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Weak and strong convergence theorems for relatively nonexpansive multi-valued mappings in Banach spaces

Fixed Point Theory and Applications20112011:73

https://doi.org/10.1186/1687-1812-2011-73

Received: 2 June 2011

Accepted: 31 October 2011

Published: 31 October 2011

Abstract

In this paper, an iterative sequence for relatively nonexpansive multi-valued mappings by using the notion of generalized projection is introduced, and then weak and strong convergence theorems are proved.

2000 Mathematics Subject Classification: 47H09; 47H10; 47J25.

Keywords

multi-valued mappingrelatively nonexpansivefixed pointiterative sequence

1 Introduction and preliminaries

Let D be a nonempty closed convex subset of a real Banach space X. A single-valued mapping T : DD is called nonexpansive if ||T(x) - T(y)|| ≤ ||x - y|| for all x, y D. Let N(D) and CB(D) denote the family of nonempty subsets and nonempty closed bounded subsets of D, respectively. The Hausdorff metric on CB(D) is defined by
H ( A 1 , A 2 ) = max sup x A 1 d ( x , A 2 ) , sup y A 2 d ( y , A 1 ) ,

for A1, A2 CB(D), where d(x, A1) = inf {||x - y||; y A1}. The multi-valued mapping T : DCB(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 : DN(D) (respectively, T : DD) if p F(T) (respectively, T(p) = p). The set of fixed points of T is represented by F(T).

Let X be a real Banach space with dual X*. We denote by J the normalized duality mapping from X to 2 X * defined by
J ( x ) : = { f * X * : x , f * = x 2 = f * 2 } ,

where 〈.,.〉 denotes the generalized duality pairing.

The Banach space X is strictly convex if ||(x + y)/2|| < 1 for all x, y X with ||x|| = ||y|| = 1 and xy. The Banach space X is uniformly convex if lim n →∞ ||x n - y n || = 0 for any two sequences {x n }, {y n } X with ||x n || = ||y n || = 1 for all n and lim n →∞ ||(x n + y n )/2|| = 1.

Lemma 1.1. [1]Let X be a uniformly convex Banach space and B r = {x X : ||x|| ≤ r}, r > 0. Then, there exists a continuous, strictly increasing, and convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that
α x + β y 2 α x 2 + β y 2 - α β g ( x - y ) ,

for all x, y B r and all α, β [0, 1] with α + β = 1.

The norm of Banach space X is said to be Gâteaux differentiable if for each x, y S(X):= {x X : ||x|| = 1} the limit
lim t 0 x + t y - x t ,
(1.1)
exists. In this case, X is called smooth. The norm of Banach space X is said to be Fréchet differentiable if for each x S(X), limit (1.1) is attained uniformly for y S(X) and the norm is uniformly Fréchet differentiable if limit (1.1) is attained uniformly for x, y S(X). In this case, X is said to be uniformly smooth. The following properties of J are well known [2]:
  1. 1.

    X (X*, resp.) is uniformly convex if and only if X* (X, resp.) is uniformly smooth;

     
  2. 2.

    If X is smooth, then J is single-valued and norm-to-weak* continuous;

     
  3. 3.

    If X is reflexive, then J is onto;

     
  4. 4.

    If X is strictly convex, then J(x) ∩ J(y) = for all xy;

     
  5. 5.

    If X has a Fréchet differentiable norm, then J is norm-to-norm continuous;

     
  6. 6.

    If X is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of X.

     

The normalized duality mapping J of a smooth Banach space X is called weakly sequentially continuous if x n x implies that J ( x n ) * J ( x ) , where denotes the weak convergence and * denotes the weak* convergence.

Let X be a smooth Banach space. The function ϕ : X × X is defined by
ϕ ( x , y ) = x 2 - 2 x , J ( y ) + y 2 , x , y X .
It is obvious from the definition of the function ϕ that
( x - y ) 2 ϕ ( x , y ) ( x + y ) 2 , x , y X .
(1.2)
In addition, the function ϕ has the following property:
ϕ ( y , x ) = ϕ ( z , x ) + ϕ ( y , z ) + 2 z - y , J ( x ) - J ( z ) , x , y , z X .
(1.3)

Lemma 1.2. [3, Remark 2.1] Let X be a strictly convex and smooth Banach space, then ϕ(x, y) = 0 if and only if x = y.

Lemma 1.3. [4]Let X be a uniformly convex and smooth Banach space and r > 0. Then
g ( y - z ) ϕ ( y , z ) ,

for all y, z B r = {x X; ||x|| ≤ r}, where g : [0, ∞) → [0, ∞) is a continuous, strictly increasing and convex function with g(0) = 0.

Let D be a nonempty closed convex subset of a smooth Banach space X. A point p D is called an asymptotic fixed point of T : DD[5], if there exists a sequence {x n } in D which converges weakly to p and lim n →∞ ||x n - T(x n )|| = 0. The set of asymptotic fixed points of T is represented by F ^ ( T ) . A mapping T : DD is called relatively nonexpansive [3, 68], if the following conditions are satisfied:
  1. 1.

    F(T) is nonempty;

     
  2. 2.

    ϕ(p, T(x)) ≤ ϕ(p, x), x D, p F(T);

     

3. F ^ ( T ) = F ( T ) .

Let D be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X. It is known that [4, 9] for any x X, there exists a unique point x0 D such that
ϕ ( x 0 , x ) = min y D ϕ ( y , x ) .

Following Alber [9], we denote such an element x0 by Π D x. The mapping Π D is called the generalized projection from X onto D. If X is a Hilbert space, then ϕ(y, x) = ||y - x||2 and Π D is the metric projection of X onto D.

Lemma 1.4. [4, 9]Let D be a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space X. Then
ϕ ( x , Π D y ) + ϕ ( Π D y , y ) ϕ ( x , y ) , x D , y X .
Lemma 1.5. [4, 9]Let D be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X. Let x X and z D, then
z = Π D x z - y , J ( x ) - J ( z ) 0 , y D .
In 2004, Matsushita and Takahashi [10] introduced the following iterative sequence for finding a fixed point of relatively nonexpansive mapping T : DD. Given x1 D,
x n + 1 = Π D J - 1 ( α n J ( x n ) + ( 1 - α n ) J ( T ( x n ) ) ) ,
(1.4)

where D is a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space X, Π D is the generalized projection onto D and {α n } is a sequence in [0, 1].

They proved weak and strong convergence theorems in uniformly convex and uniformly smooth Banach space X.

Iterative methods for approximating fixed points of multi-valued mappings in Banach spaces have been studied by some authors, see for instance [1114].

Let D be a nonempty closed convex subset of a smooth Banach space X. We define an asymptotic fixed point for a multi-valued mapping as follows.

Definition 1.6. A point p D is called an asymptotic fixed point of T : DN(D), if there exists a sequence {x n } in D which converges weakly to p and lim n →∞d(x n , T(x n )) = 0.

Moreover, we define a relatively nonexpansive multi-valued mapping as follows.

Definition 1.7. A multi-valued mapping T : DN(D) is called relatively nonexpansive, if the following conditions are satisfied:

1. F(T) is nonempty;

2. ϕ(p, z) ≤ ϕ(p, x), x D, z T(x), p F(T);

3. F ^ ( T ) = F ( T ) ,

where F ^ ( T ) is the set of asymptotic fixed points of T.

There exist relatively nonexpansive multi-valued mappings that are not nonexpansive.

Example 1.8. Let I = [0,1], X = L p (I), 1 < p < ∞ and D = {f X; f(x) ≥ 0, x I}. Let T : DCB(D) be defined by
T ( f ) = { g D ; f ( x ) - 3 4 g ( x ) f ( x ) - 1 4 , x I } , f ( x ) > 1 , x I ; { 0 } , otherwise .
It is clear that F(T) = {0}. Let h F ^ ( T ) . Then, there exists a sequence {f n } in D which converges weakly to h, and z n = d(f n , T(f n )) → 0. Let n , we have
z n = 1 4 , f n ( x ) > 1 , x I ; f n p , otherwise .
Since z n → 0, we have ||f n || p → 0. Therefore, f n → 0. Hence, h = 0. Therefore, F ^ ( T ) = F ( T ) = { 0 } . Let f D such that f(x) > 1 for all x I, and g T(f), then
ϕ ( 0 , g ) = g p 2 f p 2 = ϕ ( 0 , f ) .
Next, let f D such that there exists x I such that f(x) ≤ 1, then
ϕ ( 0 , 0 ) = 0 f p 2 = ϕ ( 0 , f ) .

Hence, T is relatively nonexpansive. However, if f(x) = 2 and g(x) = 1 for all x I, we get H ( T ( f ) , T ( g ) ) = 7 4 . Then, H(T(f), T(g)) > ||f - g|| p = 1. Hence, T is not nonexpansive.

In this article, inspired by Matsushita and Takahashi [10], we introduce the following iterative sequence for finding a fixed point of relatively nonexpansive multi-valued mapping T : DN(D). Given x1 D,
x n + 1 = Π D J - 1 ( α n J ( x n ) + ( 1 - α n ) J ( z n ) ) ,
(1.5)

where z n T(x n ) for all n , D is a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space X, Π D is the generalized projection onto D and {α n } is a sequence in [0, 1]. We prove weak and strong convergence theorems in uniformly convex and uniformly smooth Banach space X.

2 Main results

In this section, at first, concerning the fixed point set of a relatively nonexpansive multi-valued mapping, we prove the following proposition.

Proposition 2.1. Let X be a strictly convex and smooth Banach space, and D a nonempty closed convex subset of X. Suppose T : DN(D) is a relatively nonexpansive multi-valued mapping. Then, F(T) is closed and convex.

Proof. First, we show F(T) is closed. Let {x n } be a sequence in F(T) such that x n x*. Since T is relatively nonexpansive, we have
ϕ ( x n , z ) ϕ ( x n , x * ) ,
for all z T(x*) and for all n . Therefore,
ϕ ( x * , z ) = lim n ϕ ( x n , z ) lim n ϕ ( x n , x * ) = ϕ ( x * , x * ) = 0 .
(2.1)
By Lemma 1.2, we obtain x* = z. Hence, T(x*) = {x*}. So, we have x* F(T). Next, we show F(T) is convex. Let x, y F(T) and t (0, 1), put p = tx + (1 - t)y. We show p F(T). Let w T(p), we have
ϕ ( p , w ) = p 2 - 2 p , J ( w ) + w 2 = p 2 - 2 t x + ( 1 - t ) y , J ( w ) + w 2 = p 2 - 2 t x , J ( w ) - 2 ( 1 - t ) y , J ( w ) + w 2 = p 2 + t ϕ ( x , w ) + ( 1 - t ) ϕ ( y , w ) - t x 2 - ( 1 - t ) y 2 p 2 + t ϕ ( x , p ) + ( 1 - t ) ϕ ( y , p ) - t x 2 - ( 1 - t ) y 2 = p 2 - 2 t x + ( 1 - t ) y , J ( p ) + p 2 = p 2 - 2 p , J ( p ) + p 2 = 0 .
(2.2)

By Lemma 1.2, we obtain p = w. Hence, T(p) = {p}. So, we have p F(T). Therefore, F(T) is convex. □

Remark 2.2. Let X be a strictly convex and smooth Banach space, and D a nonempty closed convex subset of X. Suppose T : DN(D) is a relatively nonexpansive multi-valued mapping. If p F(T), then T(p) = {p}.

Proposition 2.3. Let X be a uniformly convex and smooth Banach space, and D a nonempty closed convex subset of X. Suppose T : DN(D) is a relatively nonexpansive multi-valued mapping. Let {α n } be a sequence of real numbers such that 0 ≤ α n ≤ 1 for all n . For a given x1 D, let {x n } be the iterative sequence defined by (1.5). Then, {Π F (T)x n } converges strongly to a fixed point of T, where Π F (T)is the generalized projection from D onto F(T).

Proof. By Proposition 2.1, F(T) is closed and convex. So, we can define the generalized projection Π F (T)onto F(T). Let p F(T). From Lemma 1.4, we have
ϕ ( p , x n + 1 ) = ϕ ( p , Π D J - 1 ( α n J ( x n ) + ( 1 - α n ) J ( z n ) ) ) ϕ ( p , J - 1 ( α n J ( x n ) + ( 1 - α n ) J ( z n ) ) ) = p 2 - 2 p , α n J ( x n ) + ( 1 - α n ) J ( z n ) + α n J ( x n ) + ( 1 - α n ) J ( z n ) 2 p 2 - 2 α n p , J ( x n ) - 2 ( 1 - α n ) p , J ( z n ) + α n x n 2 + ( 1 - α n ) z n 2 = α n ϕ ( p , x n ) + ( 1 - α n ) ϕ ( p , z n ) α n ϕ ( p , x n ) + ( 1 - α n ) ϕ ( p , x n ) = ϕ ( p , x n ) . (10)
(2.3)
Hence, lim n → ∞ϕ(p, x n ) exists. So, {ϕ(p, x n )} is bounded. Then, by (1.2) we have {x n } is bounded, and hence, {z n } is bounded. Let u n = Π F (T)x n , for all n . Then, we have
ϕ ( u n , x n + 1 ) ϕ ( u n , x n ) .
(2.4)
Therefore
ϕ ( u n , x n + m ) ϕ ( u n , x n ) ,
(2.5)
for all m . From Lemma 1.4, we obtain
ϕ ( u n + 1 , x n + 1 ) = ϕ ( Π F ( T ) x n + 1 , x n + 1 ) ϕ ( u n , x n + 1 ) - ϕ ( u n , Π F ( T ) x n + 1 ) .
(2.6)
By (2.4) and (2.6) we have
ϕ ( u n + 1 , x n + 1 ) ϕ ( u n , x n ) .
(2.7)
It follows that {ϕ(u n , x n )} converges. From u n + m = Π F (T)x n + m and Lemma 1.4, we have
ϕ ( u n , u n + m ) + ϕ ( u n + m , x n + m ) ϕ ( u n , x n + m ) .
Hence, by (2.5) we obtain
ϕ ( u n , u n + m ) ϕ ( u n , x n ) - ϕ ( u n + m , x n + m ) ,
(2.8)
for all m, n . Let r = sup n ||u n ||. From Lemma 1.3, there exists a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that
g ( u m - u n ) ϕ ( u m , u n ) ϕ ( u m , x m ) - ϕ ( u n , x n ) ,
(2.9)

for all m, n , n > m. Therefore, {u n } is a Cauchy sequence. Since X is complete and F(T) is closed, there exists q F(T) such that {u n } converges strongly to q. □

If the duality mapping J is weakly sequentially continuous, we have the following weak convergence theorem.

Theorem 2.4. Let X be a uniformly convex and uniformly smooth Banach space, and D a nonempty closed convex subset of X. Suppose T : DN(D) is a relatively nonexpansive multi-valued mapping. Let {α n } be a sequence of real numbers such that 0 ≤ α n ≤ 1 for all n and lim inf n →∞α n (1 - α n ) > 0. For a given x1 D, let {x n } be the iterative sequence defined by (1.5). If J is weakly sequentially continuous, then {x n } converges weakly to a fixed point of T.

Proof. As in the proof of Proposition 2.3, {x n } and {z n } are bounded. So, there exists r > 0 such that x n , z n B r for all n . Since X is a uniformly smooth Banach space, X* is a uniformly convex Banach space. Let p F(T). By Lemma 1.1, there exists a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that
ϕ ( p , x n + 1 ) = ϕ ( p , Π D J - 1 ( α n J ( x n ) + ( 1 - α n ) J ( z n ) ) ) ϕ ( p , J - 1 ( α n J ( x n ) + ( 1 - α n ) J ( z n ) ) ) = p 2 - 2 p , α n J ( x n ) + ( 1 - α n ) J ( z n ) + α n J ( x n ) + ( 1 - α n ) J ( z n ) 2 p 2 - 2 α n p , J ( x n ) - 2 ( 1 - α n ) p , J ( z n ) + α n x n 2 + ( 1 - α n ) z n 2 - α n ( 1 - α n ) g ( J ( x n ) - J ( z n ) ) = α n ϕ ( p , x n ) + ( 1 - α n ) ϕ ( p , z n ) - α n ( 1 - α n ) g ( J ( x n ) - J ( z n ) ) ϕ ( p , x n ) - α n ( 1 - α n ) g ( J ( x n ) - J ( z n ) ) .
(2.10)
Hence
α n ( 1 - α n ) g ( J ( x n ) - J ( z n ) ) ϕ ( p , x n ) - ϕ ( p , x n + 1 ) .
Since lim n →∞ϕ(p, x n ) exists and lim inf n →∞α n (1 - α n ) > 0, we obtain
lim n g ( J ( x n ) - J ( z n ) ) = 0 .
Therefore,
lim n J ( x n ) - J ( z n ) = 0 .
Since J-1 is uniformly norm-to-norm continuous on bounded sets, we have
lim n x n - z n = 0 .
Since d(x n , T(x n )) ≤ ||x n - z n ||, we obtain
lim n d ( x n , T ( x n ) ) = 0 .
(2.11)
Let u n = Π F ( T ) x n . By Lemma 1.5, we have
u n - w , J ( x n ) - J ( u n ) 0 ,
(2.12)
for each w F(T). From Proposition 2.3, there exists p F(T) such that {u n } converges strongly to p. Let { x n j } be a subsequence of {x n } such that { x n j } converges weakly to q. Then, by (2.11) we have q F(T). It follows from (2.12) that
u n j - q , J ( x n j ) - J ( u n j ) 0 .
(2.13)
Let j → ∞ in inequality (2.13), since J is weakly sequentially continuous we have
p - q , J ( q ) - J ( p ) 0 .
(2.14)
Since J is monotone, we have
q - p , J ( q ) - J ( p ) 0 .
(2.15)
It follows from (2.14) and (2.15) that
q - p , J ( q ) - J ( p ) = 0 .
(2.16)

Since X is strictly convex, we have p = q. Therefore, {x n } converges weakly to p. The proof is complete. □

Theorem 2.5. Let X be a uniformly convex and uniformly smooth Banach space, and D a nonempty closed convex subset of X. Suppose T : DN(D) is a relatively nonexpansive multi-valued mapping. Let {α n } be a sequence of real numbers such that 0 ≤ α n ≤ 1 for all n and lim inf n →∞α n (1 - α n ) > 0. For a given x1 D, let {x n } be the iterative sequence defined by (1.5). If the interior of F(T) is nonempty, then {x n } converges strongly to a fixed point of T.

Proof. Since the interior of F(T) is nonempty, there exists p F(T) and r > 0 such that p + rh F(T), whenever ||h|| ≤ 1. By (1.3) for any q F(T) we have
ϕ ( q , x n ) = ϕ ( x n + 1 , x n ) + ϕ ( q , x n + 1 ) + 2 x n + 1 - q , J ( x n ) - J ( x n + 1 ) .
(2.17)
Therefore,
1 2 ( ϕ ( q , x n ) - ϕ ( q , x n + 1 ) ) = 1 2 ϕ ( x n + 1 , x n ) + x n + 1 - q , J ( x n ) - J ( x n + 1 ) .
(2.18)
Since p + rh F(T), as in the proof of Proposition 2.3, we have
ϕ ( p + r h , x n + 1 ) ϕ ( p + r h , x n ) .
(2.19)
It follows from (2.18) and (2.19) that
1 2 ϕ ( x n + 1 , x n ) + x n + 1 - ( p + r h ) , J ( x n ) - J ( x n + 1 ) 0 .
(2.20)
Then, by (2.18) and (2.20) we have
h , J ( x n ) - J ( x n + 1 ) 1 r ( x n + 1 - p , J ( x n ) - J ( x n + 1 ) + 1 2 ϕ ( x n + 1 , x n ) ) = 1 2 r ( ϕ ( p , x n ) - ϕ ( p , x n + 1 ) ) ,
(2.21)
whenever ||h|| ≤ 1. Therefore, we obtain
J ( x n ) - J ( x n + 1 ) 1 2 r ( ϕ ( p , x n ) - ϕ ( p , x n + 1 ) ) .
It follows that
J ( x m ) - J ( x n ) Σ i = m n - 1 J ( x i ) - J ( x i + 1 ) Σ i = m n - 1 1 2 r ( ϕ ( p , x i ) - ϕ ( p , x i + 1 ) ) = 1 2 r ( ϕ ( p , x m ) - ϕ ( p , x n ) ) ,
(2.22)

for all m, n , n > m. As in the proof of Proposition 2.3, {ϕ(p, x n )} converges. Hence, {J(x n )} is a Cauchy sequence. Since X* is complete, {J(x n )} converges strongly to a point in X*. Since X* has a Fréchet differentiable norm, then J-1 is norm-to-norm continuous on X*. Hence, {x n } converges strongly to some point u in D. As in the proof of Theorem 2.4, lim n →∞d(x n , T(x n )) = 0. Hence, we have u F(T), where u = lim n →∞ Π F ( T )x n . □

Declarations

Acknowledgements

AR would like to thank the School of Mathematics of the Institute for Research in Fundamental Sciences, Tehran, Iran, for supporting this research (Grant No. 90470122).

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran
(2)
School of Mathematics, Institute for Research in Fundamental Sciences, Tehran, Iran

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Copyright

© Homaeipour and Razani; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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