 Review
 Open access
 Published:
Strong convergence theorems for solutions of fixed point and variational inequality problems
Journal of Inequalities and Applications volume 2014, Article number: 215 (2014)
Abstract
The purpose of this paper is to investigate viscosity approximation methods for finding a common element in the set of fixed points of a strict pseudocontraction and in the set of solutions of a generalized variational inequality in the framework of Banach spaces.
1 Introduction
Let C be a nonempty, closed and convex subset of a real Hilbert space H, and let {P}_{C} be the metric projection of H onto C. Recall that a mapping A:C\to H is said to be monotone iff
Recall that a mapping A:C\to H is said to be inversestrongly monotone iff there exists a positive real number \alpha >0 such that
For such a case, A is said to be αinversestrongly monotone.
Recall that the classical variational inequality problem, denoted by VI(C,A), is to find u\in C such that
It is clear that variational inequality problem (1.1) is equivalent to a fixed point problem. u is a solution of the above inequality iff it is a fixed point of the mapping {P}_{C}(IrA), where I is the identity and r is some positive real number.
Variational inequality problems have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, and network. Recently, many authors studied the solutions of inequality (1.1) based on iterative methods; see [1–17] and the references therein.
Let S:C\to C be a mapping. In this paper, we denote by F(S) the set of fixed points of the mapping S.
Recall that S is said to be nonexpansive iff
Recall that S is said to be a strict pseudocontraction iff there exits a positive constant λ such that
It is clear that the class of strict pseudocontractions includes the class of nonexpansive mappings as a special case.
Recently, many authors have investigated the problems of finding a common element in the set of solution of variational inequalities for an inversestrongly monotone mapping and in the set of fixed points of nonexpansive mappings or strict pseudocontractions; see [18–25] and the references therein. However, most of the results are in the framework of Hilbert spaces. In this paper, we investigate a common element problem in the framework of Banach spaces. A strong convergence theorem for common solutions to fixed point problems of strict pseudocontractions and solution problems of variational inequality (1.1) is established in uniformly convex and 2uniformly smooth Banach spaces. The results presented in this paper improve and extend the corresponding results announced by Iiduka and Takahashi [5] and Hao [26].
2 Preliminaries
Let C be a nonempty closed and convex subset of a Banach space E. Let {E}^{\ast} be the dual space of E, and let \u3008\cdot ,\cdot \u3009 denote the pairing between E and {E}^{\ast}. For q>1, the generalized duality mapping {J}_{q}:E\to {2}^{{E}^{\ast}} is defined by
for all x\in E. In particular, J={J}_{2} is called the normalized duality mapping. It is known that {J}_{q}(x)={\parallel x\parallel}^{q2}J(x) for all x\in E. If E is a Hilbert space, then J=I, the identity mapping. The normalized duality mapping J has the following properties:

(1)
if E is smooth, then J is singlevalued;

(2)
if E is strictly convex, then it is onetoone and \u3008xy,{x}^{\ast}{y}^{\ast}\u3009>0 holds for all (x,{x}^{\ast}),(y,{y}^{\ast})\in J with x\ne y;

(3)
if E is reflexive, then J is surjective;

(4)
if E is uniformly smooth, then J is uniformly normtonorm continuous on each bounded subset of E.
Let U=\{x\in X:\parallel x\parallel =1\}. A Banach space E is said to be uniformly convex if, for any \u03f5\in (0,2], there exists \delta >0 such that, for any x,y\in U,
It is known that a uniformly convex Banach space is reflexive and strictly convex. Hilbert spaces are 2uniformly convex, while {L}^{p} is max\{p,2\}uniformly convex for every p>1. A Banach space E is said to be smooth if the limit
exists for all x,y\in U. It is also said to be uniformly smooth if the limit (2.1) is attained uniformly for x,y\in U. The norm of E is said to be Fréchet differentiable if, for any x\in U, the limit (2.1) is attained uniformly for all y\in U. The modulus of smoothness of E is defined by
where \rho :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is a function. It is known that E is uniformly smooth if and only if {lim}_{\tau \to 0}\frac{\rho (\tau )}{\tau}=0. Let q be a fixed real number with 1<q\le 2. A Banach space E is said to be quniformly smooth if there exists a constant c>0 such that \rho (\tau )\le c{\tau}^{q} for all \tau >0.
We remark that all Hilbert spaces, {L}_{p} (or {l}_{p}) spaces (p\ge 2) and the Sobolev spaces {W}_{m}^{p} (p\ge 2) are 2uniformly smooth, while {L}_{p} (or {l}_{p}) and {W}_{m}^{p} spaces (1<p\le 2) are puniformly smooth. Typical examples of both uniformly convex and uniformly smooth Banach spaces are {L}^{p}, where p>1. More precisely, {L}^{p} is min\{p,2\}uniformly smooth for every p>1.
Recall that a mapping S is said to be λstrictly pseudocontractive iff there exist a constant \lambda \in (0,1) and j(xy)\in J(xy) such that
It is clear that (2.2) is equivalent to the following:
Next, we assume that E is a smooth Banach space. Let C be a nonempty closed convex subset of E. Recall that an operator A of C into E is said to be accretive iff
An accretive operator A is said to be maccretive if the range of I+rA is E for all r>0. In a real Hilbert space, an operator A is maccretive if and only if A is maximal monotone.
Recall that an operator A of C into E is said to be αinverse strongly accretive iff there exits a real constant \alpha >0 such that
Evidently, the definition of an inversestrongly accretive operator is based on that of an inversestrongly monotone operator.
Let D be a subset of C and Q be a mapping of C into D. Then Q is said to be sunny if
whenever Qx+t(xQx)\in C for x\in C and t\ge 0. A mapping Q of C into itself is called a retraction if {Q}^{2}=Q. If a mapping Q of C into itself is a retraction, then Qz=z for all z\in R(Q), where R(Q) is the range of Q. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D.
The following result describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
Proposition 2.1 [27]
Let E be a smooth Banach space, and let C be a nonempty subset of E. Let Q:E\to C be a retraction, and let J be the normalized duality mapping on E. Then the following are equivalent:

(1)
{Q}_{C} is sunny and nonexpansive;

(2)
{\parallel {Q}_{C}x{Q}_{C}y\parallel}^{2}\le \u3008xy,J({Q}_{C}x{Q}_{C}y)\u3009 \mathrm{\forall}x,y\in E;

(3)
\u3008x{Q}_{C}x,J(y{Q}_{C}x)\u3009\le 0 \mathrm{\forall}x\in E,y\in C.
Proposition 2.2 [28]
Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, and let T be a nonexpansive mapping of C into itself with F(T)\ne \mathrm{\varnothing}. Then the set F(T) is a sunny nonexpansive retract of C.
Recently, Aoyama et al. [29] considered the following generalized variational inequality problem.
Let C be a nonempty closed convex subset of E, and let A be an accretive operator of C into E. Find a point u\in C such that
Next, we use BVI(C,A) to denote the set of solutions of variational inequality problem (2.4).
Aoyama et al. [29] proved that variational inequality (2.4) is equivalent to a fixed point problem. The element u\in C is a solution of variational inequality (2.4) iff u\in C is a fixed point of the mapping {Q}_{C}(IrA), where r>0 is a constant and {Q}_{C} is a sunny nonexpansive retraction from E onto C.
The following lemmas also play an important role in this paper.
Lemma 2.3 [30]
Assume that \{{\alpha}_{n}\} is a sequence of nonnegative real numbers such that
where \{{\gamma}_{n}\} is a sequence in (0,1), \{{e}_{n}\} and \{{\delta}_{n}\} are sequences such that

(1)
{\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}=\mathrm{\infty};

(2)
{\sum}_{n=1}^{\mathrm{\infty}}{e}_{n}<\mathrm{\infty};

(3)
{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\delta}_{n}/{\gamma}_{n}\le 0 or {\sum}_{n=1}^{\mathrm{\infty}}{\delta}_{n}<\mathrm{\infty}.
Then {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0.
Lemma 2.4 [31]
Let E be a real 2uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:
Lemma 2.5 [29]
Let C be a nonempty closed convex subset of a smooth Banach space E. Let {Q}_{C} be a sunny nonexpansive retraction from E onto C, and let A be an accretive operator of C into E. Then, for all \lambda >0,
Lemma 2.6 [32]
Let C be a closed convex subset of a real strictly convex Banach space E and {S}_{i}:C\to C (i=1,2) be two nonexpansive mappings such that F=F({S}_{1})\cap F({S}_{2})\ne \mathrm{\varnothing}. Define Sx=\delta {S}_{1}x+(1\delta ){S}_{2}x, where \delta \in (0,1). Then S:C\to C is a nonexpansive mapping with F(S)=F\ne \mathrm{\varnothing}.
Lemma 2.7 [33]
Let C be a nonempty subset of a real 2uniformly smooth Banach space E, and let T:C\to C be a κstrict pseudocontraction. For \alpha \in (0,1), we define {T}_{\alpha}x=(1\alpha )x+\alpha Tx for every x\in C. Then, as \alpha \in (0,\frac{\kappa}{{K}^{2}}], {T}_{\alpha} is nonexpansive such that F({T}_{\alpha})=F(T).
Lemma 2.8 [34]
Let E be a real uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let T:C\to C be a nonexpansive mapping with a fixed point, and let f:C\to C be a contraction. For each t\in (0,1), let {z}_{t} be the unique solution of the equation x=tf(x)+(1t)Tx. Then \{{z}_{t}\} converges to a fixed point of T as t\to 0 and Q(f)=s\text{}{lim}_{t\to 0}{z}_{t} defines the unique sunny nonexpansive retraction from C onto F(T).
3 Main results
Theorem 3.1 Let E be a uniformly convex and 2uniformly smooth Banach space with the best smooth constant K, and let C be a nonempty, closed and convex subset of E. Let {Q}_{C} be a sunny nonexpansive retraction from E onto C, and let A:C\to E be an αinverse strongly accretive mapping. Let S:C\to C be a λstrict pseudocontraction with a fixed point. Assume that F:=F(S)\cap BVI(C,A)\ne \mathrm{\varnothing}. Let \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{\gamma}_{n}\} be real number sequences in (0,1). Suppose that {x}_{1}=x\in C and that \{{x}_{n}\} is given by
where {S}_{t}=(1t)x+tSx, t\in (0,\frac{\lambda}{{K}^{2}}], f:C\to C is a κcontractive mapping, \{{e}_{n}\} is a bounded computational error in E, \lambda \in (0,\alpha /{K}^{2}] and \mu \in (0,1). Assume that the following restrictions are satisfied:

(a)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n+1}{\alpha}_{n}<\mathrm{\infty};

(b)
{\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}<\mathrm{\infty}.
Then \{{x}_{n}\} converges strongly to x={Q}_{F}f(x), where {Q}_{F} is a sunny nonexpansive retraction from C onto F.
Proof Fixing {x}^{\ast}\in F, we find that {x}^{\ast}={Q}_{C}({x}^{\ast}\lambda A{x}^{\ast}) and S{x}^{\ast}={x}^{\ast}. It follows from Lemma 2.7 that {S}_{t}{x}^{\ast}={x}^{\ast}. Put {y}_{n}={Q}_{C}({x}_{n}\lambda A{x}_{n}). In view of Lemma 2.4, we find that
Since \lambda \in (0,\alpha /{K}^{2}], we have that
This implies that {Q}_{C}(I\lambda A) is a nonexpansive mapping. Hence, we have
which implies that the sequence \{{x}_{n}\} is bounded, so is \{{y}_{n}\}. Define
It follows that
On the other hand, we have
Substituting (3.1) into (3.2), we see that
In view of Lemma 2.3, we find from the restrictions (a) and (b) that
Note that
Using (3.3), we find from the restrictions (a) and (b) that
Define a mapping V by
Using Lemma 2.6, we see that the mapping V is a nonexpansive mapping with
From (3.4), we see that
Next, we show that
where x={Q}_{F}f(x), and {Q}_{F} is a sunny nonexpansive retraction from C onto F, the strong limit of the sequence {z}_{t} defined by
It follows that
For any t\in (0,1), we see that
where
It follows from (3.7) that
This implies that
Since E is 2uniformly smooth, j:E\to {E}^{\ast} is uniformly continuous on any bounded sets of E, which ensures that the {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}} and {lim\hspace{0.17em}sup}_{t\to 0} are interchangeable, hence
This shows that (3.6) holds.
Finally, we show that {x}_{n}\to x as n\to \mathrm{\infty}. Note that
It follows that
Using Lemma 2.3, we find from the restrictions (a) and (b) that
This completes the proof. □
Remark 3.2 The framework of the space in Theorem 3.1 can be applicable to {L}^{p}, p\ge 2.
4 Applications
In this section, we always assume that E is a uniformly convex and 2uniformly smooth Banach space. Let C be a nonempty, closed and convex subset of E.
First, we consider common fixed points of two strict pseudocontractions.
Theorem 4.1 Let E be a uniformly convex and 2uniformly smooth Banach space with the best smooth constant K, and let C be a nonempty closed convex subset of E. Let {Q}_{C} be a sunny nonexpansive retraction from E onto C, and let T:C\to C be an αstrict pseudocontraction. Let S:C\to C be a λstrict pseudocontraction. Assume that F:=F(S)\cap F(T)\ne \mathrm{\varnothing}. Let \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{\gamma}_{n}\} be real number sequences in (0,1). Suppose that {x}_{1}=x\in C and that \{{x}_{n}\} is given by
where {S}_{t}=(1t)x+tSx, t\in (0,\frac{\lambda}{{K}^{2}}], f:C\to C is a κcontractive mapping, \{{e}_{n}\} is a bounded computational error in E, \lambda \in (0,\alpha /{K}^{2}] and \mu \in (0,1). Assume that the following restrictions are satisfied:

(a)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n+1}{\alpha}_{n}<\mathrm{\infty};

(b)
{\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}<\mathrm{\infty}.
Then \{{x}_{n}\} converges strongly to x={Q}_{F}f(x), where {Q}_{F} is a sunny nonexpansive retraction from C onto F.
Proof Since (IT) is an αinverse strongly accretive mapping, we find from Theorem 3.1 the desired conclusion. □
Closely related to the class of pseudocontractive mappings is the class of accretive mappings. Recall that an operator B with domain D(B) and range R(B) in E is accretive if for each {x}_{i}\in D(B) and {y}_{i}\in B{x}_{i} (i=1,2),
An accretive operator B is maccretive if R(I+rB)=E for each r>0. Next, we assume that B is maccretive and has a zero (i.e., the inclusion 0\in B(z) is solvable). The set of zeros of B is denoted by Ω. Hence,
For each r>0, we denote by {J}_{r} the resolvent of B, i.e., {J}_{r}={(I+rB)}^{1}. Note that if B is maccretive, then {J}_{r}:E\to E is nonexpansive and F({J}_{r})=\mathrm{\Omega} for all r>0.
From the above, we have the following theorem.
Theorem 4.2 Let E be a uniformly convex and 2uniformly smooth Banach space with the best smooth constant K, and let C be a nonempty, closed and convex subset of E. Let {Q}_{C} be a sunny nonexpansive retraction from E onto C, and let A:C\to E be an αinverse strongly accretive mapping. Let B:C\to C be an maccretive operator. Assume that F:={A}^{1}(0)\cap {B}^{1}(0)\ne \mathrm{\varnothing}. Let \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{\gamma}_{n}\} be real number sequences in (0,1). Suppose that {x}_{1}=x\in C and that\{{x}_{n}\} is given by
where {J}_{r}={(I+rB)}^{1}, f:C\to C is a κcontractive mapping, \{{e}_{n}\} is a bounded computational error in E, \lambda \in (0,\alpha /{K}^{2}] and \mu \in (0,1). Assume that the following restrictions are satisfied:

(a)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n+1}{\alpha}_{n}<\mathrm{\infty};

(b)
{\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}<\mathrm{\infty}.
Then \{{x}_{n}\} converges strongly to x={Q}_{F}f(x), where {Q}_{F} is a sunny nonexpansive retraction from C onto F.
References
Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.
Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618. 10.1016/S02529602(12)601271
Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl. 2007, 329: 415–424. 10.1016/j.jmaa.2006.06.067
Qin X, Shang M, Su Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model. 2008, 48: 1033–1046. 10.1016/j.mcm.2007.12.008
Iiduka H, Takahashi W: Strong convergence theorems for nonexpansive mappings and inversestrongly monotone mappings. Nonlinear Anal. 2005, 61: 341–350. 10.1016/j.na.2003.07.023
Lv S, Wu C: Convergence of iterative algorithms for a generalized variational inequality and a nonexpansive mapping. Eng. Math. Lett. 2012, 1: 44–57.
Wu C: Strong convergence theorems for common solutions of variational inequality and fixed point problems. Adv. Fixed Point Theory 2014, 4: 229–244.
Wu C, Liu A: Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems. Fixed Point Theory Appl. 2012., 2012: Article ID 90
Wu C: WienerHope equations methods for generalized variational inequalities. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 3
Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017
Bnouhachem A: On LQP alternating direction method for solving variational inequality problems with separable structure. J. Inequal. Appl. 2014., 2014: Article ID 80
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., 2014: Article ID 75
Chen JH: Iterations for equilibrium and fixed point problems. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 4
Guan WB: An iterative method for variational inequality problems. J. Inequal. Appl. 2013., 2013: Article ID 574
Qin X, Cho SY, Kang SM: Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J. Comput. Appl. Math. 2009, 233: 231–240. 10.1016/j.cam.2009.07.018
He R: Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FCspaces. Adv. Fixed Point Theory 2012, 2: 47–57.
Cho SY, Qin X: On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems. Appl. Math. Comput. 2014, 235: 430–438.
Luo H, Wang Y: Iterative approximation for the common solutions of an infinite variational inequality system for inversestrongly accretive mappings. J. Math. Comput. Sci. 2012, 2: 1660–1670.
Wang ZM, Lou W: A new iterative algorithm of common solutions to quasivariational inclusion and fixed point problems. J. Math. Comput. Sci. 2013, 3: 57–72.
Lv S: Strong convergence of a general iterative algorithm in Hilbert spaces. J. Inequal. Appl. 2013., 2013: Article ID 19
Cho SY, Qin X, Wang L: Strong convergence of a splitting algorithm for treating monotone operators. Fixed Point Theory Appl. 2014., 2014: Article ID 94
Hao Y: On variational inclusion and common fixed point problems in Hilbert spaces with applications. Appl. Math. Comput. 2010, 217: 3000–3010. 10.1016/j.amc.2010.08.033
Kim KS, Kim JK, Lim WH: Convergence theorems for common solutions of various problems with nonlinear mapping. J. Inequal. Appl. 2014., 2014: Article ID 2
Wang G, Sun S: Hybrid projection algorithms for fixed point and equilibrium problems in a Banach space. Adv. Fixed Point Theory 2013, 3: 578–594.
Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199
Hao Y: Strong convergence of an iterative method for inverse strongly accretive operators. J. Inequal. Appl. 2008., 2008: Article ID 420989
Reich S: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 1973, 44: 57–70. 10.1016/0022247X(73)900243
Kitahara S, Takahashi W: Image recovery by convex combinations of sunny nonexpansive retractions. Topol. Methods Nonlinear Anal. 1993, 2: 333–342.
Aoyama K, Iiduka H, Takahashi W: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. 2006., 2006: Article ID 35390
Liu LS: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362546X(91)90200K
Bruck RE: Properties of fixed point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 1973, 179: 251–262.
Zhou H: Convergence theorems for λ strict pseudocontractions in 2uniformly smooth Banach spaces. Nonlinear Anal. 2008, 69: 3160–3173. 10.1016/j.na.2007.09.009
Qin X, Cho SY, Wang L: Iterative algorithms with errors for zero points of m accretive operators. Fixed Point Theory Appl. 2013., 2013: Article ID 148
Acknowledgements
The authors are grateful to the reviewers for useful suggestions which improved the contents of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this manuscript. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as 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.
About this article
Cite this article
Yu, L., Song, J. Strong convergence theorems for solutions of fixed point and variational inequality problems. J Inequal Appl 2014, 215 (2014). https://doi.org/10.1186/1029242X2014215
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2014215