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# On synchronal algorithm for fixed point and variational inequality problems in hilbert spaces

- L. M. Bulama
^{1}and - A. Kılıçman
^{1}Email author

**Received:**21 October 2015**Accepted:**14 January 2016**Published:**1 February 2016

## Abstract

The aim of this article is to expand and generalize some approximation methods proposed by Tian and Di (J Fixed Point Appl, 2011. doi:10.1186/1687-1812-21) to the class of \((k, \{\mu _{n}\}, \{\xi _{n}\}, \phi )\)-total asymptotically strict pseudocontraction to solve the fixed point problem as well as variational inequality problem in the frame work of Hilbert space. Further, the results presented in this paper extend, improve and also generalize several known results in the literature .

## Keywords

- Synchronal algorithm
- Total strict asymptotically pseudocontraction
- K-strict pseudo-contraction
- Nonexpansive mapping
- Fixed point and variational inequality problem

## Mathematics Subject Classification

- 47H09
- 47H10

## Background

*H*be a Hilbert space. The mapping \(T:H\rightarrow H\) is said to be; nonexpansive, if \(\left\| Tx-Ty\right\| \le \left\| x-y\right\| , \forall x,y\in H\), quasi-nonexpansive, if \(\left\| Tx-q\right\| \le \left\| x-q\right\| , \forall x\in H\) and \(q\in Fix(T)\), \(\eta\)-strongly monotone, if there exists a positive constant \(\eta >0\) such that \(\left\langle Tx-Ty,x-y\right\rangle \ge \eta \left\| x-y\right\| ^{2}, \forall x,y\in H\), uniformly

*L*-Lipschitzian, if there exists \(L>0\) such that \(\left\| T^{n}x-T^{n}y\right\| \le L\left\| x-y\right\|\), \(\forall x,y\in H\) and

*T*is said to be strongly positive bounded linear operator, if there is a constant \(\gamma >0\) such that \(\left\langle Tx,x\right\rangle \ge \gamma \left\| x\right\| ^{2}, \forall x\in H,\) and also

*T*is said to be; contraction if there exists a constant \(\beta \in [0,1)\) such that \(\left\| Tx-Ty\right\| \le \beta \left\| x-y\right\| , \forall x,y\in H\), strictly pseudocontraction if there exists a constant \(k\in [0,1)\) such that

*T*is said to be; asymptotically strict pseudocontraction if there exists a constant \(k\in [0,1)\) and a sequence \(\{k_{n}\}\subset [1,\infty )\) with \(k_{n}\rightarrow 1\) as \(n\rightarrow \infty\) such that

We now give an example of \((k,\; \{\mu _{n}\}, \{\xi _{n}\}, \phi )\)-total asymptotically strict pseudocontraction mappings .

###
*Example 1*

*B*be a unit ball in a real Hilbert space \(l_{2}\) and \(T:B\rightarrow B\) be a mapping define by

- (i)
\({\left\| Tx-Ty\right\| \le 2\left\| x-y\right\| };\)

- (ii)
\({\left\| T^{n}x-T^{n}y\right\| \le 2\prod ^{n}_{i=2}(a_{i})\left\| x-y\right\| \;\forall x, y\in B\; \mathrm{and}\; n\ge 2}\).

###
*Remark 2*

Note that, every nonexpansive mapping is k-strict pseudocontraction, k-strict pseudocontraction is asymptotically strict pseudocontraction mapping, asymptotically strict pseudocontraction mapping is (\(k,~ \{\mu _{n}\},~\{\xi _{n}\},~ \phi )\)-total asymptotically strict pseudocontraction mapping.

Throughout this paper, we adopt the notations; *I* is the identity operator, *Fix*(*T*) is the fixed point set of *T*, VIP(C,F) is the solution set of variational inequality problem [see Eq. (1)], “\(\rightarrow\)” and “\(\rightharpoonup\)” denote the strong and weak convergence respectively, and \(\omega _{\omega }(x_{n})\) denote the set of the cluster point of \(\{x_{n}\}\) in the weak topology i.e., \(\{ \exists x_{n_{j}}\) of \(\{x_{n}\}\) such that \(x_{n_{j}} \rightharpoonup x\}\).

*C*be a nonempty closed convex subset of

*H*and \(F:C \rightarrow H\) be a map. The variational inequality problem with respect to

*C*and

*F*is defined as search for \(x^{*}\in C,\) such that

*T*is nonexpansive mapping,

*F*is

*L*-Lipschitzian and \(\eta\)-strongly monotone with \({L>0, \eta >0, 0<\mu <\frac{2\eta }{L^{2}}}\) and \(\lambda _{n}\subseteq (0,1)\) satisfying the following conditions:

*f*is a contraction,

*A*is strongly positive bounded linear operator,

*T*is a nonexpansive, \(\{\alpha _{n}\}\) is a sequence in (0, 1) satisfying the conditions in Eq. (3), then they showed that, the sequence \(\{x_{n}\}\) generated by algorithm (6), converged strongly to a common fixed point \(x^{*}\) of

*T*which solve the variational inequality problem

*T*is a nonexpansive,

*f*is a contraction,

*F*is

*k*-Lipschitzian and \(\eta\)- strongly monotone with \(k>0\), \(\eta >0\), \(0<\mu < \frac{2\eta }{k^{2}}\) and \(\{\alpha _{n}\}\) is a sequence in (0, 1) satisfying the conditions in Eq. (3), then the sequence \(\{x_{n}\}\) generated by algorithm (8), converged to a common fixed point \(x^{*}\) of

*T*which solves the variational inequality

###
**Theorem 3**

*Let*

*H*

*be a real Hilbert space and*\(T_{i}:H\rightarrow H\)

*be a*\(k_{i}-\)

*strict pseudocontraction, for some*\(k_{i}\subset (0,1)\), \((i=1,2,3,\ldots ,N)\)

*such that*\(\bigcap ^{N}_{i=1} Fix(T_{i})\ne \emptyset\),

*let f be a contraction with coefficient*\(\beta \in (0,1)\)

*and*\(\lambda _{i}\)

*be a positive constant such that*\({\sum ^{N}_{i=1}\lambda _{i}=1}\).

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

*be a*\(\eta\)

*-strongly monotone and*

*L*

*-Lipschitzian operator with*\(L>0\)

*and*\(\eta >0.\)

*Assume that*\(0<\mu <\frac{2\eta }{L^{2}},\) \({0<\gamma <\mu (\eta -\frac{\mu L^{2}}{2})/\beta =\frac{\tau }{\beta }}\).

*Given the initial guess*\(x_{0}\in H\)

*chosen arbitrarily and given sequences*\(\{\alpha _{n}\}\)

*and*\(\{\beta _{n}\}\)

*in*(0, 1)

*satisfying the following conditions:*

*Let*\(\{x_{n}\}\)

*be the sequence defined by*

*Then*\(\{x_{n}\}\)

*converged strongly to a common point of*\(\{T_{i}\}^{N}_{i=1}\)

*which solves the variational inequality problem*(10).

###
**Theorem 4**

*Let*

*H*

*be a real Hilbert space and*\(T_{i}:H\rightarrow H\)

*be a*\(k_{i}-\)

*strict pseudo-contraction for some*\(k_{i}\in (0,1)\) \((i=1,2,3,\ldots ,N)\)

*such that*\(\bigcap ^{N}_{i=1} Fix(T_{i})\ne \emptyset\)

*and let f be a contraction with coefficient*\(\beta \in (0,1)\).

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

*be a*\(\eta\)

*-strongly monotone and*

*L*

*-Lipschitzian operator with*\(L>0\)

*and*\(\eta >0.\)

*Assume that*\({0<\gamma <\mu \left( \eta -\frac{\mu L^{2}}{2}\right) /\beta =\frac{\tau }{\beta }}\).

*Given the initial guess*\(x_{0}\in H\)

*chosen arbitrarily and given sequences*\(\{\alpha _{n}\}\)

*and*\(\{\beta _{n}\}\)

*in*(0, 1)

*satisfying the following conditions:*

*let*\(\{x_{n}\}\)

*be the sequence defined by*

*where*\(T_{[n]}=T_{i}\),

*with i=n(mod N),*\(1\le i\le N\),

*namely*\(T_{[n]}\)

*is one of*\(T_{1},T_{2},T_{3},\ldots ,T_{N}\)

*circularly. Then*\(\{x_{n}\}\)

*converged strongly to a common point of*\(\{T_{i}\}^{N}_{i=1}\)

*which solve the variational inequality problem*(10).

And also Auwalu et al. (2013) proved the following results in real Banach space which is the generalization of Tian and Di (2011).

###
**Theorem 5**

*Let*

*E*

*be a real*

*q*

*-uniformly smooth Banach space, and*

*C*

*be a nonempty closed convex subset of*

*E*.

*Let*\(T_{i}:C\rightarrow C\)

*be a*\(k_{i}-\)

*strict pseudocontractions for some*\(k_{i}\in (0,1)\), \((i=1,2,3,\ldots ,N)\)

*such that*\(\bigcap ^{N}_{i=1} Fix(T_{i})\ne \emptyset\).

*Let*

*f*

*be a contraction with coefficient*\(\beta \in (0,1)\)

*and*\(\{\lambda _{i}\}_{i=1}^{N}\)

*be a sequence of positive number such that*\({\sum ^{N}_{i=1}\lambda _{i}=1}\).

*Let*\(G:C\rightarrow C\)

*be an*\(\eta\)

*-strongly accretive and*

*L*

*-Lipschitzian operator with*\(L>0\) and \(\eta >0.\)

*Assume that*\(0<\mu <(q\eta /d_{q}L^{q})^{1/q-1}\), \({0<\gamma <\mu (\eta -d_{q}\mu ^{q-1}L^{q}/q)/\beta =\frac{\tau }{\beta }}\).

*Let*\(\{\alpha _{n}\}\)

*and*\(\{\beta _{n}\}\)

*be sequences in (0,1) satisfying the following conditions:*

*Let*\(\{x_{n}\}\)

*be a sequence defined by algorithm*(12),

*then*\(\{x_{n}\}\)

*converged strongly to a common fixed point of*\(\{T_{i}\}^{N}_{i=1}\)

*which solve the variational inequality problem*(10).

Motivated by these two results, in this paper, we modified the algorithms of Tian and Di (2011) to the class of total asymptotically strict pseudocontraction mapping to solve the fixed-point problem as well variational inequality problem, this will be done in the frame work of real Hilbert space. By imposing some conditions, we obtained new strong convergence results. The results presented in this paper, not only extend and improve the results of Tian and Di (2011) but also extend, improve and generalize the results of; Yamada (2001), Marino and Xu (2006), Tain (2010) and Mainge (2009).

## Preliminaries

In the sequel we shall make use of the following lemmas in proving our main results.

###
**Lemma 6**

*Let*

*H*

*be a Hilbert space, there hold the following identities;*

- (i)
\(\left\| x-y\right\| ^{2}=\left\| x\right\| ^{2}-\left\| y\right\| ^{2}-2\left\langle x-y,y\right\rangle , \quad \forall x,y\in H;\)

- (ii)
\(\left\| tx+(1-t)y\right\| ^{2}=t\left\| x\right\| ^{2}+(1-t)\left\| y\right\| ^{2}-t(1-t)\left\| x-y\right\| ^{2}, \quad \forall t\in [0,1]\mathrm{~ and~} x,y\in H\);

- (iii)if \(\{x_{n}\}\) is a sequence in
*H*such that \(x_{n}\rightharpoonup z,\) then$$\begin{aligned}\underset{n\rightarrow \infty }{\limsup }\left\| x_{n}-y\right\| ^{2}=\underset{n\rightarrow \infty }{\limsup }\left\| x_{n}-z\right\| ^{2}+\left\| z-y\right\| ^{2},\forall y\in H.\end{aligned}$$

###
**Lemma 7**

(Chang et al. 2013) *Let*
*C*
*be a nonempty closed convex subset of a real Hilbert space*
*H*
*and let*
\(T:C\rightarrow C\)
*be a* (\(k,~ \{\mu _{n}\},~\{\xi _{n}\},~ \phi )\)
*-total asymptotically strict pseudocontraction mapping and uniformly L-Lipschitzian. Then*
\(I-T\)
*is demiclosed at zero in the sense that if*
\(\{x_{n}\}\)
*is a sequence in*
*C*
*such that*
\(x_{n}\rightharpoonup x^{*}\), *and*
\(\limsup\nolimits_{n\rightarrow \infty}\left\| (T^{n}-I)x_{n}\right\| =0\), *then*
\((T-I)x^{*}=0.\)

###
**Lemma 8**

*Assume that*\(\{a_{n}\}\)

*is a sequence of nonnegative real number such that*

*where*\(\gamma _{n}\)

*is a sequence in*(0, 1)

*and*\(\sigma _{n}\)

*is a sequence of real number such that;*

- (i)
\(\mathop{\lim }\nolimits_{{n \to \infty}}\gamma _{n}=0\; \mathrm{and}\; \sum \gamma _{n}=\infty\);

- (ii)
\(\mathop{\lim }\nolimits_{{n \to \infty}}\frac{\sigma _{n}}{\gamma _{n}}\le 0\) or \(\sum |\sigma _{n}|<\infty .\) Then \(\mathop{\lim }\nolimits_{{n \to \infty}}a_{n}=0.\)

###
**Lemma 9**

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

*be a*\(\eta\)

*-strongly monotone and*

*L*

*-Lipschitzian operator with*\(L>0\)

*and*\(\eta >0\).

*Assume that*\({0<\mu <\frac{2\eta }{L^{2}}}\), \({\tau =\mu \left( \eta -\frac{2L^{2}\mu }{2}\right) }\)

*and*\(0<t<1\).

*Then*

###
**Lemma 10**

*Let*
\(S:C\rightarrow H\)
*be a uniformly*
*L*
*-Lipschitzian mapping with*
\(L\in (0,1]\). *Define*
\(T:C\rightarrow H\)
*by*
\(T^{\beta _{n}}x=\beta _{n} x + (1-\beta _{n} )S^{n}x\)
*with*
\(\beta _{n}\in (0,1)\)
*and*
\(\forall x\in C\). *Then*
\(T^{\beta _{n}}\)
*is nonexpansive and*
\(Fix(T^{\beta _{n}})= Fix(S^{n})\).

###
*Proof*

###
**Lemma 11**

*Let*

*H*

*be a real Hilbert space,*\(f:H\rightarrow H\)

*be a contraction with coefficient*\(0<\alpha <1\)

*and*\(F:H\rightarrow H\)

*be a*

*L*

*-Lipschitzian continuous operator and*\(\eta\)

*-strongly monotone operator with*\(L>0\)

*and*\(\eta >0\).

*Then for*\(0<\gamma <\frac{\mu \eta }{\alpha }\),

## Main results

In this section, we prove the following theorem which is the extension of the theorems (3) and (4).

###
**Theorem 12**

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

*be a*\((k, \{\mu _{n}\}, \{\xi _{n}\}, \phi )\)

*-total asymptotically strict pseudocontraction mapping and uniformly M-Lipschitzian with*\(\phi (t)=t^{2},\quad \forall t \ge 0\)

*and*\(M\in (0,1]\).

*Assume that*\(Fix(T^{n})\ne \emptyset ,\)

*and let*

*f*

*be a contraction with coefficient*\(\beta \in (0,1)\), \(G:H\rightarrow H\)

*be a*\(\eta\)

*-strongly monotone and*

*L*

*-Lipschitzian operator with*\(L>0\)

*and*\(\eta >0\)

*respectively. Assume that*\({0<\gamma <\mu (\eta -\frac{\mu L^{2}}{2})/\beta =\frac{\tau }{\beta }}\)

*and let*\(x_{0}\in H\)

*be chosen arbitrarily,*\(\{\alpha _{n}\}\)

*and*\(\{\beta _{n}\}\)

*be two sequences in (0,1) satisfying the following conditions:*

*Let*\(\{x_{n}\}\)

*be a sequence defined by*

*then*\(\{x_{n}\}\)

*converges strongly to a common fixed of*\(T^{n}\)

*which solve the variational inequality problem*

###
*Proof*

The proof is divided into five steps as follows.

**Step 1**. In this step, we show that

**Step 2**. In this step, we show that

*f*is a contraction, we have

*G*is

*L*-Lipschitzian,

*f*is contraction and the fact that \(\{x_{n}\}, \{T^{n}x_{n}\}\) are bounded, it is easy to see that \(\{GT^{n}x_{n}\}\) and \(\{f(x_{n})\}\) are also bounded.

**Step 3.**In this step, we show that

**Step 4**. In this step, we show that

**Step 5.**In this step, we show that

*f*is a contraction, we have

Hence by Lemma (8), it follows that \(x_{n}\rightarrow x^{*}\) as \(n\rightarrow \infty\). \(\square\)

###
**Corollary 13**

*Let*

*B*

*be a unit ball in a real Hilbert space*\(l_{2}\),

*and let the mapping*\(T:B\rightarrow B\)

*be defined by*

*where*\(\{a_{i}\}\)

*is a sequence in*(0, 1)

*such that*\({\prod ^{\infty }_{i=2}(a_{i})=\frac{1}{2}}\).

*Let,*\(f, G,\gamma , \{\alpha _{n}\}, \{\beta _{n}\}\)

*be as in theorem*(12)

*. Then the sequence*\(\{x_{n}\}\)

*define by algorithm*(17),

*converges strongly to a common fixed point of*\(~T^{n}\)

*which solve the variational inequality problem*(18)

*.*

###
*Proof*

By example (1), it follows that *T* is \((k, \{\mu \}, \{\xi _{n}\}, \phi )\)-total asymptotically strict pseudocontraction mapping and uniformly *M*-Lipschitzian with \(M= {2\prod\nolimits ^{n}_{i=2}(a_{i})}\). Hence, the conclusion of this corollary, follows directly from theorem (12). \(\square\)

###
**Corollary 14**

*Let*
*H*
*be a real Hilbert space and*
\(T:H\rightarrow H\)
*be a*
\((k, \{k_{n}\})\)
*- asymptotically strict pseudocontraction mapping and uniformly*
*M*
*-Lipschitzian with*
\(M\in (0,1]\). *Assume that*
\(Fix(T^{n})\ne \emptyset\), *and Let*
\(f, G, \gamma\)
\(\{\alpha _{n}\}\)
*and*
\(\{\beta _{n}\}\)
*be as in theorem *(12)*. Then, the sequence*
\(\{x_{n}\}\)
*generated by algorithm* (17), *converges strongly to a common fixed point of*
\(~T^{n}\)
*which solve the variational inequality problem* (18).

###
**Corollary 15**

*Let the sequence*\(\{x_{n}\}\)

*be generated by the mapping*

*where*

*T*

*is nonexpansive,*\(\alpha _{n}\)

*is a sequence in (0,1) satisfying the conditions in Eq.*(11).

*It was proved in*Tain (2010)

*that*\(\{x_{n}\}\)

*converged strongly to the common fixed point*\(x^{*}\)

*of*

*T*,

*which is the solution of variational inequality problem*

###
*Proof*

Take n=1, \(k=\mu _{n}=\xi _{n}=0\) and \(F=G\) in theorem (12). Therefore all the conditions in theorem (12) are satisfied. Hence the conclusion of this corollary follows directly from theorem (12). \(\square\)

###
**Corollary 16**

*Let the sequence*\(\{x_{n}\}\)

*be generated by*

*where*

*T*

*is nonexpansive and the sequence*\(\alpha _{n}\subset (0,1)\)

*satisfy the conditions in Eq.*(16).

*Then it was proved in*Marino and Xu (2006)

*that*\(\{x_{n}\}\)

*converged strongly to*\(x^{*}\)

*which solve the variational inequality*

###
*Proof*

Take n=1, \(\mu _{n}=\xi _{n}=0\) and \(\mu =1\) and \(G=A\) in theorem (12). Therefore all the conditions in theorem (12) are satisfied. Hence the conclusion of this corollary follows directly from theorem (12). \(\square\)

###
**Corollary 17**

*Let the sequence*\(\{x_{n}\}\)

*be generated by*

*where*

*T*

*is nonexpansive mapping on*

*H*,

*F*

*is L-Lipschitzian and*\(\eta\)

*-strongly monotone with*\(L>0, \eta >0\)

*and*\(0<\mu <\frac{2\eta }{L^{2}}\),

*if the sequence*\(\lambda _{n}\subset (0,1)\)

*satisfies the conditions in*(3).

*Then, it was proved by*Yamada (2001)

*that*\(\{x_{n}\}\)

*converged strongly to the unique solution of the variational inequality*

## Conclusion

In this paper, we modified the algorithms by Tian and Di (2011) in order to include the class of total asymptotically strict pseudocontraction mapping to solve the fixed-point problem as well variational inequality problem, this was done in the frame work of real Hilbert spaces. By imposing some conditions, we also obtained some new strong convergence results. Further we state that the results which were presented in this paper, not only extend and improve the results (Tian and Di 2011) but also extend, improve and generalize the results of; Yamada (2001), Marino and Xu (2006), Tain (2010) and Mainge (2009).

## Declarations

### Authors’ contributions

Both authors jointly worked on deriving the results. Both authors read and approved the final manuscrip t.

### Acknowledgements

The authors would like to thank the referees for valuable suggestions and comments, which helped the authors to improve this article substantially. The authors also gratefully acknowledge that this research was partially supported by the Universiti Putra Malaysia under the GP-IBT Grant Scheme having Project Number GP-IBT/2013/9420100.

### 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

- Auwalu A, Mohammed LB, Saliu A (2013) Synchronal and cyclic algorithms for fixed point problems and variational inequality problems in Banach spaces. J Fixed Point Appl 2013:1–24. doi:10.1186/1687-1812-2013-202 View ArticleGoogle Scholar
- Chang SS, Lee HWJ, Chan CK, Wang L, Qin LJ (2013) Split feasibility problem for quasi-nonexpansive multi-valued mappings and total asymptotically strict pseudo-contractive mapping. Appl Math Comput 219(20):10416–10424View ArticleGoogle Scholar
- Goebel K, Kirk WA (1972) A fixed point theory for asymptotically nonexpansive mapping. Proc Am Math Soc 35:171–174View ArticleGoogle Scholar
- Jianghua F (2008) A Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces. J Math Anal Appl 337:1041–1047View ArticleGoogle Scholar
- Kinderlehrer D, Stampacchia G (1980) An introduction to variational inequalities and their applications, vol 31. Siam, PhiladelphiaGoogle Scholar
- Mainge PE (2009) The viscosity approximation process for quasi-nonexpansive mapping in Hilbert space. Comput Math Appl 59:74–79View ArticleGoogle Scholar
- Marino G, Xu HK (2006) A general iterative method for nonexpansive mapping in Hilbert space. J Math Anal Appl 318:43–52View ArticleGoogle Scholar
- Marino G, Xu HK (2007) Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J Math Anal Appl 329(1):336–346View ArticleGoogle Scholar
- Noor MA (2007) General variational inequalities and nonexpansive mappings. J Math Anal Appl 331:810–822View ArticleGoogle Scholar
- Tain M (2010) A general iterative algorithm for nonexpansive mapping in Hilbert space. J Nonlinear Anal 73:689–694View ArticleGoogle Scholar
- Tian M, Di L (2011) Synchronal algorithm and cyclic algorithm for fixed point problems and variational inequality problems in Hilbert space. J Fixed Point Appl. doi:10.1186/1687-1812-21 Google Scholar
- Xu HK (2002) Iterative algorithms for nonlinear operators. J Lond Math Soc 66(01):240–256View ArticleGoogle Scholar
- Yamada I (2001) The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. Stud Comput Math 8:473–504View ArticleGoogle Scholar