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

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.

Background

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

The inclusion problem is finding a solution towhere T is a set-valued mapping and from E to .

It was first considered by Rockafellar (1976) by using the proximal point algorithm in a Hilbert space in 1976. For any initial point , the proximal point algorithm generates a sequence in by the following algorithmwhere and , 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 is generated as follows:where , with for , , .

And proved the sequence converges strongly to a common solution of the common zero of the operators for .

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 be a single-valued operator and be a multivalued operator. T is said to be H-monotone if T is monotone and holds for every .

Definition 2

Let be a single-valued operator and be a multivalued operator. T is said to be H-accretive if T is accretive and holds for all .

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 be a sequence and u be a point in a real Banach space with norm and let be its dual space. We use to denote strong and weak convergence to x of the sequence .

A real Banach space E is said to be uniformly convex if for every , where the modulus of convexity of E is defined byfor every with . It is well known that if E is uniformly convex, then E is reflexive and strictly convex (Goebel and Reich 1984)

Let be the unit sphere of E, we consider the limit

The norm of Banach space E is said to be Gâteaux differentiable if the limit (5) exists for each . 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 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 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 (x, h) in . In this case, the Banach space E is said to be uniformly smooth.

The dual space 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 be a strongly accretive and Lipschtiz continuous operator with constant . Let be an H-accretive operator and the resolvent operator is defined byfor each . We can define the following operators which are called Yosidaapproximation:

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

Lemma 1

(see Xu 2003) Letbe a sequence of non-negative real numbers satisfying the following relation:wherefor eachsatisfy the conditions:Thenconverges strongly to zero.

;

or ;

Lemma 2

(Reich 1980) LetEbe a uniformly smooth Banach space and letbe a nonexpansive mapping with a fixed point. For each fixedand, the unique fixed pointof the contractionconverges strongly asto a fixed point ofT. Define by . ThenQis the unique sunny nonexpansive retract fromContoF(T), that is,Qsatisfies the property

Lemma 3

(Proposition 4.1 in Liu and He 2012) Letbe a strongly accretive and Lipschtiz continuous operator with constantandbe aH-accretive operator. Then the following hold:

or

is accretive and

Lemma 4

(Proposition 4.2 in Liu and He 2012) if and only ifusatisfies the relationwhereis a constant andis the resolvent operator defined by (6).

Lemma 5

(see Petryshyn 1970) LetEbe a real Banach space. Then for all, ,

Main results

Proposition 1

LetEbe a strictly convex Banach space,be a strongly accretive and Lipschtiz continuous operator with constants. Letbe a family ofH-accretive operators with. Letbe real numbers in (0, 1) such thatand let, where. Thenis nonexpansive and.

Proof

Since every is H-accretive for , then is well defined and it is a nonexpansive mapping from Lemma 4, and we can also get that .

Hence, it is easy to obtain thatand is nonexpansive.

Next, we prove that .

Let , , we haveThe above equality can be also written as follows:soFrom (12), we also haveFrom (14), we getHence,Similarly, we can getFrom the strict convexity of E, (13) and (15), we know thatTherefore,Namely,

The proof is completed.

Theorem 1

LetEbe a strictly convex and real uniformly smooth Banach space which has a uniformly Gteaux differentiable norm,be a strongly accretive and Lipschtiz continuous operator with constants. Letbe a family ofH-accretive operators with, For given, letbe generated by the algorithmwhere, withfor, whereandsatisfy the following conditions:Thenconverges strongly to a common solution of the equations for .

,

,

or ,

First, we show that is bounded.

By the Proposition 1, we have that . Then, take a point , we getBy induction we obtain thatHence, is bounded, and so is .

Second, we will show that .

From (16) we can get thatwhere for is bounded. By applying the Lemma 1 and condition (iii), we assert thatas .

Then, we haveand so thatas .

Based on the Lemma 2, there exists the sunny nonexpansive retract Q from E onto the common zeros point set of () and it is unique, that is to say for ,and satisfies the following equationwhere is arbitrarily taken for all .

Applying the Lemma 5, we obtain thatThen, we have

Since as by (17).

Let , we obtain thatwhere M is a constant such that for all and .

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

Finally, we will show . Applying Lemma 5 to get,where is some constant such that . An application of Lemma 1 yields that

This completes the proof.

Remark 2

If we take , in Theorem 1, we can get Theorem 4.1 in Liu and He (2012).

Remark 3

If we suppose (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.