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Springerplus. 2016; 5(1): 941.
Published online 2016 June 30. doi:  10.1186/s40064-016-2646-y
PMCID: PMC4929130

Strong convergence theorems for a common zero of a finite family of H-accretive operators in Banach space

Abstract

The aim of this paper is to study a finite family of H-accretive operators and prove common zero point theorems of them in Banach space. The results presented in this paper extend and improve the corresponding results of Zegeye and Shahzad (Nonlinear Anal 66:1161–1169, 2007), Liu and He (J Math Anal Appl 385:466–476, 2012) and the related results.

Keywords: H-accretive operators, Resolvent operator, Iteration algorithms, Strong convergence

Background

Let E be a real Banach space with norm ‖ · ‖ and Let E be its dual space. The value of x ∈ E at x ∈ E will be denoted by xx.

The inclusion problem is finding a solution to

0 ∈ T(x), 
1

where T is a set-valued mapping and from E to 2E.

It was first considered by Rockafellar (1976) by using the proximal point algorithm in a Hilbert space in 1976. For any initial point x0x ∈ ℋ, the proximal point algorithm generates a sequence {xn} in by the following algorithm

xn+1Jrnxnn = 0, 1, 2, …, 
2

where Jrn = (I+rnT)-1 and {rn} ⊂ (0, ), T is maximal monotone operators.

From then on, the inclusion problem becomes a hot topic and it has been widely studied by many researchers in many ways. The mainly studies focus on the more general algorithms, the more general spaces or the weaker assumption conditions, such as Reich (1979, 1980), Benavides et al. (2003), Xu (2006), Kartsatos (1996), Kamimura and Takahashi (2000), Zhou et al. (2000), Maing (2006), Qin and Su (2007), Ceng et al. (2009), Chen et al. (2009), Song et al. (2010), Jung (2010), Fan et al. (2016) and so on. And their researches mainly contain the maximal monotone operators in Hilbert spaces and the m-accretive operators in Banach spaces.

Zegeye and Shahzad (2007) studied a finite family of m-accretive mappings and proposed the iterative sequence {xn} is generated as follows:

xn+1αnu + (1 - αn)Srxn
3

where Sr: = a0Ia1JA1a2JA2 +  ⋯  + arJAr, with JAi: = (I+Ai)-1 for 0 < ai < 1, i = 1, 2, …, r, i=0rai=1.

And proved the sequence {xn} converges strongly to a common solution of the common zero of the operators Ai for i = 1, 2, …, r.

Recently, Fang and Huang (2003, 2004) respectively firstly introduced a new class of monotone operators and accretive operators called H-monotone operators and H-accretive operators, and they discussed some properties of this class of operators.

Definition 1

Let H:ℋ → ℋ be a single-valued operator and T:ℋ → 2 be a multivalued operator. T is said to be H-monotone if T is monotone and (HλT)(ℋ) = ℋ holds for every λ > 0.

Definition 2

Let H:E → E be a single-valued operator and T:E → 2E be a multivalued operator. T is said to be H-accretive if T is accretive and (HλT)EE holds for all λ > 0.

Remark 1

The relations between H-accretive (monotone) operators and m-accretive (maximal monotone) operators are very close, for details, see Liu et al. (2013), Liu and He (2012).

From then, the study of the zero points of H-monotone operators in Hilbert space and H-accretive operators in Banach space have received much attention, see Peng (2008), Zou and Huang (2008, 2009), Ahmad and Usman (2009), Wang and Ding (2010), Li and Huang (2011), Tang and Wang (2014) and Huang and Noor (2007), Xia and Huang (2007), Peng and Zhu (2007). Especially, Very recently, Liu and He (2013, 2012) studied the strong and weak convergence for the zero points of H-monotone operators in Hilbert space and H-accretive operators in Banach space respectively.

Motivated mainly by Zegeye and Shahzad (2007) and Liu and He (2012), in this paper, we will study the zero points problem of a common zero of a finite family of H-accretive operators and establish some strong convergence theorems of them. These results extend and improve the corresponding results of Zegeye and Shahzad (2007) and Liu and He (2012).

Preliminaries

Throughout this paper, we adopt the following notation: Let {xn} be a sequence and u be a point in a real Banach space with norm ‖ · ‖ and let E be its dual space. We use xn → x to denote strong and weak convergence to x of the sequence {xn}.

A real Banach space E is said to be uniformly convex if δ(ε) > 0 for every ε > 0, where the modulus δ(ε) of convexity of E is defined by

δ(ε)=inf1-x+y2:x1,y1,x-yε
4

for every ε with 0 ≤ ε ≤ 2. It is well known that if E is uniformly convex, then E is reflexive and strictly convex (Goebel and Reich 1984)

Let S ≜ {x ∈ E:‖x‖ = 1} be the unit sphere of E, we consider the limit

limt0x+th-xt
5

The norm ‖ · ‖ of Banach space E is said to be Gâteaux differentiable if the limit (5) exists for each xh ∈ S. In this case, the Banach space E is said to be smooth.

The norm ‖ · ‖ of Banach space E is said to be uniformly Gâteaux differentiable if for each h ∈ S the limit (5) is attained uniformly for x in S.

The norm ‖ · ‖ of Banach space E is said to be Fréchet differentiable if for each x ∈ S the limit (5) is attained uniformly for h in S.

The norm ‖ · ‖ of Banach space E is said to be uniformly Fréchet differentiable if the limit (5) is attained uniformly for (xh) in S × S. In this case, the Banach space E is said to be uniformly smooth.

The dual space E of E is uniformly convex if and only if the norm of E is uniformly Fréchet differentiable, then every Banach space with a uniformly convex dual is reflexive and its norm is uniformly Gâteaux differentiable, the converse implication is false. Some related concepts can be found in Day (1993).

Let H:E → E be a strongly accretive and Lipschtiz continuous operator with constant γ. Let T:E → 2E be an H-accretive operator and the resolvent operator JH,ρT:EE is defined by

JH,ρT(u)=(H+ρT)-1(u)uE.
6

for each ρ > 0. We can define the following operators which are called Yosidaapproximation:

Aρ=1ρI-H·JH,ρTforallρ>0.
7

Some elementary properties of JH,ρT and Aρ are given as some lemmas in the following in order to establish our convergence theorems.

Lemma 1

(see Xu 2003) Let{an}be a sequence of non-negative real numbers satisfying the following relation:

an+1 ≤ (1 - γn)anσnn ≥ 0, 
8

where{γn} ⊂ (0, 1)for eachn ≥ 0satisfy the conditions:

  • (i)
    n=1γn=;
  • (ii)
    lim supnσnγn0 or n=1|σn|<;

Then{an}converges strongly to zero.

Lemma 2

(Reich 1980) LetEbe a uniformly smooth Banach space and letT:C → Cbe a nonexpansive mapping with a fixed point. For each fixedu ∈ Candt ∈ (0, 1), the unique fixed pointxt ∈ Cof the contractionC ∋ x ↦ tu + (1 - t)Txconverges strongly ast → 0to a fixed point ofT. DefineQ:C → F(T) by Qus - limt→0xt. ThenQis the unique sunny nonexpansive retract fromContoF(T), that is,Qsatisfies the property

uQuJ(zQu)⟩ ≤ 0,  u ∈ Cz ∈ F(T).
9

Lemma 3

(Proposition 4.1 in Liu and He 2012) LetH:E → Ebe a strongly accretive and Lipschtiz continuous operator with constantγandT:E → 2Ebe aH-accretive operator. Then the following hold:

  • (i) JH,ρT(x)-JH,ρT(y)1/γx-yx,yR(H+ρT);
  • (ii) H·JH,ρT(x)-H·JH,ρT(y)x-yx,yE, or JH,ρT·H(x)-JH,ρT·H(y)x-yx,yE;
  • (iii) Aρis accretive and
    Aρx-Aρy2ρx-yforallx,yR(H+ρT);
  • (iv) AρxTJH,ρT(x)forallxR(H+ρT).

Lemma 4

(Proposition 4.2 in Liu and He 2012) u ∈ T-10if and only ifusatisfies the relation

u=JH,ρT(H(u))
10

whereρ > 0is a constant andJH,ρTis the resolvent operator defined by (6).

Lemma 5

(see Petryshyn 1970) LetEbe a real Banach space. Then for allxy ∈ E, ∀ j(xy) ∈ J(xy),

x+y2 ≤ ‖x2 + 2⟨yj(xy)⟩.
11

Main results

Proposition 1

LetEbe a strictly convex Banach space,H:E → Ebe a strongly accretive and Lipschtiz continuous operator with constantsγ. LetTi:E → 2Ei = 1, 2, …, rbe a family ofH-accretive operators withi=1rN(Ti). Leta0a1a2, …, arbe real numbers in (0, 1) such thati=0rai=1and letSr:=a0I+a1JH,ρT1H+a2JH,ρT2H++arJH,ρTrH, whereJH,ρT=(H+ρT)-1. ThenSris nonexpansive andF(Sr)=i=1rN(Ti).

Proof

Since every Ti is H-accretive for i = 1, 2, …, r, then JH,ρTiH is well defined and it is a nonexpansive mapping from Lemma 4, and we can also get that F(JH,ρTiH)=N(Ti).

Hence, it is easy to obtain that

i=1rN(Ti)=i=1rFJH,ρTiHF(Sr)

and Sr is nonexpansive.

Next, we prove that F(Sr)i=1rF(JH,ρTiH).

Let z ∈ F(Sr), wi=1rF(JH,ρTiH), we have

z-w=a0z+a1JH,ρT1Hz+a2JH,ρT2Hz++arJH,ρTrHz-w=a0(z-w)+a1(JH,ρT1Hz-w)+a2(JH,ρT2Hz-w)++ar(JH,ρTrHz-w).
12

The above equality can be also written as follows:

z-w=a1i=1rai(JH,ρT1Hz-w)++ari=1rai(JH,ρTrHz-w)

so

z-w=a1i=1rai(JH,ρT1Hz-w)++ari=1rai(JH,ρTrHz-w)
13

From (12), we also have

z-w=a0(z-w)+a1(JH,ρT1Hz-w)+a2(JH,ρT2Hz-w)++ar(JH,ρTrHz-w)a0z-w+a1JH,ρT1Hz-w+a2JH,ρT2Hz-w++arJH,ρTrHz-wi=0raiz-w=z-w.
14

From (14), we get

z-w=i=0raiz-w=i=0r-1aiz-w+arJH,ρTrHz-w=(1-ar)z-w+arJH,ρTrHz-w.

Hence,

z-w=JH,ρTrHz-w.

Similarly, we can get

z-w=JH,ρT1Hz-w=JH,ρT2Hz-w==JH,ρTrHz-w.
15

From the strict convexity of E, (13) and (15), we know that

z-w=JH,ρT1Hz-w=JH,ρT2Hz-w==JH,ρTrHz-w.

Therefore,

JH,ρTiHz=z,fori=1,2,,r.

Namely,

zi=1rF(JH,ρTiH)

The proof is completed.

Theorem 1

LetEbe a strictly convex and real uniformly smooth Banach space which has a uniformly Ga^teaux differentiable norm,H:E → Ebe a strongly accretive and Lipschtiz continuous operator with constantsγ. LetTi:E → 2Ei = 1, 2, …, rbe a family ofH-accretive operators withi=1rN(Ti), For givenux0 ∈ E, let{xn}be generated by the algorithm

xn+1αnu + (1 - αn)Srxnn ≥ 0.
16

whereSr:=a0I+a1JH,ρT1H+a2JH,ρT2H++arJH,ρTrH, withJH,ρTi=(H+ρTi)-1for0<ai<1,i=1,2,,r,i=0rai=1, where∀ ρ ∈ (0, )and{αn} ⊂ [0, 1]satisfy the following conditions:

  • (i) limnαn = 0,
  • (ii) n=0αn=,
  • (iii) n=0|αn-αn-1|< or limn|αn-αn-1|αn=0,

Then{xn}converges strongly to a common solution of the equationsTix = 0 for i = 1, 2, …, r.

Proof

First, we show that {xn} is bounded.

By the Proposition 1, we have that F(Sr)=i=1rN(Ti). Then, take a point x ∈ F(Sr), we get

xn+1-xαn(u-x)+(1-αn)(Srxn-x)αnu-x+(1-αn)xn-x.

By induction we obtain that

xnx‖ ≤ max {‖ux‖, ‖x0x‖},  for n = 0, 1, 2…

Hence, {xn} is bounded, and so is {Srxn}.

Second, we will show that xn+1xn‖ → 0.

From (16) we can get that

xn+1-xn=αnu+(1-αn)Srxn-αn-1u-(1-αn-1)Srxn-1=(αn-αn-1)u+(1-αn)Srxn-(1-αn-1)Srxn-1=(αn-αn-1)u+(1-αn)Srxn-(1-αn)Srxn-1+(1-αn)Srxn-1-(1-αn-1)Srxn-1=(αn-αn-1)(u-Srxn-1)+(1-αn)(Srxn-Srxn-1)(1-αn)Srxn-Srxn-1+|αn-αn-1|u-Srxn-1(1-αn)xn-xn-1+|αn-αn-1|M

where M = sup{‖uSrxn-1‖,  n = 0, 1, 2…} for {Srxn} is bounded. By applying the Lemma 1 and condition (iii), we assert that

xn+1xn‖ → 0, 

as n → .

Then, we have

xnSrxn‖ ≤ ‖xnxn+1‖ + ‖xn+1Srxn‖, 

and so that

xnSrxn‖ → 0, 
17

as n → .

Based on the Lemma 2, there exists the sunny nonexpansive retract Q from E onto the common zeros point set of Ti (i=1rN(Ti),i=1,2,r) and it is unique, that is to say for t ∈ (0, 1),

Qu=s-limt0zt,uE,

and zt satisfies the following equation

zttu + (1 - t)Srzt

where u ∈ E is arbitrarily taken for all r > 0.

Applying the Lemma 5, we obtain that

zt-xn2=t(u-xn)+(1-t)(Srzt-xn)2(1-t)2Srzt-xn2+2tu-xn,j(zt-xn)(1-t)2(Srzt-Srxn+Srxn-xn)2+2t(zt-xn2u-xn,j(zt-xn))(1+t2)zt-xn2+Srxn-xn(2zt-xn+Srxn-xn)+2tu-xn,j(zt-xn).

Then, we have

u-xn,j(zt-xn)t2zt-xn2+Srxn-xn2t(2zt-xn+Srxn-xn).

Since Srxnxn‖ → 0 as n → 0 by (17).

Let n → , we obtain that

lim supnu-xn,j(zt-xn)t2M,
18

where M is a constant such that ztxn2 ≤ M for all t ∈ (0, 1) and n = 1, 2, ….

Since zt → Qu as t →  and the duality mapping j is norm-to weak uniformly continuous on bounded subsets of E. Let t → 0 in (18), we have that

lim supnu-Qu,j(xn+1-Qu)0.
19

Finally, we will show xn → Qu. Applying Lemma 5 to get,

xn+1-Qu2=(1-αn)(yn-Qu)+αn(u-Qu)2(1-αn)(yn-Qu)2+2αnu-Qu,j(xn+1-Qu)(1-αn)(JH,rnTH(xn)-Qu+yn-JH,rnTH(xn))2+2αnu-Qu,j(xn+1-Qu)(1-αn)(xn-Qu+δn)2+2αnu-Qu,j(xn+1-Qu)(1-αn)xn-Qu2+2αnu-Qu,j(xn+1-Qu)+Mδn,
20

where M > 0 is some constant such that 2(1 - αn)‖xnQu‖ + δn ≤ M. An application of Lemma 1 yields that xnQu‖ → 0

This completes the proof.

Remark 2

If we take r = 1, a0 = 0, a1 = 1 in Theorem 1, we can get Theorem 4.1 in Liu and He (2012).

Remark 3

If we suppose Ti (i = 1,2,...,r) is m-accretive in Theorem 1, we can get Theorem 3.3 in Zegeye and Shahzad (2007).

Conclusions

In this paper, we considered the strong convergence for a common zero of a finite family of H-accretive operators in Banach space using the Halpern iterative algorithm (16). The main results presented in this paper extend and improve the corresponding results of Zegeye and Shahzad (2007) and Liu and He (2012) and the related results.

Authors' contributions

This work was carried out by the authors HH, SL, RC, in collaboration. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by Fundamental Research Funds for the Central Universities (No. JB150703), National Science Foundation for Young Scientists of China (No. 11501431), and National Science Foundation for Tian yuan of China (No. 11426167).

Competing interests

The authors declare that they have no competing interests.

Contributor Information

Huimin He, Phone: +86 029 81891397, moc.621@ehnimiuh.

Sanyang Liu, moc.621@gnaynasuil.

Rudong Chen, nc.ude.upjt@drnehc.

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