# Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators

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## Abstract

Zero point problems of two accretive operators and fixed point problems of a nonexpansive mappings are investigated based on a Mann-like iterative algorithm. Weak convergence theorems are established in a Banach space.

## Keywords

accretive operator fixed point nonexpansive mapping resolvent zero point## MSC

47H06 47H09 90C33## 1 Introduction and preliminaries

*E*be a real Banach space and let \(E^{*}\) be the dual space of

*E*. Let \(R^{+}\) be the set of nonnegative real numbers. Given a continuous strictly increasing function: \(h:R^{+}\rightarrow R^{+}\), where \(R^{+}\) denotes the set of nonnegative real numbers, such that \(\lim_{r\rightarrow\infty}h(r)=\infty\) and \(h (0)=0\), we associate with it a (possibly multivalued) generalized duality map \(\mathfrak {J}_{\varphi}:E\rightarrow2^{E^{*}}\), defined as

The modulus of convexity of *E* is the function \(\delta_{E}(\epsilon ):(0,2]\rightarrow[0,1]\) defined by \(\delta_{E}(\epsilon)=\inf\{1-\frac{\Vert x+t\Vert }{2}:\Vert x\Vert =\Vert y\Vert =1,\Vert x-y\Vert \geq \epsilon\}\). Recall that *E* is said to be uniformly convex if \(\delta_{E}(\epsilon )>0\) for any \(\epsilon\in(0,2]\). Let \(p>1\). We say that *E* is *p*-uniformly convex if there exists a constant \(c_{q}>0\) such that \(\delta_{E}(\epsilon)\geq c_{p}\epsilon^{p}\) for any \(\epsilon\in(0,2]\).

Let \(B_{E}=\{x\in E: \Vert x\Vert =1\}\). The norm of *E* is said to be Gâteaux differentiable if the limit \(\lim_{t\rightarrow0}(\Vert x+ty\Vert -\Vert x\Vert )/t \) exists for each \(x,y\in B_{E}\). In this case, *E* is said to be smooth. The norm of *E* is said to be uniformly Gâteaux differentiable if for each \(y\in B_{E}\), the limit is attained uniformly for all \(x\in B_{E}\). The norm of *E* is said to be Fréchet differentiable if for each \(x\in B_{E}\), the limit is attained uniformly for all \(y\in B_{E}\). The norm of *E* is said to be uniformly Fréchet differentiable if the limit is attained uniformly for all \(x,y\in B_{E}\).

*E*by

*E*is said to be uniformly smooth if \(\frac{\rho _{E}(t)}{t}\rightarrow0\) as \(t\rightarrow0\). Let \(q>1\).

*E*is said to be

*q*-uniformly smooth if there exists a fixed constant \(c>0\) such that \(\rho_{E}(t)\leq ct^{q}\). It is well known that

*E*is uniformly smooth if and only if the norm of

*E*is uniformly Fréchet differentiable. If

*E*is

*q*-uniformly smooth, then \(q\leq2\) and

*E*is uniformly smooth [1], and hence the norm of

*E*is uniformly Fréchet differentiable, in particular, the norm of

*E*is Fréchet differentiable.

Typical examples of both uniformly convex and uniformly smooth Banach spaces are \(L^{p}\), where \(p>1\). To be more precise, \(L^{p}\) is mini\(\{p,2\} \)-uniformly smooth for every \(p>1\). It is well known that *E* is *p*-uniformly convex if and only if \(E^{*}\) is *q*-uniformly smooth, where *p* and *q* satisfy the relation \(\frac{1}{p}+\frac{1}{q}=1\).

*T*be a mapping on

*E*. The fixed point set of

*T*is denoted by \(\operatorname{Fix}(T)\). Recall that

*T*is said to be nonexpansive iff

*I*denote the identity operator on

*E*. An operator \(A\subset E\times E\) with domain \(D(A)=\{z\in E:Az\neq\emptyset\}\) and range \(R(A)=\cup\{Az:z\in D(A)\}\) is said to be accretive if, for \(t>0\) and \(x,y\in D(A)\),

*A*is accretive if and only if, for \(x,y\in D(A)\), there exists \(\mathfrak{j}_{q}(x-y)\in\mathfrak {J}_{q}(x-y)\) such that

*A*is said to be

*m*-accretive if \(R(I+rA)=E\) for all \(r>0\). In Hilbert spaces, an operator

*A*is

*m*-accretive if and only if

*A*is maximal monotone. In this paper, we use \(A^{-1}(0)\) to denote the set of zero points of

*A*.

*A*on

*E*is said to be

*α*-strongly accretive if there exists a constant \(\alpha>0\) and some \(\mathfrak {j}_{q}(x-y)\in\mathfrak{J}_{q}(x-y)\) such that

*A*is said to be

*α*-inverse strongly accretive if there exists a constant \(\alpha>0\) and some \(\mathfrak {j}_{q}(x-y)\in\mathfrak{J}_{q}(x-y)\) such that

For a multi-valued accretive operator *A*, we can define a nonexpansive single valued mapping \(J_{r}^{A}:R(I+rA)\rightarrow D(A)\) by \(J_{r}^{A}=(I+rA)^{-1} \) for each \(r>0\), which is called the resolvent operator of *A*.

The convex feasibility problem asks to find a point in the intersection of convex sets. This is an important problem in mathematics and engineering; see, *e.g.*, [3, 4, 5, 6] and the references therein. Oftentimes, the convex sets are given as fixed point sets of projections or (more generally) averaged nonexpansive operators. In this paper, we will focus our attention on the problem of finding a common element in \(\operatorname{Fix}(T)\cap(A+B)^{-1}(0)\), where *T* is a nonexpansive mapping, *A* is an *α*-inverse strongly accretive operator and *B* is an *m*-accretive operator, in the framework of uniformly convex and *q*-uniformly smooth Banach spaces. The problem is quite general in the sense that it includes: split feasibility problems, convexly constrained linear inverse problems, fixed point problems, variational inequalities, convexly constrained minimization problems, and Nash equilibrium problems in noncooperative games, as special cases; see, for instance, [7, 8, 9, 10, 11, 12] and the references therein. Recently, mean valued iterative algorithms have been introduced by many authors to investigate this problem; see, for instance, [13, 14, 15, 16, 17, 18] and the references therein. Related work can also be found, *e.g.*, in [19, 20, 21, 22]. However, there is little work in the existing literature in the setting of Banach spaces. The aim of this paper is to establish a weak convergence theorem in the framework of Banach spaces based on a Mann-like iterative algorithm. Applications are also provided to support the main results of this article.

In order to obtain our main results, we also need the following lemmas.

### Lemma 1.1

*Let*

*E*

*be a real Banach space*.

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

*be a single valued operator and let*\(B:E\rightarrow 2^{E}\)

*be an*

*m*-

*accretive operator*.

*Then*

*where*\(J_{r}^{B}(I-rA)\)

*is the resolvent of*

*B*

*for*\(a>0\).

### Proof

### Lemma 1.2

[1]

*Let*

*E*

*be a real*

*q*-

*uniformly smooth Banach space*.

*Then the following inequality holds*:

*and*

*where*\(K_{q}\)

*is some fixed positive constant*.

### Lemma 1.3

[1]

*Let*\(r>0\)

*and*\(q>1\)

*be two fixed real numbers*.

*Then a Banach space*

*E*

*is uniformly convex if and only if there exists a continuous strictly increasing convex function*\(\varphi:[0,\infty)\rightarrow[0,\infty)\)

*with*\(\varphi(0)=0\)

*such that*

*where*\(w(a)=a^{q}(1-a)+(1-a)^{q}a\),

*for all*\(x,y\in B_{r}(0):=\{x\in E: \Vert x\Vert \leq r\}\)

*and*\(a\in[0, 1]\).

### Lemma 1.4

[23]

*Let*

*E*

*be a real uniformly convex Banach space and let*

*C*

*be a nonempty closed convex and bounded subset of*

*E*.

*Then there is a strictly increasing and continuous convex function*\(\psi: [0,\infty)\rightarrow[0, \infty)\)

*with*\(\varphi(0)=0\)

*such that*,

*for every nonexpansive mapping*\(T:C\rightarrow C\)

*and*,

*for all*\(x,y\in C\)

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

*the following inequality holds*:

### Lemma 1.5

[24]

*Let* *E* *be a real uniformly convex Banach space*, *and let* *T* *be a nonexpansive mapping on* *E*. *Then* \(I-T\) *is demiclosed at zero*.

### Lemma 1.6

[25]

*Let* *E* *be a real uniformly convex Banach space*. *Let* \(E^{*}\) *the dual space of* *E* *such that it has the Kadec*-*Klee property*. *Suppose that* \(\{x_{n}\}\) *is a bounded sequence such that* \(\lim_{n\rightarrow\infty} \Vert (1-a)p_{1}-p_{2}+ax_{n}\Vert \) *exists for all* \(a\in[0,1]\) *and* \(p_{1},p_{2}\in\omega_{w}(x_{n})\), *where* \(\omega _{w}(x_{n}):\{x:\exists x_{n_{i}}\rightharpoonup x\}\) *denotes the weak* *ω*-*limit set of* \(\{x_{n}\}\) *Then* \(\omega_{w}(x_{n})\) *is a singleton*.

## 2 Main results

### Theorem 2.1

*Let* *E* *be a real uniformly convex and* *q*-*uniformly smooth Banach space with constant* \(K_{q}\). *Let* \(B:D(B)\subset E\rightarrow2^{E}\) *be an* *m*-*accretive operator*, \(A:E\rightarrow E\) *an* *α*-*inverse strongly accretive operator and* \(T:E\rightarrow E\) *a nonexpansive mapping such that* \(\operatorname{Fix}(T)\cap (A+B)^{-1}(0)\neq\emptyset\). *Let* \(\{r_{n}\}\) *be a positive number sequence and let* \(\{\alpha_{n}\}\) *be a real number sequence in* \((0,1)\) *such that* \(\{\alpha_{n}\}\subset[\alpha,\alpha ']\), *where* \(0<\alpha<\alpha'<1\) *and* \(\{r_{n}\}\subset[r,r']\), *where* \(0< r< r'<(\frac{q\alpha}{K_{q}})^{\frac{1}{q-1}}\). *Let* \(\{x_{n}\}\) *be a sequence generated in the following manner*: \(x_{0}\in E\) *and* \(x_{n+1}=\alpha_{n}Tx_{n}+(1-\alpha_{n})(I+r_{n}B)^{-1}(x_{n}-r_{n}Ax_{n})\), \(\forall n\geq0\), *Then* \(\{x_{n}\}\) *converges weakly to some point in* \(\operatorname{Fix}(T)\cap (A+B)^{-1}(0)\).

### Proof

*B*is

*m*-accretive, we find from Lemma 1.3 and (2.1) that

*B*is an

*m*-accretive operator, we have

Note that, in the framework of Hilbert spaces, the concept of monotonicity coincides with the concept of accretivity. Next, we apply our main results to solve variational inequality problems and minimizer problems of convex functions in the framework of Hilbert spaces.

*H*be a Hilbert space with inner product \(\langle\cdot,\cdot\rangle \) and its induced norm \(\Vert \cdot \Vert \). Let

*C*be a nonempty closed convex subset of

*H*and let \(\operatorname{Proj}_{C}^{H}\) be the metric projection from

*H*onto

*C*. Recall the following classical variational inequality: find \(x\in C\) such that \(\langle y-x,Ax\rangle\geq0\), \(\forall y\in C\). The solution set of the variational inequality is denoted by \(\operatorname{VI}(C,A)\). Projection-gradient methods have been recently investigated for solving the variational inequality. It is well known that

*x*is a solution to the variational inequality iff

*x*is a fixed point of \(\operatorname{Proj}_{C}^{H}(I-rA)\), where

*I*denotes the identity on

*H*and

*r*is a positive real number. If

*A*is inverse strongly monotone, then \(\operatorname{Proj}_{C}^{H}(I-rA)\) is a nonexpansive mapping. Moreover. If

*C*is also bounded, then the existence of solutions of the variational inequality is guaranteed by the nonexpansivity of mapping \(\operatorname{Proj}_{C}^{H}(I-rA)\). Let \(i_{C}\) be a function defined by

*H*, and the subdifferential \(\partial i_{C}\) of \(i_{C}\) is maximal monotone. Define the resolvent \(J_{r}:=(I+r\partial i_{C})^{-1}\) of subdifferential operator \(\partial i_{C}\). Letting \(x=J_{r}y\), we find that

### Corollary 2.2

*Let* *H* *be a real Hilbert space*. *Let* *C* *be a nonempty closed and convex subset of* *E* *and let* \(\operatorname{Proj}_{C}^{H}\) *be the metric projection from* *H* *onto* *C*. *Let* *A* *an* *α*-*inverse strongly monotone operator on* *H* *and* *T* *a nonexpansive mapping on* *C* *such that* \(\operatorname{Fix}(T)\cap \operatorname{VI}(C,A)\neq\emptyset\). *Let* \(\{r_{n}\}\) *be a positive number sequence and let* \(\{\alpha_{n}\}\) *be a real number sequence in* \((0,1)\) *such that* \(\{\alpha_{n}\}\subset[\alpha,\alpha ']\), *where* \(0<\alpha<\alpha'<1\) *and* \(\{r_{n}\}\subset[r,r']\), *where* \(0< r< r'<2\alpha\). *Let* \(\{x_{n}\}\) *be a sequence generated in the following manner*: \(x_{0}\in C\) *and* \(x_{n+1}=\alpha_{n}Tx_{n}+(1-\alpha_{n})\operatorname{Proj}_{C}^{H}(x_{n}-r_{n}Ax_{n})\), \(\forall n\geq0\). *Then* \(\{x_{n}\}\) *converges weakly to some point in* \(\operatorname{Fix}(T)\cap \operatorname{VI}(C,A)\).

Now, we are in a position to consider the problem of finding minimizers of proper lower semicontinuous convex functions. For a proper lower semicontinuous convex function \(g:H\rightarrow(-\infty,\infty]\), the subdifferential mapping *∂g* of *g* is defined by \(\partial g(x)=\{x^{*}\in H:g(x)+\langle y-x,x^{*}\rangle\leq g(y),\forall y\in H\}\), \(\forall x\in H\). Rockafellar [26] proved that *∂g* is a maximal monotone operator. It is easy to verify that \(0\in\partial g(v)\) if and only if \(g(v)=\min_{x\in H} g(x)\).

### Corollary 2.3

*Let* *H* *be a real Hilbert space*. *Let* \(g:H\rightarrow(-\infty,\infty]\) *be a proper convex and lower semicontinuous function and let* \(T:H\rightarrow H\) *be a nonexpansive mapping such that* \(\operatorname{Fix}(T)\cap(\partial g)^{-1}(0)\neq\emptyset\). *Let* \(\{r_{n}\}\) *be a positive number sequence and let* \(\{\alpha_{n}\}\) *be a real number sequence in* \((0,1)\) *such that* \(\{\alpha_{n}\}\subset[\alpha,\alpha ']\), *where* \(0<\alpha<\alpha'<1\) *and* \(\{r_{n}\}\subset[r,r']\), *where* \(0< r< r'<(\frac{q\alpha}{K_{q}})^{\frac{1}{q-1}}\). *Let* \(\{x_{n}\}\) *be a sequence generated in the following manner*: \(x_{0}\in H\) *and* \(x_{n+1}=\alpha_{n}Tx_{n}+(1-\alpha_{n})y_{n}\), \(\forall n\geq0\), *where* \(y_{n}=\min_{z\in H}\{g(z)+\frac{\Vert z-x_{n}+e_{n}\Vert ^{2}}{2r_{n}}\}\). *Then* \(\{x_{n}\}\) *converges weakly to some point in* \(\operatorname{Fix}(T)\cap (A+B)^{-1}(0)\).

### Proof

Since \(g:H\rightarrow(-\infty,\infty]\) is a proper convex and lower semicontinuous function, we see that subdifferential *∂g* of *g* is maximal monotone. Putting \(A=0\), we have \(y_{n}=\arg\min_{z\in H}\{g(z)+\frac{\Vert z-x_{n}\Vert ^{2}}{2r_{n}}\} \) is equivalent to \(0\in\partial g(y_{n})+\frac{1}{r_{n}}(y_{n}-x_{n})\). Hence, we have \(x_{n}\in y_{n}+r_{n}\partial g(y_{n})\). By use of Theorem 2.1, we find the desired conclusion immediately. □

## Notes

### Acknowledgements

The first author was supported by the National Natural Science Foundation of China under Grant No. 11401152. The second author was partially supported by the Grant MOST 103-2923-E-039-001-MY3.

## References

- 1.Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal.
**16**, 1127-1138 (1991) MathSciNetCrossRefMATHGoogle Scholar - 2.Kato, T: Nonlinear semigroups and evolution equations. J. Math. Soc. Jpn.
**19**, 508-520 (1967) MathSciNetCrossRefMATHGoogle Scholar - 3.Bauschke, HH, Combettes, PL: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011) CrossRefMATHGoogle Scholar
- 4.Censor, Y, Zenios, SA: Parallel Optimization. Oxford University Press, Oxford (1997) MATHGoogle Scholar
- 5.Combettes, PL: The convex feasibility problem in image recovery. Adv. Imaging Electron Phys.
**95**, 155-270 (1996) CrossRefGoogle Scholar - 6.Iiduka, H: Iterative algorithm for solving triple-hierarchical constrained optimization problem. J. Optim. Theory Appl.
**48**, 580-592 (2011) MathSciNetCrossRefMATHGoogle Scholar - 7.Aoyama, K, Iiduka, H, Takahashi, W: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl.
**2006**, Article ID 35390 (2006) MathSciNetCrossRefMATHGoogle Scholar - 8.Cho, SY, Qin, X, Wang, L: Strong convergence of a splitting algorithm for treating monotone operators. Fixed Point Theory Appl.
**2014**, Article ID 94 (2014) MathSciNetCrossRefMATHGoogle Scholar - 9.Guan, WB, Song, W: The generalized forward-backward splitting method for the minimization of the sum of two functions in Banach spaces. Numer. Funct. Anal. Optim.
**36**, 867-886 (2015) MathSciNetCrossRefMATHGoogle Scholar - 10.Kamimura, S, Takahashi, W: Weak and strong convergence of solutions to accretive operator inclusions and applications. Set-Valued Anal.
**8**, 361-374 (2000) MathSciNetCrossRefMATHGoogle Scholar - 11.Liu, M, Chang, SS: An iterative method for equilibrium problems and quasi-variational inclusion problems. Nonlinear Funct. Anal. Appl.
**14**, 619-638 (2009) MathSciNetMATHGoogle Scholar - 12.Qin, X, Cho, SY, Wang, L: Iterative algorithms with errors for zero points of m-accretive operators. Fixed Point Theory Appl.
**2013**, Article ID 148 (2013) MathSciNetCrossRefMATHGoogle Scholar - 13.Bin Dehaish, BA, Qin, X, Latif, A, Bakodah, H: Weak and strong convergence of algorithms for the sum of two accretive operators with applications. J. Nonlinear Convex Anal.
**16**, 1321-1336 (2015) MathSciNetMATHGoogle Scholar - 14.Cho, SY, Latif, A, Qin, X: Regularization iterative algorithms for monotone and strictly pseudocontractive mappings. J. Nonlinear Sci. Appl.
**9**, 3909-3919 (2016) MathSciNetMATHGoogle Scholar - 15.Duca, PM, Muu, LD: A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings. Optimization
**65**, 1855-1866 (2016). doi: 10.1080/02331934.2016.1195831 MathSciNetCrossRefGoogle Scholar - 16.Hecai, Y: On weak convergence of an iterative algorithm for common solutions of inclusion problems and fixed point problems in Hilbert spaces. Fixed Point Theory Appl.
**2013**, Article ID 155 (2013) MathSciNetCrossRefMATHGoogle Scholar - 17.Qin, X, Cho, SY, Wang, L: A regularization method for treating zero points of the sum of two monotone operators. Fixed Point Theory Appl.
**2014**, Article ID 75 (2014) MathSciNetCrossRefMATHGoogle Scholar - 18.Qin, X, Bin Dehaish, BA, Cho, SY: Viscosity splitting methods for variational inclusion and fixed point problems in Hilbert spaces. J. Nonlinear Sci. Appl.
**9**, 2789-2797 (2016) MathSciNetMATHGoogle Scholar - 19.Yao, Y, Liou, YC, Yao, JC: Split common fixed point problem for two quasi-pseudocontractive operators and its algorithm construction. Fixed Point Theory Appl.
**2015**, Article ID 127 (2015) MathSciNetCrossRefMATHGoogle Scholar - 20.Yao, Y, Agarwal, RP, Postolache, M, Liou, YC: Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem. Fixed Point Theory Appl.
**2014**, Article ID 183 (2014) MathSciNetCrossRefGoogle Scholar - 21.Yao, Y, Agarwal, RP, Liou, YC: Iterative algorithms for quasi-variational inclusions and fixed point problems of pseudocontractions. Fixed Point Theory Appl.
**2014**, Article ID 82 (2014) MathSciNetCrossRefMATHGoogle Scholar - 22.Yao, Y, Postolache, M, Liou, YC: Strong convergence of a self-adaptive method for the split feasibility problem. Fixed Point Theory Appl.
**2013**, Article ID 201 (2013) MathSciNetCrossRefMATHGoogle Scholar - 23.Bruck, RE: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Isr. J. Math.
**32**, 107-116 (1979) MathSciNetCrossRefMATHGoogle Scholar - 24.Browder, FE: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA
**54**, 1041-1044 (1965) MathSciNetCrossRefMATHGoogle Scholar - 25.Falset, JG, Kaczor, W, Kuczumow, T, Reich, S: Weak convergence theorems for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal.
**43**, 377-401 (2007) MathSciNetCrossRefMATHGoogle Scholar - 26.Rockafellar, RT: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res.
**1**, 97-116 (1976) MathSciNetCrossRefMATHGoogle Scholar

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