Key words and phrases: Asymptotically demicontractive mapping

http://jipam.vu.edu.au/

Volume 3, Issue 5, Article 83, 2002

STRONG CONVERGENCE THEOREMS FOR ITERATIVE SCHEMES WITH ERRORS FOR ASYMPTOTICALLY DEMICONTRACTIVE MAPPINGS IN

ARBITRARY REAL NORMED LINEAR SPACES

1YEOL JE CHO,²HAIYUN ZHOU, AND¹SHIN MIN KANG

1DEPARTMENT OFMATHEMATICS, GYEONGSANGNATIONALUNIVERSITY,

CHINJU660-701, KOREA

yjcho@nongae.gsnu.ac.kr

2DEPARTMENT OFMATHEMATICS,

SHIJIAZHUANGMECHANICALENGINEERINGCOLLEGE, SHIJIAZHUANG050003,

PEOPLE’SREPUBLIC OFCHINA

luyao_846@163.com

Received 30 April, 2002; accepted 5 September, 2002 Communicated by S.S. Dragomir

ABSTRACT. In the present paper, by virtue of new analysis technique, we will establish several strong convergence theorems for the modified Ishikawa and Mann iteration schemes with errors for a class of asymptotically demicontractive mappings in arbitrary real normed linear spaces.

Our results extend, generalize and improve the corresponding results obtained by Igbokwe [1], Liu [2], Osilike [3] and others.

Key words and phrases: Asymptotically demicontractive mapping; Modified Mann and Ishikawa iteration schemes with er- rors; arbitrary linear space.

2000 Mathematics Subject Classification. Primary 47H17; Secondary 47H05, 47H10.

1. INTRODUCTION

LetX be a real normed linear space and letJ denote the normalized duality mapping from Xinto2^X∗ given by

J(x) ={f ∈X^∗ :hx, fi=kxk² =kfk²}, x∈X,

where X^∗ denotes the dual space of X and h·,·i denotes the generalized duality pairing of elements betweenX andX^∗.

ISSN (electronic): 1443-5756 c

2002 Victoria University. All rights reserved.

This work was supported by grant No. (2000-1-10100-003-3) from the Basic Research Program of the Korea Science & Engineering Foundation.

040-02

LetF(T)denote the set of all fixed points of a mappingT. LetC be a nonempty subset of X.

A mapping T : C → C is said to be k-strictly asymptotically pseudocontractive with a sequence{k_n} ⊂[0,∞), k_n ≥1andk_n→1asn→ ∞if there existsk ∈[0,1)such that (1.1) kTⁿx−Tⁿyk² ≤k_n²kx−yk²+kk(x−Tⁿx)−(y−Tⁿy)k²

for alln≥1andx, y ∈C.

The mappingT is said to be asymptotically demicontractive with a sequence{k_n} ⊂[1,∞) andlimn→∞k_n= 1ifF(T)6=∅and there existsk∈[0,1)such that

(1.2) kTⁿx−pk² ≤k_n²kx−pk²+kkx−Tⁿxk² for alln≥1, x∈C andp∈F(T).

The classes of k-strictly asymptotically pseudocontractive and asymptotically demicontrac- tive mappings, as a natural extension to the class of asymptotically non-expansive mappings, were first introduced in Hilbert spaces by Liu [5] in 1996. By using the modified Mann iter- ates introduced by Schu [4, 5], he established several strong convergence results concerning an iterative approximation to fixed points of k-strictly asymptotically pseudocontractive and asymptotically demicontractive mappings in Hilbert spaces. In 1998, Osilike [3], by virtue of normalized duality mapping, first extended the concepts ofk-strictly asymptotically pseudocon- tractive and asymptotically demicontractive maps from Hilbert spaces to the much more general Banach spaces, and then proved the corresponding convergence theorems which generalized the results of Liu [2].

A mapping T : C → C is said to be k-strictly asymptotically pseudocontractive with a sequence {k_n} ⊂ [0,∞), k_n ≥ 1 and k_n → 1 as n → ∞ if there exist k ∈ [0,1) and j(x−y)∈J(x−y)such that

(1.3) h(I−Tⁿ)x−(I−Tⁿ)y, j(x−y)i

≥ 1

2(1−k)k(I−Tⁿ)x−(I−Tⁿ)yk²− 1

2(k_n² −1)kx−yk² for alln≥1andx, y ∈C.

The mappingT is called an asymptotically demicontractive mapping with a sequence{kn} ⊂ [0,∞),lim_n→∞k_n = 1 ifF(T) 6=∅and there exist k ∈ [0,1)andj(x−y)∈ J(x−y)such that

(1.4) hx−Tⁿx, j(x−p)i ≥ 1

2(1−k)kx−Tⁿxk²− 1

2(k_n² −1)kx−pk² for alln≥1,x∈Candp∈F(T).

Furthermore,T is said to be uniformlyL-Lipschitzian if there is a constantL≥1such that

(1.5) kTⁿx−Tⁿyk ≤Lkx−yk

for allx, y ∈Candn ≥1.

Remark 1.1. The definitions above may be stated in the setting of a real normed linear space. In the case ofXbeing a Hilbert space, (1.1) and (1.2) are equivalent to (1.3) and (1.4), respectively.

Recall that there are two iterative schemes with errors which have been used extensively by various authors.

LetXbe a normed linear space,Cbe a nonempty convex subset ofXandT :C →C be a given mapping. Then the modified Ishikawa iteration scheme{x_n}with errors is defined by















x₁ ∈C,

y_n= (1−β_n)x_n+β_nTⁿ(x_n) +v_n, n ≥1, x_n+1 = (1−α_n)x_n+α_nTⁿ(y_n) +u_n, n ≥1,

where {α_n}, {β_n} are some suitable sequences in [0,1] and {u_n}, {v_n} are two summable sequences inX.

WithX, C, {αn}and x1 as above, the modified Mann iteration scheme{xn}with errors is defined by

(1.6)







x₁ ∈C,

x_n+1 = (1−α_n)x_n+α_nTⁿ(x_n) +u_n, n≥1.

LetX, CandT be as in above. Let{a_n},{b_n},{c_n},{a⁰_n},{b⁰_n}and{c⁰_n}be real sequences in [0,1] satisfying a_n +b_n +c_n = 1 = a⁰_n + b⁰_n + c⁰_n and let {u_n} and {v_n} be bounded sequences in C. Define the modified Ishikawa iteration schemes {x_n} with errors generated from an arbitraryx₁ ∈Cas follows:

(1.7)







y_n =a_nx_n+b_nTⁿ(x_n) +c_nu_n, n ≥1, x_n+1 =a⁰_nx_n+b⁰_nTⁿy_n+c⁰_nv_n, n≥1.

In particular, if we setb_n =c_n = 0in (1.7), we obtain the modified Mann iteration scheme {x_n}with errors given by

(1.8)







x₁ ∈C,

x_n+1 =a⁰_nx_n+b⁰_nTⁿx_n+c⁰_nv_n, n ≥1.

Osilike [3] proved the following convergence theorems for k-strictly asymptotically demi- contractive mappings:

Theorem 1.2. [3] Letq >1and letEbe a realq-uniformly smooth Banach space. LetK be a nonempty closed convex and bounded subset ofEandT :K →K be a completely continuous and uniformlyL-Lipschitzian asymptotically demicontractive mapping with a sequence{kn} ⊂ [1,∞)for alln≥1,k_n →1asn→ ∞andP∞

n=1(k_n²−1)<∞. Let{α_n}and{β_n}be real sequences in[0,1]satisfying the conditions:

(i) 0< ≤c_qα_n^q−1 ≤ ¹₂{q(1−k)(1 +L)^−(q−2)} −for alln ≥1and for some >0, (ii) P∞

n=1β_n<∞.

Then the sequence {x_n}defined by (1.6)withu_n ≡ 0andv_n ≡ 0for alln ≥ 1converges strongly to a fixed point ofT.

Very recently, Igbokwe [2] extended the above Theorem 1.2 to Banach spaces. More pre- cisely, he proved the following results:

Theorem 1.3. [2] LetE be a real Banach space andK be a nonempty closed convex subset of E. Let T : K → K be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with a sequence{k_n} ⊂[1,∞)for alln≥1,k_n→1asn→ ∞and P∞

n=1(k²_n−1)<∞. Let the sequence{x_n}be defined by(1.7)with the restrictions that (i) a_n+b_n+c_n= 1 =a⁰_n+b⁰_n+c⁰_n,

(ii) P∞

n=0b⁰_n =∞, (iii) P∞

n=0(b⁰)² <∞,P∞

n=0c⁰_n <∞,P∞

n=0b_n<∞andP∞

n=0c_n<∞.

Then the modified Ishikawa iteration{x_n}defined by(1.6)and(1.7)converges strongly to a fixed pointpofT.

It is our purpose in this paper to extend and improve the above Theorem 1.3 from Banach spaces to real normed linear spaces. In the case of Banach spaces, we use the Condition(A)to replace the assumption thatT is completely continuous.

In the sequel, we will need the following lemmas:

Lemma 1.4. Let X be a normed linear space andC be a nonempty convex subset of X. Let T :C→Cbe a uniformlyL-Lipschitzian mapping and the sequence{x_n}be defined by(1.7).

Then we have

kT xn−xnk ≤ kTⁿxn−xnk+L(1 +L²)kTⁿ⁻¹xn−1−xn−1k (1.9)

+L(1 +L)c⁰_n−1kvn−1 −xn−1k +L²(1 +L)cn−1kun−1−x_nk

+Lc⁰_n−1kxn−1−Tⁿ⁻¹xn−1k, n≥1.

Proof. See Igbokwe [1, Lemma 1].

Lemma 1.5. Let X be a normed linear space andC be a nonempty convex subset of X. Let T :C →C be a uniformlyL-Lipschitzian and asymptotically demicontractive mapping with a sequence{k_n}such thatk_n ≥1andP∞

n=1(k_n−1)<∞. Let the sequence{x_n}be defined by (1.7)with the restrictions

∞

X

n=1

b⁰_n =∞,

∞

X

n=1

(b⁰_n)² <∞,

∞

X

n=1

c⁰_n <∞,

∞

X

n=1

c_n<∞.

Then we have the following conclusions:

(i) limn→∞kx_n−pkexists for anyp∈F(T).

(ii) limn→∞d(x_n, F(T))exists.

(iii) lim infn→∞kx_n−T x_nk= 0.

Proof. It is very clear that (1.7) is equivalent to the following:

(1.10)







y_n = (1−b_n)x_n+b_nTⁿ(x_n) +c_n(u_n−x_n), n≥1, x_n+1 = (1−b⁰_n)x_n+b⁰_nTⁿy_n+c⁰_n(v_n−x_n), n≥1.

For anyp∈F(T), letM >0be such that

M = max{sup{ku_n−pk},sup{kv_n−pk}}.

Observe first that

ky_n−pk ≤(1−b_n)kx_n−pk+b_nLkx_n−pk+c_n(M +kx_n−pk) (1.11)

≤(1 +L)kx_n−pk+M and

kTⁿy_n−x_nk ≤Lky_n−pk+kx_n−pk (1.12)

≤L[(1 +L)kxn−pk+M] +kxn−pk

≤[1 +L(1 +L)]kx_n−pk+M L.

Observe also that

kx_n+1−y_nk ≤ kx_n−y_nk+b⁰_n[kTⁿy_n−pk+ky_n−pk]

(1.13)

+c⁰_n[kvn−pk+kyn−pk]

≤[b_n(1 +L) +c_n]kx_n−pk+ (c_n+c⁰_n)M + [b⁰_n(1 +L) +c⁰_n][(1 +L)kx_n−pk+M]

≤σ_nkx_n−pk+ς_n, where

σ_n= [b_n(1 +L) +c_n] + [b⁰_n(1 +L) +c⁰_n](1 +L) and

ς_n=M[b⁰_n(1 +L) + 2c⁰_n+c_n].

Thus we have

kx_n+1−x_nk ≤b⁰_nkTⁿy_n−x_nk+c⁰_n(M+kx_n−pk) (1.14)

≤b⁰_nσ_nkx_n−pk+ς_n+c⁰_n(M +kx_n−pk).

Using iterates (1.10), we have

kxn+1−pk² ≤ kxn−pkkxn+1−pk −b⁰_nhxn−Tⁿyn, j(xn+1−p)i (1.15)

+c⁰_nhv_n−x_n, j(x_n+1−p)ik

≤ 1

2kx_n−pk²+ 1

2kx_n+1−pk²−b⁰_nhx_n+1−Tⁿx_n+1, j(x_n+1−p)i +b⁰_nhx_n+1−x_n+Tⁿy_n−Tⁿx_n+1, j(x_n+1−p)i

+c⁰_n(M +kx_n−pk)kx_n+1−pk, which implies that

kx_n+1−pk² ≤ kx_n−pk²−(1−k)b⁰_nkx_n+1−Tⁿx_n+1k² (1.16)

+b⁰_n(k²_n−1)kxn+1−pk²+ 2b⁰_n(kxn+1−xnk +kTⁿx_n+1−Tⁿy_nk)kx_n+1−pk

+ 2c⁰_n(M +kxn−pk)kxn+1−pk.

Substituting (1.12) – (1.14) in (1.16) and, after some calculations, we obtain (1.17) kx_n+1−pk² ≤(1 +γ_n)kx_n−pk² −(1−k)b⁰_nkx_n+1−Tⁿx_n+1k² for all very large n, where the sequence{γ_n}satisfies thatP∞

n=1γ_n < ∞. A direct induction of (1.17) leads to

(1.18) kx_n+1−pk² ≤ kx_n−pk²+M γ_n−b⁰_nkx_n+1−Tⁿx_n+1k²,

which implies thatlimn→∞kx_n−pkexists by Tan and Xu [7, Lemma 1] and so this proves the claim (i). The claim (ii) follows from (1.18).

Now, we prove the claim (iii). It follows from (1.18) that

∞

X

n=1

b⁰_nkx_n+1−Tⁿx_n+1k² <∞ and hence

lim inf

n→∞ kx_n+1−Tⁿx_n+1k= 0 since P

nb⁰_n = ∞. Therefore, we have lim infn→∞kx_n − Tⁿx_nk = 0 by (1.7) and so lim infn→∞kx_n−T x_nk= 0by Lemma 1.4. This completes the proof.

A mappingT :C →Cwith a nonempty fixed point setF(T)inCwill be said to satisfy the Condition(A) onC if there is a nondecreasing function f : [0,∞) → [0,∞)withf(0) = 0 andf(r)>0for allr∈(0,∞)such that

kx−T xk ≥f(d(x, F(T))) for allx∈C.

2. THEMAIN RESULTS

Now we prove the main results of this paper.

Theorem 2.1. LetXbe a real normed linear space, Cbe a nonempty closed convex subset of X andT : C → C be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with a sequence{k_n} ⊂[1,∞)such thatk_n≥1andP∞

n=1(k_n−1)<

∞. Let the sequence{x_n}be defined by(1.7)with the restrictions

∞

X

n=1

b⁰_n=∞,

∞

X

n=1

b⁰_n² <∞,

∞

X

n=1

c⁰_n<∞,

∞

X

n=1

c_n<∞.

Then{x_n}converges strongly to a fixed pointpofT. Proof. It follows from Lemma 1.5 that

lim inf

n→∞ kx_n−T x_nk= 0.

SinceT is completely continuous, we see that there exists an infinite subsequence{xn_k}such that {x_nk} converges strongly for some p ∈ C and T p = p. This shows that p ∈ F(T).

However, lim_n→∞kx_n −pk exists for any p ∈ F(T)and so we must have that the sequence

{x_n}converges strongly top. This completes the proof.

Theorem 2.2. LetX be a real Banach space,C be a nonempty closed convex subset ofXand T :C →C be a uniformlyL-Lipschitzian and asymptotically demicontractive mapping with a sequence{k_n} ⊂ [1,∞)such thatk_n≥ 1andP∞

n=1(k_n−1)<∞. Let the sequence{x_n}be defined by(1.7)with the restrictions

∞

X

n=1

b⁰_n=∞,

∞

X

n=1

b⁰_n² <∞,

∞

X

n=1

c⁰_n<∞,

∞

X

n=1

c_n<∞.

Suppose in addition that T satisfies the Condition (A), then the sequence {x_n} converges strongly to a fixed pointpofT.

Proof. By Lemma 1.5, we see that

lim inf

n→∞ kx_n−T x_nk= 0.

SinceT satisfies the Condition(A), we have lim inf

n→∞ f(d(xn, F(T))) = 0 and hence

lim inf

n→∞ d(x_n, F(T)) = 0.

By Lemma 1.5 (ii), we conclude thatd(x_n, F(T))→0asn → ∞.

Now we can take an infinite subsequence{x_nj}of{x_n}and a sequence{p_j} ⊂F(T)such thatkx_nj−p_jk ≤2^−j. SetM = exp{P∞

n=1γ_n}and writen_j+1 =n_j+lfor somel ≥1. Then we have

kx_nj+1 −p_jk=kx_nj+l−p_jk (2.1)

≤[1 +γ_nj+l−1]kx_nj+l−1−p_jk

≤exp (_l−1

X

m=0

γ_nj+m )

kx_nj −p_jk

≤ M 2^j. It follows from (2.1) that

kp_j+1−p_jk ≤ 2M + 1 2^j+1 .

Hence{p_j}is a Cauchy sequence. Assume thatpj →pasj → ∞. Thenp∈F(T)sinceF(T) is closed and this in turn implies thatx_j →pasj → ∞. This completes the proof.

Remark 2.3. We remark that, ifT : C → C is completely continuous, then it must be demi- compact (cf. [6]) and, if T is continuous and demicompact, it must satisfy the Condition (A) (cf. [6]). In view of this observation, our Theorem 2.1 improves Theorem 1.3 in the following aspects:

(i) X may be not a Banach space.

(ii) T may be not completely continuous.

(iii) Our proof methods are simpler than those of Igbokwe [1, Theorem 2].

As corollaries of Theorems 2.1 and 2.2, we have the following:

Corollary 2.4. LetXbe a real normed linear space,Cbe a nonempty closed convex subset of X andT : C → C be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with a sequence{k_n} ⊂[0,∞)such thatk_n≥1andP∞

n=1(k_n−1)<

∞. Let the sequence{x_n}be defined by(1.8)with the restrictions

∞

X

n=1

b⁰_n =∞,

∞

X

n=1

b⁰_n² <∞,

∞

X

n=1

c⁰_n<∞.

Then{x_n}converges strongly to a fixed pointpofT.

Proof. By takingb_n, c_n ≡ 0forn ≥ 1in Theorem 2.1, we can obtain the desired conclusion.

Corollary 2.5. LetX be a real Banch space,C be a nonempty closed convex subset ofXand T : C → C be a uniformly L-Lipschitzian asymptotically demicontractive mapping with a sequence{k_n} ⊂ [0,∞)such thatk_n≥ 1andP∞

n=1(k_n−1)<∞. Let the sequence{x_n}be defined by(1.8)with the restrictions

∞

X

n=1

b⁰_n=∞,

∞

X

n=1

b²_n<∞,

∞

X

n=1

c⁰_n <∞.

IfT satisfies the Condition(A)on the sequence{x_n}, then{x_n}converges strongly to a fixed pointpofT.

Proof. It follows from Theorem 2.2 by takingb_n, c_n≡0for alln ≥1.

Remark 2.6. Using the same methods as in Lemma 1.5, Theorems 2.1 and 2.2, we can prove several convergence results similar to Theorems 2.1 and 2.2 concerning on the modified Ishikawa iteration schemes with errors defined by(1.6).

Remark 2.7. Igbokwe [1, Corollary 1] has shown that, if T : C → C is asymptotically pseu- docontractive, then it must be uniformly L-Lipschitzian and hence our Theorems 2.1 and 2.2 hold for asymptotically pseudocontractive mappings with a nonempty fixed point setF(T).

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