1. Introduction

Let D be a nonempty and convex subset of a Banach spaces E. The set D is called proximinal if for each x∈E, there exists an element y∈D such that ||x-y|| = d(x,D), where d(x,D) = inf{||x-z||: z∈D}. Let CB(D), CCB(D), K(D) and P(D) denote the families of nonempty closed bounded subsets, nonempty closed convex bounded subsets, nonempty compact subsets, and nonempty proximinal bounded subsets of D, respectively. The Hausdorff metric on CB(D) is defined by

for A,B∈CB(D). A single-valued map T: D→D is called nonexpansive if ||T_x-T_y||≤||x-y|| for all x,y∈D. A multi-valued mapping T: D→CB(D) is called nonexpansive if H(T_x,T_y) ≤||x-y|| for all x,y∈D. An element p∈D is called a fixed point of T: D→D (respectively, T: D→CB(D)) if p = T_p (respectively, p∈T_p). The set of fixed points of T is denoted by F(T). The mapping T: D→CB(D) is called quasi-nonexpansive ^[1] if F(T)≠∅ and H(T_x,T_p) ≤||x-p|| for all x∈D and all p∈F(T). It is clear that every nonexpansive multi-valued mapping T with F(T)≠∅ ; is quasi-nonexpansive. But there exist quasi-nonexpansive mappings that are not nonexpansive (see ^[2]). It is known that if T is a quasi-nonexpansive multi-valued mapping, then F(T) is closed.

Throughout this paper, we denote the weak convergence and the strong convergence by – and→, respectively. The mapping T: D→CB(D) is called hemicompact if, for any sequence {x_n} in D such that d(x_n, Tx_n) →0 as n→1, there exists a subsequence {x_nk} of {x_n} such that x_nk→p∈D. We note that if D is compact, then every multi-valued mapping T: D→CB(D) is hemicompact.

A Banach space E is said to satisfy Opial's condition ^[3] if for each x∈E and a sequence {x_n} in E such that x_n→x, the following condition holds for all x≠y :

The mapping T: D→CB(D) is called demi-closed if for every sequence {x_n}⊂D and any y_n∈Tx_n such that x_n – x and y_n →y, we have x∈D and y∈Tx.

Remark 1.1 (^[4]). If the space E satisfies Opial's condition, then I-T is demi-closed at 0, where T: D→K(D) is a nonexpansive multi-valued mapping.

For a single-valued case, in 1953, Mann ^[5] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a real Hilbert space H:

(1.1)

where the initial point x₁ is taken in D arbitrarily and {α_n} is a sequence in (0,1).

However, we note that Mann's iteration process (1.1) has only weak convergence, in general; for instance, see ^{[6, 7, 8]}.

Since 1953, Mann's iteration has extensively been studied by many authors (see, for examples, [9-18]^[9]). However, the studying of multivalued nonexpansive mappings is harder than that of single-valued nonexpansive mappings in both Hilbert spaces and Banach spaces.

The result of fixed points for multi-valued contractions and nonexpansive mappings by using the Hausdorff metric was initiated by Markin ^[19]. Later, different iterative processes have been used to approximate fixed points of multi-valued nonexpansive mappings (see also [1,20-26]).

In 2009, Song and Wang ^[26] proved strong and weak convergence theorems for Mann's iteration of a multi-valued nonexpansive mapping T in a Banach space. They studied strong convergence of the modified Mann iteration which is independent of the implicit anchor-like continuous path .

Let D be a nonempty and closed subset of a Banach space E, {β_n}⊂[0,1]^{[, 1]}, {α_n}⊂[0,1]^{[, 1]} and {γ_n}⊂(0,+∞) such that .

(A) Choose x₀ ∈D,

where y_n∈Tx_n such that ||y_n+1-y_n||≤ H(Tx_n+1,Tx_n) +γ_n.

(B) For fixed u 2 D, the sequence of modified Mann iteration is defined by x₀ ∈D,

where y_n∈Tx_n such that ||y_n+1-y_n||≤ H(Tx_n+1,Tx_n) +γ_n.

Very recently, Shahzad and Zegeye ^[2] obtained the strong convergence theorems for a quasi-nonexpansive multi-valued mapping. They relaxed the compactness of domain of T and constructed an iterative scheme which removes the restriction of T namely T_p = {p} for any p∈F(T). The results provided an affirmative answer to some questions raised in ^[21]. In fact, they introduced iterations as follows:

Let D be a nonempty and convex subset of a Banach space E, let T: D→CB(D) and let {α_n}, {α’_n}⊂[0,1]^{[, 1]}.

(C) The sequence of Ishikawa's iteration is defined by x₀∈D,

where and .

(D) Let T: D→P(D) and P_Tx ={y∈Tx:||x-y|| = d(x,Tx)}, where PT is the best approximation operator. The sequence of Ishikawa's iteration ^[30] is defined by x₀ ∈D,

where and .

It is remarked that Hussain and Khan ^[27], in 2003, employed the best approximation operator PT to study fixed points of *-nonexpansive multi-valued mapping T and strong convergence of its iterates to a fixed point of T defined on a closed and convex subset of a real Hilbert space.

Let D be a nonempty, closed and convex subset of a Banach space E. Let be a family of multi-valued mappings from D into 2^D and let P_Tnx = {y_n∈T_nx ||x-y_n||=d(x,T_nx)}, n≥1. Let {α_n} be a sequence in (0,1).

(E) The sequence of the modified Ishikawa's iteration is defined by x₁ ∈ D and

(1.2)

In this paper, we modify Mann's iteration by using the best approximation operator P_Tn, n≥1 to find common fixed points of a countable family of nonexpansive multi-valued mappings . Then we prove weak and strong convergence theorems for a countable family of multi-valued mappings in Banach spaces. Finally, we apply our main result to the problem of finding a common fixed point of a family of nonexpansive multi-valued mappings.

2. Preliminaries

In this section, we give some characterizations and properties of the metric projection in a real Hilbert space.

Let H be a real Hilbert space with inner product <•,•> and norm ||•||. Let D be a closed and convex subset of H. If, for any point x 2 H, there exists a unique nearest point in D, denoted by P_Dx, such that

then PD is called the metric projection of H onto D. We know that PD is a nonexpansive mapping of H onto D.

Lemma 2.1 (^[28]). Let D be a closed and convex subset of a real Hilbert space H and PD be the metric projection from H onto D. Then, for any x∈H and z∈D, z = PDx if and only if the following holds:

Using the proof line in Lemma 3.1.3 of ^[28], we obtain the following result.

Proposition 2.2. Let D be a closed and convex subset of a real Hilbert space H. Let T: D→CCB(D) be a multi-valued mapping and PT the best approximation operator. Then, for any x∈D, z∈P_Tx if and only if the following holds:

Lemma 2.3 (^[28]). Let H be a real Hilbert space. Then the following equations hold:

(1)

(2) for all t∈[0,1]^{[, 1]} and x,y∈H.

We next show that P_T is nonexpansive under some suitable conditions imposed on T.

Remark 2.4. Let D be a closed and convex subset of a real Hilbert space H. Let T: D→CCB(D) be a multi-valued mapping. If Tx = Ty, x,y∈D, then P_T is a nonexpansive multi-valued mapping.

In fact, let x,y∈D. For each a∈P_Tx, we have

(2.1)

From Proposition 2.2, we have

It follows that

(2.2)

This implies that

(2.3)

From (2.1) and (2.3), we obtain

for every a∈P_Tx. Hence . Similarly, we can show that . Therefore .

It is clear that if a nonexpansive multi-valued mapping T satisfies the condition that Tx = Ty, x,y∈D, then PT is nonexpansive. The following example shows that if T is a nonexpansive multi-valued mapping satisfying the property that Tx = Ty, x,y∈D, then Tx is not a singleton for all x∈D.

3. Strong and Weak Convergence of the Modified Ishikawa Iteration in Banach Spaces

In this section, we first prove a strong convergence theorem for a countable family of multi-valued mappings under the SC-condition and Condition (A) and then prove a weak convergence theorem under the SC-condition in Banach spaces.

Theorem 3.1. Let D be a closed and convex subset of a uniformly convex Banach space E which satisfies Opial's condition. Let {T_n} and τ be two families of multivalued mappings from D into P(D) with . Let {α_n} be a sequence in (0,1) such that 0 . Let {x_n} be generated by (1.2). Assume that

(A1) for each ;

(A2) I-T is demi-closed at 0 for all T∈τ.

If {T_n} satisfies the SC-condition, then {x_n} converges weakly to an element in F(τ).

Proof. Since

there exists z_n∈PT_nx_n, p∈F(τ) and n∈. We note that PT_np = {p} for all. It follows from (A1) that

(3.1)

and Also

(3.2)

for every p∈F(T). Then <||x_n-p||> is a decreasing sequence and hence lim_n→∞||x_n- p|| exists for every p∈F(T). For p∈F(T), since <x_n> and <z_n> are bounded by Lemma 2.9, there exists a continuous, strictly increasing and convex function g: [0,1)→[0,1) with g(0) = 0 such that

It follows that

Since lim_n→∞||x_n- p|| exists and ,

By the properties of g, we can conclude that

Since <T_n> satisfies the SC-condition, there exists C_n∈T_xn such that

(3.3)

for every T∈τ. Since <x_n> is bounded, there exists a subsequence <x_nk> of <x_n> converges weakly to some q₁∈D. It follows from (A2) and (3.3) that q₁∈Tq₁ for every T∈τ. Next, we show that <x_n> converges weakly to q1, take another subsequence <x_mk> of <x_n> converging weakly to some q₂∈D. Again, as above we can conclude that q₂∈Tq₂ for every T∈τ. Finally, we show that q₁ = q₂. Assume q₁≠ q₂. Then by Opial's condition of E, we have

which is a contradiction. Therefore q₁ = q₂. This shows that <x_n> converges weakly to a fixed point of τ for every T ∈τ. This completes the proof.

Corollary 3.2. Let D be a closed and convex subset of a uniformly convex Banach space E which satisfies Opial's condition. Let {T_n} and τ be two families of nonexpansive multivalued mappings from D into K(D) with. Let {α_n} be a sequence in (0, 1) such that . Let {x_n} be generated by (1.2). Assume that for each n∈,

x∈D, p∈F(τ). If {T_n} satisfies the SC-condition, then {x_n} converges weakly to an element in F(τ).

Theorem 3.3. Let D be a closed and convex subset of a uniformly convex Banach space E which satisfies Opial's condition. Let {T_n} and τ be two families of multivalued mappings from D into P(D) with . Let {α_n} be a sequence in (0,1) such that 0 . Let {x_n} be generated by (1.2). Assume that

(B1) for each ;

(B2) the best approximation operator P_T is nonexpansive for every T∈τ;

(B3) F(τ) is closed.

If {T_n} satisfies the SC-condition and Condition (A), then fxng converges strongly to an element in F(τ).

Proof. It follows from the proof of Theorem 3.1 that exists for every p∈F(τ) and where . Since fTng satisfies the SC- condition, there exists c_n∈Tx_n such that

for every T∈τ. This implies that

for every T∈τ. Since that {T_n} satisfies Condition (A), we have .

It follows from (B3), there is subsequence of {x_n} and a sequence {p_k}⊂F(τ) such that

(3.4)

for all k. From (3.1), we obtain

for all p∈F(τ). This implies that

(3.5)

Next, we show that {p_k} is a Cauchy sequence in D. From (3.4) and (3.5), we have

(3.6)

This implies that {p_k} is a Cauchy sequence in D and thus converges to q∈D. Since PT is nonexpansive for every T ∈τ,

(3.7)

for every T ∈τ. It follows that d(q,Tq) = 0 for every T ∈τ and thus q∈F(τ). It implies by (3.4) that {x_nk} converges strongly to q. Since exists, it follows that {x_n} converges strongly to q. This completes the proof.

We know that if T is a quasi nonexpansive multivalued mapping, then F(T) is closed. So we have the following result:

Corollary 3.4. Let D be a closed and convex subset of a uniformly convex Banach space E. Let {T_n} and τ be two families of nonexpansive multivalued mappings from D into P(D) with . Let {α_n} be a sequence in (0,1) such that . Let {x_n} be generated by (1.2). Assume that for each and the best approximation operator P_T is nonexpansive for every T ∈τ.

If {T_n} satisfies the SC-condition and Condition (A), then {x_n} converges strongly to an element in F(τ).

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