We introduce an iterative process which converges strongly to solutions of a certain variational inequity problem for η-inverse strongly accretive mappings in the set of common fixed points of finite family of strictly pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

1. Introduction

Let E be a real normed linear space with dual E^*. For 1 < q < ∞, we denote by J_q the generalized duality mapping from E to

defined by

()

where 〈·, ·〉 denotes the duality pairing. In particular, J = J₂ is called the normalized duality map. It is well known (see e.g., [1]) that J_q is single valued if E is smooth and that

()

In the sequel, we will denote the single-valued generalized map by j_q.

A mapping A with domain D(A)⊆E and range R(A) in E is called α-strongly accretive if there exist α ∈ (0,1) and j_q(x − y) ∈ J_q(x − y) such that

()

A is called η-inverse strongly accretive if there exist η ∈ (0,1) and j_q(x − y) ∈ J_q(x − y) such that

()

Let C be a nonempty, closed, and convex subset of E and, let A : C → E be a nonlinear mapping. The variational inequality problem is to

()

for some j(v − u) ∈ J(v − u). The set of solutions of variational inequality problem is denoted by VI(C, A). If E : = H, a real Hilbert space, the variational inequality problem reduces to

()

which was introduced and studied by Stampacchia [2].

Variational inequality theory has emerged as an important tool in studying a wide class of related problems arising in mathematical, physical, regional, engineering, and nonlinear optimization sciences (see, for instance, [3–12]).

In 1976, Korpelevič [4] introduced the following well-known extragradient method:

()

where P_C is the metric projection from ℝⁿ onto its subset C, for some γ > 0, and A : C → ℝⁿ is an accretive operator. He proved that the sequence {x_n} converges to a solution of the variational inequality (6).

Furthermore, Noor [6] proved that the iterative scheme, given by

()

where A : C⊆ℝⁿ → ℝⁿ is an accretive operator, converges to a solution of the variational inequality (6).

We note that the above algorithms give strong convergence to a solution of the variational inequality (6). However, both algorithms fail, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces.

In 2006, Aoyama et al. [13] introduced and studied the following iterative algorithm in a uniformly convex and 2-uniformly smooth Banach spaces possessing weakly sequentially continuous duality mapping:

()

where Q_C is a sunny nonexpansive retraction from E onto a closed and convex C, A : C → E is an η-inverse strongly accretive mapping and {α_n} and {λ_n} subsets of real numbers, satisfy certain conditions. They proved that the sequence in (9) converges weakly to a point z ∈ V(C, A).

Recently, Yao et al. [8] introduced and considered the following iterative method for η-strongly accretive mappings in a uniformly convex and 2-uniformly smooth Banach space possessing weakly sequentially continuous duality mapping:

()

where Q_C is a sunny nonexpansive retraction from E onto C. They proved that the sequence {x_n} defined by (10) converges strongly to Q_VI(C,A)u provided that real sequences {α_n}, {β_n}, {γ_n}, and {λ_n} satisfy certain conditions.

Let C be a nonempty subset of a real Banach space E. A mapping T : C → E is called λ-strictly pseudocontractive of Browder-Petryshyn type [14] if for all x, y ∈ D(T) there exist λ > 0 and j_q(x − y) ∈ J_q(x − y) such that

()

T is called Lipschitz if there exists L ≥ 0 such that

()

If L < 1 in (12), then T is called contraction, while T is said to be nonexpansive if L = 1.

If E = H, a real Hilbert space, then (11) is equivalent to the inequality

()

and we can assume also that k ≥ 0, so that k ∈ [0,1). A point x ∈ C is a fixed point of T if Tx = x, and we denote by F(T) the set of fixed points of T; that is, F(T) = {x ∈ C : Tx = x}.

In 2001, Yamada [7] introduced a hybrid steepest descent method which relates solutions of variational inequality problems with fixed point of mappings in Hilbert spaces. He proved that if T is nonexpansive self-map on C and A is an η-strongly accretive mapping from C into E satisfying certain conditions, then the sequence defined by

()

converges strongly to the unique solution of the variational inequality

()

The above results naturally bring us to the following question.

Question. Could we produce an iterative scheme which approximates a solution of variational inequality (5) for η-inverse strongly accretive mappings in Banach spaces?

In this paper, motivated by Yao et al. [8] and Yamada [7], it is our purpose to introduce an iterative scheme which converges strongly to a solution of the variational inequality (5) for η-inverse strongly accretive mapping in the set of common fixed points of finite family of strictly pseudocontractive mappings in a uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Our results complement or improve the results of Yao et al. [8], Aoyama et al. [13], and some authors.

2. Preliminaries

Let E be a real Banach space. The modulus of smoothness of E is the function ρ_E : [0, ∞) → [0, ∞) defined by ρ_E(τ) : = sup {(1/2)(∥x + y∥ + ∥x − y∥) − 1 : ∥x∥ = 1, ∥y∥ = τ}. If ρ_E(τ) > 0 for all τ > 0, then E is said to be smooth. If there exists a constant c > 0 and a real number 1 < q < ∞, such that ρ_E(τ) ≤ cτ^q, then E is said to be q-uniformly smooth.

If E is a real q-uniformly smooth Banach space, then by [15], the following geometric inequality holds:

()

for all x, y ∈ E and some real constant c_q > 0.

It is well known (see e.g., [16]) that

()

The Banach space E is said to be uniformly convex if, given ε > 0, there exists δ > 0, such that, for all x, y ∈ E with ∥x∥ ≤ 1, ∥y∥ ≤ 1 and ∥x − y∥ ≥ ε, ∥(1/2)(x + y)∥ ≤ 1 − δ. It is well known that L_p, ℓ_p, and Sobolev spaces

, (1 < p < ∞) are uniformly convex.

Let C⊆E be closed convex and Q a mapping of E onto C. Then, Q is said to be sunny if Q(Q(x) + t(x − Q(x))) = Q(x) for all x ∈ E and t ≥ 0. A mapping Q of E into C is said to be a retraction if Q² = Q. If a mapping Q is a retraction, then Q(z) = (z) for every z ∈ R(Q), range of Q. A subset C of E is said to be a sunny nonexpansive retract of E if there exists a sunny nonexpansive retraction of E onto C, and it is said to be a nonexpansive retract of E if there exists a nonexpansive retraction of E onto C. If E = H, the metric projection P_C is a sunny nonexpansive retraction from H to any closed convex subset of H. Moreover, if C is a nonempty closed convex subset of a uniformly convex and uniformly smooth real Banach space E and T is a nonexpansive mapping of C into itself with F(T) ≠ ∅, then the set F(T) is a sunny nonexpansive retract of C.

In what follows, we will make use of the following lemmas.

Lemma 1 (see, e.g., [17]). Let E be a smooth Banach space, and let K be a nonempty subset of E. Let Q : E → K be a retraction, and let J be the normalized duality map on E. Then, the following are equivalent:

(i)
Q is sunny nonexpansive,
(ii)
〈x − Q(x), J(y − Q(x))〉 ≤ 0 for all x ∈ E and y ∈ K.

Lemma 2 (see [18].)Let {a_n} be a sequence of nonnegative real numbers satisfying the following relation:

()

where {α_n} ⊂ (0,1) and {δ_n} ⊂ ℝ satisfying the following conditions:

, and limsup _n→∞δ_n ≤ 0. Then, lim _n→∞a_n = 0.

Lemma 3 (see [13].)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 λ > 0,

()

where VI(C, A) = {x^* ∈ C : 〈Ax^*, J(x − x^*)〉 ≥ 0, ∀ x ∈ C}.

Lemma 4 (see [17].)Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E, and let T be nonexpansive mapping of C into itself. If {x_n} is a sequence of C such that x_n → x weakly and x_n − Tx_n → 0 strongly, then x is a fixed point of T.

Lemma 5 (see [19].)Let E be a real Banach space. Then, for any given x, y ∈ E, the following inequality holds:

()

Lemma 6 (see [20].)Let K be a nonempty closed convex subset of a strictly convex Banach space E. Let T_i : K → E, i = 1,2, …, r, be a family of nonexpansive mappings such that . Let α₀, α₁, α₂, …, α_r be real numbers in (0,1) such that , and let T : = α₀I + α₁T₁ + ⋯+α_rT_r. Then, T is nonexpansive, and .

Lemma 7 (see [21].)Let C be a nonempty, closed and convex subset of a real uniformly convex and smooth Banach space E. Let T_i : C → E, i = 1, …, N, be λ_i-strictly pseudocontractive mappings such that . Let T : = θ₁T₁ + θ₂T₂ + ⋯+θ_NT_N with θ₁ + θ₂ + ⋯+θ_N = 1. Then T is λ-strictly pseudocontractive with λ : = min {λ_i : i = 1, …, N} and .

Lemma 8 (see [22].)Let C be a nonempty closed and convex subset of a real q-uniformly smooth Banach space E for 1 < q < ∞. Let T : C → E be a λ-strictly pseudocontractive mapping. Then, for , where L is the Lipchitz constant of T and c_q is a constant in (16), the mapping T_μx : = (1 − μ)x + μTx is nonexpansive, and F(T_μ) = F(T).

Lemma 9. Let C be a nonempty closed and convex subset of a a real q-uniformly smooth Banach space E for 1 < q < ∞. Let A : C → E be an η-inverse strongly accretive mapping. Then, for 0 < γ < (qη/c_q) ^1/(q−1), the mapping A_μx : = (x − γAx) is nonexpansive.

Proof. Now, using inequality (16), we get that

()

The proof is complete.

Lemma 10 (see [10].)Let E be a uniformly convex Banach space, and let B_R(0) be a closed ball of E. Then, there exists a continuous strictly increasing convex function g : [0, ∞)→[0, ∞) with g(0) = 0 such that

()

for x_i ∈ B_R(0): = {x ∈ E : ∥x∥ ≤ R}, i = 0,1, 2, …, k with

Lemma 11 (see [5].)Let {a_n} be sequences of real numbers such that there exists a subsequence {n_i} of {n} such that , for all i ∈ ℕ. Then, there exists a nondecreasing sequence {m_k} ⊂ ℕ such that m_k → ∞, and the following properties are satisfied by all (sufficiently large) numbers k ∈ ℕ:

()

In fact, m_k = max {j ≤ k : a_j < a_j+1}.

3. Main Results

Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E (1 < q < ∞). Let T_i : C → C, for i = 1, …, N, be a λ_i-strictly pseudocontractive mappings, and let A : C → E be an η-inverse strongly accretive mapping. Then, in what follows, we will study the variational inequality

()

and the following iteration process:

()

where S : = [(1 − μ)I + μT], for T : = θ₁T₁ + θ₂T₂ + ⋯+θ_nT_N, such that θ₁ + θ₂ + ⋯+θ_N = 1, 0 < μ < μ₀ = min {1, (qλ/c_q) ^1/(q−1)}, for λ : = min {λ_i : i = 1,2, …, N}, and 0 < γ < (qη/c_q) ^1/(q−1), and c_q is the real number in (16). In addition, we assume {α_n}⊂(0, c)⊂(0,1) and {β_n}, {c_n}⊂[a, b]⊂(0,1) as real sequences satisfying the following control conditions: (i) lim _n→∞α_n = 0, (ii) ∑ α_n = ∞, lim _n→∞(|β_n − β_n−1|/α_n−1) = 0, lim _n→∞(|α_n − α_n−1|/α_n−1) = 0, and lim _n→∞(|c_n − c_n−1|/α_n−1) = 0.

We now prove our main theorem.

Theorem 12. Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Let T_i : C → C, for i = 1, …, N, be λ_i-strictly pseudocontractive mappings, and let A : C → E be η-inverse strongly accretive mapping. Let {x_n} be a sequence defined by (25). Assume that ℱ : = F∩VI(C, A) ≠ ∅, where . Then, {x_n} converges strongly to Q_ℱ(0), where Q_ℱ is a sunny nonexpansive retraction of E onto ℱ, which is a solution of the variational inequality (24).

Proof. By Lemmas 7 and 8 we have that S is nonexpansive. In addition, by Lemma 9 we get that Q_C[I − γA] is nonexpansive. Let p ∈ ℱ and, let y_n : = β_nx_n + (1 − β_n)Q_C[I − γA]z_n. Then from (25), Lemmas 8 and 9 we have that

()

Thus, from (25) and (27), we get that

()

Therefore, by induction,

()

which implies that {x_n} and hence {y_n}, {z_n}, and {Ax_n} are bounded. Furthermore, from (25), we obtain that

()

And, hence, from (25) and (31), we have that

()

for some M > 0. Thus, using the properties of {α_n}, {β_n}, {c_n}, (32), and Lemma 2, we obtain that ∥x_n+1 − x_n∥ → 0, as n → ∞, which implies from (31) that ∥y_n+1 − y_n∥ → 0, as n → ∞. Again from (25), we have that ∥x_n+1 − y_n∥ = ∥Q_C[(1 − α_n)y_n] − Q_Cy_n∥ = α_n∥y_n∥ → 0, as n → ∞. Consequently,

()

Now, we prove that {x_n} converges strongly to the point x^* = Q_ℱ(0). Let t_n = Q_C[I − γA]z_n, and let d_n = (1 − α_n)y_n. Then, since α_n → 0, we have that

()

Furthermore, from (25), Lemma 5, and Lemma 10, we get that

()

which implies that

()

Now, following the method of proof of Lemma 3.2 of Maingé [5], we consider two cases.

Case 1. Suppose that there exists n₀ ∈ ℕ such that {∥x_n − x^*∥} is decreasing for all n ≥ n₀. Then, we get that {∥x_n − x^*∥} is convergent. Thus, from (36) and the fact that α_n → 0, as n → ∞, we have that

()

which implies that

()

In addition, since {d_n} is bounded subset of a reflexive space E, we can choose a subsequence of {d_n} such that and . This implies from (34) and (33) that. Then, from (39) and Lemma 4, we have that . Moreover, from (39) and Lemma 4, we have that z ∈ F(Q_C[I − γA]), and by Lemma 3, we get z ∈ VI(C, A), and hence z ∈ ℱ. Therefore, using the fact that E has a weakly sequentially continuous duality mapping and Lemma 1, we immediately obtain that

()

Then, it follows from (37), (40), and Lemma 2 that ∥x_n − x^*∥ → 0, as n → ∞. Consequently, x_n → x^* = Q_ℱ0.

Case 2. Suppose that there exists a subsequence {n_i} of {n} such that

()

for all i ∈ ℕ. Then, by Lemma 11, there exists a nondecreasing sequence {m_k} ⊂ ℕ such that m_k → ∞, and

()

for all k ∈ ℕ. Now, from (36) and the fact that α_n → 0, we get that

and

, as k → ∞. Thus, like in Case 1, we obtain that

()

Moreover, from (37), we have that

()

which implies from (42) and (44) that

()

Now, since

, we obtain that

()

and using (43), we get that

. This together with (44) implies that

, as k → ∞. But

, for all k ∈ ℕ; thus, we obtain that x_k → x^*. Therefore, from both cases, we can conclude that {x_n} converges strongly to x^* = P_ℱ(0), which is a solution of the variational inequality (24), and the proof is complete.

If in Theorem 12, we consider that N = 1, we get the following corollary.

Corollary 13. Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Let T : C → C be a λ-strictly pseudocontractive mapping, and let A : C → E be an η-inverse strongly accretive mapping. Let {x_n} be a sequence defined by (25), where S : = [(1 − μ)I + μT]. Assume that F : = F(T)∩VI(C, A) ≠ ∅. Then, {x_n} converges strongly to Q_F(0) which is a solution of the variational inequality

()

If in Theorem 12, we assume that T_i, for i = 1,2, …, N, are nonexpansive, we get the following corollary.

Corollary 14. Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Let T_i : C → C, for i = 1,2, …, N, be nonexpansive mappings, and let A : C → E be an η-inverse strongly accretive mapping. For x₀ ∈ C, let the sequence {x_n} be generated iteratively by

()

where T : = θ₀I + θ₁T₁ + ⋯+θ_NT_N for

, {α_n}, {β_n}, γ are as in (24). Assume that ℱ : = F∩VI(C, A) ≠ ∅, where

. Then, {x_n} converges strongly to Q_ℱ(0), which is a solution of the variational inequality problem (24).

Proof. Lemma 6 and the method of proof of Theorem 12 provide the required assertion.

If in Corollary 14, we consider that N = 1, we get the following corollary.

Corollary 15. Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Let T : C → C be a nonexpansive mapping, and let A : C → E be an η-inverse strongly accretive mapping. For x₀ ∈ C, let the sequence {x_n} be generated iteratively by

()

Assume that F : = F(T)∩VI(C, A) ≠ ∅. Then, {x_n} converges strongly to Q_F(0), which is a solution of the variational inequality problem

()

If in Corollary 14, we assume that T = T₁ = T₂ = ⋯ = T_N = I, we obtain the following corollary.

Corollary 16. Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Let A : C → E be an η-inverse strongly accretive mapping. For x₀ ∈ C, let the sequence {x_n} be generated iteratively by

()

Assume that VI(C, A) ≠ ∅. Then, {x_n} converges strongly to Q_VI(C,A)(0), where Q_VI(C,A) is a sunny nonexpansive retraction of E onto VI(C, A).

If in Theorem 12, we assume that A is an α-strongly accretive and L-Lipschitzian continuous mapping, we obtain the following corollary.

Corollary 17. Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Let T_i : C → C, for i = 1, …, N, be λ_i-strictly pseudocontractive mappings, and Let A : C → E be an α-strongly accretive and L-Lipschitzian continuous mapping. Let {x_n} be a sequence defined by (25) for η = α/L². Assume that ℱ : = F∩VI(C, A) ≠ ∅, where . Then, {x_n} converges strongly to Q_ℱ(0), which is a solution of the variational inequality problem

()

Proof. We note that if A is an α-strongly accretive and L-Lipschitzian continuous mapping of C into E, then we have that

()

and hence, A is an η-inverse strongly accretive mapping with η = α/L². Thus, the conclusion follows from Theorem 12.

If E = H, a real Hilbert space, then E is a uniformly convex and q-uniformly smooth Banach space E for 1 < q < ∞ possessing weakly sequentially continuous duality mapping. In this case, we have that Q_C = P_C, projection mapping from H onto C. Thus, we have the following corollary.

Corollary 18. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T_i : C → C, for i = 1, …, N, be λ_i-strictly pseudocontractive mappings, and let A : C → E be an η-inverse strongly accretive mapping. For x₀ ∈ C, let the sequence {x_n} be generated iteratively by

()

where S : = [(1 − μ)I + μT], for T : = θ₁T₁ + θ₂T₂ + ⋯+θ_nT_N, such that θ₁ + θ₂ + ⋯+θ_N = 1, 0 < μ < min {1,2λ}, for λ : = min {λ_i : i = 1,2, …, N}, and 0 < γ < 2η. Assume that ℱ : = F∩VI(C, A) ≠ ∅, where

. Then, {x_n} converges strongly to P_ℱ(0), which is a solution of the variational inequality

()

Remark 19. Theorem 12 complements Theorem 3.2 of Yao et al. [8] in more general Banach spaces for η-inverse strongly accretive mappings. Moreover, Theorem 12 improves Theorem 3.1 of Aoyama et al. [13] and Theorem 3.7 of Saejung et al. [23] in the sense that our convergence is strong in the set of common fixed points of finite family of strictly pseudocontractive mappings.

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Extragradient Method for Solutions of Variational Inequality Problems in Banach Spaces

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1. Introduction

2. Preliminaries

3. Main Results

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Extragradient Method for Solutions of Variational Inequality Problems in Banach Spaces

Abstract

1. Introduction

2. Preliminaries

3. Main Results

References

Citing Literature

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