We introduce a general algorithm to approximate common fixed points for a countable family of nonexpansive mappings in a real Banach space. We prove strong convergence theorems for the sequences produced by the methods and approximate a common fixed point of a countable family of nonexpansive mappings which solves uniquely the corresponding variational inequality. Furthermore, we apply our results for finding a zero of an accretive operator. It is important to state clearly that the contribution of this paper in relation with the previous works (Marino and Xu, 2006) is a technical method to prove strong convergence theorems of a general iterative algorithm for an infinite family of nonexpansive mappings in Banach spaces. Our results improve and generalize many known results in the current literature.

1. Introduction

Viscosity approximation method for finding the fixed points of nonexpansive mappings was first proposed by Moudafi [1]. He proved the convergence of the sequence generated by the proposed method. In 2004, Xu [2] proved the strong convergence of the sequence generated by the viscosity approximation method to a unique solution of a certain variational inequality problem defined on the set of fixed points of a nonexpansive map (see also [3]). Marino and Xu [4] considered a general iterative method and proved that the sequence generated by the method converges strongly to a unique solution of a certain variational inequality problem which is the optimality condition for a particular minimization problem. Liu [5] and Qin et al. [6] also studied some applications of the iterative method considered in [4]. Yamada [7] introduced the so-called hybrid steepest-descent method for solving the variational inequality problem and also studied the convergence of the sequence generated by the proposed method. Recently, Tian [8] combined the iterative methods of [4, 7] in order to propose implicit and explicit schemes for constructing a fixed point of a nonexpansive mapping T defined on a real Hilbert space. He also proved the strong convergence of these two schemes to a fixed point of T under appropriate conditions. Related iterative methods for solving fixed point problems, variational inequalities, and optimization problems can be found in [9–14] and the references therein. By virtue of the projection, the authors in [13, 15] extended the implicit and explicit iterative schemes proposed in [8]. The approximation methods for common fixed points of a countable family of nonexpansive mappings have been recently studied by several authors; see, for example, [16, 17].

The purpose of this paper is to introduce a general algorithm to approximate common fixed points for a countable family of nonexpansive mappings in a Banach space. We prove strong convergence theorems for the sequences produced by the methods for a common fixed point of a countable family of nonexpansive mappings which solves uniquely the corresponding variational inequality. Furthermore, we apply our results for finding a zero of an accretive operator. Our results improve and generalize many known results in the current literature; see, for example, [4, 7, 8, 13–15, 18–20].

2. Preliminaries

Throughout this paper, we denote the set of real numbers and the set of positive integers by ℝ and ℕ, respectively. Let E be a Banach space with the norm ∥·∥ and the dual space E^*. When {x_n} is a sequence in E, we denote the strong convergence of {x_n} to x ∈ E by x_n → x and the weak convergence by x_n⇀x. For any sequence

in E^*, we denote the strong convergence of

to x^* ∈ E^* by

, the weak convergence by

, and the weak-star convergence by

. The normalized duality mapping

is defined by

()

The modulus δ of convexity of E is denoted by

()

for every ϵ with 0 ≤ ϵ ≤ 2. A Banach space E is said to be uniformly convex if δ(ϵ) > 0 for every ϵ > 0. Let S = {x ∈ E : ∥x∥ = 1}. The norm of E is said to be Gâteaux differentiable if for each x, y ∈ S, the limit

()

exists. In this case, E is called smooth. If the limit (3) is attained uniformly in x, y ∈ S, then E is called uniformly smooth. The Banach space E is said to be strictly convex if ∥(x + y)/2∥ < 1 whenever x, y ∈ S and x ≠ y. It is well known that E is uniformly convex if and only if E^* is uniformly smooth. It is also known that if E is reflexive, then E is strictly convex if and only if E^* is smooth; for more details, see [21]. Now, we define a mapping ρ : [0, ∞)→[0, ∞), the modulus of smoothness of E, as follows:

()

It is well known that E is uniformly smooth if and only if lim _t→0 (ρ(t)/t) = 0. Let q ∈ ℝ be such that 1 < q ≤ 2. Then a Banach space E is said to be q-uniformly smooth if there exists a constant c_q > 0 such that ρ(t) ≤ c_qt^q for all t > 0. If a Banach space E admits a sequentially continuous duality mapping J from weak topology to weak star topology, then J is single valued and also E is smooth; for more details, see [22]. In this case, the normalized duality mapping J is said to be weakly sequentially continuous; that is, if {x_n} ⊂ E is a sequence with x_n⇀x ∈ E, then J(x_n)⇀^*J(x) [22]. A Banach space E is said to satisfy the Opial property [23] if for any weakly convergent sequence {x_n} in E with weak limit x,

()

for all y ∈ E with y ≠ x. It is well known that all Hilbert spaces, all finite dimensional Banach spaces, and the Banach spaces l^p (1 ≤ p < ∞) satisfy the Opial property; for example, see [22, 23]. It is also known that if E admits a weakly sequentially continuous duality mapping, then E is smooth and enjoys the Opial property; see for more details [22].

Let E be a real Banach space and C a nonempty subset of E. Let T : C → E be a mapping. We denote by F(T) the set of fixed points of T; that is, F(T) = {x ∈ C : Tx = x}.

Definition 1. Let C be a nonempty, closed, and convex subset of a real Banach space E. An operator A : C → E is said to be

(i)
accretive if there exists j(x − y) ∈ J(x − y) such that
()
(ii)
η-strongly accretive if, for some η > 0, there exists j(x − y) ∈ J(x − y) such that
()
(iii)
l-Lipschitzian if, for some l > 0,
()
in particular, if l ∈ [0,1), then A is called a contraction;
(iv)
nonexpansive if
()

A linear bounded operator A : E → E^* is said to be strongly positive if there exists

such that

()

Remark 2. Let C be a nonempty, closed, and convex subset of a real Banach space E and let T : C → C be a nonexpansive mapping. Then I − T is an accretive operator, where I is the identity mapping. Indeed, for any x, y ∈ C we have

()

which means that I − T is accretive.

The following result has been proved in [24].

Lemma 3. Let E be a real 2-uniformly smooth Banach space. Then there exists a best uniformly smooth constant ρ > 0 such that

()

for all x, y ∈ E.

Let C and D be nonempty subsets of real Banach space E with D ⊂ C. A mapping Q_D : C → D is said to be sunny if

()

for each x ∈ E and t ≥ 0. A mapping Q_D : C → D is said to be a retraction if Q_Dx = x for each x ∈ C.

The following result has been proved in [25].

Lemma 4. Let C and D be nonempty subsets of a real Banach space E with D ⊂ C and Q_D : C → D a retraction from C into D. Then Q_D is sunny and nonexpansive if and only if

()

for all z ∈ C and y ∈ D.

Lemma 5 (demiclosedness principle [26]). Let C be a closed and convex subset of a real 2-uniformly smooth Banach space E and let the normalized duality mapping J : E → E^* be weakly sequentially continuous at zero. Suppose that T : C → E is a nonexpansive mapping with F(T) ≠ ⌀. If {x_n} is a sequence in C that converges weakly to x and if {(I − T)x_n} converges strongly to y, then (I − T)x = y; in particular, if y = 0, then x ∈ F(T).

Lemma 6 (see [27].)Let {s_n} be a sequence of nonnegative real numbers satisfying the inequality

()

where {γ_n} and {δ_n} satisfy the conditions

(i)
{γ_n}⊂[0,1] and , or equivalently, ;
(ii)
limsup _n→∞δ_n ≤ 0, or
(ii)′
.

Then, lim _n→∞s_n = 0.

Lemma 7 (see [28].)Let {x_n} and {z_n} be two sequences in a Banach space E such that

()

where {β_n} satisfies the following conditions: 0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1. If limsup _n→∞(∥z_n+1 − z_n∥ − ∥x_n+1 − x_n∥) ≤ 0, then lim _n→∞∥x_n − z_n∥ = 0.

Let C be a subset of a real Banach space E and

a family of mappings of C such that

. Then

is said to satisfy the AKTT-condition [29] if for each bounded subset K of C,

()

Lemma 8 (see [29].)Let C be a subset of a real Banach space E and a family of mappings of C into itself which satisfies the AKTT-condition. Then, for each x ∈ C, converges strongly to a point in C. Moreover, let the mapping T be defined by

()

Then for each bounded subset K of C,

()

In the sequel, one will write that

satisfies the AKKT-condition if

satisfies the AKKT-condition and T is defined by Lemma 8 with

We end this section with the following simple examples of mappings satisfying the AKTT-condition (see also Lemma 19).

Example 9. (i) Let E be a Banach space. For any n ∈ ℕ, let a mapping T_n : E → E be defined by

()

Then, T_n is a nonexpansive mapping for each n ∈ ℕ. It could easily be seen that

satisfies the AKKT-condition, where T(x) = 0 for all x ∈ E.

(ii) Let E be a smooth Banach space and let x₀ ≠ 0 be any element of E. For any j ∈ ℕ, we define a mapping T_j : E → E by

()

for all n ≥ 0. We define also a mapping T : E → E by

()

for all n ≥ 0. It is easy to verify that

satisfies the AKKT-condition.

(iii) Let E = l², where

()

Let {x_n} _{n∈ℕ∪{0}} ⊂ E be a sequence defined by

()

where

()

for all n ∈ ℕ. It is clear that the sequence {x_n} _n∈ℕ converges weakly to x₀. Indeed, for any Λ = (λ₁, λ₂, …, λ_n, …) ∈ l² = (l²) ^*, we have

()

as n → ∞. It is also obvious that

for any n ≠ m with n, m sufficiently large. Thus, {x_n} _n∈ℕ is not a Cauchy sequence. We define a countable family of mappings T_j : E → E by

()

for all j ≥ 1 and n ≥ 0. It is clear that F(T_j) = {0} for all j ≥ 1. It is obvious that T_j is a quasi-nonexpansive mapping for each j ∈ ℕ. Thus {T_j} _j∈ℕ is a countable family of quasi-nonexpansive mappings.

Let Tx = lim _j→∞T_jx for all x ∈ E. It is easy to see that

()

Then, we obtain that T is a quasi-nonexpansive mapping with

. Let D be a bounded subset of E. Then there exists r > 0 such that D ⊂ B_r = {z ∈ E : ∥z∥ < r}. On the other hand, for any j ∈ ℕ, we have

()

Furthermore, we have

()

Therefore,

satisfies the AKKT-condition.

3. Fixed Point and Convergence Theorems

Let E be a 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and let C be a closed and convex subset of E. Let A : C → E be a k-Lipschitzian and η-strongly accretive operator with constants k, η > 0, let B : C → E be an l-Lipschitzian mapping with constant l ≥ 0, and let T : C → C be a nonexpansive mapping with F(T) ≠ ⌀. Suppose that

, 0 < μ < η/k²ρ². Define a mapping f : [0,1] → ℝ by

()

From the definition of f we deduce that

()

Indeed, for any t ∈ (0,1], in view of (31) we obtain

()

On the other hand, it is easy to see that f is continuous on compact interval [0,1]. In fact, employing L’Hôpital’s Rule, we conclude that lim _t→0f(t) = μη. Thus,

()

Set

and τ_t : = f(t) if t ∈ [0,1]. Then we have

()

Assume now that γ satisfies 0 ≤ γl < τ₀. Then we get

()

In this section, we introduce the following implicit scheme that generates a net {x_t} _t∈(0,1) in an implicit way:

()

We prove the strong convergence of {x_t} to a fixed point

of T which solves the variational inequality

()

We first prove the following extension of Lemma 3.1 in [7] in a 2-uniformly smooth Banach space.

Lemma 10. Let E be a 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and let C be a closed and convex subset of E. Let A : C → E be a k-Lipschitzian and η-strongly accretive operator with , 0 < μ < η/k²ρ², and t ∈ (0,1). In association with a nonexpansive mapping T : C → C, define the mapping S_t : C → E by

()

Then, S_t is a contraction with contraction constant τ_t = 1 − c_t, where

Proof. In view of Lemma 3, we conclude that

()

for all x, y ∈ C. Put

. Then by the assumptions t ∈ (0,1) and

, we infer that

()

Let τ_t = (1 − c_t)∈(0,1). Then we have

()

Therefore, S_t is a contraction with contraction constant 1 − τ_t, which completes the proof.

Remark 11. Let E be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and C a closed convex subset of E. Let A : C → E be a k-Lipschitzian and η-strongly accretive operator with constants κ, η > 0 and let B : C → H be an l-Lipschitzian mapping with constant l ≥ 0. Assume T : C → C is a nonexpansive mapping with F(T) ≠ ⌀. Let , 0 < μ < η/k²ρ², and 0 ≤ γl < τ₀, where satisfies (34). For any t ∈ (0,1), let the mapping R_t : C → E be defined by

()

Using Remark 11, it could easily be seen that

()

Thus in view of Banach contraction principle, the contraction mapping R_t : C → E has a unique fixed point x_t in C, which uniquely solves the fixed point equation (37).

Remark 12. Let E be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and C a closed convex subset of E. Let A : C → E be a k-Lipschitzian and η-strongly accretive operator with constants κ, η > 0 and let B : C → E be an l-Lipschitzian mapping with constant l ≥ 0. Assume T : C → C is a nonexpansive mapping with F(T) ≠ ⌀. Let , 0 < μ < η/k²ρ², and 0 ≤ γl < τ₀, where satisfies (34). Then

()

That is, μA − γB is strongly accretive with coefficient μη − γl.

In the following result, we drive some important properties of the net {x_t} _t∈(0,1) which will be used in the sequel.

Proposition 13. Let E be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and let C be a closed and convex subset of E. Let A : C → E be a k-Lipschitzian and η-strongly accretive operator with constants κ, η > 0 and let B : C → H be an l-Lipschitzian mapping with constant l ≥ 0. Assume T : C → C is a nonexpansive mapping with F(T) ≠ ⌀. Let , 0 < μ < η/k²ρ², and 0 ≤ γl < τ₀, where satisfies (34). For each t ∈ (0,1), let x_t denote a unique solution of the fixed point equation (37). Then, the following properties hold for the net {x_t} _t∈(0,1):

(1)
{x_t} _t∈(0,1) is bounded;
(2)
lim _t→0∥x_t − Tx_t∥ = 0;
(3)
x_t defines a continuous curve from (0,1) into C.

Proof. (1) Let p ∈ F(T) be taken arbitrarily. Then, in view of Lemma 10 we obtain

()

This implies that

()

This shows that {x_t} is bounded.

(2) Since {x_t} is bounded, we have that {Bx_t} and {ATx_t} are bounded too. In view of the definition of {x_t} we conclude that

()

as t → 0.

(3) Take t₁, t₂ ∈ (0,1) arbitrarily. Then, we have

()

This implies that

()

The boundedness of {x_t} implies that x_t defines a continuous curve from (0,1) into C.

Theorem 14. Let E be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and let C be a closed and convex subset of E. Let A : C → E be a k-Lipschitzian and η-strongly accretive operator with constants κ, η > 0 and let B : C → H be an l-Lipschitzian mapping with constant l ≥ 0. Assume T : C → C is a nonexpansive mapping with F(T) ≠ ⌀. Let , 0 < μ < η/k²ρ², and 0 ≤ γl < τ₀, where satisfies (34). For each t ∈ (0,1), let {x_t} denote a unique solution of the fixed point equation (37). Then the net {x_t} converges strongly, as t → 0, to a fixed point of T which solves the variational inequality (38), or equivalently, .

Proof. In view of Remark 11 the variational inequality (38) has a unique solution, say . We show that as t → 0. To this end, let z ∈ F(T) be given arbitrary. Set

()

Then we have x_t = Q_Cy_t and hence

()

Since Q_C is a nonexpansive mapping from E onto C, in view of Lemma 4, we conclude that

()

Exploiting Lemma 10, (37), and (52), we obtain

()

This implies that

()

Let {t_n}⊂(0,1) be such that t_n → 0⁺ as n → ∞. Letting

, it follows from Proposition 13(2) that

. The boundedness of {x_t} implies that there exists x^* ∈ C such that

as n → ∞. In view of Lemma 5, we deduce that x^* ∈ F. Since

as n → ∞, it follows from (55) that

. Thus we have

well defined. Next, we show that x^* solves the variational inequality (38). We first notice that

()

This, together with (52), implies that

()

Since T is nonexpansive, in view of Remark 2, we conclude that I − T is accretive. This implies that

()

Replacing t by t_n in (58), taking the limit n → ∞, and noticing that {x_t − z} _t∈(0,1) is bounded for z ∈ F(T), we obtain

()

Thus, we have x^* ∈ F(T) a solution of the variational inequality (38). Consequently,

by uniqueness. Therefore,

as t → 0. The variational inequality (38) can be written as

()

Thus, in view of Lemma 4, it is equivalent to the following fixed point equation:

()

This completes the proof.

Theorem 15. Let E be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and let C be a nonempty, closed and convex subset of E. Suppose that the normalized duality mapping J : E → E^* is weakly sequentially continuous at zero. Let A : C → E be a k-Lipschitzian and η-strongly accretive operator with constants κ, η > 0 and let B : C → H be an l-Lipschitzian mapping with constant l ≥ 0. Let , 0 < μ < η/k²ρ², and 0 ≤ γl < τ₀, where satisfies (34). Assume is a sequence of nonexpansive mappings from C into itself such that . Suppose in addition that T : C → C is a nonexpansive mapping such that satisfies the AKTT-condition. For given x₁ ∈ C arbitrarily, let the sequence {x_n} be generated iteratively by

()

where Q_C is the sunny nonexpansive retraction from E onto C and {α_n} and {β_n} are two real sequences in (0,1) satisfying the following control conditions:

()

Then, the sequence {x_n} converges strongly to

which solves the variational inequality

()

Proof. We divide the proof into several steps.

Step 1. We claim that the sequence {x_n} is bounded. Let p ∈ F be fixed. In view of (62)–(64) and Lemma 10, we obtain

()

Since T_n is nonexpansive, for all n ∈ ℕ, it follows from (62) and (65) that

()

By induction, we conclude that {x_n} is bounded. This implies that the sequences {Ax_n}, {Bx_n}, {y_n}, and {T_ny_n} are bounded too. Let M₁ = sup {∥x_n∥, ∥Ax_n∥, ∥Bx_n∥, ∥y_n∥, ∥T_ny_n∥ : n ∈ ℕ} < ∞ and set K = {z ∈ E : ∥z∥ ≤ M}. Then we have K a bounded subset of E and {x_n, Ax_n, Bx_n, y_n, T_ny_n} ⊂ K.

Step 2. We claim that lim _n→∞∥y_n − Ty_n∥ = 0. For this purpose, we denote a sequence {z_n} by z_n = T_ny_n. Then we have

()

This implies that

()

In view of Lemma 8 and (63)(a) we conclude that

()

Utilizing Lemma 7, we deduce that

()

It follows from (63)(b) and (70) that

()

Observe now that

()

This implies that

()

Utilizing Lemma 7 and taking into account α_n → 0, we deduce that

()

On the other hand, we have

()

Employing Lemma 8, we obtain

()

Step 3. We prove that there exists x^* ∈ F such that

()

where x^* is as in Theorem 14. We first note that there exists a subsequence

of {y_n} such that

()

Since {y_n} is bounded, without loss of generality, we may assume that

as i → ∞. In view of Lemma 6 and Step 2, we conclude that u ∈ F. This, together with (78), implies that

()

Step 4. We claim that lim _n→∞∥x_n − x^*∥ = 0.

For each n ∈ ℕ ∪ {0}, by Lemma 10 and (36) we obtain

()

This implies that

()

where

()

In view of (81), we conclude that

()

where γ_n = β_nα_n(τ₀ − γl). It is easy to show that lim _n→∞γ_n = 0,

, and limsup _n→∞ξ_n ≤ 0. Hence, in view of Lemma 6 and (83), we conclude that the sequence {x_n} converges strongly to x^* ∈ F(T). This completes the proof.

Remark 16. Theorem 15 improves and extends [19, Theorems 3.1 and 3.2] in the following aspects.

(i)
The self-contractive mapping f : C → C in [19, Theorems 3.1 and 3.2] is extended to the case of a Lipschitzian (possibly nonself-) mapping B : C → E on a nonempty closed convex subset C of a Banach space E.
(ii)
The identity mapping I is extended to the case of I − A : C → E, where A : C → E is a k-Lipschitzian and η-strongly accretive (possibly nonself-) mapping.
(iii)
The contractive coefficient α ∈ (0,1) in [19, Theorems 3.1 and 3.2] is extended to the case where the Lipschitzian constant l lies in [0, ∞).
(iv)
In order to find a common fixed point of a countable family of nonexpansive self-mappings T_n : C → C, the Mann type iterations in [19, Theorems 3.1 and 3.2] are extended to develop the new Mann type iteration (62).
(v)
The new technique of argument is applied in deriving Theorem 14. For instance the characteristic properties (Lemma 4) of sunny nonexpansive retraction play an important role in proving the strong convergence of the net {x_t} _t∈(0,1) in Theorem 14.
(vi)
Whenever we have C = E, B = f a contraction mapping with coefficient α ∈ (0,1), A = I the identity mapping on C, and l = α with , Theorem 14 reduces to [19, Theorems 3.1 and 3.2]. Thus, Theorem 14 covers [19, Theorems 3.1 and 3.2] as special cases.

Remark 17. Proposition 13 and Theorems 14 and 15 improve and generalize the corresponding results of [4] from Hilbert spaces to Banach spaces.

4. Applications

In this section, we apply Theorem 15 for finding a zero of an accretive operator. Let E be a real Banach space and let S : E → 2^E be a mapping. The effective domain of S is denoted by dom (S); that is, dom (S) = {x ∈ E : Sx ≠ ⌀}. The range of S is denoted by R(S). A multivalued mapping S is said to be accretive if for all x, y ∈ E there exists j ∈ J(x − y) such that 〈x − y, j〉≥0, where is the duality mapping. An accretive operator S is m-accretive if R(I + rS) = E for each r ≥ 0. Throughout this section, we assume that S : E → 2^E is m-accretive and has a zero. For an accretive operator S on E and r > 0, we may define a single-valued operator J_r = (I + rS) ⁻¹ : E → dom (S), which is called the resolvent of S for r > 0. Assume S⁻¹0 = {x ∈ E : 0 ∈ Sx}. It is known that S⁻¹0 = F(J_r) for all r > 0.

The following lemma has been proved in [21].

Lemma 18. Let E be a real Banach space and let S be an m-accretive operator on E. For r > 0, let J_r be the resolvent operator associated with S and r. Then

()

for all r, s > 0 and x ∈ E.

We also know the following lemma from [29].

Lemma 19. Let C be a nonempty, closed, and convex subset of a real Banach space E and let S be an accretive operator on E such that S⁻¹0 ≠ ⌀ and . Suppose that {r_n} is a sequence of (0, ∞) such that inf {r_n : n ∈ ℕ} > 0 and . Then

(i)
for any bounded subset B of C;
(ii)
for all z ∈ C and , where r_n → r as n → ∞.

As an application of our main result, we include a concrete example in support of Theorem 15. Using Theorem 15, we obtain the following strong convergence theorem for m-accretive operators.

Theorem 20. Let E be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and C a nonempty, closed, and convex subset of E. Suppose that the normalized duality mapping J : E → E^* is weakly sequentially continuous at zero. Let A : C → E be a k-Lipschitzian and η-strongly accretive operator with constants κ, η > 0 and let B : C → H be an l-Lipschitzian mapping with constant l ≥ 0. Let , 0 < μ < η/k²ρ², and 0 ≤ γl < τ₀, where satisfies (34). Let S be an m-accretive operator from E to E^* such that S⁻¹(0) ≠ ⌀. Let r_n > 0 such that liminf _n→∞r_n > 0, and let be the resolvent of S. Let and be sequences in [0,1] satisfying the following control conditions:

(a)
lim _n→∞α_n = 0;
(b)
;
(c)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1.

Let be a sequence generated by

()

where Q_C is the sunny nonexpansive retraction from E onto C. Then, the sequence {x_n} defined in (85) converges strongly to x^* ∈ S⁻¹(0).

Proof. Letting , in Theorem 15, from (62), we obtain (85). It is easy to see that T_n satisfies all the conditions in Theorem 15 for all n ∈ ℕ. Therefore, in view of Theorem 15 we have the conclusions of Theorem 20. This completes the proof.

Remark 21. Theorem 20 improves and extends Theorems 4.2, 4.3, 4.4, and 4.5 in [19].

Acknowledgments

The authors would like to thank the editor and the referees for sincere evaluation and constructive comments which improved the paper considerably. The work of Eskandar Naraghirad was conducted with a postdoctoral fellowship at the National Sun Yat-sen University of Kaohsiung, Taiwan.

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All articles

Strong Convergence of a General Iterative Method for a Countable Family of Nonexpansive Mappings in Banach Spaces

Abstract

1. Introduction

2. Preliminaries

3. Fixed Point and Convergence Theorems

4. Applications

Acknowledgments

References

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