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

- Huimin He
^{1}Email author, - Sanyang Liu
^{1}and - Rudong Chen
^{2}

**Received: **31 March 2016

**Accepted: **22 June 2016

**Published: **30 June 2016

## 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

### Mathematics Subject Classification

47H06 47H10## Background

Let *E* be a real Banach space with norm \(\Vert \cdot \Vert\) and Let \(E^*\) be its dual space. The value of \(x^*\in E^*\) at \(x\in E\) will be denoted by \(\langle x,x^*\rangle\).

*T*is a set-valued mapping and from

*E*to \(2^E\).

*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.

*m*-accretive mappings and proposed the iterative sequence \(\{x_n\}\) is generated as follows:

And proved the sequence \(\{x_n\}\) converges strongly to a common solution of the common zero of the operators \(A_i\) for \(i=1,2,\ldots ,r\).

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 \(H:\mathscr {H}\rightarrow \mathscr {H}\) be a single-valued operator and \(T:\mathscr {H}\rightarrow 2^{\mathscr {H}}\) be a multivalued operator. *T* is said to be *H*-monotone if *T* is monotone and \((H+\lambda T)(\mathscr {H})=\mathscr {H}\) holds for every \(\lambda >0\).

###
**Definition 2**

Let \(H:E\rightarrow E\) be a single-valued operator and \(T:E\rightarrow 2^E\) be a multivalued operator. *T* is said to be *H*-accretive if *T* is accretive and \((H+\lambda T)E=E\) holds for all \(\lambda >0\).

###
*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 \(\{x_n\}\) be a sequence and *u* be a point in a real Banach space with norm \(\Vert \cdot \Vert\) and let \(E^*\) be its dual space. We use \(x_n\rightarrow x\) to denote strong and weak convergence to *x* of the sequence \(\{x_n\}\).

*E*is said to be uniformly convex if \(\delta (\varepsilon )>0\) for every \(\varepsilon >0\), where the modulus \(\delta (\varepsilon )\) of convexity of

*E*is defined by

*E*is uniformly convex, then

*E*is reflexive and strictly convex (Goebel and Reich 1984)

*E*, we consider the limit

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

The norm \(\Vert \cdot \Vert\) of Banach space *E* is said to be uniformly Gâteaux differentiable if for each \(h \in S\) the limit (5) is attained uniformly for *x* in *S*.

The norm \(\Vert \cdot \Vert\) of Banach space *E* is said to be Fréchet differentiable if for each \(x \in S\) the limit (5) is attained uniformly for *h* in *S*.

The norm \(\Vert \cdot \Vert\) of Banach space *E* is said to be uniformly Fréchet differentiable if the limit (5) is attained uniformly for (*x*, *h*) in \(S\times S\). 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).

*H*-accretive operator and the resolvent operator \(J_{H,\rho }^T{:}E\rightarrow E\) is defined by

*Yosida*

*approximation*:

Some elementary properties of \(J_{H,\rho }^T\) and \(A_\rho\) are given as some lemmas in the following in order to establish our convergence theorems.

###
**Lemma 1**

*Let*\(\{a_n\}\)

*be a sequence of non-negative real numbers satisfying the following relation:*

*where*\(\{\gamma _n\}\subset (0,1)\)

*for each*\(n\ge 0\)

*satisfy the conditions:*

- (i)
\(\sum _{n=1}^\infty \gamma _n=\infty\);

- (ii)
\(\limsup \nolimits _{n\rightarrow \infty }\frac{\sigma _n}{\gamma _n}\le 0\) or \(\sum _{n=1}^\infty |\sigma _n|<\infty\);

*Then*\(\{a_n\}\)

*converges strongly to zero.*

###
**Lemma 2**

*Let*

*E*

*be a uniformly smooth Banach space and let*\(T:C\rightarrow C\)

*be a nonexpansive mapping with a fixed point. For each fixed*\(u\in C\)

*and*\(t\in (0,1)\),

*the unique fixed point*\(x_t\in C\)

*of the contraction*\(C\ni x \mapsto tu+(1-t)Tx\)

*converges strongly as*\(t\rightarrow 0\)

*to a fixed point of*

*T*.

*Define*\(Q:C\rightarrow F(T)\) by \(Qu=s-\lim \nolimits _{t\rightarrow 0}x_t\).

*Then*

*Q*

*is the unique sunny nonexpansive retract from*

*C*

*onto*

*F*(

*T*),

*that is,*

*Q*

*satisfies the property*

###
**Lemma 3**

*Let*\(H: E\rightarrow E\)

*be a strongly accretive and Lipschtiz continuous operator with constant*\(\gamma\)

*and*\(T: E\rightarrow 2^E\)

*be a*

*H*-

*accretive operator. Then the following hold:*

- (i)
\(\Vert J_{H,\rho }^T(x)-J_{H,\rho }^T(y)\Vert \le 1/\gamma \Vert x-y\Vert \quad \forall x,y\in R(H+\rho T);\)

- (ii)
\(\Vert H\cdot J_{H,\rho }^T(x)-H\cdot J_{H,\rho }^T(y)\Vert \le \Vert x-y\Vert \quad \forall x,y\in E,\) or \(\Vert J_{H,\rho }^T\cdot H(x)-J_{H,\rho }^T\cdot H(y)\Vert \le \Vert x-y\Vert \quad \forall x,y\in E;\)

- (iii)\(A_\rho\)
*is accretive and*$$\Vert A_\rho x-A_\rho y\Vert \le \frac{2}{\rho }\Vert x-y\Vert \quad for\,all\,x,y\in R(H+\rho T);$$ - (iv)
\(A_\rho x\in T J_{H,\rho }^T(x)\quad for\,all\,x\in R(H+\rho T).\)

###
**Lemma 4**

###
**Lemma 5**

*Let*

*E*

*be a real Banach space. Then for all*\(x,y\in E\), \(\forall j(x+y)\in J(x+y)\),

## Main results

###
**Proposition 1**

*Let*
*E*
*be a strictly convex Banach space,*
\(H{:}E \rightarrow E\)
*be a strongly accretive and Lipschtiz continuous operator with constants*
\(\gamma\). *Let*
\(T_i{:}E\rightarrow 2^E,i=1,2,\ldots ,r\)
*be a family of*
*H*-*accretive operators with*
\(\cap _{i=1}^rN(T_i)\ne \emptyset\). *Let*
\(a_0,a_1,a_2,\ldots ,a_r\)
*be real numbers in* (0, 1) *such that*
\(\sum _{i=0}^r a_i=1\)
*and let*
\(S_r:=a_0I+a_1J_{H,\rho }^{T_1}H+a_2J_{H,\rho }^{T_2}H+\cdots +a_rJ_{H,\rho }^{T_r}H\), *where*
\(J_{H,\rho }^T=(H+\rho T)^{-1}\). *Then*
\(S_r\)
*is nonexpansive and*
\(F(S_r)=\cap _{i=1}^rN(T_i)\).

###
*Proof*

Since every \(T_i\) is *H*-accretive for \(i=1,2,\ldots ,r\), then \(J_{H,\rho }^{T_i}H\) is well defined and it is a nonexpansive mapping from Lemma 4, and we can also get that \(F(J_{H,\rho }^{T_i}H)=N(T_i)\).

Next, we prove that \(F(S_r)\subseteq \cap _{i=1}^rF(J_{H,\rho }^{T_i}H)\).

*E*, (13) and (15), we know that

The proof is completed. \(\square\)

###
**Theorem 1**

*Let*

*E*

*be a strictly convex and real uniformly smooth Banach space which has a uniformly G*\(\hat{a}\)

*teaux differentiable norm,*\(H{:}E \rightarrow E\)

*be a strongly accretive and Lipschtiz continuous operator with constants*\(\gamma\).

*Let*\(T_i{:}E\rightarrow 2^E,i=1,2,\ldots ,r\)

*be a family of*

*H*-

*accretive operators with*\(\cap _{i=1}^rN(T_i)\ne \emptyset\),

*For given*\(u,x_0\in E\),

*let*\(\{x_n\}\)

*be generated by the algorithm*

*where*\(S_r:=a_0I+a_1J_{H,\rho }^{T_1}H+a_2J_{H,\rho }^{T_2}H+\cdots +a_rJ_{H,\rho }^{T_r}H\),

*with*\(J_{H,\rho }^{T_i}=(H+\rho {T_i})^{-1}\)

*for*\(0<a_i<1,\,i=1,2,\ldots ,r,\,\sum _{i=0}^r a_i=1\),

*where*\(\forall \rho \in (0,\infty )\)

*and*\(\{\alpha _n\}\subset [0,1]\)

*satisfy the following conditions:*

- (i)
\(\lim \nolimits _{n\rightarrow \infty }\alpha _n=0\),

- (ii)
\(\sum _{n=0}^\infty \alpha _n=\infty\),

- (iii)
\(\sum _{n=0}^\infty |\alpha _n-\alpha _{n-1}|<\infty\) or \(\lim \nolimits _{n\rightarrow \infty }\frac{|\alpha _n-\alpha _{n-1}|}{\alpha _n}=0\),

*Then*\(\{x_n\}\)

*converges strongly to a common solution of the equations*\(T_ix=0\) for \(i=1,2,\ldots ,r\).

###
*Proof*

First, we show that \(\{x_n\}\) is bounded.

Second, we will show that \(\Vert x_{n+1}-x_n\Vert \rightarrow 0\).

*Q*from

*E*onto the common zeros point set of \(T_i\) (\(\cap _{i=1}^rN(T_i), \,i=1,2,\ldots r\)) and it is unique, that is to say for \(t\in (0,1)\),

Since \(\Vert S_rx_n-x_n\Vert \rightarrow 0\) as \(n\rightarrow 0\) by (17).

*M*is a constant such that \(\Vert z_t-x_n\Vert ^2\le M\) for all \(t\in (0,1)\) and \(n=1,2,\ldots\).

*j*is norm-to weak\(^*\) uniformly continuous on bounded subsets of

*E*. Let \(t\rightarrow 0\) in (18), we have that

This completes the proof. \(\square\)

###
*Remark 2*

If we take \(r=1\), \(a_0=0,a_1=1\) in Theorem 1, we can get Theorem 4.1 in Liu and He (2012).

###
*Remark 3*

If we suppose \(T_i\) (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.

## Declarations

### 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.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Ahmad R, Usman F (2009) System of generalized variational inclusions with \(H\)-accretive operators in uniformly smooth Banach spaces. J Comput Appl Math 230:424–432View ArticleGoogle Scholar
- Benavides TD, Acedo GL, Xu H-K (2003) Iterative solutions for zeros of accretive operators. Math Nachr 248–249:62–71. doi:https://doi.org/10.1002/mana.200310003 View ArticleGoogle Scholar
- Ceng L-C, Khan AR, Ansari QH, Yao J-C (2009) Strong convergence of composite iterative schemes for zeros of \(m\)-accretive operators in Banach spaces. Nonlinear Anal 70:1830–1840View ArticleGoogle Scholar
- Chen R, Liu Y, Shen X (2009) Iterative approximation of a zero of accretive operator in Banach space. Nonlinear Anal 71:e346–e350View ArticleGoogle Scholar
- Day MM (1993) Normed Linear Spaces, 3rd edn. Springer, BerlinGoogle Scholar
- Fan QW, Wu W, Zurada JM (2016) Convergence of batch gradient learning with smoothing regularization and adaptive momentum for neural networks. SpringerPlus 5(295):1–17Google Scholar
- Fang YP, Huang NJ (2003) \(H\)-monotone operator and resolvent operator technique for variational inclusions. Appl Math Comput 145:795–803Google Scholar
- Fang Y, Huang N (2004) \(H\)-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces. Appl Math Lett 17:647–653View ArticleGoogle Scholar
- Goebel K, Reich S (1984) Uniform convexity. Hyperbolic geometry and nonexpansive mappings. Marcel Dekker, New YorkGoogle Scholar
- Huang Z, Noor MA (2007) Equivalency of convergence between one-step iteration algorithm and two-step iteration algorithm of variational inclusions for \(H\)-monotone mappings. Comput Math Appl 53:1567–1571View ArticleGoogle Scholar
- Jung JS (2010) Strong convergence of iterative schemes for zeros of accretive operators in reflexive banach spaces. Fixed Point Theory Appl. doi:https://doi.org/10.1155/2010/103465 Google Scholar
- Kamimura S, Takahashi W (2000) Weak and strong convergence of solutions to accretive operator inclusions and applications. Set-Valued Anal 8:361–374View ArticleGoogle Scholar
- Kartsatos Athanassios G (1996) On the perturbation theory of \(m\)-accretive operators in Banach spaces. Proc Am Math Soc 124(6):1811–1820View ArticleGoogle Scholar
- Li X, Huang N-J (2011) Graph convergence for the \(H(\cdot,\cdot )\)-accretive operator in Banach spaces with an application. Appl Math Comput 217:9053–9061Google Scholar
- Liu S, He H (2012) Approximating solution of \(0\in T(x)\) for an \(H\)-accretive operator in Banach spaces. J Math Anal Appl 385:466–476View ArticleGoogle Scholar
- Liu S, He H, Chen R (2013) Approximating solution of \(0\in T(x)\) for an \(H\)-monotone operator in Hilbert spaces. Acta Math Sci (English Seiries) 33(5):1347–1360View ArticleGoogle Scholar
- Maing P-E (2006) Viscosity methods for zeroes of accretive operators. J Approx Theory 140:127–140View ArticleGoogle Scholar
- Peng J-W (2008) On a new system of generalized mixed quasi-variational-like inclusions with \(H,\eta\)-accretive operators in real \(q\)-uniformly smooth Banach spaces. Nonlinear Anal (TMA) 68:981–993View ArticleGoogle Scholar
- Peng J, Zhu D (2007) A new system of generalized mixed quasi-variational inclusions with \(H,\eta\)-monotone operators. J Math Anal Appl 327:175–187View ArticleGoogle Scholar
- Petryshyn WV (1970) A characterization of strict convexity of Banach spaces and other uses of duality mappings. J Func Anal 6(2):282–291View ArticleGoogle Scholar
- Qin X, Su Y (2007) Approximation of a zero point of accretive operator in Banach spaces. J Math Anal Appl 329:415–424View ArticleGoogle Scholar
- Reich S (1979) Weak convergence theorems for nonexpansive mappings in Banach spaces. J Math Anal Appl 67:274–276View ArticleGoogle Scholar
- Reich S (1980) Strong convergence theorems for resolvents of accretive operators in Banach spaces. J Math Anal Appl 75:287–292View ArticleGoogle Scholar
- Rockafellar RT (1976) Monotone operators and the proximal point algorithm. SIAM J Control Optim 14(5):877–896View ArticleGoogle Scholar
- Song Y, Kang JI, Cho YJ (2010) On iterations methods for zeros of accretive operators in Banach spaces. Appl Math Comput 216:1007–1017Google Scholar
- Tang G-j, Wang X (2014) A perturbed algorithm for a system of variational inclusions involving \(H\)-accretive operators in Banach spaces. J Comput Appl Math 272:1–7View ArticleGoogle Scholar
- Wang ZB, Ding XP (2010) \(H(\cdot,\cdot ),\eta\)-accretive operators with an application for solving set-valued variational inclusions in Banach spaces. Comput Math Appl 59:1559–1567View ArticleGoogle Scholar
- Xia F-Q, Huang N-J (2007) Variational inclusions with a general \(H\)-monotone operator in Banach spaces. Comput Math Appl 54:24–30View ArticleGoogle Scholar
- Xu H-K (2003) An iterative approach to quadratic optimization. J Optim Theory Appl 116:659–678View ArticleGoogle Scholar
- Xu H-K (2006) Strong convergence of an iterative method for nonexpansive and accretive operators. J Math Anal Appl 314:631–643View ArticleGoogle Scholar
- Zegeye H, Shahzad N (2007) Strong convergence theorems for a common zero of a finite family of \(m\)-accretive mappings. Nonlinear Anal 66:1161–1169View ArticleGoogle Scholar
- Zhou H, Cho YJ, Kang SM (2000) Characteristic Conditions for convergence of generalized steepest descent approximation to multivalued accretive operator equations. Comput Math Appl 39:1–11View ArticleGoogle Scholar
- Zou Y-Z, Huang N-J (2008) \(H(\cdot,\cdot )\)-accretive operator with an application for solving variational inclusions in Banach spaces. Appl Math Comput 204:809–816Google Scholar
- Zou Y-Z, Huang N-J (2009) A new system of variational inclusions involving \(H(\cdot,\cdot )\)-accretive operator in Banach spaces. Appl Math Comput 212:135–144Google Scholar