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Strong convergence theorems for a common zero of a finite family of Haccretive operators in Banach space
SpringerPlus volume 5, Article number: 941 (2016)
Abstract
The aim of this paper is to study a finite family of Haccretive 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 \(\Vert \cdot \Vert\) and Let \(E^*\) be its dual space. The value of \(x^*\in E^*\) at \(x\in E\) will be denoted by \(\langle x,x^*\rangle\).
The inclusion problem is finding a solution to
where T is a setvalued mapping and from E to \(2^E\).
It was first considered by Rockafellar (1976) by using the proximal point algorithm in a Hilbert space \({\mathscr {H}}\) in 1976. For any initial point \(x_0=x\in {\mathscr {H}}\), the proximal point algorithm generates a sequence \(\{x_n\}\) in \({\mathscr {H}}\) by the following algorithm
where \(J_{r_n}=(I+r_nT)^{1}\) and \(\{r_n\}\subset (0,\infty )\), 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 maccretive operators in Banach spaces.
Zegeye and Shahzad (2007) studied a finite family of maccretive mappings and proposed the iterative sequence \(\{x_n\}\) is generated as follows:
where \(S_r:=a_0I+a_1J_{A_1}+a_2J_{A_2}+\cdots +a_rJ_{A_r}\), with \(J_{A_i}:=(I+A_i)^{1}\) for \(0<a_i<1\), \(i=1,2,\ldots ,r\), \(\sum _{i=0}^r a_i=1\).
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 Hmonotone operators and Haccretive operators, and they discussed some properties of this class of operators.
Definition 1
Let \(H:\mathscr {H}\rightarrow \mathscr {H}\) be a singlevalued operator and \(T:\mathscr {H}\rightarrow 2^{\mathscr {H}}\) be a multivalued operator. T is said to be Hmonotone 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 singlevalued operator and \(T:E\rightarrow 2^E\) be a multivalued operator. T is said to be Haccretive if T is accretive and \((H+\lambda T)E=E\) holds for all \(\lambda >0\).
Remark 1
The relations between Haccretive (monotone) operators and maccretive (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 Hmonotone operators in Hilbert space and Haccretive 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 Hmonotone operators in Hilbert space and Haccretive 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 Haccretive 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 \(\{x_n\}\).
A real Banach space 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
for every \(\varepsilon\) with \(0\le \varepsilon \le 2\). It is well known that if E is uniformly convex, then E is reflexive and strictly convex (Goebel and Reich 1984)
Let \(S\triangleq \{x\in E:\Vert x\Vert =1\}\) be the unit sphere of 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).
Let \(H{:}E\rightarrow E\) be a strongly accretive and Lipschtiz continuous operator with constant \(\gamma\). Let \(T{:}E\rightarrow 2^E\) be an Haccretive operator and the resolvent operator \(J_{H,\rho }^T{:}E\rightarrow E\) is defined by
for each \(\rho >0\). We can define the following operators which are called 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
(see Xu 2003) Let \(\{a_n\}\) be a sequence of nonnegative 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
(Reich 1980) 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 \(C\ni x \mapsto tu+(1t)Tx\) 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
(Proposition 4.1 in Liu and He 2012) Let \(H: E\rightarrow E\) be a strongly accretive and Lipschtiz continuous operator with constant \(\gamma\) and \(T: E\rightarrow 2^E\) be a Haccretive operator. Then the following hold:

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

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

(iii)
\(A_\rho\) is accretive and
$$\Vert A_\rho xA_\rho y\Vert \le \frac{2}{\rho }\Vert xy\Vert \quad for\,all\,x,y\in R(H+\rho T);$$ 
(iv)
\(A_\rho x\in T J_{H,\rho }^T(x)\quad for\,all\,x\in R(H+\rho T).\)
Lemma 4
(Proposition 4.2 in Liu and He 2012) \(u\in T^{1}0\) if and only if u satisfies the relation
where \(\rho >0\) is a constant and \(J_{H,\rho }^T\) is the resolvent operator defined by (6).
Lemma 5
(see Petryshyn 1970) 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 Haccretive 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 Haccretive 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 \(F(J_{H,\rho }^{T_i}H)=N(T_i)\).
Hence, it is easy to obtain that
and \(S_r\) is nonexpansive.
Next, we prove that \(F(S_r)\subseteq \cap _{i=1}^rF(J_{H,\rho }^{T_i}H)\).
Let \(z\in F(S_r)\), \(w\in \cap _{i=1}^rF(J_{H,\rho }^{T_i}H)\), we have
The above equality can be also written as follows:
so
From (12), we also have
From (14), we get
Hence,
Similarly, we can get
From the strict convexity of E, (13) and (15), we know that
Therefore,
Namely,
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 Haccretive 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)
\(\sum _{n=0}^\infty \alpha _n\alpha _{n1}<\infty\) or \(\lim \nolimits _{n\rightarrow \infty }\frac{\alpha _n\alpha _{n1}}{\alpha _n}=0\),
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.
By the Proposition 1, we have that \(F(S_r)=\cap _{i=1}^rN(T_i)\ne \emptyset\). Then, take a point \(x^*\in F(S_r)\), we get
By induction we obtain that
Hence, \(\{x_n\}\) is bounded, and so is \(\{S_rx_n\}\).
Second, we will show that \(\Vert x_{n+1}x_n\Vert \rightarrow 0\).
From (16) we can get that
where \(M=\sup \{\Vert uS_rx_{n1}\Vert ,\, n=0,1,2\ldots \}\) for \(\{S_rx_n\}\) is bounded. By applying the Lemma 1 and condition (iii), we assert that
as \(n\rightarrow \infty\).
Then, we have
and so that
as \(n\rightarrow \infty\).
Based on the Lemma 2, there exists the sunny nonexpansive retract 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 \(t\in (0,1)\),
and \(z_t\) satisfies the following equation
where \(u\in E\) is arbitrarily taken for all \(r>0\).
Applying the Lemma 5, we obtain that
Then, we have
Since \(\Vert S_rx_nx_n\Vert \rightarrow 0\) as \(n\rightarrow 0\) by (17).
Let \(n\rightarrow \infty\), we obtain that
where M is a constant such that \(\Vert z_tx_n\Vert ^2\le M\) for all \(t\in (0,1)\) and \(n=1,2,\ldots\).
Since \(z_t\rightarrow Qu\) as \(t\rightarrow \infty\) and the duality mapping j is normto weak\(^*\) uniformly continuous on bounded subsets of E. Let \(t\rightarrow 0\) in (18), we have that
Finally, we will show \(x_n\rightarrow Qu\). Applying Lemma 5 to get,
where \(M>0\) is some constant such that \(2(1\alpha _n)\Vert x_nQu\Vert +\delta _n\le M\). An application of Lemma 1 yields that \(\Vert x_nQu\Vert \rightarrow 0\)
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 maccretive 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 Haccretive 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.
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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).
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The authors declare that they have no competing interests.
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He, H., Liu, S. & Chen, R. Strong convergence theorems for a common zero of a finite family of Haccretive operators in Banach space. SpringerPlus 5, 941 (2016). https://doi.org/10.1186/s400640162646y
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Keywords
 Haccretive operators
 Resolvent operator
 Iteration algorithms
 Strong convergence
Mathematics Subject Classification
 47H06
 47H10