Journal of Applied Mathematics
Volume 2011, Issue 1 953903
Research Article
Open Access

Common Solutions of Generalized Mixed Equilibrium Problems, Variational Inclusions, and Common Fixed Points for Nonexpansive Semigroups and Strictly Pseudocontractive Mappings

Poom Kumam, 

Poom Kumam

Department of Mathematics, Faculty of Science, King Mongkut′s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand kmutt.ac.th

Search for more papers by this author
Usa Hamphries, 

Usa Hamphries

Department of Mathematics, Faculty of Science, King Mongkut′s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand kmutt.ac.th

Search for more papers by this author
Phayap Katchang, 

Corresponding Author

Phayap Katchang

Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand rmutl.ac.th

Search for more papers by this author
First published: 11 September 2011
https://doi.org/10.1155/2011/953903
Citations: 8
Academic Editor: Yansheng Liu

Abstract

We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. Our results improve and extend the corresponding results of Martinez-Yanes and Xu (2006), Shehu (2011), Zhang et al. (2008), and many authors.

1. Introduction

Throughout this paper, we assume that H is a real Hilbert space with inner product and norm are denoted by 〈·, ·〉 and ∥·∥, respectively. Let 2H denote the family of all subsets of H, and let C be a closed-convex subset of H. Recall that a mapping T : C → C is said to be a k-strict pseudocontraction [1] if there exists 0 ≤ k < 1 such that
()
where I denotes the identity operator on C. When k = 0, T : C → C is said to be nonexpansive [2] if
()
And when k = 1, T : C → C is said to be pseudocontraction if
()
Clearly, the class of k-strict pseudocontraction falls into the one between classes of nonexpansive mappings and pseudocontraction mapping. We denote the set of fixed points of T by F(T).
A family 𝒮 = {S(s) : 0 ≤ s < ∞} of mappings of C into itself is called a nonexpansive semigroup on C if it satisfies the following conditions:
  • (i)

    S(0)x = x for all x ∈ C,

  • (ii)

    S(s + t) = S(s)S(t) for all s, t ≥ 0,

  • (iii)

    ∥S(s)x − S(s)y∥ ≤ ∥x − y∥ for all x, y ∈ C and s ≥ 0,

  • (iv)

    for all x ∈ C, s ↦ S(s)x is continuous.

We denote by F(𝒮) the set of all common fixed points of 𝒮 = {S(s) : s ≥ 0}, that is, F(S) = ⋂s≥0F(S(s)). It is known that F(𝒮) is closed and convex.

Let A : H → H be a single-valued nonlinear mapping, and let M : H → 2H be a set-valued mapping. We consider the following variational inclusion problem, which is to find a point u ∈ H such that
()
where θ is the zero vector in H. The set of solutions of problem (1.4) is denoted by I(A, M).
Let the set-valued mapping M : H → 2H be a maximal monotone. We define the resolvent operator JM,λ associate with M and λ as follows:
()
where λ is a positive number. It is worth mentioning that the resolvent operator JM,λ is single-valued, nonexpansive, and 1-inverse-strongly monotone ([3, 4]).
Let F be a bifunction of C × C into ℝ, where ℝ is the set of real numbers, let A : C → H be a mapping, and let φ : C → ℝ be a real-valued function. The generalized mixed equilibrium problem is for finding x ∈ C such that
()
The set of solutions of (1.6) is denoted by GMEP (F, φ, A), that is,
()
If A ≡ 0, then the problem (1.6) is reduced into the mixed equilibrium problem for finding x ∈ C such that
()
The set of solutions of (1.8) is denoted by MEP (F, φ). The (generalized) mixed equilibrium problems include fixed-point problems, variational inequality problems, optimization problems, Nash equilibrium problems, noncooperative games, economics, and the equilibrium problem as special cases ([5–15]). In the last two decades, many papers have appeared in the literature on the existence of solutions of equilibrium problems; see, for example, [9] and references therein. Some solution methods have been proposed to solve the mixed equilibrium problems; see, for example, ([7–10, 12–20]) and references therein.
In 2006, Martinez-Yanes and Xu [21] introduced the following iterative:
()
where T is a nonexpansive mapping in a Hilbert space H, and PC is metric projection of H onto a closed and convex subset C of H. They proved that if the sequence {αn} of parameters satisfies appropriate conditions, then the sequence {xn} converges strongly to PF(T)x0.
In 2008, Zhang et al. [4] introduced an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with a multivalued maximal monotone mapping and an inverse-strongly monotone mapping and the set of fixed points of nonexpansive mapping in Hilbert spaces. The following iterative scheme x0 = x ∈ H and
()
for all n ≥ 0. They proved the strong convergence theorem under some mind conditions.
Recently, Shehu [19] introduced a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of k-strictly pseudocontractive mappings, the set of common solutions to a system of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in Hilbert spaces. Starting with an arbitrary , and define sequence {xn}, {wn}, {un}, {zn}, and {yn,i} as follows:
()
where Ti is a ki-strictly pseudocontractive mapping and for some 0 ≤ ki < 1, A, B is α, β-inverse-strongly monotone mapping of C into H. He proved that if the sequence {αn,i}, {rn}, {sn}, and {λn} of parameters satisfies appropriate conditions, then {xn} generated by (1.11) converges strongly to PΩx0.

In this paper, motivated by the above results, we present a new general iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroup mappings, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. Then, we prove strong convergence theorem under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. The results presented in this paper extend and improve the results of Martinez-Yanes and Xu [21], Shehu [19], Zhang et al. [4], and many authors.

2. Preliminaries

Let H be a real Hilbert space with norm ∥·∥ and inner product 〈·, ·〉, and let C be a closed-convex subset of H. When {xn} is a sequence in H, xn⇀x means that {xn} converges weakly to x, and xn → x means that {xn} converges strongly to x. In a real Hilbert space H, we have
()
()
and λ ∈ ℝ. For every point x ∈ H, there exists a unique nearest point in C, denoted by PCx, such that
()
PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive mapping of H onto C and satisfies
()
Moreover, PCx is characterized by the following properties: PCx ∈ C and
()
()
Recall that a mapping A of H into itself is called α-inverse-strongly monotone if there exists a positive real number α such that
()
It is obvious that any α-inverse-strongly monotone mapping A is (1/α)-Lipschitz monotone and continuous mapping.

In order to prove our main results, we need the following Lemmas.

Lemma 2.1 (see [22].)Let V : C → H be a k-strict pseudocontraction, then

  • (1)

    the fixed-point set F(V) of V is closed convex, so that the projection PF(V) is well defined;

  • (2)

    define a mapping T : C → H by

()
If t ∈ [k, 1), then T is a nonexpansive mapping such that F(V) = F(T).

A family of mappings is called a family of uniformly k-strict pseudocontractions if there exists a constant k ∈ [0,1) such that
()
Let be a countable family of uniformly k-strict pseudocontractions. Let be the sequence of nonexpansive mappings defined by (2.8), that is,
()
Let {Ti} be a sequence of nonexpansive mappings of C into itself defined by (2.10), and let {μi} be a sequence of nonnegative numbers in [0,1]. For each n ≥ 1, define a mapping Wn of C into itself as follows:
()
Such a mapping Wn is nonexpansive from C to C and it is called the W-mapping generated by T1, T2, …, Tn and μ1, μ2, …, μn.

For each n, k ∈ ℕ, let the mapping Un,k be defined by (2.11), then we can have the following crucial conclusions concerning Wn. You can find them in [23]. Now, we only need the following similar version in Hilbert spaces.

Lemma 2.2 (see [23].)Let C be a nonempty closed-convex subset of a real Hilbert space H. Let T1, T2, … be nonexpansive mappings of C into itself such that is nonempty, and let μ1, μ2, … be real numbers such that 0 ≤ μn ≤ b < 1 for every n ≥ 1, then

  • (1)

    Wn is nonexpansive and , ∀n ≥ 1,

  • (2)

    for every x ∈ C and k ∈ ℕ, the limit lim n→∞Un,kx exists,

  • (3)

    a mapping W : C → C defined by

()
is a nonexpansive mapping satisfying , and it is called the W-mapping generated by T1, T2, … and μ1, μ2, ….

Lemma 2.3 (see [24].)Let C be a nonempty closed-convex subset of a Hilbert space H, let {Ti : C → C} be a countable family of nonexpansive mappings with , and let {μi} be a real sequence such that 0 < μi ≤ b < 1,   ∀ i ≥ 1. If D is any bounded subset of C, then

()

Lemma 2.4 (see [25].)Each Hilbert space H satisfies Opials condition, that is, for any sequence {xn} ⊂ H with xn⇀x, the inequality

()
holds for each y ∈ H with y ≠ x.

Lemma 2.5 (see [3].)Let M : H → 2H be a maximal monotone mapping, and let A : H → H be a monotone mapping, then the mapping S = M + A : H → 2H is a maximal monotone mapping.

Remark 2.6. Lemma 2.5 implies that I(A, M) is closed and convex if M : H → 2H is a maximal monotone mapping and A : H → H is a monotone mapping.

Lemma 2.7 (see [4].)Let u ∈ H be a solution of variational inclusion (1.4) if and only if u = JM,λ(u − λAu), ∀λ > 0, that is,

()

Lemma 2.8 (see [20].)Let C be a nonempty bounded closed-convex subset of a Hilbert space H, and let 𝒮 = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C, then for any h ≥ 0,

()

Lemma 2.9 (see [26].)Let C be a nonempty bounded closed-convex subset of H, let {xn} be a sequence in C, and let 𝒮 = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C. If the following conditions are satisfied:

  • (1)

    xn⇀z,

  • (2)

    limsup s→∞limsup n→∞∥S(s)xn − xn∥ = 0,   then  z ∈ F(S).

For solving the generalized mixed equilibrium problem for F : C × C → ℝ, one gives the following assumptions for the bifunction F, φ and the set C:

  • (A1)

    F(x, x) = 0 for all x ∈ C,

  • (A2)

    F is monotone, that is, F(x, y) + F(y, x) ≤ 0 for all x, y ∈ C,

  • (A3)

    for each x, y, z ∈ C, lim t→0F(tz + (1 − t)x, y) ≤ F(x, y),

  • (A4)

    for each x ∈ C, y ↦ F(x, y) is convex and lower semicontinuous,

  • (A5)

    for each y ∈ C, x ↦ F(x, y) is weakly upper semicontinuous,

  • (B1)

    for each x ∈ H and r > 0, there exist a bounded subset Dx⊆C and yx ∈ C such that for any z ∈ C∖Dx,

    ()

  • (B2)

    C is a bounded set,

then one has the following lemma.

Lemma 2.10 (see [18].)Let C be a nonempty closed-convex subset of H. Let F : C × C → ℝ be a bifunction that satisfies (A1)–(A5), and let φ : C → ℝ ∪ {+∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r > 0 and x ∈ H, define a mapping Tr : H → C as follows:

()
for all z ∈ H, then the following hold:
  • (1)

    for each ,

  • (2)

    is single valued,

  • (3)

    is firmly nonexpansive, that is, for any ,

  • (4)

    ,

  • (5)

    MEP(F, φ) is closed and convex.

3. Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroup mappings, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space.

Theorem 3.1. Let C be a nonempty closed-convex subset of a real Hilbert Space H. Let F1, F2 be bifunctions of C × C into real numbers ℝ satisfying (A1)–(A5), and let φ1, φ2 : C → ℝ ∪ {+∞} be proper lower semicontinuous and convex functions with assumption (B1) or (B2). Let A, B, E1, E2 be α, β, η1, η2-inverse-strongly monotone mappings of C into H, respectively, and let M1, M2 : H → 2H be maximal monotone mappings. Let 𝒮 = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C, and let {tn} be a positive real divergent sequence. Let be a countable family of uniformly k-strict pseudocontractions, let be a countable family of nonexpansive mappings defined by Tix = tx + (1 − t)Vix,   ∀ x ∈ C, ∀i ≥ 1,   t ∈ [k, 1), and let Wn be the W-mapping defined by (2.11) and W a mapping defined by (2.12) with F(W) ≠ ∅. Suppose that Θ : = F(S)∩F(W)∩GMEP(F1, φ1, A)∩GMEP(F2, φ2, B)∩I(E1, M1)∩I(E2, M2) ≠ ∅. Let {xn} be a sequence generated by , and

()
for every n ≥ 0, where , {rn}, {qn}⊂(0, ∞), λ1 ∈ (0,2η1), and λ2 ∈ (0,2η2) satisfy the following conditions:
  • (i)

    0 < a ≤ rn ≤ b < 2α,

  • (ii)

    0 < c ≤ qn ≤ d < 2β,

  • (iii)

    lim n→∞αn,i = 0,

  • (iv)

    0 < e ≤ λ1 ≤ f < 2η1,

  • (v)

    0 < g ≤ λ2 ≤ j < 2η2,

then {xn} converges strongly to PΘx0.

Proof. First, we show that I − λ1E1 and I − λ2E2 are nonexpansive. Indeed, for all x, y ∈ C and λ1 ∈ (0,2η1), we obtain

()
which implies that the mapping I − λ1E1 is nonexpansive, so is I − λ2E2. Let p ∈ Θ. We observe that
()
Since both I − rnA and I − qnB are nonexpansive for each n ≥ 1, let p ∈ Θ, then and ; by conditions (i) and (ii), we have
()
Therefore, we get
()
Next, we will divide the proof into five steps.

Step 1. We show that {xn} is well defined. Let n = 1, then C1,i = C is closed and convex for each i ≥ 1. Suppose that Cn,i is closed convex for some n > 1, then, from the definition of Cn+1,i, we know that Cn+1,i is closed convex for the same n ≥ 1. Hence, Cn,i is closed convex for n ≥ 1 and for each i ≥ 1. This implies that Cn is closed convex for n ≥ 1. Furthermore, we show that Θ ⊂ Cn. For n = 1, Θ ⊂ C = C1,i. For n ≥ 2, let p ∈ Θ, then

()
which shows that p ∈ Cn,i,   ∀ n ≥ 2,   ∀ i ≥ 1. Thus, Θ ⊂ Cn,i,   ∀ n ≥ 1,   ∀ i ≥ 1. Hence, it follows that ∅ ≠ Θ ⊂ Cn,   ∀ n ≥ 1. This implies that {xn} is well defined.

Step 2. We claim that lim n→∞∥xn+1 − xn∥ = 0 and lim n→∞∥yn,i − xn∥ = 0, for i ≥ 1. Since and , we have

()
Also, as Θ ⊂ Cn by (2.1), it follows that
()
Form (3.7) and (3.8), we have that lim n→∞∥xn − x0∥ exists. Hence, {xn} is bounded and so are {yn,i}, ∀i ≥ 1, {wn}, {vn}, {un}, {tn}, {Axn}, {Btn}, {E1un}, {E2vn}, {Wnwn}, and . For m > n ≥ 1, we have that . By (2.5), we obtain
()
Letting m, n → ∞ and taking the limit in (3.9), we have ∥xm − xn∥ → 0, which shows that {xn} is Cauchy. In particular,
()
Since {xn} is Cauchy, we assume that xn → z ∈ C. Since , then
()
and it follows that
()
Therefore,
()

Step 3. We claim that the following statements hold:

  • (1)

    lim n→∞∥un − tn∥ = 0,

  • (2)

    lim n→∞∥tn − xn∥ = 0,

  • (3)

    lim n→∞∥wn − vn∥ = 0,

  • (4)

    lim n→∞∥vn − un∥ = 0.

For p ∈ Θ, from (3.4), and (3.6), we obtain

()
Since 0 < c ≤ qn ≤ d < 2β, we have
()
Hence, by condition (iii) and (3.13), we have
()
From (3.6), we have
()
On the other hand,
()
and hence,
()
Putting (3.19) into (3.17), for i ≥ 1, we have
()
It follows that
()
Therefore, from condition (iii), (3.13), and (3.16), we have
()
Furthermore, from (3.4), and (3.6), we get
()
Since 0 < a ≤ rn ≤ b < 2α, we have
()
Then, by condition (iii) and (3.13), we obtain that
()
From (3.6), we have
()
On the other hand, we note that
()
and hence,
()
Putting (3.28) into (3.26), we have
()
It follows that
()
Therefore, by condition (iii), (3.13), and (3.25), we have
()
Condition (iii) implies that
()
It follows that
()
From (3.6), we have
()
Since 0 < g ≤ λ2 ≤ j < 2η2, we have
()
Then, by condition (iii) and (3.13), we obtain that
()
From (3.6), we have
()
On the other hand, we note that
()
and hence,
()
Putting (3.39) into (3.37),
()
this implies that
()
Therefore, by condition (iii), (3.13), and (3.36), we have
()
Furthermore, from (3.6), we have
()
Since 0 < e ≤ λ1 ≤ f < 2η1, we have
()
Then, by condition (iii) and (3.13), we obtain that
()
From (3.6), we have
()
On the other hand, we note that
()
and hence,
()
Putting (3.48) into (3.46),
()
this implies that
()
Therefore, by condition (iii), (3.13), and (3.45), we have
()

Step 4. We show that z ∈ Θ : = F(𝒮)∩F(W)∩GMEP (F1, φ1, A)∩GMEP (F2, φ2, B)∩I(E1, M1)∩I(E2, M2). Since is bounded, there exists a subsequence of which converges weakly to z ∈ C. Without loss of generality, we can assume that .

(1) First, we prove that z ∈ F(𝒮). From (3.22), (3.31), (3.33), (3.42), and (3.51), we get

()
Since {Wnwn} is bounded and from Lemma 2.8 for all s ≥ 0, we have
()
and since
()
It follows from (3.52) and (3.53) that
()
Indeed, from Lemma 2.9 and (3.55), we get z ∈ F(𝒮), that is, z = S(s)z, ∀s ≥ 0.

(2) Next, we show that , where , and F(Wn+1) ⊂ F(Wn). Assume that z ∉ F(W), then there exists a positive integer m such that z ∉ F(Tm), and so . Hence, for any , that is, z ≠ Wnz. This together with z = S(s)z, ∀s ≥ 0 shows that z = S(s)z ≠ S(s)Wnz, ∀s ≥ 0; therefore, we have . It follows from the Opial′s condition and (3.52) that

()
which is a contradiction. Thus, we get z ∈ F(W).

(3) Now, we prove that z ∈ GMEP (F1, φ, A). Since , we have for any y ∈ C that

()
From (A2), we also have
()
For t with 0 < t ≤ 1 and y ∈ C, let yt = ty + (1 − t)z. Since y ∈ C and z ∈ C, we have yt ∈ C. Then, we have
()
Since , we have . Furthermore, from the inverse-strongly monotonicity of A, we have . So, from (A4), (A5), and the weak lower semicontinuity of and , we have at the limit
()
as i → ∞. From (A1), (A4), and (3.60), we also get
()
and hence,
()
Letting t → 0, we have, for each y ∈ C,
()
This implies that z ∈ GMEP (F1, φ, A). By following the same arguments, we can show that z ∈ GMEP (F2, φ, B).

(4) At last, we show that z ∈ I(E2, M2). Infact, since E2 is η2-inverse-strongly monotone, this implies that E2 is(1/η2)-Lipschitz continuous monotone mapping and domain of E2 equal to H. It follows from Lemma 2.5 that M2 + E2 is a maximal monotone. Let (y, g) ∈ G(M2 + E2), that is, g − E2y ∈ M2(y). Since , we have , that is,

()
Since M2 + E2 is a maximal monotone, we have
()
and so
()
It follows from ∥vn − wn∥ → 0, ∥E2vn − E2wn∥ → 0, and that
()
It follows from the maximal monotonicity of M2 + E2 that 0 ∈ (M2 + E2)(z), that is, z ∈ I(E2, M2). By following the same arguments, we can show that z ∈ I(E1, M1). Hence, by (1)–(4), we have z ∈ Θ.

Step 5. Noting that , by (2.5), we have

()
Since Θ ⊂ Cn and by the continuity of inner product, we obtain from the above inequality that
()
By (2.5) again, we conclude that z = PΘx0. This completes the proof.

Using Theorem 3.1, we obtain the following corollaries.

Corollary 3.2. Let C be a nonempty closed-convex subset of a real Hilbert Space H. Let F1, F2 be bifunctions of C × C into real numbers ℝ satisfying (A1)–(A5), and let φ1, φ2 : C → ℝ ∪ {+∞} be proper lower semicontinuous and convex functions with assumption (B1) or (B2). Let A, B, E1, E2 be α, β, η1, η2-inverse-strongly monotone mappings of C into H, respectively. Let 𝒮 = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C, and let {tn} be a positive real divergent sequence. Let be a countable family of uniformly k-strict pseudocontractions, let be a countable family of nonexpansive mappings defined by Tix = tx + (1 − t)Vix, ∀x ∈ C, ∀i ≥ 1, t ∈ [k, 1), and let Wn be the W-mapping defined by (2.11) and W a mapping defined by (2.12) with F(W) ≠ ∅. Suppose that Θ : = F(𝒮)∩F(W)∩GMEP(F1, φ1, A)∩GMEP(F2, φ2, B)∩VI(C, E1)∩VI(C, E2) ≠ ∅. Let {xn} be a sequence generated by , and

()
for every n ≥ 0, where , {rn}, {qn}⊂(0, ∞), λ1 ∈ (0,2η1), and λ2 ∈ (0,2η2) satisfy the following conditions:
  • (i)

    0 < a ≤ rn ≤ b < 2α,

  • (ii)

    0 < c ≤ qn ≤ d < 2β,

  • (iii)

    lim n→∞αn,i = 0,

  • (iv)

    0 < e ≤ λ1 ≤ f < 2η1,

  • (v)

    0 < g ≤ λ2 ≤ j < 2η2,

then {xn} converges strongly to PΘx0.

Proof. From Theorem 3.1, put M = ∂δC, then and . So we have vn = PC(un − λ1E1un) and wn = PC(vn − λ2E2vn). The conclusion of Corollary 3.2 can be obtained from Theorem 3.1 immediately.

4. Applications

In this section, we study a kind of multiobjective optimization problem by using the result of this paper. We will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions:
()
where h(x) is a convex and lower semicontinuous functional, and define C as a closed-convex subset of a real Hilbert space H. We denote the set of solutions of (4.1) by M(h1) and M(h2). Let Fi : C × C → ℝ be a bifunction defined by Fi(x, y) = hi(y) − hi(x). We consider the equilibrium problem, and it is obvious that EP(Fi) = M(hi), i = 1,2. Therefore, from Theorem 3.1, we obtain the following Corollaries.

Corollary 4.1. Let C be a nonempty closed-convex subset of a real Hilbert Space H. Let h1, h2 : C → ℝ ∪ {+∞} be proper lower semicontinuous and convex functions. Let E1, E2 be η1, η2-inverse-strongly monotone mappings of C into H, respectively, and let M1, M2 : H → 2H be maximal monotone mappings. Let 𝒮 = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C, and let {tn} be a positive real divergent sequence. Let be a countable family of uniformly k-strict pseudocontractions, let be a countable family of nonexpansive mappings defined by Tix = tx + (1 − t)Vix,   ∀ x ∈ C,   ∀ i ≥ 1,   t ∈ [k, 1), and let Wn be the W-mapping defined by (2.11) and W a mapping defined by (2.12) with F(W) ≠ ∅. Suppose that Θ : = F(𝒮)∩F(W)∩M(h1)∩M(h2)∩I(E1, M1)∩I(E2, M2) ≠ ∅. Let {xn} be a sequence generated by , and

()
for every n ≥ 0, where , {rn}, {qn}⊂(0, ∞), λ1 ∈ (0,2η1), and λ2 ∈ (0,2η2) satisfy the following conditions:
  • (i)

    liminf n→∞rn > 0,

  • (ii)

    liminf n→∞qn > 0,

  • (iii)

    lim n→∞αn,i = 0,

  • (iv)

    0 < e ≤ λ1 ≤ f < 2η1,

  • (v)

    0 < g ≤ λ2 ≤ j < 2η2,

then {xn} converges strongly to PΘx0.

Proof. From Theorem 3.1, put F1(tn, t) = h1(t) − h1(tn), F2(un, u) = h2(u) − h2(un), and A, B, φ1, φ2 ≡ 0. The conclusion of Corollary 4.1 can be obtained from Theorem 3.1 immediately.

Corollary 4.2. Let C be a nonempty closed-convex subset of a real Hilbert Space H. Let h1, h2 : C → ℝ ∪ {+∞} be proper lower semicontinuous and convex functions. Let E1, E2 be η1, η2-inverse-strongly monotone mappings of C into H, respectively, and let M1, M2 : H → 2H be maximal monotone mappings. Let 𝒮 = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C, and let {tn} be a positive real divergent sequence. Let be a countable family of uniformly k-strict pseudocontractions, let be a countable family of nonexpansive mappings defined by Tix = tx + (1 − t)Vix,   ∀ x ∈ C,   ∀ i ≥ 1,   t ∈ [k, 1), and let Wn be the W-mapping defined by (2.11) and W a mapping defined by (2.12) with F(W) ≠ ∅. Suppose that Θ : = F(𝒮)∩F(W)∩M(h1)∩M(h2)∩VI(C, E1)∩VI(C, E2) ≠ ∅. Let {xn} be a sequence generated by , and

()
for every n ≥ 0, where , {rn}, {qn} ⊂ (0, ∞), λ1 ∈ (0,2η1), and λ2 ∈ (0,2η2) satisfy the following conditions:

  • (i)

    liminf n→∞rn > 0,

  • (ii)

    liminf n→∞qn > 0,

  • (iii)

    lim n→∞αn,i = 0,

  • (iv)

    0 < e ≤ λ1 ≤ f < 2η1,

  • (v)

    0 < g ≤ λ2 ≤ j < 2η2,

then {xn} converges strongly to PΘx0.

Acknowledgments

The authors thank the referees for their appreciation, valuable comments, and suggestions. They would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission for financial support. Furthermore, they would like to thank the Faculty of science (KMUTT) and the National Research Council of Thailand. This work was completed with the support of the National Research Council of Thailand (NRCT 2010-2011).

      The full text of this article hosted at iucr.org is unavailable due to technical difficulties.