∗
= lim n→∞xn–ω
∗
=lim inf j→∞
xnj–ω∗
<lim inf
j→∞ xnj–ω=d.
This is a contradiction. This shows thatω=ω∗. This proves that the sequence{xn}
con-verges weakly toω∈F. This completes the proof.
We obtain from Theorem . the following inclusion problem.
Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H, let A:C→H be anα-inverse-strongly monotone mapping,and let B be a maximal monotone operator on H such that the domain of B is included in C.Assume that(A+B)–()=∅.Let
{αn},{βn},and{γn}be real number sequences in(, )such thatαn+βn+γn= .Let{λn} be a positive real number sequence and let{en}be a bounded error sequence in C.Let{xn} be a sequence in C generated in the following iterative process:
x∈C, xn+=αnxn+βnJλn(xn–λnAxn) +γnen
for all n∈N,where Jλn= (I+λnB)–.Assume that the sequences{αn},{βn},{γn},and{λn}
satisfy the following restrictions:
(a) <a≤βn≤b< , (b) <c≤λn≤d< α,
(c) ∞n=γn<∞,
where a,b,c,and d are some real numbers.Then the sequence{xn}converges weakly to some point in(A+B)–().
Letf :H→(–∞,∞] be a proper lower semicontinuous convex function. Define the subdifferential
for allx∈H. Then∂fis a maximal monotone operator ofHinto itself; for more details, see []. LetCbe a nonempty closed convex subset ofHand letiCbe the indicator function ofC, that is,
iCx= ⎧ ⎨ ⎩
, x∈C,
∞, x∈/C.
Furthermore, we define the normal coneNC(v) ofCatvas follows:
NCv=z∈H:z,y–v ≤,∀y∈H
for anyv∈C. TheniC:H→(–∞,∞] is a proper lower semicontinuous convex function onH and∂iCis a maximal monotone operator. LetJλx= (I+λ∂iC)–xfor anyλ> and x∈H. From∂iCx=NCxandx∈C, we get
v=Jλx ⇔ x∈v+λNCv
⇔ x–v,y–v ≤, ∀y∈C,
⇔ v=PCx,
where PC is the metric projection from H into C. Similarly, we can get thatx∈(A+
∂iC)–()⇔x∈VI(A,C). PuttingB=∂iCin Theorem ., we find thatJλn=PC. The fol-lowing results are not hard to derive.
Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H,let A:
C→H be anα-inverse-strongly monotone mapping and let S:C→C be a nonexpansive mapping.Assume thatF=F(S)∩VI(C,A)=∅.Let{αn},{βn},and{γn}be real number sequences in(, )such thatαn+βn+γn= .Let{λn}be a positive real number sequence and let{en}be a bounded error sequence in C.Let{xn}be a sequence in C generated in the following iterative process:
x∈C, xn+=αnSxn+βnPC(xn–λnAxn) +γnen
for all n∈N. Assume that the sequences{αn}, {βn}, {γn},and{λn}satisfy the following restrictions:
(a) <a≤βn≤b< , (b) <c≤λn≤d< α,
(c) ∞n=γn<∞,
where a,b,c,and d are some real numbers.Then the sequence{xn}converges weakly to some point inF.
In view of Theorem ., we have the following result.
Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H and let A:C→H be anα-inverse-strongly monotone mapping such thatVI(C,A)=∅.Let{αn},
positive real number sequence and let{en}be a bounded error sequence in C.Let{xn}be a sequence in C generated in the following iterative process:
x∈C, xn+=αnxn+βnPC(xn–λnAxn) +γnen
for all n∈N. Assume that the sequences{αn}, {βn}, {γn},and{λn}satisfy the following restrictions:
(a) <a≤βn≤b< , (b) <c≤λn≤d< α,
(c) ∞n=γn<∞,
where a,b,c,and d are some real numbers.Then the sequence{xn}converges weakly to some point inVI(C,A).
LetFbe a bifunction ofC×CintoR, whereRdenotes the set of real numbers. Recall the following equilibrium problem.
Findx∈Csuch thatF(x,y)≥, ∀y∈C.
In this paper, we useEP(F) to denote the solution set of the equilibrium problem. To study the equilibrium problems, we may assume thatFsatisfies the following condi-tions:
(A) F(x,x) = for allx∈C;
(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤for allx,y∈C; (A) for eachx,y,z∈C,
lim sup t↓
Ftz+ ( –t)x,y≤F(x,y);
(A) for eachx∈C,y→F(x,y)is convex and weakly lower semi-continuous.
PuttingF(x,y) =Ax,y–xfor everyx,y∈C, we see that the equilibrium problem is reduced to the variational inequality (.).
The following lemma can be found in [].
Lemma . Let C be a nonempty closed convex subset of H and let F:C×C→Rbe a bifunction satisfying(A)-(A).Then,for any r> and x∈H,there exists z∈C such that
F(z,y) +
ry–z,z–x ≥, ∀y∈C. Further,define
Trx=
z∈C:F(z,y) +
ry–z,z–x ≥,∀y∈C
(.)
for all r> and x∈H.Then the following hold:
(a) Tris single-valued,
(b) Tris firmly nonexpansive,i.e.,for anyx,y∈H,
(c) F(Tr) =EP(F),
(d) EP(F)is closed and convex.
Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H,let F a bifunction from C×C toRwhich satisfies(A)-(A)and let AFbe a multivalued mapping of H into itself defined by
AFx= ⎧ ⎨ ⎩
{z∈H:F(x,y)≥ y–x,z,∀y∈C}, x∈C,
∅, x∈/C.
Then AFis a maximal monotone operator with the domain D(AF)⊂C,EP(F) =A–
F ()and
Trx= (I+rAF)–x, ∀x∈H,r> ,
where Tris defined as in(.).
Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H,let S:
C→C be a nonexpansive mapping and let F be a bifunction from C×C toRwhich satisfies
(A)-(A).Assume thatF =F(S)∩EP(F)=∅.Let {αn}, {βn},and{γn} be real number sequences in(, )such thatαn+βn+γn= .Let{λn}be a positive real number sequence and let{en}be a bounded error sequence in C.Let{xn}be a sequence in C generated in the following iterative process:
x∈C, xn+=αnSxn+βnyn+γnen
for all n∈N,where yn∈C such that
F(yn,u) + λn
u–yn,yn–xn ≥, ∀u∈C.
Assume that the sequences{αn},{βn},{γn},and{λn}satisfy the following restrictions: (a) <a≤βn≤b< ,
(b) <c≤λn≤d<∞, (c) ∞n=γn<∞,
where a,b,c,and d are some real numbers.Then the sequence{xn}converges weakly to some point inF.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author is grateful to the reviewers for useful suggestions which improved the contents of the article.
Received: 10 March 2013 Accepted: 28 May 2013 Published: 17 June 2013 References
1. Douglas, J, Rachford, HH: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc.82, 421-439 (1956)
3. Lions, PL, Mercier, B: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal.16, 964-979 (1979)
4. Qin, X, Chang, SS, Cho, YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal.11, 2963-2972 (2010)
5. Takahashi, S, Takahashi, W, Toyoda, M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl.147, 27-41 (2010)
6. Abdel-Salam, HS, Al-Khaled, K: Variational iteration method for solving optimization problems. J. Math. Comput. Sci.2, 1475-1497 (2012)
7. Cho, SY, Kang, SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci.32, 1607-1618 (2012)
8. Noor, MA, Noor, KI, Waseem, M: Decomposition method for solving system of linear equations. Eng. Math. Lett.2, 31-41 (2013)
9. Cho, SY, Li, W, Kang, SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl.
2013, Article ID 199 (2013)
10. Zegeye, H, Shahzad, N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory2, 374-397 (2012)
11. Cho, SY, Kang, SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett.24, 224-228 (2011)
12. He, RH: Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC-spaces. Adv. Fixed Point Theory2, 47-57 (2012)
13. Kellogg, RB: Nonlinear alternating direction algorithm. Math. Comput.23, 23-27 (1969)
14. Cho, SY, Kang, SM: Zero point theorems ofm-accretive operators in a Banach space. Fixed Point Theory13, 49-58 (2012)
15. Browder, FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math.18, 78-81 (1976)
16. Takahahsi, W, Toyoda, M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl.118, 417-428 (2003)
17. Qin, X, Cho, YJ, Kang, SM: Approximating zeros of monotone operators by proximal point algorithms. J. Glob. Optim.
46, 75-87 (2010)
18. Eshita, K, Takahashi, W: Approximating zero points of accretive operators in general Banach spaces. Fixed Point Theory Appl.2, 105-116 (2007)
19. Kammura, S, Takahashi, W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory106, 226-240 (2000)
20. Qin, X, Su, Y: Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl.329, 415-424 (2007)
21. Yuan, Q, Shang, M: Convergence of an extragradient-like iterative algorithm for monotone mappings and nonexpansive mappings. Fixed Point Theory Appl.2013, Article ID 67 (2013)
22. Luo, H, Wang, Y: Iterative approximation for the common solutions of a infinite variational inequality system for inverse-strongly accretive mappings. J. Math. Comput. Sci.2, 1660-1670 (2012)
23. Tada, A, Takahashi, W: Weak and strong convergence theorems for a nonexpansive mappings and an equilibrium problem. J. Optim. Theory Appl.133, 359-370 (2007)
24. Noor, MA, Huang, Z: Some resolvent iterative methods for variational inclusions and nonexpansive mappings. Appl. Math. Comput.194, 267-275 (2007)
25. Qin, X, Cho, YJ, Kang, SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math.225, 20-30 (2009)
26. Qin, X, Cho, SY, Kang, SM: Strong convergence of shrinking projection methods for quasi-φ-nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math.234, 750-760 (2010)
27. Shehu, Y: An iterative method for fixed point problems, variational inclusions and generalized equilibrium problems. Math. Comput. Model.54, 1394-1404 (2011)
28. Saejung, S, Yotkae, P: Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal.75, 742-750 (2012)
29. Saejung, S, Wongchan, K, Yotkae, P: Another weak convergence theorems for accretive mappings in Banach spaces. Fixed Point Theory Appl.2011, Article ID 26 (2011)
30. Cho, YJ, Kang, SM, Zhou, H: Approximate proximal point algorithms for finding zeroes of maximal monotone operators in Hilbert spaces. J. Inequal. Appl.2008, Article ID 598191 (2008)
31. Wu, C, Liu, A: Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems. Fixed Point Theory Appl.012, Article ID 90 (2012)
32. Qin, X, Cho, SY, Kang, SM: On hybrid projection methods for asymptotically quasi-φ-nonexpansive mappings. Appl. Math. Comput.215, 3874-3883 (2010)
33. Kim, JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-φ-nonexpansive mappings. Fixed Point Theory Appl.2011, Article ID 10 (2011) 34. Aoyama, K, et al.: On a strongly nonexpansive sequence in Hilbert spaces. J. Nonlinear Convex Anal.8, 471-489 (2007) 35. Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am.
Math. Soc.73, 591-597 (1967)
36. Tan, KK, Xu, HK: The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc.114, 399-404 (1992)
37. Schu, J: Weak and strong convergence to fixed points of a asymptotically nonexpansive mappings. Bull. Aust. Math. Soc.43, 153-159 (1991)
38. Rockafellar, RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim.14, 877-898 (1976) 39. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud.63, 123-145
doi:10.1186/1687-1812-2013-155
