It is our purpose in this paper to prove two convergents of viscosity approximation scheme to a common fixed point of a family of multivalued nonexpansive mappings in Banach spaces. Moreover, it is the unique solution in F to a certain variational inequality, where stands for the common fixed-point set of the family of multivalued nonexpansive mapping {T_n}.

1. Introduction

Let X be a Banach space with dual X^*, and let K be a nonempty subset of X. A gauge function is a continuous strictly increasing function φ : R⁺ → R⁺ such that φ(0) = 0 and lim _t→∞φ(t) = ∞. The duality mapping J_φ : X → X^* associated with a gauge function φ is defined by J_φ(x): = {f ∈ X^* : 〈x, f〉 = ∥x∥∥f∥, ∥f∥ = φ(∥x∥)}, x ∈ X, where 〈·, ·〉 denotes the generalized duality pairing. In the particular case φ(t) = t, the duality map J = J_φ is called the normalized duality map. We note that J_φ(x) = (φ(∥x∥)/∥x∥)J(x). It is known that if X is smooth, then J_φ is single valued and norm to weak* continuous (see [1]). When {x_n} is a sequence in X, then x_n → x (x_n⇀x, x_n⇁x) will denote strong (weak, weak*) convergence of the sequence {x_n} to x. s

Following Browder [2], we say that a Banach space X has the weakly continuous duality mapping if there exists a gauge function φ for which the duality map J_φ is single valued and weak to weak* sequentially continuous, that is, if {x_n} is a sequence in X weakly convergent to a point x, then the sequence {J_φ(x_n)} converges weak* to J_φ(x). It is known that l_p (1 < p < 1) spaces have a weakly continuous duality mapping J_φ with a gauge φ(t) = t^p−1. Setting

(1.1)

it is easy to see that Φ(t) is a convex function and J_φ(x) = ∂Φ(∥x∥), for x ∈ X, where ∂ denotes the subdifferential in the sense of convex analysis. We will denote by 2^X the family of all subsets of X, by CB(X) the family of all nonempty closed bounded subsets of X, and by C(X) the family of all nonempty compact subsets of X. A multivalued mapping T : K → 2^X is said to be nonexpansive (resp., contractive) if

(1.2)

where H(·, ·) denotes the Hausdorff metric on CB(X) defined by

(1.3)

Since Banach′s contraction mapping principle was extended nicely to multivalued mappings by Nadler in 1969 (see [3]), many authors have studied the fixed-point theory for multivalued mappings.

In this paper, we construct two viscosity approximation sequences for a family of multivalued nonexpansive mappings in Banach spaces. Let K be a nonempty closed convex subset of Banach space X and let T_n : K → C(K), n = 1,2… be a family of multivalued nonexpansive mapping, f : K → K is a contraction mapping with constant α ∈ (0,1). Let α_n ∈ (0,1), β_n ∈ (0,1). For any given x₀ ∈ K, let y₀ ∈ T₀x₀ such that

(1.4)

From Nadler Theorem (see [3]), we can choose y₁ ∈ T₁x₁ such that

(1.5)

Inductively, we can get the sequence {x_n} as follows:

(1.6)

where, for each n ∈ N, y_n ∈ T_nx_n such that

(1.7)

Similarly, we also have the following multivalued version of the modified Mann iteration:

(1.8)

and y_n ∈ T_nx_n such that ∥y_n+1 − y_n∥≤H(T_n+1x_n+1, T_nx_n). Then, {x_n} is said to satisfy Condition (A^′) if for any subsequence

and d(x_n+1, T_n(x_n)) → 0 implies that x ∈ F, where

is the common fixed-point set of the family of multivalued mapping {T_n}. We give an example of a family of multivalued nonexpansive mappings with Condition (A^′) as follows.

Example 1.1. Take X = R and T_n = T (for all n ≥ 0), where T is defined by

(1.9)

Let f : R → {0} and α_n = 1/n, n ≥ 2, then F = {0} and the iteration (1.6), reduced to

(1.10)

where y_n ∈ Tx_n, and it satisfies Condition (A^′). In fact, if x₀ ≤ 1, then (for all n ∈ ℕ, n > 0)x_n = 0 and Condition (A^′) is automatically satisfied. If x₀ > 1, then there exists an integer p ≥ 2, such that

(1.11)

Then, y_p ∈ Tx_p−1 = {0}; hence, x_n = 0 (for all n ≥ p), from which we deduce that Condition (A^′) is satisfied.

2. Preliminaries

Let K ⊂ X be a closed convex and Q a mapping of X onto K, then Q is said to be sunny if Q(Q(x) + t(x − Q(x))) = Q(x) for all x ∈ X and t ≥ 0. A mapping Q of X into X is said to be a retraction if Q² = Q. A subset K of X is said to be a sunny nonexpansive retract of X if there exists a sunny nonexpansive retraction of X onto K, and it is said to be a nonexpansive retract of X. If X = H, the metric projection P is a sunny nonexpansive retraction from H to any closed convex subset of H. The following Lemmas will be useful in this paper.

Lemma 2.1 (see [4].)Let K be a nonempty convex subset of a smooth Banach space X, let J : X → X^* be the (normalized) duality mapping of X, and let Q : X → K be a retraction, then the following are equivalent:

(1)
〈x − Px, j(y − Px)〉≤0 for all x ∈ X and y ∈ K,
(2)
Q is both sunny and nonexpansive.

We note that Lemma 2.1 still holds if the normalized duality map J is replaced with the general duality map J_φ, where φ is a gauge function.

Lemma 2.2 (see [5].)Let {a_n} be a sequence of nonnegative real numbers satisfying the property

(2.1)

where {γ_n}⊂(0,1) and {β_n} is a real number sequence such that

(i)
,
(ii)
either limsup _n→∞(β_n/γ_n) ≤ 0 or ,

then {a_n} converges to zero, as n → ∞.

Lemma 2.3 (see [1].)Let X be a real Banach space, then for all x, y ∈ X, one gets that

(2.2)

Lemma 2.4 (see [6].)Let {x_n} and {y_n} be bounded sequences in a Banach space X such that

(2.3)

where {γ_n} is a sequence in [0,1] such that

(2.4)

Assume that limsup _n→∞(∥y_n+1 − y_n∥−∥x_n+1 − x_n∥) ≤ 0, then lim _n→∞∥y_n − x_n∥ = 0.

3. Main Results

Theorem 3.1. Let X be a reflexive Banach space with weakly sequentially continuous duality mapping J_φ for some gauge φ, let K be a nonempty closed convex subset of X, and let T_n : K → C(K), n = 0,1, 2…, be a family of multivalued nonexpansive mappings with F ≠ ∅ which is sunny nonexpansive retract of K with Q a nonexpansive retraction. Furthermore, T_n(p) = {p} for any fixed-point p ∈ F, {x_n} is defined by (1.6), and α_n ∈ (0,1) satisfies the following conditions:

(1)
α_n → 0 as n → ∞,
(2)
,
(3)
{x_n} satisfies Condition (A′).

Then, {x_n} converges strongly to a common fixed-point

of a family T_n, n = 0,1, 2…, as n → ∞. Moreover,

is the unique solution in F to the variational inequality

(3.1)

Proof. First, we show the uniqueness of the solution to the variational inequality (3.1) in X. In fact, let be another solution of (3.1) in F, then we have

(3.2)

From (3.2), we have that

(3.3)

We must have

and the uniqueness is proved. Let p ∈ F, then, from iteration (1.6), we obtain that

(3.4)

Using an induction, we obtain ∥x_n − p∥≤max {∥x₀ − p∥, (1/(1 − α))∥f(p) − p∥}, for all integers n, thus, {x_n} is bounded and so are {T_nx_n} and {f(x_n)}. This implies that

(3.5)

Next, we will show that

(3.6)

Since X is reflexive and {x_n} is bounded, we may assume that

such that

(3.7)

From (3.5) and {x_n} satisfying Condition (A^′), we obtain that q ∈ F. On the other hand, we notice that the assumption that the duality mapping J_φ is weakly continuous implies that X is smooth; from Lemma 2.1, we have

(3.8)

Finally, we will show that

as n → ∞. From iteration (1.6) and Lemma 2.3, we get that

(3.9)

Lemma 2.2 gives that

as n → ∞. Moreover,

satisfying the variational inequality follows from the property of Q.

Let f ≡ u ∈ K in iteration (1.6) be a constant mapping, then . In fact, we have the following corollary.

Corollary 3.2. Let {x_n} and T_n be as in Theorem 3.1, f ≡ u ∈ K, then {x_n} converges strongly to a common fixed-point of a family T_n, n = 0,1, 2…, as n → ∞. Moreover, is the unique solution in F to the variational inequality

(3.10)

If X = H, then the condition that F is a sunny nonexpansive retract of K in Theorem 3.1 is not necessary, and one has the following Corollary.

Corollary 3.3. Let H be a Hilbert space with weakly sequentially continuous duality mapping J_φ for some gauge φ, and let {x_n} and T_n be as in Theorem 3.1, then {x_n} converges strongly to a common fixed-point of a family of T_n, n = 0,1, 2…, where P_F is the metric projection from K onto F.

Proof. It is well known that H is reflexive; by Propositions 2.3 and 2.6(ii) of [7], we get that F is closed and convex, and hence the projection mapping P_F is sunny nonexpansive retraction mapping, and the result follows from Theorem 3.1.

Corollary 3.4. Let X be a real smooth Banach space, let K be a nonempty compact subset of X, and let T_n and {x_n} be as in Theorem 3.1, then {x_n} converges strongly to a common fixed-point of a family of T_n, n = 0,1, 2…, as n → ∞. Moreover, is the unique solution in F to the variational inequality

(3.11)

Proof. Following the method of the proof of Theorem 3.1, we get that

(3.12)

Next, we will show that

(3.13)

Since K is compact and {x_n} is bounded, we can assume that there exists a subsequence

of {x_n} such that

(3.14)

From (3.12) and {x_n} satisfying Condition (A^′), we obtain that q ∈ F. On the other hand, from the fact that X is smooth, the duality being norm to weak* continuous, and the standard characterization of retraction on F, we obtain that

(3.15)

Now, following the method of the proof of Theorem 3.1, we get the required result.

Theorem 3.5. Let X be a reflexive Banach space with weakly sequentially continuous duality mapping J_φ for some gauge φ, let K be a nonempty closed convex subset of X, and let T_n : K → C(K), n = 0,1, 2…, be a family of multivalued nonexpansive mappings with F ≠ ∅ which is sunny nonexpansive retract of K with Q a nonexpansive retraction. H(T_n+1x, T_ny)≤∥x − y∥ for arbitrary n ∈ ℕ. Furthermore, T_n(p) = {p} for any fixed-point p ∈ F. {x_n} is defined by (1.8) and α_n, β_n satisfy the following conditions:

(i)
β_n → 0 as n → ∞,
(ii)
,
(iii)
0 < liminf _n→∞α_n ≤ lim sup _n→∞α_n < 1.

If {x_n} satisfies Condition (A^′), then {x_n} converges strongly to a common fixed-point

of a family of T_n, n = 0,1, 2…, as n → ∞. Moreover,

is the unique solution in F to the variational inequality

(3.16)

Proof. We first show that the sequence {x_n} defined by (1.8) is bounded. In fact, take p ∈ F, noting that T_n(p) = {p}, we have

(3.17)

It follows from induction that

(3.18)

so are {y_n} and {f(x_n)}. Thus, we have that

(3.19)

Next, we show that

(3.20)

Let λ_n = β_n/(1 − α_n) and z_n = λ_nf(x_n)+(1 − λ_n)y_n, then

(3.21)

Therefore, we have for some appropriate constant M > 0 that the following inequality:

(3.22)

holds. Thus, limsup _n→∞(∥z_n+1 − z_n∥−∥x_n+1 − x_n∥) ≤ lim _n→∞(|λ_n+1 − λ_n | ∥f(x_n+1) − f(x_n)∥+(λ_n + λ_n+1)M) = 0. By Lemma 2.4, we obtain

(3.23)

Therefore, we have

(3.24)

Using (3.20) and {x_n} satisfying Condition (A^′), we can use the same argumentation as Theorem 3.1 proves that

and

(3.25)

Finally, we show that

as n → ∞. In fact, from iteration (1.8) and Lemma 2.3, we have

(3.26)

From (ii) and (3.25), it then follows that

(3.27)

Apply Lemma 2.2 to conclude that

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Common Fixed-Point Problem for a Family Multivalued Mapping in Banach Space

Abstract

1. Introduction

2. Preliminaries

3. Main Results

References

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Common Fixed-Point Problem for a Family Multivalued Mapping in Banach Space

Abstract

1. Introduction

2. Preliminaries

3. Main Results

References

References

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