Let E be a reflexive Banach space having a weakly sequentially continuous duality mapping J_φ with gauge function φ, C a nonempty closed convex subset of E, and T : C → 𝒦(E) a multivalued nonself-mapping such that P_T is nonexpansive, where P_T(x) = {u_x ∈ Tx : ∥x − u_x∥ = d(x, Tx)}. Let f : C → C be a contraction with constant k. Suppose that, for each v ∈ C and t ∈ (0,1), the contraction defined by S_tx = tP_Tx + (1 − t)v has a fixed point x_t ∈ C. Let {α_n}, {β_n}, and {γ_n} be three sequences in (0,1) satisfying approximate conditions. Then, for arbitrary x₀ ∈ C, the sequence {x_n} generated by x_n ∈ α_nf(x_n−1) + β_nx_n−1 + γ_nP_T(x_n) for all n ≥ 1 converges strongly to a fixed point of T.

1. Introduction

Let E be a Banach space and C a nonempty closed subset of E. We will denote by ℱ(E) the family of nonempty closed subsets of E, by 𝒞ℬ(E) the family of nonempty closed bounded subsets of E, by 𝒦(E) the family of nonempty compact subsets of E, and by 𝒦𝒞(E) the family of nonempty compact convex subsets of E. Let H(·, ·) be the Hausdorff distance on 𝒞ℬ(E); that is,

()

for all A, B ∈ 𝒞ℬ(E), where d(a, B) = inf {∥a − b∥:b ∈ B} is the distance from the point a to the subset B. Recall that a mapping f : C → C is a contraction on C if there exists a constant k ∈ (0,1) such that ∥f(x) − f(y)∥≤k∥x − y∥, x, y ∈ C.

A multivalued mapping T : C → ℱ(E) is said to be a contraction if there exists a constant k ∈ [0,1) such that

()

If (2) is valid when k = 1, the T is called nonexpansive. A point x is a fixed point for a multivalued mapping T if x ∈ Tx. Banach′s contraction principle was extended to a multivalued contraction by Nadler [1] in 1969. The set of fixed points of T is denoted by F(T).

Given a contraction f with constant k and t ∈ (0,1), we can define a contraction G_t : C → 𝒦(C) by

()

Then G_t is a multivalued, and hence it has a (nonunique, in general) fixed point

(see [1]); that is,

()

If T is single valued, we have

()

which was studied by Moudafi [2] (see also Xu [3]). As a special case of (5),

()

has been considered by Browder [4], Halpern [5], Jung and Kim [6, 7], Kim and Takahashi [8], Reich [9], Singh and Watson [10], Takahashi and Kim [11], Xu [12], and Xu and Yin [13] in a Hilbert space and Banach spaces.

In 2007, Jung [14] established the strong convergence of {x_t} defined by x_t ∈ tTx_t + (1 − t)u, u ∈ C for the multivalued nonexpansive nonself-mapping T in a reflexive Banach space having a uniformly Gâteaux differentiable norm under the assumption Ty = {y}.

In order to give a partial answer to Jung′s open question [14] Can the assumption Ty = {y} be omitted?, in 2008, Shahzad and Zegeye [15] considered a class of multivalued mapping under some mild conditions as follows.

Let C be a closed convex subset of a Banach space E. Let T : C → 𝒦(E) be a multivalued nonself-mapping and

()

Then P_T : C → 𝒦(E) is multivalued, and P_Tx is nonempty and compact for every x ∈ C. Instead of

()

we consider, for t ∈ (0,1), that

()

It is clear that S_tx⊆G_tx, and if P_T is nonexpansive and T is weakly inward, then S_t is weakly inward contraction. Theorem 1 of Lim [16] guarantees that S_t has a fixed point, point x_t; that is,

()

If T is single valued, then (10) is reduced to (6).

Shahzad and Zegeye [15] also gave the strong convergence result of {x_t} defined by (10) in a reflexive Banach space having a uniformly Gâteaux differentiable norm, which unified, extended, and complemented several known results including those of Jung [14], Jung and Kim [6, 7], Kim and Jung [17], López Acedo and Xu [18], Sahu [19], and Xu and Yin [13].

In 2009, motivated by the results of Rafiq [20] and Yao et al. [21], Ceng and Yao [22] considered the following iterative scheme.

Theorem CY (see [22], Theorem 3.1.)Let E be a uniformly convex Banach space E having a uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of E, and T : C → 𝒦(E) a multivalued nonself-mapping such that P_T is nonexpansive. Suppose that C is a nonexpansive retract of E and that for each v ∈ C and t ∈ (0,1) the contraction S_t defined by S_tx = tP_Tx + (1 − t)v has a fixed point x_t ∈ C. Let {α_n}, {β_n}, and {γ_n} be three sequences in (0,1) satisfying the following conditions:

(i)
α_n + β_n + γ_n = 1;
(ii)
lim _n→∞α_n = 0 and lim _n→∞(β_n/α_n) = 0.

For arbitrary initial value x₀ ∈ C and a fixed element u ∈ C, let the sequence {x_n} be generated by

()

Then {x_n} converges strongly to a fixed point of T.

Theorem CY also improves, develops, and complements the corresponding results in Jung [14], Jung and Kim [6, 7], Kim and Jung [17], López Acedo and Xu [18], Shahzad and Zegeye [15], and Xu and Yin [13] to the iterative scheme (11). For convergence of related iterative schemes for several nonlinear mappings, we can refer to [23–26] and the references therein.

In this paper, inspired and motivated by the above-mentioned results, we consider a viscosity iterative method for a multivalued nonself-mapping in a reflexive Banach space having a weakly sequentially continuous duality mapping and establish the strong convergence of the sequence generated by the proposed iterative method. Our results improve and develop the corresponding results of Ceng and Yao [22], as well as some known results in the earlier and recent literature, including those of Jung [14], Jung and Kim [6, 7], Kim and Jung [17], López Acedo and Xu [18], Sahu [19], Shahzad and Zegeye [15], Xu [12], and Xu and Yin [13], to the viscosity iterative scheme in different Banach space.

2. Preliminaries

Let E be a real Banach space with norm ∥·∥, and let E^* be its dual. The value of x^* ∈ E^* at x ∈ E will be denoted by 〈x, x^*〉.

A Banach space E is called uniformly convex if δ(ε) > 0 for every ε > 0, where the modulus δ(ε) of convexity of E is defined by

()

for every ε with 0 ≤ ε ≤ 2. It is well known that if E is uniformly convex, then E is reflexive and strictly convex (cf. [27]).

By a gauge function we mean a continuous strictly increasing function φ defined on ℝ⁺ : = [0, ∞) such that φ(0) = 0 and lim _r→∞φ(r) = ∞. The mapping

defined by

()

is called the duality mapping with gauge function φ. In particular, the duality mapping with gauge function φ(t) = t denoted by J is referred to as the normalized duality mapping. It is known that a Banach space E is smooth if and only if the normalized duality mapping J is single valued. The following property of duality mapping is also well known:

()

where ℝ is the set of all real numbers; in particular, J(−x) = −J(x) for all x ∈ E ([28]).

We say that a Banach space E has a weakly sequentially continuous duality mapping if there exists a gauge function φ such that the duality mapping J_φ is single valued and continuous from the weak topology to the weak^* topology; that is, for any {x_n} ∈ E with

. For example, every l^p space (1 < p < ∞) has a weakly continuous duality mapping with gauge function φ(t) = t^p−1. It is well known that if E is a Banach space having a weakly sequentially continuous duality mapping J_φ with gauge function φ, then E has the opial condition [29]; this is, whenever a sequence {x_n} in E converges weakly to x ∈ E, then

()

A mapping T : C → 𝒞ℬ(E) is *-nonexpansive [30] if, for all x, y ∈ C and u_x ∈ Tx with ∥x − u_x∥ = inf {∥x − z∥:z ∈ Tx}, there exists u_y ∈ Ty with ∥y − u_y∥ = inf {∥y − w∥:w ∈ Ty} such that

()

It is known that *-nonexpansiveness is different from nonexpansiveness for multivalued mappings. There are some *-nonexpansiveness multivalued mappings which are not nonexpansive and some nonexpansive multivalued mappings which are not *-nonexpansive [31].

We introduce some terminology for boundary conditions for nonself-mappings. The inward set of C at x is defined by

()

Let

with

()

for any x ∈ C. Note that, for a convex set C, we have

, the closure of I_C(x). A multivalued mapping T : C → ℱ(E) is said to satisfy the weak inwardness condition if

for all x ∈ C.

Finally, the following lemma was given by Xu [32] (also see Xu [33]).

Lemma 1. If C is a closed bounded convex subset of a uniformly convex Banach space E and T : C → 𝒦(E) is a nonexpansive mapping satisfying the weak inwardness condition, then T has a fixed point.

3. Main Results

Now, we establish strong convergence of a viscosity iterative scheme for a multivalued nonself-mapping.

Theorem 2. Let E be a reflexive Banach space having a weakly sequentially continuous duality mapping J_φ with gauge function φ. Let C be a nonempty closed convex subset of E and T : C → 𝒦(E) a multivalued nonself-mapping such that F(T) ≠ ∅ and P_T is nonexpansive. Let f : C → C be a contraction with constant k. Suppose that for each v ∈ C and t ∈ (0,1), the contraction S_t defined by S_tx = tP_Tx + (1 − t)v has a fixed point x_t ∈ C. Let {α_n}, {β_n}, and {γ_n} be three sequences in (0,1) satisfying the following conditions:

(i)
α_n + β_n + γ_n = 1;
(ii)
lim _n→∞α_n = 0 and lim _n→∞(β_n/α_n) = 0.

For arbitrary initial value x₀ ∈ C, let the sequence {x_n} be defined by

()

Then {x_n} converges strongly to a fixed point of T.

Proof. First, observe that, for each n ≥ 1,

()

From x_n−1, f(x_n−1) ∈ C, it follows that (α_n/(1 − γ_n))f(x_n−1)+(β_n/(1 − γ_n))x_n−1 ∈ C. Also, notice that, for each v ∈ C and t ∈ (0,1), the contraction S_t defined by S_tx = tP_Tx + (1 − t)v has a fixed point x_t ∈ C. Thus, for (α_n/(1 − γ_n))f(x_n−1)+(β_n/(1 − γ_n))x_n−1 ∈ C and γ_n ∈ (0,1), there exists x_n ∈ C such that

()

This shows that the sequence {x_n} can be defined well via the following:

()

Therefore, for any n ≥ 1, we can find some z_n ∈ P_T(x_n) such that

()

Next, we divide the proof into several steps.

Step 1. We show that {x_n} is bounded. Indeed, notice that P_T(y) = {y} whenever y is a fixed point of T. Let p ∈ F(T). Then p ∈ P_T(p) and we have

()

It follows that

()

and so

()

By induction, we have

()

Hence {x_n} is bounded and so are {z_n} and {f(x_n)}.

Step 2. We show that lim _n→∞∥x_n − z_n∥ = 0. In fact, since x_n = α_nf(x_n−1) + β_nx_n−1 + γ_nz_n for some z_n ∈ P_T(x_n), by conditions (i) and (ii), we have

()

Step 3. We show that there exists p ∈ P_T(p) ⊂ Tp. In fact, since {x_n} and {z_n} are bounded and E is reflexive, there exists a subsequence

of {x_n} such that

. For this p, by compactness of P_T(p), we can find w_n ∈ P_T(p), ∀n ≥ 1, such that

()

Now suppose that

and x_n⇀p. The sequence {w_n} has a convergent subsequence, which is denoted again by {w_n} with w_n → w ∈ P_T(p). Assume that w ≠ p. Since a Banach space having the weakly sequentially continuous duality mapping satisfies the opial condition [29], by Step 2, we have

()

which is a contradiction. Hence we have w = p, and so p = w ∈ P_T(p) ⊂ Tp.

Step 4. We show that lim _n→∞〈z_n − f(x_n−1), J_φ(x_n − y)〉≤0 for y ∈ F(T). Indeed, for y ∈ F(T), by (24), we have

()

So,

()

Thus it follows that

()

Hence, from condition (ii), we conclude that

()

Step 5. We show that lim _n→∞∥x_n − p∥ = 0, where p is defined as in Step 3. Indeed, for y ∈ F(T),

()

Interchanging p and y in (35), we obtain

()

and so

()

Using the fact that J_φ is weakly sequentially continuous, Step 4, and condition (ii), we have

()

This implies that x_n → p as n → ∞. In fact, if lim _n→∞∥x_n − p∥ = η ≠ 0, then lim _n→∞φ(∥x_n − p∥) ≠ 0, and by (38) lim _n→∞(∥x_n − p∥−k∥x_n−1 − p∥) = 0. This means that lim _n→∞∥x_n − p∥ = η = kη, which is a contradiction. Thus we proved that there exists a subsequence

of {x_n} which converges strongly to a fixed point p of T.

Step 6. We show that the entire sequence {x_n} converges strongly to p ∈ F(T). Suppose that there exists another subsequence of {x_n} such that as j → ∞. Since as j → ∞, it follows that d(q, P_T(q)) = 0 and so q ∈ P_T(q) ⊂ T(q); that is, q ∈ F(T). Notice that P_T(q) = {q}. Since {x_n} is bounded and the duality mapping J_φ is single valued and weakly sequentially continuous from E to E^*, we have

()

Thus, from Steps 2 and 4, it follows that

()

By the same argument, we also have

()

Therefore, from (40) and (41), we obtain

()

and so (1 − k)∥p − q∥φ(∥p − q∥) ≤ 0. Thus p = q. This completes the proof.

Remark 3. (1) In Theorem 2, if f(x) = u ∈ C, x ∈ E, is a constant mapping, then the iterative scheme (19) is reduced to the iterative scheme (11) in Theorem CY of Ceng and Yao [22] in the Introduction section. Therefore Theorem 2 improves Theorem CY to the viscosity iterative scheme in different Banach space.

(2) In Theorem 2, we remove the assumption that C is a nonexpansive retract of E in Theorem CY.

(3) In Theorem 2, if β_n = 0 for n ≥ 0, then the iterative scheme (19) becomes the following scheme:

()

which is a viscosity iterative scheme for those in Shahzad and Zegeye [15]. Therefore Theorem 2 develops Theorem 3.1 of Shahzad and Zegeye [15], as well as Theorem 1 of Jung [14], to the viscosity iterative method in different Banach space.

(4) Theorem 2 also improves and complements the corresponding results of Kim and Jung [17] and Sahu [19] as well as Jung and Kim [6, 7], López Acedo and Xu [18], and Xu and Yin [13].

By definition of the Hausdorff metric, we obtain that if T is *-nonexpansive, then P_T is nonexpansive. Hence, as a direct consequence of Theorem 2, we have the following result.

Corollary 4. Let E be a reflexive Banach space having a weakly sequentially continuous duality mapping J_φ with gauge function φ. Let C be a nonempty closed convex subset of E and T : C → 𝒦(E) a multivalued *-nonexpansive nonself-mapping such that F(T) ≠ ∅. Let f : C → C be a contraction with constant k. Suppose that, for each v ∈ C and t ∈ (0,1), the contraction S_t defined by S_tx = tP_Tx + (1 − t)v has a fixed point x_t ∈ C. Let {α_n}, {β_n}, and {γ_n} be three sequences in (0,1) satisfying the following conditions:

(i)
α_n + β_n + γ_n = 1;
(ii)
lim _n→∞α_n = 0 and lim _n→∞(β_n/α_n) = 0.

For arbitrary initial value x₀ ∈ C, let the sequence {x_n} be generated by (19). Then {x_n} converges strongly to a fixed point of T.

It is well known that every nonempty closed convex subset C of a strictly convex and reflexive Banach space E is Chebyshev; that is, for any x ∈ E, there is a unique element u ∈ C such that ∥x − u∥ = inf {∥x − v∥:v ∈ C}. Thus, we have the following corollary.

Corollary 5. Let E be a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping J_φ with gauge function φ. Let C be a nonempty closed convex subset of E and T : C → 𝒦𝒞(E) a multivalued nonself-mapping such that F(T) ≠ ∅ and P_T is nonexpansive. Let f : C → C be a contraction with constant k. Suppose that, for each v ∈ C and t ∈ (0,1), the contraction S_t defined by S_tx = tP_Tx + (1 − t)v has a fixed point x_t ∈ C. Let {α_n}, {β_n}, and {γ_n} be three sequences in (0,1) satisfying the following conditions:

(i)
α_n + β_n + γ_n = 1;
(ii)
lim _n→∞α_n = 0 and lim _n→∞(β_n/α_n) = 0.

For arbitrary initial value x₀ ∈ C, let the sequence {x_n} be generated by (19). Then {x_n} converges strongly to a fixed point of T.

Proof. In this case, Tx is Chebyshev for each x ∈ C. So P_T is a selector of T and P_T is single valued. Thus the result follows from Theorem 2.

Corollary 6. Let E be a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping J_φ with gauge function φ. Let C be a nonempty closed convex subset of E and T : C → 𝒦𝒞(E) a multivalued *-nonexpansive nonself-mapping such that F(T) ≠ ∅. Let f : C → C be a contraction with constant k. Suppose that, for each v ∈ C and t ∈ (0,1), the contraction S_t defined by S_tx = tP_Tx + (1 − t)v has a fixed point x_t ∈ C. Let {α_n}, {β_n}, and {γ_n} be three sequences in (0,1) satisfying the following conditions:

(i)
α_n + β_n + γ_n = 1;
(ii)
lim _n→∞α_n = 0 and lim _n→∞(β_n/α_n) = 0.

For arbitrary initial value x₀ ∈ C, let the sequence {x_n} be generated by (19). Then {x_n} converges strongly to a fixed point of T.

Corollary 7. Let E be a uniformly convex Banach space having a weakly sequentially continuous duality mapping J_φ with gauge function φ. Let C be a nonempty closed convex subset of E and T : C → 𝒦(E) a multivalued nonself-mapping satisfying the weak inwardness condition such that P_T is nonexpansive. Let f : C → C be a contraction with constant k. Let {α_n}, {β_n}, and {γ_n} be three sequences in (0,1) satisfying the following conditions:

(i)
α_n + β_n + γ_n = 1;
(ii)
lim _n→∞α_n = 0 and lim _n→∞(β_n/α_n) = 0.

For arbitrary initial value x₀ ∈ C, let the sequence {x_n} be generated by (19). Then {x_n} converges strongly to a fixed point of T.

Proof. Define, for each v ∈ C and t ∈ (0,1), the contraction S_t : C → 𝒦(E) by

()

As it is easily seen that S_t also satisfies the weak inwardness condition:

for all x ∈ C, it follows from Lemma 1 that S_t has a fixed point denoted by x_t. Thus the result follows from Theorem 2.

Remark 8. (1) As in [31], Shahzad and Zegeye [15] gave the following example of a multivalued T such that P_T is nonexpansive. Let C = [0, ∞), and let T be defined by Tx = [x, 2x] for x ∈ C. Then P_Tx = {x} for x ∈ C. Also T is *-nonexpansive but not nonexpansive (see [31]).

(2) Corollaries 4–7 develop Corollaries 3.3–3.6 of Ceng and Yao [22] to the viscosity iterative method in different Banach spaces.

(3) By replacing the iterative scheme (11) in Theorem CY with the iterative scheme (19) in Theorem 2 and using the same proof lines as Theorem CY together with our method, we can also establish the viscosity iteration version of Theorem CY.

Acknowledgments

The author thanks the anonymous referees for their valuable comments and suggests, which improved the presentation of this paper and for providing some recent related papers. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012000895).

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Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces

Abstract

1. Introduction

2. Preliminaries

3. Main Results

Acknowledgments

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Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces

Abstract

1. Introduction

2. Preliminaries

3. Main Results

Acknowledgments

References

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