The aim of this paper is to introduce an iterative algorithm for finding a common solution of the sets (0) and (0), where M is a maximal accretive operator in a Banach space and, by using the proposed algorithm, to establish some strong convergence theorems for common solutions of the two sets above in a uniformly convex and 2-uniformly smooth Banach space. The results obtained in this paper extend and improve the corresponding results of Qin et al. 2011 from Hilbert spaces to Banach spaces and Petrot et al. 2011. Moreover, we also apply our results to some applications for solving convex feasibility problems.

1. Introduction

Let E be a real Banach space with norm ∥·∥ with the dual space E^* and let 〈·, ·〉 denote the pairing between E and E^*. Let C be a nonempty closed convex subset of E. We define the generalized duality mapping

(1.1)

for all q > 1. In the special case, for q = 2, we called the mapping J₂ as the normalized duality mapping and as usual we write J₂ = J. The following is the well-known properties of the generalized duality mapping J_q:

(1)
J_q(x) = ∥x∥^q−2J₂(x) for all x ∈ E with x ≠ 0;
(2)
J_q(tx) = t^q−1J_q(x) for all x ∈ E and t ∈ [0, ∞);
(3)
J_q(−x) = −J_q(x) for all x ∈ E.

It is well known that if X is smooth, then J is single valued, which is denoted by j. Recall that the duality mapping j is said to be weakly sequentially continuous if, for each sequence {x_n} with x_n → x weakly, we have j(x_n) → j(x) weakly*. We know that, if X admits a weakly sequentially continuous duality mapping, then X is smooth. For the details, see [1–3].

Let U = {x ∈ E : ∥x∥ = 1}. A Banach space E is said to be

(1)
uniformly convex if there exists δ > 0 such that, for any x, y ∈ U and, for any ϵ ∈ (0,2], ∥x − y∥≥ϵ implies ∥(x + y)/2∥≤1 − δ.

We can see that every uniformly convex Banach space is also reflexive and strictly convex.

(2)
Smooth if lim _t→0(∥x + ty∥−∥x∥)/t exists for all x, y ∈ U.
(3)
Uniformly smooth if the limit is attained uniformly for x, y ∈ U. The modulus of smoothness of E is defined by
(1.2)
where ρ : [0, ∞)→[0, ∞) is a function. In the other way, E is uniformly smooth if and only if lim _τ→0 ρ(τ)/τ = 0.
(4)
q-uniformly smooth if there exists a constant c > 0 such that ρ(τ) ≤ cτ^q for all τ > 0 where q is a fixed real number with 1 < q ≤ 2. (see, for instance, [1, 4]).

We note that E is a uniformly smooth Banach space if and only if J_q is single valued and uniformly continuous on any bounded subset of E. 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 any p > 1. Note also that no Banach space is q-uniformly smooth for q > 2 (see [1, 5] for more details).

Let A : C → E be a nonlinear mapping. The mapping A is said to be

(1)
accretive if
(1.3)
(2)
λ-strongly accretive if there exists a constant α > 0 such that
(1.4)
(3)
λ-inverse-strongly accretive if there exists a constant α > 0 such that

(1.5)

Definition 1.1. Let M : E → 2^E be a multivalued maximal accretive mapping. The single-valued mapping J_M,ρ : E → E defined by

(1.6)

is called the resolvent operator associated with M, where ρ is any positive number and I is the identity mapping.

Let T be a mapping from E into itself. We use F(T) to denote the set of fixed points of the mapping T. Recall that the mapping T is said to be nonexpansive if

(1.7)

A mapping f : C → C is said to be contractive if there exists a constant α ∈ (0,1) such that

(1.8)

Recently, Aoyama et al. [4] considered the following generalized variational inequality problem in a smooth Banach space: Find a point x ∈ C such that

(1.9)

where A is an accretive operator of C into E. This problem is related to the fixed point problem for nonlinear mappings, the problem of finding a zero point of an accretive operator, and so on. For the problem of finding a zero point of an accretive operator by the proximal point algorithm, see Agarwal et al. [6], Cho et al. [7, 8], Kamimura and Takahashi [9, 10], Qin et al. [11], Song et al. [12], and Wei and Cho [13]. In order to find a solution of the variational inequality (1.9), Aoyama et al. [4] studied the weak convergence theorem for accretive operators in Banach spaces, which is a generalization of the result by Iiduka et al. [14] from the class of Hilbert spaces.

Theorem AIT (see [4], Aoyama et al. Theorem 3.1.)Let E be a uniformly convex and 2-uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let Q_C be a sunny nonexpansive retraction from E onto C, α > 0 and A be an α-inverse strongly accretive operator of C into E with S(C, A) ≠ ∅, where

(1.10)

If {λ_n} and {α_n} are chosen such that λ_n ∈ [a, α/K²] for some a > 0 and α_n ∈ [b, c] for some b, c with 0 < b < c < 1, then the sequence {x_n} defined by the following manners: x₁ = x ∈ C and

(1.11)

converges weakly to an element z of S(C, A), where K is the 2-uniformly smoothness constant of E and Q_C is a sunny nonexpansive retraction.

In 2011, Katchang and Kumam [15] presented an iterative algorithm for finding a common solution of fixed point problems and a general system of variational inequality problems for two accretive operators as shown in the following: for all n ≥ 0,

(1.12)

They proved that the sequence {x_n} generated by the above algorithm converges strongly to a point

. Moreover, they apply their theorem to find zeros of accretive operators and the class of k-strictly pseudocontractive mappings.

Recently, Petrot et al. [16] considered the problem so-called quasivariational inclusion problem, that is, determine an element u ∈ H such that

(1.13)

where A : H → H is a single-valued nonlinear mapping and M : H → 2^H is a multivalued mapping. The set of solutions of the above problem is denoted by VI (H, A, M). Therefore, they presented a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inclusion problem with a multivalued maximal monotone mapping and an α-inverse-strongly monotone mapping by using the iterative sequence {x_n} defined as follows:

(1.14)

and, under appropriated conditions, they proved the the sequence {x_n} generated by (1.14) converges strongly to a point z₀ ∈ H, which is the unique solution in F(S)∩VI (H, A, M) to the following variational inequality:

(1.15)

Very recently, Qin et al. [17] introduced an iterative scheme for a general variational inequality (VI ) and proved the strong convergence theorems of common solutions of two variational inequalities in a uniformly convex and 2-uniformly smooth Banach space by using the following iterative sequence {x_n}:

(1.16)

They proved that the sequence {x_n} generated by the above algorithm converges strongly to a point q = Q_VIu, where Q_VI is the unique sunny nonexpansive retraction from C onto VI.

Motivated and inspired by the above recent works, in this paper, we introduce an iterative scheme for finding zeros of maximal accretive operators. Furthermore, we prove some strong convergence theorems and also propose applications for solving the convex feasibility problems. Our results improve and extend the corresponding results of Qin et al. [17] and Katchang and Kumam [15], Petrot et al. [16], and many others.

2. Preliminaries

Note that, if C and D are nonempty subsets of a Banach space E such that D is a subset of a closed convex subset C and Q : C → D. Then Q is said to be sunny if

(2.1)

whenever Qx + t(x − Qx) ∈ C for any x ∈ C and t ≥ 0. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction Q of C onto D. A mapping Q : C → C is called a retraction if Q² = Q. If a mapping Q : C → C is a retraction, then Qz = z for all z is in the range of Q (see [4, 18] for more details).

The following result describes a characterization of sunny nonexpansive retractions on a smooth Banach space.

Proposition 2.1 (see [19].)Let E be a smooth Banach space and let C be a nonempty subset of E. Let Q : E → C be a retraction and let J be the normalized duality mapping on E. Then the following are equivalent:

(1)
Q is sunny and nonexpansive;
(2)
∥Qx − Qy∥² ≤ 〈x − y, J(Qx − Qy)〉 for all x, y ∈ E;
(3)
〈x − Qx, J(y − Qx)〉≤0 for all x ∈ E and y ∈ C.

Proposition 2.2 (see [20].)Let C be a nonempty closed convex subset of a uniformly convex and let uniformly smooth Banach space E and T be a nonexpansive mapping of C into itself with F(T) ≠ ∅. Then the set F(T) is a sunny nonexpansive retract of C.

We need the following lemmas in order to prove our main results.

Lemma 2.3 (see [5].)Let E be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:

(2.2)

Lemma 2.4 (see [21].)Let {x_n} and {y_n} be bounded sequences in a Banach space E and {β_n} be a sequence in [0,1] with

(2.3)

Suppose that x_n+1 = (1 − β_n)y_n + β_nx_n for all n ≥ 0 and

(2.4)

Then lim _n→∞∥y_n − x_n∥ = 0.

Lemma 2.5 (see [22].)Assume that {a_n} is a sequence of nonnegative real numbers such that

(2.5)

where {α_n} is a sequence in (0,1) and {δ_n} is a sequence in ℝ such that

(1)
;
(2)
limsup _n→∞δ_n/α_n ≤ 0 or .

Then lim _n→∞a_n = 0.

Lemma 2.6 (see [23].)Let C be a closed convex subset of a strictly convex Banach space E. Let T_m : C → C be a nonexpansive mappings for each 1 ≤ m ≤ r, where r is some integer. Suppose that is nonempty. Let {λ_n} be a sequence of positive numbers with . Then the mapping S : C → C defined by

(2.6)

is well defined, nonexpansive, and

holds.

Lemma 2.7 (see [24].)Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E and T be nonexpansive mapping of C into itself. If {x_n} is a sequence in C such that x_n → x weakly and x_n − Tx_n → 0 strongly, then x is a fixed point of T.

Lemma 2.8 (see [3], [4].)Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E. Let a mapping A : C → E be λ-inverse-strongly accretive. Then one has

(2.7)

If λ ≥ ρ₂K², then I − ρ₂A is nonexpansive.

Proof. For any x, y ∈ C, it follows from Lemma 2.3 that

(2.8)

If λ ≥ ρ₂K², then I − ρ₂A is nonexpansive. This completes the proof.

Lemma 2.9. Let C be a nonempty subset of a Banach space E. Let A be a mapping of C into E, M be a maximal accretive operator on E and J_M,ρ = (I + ρM) ⁻¹ be the resolvent of M for any ρ > 0. Then F(J_M,ρ(I − ρA)) = (A + M) ⁻¹(0) for all ρ > 0.

Proof. Let ρ > 0 be fixed. Then we have

(2.9)

This completes the proof.

Lemma 2.10. Let E be a Banach space. Then for all x, y ∈ E,

(2.10)

3. Main Results

In this section, we prove strong convergence theorems for a λ-inverse-strongly accretive mapping A : C → E and a β-inverse-strongly accretive B : C → E in a real 2-uniformly smooth Banach space E.

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

Lemma 3.1. Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E with the best smooth constant K. Let be a resolvent operator associated with M₁, M₂, where M₁, M₂ : E → 2^E is a multivalued maximal accretive mapping. Let the mappings A, B : C → E be λ-inverse-strongly accretive and β-inverse-strongly accretive, respectively. Let G : C → C be a mapping defined by

(3.1)

If λ ≥ ρ₂K² and β ≥ ρ₁K², then G is nonexpansive.

Proof. Since and are nonexpansive, for any x, y ∈ C, it follows from Lemma 2.8 that

(3.2)

Therefore, G is nonexpansive. This completes the proof.

Next, we state the main result of this work.

Theorem 3.2. Let E be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and C be a nonempty closed convex subset of E. Let A, B : C → E be λ-inverse-strongly accretive and β-inverse-strongly accretive, respectively, and K be the best smooth constant. Let f be a contraction of E into itself with coefficient α ∈ [0,1). Suppose that Ω : = (A + M₂) ⁻¹(0)∩(B + M₁) ⁻¹(0) ≠ ∅ and G is a mapping defined by Lemma 3.1. Let ρ₁, ρ₂ be any positive real numbers such that ρ₁ ≤ β/K² and ρ₂ ≤ λ/K². For arbitrary x₀ = x ∈ C, define the iterative sequence {x_n} as follows:

(3.3)

where the sequences {α_n}, {β_n}, and {γ_n} in (0,1) satisfy the following conditions:

(C1)
α_n + β_n + γ_n = 1;
(C2)
lim _n→∞α_n = 0 and ;
(C3)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1;
(C4)
lim _n→∞δ_n = δ ∈ (0,1).

Then the sequence {x_n} generated by (3.3) converges strongly to a point q = Q_Ωf(q), where Q_Ω is a sunny nonexpansive retraction on Ω.

Proof. First, we prove that and are nonexpansive mappings. Consider the following:

(3.4)

Thus, it follows that

is nonexpansive and so is

Step 1. We show that {x_n} is bounded. For any p ∈ Ω, we have

(3.5)

It follows by induction that

(3.6)

Thus the sequence {x_n} is bounded and so is {y_n}.

Step 2. We show that lim _n→∞∥x_n+1 − x_n∥ = 0. Let and for each n ≥ 0. Then we have

(3.7)

and so

(3.8)

where M₁ ia an appropriate constant such that M₁ ≥ sup _n≥0{∥u_n − v_n∥}.

Next, let z_n = (x_n+1 − β_nx_n)/(1 − β_n) for all n ≥ 0. Then we have x_n+1 = (1 − β_n)z_n + β_nx_n for all n ≥ 0. Now, we compute

(3.9)

and so

(3.10)

Substituting (3.8) into (3.10), we get

(3.11)

that is,

(3.12)

From the conditions (C2) and (C3), it follows that

(3.13)

Thus, from Lemma 2.4, it follows that

(3.14)

From the definition of x_n+1 in this step, we observe that x_n+1 − x_n = (1 − β_n)(z_n − x_n). Then we have

(3.15)

Step 3. We show that limsup _n→∞〈(f − I)q, J(x_n − q)〉≤0, where q = Q_Ωf(q). Define a mapping G : C → C by Lemma 3.1 Then, it follows that G is a nonexpansive mapping such that

(3.16)

Consider the following:

(3.17)

From the condition (C4), we have

(3.18)

Next, we consider

(3.19)

Therefore, we have

(3.20)

From the conditions (C2), (C3), (3.15), (3.18), and the inequality above, we obtain

(3.21)

Thus, since Q_Ωf(q) is a contraction, there exists a unique fixed point. We denote that q is the unique fixed point to the mapping Q_Ωf(q) which means that q = Q_Ωf(q).

Since {x_n} is bounded, there exists a subsequence of {x_n} such that , it follows from (3.21) that

(3.22)

Since G is nonexpansive, it follows from Lemma 2.7 that p = Gp we obtain that p ∈ F(G). By (3.22), we have p ∈ Ω.

Furthermore, with the reason that {x_n} is bounded, we can choose the sequence of {x_n} which such that

(3.23)

Now, from (3.23) and Proposition 2.1(3) and the weakly sequential continuity of the duality mapping J, we have

(3.24)

From (3.15), it follows that

(3.25)

Step 4. We show that {x_n} converges strongly to a point q = Q_Ωf(q). In fact, observe that

(3.26)

Thus it follows that

(3.27)

Therefore, from Condition (C2), (3.25), and Lemma 2.5, we get ∥x_n − q∥→0 as n → ∞. This completes the proof.

Corollary 3.3. Let E be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and let C be a nonempty closed convex subset of E. Let A, B : C → E be λ-inverse-strongly accretive and β-inverse-strongly accretive, respectively, and K be the best smooth constant. Suppose that Ω : = (A + M₂) ⁻¹(0)∩(B + M₁) ⁻¹(0) ≠ ∅, where G is a mapping defined by Lemma 3.1. Let ρ₁, ρ₂ be any positive real numbers such that ρ₁ ≤ β/K² and ρ₂ ≤ λ/K². For arbitrary x₀ = x ∈ C, define the iterative sequence {x_n} by

(3.28)

where the sequences {α_n}, {β_n}, and {γ_n} in (0,1) satisfy the following conditions:

(C1)
α_n + β_n + γ_n = 1;
(C2)
lim _n→∞α_n = 0 and ;
(C3)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1;
(C4)
lim _n→∞δ_n = δ ∈ (0,1).

Then the sequence {x_n} generated by (3.28) converges strongly to a point q = Q_Ωf(q), where Q_Ω is a sunny nonexpansive retraction on Ω.

Proof. Take f(x_n) = u for all n ≥ 1 for any fixed u ∈ C in (3.3). Then, by Theorem 3.2, we can conclude the desired conclusion easily.

4. Applications

4.1. Application to Convex Feasibility Problems

In this part, we consider the following convex feasibility problem (CFP): find , where j ∈ {1,2, …, N} and C_j denotes the set of zeros of a maximal accretive operator.

The following result can be obtained from Theorem 3.2.

Theorem 4.1. Let E be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and let C be a nonempty closed convex subset of E. Let be an λ_i-inverse-strongly accretive and K be the best smooth constant. Let f be a contraction of E into itself with coefficient α ∈ [0,1). Suppose that , where G is a mapping defined by Lemma 3.1. Let ρ_i be any positive real numbers such that ρ_i ≤ λ_i/K², i = 1,2, 3, …, N. For arbitrary x₀ = x ∈ C, define the iterative sequence {x_n} by

(4.1)

where the sequences {α_n}, {β_n}, and {γ_n} in (0,1) satisfy the following conditions:

(C1)
α_n + β_n + γ_n = 1 and ;
(C2)
lim _n→∞α_n = 0 and ;
(C3)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1;
(C4)
lim _n→∞δ_i,n = δ ∈ (0,1).

Then the sequence {x_n} generated by (4.1) converges strongly to a point q = Q_Ω, where Q_Ω is a sunny nonexpansive retraction on Ω.

4.2. Application to Hilbert Spaces

Assume that H is a real Hilbert space with inner product 〈·, ·〉 and norm ∥·∥. Let A : H → H be a single-valued nonlinear mapping and M : H → 2^H be a multivalued mapping. The problem of finding u ∈ H such that

(4.2)

is called the quasivariational inclusion problem, and we denote the set of solutions of the above variational inclusion by VI (H, A, M).

If M = ∂δ_C, where C is a nonempty closed convex subset of H and δ_C : H → [0, ∞] is the indicator function of C, that is,

(4.3)

Then the variational inclusion problem (4.2) is equivalent to the problem of finding u ∈ C such that

(4.4)

which is the well-known Hartman-Stampacchia variational inequality problem [25].

Theorem 4.2. Let C be a closed convex subset of a real Hilbert space H. Let A, B : C → H be λ-inverse-strongly monotone and β-inverse-strongly monotone, respectively. Let f be a contraction of E into itself with coefficient α ∈ [0,1). Suppose that Ω : = VI (C, A)∩VI (C, B) ≠ ∅, where VI (C, A) and VI (C, B) are the sets of solutions of variational inequality (4.4). For arbitrary x₀ = x ∈ C, define the iterative sequence {x_n} by

(4.5)

where the sequences {α_n}, {β_n}, and {γ_n} in (0,1) satisfy the following conditions:

(C1)
α_n + β_n + γ_n = 1;
(C2)
lim _n→∞α_n = 0 and ;
(C3)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1;
(C4)
lim _n→∞δ_n = δ ∈ (0,1).

Then the sequence {x_n} generated by (4.5) converges strongly to a point P_Ωx₀.

Proof. Take M = ∂δ_C : H → 2^H, where δ_C : H → [0, ∞] is the indicator function of C. Let J(M, ρ) = I. Then we get

(4.6)

Thus the conclusion can be obtained from Theorem 3.2 immediately.

Acknowledgments

This paper was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under the project NRU-CSEC no. 54000267) for financial support during the preparation of this paper. Furthermore, the first author would like to thank the Office of the Higher Education Commission, Thailand, for the financial support of the Ph.D. program at KMUTT. This research was partially finished at Department of Mathematics Education, Gyeongsang National University, Republic of Korea. Also, the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (Grant no. 2011-0021821).

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Convergence of an Iterative Algorithm for Common Solutions for Zeros of Maximal Accretive Operator with Applications

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Applications

4.1. Application to Convex Feasibility Problems

4.2. Application to Hilbert Spaces

Acknowledgments

References

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Convergence of an Iterative Algorithm for Common Solutions for Zeros of Maximal Accretive Operator with Applications

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Applications

4.1. Application to Convex Feasibility Problems

4.2. Application to Hilbert Spaces

Acknowledgments

References

Citing Literature

References

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