JJ II

(1)

volume 3, issue 5, article 83, 2002.

Received 30 April, 2002;

accepted 5 September, 2002.

Communicated by:S.S. Dragomir

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

STRONG CONVERGENCE THEOREMS FOR ITERATIVE SCHEMES WITH ERRORS FOR ASYMPTOTICALLY DEMICONTRACTIVE MAPPINGS IN ARBITRARY REAL NORMED LINEAR SPACES

YEOL JE CHO¹, HAIYUN ZHOU² AND SHIN MIN KANG¹

1Department of Mathematics, Gyeongsang National University, Chinju 660-701, Korea.

EMail:yjcho@nongae.gsnu.ac.kr

2Department of Mathematics,

Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003,

People’s Republic of China.

EMail:luyao_846@163.com

c

2000Victoria University ISSN (electronic): 1443-5756 040-02

(2)

Strong Convergence Theorems for Iterative Schemes with

Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear

Spaces

Yeol Je Cho, Haiyun Zhou and Shin Min Kang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of16

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.

2000 Mathematics Subject Classification:Primary 47H17; Secondary 47H05, 47H10 Key words: Asymptotically demicontractive mapping; Modified Mann and Ishikawa

iteration schemes with errors; arbitrary linear space.

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

1. Introduction

Let X be a real normed linear space and let J denote the normalized duality mapping fromXinto2^X^∗ given by

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

whereX^∗ denotes the dual space ofXandh·,·idenotes the generalized duality pairing of elements betweenXandX^∗.

Let F(T) denote the set of all fixed points of a mapping T. Let C be a nonempty subset ofX.

A mapping T : C → C is said to be k-strictly asymptotically pseudocon- tractive with a sequence{kn} ⊂[0,∞), kn≥1andkn→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 ∈Candp∈F(T).

The classes of k-strictly asymptotically pseudocontractive and asymptotically demicontractive mappings, as a natural extension to the class of asymptotically non-expansive mappings, were first introduced in Hilbert spaces by Liu

(4)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of16

[5] in 1996. By using the modified Mann iterates 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 of k-strictly asymptotically pseudocontractive 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 pseudocon- tractive with a sequence{k_n} ⊂[0,∞),k_n≥1andk_n→1asn→ ∞if there existk ∈[0,1)andj(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{k_n} ⊂[0,∞),limn→∞k_n= 1ifF(T)6=∅and there existk∈[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∈C andp∈F(T).

(5)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of16

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

(1.5) kTⁿx−Tⁿyk ≤Lkx−yk

for allx, y ∈C andn≥1.

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

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

LetX be a normed linear space, C be a nonempty convex subset ofX and T :C → Cbe 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, xn+1 = (1−αn)xn+αnTⁿ(yn) +un, 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,{α_n}andx₁as above, the modified Mann iteration scheme{x_n} with errors is defined by

(1.6)







x₁ ∈C,

xn+1 = (1−αn)xn+αnTⁿ(xn) +un, n≥1.

(6)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of16

LetX, C andT be as in above. Let{a_n},{b_n},{c_n},{a⁰_n},{b⁰_n}and{c⁰_n} be real sequences in[0,1]satisfyingan+bn+cn= 1 =a⁰_n+b⁰_n+c⁰_nand let{un} 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)







yn =anxn+bnTⁿ(xn) +cnun, 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 fork-strictly asymptotically demicontractive mappings:

Theorem 1.1. [3] Let q > 1 and let E be a real q-uniformly smooth Ba- nach space. Let K be a nonempty closed convex and bounded subset of E and T : K → K be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with a sequence{kn} ⊂[1,∞)for all n ≥ 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,

(7)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of16

(ii) P∞

n=1β_n <∞.

Then the sequence {x_n} defined by (1.6) with u_n ≡ 0 and v_n ≡ 0 for all n ≥1converges strongly to a fixed point ofT.

Very recently, Igbokwe [2] extended the above Theorem1.1to Banach spaces.

More precisely, he proved the following results:

Theorem 1.2. [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 uni- formlyL-Lipschitzian asymptotically demicontractive mapping with a sequence {k_n} ⊂[1,∞)for alln ≥1,k_n →1asn → ∞andP∞

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) con- verges strongly to a fixed pointpofT.

It is our purpose in this paper to extend and improve the above Theorem 1.2 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:

(8)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of16

Lemma 1.3. Let X be a normed linear space and C be a nonempty convex subset of X. LetT : C → C be a uniformly L-Lipschitzian mapping and the sequence{x_n}be defined by(1.7). Then we have

kT x_n−x_nk ≤ kTⁿx_n−x_nk+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 1.4. Let X be a normed linear space and C be a nonempty convex subset of X. Let T : C → C be a uniformly L-Lipschitzian and asymptot- ically demicontractive mapping with a sequence {k_n} such that k_n ≥ 1 and P∞

n=1(kn−1)<∞. Let the sequence{xn}be defined by(1.7)with the restric- tions

∞

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→∞kxn−T xnk= 0.

(9)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of16

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)kx_n−pk+M] +kx_n−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,

(10)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of16

where

σn= [bn(1 +L) +cn] + [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

kx_n+1−pk² ≤ kx_n−pkkx_n+1−pk −b⁰_nhx_n−Tⁿy_n, j(x_n+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

kxn+1−pk² ≤ kxn−pk²−(1−k)b⁰_nkxn+1−Tⁿxn+1k² (1.16)

+b⁰_n(k_n² −1)kx_n+1−pk² + 2b⁰_n(kx_n+1−x_nk +kTⁿx_n+1−Tⁿy_nk)kx_n+1−pk

+ 2c⁰_n(M +kx_n−pk)kx_n+1−pk.

(11)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of16

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 thatlim_n→∞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→∞ kxn+1−Tⁿxn+1k= 0 sinceP

nb⁰_n = ∞. Therefore, we havelim infn→∞kxn−Tⁿxnk = 0by (1.7) and solim infn→∞kx_n−T x_nk= 0 by Lemma1.3. This completes the proof.

A mapping T : C → C with a nonempty fixed point set F(T) in C will be said to satisfy the Condition (A) onC if there is a nondecreasing function f : [0,∞)→[0,∞)withf(0) = 0andf(r)>0for allr∈(0,∞)such that

kx−T xk ≥f(d(x, F(T))) for allx∈C.

(12)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of16

2. The Main Results

Now we prove the main results of this paper.

Theorem 2.1. Let X be a real normed linear space, C be a nonempty closed convex subset of X and T : C → C be a completely continuous and uni- formlyL-Lipschitzian asymptotically demicontractive mapping with a sequence {k_n} ⊂ [1,∞) such thatk_n ≥ 1and P∞

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 Lemma1.4that

lim inf

n→∞ kx_n−T x_nk= 0.

Since T is completely continuous, we see that there exists an infinite subsequence{xn_k}such that{xn_k}converges strongly for somep∈C andT p=p.

This shows thatp∈F(T). However,limn→∞kx_n−pkexists for anyp∈F(T) and so we must have that the sequence {x_n} converges strongly to p. This completes the proof.

Theorem 2.2. Let X be a real Banach space,C be a nonempty closed convex subset ofXandT : C → Cbe a uniformlyL-Lipschitzian and asymptotically demicontractive mapping with a sequence {kn} ⊂ [1,∞) such that kn ≥ 1

(13)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of16

and P∞

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 Lemma1.4, 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 Lemma1.4(ii), we conclude thatd(x_n, F(T))→0asn→ ∞.

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

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

kx_n_j+1 −p_jk=kx_n_j_+l−p_jk (2.1)

≤[1 +γ_n_j+l−1]kx_n_j+l−1−p_jk

≤exp (_l−1

X

m=0

γ_n_j_+m )

kx_n_j −p_jk ≤ M 2^j.

(14)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of16

It follows from (2.1) that

kp_j+1−p_jk ≤ 2M+ 1 2^j+1 .

Hence {p_j} is a Cauchy sequence. Assume that p_j → p as j → ∞. Then p∈F(T)sinceF(T)is closed and this in turn implies thatx_j →pasj → ∞.

This completes the proof.

Remark 2.1. We remark that, if T : C → C is completely continuous, then it must be demicompact (cf. [6]) and, ifT is continuous and demicompact, it must satisfy the Condition(A)(cf. [6]). In view of this observation, our Theorem2.1 improves Theorem1.2in the following aspects:

(i) Xmay 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 Theorems2.1and2.2, we have the following:

Corollary 2.3. Let X be a real normed linear space,C be a nonempty closed convex subset of X and T : C → C be a completely continuous and uni- formlyL-Lipschitzian asymptotically demicontractive mapping with a sequence {k_n} ⊂ [0,∞) such thatk_n ≥ 1and P∞

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{xn}converges strongly to a fixed pointpofT.

(15)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of16

Proof. By taking b_n, c_n ≡ 0 for n ≥ 1 in Theorem 2.1, we can obtain the desired conclusion.

Corollary 2.4. LetX be a real Banch space, C be a nonempty closed convex subset of X and T : C → C be a uniformly L-Lipschitzian asymptotically demicontractive mapping with a sequence {kn} ⊂ [0,∞) such that kn ≥ 1 and P∞

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<∞.

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

Proof. It follows from Theorem2.2by takingb_n, c_n ≡0for alln≥1.

Remark 2.2. Using the same methods as in Lemma 1.4, Theorems 2.1 and 2.2, we can prove several convergence results similar to Theorems2.1and2.2 concerning on the modified Ishikawa iteration schemes with errors defined by (1.6).

Remark 2.3. Igbokwe [1, Corollary 1] has shown that, ifT :C →Cis asymp- totically pseudocontractive, then it must be uniformlyL-Lipschitzian and hence our Theorems2.1and2.2hold for asymptotically pseudocontractive mappings with a nonempty fixed point setF(T).

(16)

Spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of16

References

[1] D.I. IGBOKWE, Approximation of fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces, J.

Ineq. Pure and Appl. Math., 3(1) (2002), Article 3. [ONLINE:

http://jipam.vu.edu.au/v3n1/043_01.html]

[2] Q.H. LIU, Convergence theorems of sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear Anal. Appl., 26 (1996), 1835–1842.

[3] M.O. OSILIKE, Iterative approximations of fixed points asymptotically demicontractive mappings, Indian J. Pure Appl. Math., 29 (1998), 1291–

1300.

[4] J. SCHU, Iterative construction of fixed points of asymptotically non- expansive mappings, J. Math. Anal.Appl., 158 (1991), 407–413.

[5] J. SCHU, Weak and strong convergence of fixed points of asymptotically non-expansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153–159.

[6] H.F. SENTER AND W.G. DOTSON, JR., Approximating fixed points of non-expansive mappings, Proc. Amer. Math. Soc., 44 (1974), 375–380.

[7] K.K. TANANDH.K. XU, Fixed point iteration processes for asymptotically non-expansive mappings, Proc. Amer. Math. Soc., 122 (1994), 733–739.