PMCCPMCCPMCC

Search tips
Search criteria 

Advanced

 
Logo of splusSpringerOpen.comSubmit OnlineRegisterThis journalThis article
 
Springerplus. 2016; 5(1): 803.
Published online 2016 June 21. doi:  10.1186/s40064-016-2389-9
PMCID: PMC4916130

Hybrid algorithm for common solution of monotone inclusion problem and fixed point problem and applications to variational inequalities

Abstract

The aim of this paper is to investigate hybrid algorithm for a common zero point of the sum of two monotone operators which is also a fixed point of a family of countable quasi-nonexpansive mappings. We point out two incorrect proof in paper (Hecai in Fixed Point Theory Appl 2013:11, 2013). Further, we modify and generalize the results of Hecai’s paper, in which only a quasi-nonexpansive mapping was considered. In addition, two family of countable quasi-nonexpansive mappings with uniform closeness examples are provided to demonstrate our results. Finally, the results are applied to variational inequalities.

Keywords: Quasi-nonexpansive mappings, Inverse-strongly monotone mapping, Maximal monotone operator, Fixed point

Introduction and preliminaries

The monotone inclusion problem is to

find anxHsuch that0i=1mAix,

where H is a real Hilbert space with inner product ⟨ · ,  · ⟩ and Ai are set-valued maximal monotone operators (Hui and Lizhi 2013). Such problem is very important in many areas, such as convex optimization and monotone variational inequalities, for instance. There is an extensive literature to approach the inclusion problem, all of which can essentially be divided into two classes according to the number of operators involved: single operator class (m = 1) and multiple operator class (m ≥ 2). The latter class can always be reduced to the case of m = 2 via Spingarn’s method (Spingarn 1983). Based on a series of studies in the next decades, splitting methods for monotone operators were inspired and studied extensively. Splitting methods for linear equations were introduced by Peaceman and Rachford (1995) and Douglas and Rachford (1956). Extensions to nonlinear equations in Hilbert spaces were carried out by Kellogg (1969) and Lions and Mercier (1979). The central problem is to iteratively find a zero of the sum of two monotone operators A and B in a Hilbert space H. Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as signal processing, image recovery and machine learning are mathematically modeled as a nonlinear operator equation (Shehu et al. 2016a, b; Shehu 2015). And the operator is decomposed into the sum of two nonlinear operators.

In this paper, we consider the problem of finding a solution for the following problem: find an x in the fixed point set of a family of countable quasi-nonexpansive mappings Sn such that

x ∈ (A+B)-1(0), 

where A and B are two monotone operators. The similar problem has been addressed by many authors in view of the applications in signal processing and image recovery; see, for example, Qin et al. (2010), Zhang (2012), Takahashi et al. (2010), Kamimura and Takahashi (2010) and the references therein.

Throughout this paper, we always assume that H is a real Hilbert space with the inner product ⟨ · ,  · ⟩ and norm ‖ · ‖, respectively. Let C be a nonempty closed convex subset of HPC be the metric projection from H onto C,  and S:C → C be a mapping. We use F(S) to denote the fixed point set of Sn below, i.e., F(S): = {x ∈ C:xSx}. Recall that S is said to be nonexpansive if

SxSy‖ ≤ ‖xy‖,  ∀  xy ∈ C.

If C is a bounded closed and convex subset of H, then F(S) is nonempty closed and convex; see Browder (1976). S is said to be quasi-nonexpansive if F(S) ≠ ∅ and

Sxp‖ ≤ ‖xp‖,  ∀  x ∈ Cp ∈ F(S).

It is easy to see that nonexpansive mappings are Lipschitz continuous, however, the quasi-nonexpansive mapping is discontinuous on its domain generally. Indeed, the quasi-nonexpansive mapping is only continuous in its fixed point set.

Let A:C → H be a mapping. Recall that A is said to be monotone if

AxAyxy⟩ ≥ 0,  ∀  xy ∈ C.

A is said to be α-strongly monotone if there exists a constant α > 0 such that

AxAyxy⟩ ≥ αx-y2,  ∀  xy ∈ C.

A is said to be α-inverse strongly monotone if there exists a constant α > 0 such that

AxAyxy⟩ ≥ αAx-Ay2,  ∀  xy ∈ C.

Notice that, a α-inverse strongly monotone operator must be 1α-Lipschitz continuous.

Recall that the classical variational inequality is to find an x ∈ C such that

Axyx⟩ ≥ 0,  ∀  y ∈ C.
1

In this paper, we use VI(CA) to denote the solution set of (1). It is known that x ∈ C is a solution to (1) if x is a fixed point of the mapping PC(IλA), where λ > 0 is a constant, I is the identity mapping, and PC is the metric projection from H onto C. Next we recall some well-known definitions.

Definition 1

(Takahashi et al. 2010) A multi-valued operator T:H → H with the domain D(T) = {x ∈ H:Tx ≠ 0} and the range R(T) = {Tx:x ∈ D(T)} is said to be monotone if for x1x2 ∈ D(T), y1y2 ∈ R(T), the following inequality holds x1x2y1y2⟩ ≥ 0.

Definition 2

(Takahashi et al. 2010) A monotone operator T is said to be maximal if its graph G(T) = {(xy):y ∈ Tx} is not properly contained in the graph of any other monotone operator.

Definition 3

(Takahashi et al. 2010) Let I denote the identity operator on H and T:H → H be a maximal monotone operator. For each λ > 0, a nonexpansive single-valued mapping Jλ = (I-λA)-1 is called the resolvent of T.

And it is known that T-1(0) = F(Jλ) for all λ > 0 and Jλ is firmly nonexpansive.

Three classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one was introduced in 1953 by Mann (1953) and is well known as Manns iteration process defined as follows:

x0chosenarbitrarily,xn+1=αnxn+(1-αn)Txn,n0,
2

where the sequence {αn} is chosen in [0,1]. Fourteen years later, Halpern (1967) proposed the new innovation iteration process which resembled Manns iteration (2). It is defined by

x0chosenarbitrarily,xn+1=αnu+(1-αn)Txn,n0,
3

where the element u ∈ C is fixed. Seven years later, Ishikawa (1974) enlarged and improved Mann’s iteration (2) to the new iteration method, which is often cited as Ishikawa’s iteration process and defined recursively by

x0chosenarbitrarily,yn=βnxn+(1-βn)Txn,xn+1=αnxn+(1-αn)Tyn,n0,
4

where {αn} and {βn} are sequences in the interval [0,1].

Moreover, many authors have studied the common solution problem, that is, find a point in a solution set and a fixed (zero) point set of some nonlinear problems; see, for example, Kamimura and Takahashi (2000), Takahashi and Toyoda (2003), Ye and Huang (2011), Cho and Kang (2011), Zegeye and Shahzad (2012), Qin et al. (2010), Lu and Wang (2012), Husain and Gupta (2012), Noor and Huang (2007), Qin et al. (2009), Kim and Tuyen (2011), Wei and Shi (2012), Qin et al. (2010), Qin et al. (2008), He et al. (2011), Wu and Liu (2012), Qin and Su (2007), Abdel-Salam and Al-Khaled (2012), Qin et al. (2010), Zegeye et al. (2012) and the references therein. In Kamimura and Takahashi (2000), in the framework of real Hilbert spaces, Kamimura and Takahashi investigated the problem of finding zero points of a maximal monotone operator by considering the following iterative algorithm:

x0 ∈ Hxn+1αnxn + (1 - αn)Jλnxnn = 0, 1, 2,  ·  ·  · 
5

where {αn} is a sequence in (0,1), {λn} is a positive sequence, T:H → H is a maximal monotone, and Jλn = (I+λnT)-1. They showed that the sequence {xn} generated in (5) converges weakly to some z ∈ T-1(0) provided that the control sequence satisfies some restrictions. Further, using this result, they also investigated the case that Tf where f:H → H is a proper lower semicontinuous convex function.

Takahashi and Toyoda (2003) investigated the problem of finding a common solution of the variational inequality problem (1) and a fixed point problem involving nonexpansive mappings by considering the following iterative algorithm:

x0 ∈ Cxn+1αnxn + (1 - αn)SPC(xnλnAxn),  ∀ n ≥ 0, 
6

where {αn} is a sequence in (0,1), {λn} is a positive sequence, S:C → C is a nonexpansive mapping, and A:C → H is an inverse-strongly monotone mapping. They showed that the sequence {xn} generated in (6) converges weakly to some z ∈ VI(CA) ∩ F(S) provided that the control sequence satisfies some restrictions.

Hecai (2013) studied the common solution for two monotone operators and a quasi-nonexpansive mapping in the framework of Hilbert spaces. The aim of this paper is to investigate hybrid algorithm for a common zero point of the sum of two monotone operators which is also a fixed point of a family of countable quasi-nonexpansive mappings. We point out two incorrect justifications in the proof of Theorem 2.1 in paper Hecai (2013). Further, we modify and generalize the results of Hecai’s paper, in which only a quasi-nonexpansive mapping was considered. In addition, two family of countable quasi-nonexpansive mappings with uniform closeness examples are provided to demonstrate our results. Finally, we apply the results to variational inequalities.

To obtain our main results in this paper, we need the following lemmas and definitions.

Let C be a nonempty, closed, and convex subset of H. Let {Sn}n=1:CC be a sequence of mappings of C into C such that n=1F(Sn) is nonempty. Then {Sn}n=1 is said to be uniformly closed, if pn=1F(Sn), whenever {xn} ⊂ C converges strongly to p and xnSnxn‖ → 0 as n → .

Lemma 4

(Aoyama et al. 2007) LetCbe a nonempty, closed, and convex subset ofHA:C → Hbe a mapping, andB:H → 2Hbe a maximal monotone operator. ThenF(Jr(IλA)) = (A+B)-1(0).

Let C be a nonempty, closed, and convex subset of H,  the projection operator PC:E → C is a map that assigns to an arbitrary point x ∈ H the minimum point of the norm xy, that is, PCx=x¯, where x¯ is a unique solution to the minimization problem

x¯-x=minyCy-x.

It is well-known that

xPCxPCxy⟩ ≥ 0,  ∀  y ∈ C.

Abdel-Salam and Al-Khaled (2012) proved the following result.

Theorem 5

LetCbe a nonempty closed convex subset of a real Hilbert spaceHA:C → Hbe anα-inverse-strongly monotone mapping, S:C → Cbe a quasi-nonexpansive mapping such thatISis demiclosed at zero andBbe a maximal monotone operator onHsuch that the domain ofBis included inC. Assume thatFF(S) ∩ (A+B)-1(0) ≠ ∅.Let{λn}be a positive real number sequence and{αn}be a real number sequence in [0,1]. Let{xn}be a sequence ofCgenerated by

x1C,C1=C,yn=αnxn+(1-αn)SJrn(xn-λnAxn),Cn+1={zCn:yn-zxn-z},xn+1=PCn+1x1,n1,

whereJrn = (I+rnB)-1.Suppose that the sequencesλnandαnsatisfy the following restrictions:

(a) 0 ≤ αn ≤ a < 1;

(b) 0 < b ≤ λn ≤ c < 2α

Then the sequence{xn}converges strongly toqPFx0.

However, the proof of above Theorem 5 is not correct. First mistake: in page 6, line 16–17, there is a mistake inequality:

zn-p2=Jλn(xn-λnAxn)-Jλn(p-λnAp)2xn-λnAxn-p-λnAp,zn-p.

Second mistake: in page 7, -line 5–7, there is a mistake ratiocination:

Since B is monotone, we get for any (uv) ∈ B that

zn-u,xn-znλn-Axn-v0.
7

Replacing n by ni and letting i → , we obtain from (7) that

ωu,  - Aωv⟩ ≥ 0.

Our comments: Notice that, the inner product ⟨ · ,  · ⟩ is not weakly continuous. For example: in Hilbert space l2, let

x0=(1,0,0,0,0,),x1=(1,1,0,0,0,),x2=(1,0,1,0,0,),x3=(1,0,0,1,0,),.

It is well-known that {xn} converges weakly to x0, but

xnxn⟩ = 2,  ⟨x0x0⟩ = 1, 

so the inner product xnxn does not converges to x0x0. Therefore,

zn-u,xn-znλn-Axn-v

does not converges to

ωu,  - Aωv⟩.

In order to modify the iterative algorithm of Theorem 5 and to get more generalized results, we present a new iterative algorithm in this paper. Moreover, the results are applied to variational inequalities.

Main results

Now we are in the position to give our main results.

Theorem 6

LetCbe a nonempty closed convex subset of a real Hilbert spaceHA:C → Hbe anα-inverse-strongly monotone mapping, andBbe a maximal monotone operator onHsuch that the domain ofBis included inC. Let{Sn}:C → Cbe a family of countable quasi-nonexpansive mappings which are uniformly closed. Assume thatFF(S) ∩ (A+B)-1(0) ≠ ∅.Let{rn}be a positive real number sequence and{αn}be a real number sequence in [0,1). Let{xn}be a sequence ofCgenerated by

x1C1=C,chosenarbitrarily,zn=Jrn(xn-rnAxn),yn=αnzn+(1-αn)Snzn,Cn+1=zCn:zn-zyn-zxn-z,xn+1=PCn+1x1,n1,

whereJrn = (I+rnB)-1,  lim infnrn > 0,  rn ≤ 2αandlim supnαn < 1.Then the sequence{xn}converges strongly toqPFx0.

Proof

We divide the proof into six steps.

Step 1. We show that Cn is closed and convex. Notice that C1C is closed and convex. Suppose that Ci is closed and convex for some i ≥ 1. Next we show that Ci+1 is closed and convex for the same i. Since

Ci+1=CizE:yi-zzi-z}{zE:zi-zxi-z=CizE:z,yi-zi12yi2-zi2zE:z,zi-xi12zi2-xi2.

It is obvious that

zE:z,yi-zi12yi2-zi2,zE:z,zi-xi12zi2-xi2

are all closed and convex, so Ci+1 is closed and convex. This shows that Cn is closed and convex for all n ≥ 1.

Step 2. We show that F ⊂ Cn for all n ≥ 1. By the assumption, we see that F ⊂ C1. Assume that F ⊂ Ci for some i ≥ 1. For any p ∈ F ⊂ Ci, we find from the Lemma that

pSipJri(priAp).

Since Jri is nonexpansive, we have

zi-p2=Jri(xi-riAxi)-Jri(p-riAp)2(xi-riAxi)-(p-riAp)2=(xi-p)-ri(Axi-Ap)2=xi-p2-2rixi-p,Axi-Ap+ri2Axi-Ap2xi-p2-ri(2α-ri)Axi-Ap2,

which implies that

zip‖ ≤ ‖xip‖.
8

On the other hand, we have

yi-p=αizi+(1-αi)Sizi-p=αi(zi-p)+(1-αi)(Sizi-p)αizi-p+(1-αi)Sizi-pαizi-p+(1-αi)zi-p=zi-p.
9

From (8) and (9), we know that p ∈ Ci+1. This show F ⊂ Cn for all n ≥ 1.

Step 3. We show that {xn} is a Cauchy sequence, so it is convergent in C.

Since xnPCnx0 and Cn+1 ⊂ Cn, then we obtain

xnx0‖ ≤ ‖xn+1x0‖,  for all n ≥ 1.
10

Therefore xnx0 is nondecreasing. On the other hand, we have

xnx0‖ = ‖PCnx0x0‖ ≤ ‖px0‖, 

for all p ∈ F ⊂ Cn and for all n ≥ 1. Therefore, xnx0 is also bounded. This together with (10) implies that the limit of xnx0 exists. Put

limnxn-x0=d.
11

It is known that for any positive integer m,

xn+m-xn2=xn+m-PCnx02xn+m-x02-PCnx0-x02=Df(xn+m,x0)-Df(xn,x0),

for all n ≥ 1. This together with (11) implies that

limnDf(xn+m,xn)=0,

uniformly for all m, holds. Therefore, we get that

limnxn+m-xn=0,

uniformly for all m, holds. Then {xn} is a Cauchy sequence, hence there exists a point p ∈ C such that xn → p.

Step 4. We prove that the limit of {xn} belongs to F.

Let limnxnq. Sine xn+1 ∈ Cn+1, so we have

ynxn+1‖ ≤ ‖znxn+1‖ ≤ ‖xnxn+1‖ → 0, 
12

as n → . Hence

limnyn=q,limnzn=q.
13

From

ynαnzn + (1 - αn)Snzn

we have that

ynzn‖ = (1 - αn)‖Snznzn‖.

The condition lim supnαn < 1 and (13) imply that

limnSnzn-zn=0.
14

Because {Sn} is an uniformly closed family of countable quasi-nonexpansive mappings, therefore this together with the (14) implies that qn=1F(Sn).

Step 5. We show that q ∈ (A+B)-1(0).

Notice that znJrn(xnrnAxn). This means that

xnrnAxn ∈ znrnBzn

Actually, that is,

xn-znrn-AxnBzn,

For B is monotone, so we get for any (uv) ∈ B that

zn-u,xn-znrn-Axn-v0.
15

Letting n → , we obtain from (15) that

qu,  - Aqv⟩ ≥ 0.

Since B is a maximal monotone operator, so we have Aq ∈ Bq, that is, 0 ∈ (AB)(q). Hence, q ∈ (A+B)-1(0). This completes the proof that q ∈ F.

Step 6. We show that qPFx0.

Observe that PFx0 ∈ Cn+1 and xn+1PCn+1x0, thus we have

xn+1x0‖ ≤ ‖PFx0x0‖.

On the other hand, we have

x0-PFx0x0-q=limnx0-xn+1x0-PFx0.

Since F is closed and convex, so the projection PFx0 is unique. Therefore we get that qPFx0. This completes the proof.

Application

In this section, we apply our results to variational inequalities.

Let f:H → (,  + ] be a proper lower semicontinuous convex function. For all x ∈ H define the subdifferential

f(x) = {z ∈ H:f(x) + ⟨yxz⟩ ≤ f(y),  ∀ y ∈ H}.

Then f is a maximal monotone operator of H into itself (Noor and Huang 2007). Let C be a nonempty closed convex subset of H and iC be the indicator function of C,  that is,

iCx=0,xC,,xC.

Furthermore, for any ν ∈ C we define the normal cone NC(ν) of C at ν as follows:

NCν = {z ∈ H:⟨zyν⟩ ≤ 0,  ∀ y ∈ H}.

Then iC:H → (] is a proper lower semicontinuous convex function on H and iC is a maximal monotone operator. Let Jx = (I+λiC)-1x for any λ > 0 and x ∈ H. From iCxNCx and x ∈ C we get

ν=Jλxxν+λNCν,x-ν,y-ν,yC,ν=PCx,

where PC is the projection operator from H into C. In the same way, we can get that x ∈ (A+iC)-1(0) ⇔ x ∈ VI(AC). Putting BiC in Theorem 6, we can see that JλnPC. Naturally, we can obtain the following consequence.

Theorem 7

LetCbe a nonempty closed convex subset of a real Hilbert spaceHA:C → Hbe anα-inverse-strongly monotone mapping, andSn:C → Cbe a family of countable quasi-nonexpansive mappings which are uniformly closed. Assume thatFF(S) ∩ VI(CA) ≠ ∅.Let{rn}be a positive real number sequence and{αn}be a real number sequence in [0,1). Let{xn}be a sequence ofCgenerated by

x1C1=C,chosenarbitrarily,zn=PC(xn-rnAxn),yn=αnzn+(1-αn)Snzn,Cn+1={zCn:zn-zyn-zxn-z},xn+1=PCn+1x1,n1,

whereJrn = (I+rnB)-1,  lim infnrn > 0,  rn ≤ 2αandlim supnαn < 1.Then the sequence{xn}converges strongly toqPFx0.

Based on Theorem 7, we have the following corollary on variational inequalities.

Corollary 8

LetCbe a nonempty closed convex subset of a real Hilbert spaceHA:C → Hbe anα-inverse-strongly monotone mapping. Assume thatFVI(CA) ≠ ∅. Let {rn} be a positive real number sequence. Let{xn}be a sequence ofCgenerated by

x1C1=C,chosenarbitrarily,zn=PC(xn-rnAxn),Cn+1={zCn:zn-zxn-z},xn+1=PCn+1x1,n1,

whereJrn = (I+rnB)-1andlim infnrn > 0,  rn ≤ 2α.Then the sequence{xn}converges strongly toqPVI(C,A)x0.

Examples

Let H be a Hilbert space and C be a nonempty closed convex and balanced subset of H. Let {xn} be a sequence in C such that xn‖ = r > 0, {xn} converges weakly to x0 ≠ 0 and xnxm‖ ≥ r > 0 for all n ≠ m. Define a family of countable mappings {Tn}:C → C as follows

Tn(x)=nn+1xnifx=xn(n1),-xifxxn(n1).

Conclusion 9

{Tn}has a unique common fixed point 0, i.e.,F=n=1F(Tn)={0}, for alln ≥ 0.

Proof

The conclusion is obvious.

Conclusion 10

{Tn}is a uniformly closed family of countable quasi-nonexpansive mappings.

Proof

First, we have

Tnx-0=nn+1xn-0,ifx=xn,x-0ifxxn.

Therefore

Tnx-0‖≤‖x-0‖2

for all x ∈ C. On the other hand, for any strong convergent sequence {zn} ⊂ E such that zn → z0 and znTnzn‖ → 0 as n → , it is easy to see that there exists sufficiently large nature number N such that zn ≠ xm, for any nm > N. Then Tzn =  - zn for n > N. It follows from znTnzn‖ → 0 that 2zn → 0. Hence zn → z0 = 0,  that is z0 ∈ F.

Example 11

Let El2, where

l2=ξ=(ξ1,ξ2,ξ3,,ξn,):n=1|xn|2<,ξ=n=1|ξn|212,ξl2,ξ,η=n=1ξnηn,ξ=(ξ1,ξ2,ξ3,,ξn,),η=(η1,η2,η3,,ηn)l2.

Let {xn} ⊂ E be a sequence defined by

x0=(1,0,0,0,),x1=(1,1,0,0,),x2=(1,0,1,0,0,),x3=(1,0,0,1,0,0,),......................................xn=(ξn,1,ξn,2,ξn,3,,ξn,k,)......................................,

where

ξn,k=1ifk=1,n+1,0ifk1,kn+1,

for all n ≥ 1. It is well-known that xn=2,n1 and {xn} converges weakly to x0. Define a countable family of mappings Tn:E → E as follows

Tn(x)=nn+1xnifx=xn,-xifxxn,

for all n ≥ 0. By using Conclusion 9 and 10, {Tn} is a uniformly closed family of countable quasi-nonexpansive mappings.

Example 12

Let ELp[0, 1] (1 < p < +) and

xn=1-12n,n=1,2,3,···.

Define a sequence of functions in Lp[0, 1] as the following expression

fn(x)=2xn+1-xnifxnx<xn+1+xn2,-2xn+1-xnifxn+1+xn2x<xn+10otherwise

for all n ≥ 1. Firstly, we can see for any x ∈ [0, 1] that

0xfn(t)dt0=0xf0(t)dt,
16

where f0(x) ≡ 0. It is well-known that the above relation (16) is equivalent to {fn(x)} converges weakly to f0(x) in uniformly smooth Banach space Lp[0, 1](1 < p < +). On the other hand, for any n ≠ m, we have

fn-fm=01|fnx-fmx|pdx1p=xnxn+1|fnx-fmx|pdx+xmxm+1|fnx-fmx|pdx1p=xnxn+1|fnx|pdx+xmxm+1|fmx|pdx1p=2xn+1-xnpxn+1-xn+2xm+1-xmpxm+1-xm1p=2pxn+1-xnp-1+2pxm+1-xmp-11p2p+2p1p>0.

Let

un(x) = fn(x) + 1,  ∀  n ≥ 1.

It is obvious that un converges weakly to u0(x) ≡ 1 and

un-um=fn-fm(2p+2p)1p>0,n1.
17

Define a mapping T:E → E as follows

Tn(x)=nn+1unifx=un(n1),-xifxun(n1).

Since (17) holds, by using Conclusion 9 and 10, we know that {Tn} is a uniformly closed family of countable quasi-nonexpansive mappings.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgements

This work has been supported by the National Natural Science Foundation of China (Grant Nos. 11332006, 1127223311572221), National key basic research and development program (plan 973) (Nos. 2012CB720101, 2012CB720103).

Competing interests

The authors declare that they have no competing interests.

Contributor Information

Jingling Zhang, moc.621@70_shtam, nc.ude.ujt@90gnahzlj.

Nan Jiang, nc.ude.ujt@jnan.

References

  • Abdel-Salam HS, Al-Khaled K. Variational iteration method for solving optimization problems. J Math Comput Sci. 2012;2:1457–1497.
  • Aoyama K, Kimura Y, Takahashi W, Toyoda M. On a strongly nonexpansive sequence in Hilbert spaces. J Nonlinear Convex Anal. 2007;8:471–489.
  • Browder FE. Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc Symp Pure Math. 1976;18:78–81.
  • Cho SY, Kang SM. Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl Math Lett. 2011;24:224–228. doi: 10.1016/j.aml.2010.09.008. [Cross Ref]
  • Douglas J, Rachford HH. On the numerical solution of heat conduction problems in two and three space variables. Trans Am Math Soc. 1956;82:421–439. doi: 10.1090/S0002-9947-1956-0084194-4. [Cross Ref]
  • Halpern B. Fixed points of nonexpanding maps. Bull Am Math Soc. 1967;73:957–961. doi: 10.1090/S0002-9904-1967-11864-0. [Cross Ref]
  • He XF, Xu YC, He Z. Iterative approximation for a zero of accretive operator and fixed points problems in Banach space. Appl Math Comput. 2011;217:4620–4626.
  • Hecai On solutions of inclusion problems and fixed point problems. Fixed Point Theory Appl. 2013;2013:11. doi: 10.1186/1687-1812-2013-11. [Cross Ref]
  • Husain S, Gupta S. A resolvent operator technique for solving generalized system of nonlinear relaxed cocoercive mixed variational inequalities. Adv Fixed Point Theory. 2012;2:18–28.
  • Ishikawa S. Fixed points by a new iteration method. Proc Am Math Soc. 1974;44:147–150. doi: 10.1090/S0002-9939-1974-0336469-5. [Cross Ref]
  • Kamimura S, Takahashi W. Approximating solutions of maximal monotone operators in Hilbert spaces. J Approx Theory. 2000;106:226–240. doi: 10.1006/jath.2000.3493. [Cross Ref]
  • Kamimura S, Takahashi W. Weak and strong convergence of solutions to accretive operator inclusions and applications. Set Valued Anal. 2010;8:361–374. doi: 10.1023/A:1026592623460. [Cross Ref]
  • Kellogg RB. Nonlinear alternating direction algorithm. Math Comput. 1969;23:23–28. doi: 10.1090/S0025-5718-1969-0238507-3. [Cross Ref]
  • Kim JK, Tuyen TM (2011) Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 52
  • Lions PL, Mercier B. Splitting algorithms for the sum of two nonlinear operators. SIAM J Numer Anal. 1979;16:964–979. doi: 10.1137/0716071. [Cross Ref]
  • Lu H, Wang Y. Iterative approximation for the common solutions of a infinite variational inequality system for inverse-strongly accretive mappings. J Math Comput Sci. 2012;2:1660–1670.
  • Mann WR. Mean value methods in iteration. Proc Am Math Soc. 1953;4:506–510. doi: 10.1090/S0002-9939-1953-0054846-3. [Cross Ref]
  • Noor MA, Huang Z. Some resolvent iterative methods for variational inclusions and nonexpansive mappings. Appl Math Comput. 2007;194:267–275.
  • Peaceman DH, Rachford HH. The numerical solution of parabolic and elliptic differential equations. J Soc Ind Appl Math. 1995;3:28–415. doi: 10.1137/0103003. [Cross Ref]
  • Qin X, Shang M, Su Y. Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math Comput Model. 2008;48:1033–1046. doi: 10.1016/j.mcm.2007.12.008. [Cross Ref]
  • 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. 2009;225:20–30. doi: 10.1016/j.cam.2008.06.011. [Cross Ref]
  • Qin X, Kang JL, Cho YJ. On quasi-variational inclusions and asymptotically strict pseudo-contractions. J Nonlinear Convex Anal. 2010;11:441–453.
  • Qin X, Cho SY, Kang SM. Strong convergence of shrinking projection methods for quasi-- --nonexpansive mappings and equilibrium problems. J Comput Appl Math. 2010;234:750–760. doi: 10.1016/j.cam.2010.01.015. [Cross Ref]
  • Qin X, Chang SS, Cho YJ. Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010;11:2963–2972. doi: 10.1016/j.nonrwa.2009.10.017. [Cross Ref]
  • Qin X, Cho SY, Kang SM. On hybrid projection methods for asymptotically quasi-ϕϕ-nonexpansive mappings. Appl Math Comput. 2010;215:3874–3883.
  • Qin X, Su YF. Approximation of a zero point of accretive operator in Banach spaces. J Math Anal Appl. 2007;329:415–424. doi: 10.1016/j.jmaa.2006.06.067. [Cross Ref]
  • Shehu Y (2015) Iterative approximations for zeros of sum of accretive operators in Banach spaces. J Funct Spaces. Article ID 5973468, 9 pages
  • Shehu Y, Ogbuisi FU, Iyiola OS. Convergence analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces. Optimization. 2016;65(2):299–323. doi: 10.1080/02331934.2015.1039533. [Cross Ref]
  • Shehu Y, Iyiola OS, Enyi CD. Iterative algorithm for split feasibility problems and fixed point problems in Banach Spaces. Numer Algorithms. 2016
  • Spingarn JE. Partial inverse of a monotone operator. Appl Math Optim. 1983;10:247–265. doi: 10.1007/BF01448388. [Cross Ref]
  • Takahashi S, Takahashi W, Toyoda M. Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J Optim Theory Appl. 2010;147:27–41. doi: 10.1007/s10957-010-9713-2. [Cross Ref]
  • Takahashi W, Toyoda M. Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl. 2003;118:417–428. doi: 10.1023/A:1025407607560. [Cross Ref]
  • Wei Z, Shi G (2012) Convergence of a proximal point algorithm for maximal monotone operators in Hilbert spaces. J Inequal Appl 137
  • Wu C, Liu A (2012) Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems. Fixed Point Theory Appl 90
  • Ye J, Huang J. Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. J Math Comput Sci. 2011;1:1–18. doi: 10.9734/BJMCS/2011/120. [Cross Ref]
  • Zegeye H, Shahzad N. Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv Fixed Point Theory. 2012;2:374–397.
  • Zegeye H, Shahzad N, Alghamdi M (2012) Strong convergence theorems for a common point of solution of variational inequality, solutions of equilibrium and fixed point problems. Fixed Point Theory Appl 119
  • Zhang M. Iterative algorithms for common elements in fixed point sets and zero point sets with applications. Fixed Point Theory Appl. 2012;2012:21. doi: 10.1186/1687-1812-2012-21. [Cross Ref]
  • Zhang H, Cheng L. Projective splitting methods for sums of maximal monotone operators with applications. J Math Anal Appl. 2013;406:323–334. doi: 10.1016/j.jmaa.2013.04.072. [Cross Ref]

Articles from SpringerPlus are provided here courtesy of Springer-Verlag