Skip to main content

Strong convergence theorems by Halpern-Mann iterations for multi-valued relatively nonexpansive mappings in Banach spaces with applications

Abstract

In this article, an iterative sequence for relatively nonexpansive multi-valued mapping by modifying Halpern and Mann's iterations is introduced, and then some strong convergence theorems are proved. At the end of the article some applications are given also.

AMS Subject Classification: 47H09; 47H10; 49J25.

1 Introduction

Throughout this article, we denote by and the sets of positive integers and real numbers, respectively. Let D be a nonempty closed subset of a real Banach space E. A single-valued mapping T : DD is called nonexpansive if Tx - Tyx - 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) if p T(p). The set of fixed points of T is represented by F(T).

Let E be a real Banach space with dual E*. We denote by J the normalized duality mapping from E to 2E*defined by

J ( x ) = x * E * : x , x * = x 2 = x * 2 , x E .

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

A Banach space E is said to be strictly convex if x + y 2 <1 for all x, y U = {z E : z = 1} with xy. E is said to be uniformly convex if, for each ϵ (0, 2], there exists δ > 0 such that x + y 2 <1-δ for all x, y U with x - yϵ. E is said to be smooth if the limit

lim t 0 x + t y - x t

exists for all x, y U. E is said to be uniformly smooth if the above limit exists uniformly in x, y U.

Remark 1.1. The following basic properties for Banach space E and for the normalized duality mapping J can be found in Cioranescu [1].

  1. (i)

    If E is an arbitrary Banach space, then J is monotone and bounded;

  2. (ii)

    If E is a strictly convex Banach space, then J is strictly monotone;

  3. (iii)

    If E is a a smooth Banach space, then J is single-valued, and hemi-continuous, i.e., J is continuous from the strong topology of E to the weak star topology of E*;

  4. (iv)

    If E is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E;

  5. (v)

    If E is a reflexive and strictly convex Banach space with a strictly convex dual E* and J*: E*E is the normalized duality mapping in E*, then J-1 = J*, J J* = IE*, and J* J = I E ;

  6. (vi)

    If E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping J is single-valued, one-to-one and onto;

  7. (vii)

    A Banach space E is uniformly smooth if and only if E* is uniformly convex. If E is uniformly smooth, then it is smooth and reflexive.

Next we assume that E is a smooth, strictly convex, and reflexive Banach space and C is a nonempty closed convex subset of E. In the sequel, we always use ϕ : E × E+ to denote the Lyapunov functional defined by

ϕ ( x , y ) = x 2 - 2 x , J y + y 2 , x , y E .
(1.2)

It is obvious from the definition of ϕ that

x - y 2 ϕ ( x , y ) x - y 2 , x , y E .
(1.3)
ϕ x , J - 1 λ J y + ( 1 - λ ) J z λ ϕ ( x , y ) + ( 1 - λ ) ϕ ( x , z ) ,
(1.4)

for all λ [0,1] and x,y,z E.

Following Alber [2], the generalized projection Π C : EC is defined by

C ( x ) = arg inf y C ϕ ( y , x ) , x E .

Let D be a nonempty subset of a smooth Banach space. A mapping T : DE is relatively expansive [35], if the following properties are satisfied:

(R1) F ( T ) ;

(R2) ϕ(p,Tx) ≤ ϕ(p,x) for all p F(T) and x D;

(R3) I - T is demi-closed at zero, that is, whenever a sequence {x n } in D converges weakly to p and {x n - Tx n } converges strongly to 0, it follows that p F(T).

If T satisfies (R1) and (R2), then T is called quasi-ϕ-nonexpansive [6].

Recently, Nilsrakoo and Saejung [7] introduced the following iterative sequence for finding a fixed point of relatively nonexpansive mapping T : DE. Given x1 D,

x n + 1 = D J - 1 α n J u + ( 1 - α n ) β n J x n + ( 1 - β n ) J T x n

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

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

Iterative methods for approximating fixed points of multi-valued mappings in Banach spaces have been studied by some authors, see for instance [815].

Let D be a nonempty closed convex subset of a smooth Banach space E. We define a relatively nonexpansive multi-valued mapping as follows.

Definition 1.2. A multi-valued mapping T : DN(D) is called relatively nonexpansive, if the following conditions are satisfied:

(S1) F ( T )

(S2) ϕ(p,z) ≤ ϕ(p, x), x D, z T(x), p F(T);

(S3) I - T is demi-closed at zero, that is, whenever a sequence {x n } in D which weakly to p and limn→∞d(x n , T(x n )) = 0, it follows that p F(T).

If T satisfies (S1) and (S2), then multi-valued mapping T is called quasi-ϕ-nonexpansive.

In this article, inspired by Nilsrakoo and Saejung [7], we introduce the following iterative sequence for finding a fixed point of relatively nonexpansive multi-valued mapping T : DN(D). Given u E,x i D,

x n + 1 = D J - 1 α n J u + ( 1 - α n ) β n J x n + ( 1 - β n ) J w n

where w n Tx n for all n , D is a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, Π D is the generalized projection of E onto D and {α n }, {β n } are sequences in [0,1]. We proved the strong convergence theorems in uniformly convex and uniformly smooth Banach space E.

2 Preliminaries

In the sequel, we denote the strong convergence and weak convergence of the sequence {x n } by x n x and x n x, respectively.

First, we recall some conclusions.

Lemma 2.1 [16, 17]. Let E be a smooth, strictly convex, and reflexive Banach space and C be a nonempty closed convex subset of E. Then the following conclusions hold:

  1. (a)

    ϕ(x, Π C y) + ϕ C y, y) ≤ ϕ(x, y) for all x C and y E;

  2. (b)

    If x E and z C, then

    z = C x z - y , J x - J z 0 , y C ;
  3. (c)

    For x, y E, ϕ(x, y) = 0 if and only x = y.

Remark 2.2. If E is a real Hilbert space H, then ϕ(x, y) = x - y2 and Π C is the metric projection P C of H onto C.

Lemma 2.3 [18]. Let E be a uniformly convex Banach space, r > 0 be a positive number and B r (0) be a closed ball of E. Then, for any given sequence x i i = 1 B r ( 0 ) and for any given sequence λ i i = 1 of positive numbers with i = 1 λ i = 1 , then there exists a continuous, strictly increasing, and convex function g : [0, 2r) → [0, ∞) with g(0) = 0 such that for any positive integers i, j with i < j,

n = 1 λ n x n 2 n = 1 λ n x n 2 - λ i λ j g x i - x j
(2.1)

In what follows, we need the following lemmas for proof of our main results.

Lemma 2.4 [17]. Let E be a uniformly convex and smooth Banach space and let {x n } and {y n } be two sequences of E such that {x n } or {y n } is bounded. If limn→∞ϕ(x n , y n ) = 0. Then limn→∞x n -y n = 0.

Let E be a reflexive, strictly convex, and smooth Banach space. The duality mapping J* from E* onto E** = E coincides with the inverse of the duality mapping J from E onto E*, that is, J* = J-1. We make use the following mapping V : E × E* studied in Alber [19]:

V ( x , x * ) = x 2 - 2 x , x * + x * 2
(2.2)

for all x E and x* E*. Obviously, V(x, x*) = ϕ(x, J-1(x*)) for all x E and x* E*. We know the following lemma.

Lemma 2.5 [20]. Let E be a reflexive, strictly convex, and smooth Banach space, and let V as in (2.2). Then

V ( x , x * ) + 2 J - 1 ( x * ) - x , y * V ( x , x * + y * ) ,
(2.3)

for all x E and x*,y* E*.

Lemma 2.6 [21]. Assume that {α n } is a sequence of nonnegative real numbers such that

α n + 1 ( 1 - γ n ) α n + γ n δ n ,

where {γ n } is a sequence in (0,1) and {δ n } is a sequence such that

  1. (a)

    lim n γ n =0, n = 1 γ n =;

  2. (b)

    lim supn→∞≤ 0.

Then limn→∞α n = 0.

Lemma 2.7 [22]. Let {α n } be a sequence of real numbers such that there exists a subsequence {n i } of {n} such that α n i < α n i + 1 for all i . Then there exists a nondecreasing sequence {m k } such that m k → ∞ and the following properties are satisfied for all (sufficiently large) numbers k :

α m k α m k + 1 a n d α k α m k + 1 .

In fact, m k = max{jk : α j < αj+1}.

3 Main results

Lemma 3.1 Let E be a strictly convex and smooth Banach space, and D a nonempty closed subset of E. Suppose T : DN(D) is a quasi-ϕ-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 quasi-ϕ-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 .

By Lemma 2.1(c), 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 F(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 , p ) - t x 2 - t ( 1 - t ) p 2 = p 2 - 2 t x + ( 1 - t ) y , J p + p 2 = p 2 - 2 p , J p + p 2 = 0 .

By Lemma 2.1(c), we obtain p = w. Hence, T(p) = {p}. So, we have p F(T). Therefore, F(T) is convex.

Lemma 3.2. Let D be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E and T : DN(D) be a relatively nonexpansive multi-valued mapping. If {x n } is a bounded sequence such that limn→∞d(x n ,Tx n ) and x* = ΠF(T)x, then

lim n sup x n - x * , J x - J x * 0 .

Proof. From (S3) of the mapping T, we choose a subsequence x n i of {x n } such that x n i yF ( T ) and

lim n sup x n - x * , J x - J x * = lim i x n i - x * , J x - J x * .

By Lemma 2.1(b), we immediately obtain that

lim n sup x n - x * , J x - J x * = y - x * , J x - J x * 0 .

Lemma 3.3. Let D be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E and T : DN(D) be a relatively nonexpansive multi-valued mapping. Let {x n } be a sequence in D defined as follows: u E, x1 D and

x n + 1 = D J - 1 α n J u + ( 1 - α n ) β n J x n + ( 1 - β n ) J w n ,
(3.1)

where w n Tx n for all n , {α n }, {β n } are sequences in [0,1]. Then {x n } is bounded.

Proof. Let p F(T) and y n = J - 1 β n J x n + ( 1 - β n ) J w n for all n . Then

x n + 1 D J - 1 α n J u + ( 1 - α n ) J y n

for all n . By using (1.4), we have

ϕ ( p , y n ) = ϕ p , J - 1 β n J x n + ( 1 - β n ) J w n β n ϕ ( p , x n ) + ( 1 - β n ) ϕ ( p , w n ) β n ϕ ( p , x n ) + ( 1 - β n ) ϕ ( p , x n ) = ϕ ( p , x n )

and

ϕ ( p , x n + 1 ) = ϕ p , D J - 1 α n J u + ( 1 - α n ) J y n ϕ p , J - 1 α n J u + ( 1 - α n ) J y n α n ϕ ( p , u ) + ( 1 - α n ) ϕ ( p , y n ) α n ϕ ( p , u ) + ( 1 - α n ) ϕ ( p , x n ) max ϕ ( p , u ) , ϕ ( p , x n ) max ϕ ( p , u ) , ϕ ( p , x 1 ) .

This implies that {x n } is bounded.

Theorem 3.4 Let D be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and T : DN(D) be a relatively nonexpansive multivalued mapping. Let {α n } and {β n } be sequences in (0,1) satisfying

(C1) limn→∞, α n = 0;

(C2) n = 1 α n =;

(C3) lim infn→∞β n (1- β n ) > 0.

Then {x n } defined by (3.1) converges strongly to ΠF(T)u, where ΠF(T)is the generalized projection from E onto F(T).

Proof. By Lemma 3.1, F(T) is closed and convex. So, we can define the generalized projection ΠF(T)onto F(T). Putting u* = ∏F(T)u, by Lemma 3.3 we know that {x n } is bounded and hence, {w n } is bounded. Let g : [0,2r] → [0,∞) be a function satisfying the properties of Lemma 2.3, where r = sup{u, x n , w n : n }. Put

y n J - 1 β n J u + ( 1 - β n ) J w n

Then

ϕ ( u * , y n ) = ϕ u * , J - 1 β n J x n + ( 1 - β n ) J w n = u * 2 - 2 u * , β n J x n + ( 1 - β n ) J w n + β n J x n + ( 1 - β n ) J w n 2 u * 2 - 2 β n u * , J x n - 2 ( 1 - β n ) u * , J w n + β n x n 2 + ( 1 - β n ) w n 2 - β n ( 1 - β n ) g J x n - J w n = ϕ u * , x n - β n 1 - β n g J x n - J w n
(3.2)

and

ϕ u * , x n + 1 = ϕ u * , D J - 1 α n J u + ( 1 - α n ) J y n ϕ u * , J - 1 α n J u + ( 1 - α n ) J y n α n ϕ ( u * , u ) + ( 1 - α n ) ϕ ( u * , y n ) α n ϕ ( u * , u ) + ( 1 - α n ) ϕ ( u * , x n ) - β n ( 1 - β n ) g J x n - J w n .
(3.3)

for all n . Put

M = sup ϕ ( u * , u ) - ϕ ( u * , x n ) + β n ( 1 - β n ) g | | J x n - J w n : n

It follows from (3.3) that

β n ( 1 - β n ) g J x n - J w n ϕ ( u * , x n ) - ϕ u * , x n + 1 + α n M .
(3.4)

Let z n J - 1 α n J u + ( 1 - α n ) J y n . Then x n + 1 = C z n for all n . It follows from (2.3) and (3.2) that

ϕ u * , x n + 1 ϕ u * , J - 1 α n J u + ( 1 - α n ) J y n = V u * , α n J u + ( 1 - α n ) J y n V u * , α n J u + ( 1 - α n ) J y n - α n ( J u - J u * ) - 2 J - 1 α n J u + ( 1 - α n ) J y n - u * , - α n J u - J u * = V u * , α n J u * + ( 1 - α n ) J y n + 2 α n z n - u * , J u - J u * = ϕ u * , J - 1 α n J u * + ( 1 - α n ) J y n + 2 α n z n - u * , J u - J u * = u * 2 - 2 u * , α n J u * + ( 1 - α n ) J y n + α n J u * + ( 1 - α n ) J y n 2 + 2 α n z n - u * , J u - J u * u * 2 - 2 α n u * , J u * - 2 ( 1 - α n ) u * , J y n + α n u * 2 + ( 1 - α n ) y n 2 + 2 α n z n - u * , J u - J u * = α n ϕ ( u , u * ) + ( 1 - α n ) ϕ ( u * , y n ) + 2 α n z n - u * , J u - J u * ( 1 - α n ) ϕ ( u * , x n ) + 2 α n z n - u * , J u - J u * .
(3.5)

The rest of proof will be divided into two parts:

Case (1). Suppose that there exists n0 such that ϕ ( u * , x n ) n = n 0 is nonincreasing. In this situation, {ϕ(u*, x n )} is then convergent. Then limn→∞(ϕ(u*, x n ) - ϕ(u*, xn+1)) = 0. This together with (C1), (C3), and (3.4), we obtain

lim n g J x n - J w n = 0 .

Therefore,

lim n J x n - J w n = 0 .

Since J-1 is uniformly norm-to-norm continuous on every bounded subset of E, we have

lim n x n - w n = 0 .
(3.6)

Since d x n , T x n x n - w n , we obtain

lim n d ( x n , T x n ) = 0
(3.7)

Then,

ϕ ( w n , y n ) = ϕ w n , J - 1 β n J x n + ( 1 - β n ) J w n β n ϕ ( w n , x n ) + ( 1 - β n ) ϕ ( w n , w n ) = β n ϕ ( w n , x n ) 0 .
(3.8)

and

ϕ ( y n , z n ) α n ϕ ( y n , u ) + ( 1 - α n ) ϕ ( y n , y n ) = α n ϕ ( y n , u ) 0 .
(3.9)

From (3.8), (3.9) and Lemma 2.3, we have

lim n w n - y n = 0

and

lim n y n - z n = 0

This together with (3.6) gives

lim n x n - z n = 0
(3.10)

From (3.7), (3.10) and invoking Lemma 3.2, we have

lim n z n - u * , J u - J u * = lim n x n - u * , J u - J u * 0

Hence the conclusion follows by Lemma 2.5.

Case (2). Suppose that there exists a subsequence {n i } of {n} such that

ϕ ( u * , x n i ) < ϕ ( u * , x n i + 1 )

for all i . Then, by Lemma 2.7, there exists a nondecreasing sequence {m k } , m k → ∞ such that

ϕ u * , x m k ϕ u * , x m k + 1 a n d ϕ u * , x k ϕ u * , x m k + 1

for all k . This together with (3.4) gives

β m k ( 1 - β m k ) g J x m k - J w m k ϕ u * , x m k - ϕ u * , x m k + 1 + α m k M α m k M

for all k N. Then, by conditions (C1) and (C3)

lim k g J x m k - J w m k = 0

By the same argument as Case (1), we get

lim k sup z m k - u * , J u - J u * 0 .
(3.11)

From (3.5), we have

ϕ u * , x m k + 1 1 - α m k ϕ u * , x m k + 2 α m k z m k - u * , J u - J u *
(3.12)

Since ϕ u * , x m k ϕ u * , x m k + 1 , we have

α m k ϕ u * , x m k ϕ u * , x m k - ϕ u * , x m k + 1 + 2 α m k z m k - u * , J u - J u * 2 α m k z m k - u * , J u - J u *

In particular, since α m k >0, we get

ϕ u * , x m k 2 z m k - u * , J u - J u *

It follows from (3.11) that lim k ϕ u * , x m k =0. This together with (3.12) gives

lim k ϕ u * , x m k + 1 = 0

But ϕ u * , x k ϕ u * , x m k + 1 for all k . We conclude that x k u*.

This implies that limn→∞x n = u* and the proof is finished.

Letting β n = β gives the following result.

Corollary 3.5. Let D be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and T : DN(D) be a relatively nonexpansive multivalued mapping. Let {x n } be a sequence in D defined as follows: u E,x1 D and

x n + 1 = D J - 1 α n J u + ( 1 - α n ) β J x n + ( 1 - β ) J w n ,

where w n Tx n for all n , {α n } is a sequence in [0,1] satisfying condition (C1) and (C2), and β (0,1). Then {x n } converges strongly to ∏F(T)u.

4 Application to zero point problem of maximal monotone mappings

Let E be a smooth, strictly convex, and reflexive Banach space. An operator A : E → 2E*is said to be monotone, if 〈x - y, x* - y*〉 ≥ 0 whenever x, y E, x* Ax, y* Ay. We denote the zero point set {x E : 0 Ax} of A by A-10. A monotone operator A is said to be maximal, if its graph G(A) := {(x, y) : y Ax} is not properly contained in the graph of any other monotone operator. If A is maximal monotone, then A-10 is closed and convex. Let A be a maximal monotone operator, then for each r > 0 and x E, there exists a unique x r D(A) such that J(x) J(x r ) + rA(x r ) (see, for example, [19]). We define the resolvent of A by J r x = x r . In other words J r = (J + rA)-1 J, r > 0. We know that J r is a single-valued relatively nonexpansive mapping and A-10 = F(J r ),r > 0, where F(J r ) is the set of fixed points of J r .

We have the following

Theorem 4.1 Let E, {α n }, and {β n } be the same as in Theorem 3.4. Let A : E → 2E*be a maximal monotone operator and J r = (J + rA)-1J for all r > 0 such that A - 1 0. Let {x n } be the sequence generated by

x n + 1 = J - 1 α n J x 1 + ( 1 - α n ) β n J x n + ( 1 - β n ) J J r n x n ,

then {x n } converges strongly to Π A -10x1.

Proof. In Theorem 3.4 taking D = E, T = J r , r > 0, then T : EE is a single-valued relatively nonexpansive mapping and A-10 = F(T) = F(J r ),r > 0 is a nonempty closed convex subset of E. Therefore all the conditions in Theorem 3.4 are satisfied. The conclusion of Theorem 4.1 can be obtained from Theorem 3.4 immediately.

References

  1. Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Press, Dordrecht; 1990.

    Chapter  Google Scholar 

  2. Alber YI: Metric and Generalized Projection Operators in Banach Spaces. Marcel Dekker, New York; 1996:15–50.

    Google Scholar 

  3. Matsushita S, Takahashi W: Weak and strong convergence theorems for relatively nonexpansive mappings in a Banach spaces. Fixed point Theory Appl 2004, 2004: 37–47.

    Article  MathSciNet  MATH  Google Scholar 

  4. Matsushita S, Takahashi W: An iterative algorithm for relatively nonexpansive mappings by hybrid method and applications. Proceedings of the Third International Conference on Nonlinear Analysis and Convex Analysis 2004, 305–313.

    Google Scholar 

  5. Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach spaces. J Approx Theory 2005, 134: 257–266. 10.1016/j.jat.2005.02.007

    Article  MathSciNet  MATH  Google Scholar 

  6. Nilsrakoo W, Saejung S: Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings. Fixed Point Theory and Applications 2008: 19. Article ID 312454

  7. Nilsrakoo W, Saejung S: Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces. Appl Math Comput 2011, 217(14):6577–6584. 10.1016/j.amc.2011.01.040

    Article  MathSciNet  MATH  Google Scholar 

  8. Jung JS: Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces. Nonlinear Anal 2007, 66: 2345–2354. 10.1016/j.na.2006.03.023

    Article  MathSciNet  MATH  Google Scholar 

  9. Shahzad N, Zegeye H: Strong convergence results for nonself multimaps in Banach spaces. Proc Am Soc 2008, 136: 539–548.

    Article  MathSciNet  MATH  Google Scholar 

  10. Shahzad N, Zegeye H: On Mann and Ishikawa iteration schems for multi-valued maps in Banach spaces. Nonlinear Anal 2009, 71: 838–844. 10.1016/j.na.2008.10.112

    Article  MathSciNet  MATH  Google Scholar 

  11. Song Y, Wang H: Convergence of iterative algorithms for multivalued mappings in Banach spaces. Nonlinear Anal 2009, 70: 1547–1556. 10.1016/j.na.2008.02.034

    Article  MathSciNet  MATH  Google Scholar 

  12. Cho YJ, Qin X, Kang SM: Strong convergence of the modified Halpern-type iteration algorithms in Banach spaces. An St Univ Ovidius Constanta Ser Mat 2009, 17: 51–68.

    MathSciNet  MATH  Google Scholar 

  13. Qin X, Cho YJ, Kang SM, Zhou H: Convergence of a modified Halpern-type iteration algorithm for quasi-ϕ-nonexpansive mappings. Appl Math Lett 2009, 22: 1051–1055. 10.1016/j.aml.2009.01.015

    Article  MathSciNet  MATH  Google Scholar 

  14. Song Y, Cho YJ: Some notes on Ishikawa iteration for multi-valued mappings. Bull Korean Math Soc 2011, 48: 575–584. 10.4134/BKMS.2011.48.3.575

    Article  MathSciNet  MATH  Google Scholar 

  15. Yao Y, Cho YJ: A strong convergence of a modified Krasnoselskii-Mann method for non-expansive mappings in Hilbert spaces. Math Model Anal 2010, 15: 265–274. 10.3846/1392-6292.2010.15.265-274

    Article  MathSciNet  MATH  Google Scholar 

  16. Alber YI: Metric and Generalized Projection Operators in Banach Spaces. Marcel Dekker, New York; 1996:15–50.

    Google Scholar 

  17. Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J Optim 2002, 13(3):938–945. 10.1137/S105262340139611X

    Article  MathSciNet  MATH  Google Scholar 

  18. Chang SS, Kim JK, Wang XR: Modified Block Iterative Algorithm for Solving Convex feasibility Problems in Banach spaces. J Inequal Appl 2010, 2010: 14. Article ID 869684

    MathSciNet  MATH  Google Scholar 

  19. Alber YI: Metric and generalized projection operators in Banach spaces:properties and applications. In Theory and Applications of Nonlinear operators Of Accretive And Monotone Type, Lecture Notes in Pure and Applied Mathematics. Volume 178. Dekker, New York; 1996:15–50.

    Google Scholar 

  20. Kohsaka F, Takahashi W: Strong convergence of an iterative sequence for maximal monotone operators in a Banach spaces. Abstr Appl Anal 2004, 2004: 239–249. 10.1155/S1085337504309036

    Article  MathSciNet  MATH  Google Scholar 

  21. Xu HK: Another control condition in an iterative method for nonexpansive mappings. Bull Austral Math Soc 2002, 65: 109–113. 10.1017/S0004972700020116

    Article  MathSciNet  MATH  Google Scholar 

  22. Mainge PE: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Jset-Valued Anal 2008, 16: 899–912. 10.1007/s11228-008-0102-z

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This study was supported by Scientific Research Fund of Sichuan Provincial Education Department (11ZB146) and Yunnan University of Finance and Economics.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Shih-sen Chang.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All the authors contributed equally to the writing of the present article. And they also read and approved the final manuscript.

Rights and permissions

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

Reprints and permissions

About this article

Cite this article

Zhu, Jh., Chang, Ss. & Liu, M. Strong convergence theorems by Halpern-Mann iterations for multi-valued relatively nonexpansive mappings in Banach spaces with applications. J Inequal Appl 2012, 73 (2012). https://doi.org/10.1186/1029-242X-2012-73

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2012-73

Keywords