We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others.

1. Introduction

Throughout this paper we denoted by ℕ and ℝ⁺ the set of all positive integers and all positive real numbers, respectively. Let X be a real Banach space, and let C be a nonempty closed convex subset of X. A mapping T of C into itself is said to be nonexpansive if ∥Tx − Ty∥≤∥x − y∥ for each x, y ∈ C. We denote by F(T) the set of fixed points of T. We know that F(T) is nonempty if C is bounded; for more detail see [1]. A one-parameter family 𝒮 = {T(t) : t ∈ ℝ⁺} from C of X into itself is said to be a nonexpansive semigroup on C if it satisfies the following conditions:

(i)
T(0)x = x for all x ∈ C;
(ii)
T(s + t) = T(s)∘T(t) for all s, t ∈ ℝ⁺;
(iii)
for each x ∈ C the mapping t ↦ T(t)x is continuous;
(iv)
∥T(t)x − T(t)y∥≤∥x − y∥ for all x, y ∈ C and t ∈ ℝ⁺.

We denote by F(𝒮) the set of all common fixed points of 𝒮, that is, . We know that F(𝒮) is nonempty if C is bounded; see [2]. Recall that a self-mapping f : C → C is a contraction if there exists a constant α ∈ (0,1) such that ∥f(x) − f(y)∥≤α∥x − y∥ for each x, y ∈ C. As in [3], we use the notation ∏_C to denote the collection of all contractions on C, that is, ∏_C = {f : C → C a contraction}. Note that each f ∈ ∏_C has a unique fixed point in C.

In the last ten years, the iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [3–5]. Let H be a real Hilbert space, whose inner product and norm are denoted by 〈·, ·〉 and ∥·∥, respectively. Let A be a strongly positive bounded linear operator on H: that is, there is a constant

with property

(1.1)

A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:

(1.2)

where C is the fixed point set of a nonexpansive mapping T on H and b is a given point in H.

In 2003, Xu [3] proved that the sequence {x_n} generated by

(1.3)

converges strongly to the unique solution of the minimization problem (1.2) provided that the sequence {α_n} satisfies certain conditions. Using the viscosity approximation method, Moudafi [6] introduced the iterative process for nonexpansive mappings (see [3, 7] for further developments in both Hilbert and Banach spaces) and proved that if H is a real Hilbert space, the sequence {x_n} generated by the following algorithm:

(1.4)

where f : C → C is a contraction mapping with constant α ∈ (0,1) and {α_n}⊂(0,1) satisfies certain conditions, converges strongly to a fixed point of T in C which is unique solution x^* of the variational inequality:

(1.5)

In 2006, Marino and Xu [8] combined the iterative method (1.3) with the viscosity approximation method (1.4) considering the following general iterative process:

(1.6)

where

. They proved that the sequence {x_n} generated by (1.6) converges strongly to a unique solution x^* of the variational inequality:

(1.7)

which is the optimality condition for the minimization problem:

(1.8)

where C is the fixed point set of a nonexpansive mapping T and h is a potential function for γf (i.e., h′(x) = γf(x) for x ∈ H). Kim and Xu [9] studied the sequence generated by the following algorithm:

(1.9)

and proved strong convergence of scheme (1.9) in the framework of uniformly smooth Banach spaces. Later, yao, et al. [10] introduced a new iteration process by combining the modified Mann iteration [9] and the viscosity approximation method introduced by Moudafi [6]. Let C be a closed convex subset of a Banach space, and let T : C → C be a nonexpansive mapping such that F(T) ≠ ∅ and f ∈ ∏_C. Define {x_n} in the following way:

(1.10)

where {α_n} and {β_n} are two sequences in (0,1). They proved under certain different control conditions on the sequences {α_n} and {β_n} that {x_n} converges strongly to a fixed point of T. Recently, Chen and Song [11] studied the sequence generated by the algorithm in a uniformly convex Banach space, as follows:

(1.11)

and they proved that the sequence {x_n} defined by (1.11) converges strongly to the unique solution of the variational inequality:

(1.12)

In 2010, Sunthrayuth and Kumam [12] introduced the a general iterative scheme generated by

(1.13)

for the approximation of common fixed point of a one-parameter nonexpansive semigroup in a Banach space under some appropriate control conditions. They proved strong convergence theorems of the iterative scheme which solve some variational inequality. Very recently, Kumam and Wattanawitoon [13] studied and introduced a new composite explicit viscosity iteration method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces. They proved strong convergence theorems of the composite iterative schemes which solve some variational inequalities under some appropriate conditions. In the same year, Sunthrayuth et al. [14] introduced a general composite iterative scheme for nonexpansive semigroups in Banach spaces. They established some strong convergence theorems of the general iteration scheme under different control conditions.

In this paper, motivated by Yao et al. [10], Sunthrayuth, and Kumam [12] and Kumam and Wattanawitoon [13] we introduce a new general iterative algorithm (3.23) for finding a common point of the set of solution of some variational inequality for nonexpansive semigroups in Banach spaces which admit a weakly continuous duality mapping and then proved the strong convergence theorem generated by the proposed iterative scheme. The results presented in this paper improve and extend some others from Hilbert spaces to Banach spaces and some others as special cases.

2. Preliminaries

Throughout this paper, we write x_n⇀x (resp., x_n⇀^*x) to indicate that the sequence {x_n} weakly (resp., weak*) converges to x; as usual x_n → x will symbolize strong convergence; also, a mapping I denote the identity mapping. Let X be a real Banach space, and let X^* be its dual space. Let U = {x ∈ X : ∥x∥ = 1}. A Banach space X is said to be uniformly convex if, for each ϵ ∈ (0,2], there exists a δ > 0 such that for each x, y ∈ U, ∥x − y∥≥ϵ implies ∥x + y∥/2 ≤ 1 − δ. It is known that a uniformly convex Banach space is reflexive and strictly convex (see also [15]). A Banach space is said to be smooth if the limit lim _t→0∥x + ty∥−∥x∥/t exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ U.

Let φ : [0, ∞)∶ = ℝ⁺ → ℝ⁺ be a continuous strictly increasing function such that φ(0) = 0 and φ(t) → ∞ as t → ∞. This function φ is called a gauge function . The duality mapping

associated with a gauge function φ is defined by

(2.1)

where 〈·, ·〉 denotes the generalized duality paring. In particular, the duality mapping with the gauge function φ(t) = t, denoted by J, is referred to as the normalized duality mapping. Clearly, there holds the relation J_φ(x) = (φ(∥x∥)/∥x∥)J(x) for each x ≠ 0 (see [16]).

Browder [16] initiated the study of certain classes of nonlinear operators by means of the duality mapping J_φ. Following Browder [16], we say that Banach space X has a weakly continuous duality mapping if there exists a gauge function φ for which the duality mapping J_φ(x) is single-valued and continuous from the weak topology to the weak^* topology; that is, for each {x_n} with x_n⇀x, the sequence {J(x_n)} converges weakly* to J_φ(x). It is known that l^p has a weakly continuous duality mapping with a gauge function φ(t) = t^p−1 for all 1 < p < ∞. Set , for all t ≥ 0; then J_φ(x) = ∂Φ(∥x∥), where ∂ denotes the subdifferential in the sense of convex analysis (recall that the subdifferential of the convex function ϕ : X → ℝ at x ∈ X is the set ∂ϕ(x) = {x^* ∈ X; ϕ(y) ≥ ϕ(x)+〈x^*, y − x〉, for all y ∈ X}).

In a Banach space having a weakly continuous duality mapping J_φ with a gauge function φ, we defined an operator A is to be strongly positive (see [17]) if there exists a constant

with the property

(2.2)

(2.3)

If X∶ = H is a real Hilbert space, then the inequality (2.2) reduces to (1.1).

The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in [18].

Lemma 2.1 (see [18].)Assume that a Banach space X has a weakly continuous duality mapping J_φ with gauge φ.

(i)
For all x, y ∈ X, the following inequality holds:
(2.4)
In particular, for all x, y ∈ X,

(2.5)

(ii)
Assume that a sequence {x_n} in X converges weakly to a point x ∈ X. Then the following identity holds:

(2.6)

Lemma 2.2 (see [17].)Assume that a Banach space X has a weakly continuous duality mapping J_φ with gauge φ. Let A be a strongly positive linear bounded operator on X with a coefficient and 0 < ρ ≤ φ(1)∥A∥⁻¹. Then .

Lemma 2.3 (see [11].)Let C be a closed convex subset of a uniformly convex Banach space X and let 𝒮 = {T(t) : t ∈ ℝ⁺} be a nonexpansive semigroup on C such that F(𝒮) ≠ ∅. Then, for each r > 0 and h ≥ 0,

(2.7)

Lemma 2.4 (see [19].)Assume that {a_n} is a sequence of nonnegative real numbers such that

(2.8)

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

(i)
;
(ii)
limsup _n→∞(δ_n/μ_n) ≤ 0 or .

Then, lim _n→∞a_n = 0.

3. Main Results

Let X be a Banach space which admits a weakly continuous duality mapping J_φ with gauge φ such that φ is invariant on [0,1], and let C be a nonempty closed convex subset of X such that C ± C ⊂ C. Let 𝒮 = {T(t) : t ∈ ℝ⁺} be a nonexpansive semigroup from C into itself, let f be a contraction mapping with a coefficient α ∈ (0,1), let A be a strongly positive linear bounded operator with a coefficient

such that

, and let t ∈ (0,1) such that t ≤ φ(1)∥A∥⁻¹ which satisfies t → 0. Define the mapping

(3.1)

to be a contraction mapping. Indeed, for each x, y ∈ C,

(3.2)

Thus, by Banach contraction mapping principle, there exists a unique fixed point x_t ∈ C, that is,

(3.3)

Remark 3.1. We note that space l_p has a weakly continuous duality mapping with a gauge function φ(t) = t^p−1 for all 1 < p < ∞. This shows that φ is invariant on [0,1].

Lemma 3.2. Let X be a uniformly convex Banach space which admits a weakly continuous duality mapping J_φ with gauge φ such that φ is invariant on [0,1], and let C be a nonempty closed convex subset of X such that C ± C ⊂ C. Let 𝒮 = {T(t) : t ∈ ℝ⁺} be a nonexpansive semigroup from C into itself such that F(𝒮) ≠ ∅, let f be a contraction mapping with a coefficient α ∈ (0,1), let A be a strongly positive linear bounded operator with a coefficient such that , and let t ∈ (0,1) such that t ≤ φ(1)∥A∥⁻¹ which satisfies t → 0. Then the net {x_t} defined by (3.3) with {λ_t} _0<t<1 is a positive real divergent sequence, converges strongly as t → 0 to a common fixed point x^*, in which x^* ∈ F(𝒮), and is the unique solution of the variational inequality:

(3.4)

Proof. Firstly, we show the uniqueness of a solution of the variational inequality (3.4). Suppose that are solutions of (3.4); then

(3.5)

Adding up (3.5), we obtain

(3.6)

which is a contradiction, we must have

, and the uniqueness is proved. Here in after, we use

to denote the unique solution of the variational inequality (3.4).

Next, we show that {x_t} is bounded. Indeed, for each p ∈ F(𝒮), we have

(3.7)

It follows that

(3.8)

Hence, {x_t} is bounded, so are {f(x_t)} and

Next, we show that ∥x_t − T(h)x_t∥→0 as t → 0. We note that

(3.9)

Moreover, we note that

(3.10)

for all h ≥ 0. Define the set

; then K is a nonempty bounded closed convex subset of C which is T(s)-invariant for each h ≥ 0. Since {x_t} ⊂ K and K is bounded, there exists r > 0 such that K ⊂ B_r, and it follows by Lemma 2.3 that

(3.11)

for each h ≥ 0. From (3.9)-(3.10), letting t → 0 and noting (3.11) then, for each h ≥ 0, we obtain

(3.12)

Assume that

is such that t_n → 0 as n → ∞. Put

and

. We will show that {x_n} contains a subsequence converging strongly to

. Since {x_n} is bounded sequence and Banach space X is a uniformly convex, hence it is reflexive, and there exists a subsequence

of {x_n} which converges weakly to some

as j → ∞. Again, since J_φ is weakly sequentially continuous, we have by Lemma 2.1 that

(3.13)

Let

, for all z ∈ C.

It follows that , for all z ∈ C. From (3.12), we have

(3.14)

On the other hand, we note that

(3.15)

Combining (3.14) with (3.15), we obtain

. This implies that

, that is,

. In fact, since

, for all t ≥ 0 and φ : ℝ⁺ → ℝ⁺ is the gauge function, then for 1 ≥ k ≥ 0, φ(ky) ≤ φ(y) and

(3.16)

By Lemma 2.1, we have

(3.17)

This implies that

(3.18)

In particular, we have

(3.19)

Since the mapping J_φ is single-valued and weakly continuous, it follows from (3.19) that

as j → ∞. This implies that

as j → ∞.

Next, we show that solves the variational inequality (3.4), for each x ∈ F(𝒮). From (3.3), we derive that

(3.20)

Now, we observe that

(3.21)

It follows from (3.20) that

(3.22)

Now, replacing t and λ_t with

and

, respectively, in (3.22), and letting j → ∞, and we notice that

for

, we obtain that

. That is,

is a solution of the variational inequality (3.4). By uniqueness, as

, we have shown that each cluster point of the net {x_t} is equal to x^*. Then, we conclude that x_t → x^* as t → 0. This proof is complete.

Theorem 3.3. Let X be a uniformly convex Banach space which admits a weakly continuous duality mapping J_φ with the gauge function φ such that φ is invariant in [0,1], and let C be a nonempty closed convex subset of X such that C ± C ⊂ C. Let 𝒮 = {T(t) : t ∈ ℝ⁺} be a nonexpansive semigroup from C into itself such that F(𝒮) ≠ ∅, let f be a contraction mapping with a coefficient α ∈ (0,1), and let A be a strongly positive linear bounded operator with a coefficient such that . Let be the sequences in (0,1) and let be a positive real divergent sequence. Assume that the following conditions hold:

(C1)
lim _n→∞ α_{_n} = 0 and ,
(C2)
lim _n→∞γ_n = 0,
(C3)
β_n = o(α_n),

Then the sequence {x_n} defined by

(3.23)

converges strongly to the common fixed point x^* that is obtained in Lemma 3.2.

Proof. From the condition (C1), we may assume, with no loss of generality, that α_n ≤ φ(1)∥A∥⁻¹ for each n ≥ 0. From Lemma 2.2, we have .

Firstly, we show that {x_n} is bounded. Let p ∈ F(𝒮); we get

(3.24)

It follows that

(3.25)

By induction on n, we have

(3.26)

Thus, {x_n} is bounded. Since {x_n} is bounded, then

and

and {f(z_n)} are also bounded.

Next, we show that lim _n→∞∥x_n − T(h)x_n∥ = 0, for all h ≥ 0. From (3.23), we note that

(3.27)

By the conditions (C1)–(C3), then (3.27), we obtain

(3.28)

Moreover, we note that

(3.29)

Define the set

. Then K is a nonempty bounded closed convex subset of C, which is T(s)—invariant for each s ≥ 0 and contains {x_n}; it follows from Lemma 2.3 that

(3.30)

Then, for all h ≥ 0, from (3.28) and (3.30), into (3.29), we obtain lim _n→∞∥x_n+1 − T(h)x_n+1∥ = 0, and hence

(3.31)

Next, we show that limsup _n→∞〈γf(x^*) − Ax^*, J_φ(x_n − x^*)〉≤0. We can take subsequence

such that

(3.32)

By the assumption that X is uniformly convex, hence it is reflexive and {x_n} is bounded; then there exists a subsequence

which converges weakly to some x ∈ C as j → ∞. Since J_φ is weakly continuous, from Lemma 2.1, we have

(3.33)

Let

, for all z ∈ C.

It follows that H(z) = H(x) + Φ(∥z − x∥), for all z ∈ C.

From (3.31), we have

(3.34)

On the other hand, we note that

(3.35)

Combining (3.34) with (3.35), we obtain Φ(∥T(h)x − x∥) ≤ 0.

This implies that T(h)x = x; that is, x ∈ F(𝒮).

Since the duality map J_φ is single-valued and weakly continuous, we get that

(3.36)

as required. Hence,

(3.37)

Since ∥x_n+1 − y_n∥ = β_n∥x_n − y_n∥, by condition (C3), we obtain that lim _n→∞∥x_n+1 − y_n∥ = 0. It follows from (3.37), that

(3.38)

Finally, we show that x_n → x^* as n → ∞. Now, from Lemma 2.1, we have

(3.39)

On the other hand, we note that

(3.40)

It follows from (3.40) that

(3.41)

where M = sup _n≥0{∥x_n − x^*∥φ(∥x_n+1 − x^*∥)}.

Put and δ_n∶ = α_n[〈γf(x^*) − Ax^*, J_φ(y_n − x^*)〉+(β_n/α_n)M]. Then (3.41) reduces to formula Φ(∥x_n+1 − x^*∥)≤(1 − μ_n)Φ(∥x_n − x^*∥) + δ_n. By conditions (C1) and (C3) and noting (3.38), it is easy to see that and . Applying Lemma 2.4, we obtain Φ(∥x_n − x^*∥) → 0 as n → ∞ this implies that x_n → x^* as n → ∞. This completes the proof.

Taking γ_n = 0 in (3.23), we can get the following corollary easily.

Corollary 3.4. Let X be a uniformly convex Banach space which admits a weakly continuous duality mapping J_φ with the gauge function φ such that φ invariant in [0,1], C be a nonempty closed convex subset of X such that C ± C ⊂ C. Let 𝒮 = {T(t) : t ∈ ℝ⁺} be a nonexpansive semigroup from C into itself such that F(𝒮) ≠ ∅, f be a contraction mapping with a coefficient α ∈ (0,1) and A be a strongly positive linear bounded operator with a coefficient such that . Let be the sequences in (0,1) and be a positive real divergent sequence. Assume the following conditions are hold:

(C1)
lim _n→∞α_n = 0 and ;
(C2)
β_n = o(α_n).

Then the sequence {x_n} defined by

(3.42)

converges strongly to the common fixed point x^*, in which x^* ∈ F(𝒮) is the unique solution of the variational inequality:

(3.43)

A strong mean convergence theorem for nonexpansive mapping was first established by Baillon [20] and it was generalized to that for nonlinear semigroups by Reich et al. [21–23]. It is clear that Theorem 3.3 are valid for nonexpansive mappings. Thus, we have the following mean ergodic theorem of viscosity iteration process for nonexpansive mappings in Hilbert spaces.

Corollary 3.5. Let H be a real Hilbert space, and letC be a nonempty closed convex subset of H such that C ± C ⊂ C. Let T be a nonexpansive mapping from C into itself such that F(T) ≠ ∅, f be a contraction mapping with a coefficient α ∈ (0,1), and let A be a strongly positive linear bounded operator with a coefficient such that . Let be the sequences in (0,1) and let be a positive real divergent sequence. Assume that the following conditions are hold:

(C1)
lim _n→∞α_n = 0 and ;
(C2)
lim _n→∞γ_n = 0;
(C3)
β_n = o(α_n).

Then the sequence {x_n} defined by

(3.44)

converges strongly to the common fixed point x^*, in which x^* ∈ F(T) is the unique solution of the variational inequality:

(3.45)

Acknowledgments

The authors are grateful for the reviewers for the careful reading of the paper and for the suggestions which improved the quality of this work. They would like to thank the National Research University Project of Thailand′s Office of the Higher Education Commission for financial support under NRU-CSEC project no. 54000267.

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A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces

Abstract

1. Introduction

2. Preliminaries

3. Main Results

Acknowledgments

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A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces

Abstract

1. Introduction

2. Preliminaries

3. Main Results

Acknowledgments

References

Citing Literature

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