Abstract
In this article, we introduce a new iterative scheme for finding a common element of the set of fixed points for a continuous pseudocontractive mapping and the solution set of a variational inequality problem governed by continuous monotone mappings. Strong convergence for the proposed iterative scheme is proved. Our results improve and extend some recent results in the literature.
2000 Mathematics Subject Classification: 46C05; 47H09; 47H10.
Keywords:
monotone mapping; nonexpansive mapping; pseudocontractive mappings; variational inequality1. Introduction
The theory of variational inequalities represents, in fact, a very natural generalization of the theory of boundary value problems and allows us to consider new problems arising from many fields of applied mathematics, such as mechanics, physics, engineering, the theory of convex programming, and the theory of control. While the variational theory of boundary value problems has its starting point in the method of orthogonal projection, the theory of variational inequalities has its starting point in the projection on a convex set.
Let C be a nonempty closed and convex subset of a real Hilbert space H. The classical variational inequality problem is to find a u ∈ C such that 〈vu, Au〉 ≥ 0 for all v ∈ C, where A is a nonlinear mapping. The set of solutions of the variational inequality is denoted by VI(C, A). The variational inequality problem has been extensively studied in the literature, see [15] and the reference therein. In the context of the variational inequality problem, this implies that u ∈ VI(C, A) ⇔ u = P_{C}(u  λAu), ∀λ > 0, where P_{C }is a metric projection of H into C.
Let A be a mapping from C to H, then A is called monotone if and only if for each x, y ∈ C,
An operator A is said to be strongly positive on H if there exists a constant such that
A mapping A of C into itself is called LLipschitz continuous if there exits a positive and number L such that
A mapping A of C into H is called αinversestrongly monotone if there exists a positive real number α such that
for all x, y ∈ C; see [2,610]. If A is an αinverse strongly monotone mapping of C into H, then it is obvious that A is Lipschitz continuous, that is, for all x, y ∈ C. Clearly, the class of monotone mappings include the class of αinverse strongly monotone mappings.
Recall that a mapping T of C into H is called pseudocontractive if for each x, y ∈ C, we have
T is said to be a kstrict pseudocontractive mapping if there exists a constant 0 ≤ k ≤ 1 such that
A mapping T of C into itself is called nonexpansive if ∥Tx  Ty∥ ≤ ∥x  y∥, for all x, y ∈ C. We denote by F(T) the set of fixed points of T. Clearly, the class of pseudocontractive mappings include the class of nonexpansive and strict pseudocontractive mappings.
For finding an element of F(T), where T is a nonexpansive mapping of C into itself, Halpern [11] was the first to study the convergence of the following scheme:
where u, x_{0 }∈ C and a sequence {α_{n}} of real numbers in (0,1) in the framework of Hilbert spaces. Lions [12] improved the result of Halpern by proving strong convergence of {x_{n}} to a fixed point of T provided that the real sequence {α_{n}} satisfies certain mild conditions. In 2000, Moudafi [13] introduced viscosity approximation method and proved that if H is a real Hilbert space, for given x_{0 }∈ C, the sequence {x_{n}} generated by the algorithm
where f : C → C is a contraction mapping with a constant β ∈ (0,1) and {α_{n}} ⊂ (0,1) satisfies certain conditions, converges strongly to fixed point of Moudafi [13] generalizes Halpern's theorems in the direction of viscosity approximations. In [14,15], Zegeye and Shahzad extended Moudafi's result to Banach spaces which more general than Hilbert spaces. For other related results, see [1618]. Viscosity approximations are very important because they are applied to convex optimization, linear programming, monotone inclusion and elliptic differential equations. Marino and Xu [19], studied the viscosity approximation method for nonexpansive mappings and considered the following general iterative method:
They proved that if the sequence {α_{n}} of parameters satisfies appropriate conditions, then the sequence {x_{n}} generated by (1.5) converges strongly to the unique solution of the variational inequality
which is the optimality condition for the minimization problem
where h is a potential function for γf (i.e., h'(x) = γf(x) for x ∈ H).
For finding an element of F(T) ∩ VI(C, A), where T is nonexpansive and A is αinverse strongly monotone, Takahashi and Toyoda [20] introduced the following iterative scheme:
where x_{0 }∈ C, {α_{n}} is a sequence in (0,1), and {λ_{n}} is a sequence in (0, 2α), and obtained weak convergence theorem in a Hilbert space H. Iiduka and Takahashi [7] proposed a new iterative scheme x_{1 }= x ∈ C and
and obtained strong convergence theorem in a Hilbert space.
Motivated and inspired by the work mentioned above which combined from Equations (1.5) and (1.6), in this article, we introduced a new iterative scheme (3.1) below which converges strongly to common element of the set of fixed points of continuous pseudocontractive mappings which more general than nonexpansive mappings and the solution set of the variational inequality problem of continuous monotone mappings which more general than αinverse strongly monotone mappings. As a consequence, we provide an iterative scheme which converges strongly to a common element of set of fixed points of finite family continuous pseudocontractive mappings and the solutions set of finite family of variational inequality problems for continuous monotone mappings. Our theorems extend and unify most the results that have been proved for these important class of nonlinear operators.
2. Preliminaries
Let H be a nonempty closed and convex subset of a real Hilbert space H. Let A be a mapping from C into H. For every point x ∈ H, there exists a unique nearest point in C, denoted by P_{C}x, such that
PC is called the metric projection of H onto C. We know that P_{C }is a nonexpansive mapping of H onto C.
Lemma 2.1.Let H be a real Hilbert space. The following identity holds:
Lemma 2.2.Let C be a closed convex subset of a Hilbert space H. Let x ∈ H and x_{0 }∈ C. Then x_{0 }= P_{C}x if and only if
Lemma 2.3.[21]Let {a_{n}} be a sequence of nonnegative real numbers satisfying the following relation
where,
Then, the sequence {a_{n}} → 0 as n → ∞.
Lemma 2.4.[22]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A : C → H be a continuous monotone mapping. Then, for r > 0 and x ∈ H, there exist z ∈ C such that
Moreover, by a similar argument of the proof of Lemmas 2.8 and 2.9 in[23], Zegeye[22]obtained the following lemmas:
Lemma 2.5.[22]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A : C → H be a continuous monotone mapping. For r > 0 and x ∈ H, define a mapping F_{r }: H → C as follows:
for all x ∈ H. Then the following hold:
(1) F_{r }is singlevalued;
(2) F_{r }is a firmly nonexpansive type mapping, i.e., for all x, y ∈ H,
(3) F(F_{r}) = VI(C,A);
(4) VI(C, A) is closed and convex.
In the sequel, we shall make use of the following lemmas:
Lemma 2.6.[22]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let T : C → H be a continuous pseudocontractive mapping. Then, for r > 0 and x ∈ H, there exist z ∈ C such that
Lemma 2.7.[22]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let T : C → C be a continuous pseudocontractive mapping. For r > 0 and x ∈ H, define a mapping T_{r }: H → C as follows:
for all x ∈ H. Then the following hold:
(1) T_{r }is single  valued;
(2) T_{r }is a firmly nonexpansive type mapping, i.e., for all x, y ∈ H,
(3) F(T_{r}) = F(T);
(4) F(T) is closed and convex.
Lemma 2.8.[19]Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient and 0 < ρ ≤ ∥A∥^{1}. Then .
Let C be a nonempty closed and convex subset of a real Hilbert space H. Let T : C → C be a continuous pseudocontractive mapping and A : C → H be a continuous monotone mapping. Then in what follows, and will be defined as follows: For x ∈ H and {r_{n}} ⊂ (0, ∞), defined
and
3. Strong convergence theorems
In this section, we will prove a strong convergence theorem for finding a common element of the set of fixed points for a continuous pseudocontractive mapping and the solution set of a variational inequality problem governed by continuous monotone mappings.
Theorem 3.1.Let C be a nonempty closed and convex subset of a real Hilbert space H. Let T : C → C be a continuous pseudocontractive mapping and A : C → H be a continuous monotone mapping such that . Let f be a contraction of H into itself with a contraction constant β and let B : H → H be a strongly positive linear bounded selfadjoint operator with coefficients and let {x_{n}} be a sequence generated by x_{1 }∈ C and
where {α_{n}} ⊂ [0,1] and {r_{n}} ⊂ (0, ∞) such that
Then, the sequence {x_{n}} converges strongly to , which is the unique solution of the variational inequality:
Equivalently, , which is the optimality condition for the minimization problem
where h is a potential function for γf (i.e.,h'(z) = γf(z) for z ∈ H).
Remark: (1) The variational inequality (3.2) has the unique solution; (see [19]). (2) It follows from condition (C1) that (1  δ_{n})I  α_{n}B is positive and for all n ≥ 1; (see [24]).
Proof. We processed the proof with following four steps:
Step 1. First, we will prove that the sequence {x_{n}} is bounded.
Let and let and . Then, from Lemmas 2.5 and 2.7 that
Moreover, from (3.1) and (3.2), we compute
Therefore, by the simple introduction, we have
which show that {x_{n}} is bounded, so {y_{n}}, {u_{n}}, and {f(x_{n})} are bounded.
Step 2. We will show that ∥x_{n+1 } x_{n}∥ → 0 and ∥u_{n } y_{n}∥ → 0 as n → ∞.
Notice that each and are firmly nonexpansive. Hence, we have
From (3.1), we note that
where K = ∥γf(x_{n1})  Bx_{n1}∥ = 2 sup{∥f(x_{n}) ∥ + ∥u_{n}∥:n ∈ N}. Moreover, since and , we get
and
Putting y = y_{n+1 }in (3.5) and y = y_{n }in (3.6), we obtain
and
Adding (3.7) and (3.8), we have
which implies that
Using the fact that A is monotone, we get
and hence
We observe that
Without loss of generality, let k be a real number such that r_{n }> k > 0 for all n ∈ N. Then, we have
where M = sup{∥y_{n } x_{n}∥: n ∈ N}. Furthermore, from (3.4) and (3.10), we have
Using Lemma 2.3, and by the conditions (C1) and (C3), we have
Consequently, from (3.10), we obtain
and
Putting y := u_{n+1 }in (3.12) and y := u_{n }in (3.13), we get
and
Adding (3.14) and (3.15), we have
Using the fact that T is pseudocontractive, we get
and hence
Thus, using the methods in (3.9) and (3.10), we can obtain
where M_{1 }= sup{∥u_{n } y_{n}∥: n ∈ N}. Therefore, from (3.11) and property of {r_{n}}, we get
Thus, by (C1) and (C2), we obtain
For , using Lemma 2.5, we obtain
and
Therefore, from (3.1), the convexity of ∥·∥^{2}, (3.2) and (3.18), we get
and hence
So, we have ∥y_{n } v∥ → 0 as n → ∞. Consequently, from (3.16) and (3.18), we obtain
Step 3. We will show that
Let , and since, Q(I  B + γf) is contraction on H into C (see also [[25], pp. 18]) and H is complete. Thus, by Banach Contraction Principle, then there exist a unique element z of H such that z = Q(I  B + γf)z.
We choose subsequence of {x_{n}} such that
Since is bounded, there exists a sequence of and y ∈ C such that . Without loss of generality, we may assume that . Since C is closed and convex it is weakly closed and hence y ∈ C. Since x_{n } y_{n }→ 0 as n → ∞ we have that . Now, we show that . Since , Lemma 2.5 and using (3.5), we get
and
Set v_{t }= tv + (1 t)y for all t ∈ (0,1] and v ∈ C. Consequently, we get v_{i }∈ C. From (3.22), it follows that
from the fact that as i → ∞, we obtain that as i → ∞. Since A is monotone, we also have that . Thus, if follows that
If t → 0, the continuity of A gives that
This implies that y ∈ VI(C, A).
Furthermore, since , Lemma 2.5 and using (3.12), we get
Put z_{t }= t(v) + (1  t)y for all t ∈ (0,1] and v ∈ C. Then, z_{t }∈ C and from (3.23) and pseudocontractivity of T, we get
Thus, since u_{n } y_{n }→ 0, as n → ∞ we obtain that as i → ∞. Therefore, as i → ∞, it follows that
and hence
Taking t → 0 and since T is continuous we obtain
Now, we get v = Ty. Then we obtain that y = Ty and hence y ∈ F(T). Therefore, y ∈ F(T) ∩ VI(C, A) and since , Lemma 2.2 implies that
Step 4. Finally, we will show that x_{n }→ z as n → ∞, where .
From (3.1) and (3.2) we observe that
which implies that
By the condition (C1), (3.24) and using Lemma 2.3, we see that lim_{n→∞ }∥x_{n } z∥ = 0. This complete to proof. □
If we take f(x) = u, ∀x ∈ H and γ = 1, then by Theorem 3.1, we have the following corollary:
Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a continuous pseudocontractive mapping and A : C → H be a continuous monotone mapping such that . let B : H → H be a strongly positive linear bounded selfadjoint operator with coefficients and let {x_{n}} be a sequence generated by x_{1 }∈ H and
where {α_{n}} ⊂ [0,1] and {r_{n}} ⊂ (0, ∞) such that
Then, the sequence {x_{n}} converges strongly to , which is the unique solution of the variational inequality:
If we take T ≡ 0, then (the identity map on C). So by Theorem 3.1, we obtain the following corollary.
Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C → H be a continuous monotone mapping such that . Let f be a contraction of H into itself and let B : H → H be a strongly positive linear bounded selfadjoint operator with coefficients and let {x_{n}} be a sequence generated by x_{1 }∈ H and
where {α_{n}} ⊂ [0,1] and {r_{n}} ⊂ (0, ∞) such that
Then, the sequence {x_{n}} converges strongly to , which is the unique solution of the variational inequality:
If we take A ≡ 0, then (the identity map on C). So by Theorem 3.1, we obtain the following corollary.
Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a continuous pseudocontractive mapping such that . Let f be a contraction of H into itself and let B : H → H be a strongly positive linear bounded selfadjoint operator with coefficients and let {x_{n}} be a sequence generated by x_{1 }∈ H and
where {α_{n}} ⊂ [0,1] and {r_{n}} ⊂ (0, ∞) such that
Then, the sequence {x_{n}}_{n≥1 }converges strongly to , which is the unique solution of the variational inequality:
If we take C ≡ H in Theorem 3.1, then we obtain the following corollary.
Corollary 3.5. Let H be a real Hilbert space. Let T_{n }: H → H be a continuous pseudocontractive mapping and A : H → H be a continuous monotone mapping such that . Let f be a contraction of C into itself and let B : H → H be a strongly positive linear bounded selfadjoint operator with coefficients and let {x_{n}} be a sequence generated by x_{1 }∈ H and
where {α_{n}} ⊂ [0,1] and {r_{n}} ⊂ (0, ∞) such that
Then, the sequence {x_{n}} converges strongly to , which is the unique solution of the variational inequality:
Proof. Since D(A) = H, we note that VI(H, A) = A^{1}(0). So, by Theorem 3.1, we obtain the desired result. □
Remark 3.6. Our results extend and unify most of the results that have been proved for these important classes of nonlinear operators. In particular, Theorem 3.1 extends Theorem 3.1 of Iiduka and Takahashi [7] and Zegeye et al. [26], Corollary 3.2 of Su et al. [27] in the sense that our convergence is for the more general class of continuous pseudocontractive and continuous monotone mappings. Corollary 3.4 also extends Theorem 4.2 of Iiduka and Takahashi [7] in the sense that our convergence is for the more general class of continuous pseudocontractive and continuous monotone mappings.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
This study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRUCSEC No. 54000267).
References

Borwein, FE: Nonlinear monotone operators and convex sets in Banach spaces. Bull Am Math Soc. 71, 780–785 (1965). Publisher Full Text

Zegeye, H, Shahzad, N: Strong convergence for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Anal. 70, 2707–2716 (2009). Publisher Full Text

Bruck, RE: On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J Math Anal Appl. 61, 159–164 (1977). Publisher Full Text

Takahashi, W: Nonlinear complementarity problem and systems of convex inequalities. J Optim Theory Appl. 24, 493–508 (1978)

Noor, MA, Noor, KI, AlSaid, E: Iterative medthods for solving general quasivarational inequalities. Optim Lett. 4, 513–530 (2010). Publisher Full Text

Borwein, JM: Fifty years of maximal monotonicity. Optim Lett. 4, 473–490 (2010). Publisher Full Text

Iiduka, H, Takahashi, W, Toyoda, M: Approximation of solutions of variational inequalities for monotone mappings. Pan Am Math J. 14, 49–61 (2004)

Browder, FE, Petryshyn, WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J Math Anal Appl. 20, 197–228 (1967). Publisher Full Text

Liu, F, Nashed, MZ: Regularization of nonlinear illposed variational inequalities and convergence rates. SetValued Anal. 6, 313–344 (1998). Publisher Full Text

Nakajo, K, Takahashi, W: Strong and weak convergence theorems by an improved spliting method. Commun Appl Nonlinear Anal. 9, 99–107 (2002)

Halpern, B: Fixed points of nonexpanding maps. Bull Am Math Soc. 73, 957–961 (1967). Publisher Full Text

Lions, PL: Approximation de points fixes de contractions. C R Acad Sci Ser A  B Paris. 284, 1357–1359 (1977)

Moudafi, MA: Viscosity approximation methods for fixed point problems. J Math Anal Appl. 241, 46–55 (2000). Publisher Full Text

Zegeye, H, Shahzad, N: Approximation methods for common fixed point of finite family of nonexpansive mappings. Numer Funct Anal. 28(1112), 1405–1419 (2007). Publisher Full Text

Zegeye, H, Shahzad, N: Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings. Appl Math Comput. 191, 155–163 (2007). Publisher Full Text

Nilsrakoo, W, Saejung, S: Equilibruim problems and Moudafi's viscosity approximaion methods in Hilbert spaces. Dyn Contin Discrete Impuls Syst Ser A Math Anal. 17, 195–213 (2010)

Petrusel, A, Yao, JC: Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings. Nonlinear Anal. 69, 1100–1111 (2008). Publisher Full Text

Yao, Y, Yao, JC: On modified iterative method for nonexpansive mappings and monotone mappings. Appl Math Comput. 186, 1551–1558 (2007). Publisher Full Text

Marino, G, Xu, HK: A general iterative method for nonexpansive mapping in Hilbert space. J Math Anal Appl. 318, 43–52 (2006). Publisher Full Text

Takahashi, W, Toyoda, M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl. 118, 417–428 (2003). Publisher Full Text

Xu, HK: Iterative algorithms for nonlinearoperator. J Lond Math Soc. 66, 240–256 (2002). Publisher Full Text

Zegeye, H: An iterative approximation methods for a common fixed point of two pseudocontractive mappings. ISRN Math Anal. 2011, 14 (2011) Article ID 621901, doi:10.5402/2011/621901

Takahashi, W, Zembayashi, K: Strong and weak convergence theorems for equilibriums problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70, 45–57 (2009). Publisher Full Text

Qin, XL, Cho, SY, Kang, SM: Iterative algorithm for variational inequality and equilibrium problems with applications. J Glob Optim. 48, 423–445 (2010). Publisher Full Text

Katchang, P, Kumam, P: A General Iterative Method of Fixed Points for Mixed Equilibrium Problems and Variational Inclusion Problems. Journal of Inequalities and Applications. 2010, 25 (2010) Article ID 370197, doi:10.1155/2010/370197

Zegeye, H, Shahzad, N: Strong convergence of an iterative method for pseudocontractive and monotone mappings. J Glob Optim (2011) doi:10.1007/s1089801197555

Su, Y, Shang, M, Qin, X: An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal. 69, 2709–2719 (2008). Publisher Full Text