Paper-19-20-2009. 117KB Jun 04 2011 12:03:11 AM

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CONVERGENCE THEOREMS FOR UNIFORMLY L-LIPSCHITZIAN ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

Arif Rafiq

Abstract. LetKbe a nonempty closed convex subset of a real Banach spaceE, T :K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n≥0 ⊂[1,∞), lim

n→∞kn= 1 such that p∈F(T) ={x∈K :T x= x}. Let{an}n≥0,{bn}n≥0,{cn}n≥0 be real sequences in [0,1] satisfying the following

conditions: (i)an+bn+cn= 1; (ii)P_n≥0bn=∞; (iii)cn=o(bn); (iv) lim

n→∞bn= 0. For arbitraryx0∈K let {xn}n≥0 be iteratively defined by

xn+1 =anxn+bnTnxn+cnun, n≥0,

where {un}n≥0 is a bounded sequence of error terms in K. Suppose there exists a

strictly increasing function ψ: [0,∞)→[0,∞), ψ(0) = 0 such that

hTnx−p, j(x−p)i ≤kn||x−p||2−ψ(||x−p||), ∀x∈K.

Then {xn}n≥0 converges strongly to p∈F(T).

The results proved in this paper significantly improve the results of Ofoedu [11]. The remark 3 is important.

2000 Mathematics Subject Classification: Primary 47H10, 47H17: Secondary 54H25.

1.Introduction

LetE be a real normed space andK be a nonempty convex subset of E. LetJ denote the normalized duality mapping fromE to 2E∗

defined by J(x) ={f∗ ∈E∗ :hx, f∗i=||x||2 and ||f∗||=||x||},

where E∗ _{denotes the dual space of} _E _and _h·_,_·i _{denotes the generalized duality} pairing. We shall denote the single-valued duality map by j.

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Definition 1 The mapping T is said to be uniformly L-Lipschitzian if there exists

L >0 such that for all x, y∈D(T)

kTnx−Tnyk ≤Lkx−yk.

Definition 2 T is said to be nonexpansive if for all x, y ∈ D(T), the following inequality holds:

kT x−T yk ≤ kx−y k for all x, y∈D(T).

Definition 3 T is said to be asymptotically nonexpansive [6], if there exists a se-quence {kn}n≥0 ⊂[1,∞) with lim

n→∞kn= 1 such that

kTnx−Tnyk ≤knkx−y k for all x, y∈D(T), n≥1.

Definition 4 T is said to be asymptotically pseudocontractive if there exists a se-quence {kn}n≥0 ⊂[1,∞) with lim

n→∞kn= 1 and there exists j(x−y)∈J(x−y) such

that

hTnx−Tny, j(x−y)i ≤knkx−y k 2

for allx, y∈D(T), n≥1.

Remark 1 1. It is easy to see that every asymptotically nonexpansive mapping is uniformly L-Lipschitzian.

2. If T is asymptotically nonexpansive mapping then for all x, y ∈ D(T) there exists j(x−y)∈J(x−y) such that

hTnx−Tny, j(x−y)i ≤ kTnx−Tnyk kx−y k ≤ knkx−y k

2

, n≥1.

Hence every asymptotically nonexpansive mapping is asymptotically pseudocontrac-tive.

3. Rhoades in [12] showed that the class of asymptotically pseudocontractive mappings properly contains the class of asymptotically nonexpansive mappings.

The asymptotically pseudocontractive mappings were introduced by Schu [13] who proved the following theorem:

Theorem 1 LetKbe a nonempty bounded closed convex subset of a Hilbert spaceH, T : K → K a completely continuous, uniformly L-Lipschitzian and asymptotically pseudocontractive with sequence {kn} ⊂[1,∞);qn= 2kn−1,∀n∈N;P(qn2−1)< ∞; {αn},{βn} ⊂ [0,1]; ǫ < αn < βn ≤ b, ∀n ∈ N, and some ǫ > 0 and some

b∈(0, L−2

[(1 +L2

)12 −1]);x

1∈K for alln∈N, define

xn+1= (1−αn)xn+αnTnxn.

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The recursion formula of theorem 1 is a modification of the well-known Mann iteration process (see [9]).

Recently, Chang [1] extended Theorem 1 to real uniformly smooth Banach space; in fact, he proved the following theorem:

Theorem 2 Let K be a nonempty bounded closed convex subset of a real uniformly smooth Banach space E, T :K → K an asymptotically pseudocontractive mapping with sequence {kn}n≥0 ⊂[1,∞), lim

n→∞kn = 1, and x

∗ _∈_F_{(T) =} _{_x _∈_K _:_{T x}₌_x_}_.

Let {αn} ⊂ [0,1] satisfying the following conditions: lim

n→∞αn = 0,

P

αn = ∞. For

arbitrary x0 ∈K let {xn} be iteratively defined by

xn+1 = (1−αn)xn+αnTnxn, n≥0.

If there exists a strictly increasing function ψ: [0,∞)→[0,∞), ψ(0) = 0 such that

hTnx−x∗, j(x−x∗)i ≤kn||x−x∗||2−ψ(||x−x∗||), ∀n∈N,

then xn→x∗ ∈F(T).

Remark 2 Theorem 2, as stated is a modification of Theorem 2.4 of Chang [1] who actually included error terms in his algorithm.

In [11], E. U. Ofoedu proved the following results.

Theorem 3 Let K be a nonempty closed convex subset of a real Banach space E, T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n≥0 ⊂ [1,∞), lim

n→∞kn = 1 such that x

∗ _∈ _F_(T_{) =} _{_x _∈ _K _: T x = x}. Let {αn}n≥0 ⊂ [0,1] be such that

P

n≥0αn = ∞,

P

n≥0α 2

n < ∞ and

P

n≥0αn(kn−1)<∞. For arbitraryx0∈K let {xn}n≥0 be iteratively defined by

Suppose there exists a strictly increasing functionψ: [0,∞)→[0,∞), ψ(0) = 0such that

hTnx−x∗, j(x−x∗)i ≤kn||x−x∗||2−ψ(||x−x∗||), ∀x∈K.

Then {xn}n≥0 is bounded.

Theorem 4 Let K be a nonempty closed convex subset of a real Banach space E, T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n≥0 ⊂ [1,∞), lim

n→∞kn = 1 such that x

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T x = x}. Let {αn}n≥0 ⊂ [0,1] be such that

P

n≥0αn = ∞,

P

n≥0α 2

n < ∞ and

P

n≥0αn(kn−1)<∞. For arbitraryx0∈K let {xn}n≥0 be iteratively defined by

hTnx−x∗, j(x−x∗)i ≤kn||x−x∗||2−ψ(||x−x∗||), ∀x∈K.

Then {xn}n≥0 converges strongly to x∗ ∈F(T).

Theorem 5 Let K be a nonempty closed convex subset of a real Banach space E, T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n≥0 ⊂[1,∞), lim

n→∞kn= 1 such that x

∗ _∈_F_{(T) =}_{_x_∈_K_:_{T x}₌ x}. Let {an}n≥0,{bn}_n≥0,{cn}n≥0 be real sequences in [0,1] satisfying the following conditions:

i) an+bn+cn= 1;

ii) P

n≥0(bn+cn) =∞;

iii) P

n≥0(bn+cn) 2 _<

∞; iv) P

n≥0(bn+cn)(kn−1)<∞;and

v) P

n≥0cn<∞.

For arbitrary x0 ∈K let {xn}n≥0 be iteratively defined by

xn+1 =anxn+bnTnxn+cnun, n≥0,

where {un}n≥0 is a bounded sequence of error terms in K. Suppose there exists a

strictly increasing function ψ: [0,∞)→[0,∞), ψ(0) = 0 such that

hTnx−x∗, j(x−x∗)i ≤kn||x−x∗|| 2

−ψ(||x−x∗||), ∀x∈K.

Then {xn}n≥0 converges strongly to x∗ ∈F(T).

Remark 3 One can easily see that if we take in theorems 3 and 4, αn = _n1σ; 0 < σ <1,thenP

αn=∞,butPα2n=∞. Hence the conclusions of theorems 3, 4 and

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In this paper our purpose is to improve the results of Ofoedu [11] in a significantly more general context by removing the conditionsP

n≥0α 2

n<∞and

P

n≥0αn(kn−

1) < ∞ from the theorems 3 −4. We also significantly extend theorem 2 from uniformly smooth Banach space to arbitrary real Banach space. The boundedness assumption imposed on K in the theorem is also dispensed with.

2.Main Results The following lemmas are now well known.

Lemma 6 [14] Let J :E → 2E _{be the normalized duality mapping. Then for any}

x, y∈E,we have

||x+y||2

≤ ||x||2

+ 2hy, j(x+y)i, ∀j(x+y)∈J(x+y).

Suppose there exists a strictly increasing functionψ: [0,∞)→[0,∞)withψ(0) = 0. Lemma 7 [10] Let {θn} be a sequence of nonnegative real numbers, {λn} be a real

sequence satisfying

0≤λn≤1,

∞

X

n=0

λn=∞

and let ψ∈Ψ. If there exists a positive integer n0 such that

θ2_n+1 ≤θ 2

n−λnψ(θn+1) +σn,

for all n≥n0,withσn≥0,∀n∈N,and σn= 0(λn), thenlimn→∞θn= 0.

Theorem 8 Let K be a nonempty closed convex subset of a real Banach space E, T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n≥0 ⊂[1,∞), lim

n→∞kn = 1 such that p∈F(T) = {x ∈K :T x= x}. Let {an}n≥0,{bn}n≥0,{cn}n≥0 be real sequences in [0,1] satisfying the following conditions:

i) an+bn+cn= 1;

ii) P

n≥0bn=∞;

iii) cn=o(bn);

iv) lim

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For arbitrary x0 ∈K let {xn}n≥0 be iteratively defined by

xn+1 =anxn+bnTnxn+cnun, n≥0, (2.1)

where {un}n≥0 is a bounded sequence of error terms in K. Suppose there exists a

strictly increasing function ψ: [0,∞)→[0,∞), ψ(0) = 0 such that

hTnx−p, j(x−p)i ≤kn||x−p||2−ψ(||x−p||), ∀x∈K. (2.2)

Then {xn}n≥0 converges strongly to p∈F(T).

Proof. From the conditioncn=o(bn) impliescn=tnbn,wheretn→0 as n→ ∞.

Since p is a fixed point ofT, then the set of fixed pointsF(T) of T is nonempty. Since the sequence {un}n≥0 is bounded, we set

M = sup

n≥0

kun−pk.

By lim

n→∞bn= 0 = limn→∞tnimply there exists n0 ∈

N _{such that}_∀_n_≥_n₀_{, b}_n_≤_δ;

0 < δ= min

1 18[φ−1

(a0)]2

, φ

−1

(a0)

2(2 +L)φ−1

(a0) +M

,

φ(2φ−1

(a0))

12(1 +L)φ−1(a0)

2(2 +L)φ−1(a0) +M

)

,

and

tn≤

φ(2φ−1(a0))

12φ−1

(a0)(M+ 2φ−1(a0))

.

Define a0 :=kxn0 −T

n0_x

n0kkxn0 −pk+ (kn0 −1)kxn0 −pk

2

. Then from (2.2), we obtain that kxn0 −pk ≤φ

−1

(a0).

CLAIM. kxn−pk ≤2φ−1(a0) ∀n≥n0.

The proof is by induction. Clearly, the claim holds forn=n0. Suppose it holds

for some n≥n0, i.e.,kxn−pk ≤2φ−1(a0). We prove that kxn+1−pk ≤2φ−1(a0).

Suppose that this is not true. Thenkxn+1−pk>2φ−1(a0)), so thatφ(kxn+1−pk)>

φ(2φ−1

(a0)). Using the recursion formula (2.1), we have the following estimates kxn−Tnxnk ≤ kxn−pk+kp−Tnxnk

≤ (1 +L)kxn−pk ≤ 2(1 +L)φ−1

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kxn+1−pk = kanxn+bnTnxn+cnun−pk

= kxn−p−bn(xn−Tnxn) +cn(un−xn)k ≤ kxn−pk+bnkxn−Tnxnk+cnkun−xnk ≤ 2φ−1

(a0) + 2(1 +L)φ−1(a0)bn+ (M+ 2φ−1(a0))cn ≤ 2φ−1

(a0) + [2(2 +L)φ−1(a0) +M]bn ≤ 3φ−1(a0).

With these estimates and again using the recursion formula (2.1), we obtain by lemma 1 that

kxn+1−pk2 = kanxn+bnTnxn+cnun−pk2

= kxn−p−bn(xn−Tnxn) +cn(un−xn)k ≤ kxn−pk2−2bnhxn−Tnxn, j(xn+1−p)i

+2cnhun−xn, j(xn+1−p)i

= kxn−pk2+ 2bnhTnxn+1−p, j(xn+1−p)i −2bnhxn+1−p, j(xn+1−p)i

+2bnhTnxn−Tnxn+1, j(xn+1−p)i

+2bnhxn+1−xn, j(xn+1−p)i

+2cnhun−xn, j(xn+1−p)i

≤ kxn−pk2+ 2bn kn||xn+1−p||2−φ(||xn+1−p||)

−2bnkxn+1−pk2+ 2bnkTnxn−Tnxn+1k ||xn+1−p||

+2bnkxn+1−xnk ||xn+1−p||

+2cn(M+kxn−pk)kxn+1−pk

≤ kxn−pk2+ 2bn(kn−1)||xn+1−p||2−2bnφ(||xn+1−p||)

+2bn(1 +L)kxn+1−xnk ||xn+1−p||

+2cn(M+kxn−pk)kxn+1−pk, (2.3)

where

kxn+1−xnk = kanxn+bnTnxn+cnun−xnk

= kbn(Tnxn−xn) +cn(un−xn)k ≤ bnkxn−Tnxnk+cnkun−xnk

≤ 2(1 +L)φ−1(a0)bn+ (M+ 2φ−1(a0))cn ≤ [2(2 +L)φ−1

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Substituting (2.4) in (2.3), we get

kxn+1−pk2 ≤ kxn−pk2−2bnφ(2φ−1(a0))

+18[φ−1

(a0)]2(kn−1)bn

+6(1 +L)φ−1

(a0)[2(2 +L)φ−1(a0) +M]b2_n

+6φ−1

(a0)(M+ 2φ−1(a0))cn

≤ kxn−pk2+ (kn−1)−φ(2φ−1(a0))bn.

Thus

φ(2φ−1(a0))bn≤ kxn−pk2− kxn+1−pk2+ (kn−1),

implies

φ(2φ−1

(a0)) j

X

n=n0

bn ≤ j

X

n=n0

(kxn−pk2− kxn+1−pk2) + j

X

n=n0

(kn−1)

= kxn0−pk

2

+

j

X

n=n0

(kn−1),

so that as j→ ∞ we have

φ(2φ−1

(a0))

∞

X

n=n0

bn≤ kxn0−pk

2

+

j

X

n=n0

(kn−1)<∞,

which implies that P

bn < ∞, a contradiction. Hence, kxn+1−x∗k ≤ 2φ−1(a0);

thus{xn} is bounded.

Now with the help (2.4) and the conditioncn=o(bn), (2.3) takes the form kxn+1−pk2 ≤ ||xn−p||2−2bnφ(kxn+1−pk)

+2bn[4[φ−1(a0)]2(kn−1)

+2(1 +L)φ−1

(a0)[2(2 +L)φ−1(a0) +M]bn

+2φ−1

(a0)(M+ 2φ−1(a0))tn]. (2.5)

Denote

θn = ||xn−p||,

λn = 2bn,

σn = 2bn[4[φ−1(a0)] 2

(kn−1)

+2(1 +L)φ−1(a0)[2(2 +L)φ−1(a0) +M]bn

+2φ−1

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Condition lim

n→∞bn= 0 assures the existence of a rankn0∈

N_{such that}_λ_n_{= 2b}_n_≤_1, for all n ≥ n0. Now with the help of P_n_≥0bn = ∞, cn = o(bn), lim

n→∞bn = 0 and Lemma 2, we obtain from (2.5) that

lim

n→∞||xn−p||= 0, completing the proof.

Corollary 9 Let K be a nonempty closed convex subset of a real Banach space E, T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n≥0 ⊂[1,∞), lim

n→∞kn = 1 such that p∈F(T) = {x ∈K :T x= x}. Let {αn}n≥0 ⊂[0,1]be such that

P

n≥0αn=∞ and lim

n→∞αn= 0. For arbitrary x0∈K let {xn}n≥0 be iteratively defined by

hTnx−p, j(x−p)i ≤kn||x−p||2−ψ(||x−p||), ∀x∈K.

Then {xn}n≥0 converges strongly to p∈F(T).

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Arif Rafiq

Dept. of Mathematics,

COMSATS Institute of Information Technology, Lahore, Pakistan