Throughout the article, ℕ denotes the set of positive integers. Let E be a real Banach space. A subset K is called proximinal if for each x ∈ E, there exists an element k ∈ K such that

d ( x , k ) = inf { | | x - y | | : y ∈ K } = d ( x , K )

It is known that a weakly compact convex subsets of a Banach space and closed convex subsets of a uniformly convex Banach space are proximinal. We shall denote the family of nonempty bounded proximinal subsets of K by P(K). Consistent with 1, let CB(K) be the class of all nonempty bounded and closed subsets of K. Let H be a Hausdorff metric induced by the metric d of E, that is

H ( A , B ) = max { sup  x ∈ A d ( x , B ) , sup  y ∈ B d ( y , A ) }

for every A, B ∈ CB(E). A multivalued mapping T : K → P (K) is said to be a contraction if there exists a constant k ∈ [0, 1) such that for any x, y ∈ K,

H ( T x , T y ) ≤ k | | x - y | | ,

and T is said to be nonexpansive if

H ( T x , T y ) ≤ | | x - y | |

for all x, y ∈ K. A point x ∈ K is called a fixed point of T if x ∈ Tx.

The study of fixed points for multivalued contractions and nonexpansive mappings using the Hausdorff metric was initiated by Markin 2 (see also 1). Later, an interesting and rich fixed point theory for such maps was developed which has applications in control theory, convex optimization, differential inclusion, and economics (see, 3 and references cited therein). Moreover, the existence of fixed points for multivalued nonexpansive mappings in uniformly convex Banach spaces was proved by Lim 4.

The theory of multivalued nonexpansive mappings is harder than the corresponding theory of single valued nonexpansive mappings. Different iterative processes have been used to approximate the fixed points of multivalued nonexpansive mappings. Among these iterative processes, Sastry and Babu 5 considered the following.

Let K be a nonempty convex subset of E, T : K → P(K) a multivalued mapping with p ∈ Tp.

(i) The sequences of Mann iterates is defined by x₁ ∈ K,

x n + 1 = ( 1 - a n ) x n + a n y n ,

where y_n ∈ Tx_n is such that ||y_n - p|| = d(p, Tx_n), and {a_n} is a sequence of numbers in (0, 1) satisfying lim n → ∞ a n = 0 and  ∑ a n = ∞ .

(ii) The sequence of Ishikawa iterates is defined by x₁ ∈ K,

y n = ( 1 - b n ) x n + b n z n , x n + 1 = ( 1 - a n ) x n + a n u n ,

where z_n ∈ Tx_n, u_n ∈ Ty_n are such that ||z_n - p|| = d(p, Tx_n) and ||u_n - p|| = d(p, Ty_n), and {a_n}, {b_n} are real sequences of numbers with 0 ≤ a_n, b_n < 1 satisfying lim n → ∞ b n = 0 and ∑a_nb_n = ∞.

Panyanak 6 generalized the results proved by Sastry and Babu 5.

The following is a useful Lemma due to Nadler 1.

Lemma 1. Let A, B ∈ CB(E) and a ∈ A. If η > 0, then there exists b ∈ B such that d(a, b) ≤ H(A, B) + η.

Based on the above Lemma, Song and Wang 7 modified the iteration scheme due to Panyanak 6 and improved the results presented therein. Their scheme is given as follows:

Let K be a nonempty convex subset of E, a_n ∈ [0, 1], b_n ∈ [0, 1] and η_n ∈ (0, ∞) such that lim_{n → ∞}η_n = 0. Choose x₁ ∈ K and z₁ ∈ Tx₁. Let

y 1 = ( 1 - b 1 ) x 1 + b 1 z 1 .

Choose u₁ ∈ Ty₁ such that || z₁ - u₁ || ≤ H(Tx₁, Ty₁) + η₁ (see 18). Let

x 2 = ( 1 - a 1 ) x 1 + a 1 u 1 .

Choose z₂ ∈ Tx₂ such that || z₂ - u₁ || ≤ H(Tx₂, Ty₁) + η₂. Take

y 2 = ( 1 - b 2 ) x 2 + b 2 z 2 .

Choose u₂ ∈ Ty₂ such that || z₂ - u₂ || ≤ H(Tx₂, Ty₂) + η₂. Let

x 3 = ( 1 - a 2 ) x 2 + a 2 u 2 .

Inductively, we have

y n = ( 1 - b n ) x n + b n z n x n + 1 = ( 1 - a n ) x n + a n u n

where z_n ∈ Tx_n, u_n ∈ Ty_n are such that ||z_n - u_n|| ≤ H(Tx_n, Ty_n) + η_n and ||z_n+1 -u_n|| ≤ H(Tx_n+1, Ty_n) + η_n, and {a_n},{b_n} are real sequences of numbers with 0 ≤ a_n, b_n < 1 satisfying lim n → ∞ b n = 0 and ∑a_nb_n = ∞.

It is to be noted that Song and Wang 7 need the condition Tp = {p} in order to prove their Theorem 1. Actually, Panyanak 6 proved some results using Ishikawa type iteration process without this condition. Song and Wang 7 showed that without this condition his process was not well-defined. They reconstructed the process using the condition Tp = {p} which made it well-defined. Such a condition was also used by Jung 9.

Recently, Shahzad and Zegeye 10 remarked as follows:

"We note that the iteration scheme constructed by Song and Wang 7 involves the estimates which are not easy to be computed and the scheme is more time consuming. We also observe that Song and Wang 7 did not use the above estimates in their proofs and applied Lemma 2.1 (of 10) without showing x_n - p, y_n - p ∈ B_R(0). The assumption on T namely "Tp = {p} for any p ∈ F(T)" is quite strong.... Then we construct an iteration scheme which removes the restriction of T namely Tp = {p} for any p ∈ F(T)."

To do this, they defined P_T(x) = {y ∈ Tx : ||x - y|| = d(x, Tx)} for a multivalued mapping T : K → P(K). They also proved a couple of strong convergence results using Ishikawa type iteration process.

On the other hand, Agarwal et al. 11 introduced the following iteration scheme for single-valued mappings:

x 1 = x ∈ C , x n + 1 = ( 1 - α n ) T x n + α n T y n , y n = ( 1 - β n ) x n + β n T x n , n ∈ ℕ

where {α_n} and {β_n} are in (0, 1). This scheme is independent of both Mann and Ishikawa schemes. They proved that this scheme converges at a rate faster than both Picard iteration scheme x_n+1 = Tx_n and Mann iteration scheme for contractions. Following their method, it was observed in [12, Example 3.7] that this scheme also converges faster than Ishikawa iteration scheme.

In this paper, we first give a multivalued version of the iteration scheme (1.4) of Agarwal et al. 11 and then use the idea of removal of "Tp = {p} for any p ∈ F(T)" due to Shahzad and Zegeye 10 to approximate fixed points of a multivalued nonexpansive mapping T. We also use a result of Song and Cho 13 saying that set of fixed points of T is same as that of P_T , see Lemma 2 below. Moreover, we use the method of direct construction of Cauchy sequence as indicated by Song and Cho 13 (and opposed to 10) but also used by many other authors including 121415. Keeping above in mind, we define our iteration scheme as follows:

x 1 ∈ K , x n + 1 = ( 1 - λ ) v n + λ u n y n = ( 1 - η ) x n + η v n , n ∈ ℕ

where v_n ∈ P_T(x_n), u_n ∈ P_T(y_n) and 0 < λ, η < 1. We have used λ, η only for the sake of simplicity but α_n, β_n could be used equally well under suitable conditions. In this way, we approximate fixed points of a multivalued nonexpansive mapping by an iteration scheme which is independent of but faster than Ishikawa scheme. Thus our results improve corresponding results of Shahzad and Zegeye 10, Song and Cho 13 and the results generalized therein.

Now, we give the following definitions.

Definition 1. A Banach space E is said to satisfy Opial's condition 16 if for any sequence {x_n} in E, x_n ⇀ x implies that

lim sup n → ∞ | | x n - x | | < lim sup n → ∞ | | x n - y | |

for all y ∈ E with y ≠ x.

Examples of Banach spaces satisfying this condition are Hilbert spaces and all l^p spaces (1 < p < ∞). On the other hand, L^p[0, 2π] with 1 < p ≠ 2 fail to satisfy Opial's condition.

Definition 2. A multivalued mapping T : K → P(E) is called demiclosed at y ∈ K if for any sequence {x_n} in K weakly convergent to an element x and y_n ∈ Tx_n strongly convergent to y, we have y ∈ Tx.

The following is the multivalued version of condition (I) of Senter and Dotson 17.

Definition 3. A multivalued nonexpansive mapping T : K → CB(K) where K a subset of E, is said to satisfy condition (I) if there exists a nondecreasing function f : [0, ∞) → [0, ∞) with f(0) = 0, f(r) > 0 for all r ∈ (0, ∞) such that d(x, Tx) ≥ f(d(x, F(T)) for all x ∈ K.

The following very useful theorem is due to Song and Cho 13.

Lemma 2. 13 Let T : K → P (K) be a multivalued mapping and P_T(x) = {y ∈ Tx : ||x - y|| = d(x, Tx)}. Then the following are equivalent.

(1) x ∈ F(T);

(2) P_T (x) = {x};

(3) x ∈ F(P_T).

Moreover, F(T) = F(P_T).

Next, we state the following helpful lemma.

Lemma 3. 18 Let E be a uniformly convex Banach space and 0 < p ≤ t_n ≤ q < 1 for all n ∈ ℕ. Suppose that {x_n} and {y_n} are two sequences of E such that lim sup_{n→ ∞} ||x_n|| ≤ r, lim sup_{n→ ∞} ||y_n|| ≤ r and lim_{n→ ∞} ||t_nx_n + (1 - t_n)y_n|| = r hold for some r ≥ 0. Then lim_{n→ ∞} ||x_n - y_n|| = 0.

We start with the following couple of important lemmas.

Lemma 4. Let E be a normed space and K a nonempty closed convex subset of E. Let T : K → P (K) be a multivalued mapping such that F(T) ≠ ∅ and P_T is a nonexpansive mapping. Let {x_n} be the sequence as defined in (1.5). Then lim_{n→ ∞} ||x_n - p|| exists for all p ∈ F (T).

Proof. Let p ∈ F(T). Then p ∈P_T (p) = {p} by Lemma 2. It follows from (1.5) that

| | x n + 1 - p | | = | | ( 1 - λ ) v n + λ u n - p | | ≤ ( 1 - λ ) | | v n - p | | + λ | | u n - p | | ≤ ( 1 - λ ) H ( P T ( x n ) , P T ( p ) ) + λ H ( P T ( y n ) , P T ( p ) ) ≤ ( 1 - λ ) | | x n - p | | + λ | | y n - p | | .

But

| | y n - p | | = | | ( 1 - η ) x n + η v n - p | | ≤ ( 1 - η ) | | x n - p | | + η | | v n - p | | ≤ ( 1 - η ) | | x n - p | | + η H ( P T ( x n ) , P T ( p ) ) ≤ ( 1 - η ) | | x n - p | | + η | | x n - p | | = | | x n - p | | .

Thus (2.1) becomes

| | x n + 1 - p | | ≤ ( 1 - λ ) | | x n - p | | + λ | | x n - p | | = | | x n - p | | ,

and lim_{n→ ∞} ||x_n - p|| exists for each p ∈ F (T). □

Lemma 5. Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let T : K → P(K) be a multivalued mapping such that F(T) ≠ ∅ and P_T is a nonexpansive mapping. Let {x_n} be the sequence as defined in (1.5). Then lim n → ∞ d ( x n , T x n ) = 0 .

Proof. From Lemma 4, lim n → ∞ | | x n - p | | exists for each p ∈ F (T). We suppose that lim n → ∞ | | x n - p | | = c for some c ≥ 0.

Since lim sup_{n→ ∞} ||v_n - p|| ≤ lim sup_{n→ ∞} H (P_T (x_n), P_T (p)) ≤ lim sup_{n→ ∞} || x_n - p || = c,

so

lim sup n → ∞ | | v n - p | | ≤ c .

Similarly,

lim sup n → ∞ | | u n - p | | ≤ c .

Applying Lemma 3, we get

lim n → ∞ | | v n - u n | | = 0 .

Taking lim sup on both sides of (2.2), we obtain

lim sup n → ∞ | | y n - p | | ≤ c .

Also

| | x n + 1 - p | | = | | ( 1 - λ ) v n + λ u n - p | | = | | ( v n - p ) + λ ( u n - v n ) | | ≤ | | v n - p | | + | | v n - u n | |

implies that

c ≤ lim inf | | v n - p | | .

Combining (2.3) and (2.5), we have

lim n → ∞ | | v n - p | | = c .

Thus

| | v n - p | | ≤ | | v n - u n | | + | | u n - p | | ≤ | | v n - u n | | + H ( P T ( y n ) , P T ( p ) ) ≤ | | v n - u n | | + | | y n - p | |

gives

c ≤ lim inf | | y n - p | |

and, in turn, by (2.4), we have

lim n → ∞ | | y n - p | | = c .

Applying Lemma 3 once again,

lim n → ∞ | | x n - v n | | = 0 .

Since d(x_n, Tx_n) ≤ ||x_n - v_n||, we have

lim n → ∞ d ( x n , T x n ) = 0 .    □

Now we approximate fixed points of the mapping T through weak convergence of the sequence {x_n} defined in (1.5).

Theorem 1. Let E be a uniformly convex Banach space satisfying Opial's condition and K a nonempty closed convex subset of E. Let T : K → P(K) be a multivalued mapping such that F(T) ≠ ∅ and P_T is a nonexpansive mapping. Let {x_n} be the sequence as defined in (1.5). Let I - P_T be demiclosed with respect to zero, then {x_n} converges weakly to a fixed point of T.

Proof. Let p ∈ F(T) = F(P_T). From the proof of Lemma 4, lim n → ∞ | | x n - p | | exists. Now we prove that {x_n} has a unique weak subsequential limit in F(T). To prove this, let z₁ and z₂ be weak limits of the subsequences { x n i } and { x n j } of {x_n}, respectively. By (2.7), there exists v_n ∈ Tx_n such that lim n → ∞ | | x n - v n | | = 0 . Since I - P_T is demiclosed with respect to zero, therefore we obtain z₁ ∈ F(P_T ) = F(T). In the same way, we can prove that z₂ ∈ F(T).

Next, we prove uniqueness. For this, suppose that z₁ ≠ z₂. Then by Opial's condition, we have

lim n → ∞ | | x n - z 1 | | = lim n i → ∞ | | x n i - z 1 | | < lim n i → ∞ | | x n i - z 2 | | = lim n → ∞ | | x n - z 2 | | = lim n j → ∞ | | x n j - z 2 | | < lim n j → ∞ | | x n j - z 1 | | = lim n → ∞ | | x n - z 1 | | ,

which is a contradiction. Hence {x_n} converges weakly to a point in F(T). □

We now give some strong convergence theorems. Our first strong convergence theorem is valid in general real Banach spaces. We then apply this theorem to obtain a result in uniformly convex Banach spaces. We also use the method of direct construction of Cauchy sequence as indicated by Song and Cho 13 (and opposed to 10) but used also by many other authors including 121415.

Theorem 2. Let E be a real Banach space and K a nonempty closed convex subset of E. Let T : K → P(K) be a multivalued mapping such that F(T) ≠ ∅ and P_T is a nonexpansive mapping. Let {x_n} be the sequence as defined in (1.5), then {x_n} converges strongly to a point of F(T) if and only if lim inf_{n→ ∞}d(x_n, F(T)) = 0.

Proof. The necessity is obvious. Conversely, suppose that lim inf_{n→ ∞}d(x_n, F(T)) = 0. As proved in Lemma 4, we have

| | x n + 1 - p | | ≤ | | x n - p | | ,

which gives

d ( x n + 1 , F ( T ) ) ≤ d ( x n , F ( T ) ) .

This implies that lim n → ∞ d ( x n , F ( T ) ) exists and so by the hypothesis, lim inf n → ∞ d ( x n , F ( T ) ) = 0 . Therefore we must have lim n → ∞ d ( x n , F ( T ) ) = 0 .

Next, we show that {x_n} is a Cauchy sequence in K. Let ε > 0 be arbitrarily chosen. Since limn→∞d(xn,F(T))=0., there exists a constant n₀ such that for all n ≥ n₀, we have

d ( x n , F ( T ) ) < ε 4 .

In particular, inf { | | x n 0 - p | | : p ∈ F ( T ) } < ε 4 . There must exist a p* ∈ F(T) such that

| | x n 0 - p * | | < ε 2 .

Now for m, n ≥ n₀, we have

| | x n + m - x n | | ≤ | | x n + m - p * | | + | | x n - p * | | ≤ 2 | | x n 0 - p * | | < 2 ε 2 = ε .

Hence {x_n} is a Cauchy sequence in a closed subset K of a Banach space E, and so it must converge in K. Let lim n → ∞ x n = q . Now

d ( q , P T q ) ≤ | | x n - q | | + d ( x n , P T x n ) + H ( P T x n , P T q ) ≤ | | x n - q | | + | | x n - v n | | + | | x n - q | | → 0 as n → ∞

which gives that d(q, P_Tq) = 0. But P_T is a nonexpansive mapping so F(P_T) is closed. Therefore, q ∈ F(P_T) = F(T). □

We now apply the above theorem to obtain the following theorem in uniformly convex Banach spaces where T : K → P (K) satisfies condition (I).

Theorem 3. Let E be a uniformly convex Banach space and K a nonempty closed convex subset of E. Let T : K → P (K) be a multivalued mapping satisfying condition (I) such that F(T ) ≠ ∅ and P_T is a nonexpansive mapping. Let {x_n} be the sequence as defined in (1.5), then {x_n} converges strongly to a point of F(T).

Proof. By Lemma 5, lim_{n→ ∞} ||x_n - p|| exists for all p ∈ F(T). Let this limit be c for some c ≥ 0.

If c = 0, there is nothing to prove.

Suppose c > 0. Now ||x_n+1-p|| ≤ ||x_n - p|| implies that

inf p ∈ F ( T ) | | x n + 1 - p | | ≤ inf p ∈ F ( T ) | | x n - p | | ,

which means that d(x_n+1, F(T)) ≤ d(x_n, F(T)) and so limn→∞d(xn,F(T)) exists. By using condition (I) and Lemma 5, we have

lim n → ∞ f ( d ( x n , F ( T ) ) ) ≤ lim n → ∞ d ( x n , T x n ) = 0 .

That is,

lim n → ∞ f ( d ( x n , F ( T ) ) ) = 0 .

Since f is a nondecreasing function and f(0) = 0, it follows that limn→∞d(xn,F(T))=0.. Now applying Theorem 2, we obtain the result. □

The authors declare that they have no competing interests.

SHK gave the idea and wrote the initial draft. IY read and agreed upon the draft. SHK then finalized the manuscript. Correspondence was mainly done by IY. All authors read and approved the final manuscript.