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Convergence of the modified Mann's iterative method for asymptotically kappa-strictly pseudocontractive mappings Fixed Point Theory and Applications 2011, 2011:100 doi:10.1186/1687-1812-

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Convergence of the modified Mann's iterative method for asymptotically

kappa-strictly pseudocontractive mappings

Fixed Point Theory and Applications 2011, 2011:100 doi:10.1186/1687-1812-2011-100

Ying Zhang (spzhangying@126.com) Zhiwei Xie (betterwill@gmail.com)

ISSN 1687-1812

Article type Research

Submission date 4 May 2011

Acceptance date 9 December 2011

Publication date 9 December 2011

Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/100

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Convergence of the modified Mann’s iterative

method for asymptotically κ-strictly

pseudocontractive mappings

Ying Zhang∗,1,2 and Zhiwei Xie3

1School of Mathematics and Physics, North China Electric Power University, Baoding, Hebei 071003, P.R China

2School of Economics, Renmin University of China,

Beijing 100872, P.R China

3Easyway Company Limited, Beijing 100872, P.R China

*Corresponding author: spzhangying@126.com

Email address:

ZX: betterwill@gmail.com

Abstract

Let E be a real uniformly convex Banach space which has the Fr´ echet differentiable

norm, and K a nonempty, closed, and convex subset of E Let T : K → K be an asymp-totically κ-strictly pseudocontractive mapping with a nonempty fixed point set We prove that (I − T ) is demiclosed at 0 and obtain a weak convergence theorem of the modi-fied Mann’s algorithm for T under suitable control conditions Moreover, we also elicit

a necessary and sufficient condition that guarantees strong convergence of the modified

Mann’s iterative sequence to a fixed point of T in a real Banach spaces with the Fr´ echet

differentiable norm

2000 AMS Subject Classification: 47H09; 47H10

Keywords: asymptotically κ-strictly pseudocontractive mappings; demiclosedness

prin-ciple; the modified Mann’s algorithm; fixed points

1 Introduction

Let E and E ∗ be a real Banach space and the dual space of E, respectively Let K be a nonempty subset of E Let J denote the normalized duality mapping from E into 2 E ∗

given

by J(x) = {f ∈ E ∗ : hx, f i = kxk2 = kf k2}, for all x ∈ E, where h·, ·i denotes the duality

1

pairing between E and E ∗ In the sequel, we will denote the set of fixed points of a mapping

T : K → K by F (T ) = {x ∈ K : T x = x}.

A mapping T : K → K is called asymptotically κ-strictly pseudocontractive with sequence

{κ n } ∞

n=1 ⊆ [1, ∞) such that lim n→∞ κ n = 1 (see, e.g., [1–3]) if for all x, y ∈ K, there exist a constant κ ∈ [0, 1) and j(x − y) ∈ J(x − y) such that

hT n x − T n y, j(x − y)i ≤ κnkx − yk2− κkx − y − (T n x − T n y)k2, ∀n ≥ 1. (1)

If I denotes the identity operator, then (1) can be written as

h(I −T n )x−(I −T n )y, j(x−y)i ≥ κk(I −T n )x−(I −T n )yk2−(κ n −1)kx−yk2, ∀n ≥ 1 (2)

The class of asymptotically κ-strictly pseudocontractive mappings was first introduced in Hilbert spaces by Qihou [3] In Hilbert spaces, j is the identity and it is shown by Osilike

et al [2] that (1) (and hence (2)) is equivalent to the inequality

kT n x − T n yk2 ≤ λnkx − yk2+ λkx − y − (T n x − T n y)k2,

where limn→∞ λn = limn→∞ [1 + 2(κ n − 1)] = 1, λ = (1 − 2κ) ∈ [0, 1).

A mapping T with domain D(T ) and range R(T ) in E is called strictly pseudocontractive of Browder–Petryshyn type [4], if for all x, y ∈ D(T ), there exists κ ∈ [0, 1) and j(x−y) ∈ J(x − y)

such that

If I denotes the identity operator, then (3) can be written as

In Hilbert spaces, (3) (and hence (4)) is equivalent to the inequality

kT x − T yk2 ≤ kx − yk2+ kkx − y − (T x − T y)k2, k = (1 − 2κ) < 1,

It is shown in [5] that the class of asymptotically κ-strictly pseudocontractive mappings and the class of κ-strictly pseudocontractive mappings are independent.

A mapping T is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such

that

kT n x − T n yk ≤ Lkx − yk, n ≥ 1

for all x, y ∈ K and is said to be demiclosed at a point p if whenever {x n} ⊂ D(T ) such that {xn} converges weakly to x ∈ D(T ) and {T xn} converges strongly to p, then T x = p.

Kim and Xu [6] studied weak and strong convergence theorems for the class of asymptotically

κ-strictly pseudocontractive mappings in Hilbert space They obtained a weak convergence

theorem of modified Mann iterative processes for this class of mappings Moreover, a strong convergence theorem was also established in a real Hilbert space by hybrid projection method They proved the following

Theorem KX [6] Let K be a closed and convex subset of a Hilbert space H Let T : K → K

be an asymptotically κ-strictly pseudocontractive mapping for some 0 ≤ κ < 1 with sequence

{κ n } ⊂ [1, ∞) such thatP∞ n=1 (κ n −1) < ∞ and F (T ) 6= ∅ Let {x n } ∞

n=1be a sequence generated

by the modified Mann’s iteration method:

x n+1 = α n x n + (1 − α n )T n x n , n ≥ 1,

Assume that the control sequence {α n} ∞

n=1 is chosen in such a way that κ + λ ≤ α n ≤ 1 − λ

for all n, where λ ∈ (0, 1) is a small enough constant Then, {x n} converges weakly to a fixed

point of T.

The modified Mann’s iteration scheme was introduced by Schu [7, 8] and has been used by several authors (see, for example, [1–3, 9–11]) One question is raised naturally: is the result

in Theorem KX true in the framework of the much general Banach space?

Osilike et al [5] proved the convergence theorems of modified Mann iteration method in

the framework of q-uniformly smooth Banach spaces which are also uniformly convex They also obtained that a modified Mann iterative process {x n } converges weakly to a fixed point

of T under suitable control conditions However, the control sequence {α n } ⊂ [0, 1] depended

on the Lipschizian constant L and excluded the natural choice α n = 1

n , n ≥ 1 These are

motivations for us to improve the results We prove the demiclosedness principle and weak

convergence theorem of the modified Mann’s algorithm for T in the framework of uniformly convex Banach spaces which have the Fr´echet differentiable norm Moreover, we also elicit a

necessary and sufficient condition that guarantees strong convergence of the modified Mann’s

iterative sequence to a fixed point of T in a real Banach spaces with the Fr´echet differentiable

norm

We will use the notation:

1 * for weak convergence.

2 ω W (x n ) = {x : ∃x n j * x} denotes the weak ω-limit set of {x n }.

2 Preliminaries

Let E be a real Banach space The space E is called uniformly convex if for each ² > 0, there exists a δ > 0 such that for x, y ∈ E with kxk ≤ 1, kyk ≤ 1, kx − yk ≥ ², we have

k1

2(x + y)k ≤ 1 − δ The modulus of convexity of E is defined by

δ E (²) = inf{1 − k1

2(x + y)k : kxk ≤ 1, kyk ≤ 1, kx − yk ≥ ², } ∀x, y ∈ E

for all ² ∈ [0, 2] E is uniformly convex if δ E (0) = 0 and δ E (²) > 0 for all ² ∈ (0, 2] The modulus

of smoothness of E is the function ρE : [0, ∞) → [0, ∞) defined by

ρE (τ ) = sup{1

2(kx + yk + kx − yk) − 1 : kxk ≤ 1, kyk ≤ τ }, ∀x, y ∈ E.

E is uniformly smooth if and only if lim τ →0 ρ E(τ ) τ = 0.

E is said to have a Fr´ echet differentiable norm if for all x ∈ U = {x ∈ E : kxk = 1}

lim

t→0

kx + tyk − kxk

t

exists and is attained uniformly in y ∈ U In this case, there exists an increasing function

b : [0, ∞) → [0, ∞) with lim t→0 [b(t)/t] = 0 such that for all x, h ∈ E

1

2kxk

2+ hh, j(x)i ≤ 1

2kx + hk

2 ≤ 1

2kxk

It is well known (see, for example, [12, p 107]) that uniformly smooth Banach space has a Fr´echet differentiable norm.

Lemma 2.1 [2, p 80] Let {a n } ∞

n=1 , {b n } ∞

n=1 , {δ n } ∞

n=1 be nonnegative sequences of real numbers satisfying the following inequality

a n+1 ≤ (1 + δ n )a n + b n , ∀n ≥ 1.

If P∞ n=1 δ n < ∞ and P∞ n=1 b n < ∞, then lim n→∞ a n exists If in addition {a n } ∞

n=1 has a subsequence which converges strongly to zero, then limn→∞ a n = 0.

Lemma 2.2 [2, p 78] Let E be a real Banach space, K a nonempty subset of E, and

T : K → K an asymptotically κ-strictly pseudocontractive mapping Then, T is uniformly L-Lipschitzian.

Lemma 2.3 [13, p 29] Let K be a nonempty, closed, convex, and bounded subset of a uniformly convex Banach space E, and let T : K → E be a nonexpansive mappings Let {x n}

be a sequence in K such that {x n} converges weakly to some point x ∈ K Then, there exists

an increasing continuous function h : [0, ∞) → [0, ∞) with h(0) = 0 depending on the diameter

of K such that

h(kx − T xk) ≤ lim inf

n→∞ kx n − T x n k.

Lemma 2.4 [14, p 9] Let E be a real Banach space with the Fr´echet differentiable norm For x ∈ E, let β ∗ (t) be defined for 0 < t < ∞ by

β ∗ (t) = sup

y∈U

¯

¯

¯

¯kx + tyk

2− kxk2

¯

¯

¯

¯

Then, limt→0+β ∗ (t) = 0 and

Remark 2.5 In a real Hilbert space, we can see that β ∗ (t) = t for t > 0 In our more general

setting, throughout this article we will still assume that

β ∗ (t) ≤ 2t,

where β ∗ is a function appearing in (6)

Then, we prove the demiclosedness principle of T in the uniformly convex Banach space which has the Fr´echet differentiable norm.

Lemma 2.6 Let E be a real uniformly convex Banach space which has the Fr´echet differ-entiable norm Let K be a nonempty, closed, and convex subset of E and T : K → K an asymptotically κ-strictly pseudocontractive mapping with F (T ) 6= ∅ Then, (I − T ) is demi-closed at 0.

Proof Let {x n } be a sequence in K which converges weakly to p ∈ K and {x n − T x n }

converges strongly to 0 We prove that (I − T )(p) = 0 Let x ∗ ∈ F (T ) Then, there exists a

constant r > 0 such that kx n − x ∗ k ≤ r, ∀n ≥ 1 Let ¯ Br = {x ∈ E : kx − x ∗ k ≤ r}, and let

C = K ∩ ¯ Br Then, C is nonempty, closed, convex, and bounded, and {xn} ⊆ C Choose any

α ∈ (0, κ) and let Tα,n : K → K be defined for all x ∈ K by

T α,n x = (1 − α)x + αT n x, n ≥ 1,

Then for all x, y ∈ K,

kT α,n x − T α,n yk2 = k(x − y) − α[(I − T n )x − (I − T n )y]k2

≤ kx − yk2− 2αh(I − T n )x − (I − T n )y, j(x − y)i +αkx − y − (T n x − T n y)kβ ∗ [αkx − y − (T n x − T n y)k] (7)

≤ kx − yk2− 2α[κkx − y − (T n x − T n y)k2− (κn − 1)kx − yk2]

+2α2kx − y − (T n x − T n y)k2

= [1 + 2α(κ n − 1)]kx − yk2− 2α(κ − α)kx − y − (T n x − T n y)k2

n kx − yk2,

where τ n = [1+2α(κ n −1)]12 (In fact, in (7) the domain of β ∗ (·) requires kx−y−(T n x−T n y)k 6=

0 But when kx−y −(T n x−T n y)k = 0, we have kT α,n x−T α,n yk2 = kx−yk2, which still satisfies

the inequality kT α,n x − T α,n yk2 ≤ τ2

n kx − yk2 So we do not specially emphasize the situation

that the argument of β ∗ (·) equals 0 in this inequality and the following proof of Theorem 3.1.) Define G α,m : K → E by

G α,m x = 1

τm T α,m x, m ≥ 1.

Then, G α,m is nonexpansive and it follows from Lemma 2.3 that there exists an increasing

continuous function h : [0, ∞) → [0, ∞) with h(0) = 0 depending on the diameter of K such

that

h(kp − Gα,mpk) ≤ lim inf

Observe that

τ m Tα,mxnk

≤ kx n − T α,m x n k + (1 − 1

τm )(τ m kx n − x

∗ k + kx ∗ k)

≤ kx n − T α,m x n k + (1 − 1

τ m )(τ m r + kx

and as n → ∞

m

X

j=1

kT j−1 xn −T j xnk ≤ [1 + L(m − 1)]kxn−T xnk → 0 (10)

Thus, it follows from (9) and (10) that

lim sup

n→∞

kx n − G α,m x n k ≤ (1 − 1

τm )(τ m r + kx

∗ k),

so that (8) implies that

h(kp − G α,m pk) ≤ (1 − 1

τ m )(τ m r + kx

∗ k).

Observe that

τ m )kT α,mpk

≥ kp − T α,m pk − (1 − 1

τm )(τ m r + kx

∗ k),

so that

kp − T α,m pk ≤ kp − G α,m pk + (1 − 1

τ m )(τ m r + kx

∗ k)

≤ h −1 [(1 − 1

τ m )(τ mr + kx

∗ k)] + (1 − 1

τ m )(τ mr + kx

∗ k) → 0, as m → ∞.

Since T is continuous, we have (I − T )(p) = 0, completing the proof of Lemma 2.6. ¤

Lemma 2.7 Let E be a real uniformly convex Banach space which has the Fr´echet differ-entiable norm, and let K be a nonempty, closed, and convex subset of E Let T : K → K be

an asymptotically κ-strictly pseudocontractive mapping with F (T ) 6= ∅ Let {x n} ∞

n=1 be the sequence satisfying the following conditions:

(a) lim

n→∞ kx n − pk exists f or every p ∈ F (T );

(b) lim

n→∞ kx n − T x n k = 0;

(c) lim

n→∞ ktx n + (1 − t)p1− p2k exists f or all t ∈ [0, 1] and f or all p1, p2 ∈ F (T ).

Then, the sequence {x n } converges weakly to a fixed point of T.

Proof Since lim n→∞ kx n − pk exists, then {x n } is bounded By (b) and Lemma 2.6, we have

ω W (x n ) ⊂ F (T ) Assume that p1, p2 ∈ ω W (x n ) and that {x n i } and {x m j } are subsequences of {x n } such that x n i * p1 and x m j * p2, respectively Since E has the Fr´ echet differentiable

norm, by setting x = p1− p2, h = t(x n − p1) in (5) we obtain

1

2kp1− p2k

2+ thx n − p1, j(p1− p2)i ≤ 1

2ktxn + (1 − t)p1− p2k

2

2kp1− p2k

2+ thx n − p1, j(p1− p2)i + b(tkx n − p1k),

where b is an increasing function Since kx n − p1k ≤ M, ∀n ≥ 1, for some M > 0, then

1

2kp1− p2k

2+ thx n − p1, j(p1− p2)i ≤ 1

2ktx n + (1 − t)p1− p2k

2

2kp1− p2k

2+ thx n − p1, j(p1− p2)i + b(tM).

1

2kp1− p2k

2+ t lim sup

n→∞ hxn − p1, j(p1− p2)i ≤ 1

2n→∞lim ktxn + (1 − t)p1− p2k2

2kp1− p2k

2 + t lim inf

n→∞ hx n − p1, j(p1− p2)i + b(tM).

Hence, lim supn→∞ hx n − p1, j(p1− p2)i ≤ lim inf n→∞ hx n − p1, j(p1− p2)i+ b(tM ) t Since lim

t→0+

b(tM )

0, then lim n→∞hxn − p1, j(p1− p2)i exists Since lim n→∞hxn − p1, j(p1− p2)i = hp − p1, j(p1− p2)i, for all p ∈ ω W (x n ) Set p = p2 We have hp2− p1, j(p1− p2)i = 0, that is, p2 = p1 Hence,

ωW (x n ) is singleton, so that {x n} converges weakly to a fixed point of T. ¤

3 Main results

Theorem 3.1 Let E be a real uniformly convex Banach space which has the Fr´echet differ-entiable norm, and let K be a nonempty, closed, and convex subset of E Let T : K → K

be an asymptotically κ-strictly pseudocontractive mapping for some 0 ≤ κ < 1 with sequence

{κ n } ∞

n=1 ⊂ [1, ∞) such that P∞ n=1 (κ n − 1) < ∞, and let F (T ) 6= ∅ Assume that the control

sequence {α n } ∞

n=1 is chosen so that (i∗ ) 0 < α n < κ, n ≥ 1;

(ii∗) P∞

n=1

Given x1 ∈ K, then the sequence {x n } ∞

n=1 is generated by the modified Mann’s algorithm:

converges weakly to a fixed point of T.

Proof Pick a p ∈ F (T ) We firstly show that lim n→∞ kx n − pk exists To see this, using (2)

and (6), we obtain

kxn+1 − pk2 = k(x n − p) − αn (x n − T n xn )k2

≤ kx n − pk2− 2α n hx n − T n x n , j(x n − p)i + α n kx n − T n x n kβ ∗ (α n kx n − T n x n k)

≤ kxn − pk2− 2αn [κkx n − T n xnk2 − (κn − 1)kxn − pk2] + 2α2n kxn − T n xnk2

= [1 + 2α n (κ n − 1)]kx n − pk2− 2α n (κ − α n )kx n − T n x n k2. (13) Obviously,

Let δ n = 1 + 2α n (κ n − 1) Since P∞ n=1 (κ n − 1) < ∞, we have

∞

X

n=1

(δ n − 1) ≤ 2

∞

X

n=1

(κ n − 1) < ∞,

then (14) implies limn→∞ kx n − pk exists by Lemma 2.1 (and hence the sequence {kx n − pk} is

bounded, that is, there exists a constant M > 0 such that kx n − pk < M ).

Then, we prove limn→∞ kx n − T x n k = 0 In fact, it follows from (13) that

j

X

n=1

2α n (κ − α n )kx n − T n x n k2 ≤

j

X

n=1

(kx n − pk2− kx n+1 − pk2) +

j

X

n=1

[2α n (κ n − 1)]kx n − pk2

≤

j

X

n=1

(kx n − pk2− kx n+1 − pk2) +

j

X

n=1

(δ n − 1)M2.

Then,

∞

X

n=1

2α n (κ − α n )kx n − T n x n k2 < kx1− pk2+ M2

∞

X

n=1

Since P∞ n=1 α n (κ − α n ) = ∞, then (15) implies that lim inf n→∞ kx n − T n x n k = 0 Thus

limn→∞ kx n − T n x n k = 0.

By Lemma 2.2 we know that T is uniformly L-Lipschitzian, then there exists a constant

L > 0, such that

kxn − T xnk ≤ kxn − T n xnk + kT n xn − T xnk ≤ kxn − T n xnk + LkT n−1 xn − xnk

≤ kx n − T n x n k + LkT n−1 x n − T n−1 x n−1 k + LkT n−1 x n−1 − x n k

≤ kx n − T n x n k + L2kx n − x n−1 k + LkT n−1 x n−1 − x n−1 k + Lkx n − x n−1 k

< kx n − T n x n k + L(2 + L)kT n−1 x n−1 − x n−1 k

Hence, limn→∞ kx n − T x n k = 0.

Now we prove that for all p1, p2 ∈ F (T ), lim n→∞ ktx n + (1 − t)p1 − p2k exists for all t ∈

[0, 1] Let σ n (t) = ktx n + (1 − t)p1 − p2k It is obvious that lim n→∞ σ n (0) = kp1 − p2k and

limn→∞ σ n(1) = limn→∞ kx n − p2k exist So, we only need to consider the case of t ∈ (0, 1).

Define T n : K → K by

Tnx = (1 − αn )x + α nT n x, x ∈ K.

Then for all x, y ∈ K,

kTnx − Tnyk2 = k(x − y) − α n [(I − T n )x − (I − T n )y]k2

≤ kx − yk2− 2α n h(I − T n )x − (I − T n )y, j(x − y)i +α n kx − y − (T n x − T n y)kβ ∗ [α n kx − y − (T n x − T n y)k]

≤ kx − yk2− 2α n [κkx − y − (T n x − T n y)k2− (κ n − 1)kx − yk2]

+2α2

n kx − y − (T n x − T n y)k2

= [1 + 2α n (κ n − 1)]kx − yk2− 2α n (κ − α n )kx − y − (T n x − T n y)k2.

By the choice of α n , we have kT n x − T n yk2 ≤ [1 + 2α n (κ n − 1)]kx − yk2 For the convenience

of the following discussing, set λ n = [1 + 2α n (κ n − 1)]12, then kT n x − T n yk ≤ λ n kx − yk.

Set S n,m = T n+m−1 T n+m−2 · · · T n , m ≥ 1 We have

kS n,m x − S n,m yk ≤ ( n+m−1Q

j=n

λ j )kx − yk f or all x, y ∈ K,

S n,m x n = x n+m , S n,m p = p f or all p ∈ F (T ).

Set b n,m = kS n,m (tx n + (1 − t)p1) − tS n,m x n − (1 − t)S n,m p1k If kx n − p1k = 0 for some n0,

then x n = p1 for any n ≥ n0 so that limn→∞ kx n − p1k = 0, in fact {x n } converges strongly to

p1 ∈ F (T ) Thus, we may assume kxn − p1k > 0 for any n ≥ 1 Let δ denote the modulus of convexity of E It is well known (see, for example, [15, p 108]) that

ktx + (1 − t)yk ≤ 1 − 2 min{t, (1 − t)}δ(kx − yk)

for all t ∈ [0, 1] and for all x, y ∈ E such that kxk ≤ 1, kyk ≤ 1 Set

w n,m = S n,m p1− S n,m (tx n + (1 − t)p1)

t(Qn+m−1 j=n λj )kx n − p1k

z n,m = S n,m (tx n + (1 − t)p1) − S n,m x n

(1 − t)(Qn+m−1 j=n λj )kx n − p1k

Then, kw n,mk ≤ 1 and kzn,mk ≤ 1 so that it follows from (16) that

2t(1 − t)δ(kw n,m − z n,m k) ≤ 1 − ktw n,m + (1 − t)z n,m k. (17) Observe that

t(1 − t)(Qn+m−1 j=n λj )kx n − p1k

and

ktw n,m + (1 − t)z n,m k = kS n,m x n − S n,m p1k

(Qn+m−1 j=n λj )kx n − p1k ,

it follows from (17) that

2t(1 − t)

Ã

n+m−1Y

j=n

λ j

!

kx n − p1kδ











b n,m t(1 − t)( n+m−1Q

j=n

λj )kx n − p1k











≤

Y

j=n

λj

!

kxn − p1k − kSn,mxn − Sn,mp1k =

Y

j=n λj

!

kxn − p1k − kxn+m − p1k.(18)

Since E is uniformly convex, then δ(s) s is nondecreasing, and since (Qn+m−1 j=n λ j )kx n − p1k ≤

(Qn+m−1 j=n λ j )λ n−1 kx n−1 − p1k ≤ · · · ≤ (Qn+m−1 j=n λ j)(Qn−1 j=1 λ j )kx1− p1k = (Qn+m−1 j=1 λ j )kx1− p1k,