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volume 3, issue 1, article 3, 2002.

Received 14 May, 2001;

accepted 17 July, 2001.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

APPROXIMATION OF FIXED POINTS OF ASYMPTOTICALLY DEMICONTRACTIVE MAPPINGS IN ARBITRARY

BANACH SPACES

D.I. IGBOKWE

Department of Mathematics University of Uyo

Uyo, NIGERIA.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 043-01

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Approximation of Fixed Points of Asymptotically Demicontractive Mappings in

Arbitrary Banach Spaces D.I. Igbokwe

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Abstract

LetEbe a real Banach Space andKa nonempty closed convex (not necessar- ily bounded) subset ofE. Iterative methods for the approximation of fixed points of asymptotically demicontractive mappingsT :K →Kare constructed using the more general modified Mann and Ishikawa iteration methods with errors.

Our results show that a recent result of Osilike [3] (which is itself a generalization of a theorem of Qihou [4]) can be extended from realq-uniformly smooth Banach spaces,1< q <∞, to arbitrary real Banach spaces, and to the more general Modified Mann and Ishikawa iteration methods with errors. Fur- thermore, the boundedness assumption imposed on the subsetKin ([3,4]) are removed in our present more general result. Moreover, our iteration parameters are independent of any geometric properties of the underlying Banach space.

2000 Mathematics Subject Classification:47H06, 47H10, 47H15, 47H17.

Key words: Asymptotically Demicontractive Maps, Fixed Points, Modified Mann and Ishikawa Iteration Methods with Errors.

1. Introduction

LetE be an arbitrary real Banach space and letJ denote the normalized duality mapping fromE into2^E^∗ given byJ(x) ={f ∈E^∗ :hx, fi=kxk² =kfk²}, whereE^∗denotes the dual space ofE andh, idenotes the generalized duality pairing. IfE^∗is strictly convex, thenJ is single-valued. In the sequel, we shall denote the single-valued duality mapping byj.

Let K be a nonempty subset of E. A mapping T : K → K is called k-strictly asymptotically pseudocontractive mapping, with sequence {k_n} ⊆ [1,∞), lim

n k_n = 1 (see for example [3, 4]), if for all x, y ∈ K there exists j(x−y)∈J(x−y)and a constantk ∈[0,1)such that

(1.1) 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 all n ∈ N. T is called an asymptotically demicontractive mapping with sequencek_n⊆[0,∞),lim

n k_n= 1(see for example [3,4]) ifF(T) ={x∈K : T x = x} 6= ∅ and for allx ∈ K andx^∗ ∈ F(T), there exists k ∈ [0,1) and j(x−x^∗)∈J(x−x^∗)such that

(1.2) hx−Tⁿx, j(x−x^∗)i ≥ 1

2(1−k)kx−Tⁿxk²− 1

2(k_n² −1)kx−x^∗k² for all n ∈ N. Furthermore, T is uniformly L-Lipschitzian, if there exists a constantL >0, such that

(1.3) kTⁿx−Tⁿyk ≤Lkx−yk,

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for allx , y ∈K andn ∈N.

It is clear that a k-strictly asymptotically pseudocontrative mapping with a nonempty fixed point set F(T) is asymptotically demicontrative. The classes ofk-strictly asymptotically pseudocontractive and asymptotically demicontractive maps were first introduced in Hilbert spaces by Qihou [4]. In a Hilbert space,jis the identity and it is shown in [3] that (1.1) and (1.2) are respectively equivalent to the inequalities:

(1.4) kTⁿx−Tⁿyk ≤k_n²kx−yk²+kk(I−Tⁿ)x−(I−Tⁿ)yk² and

(1.5) kTⁿx−Tⁿyk² ≤k²_nkx−yk²+kx−Tⁿxk² which are the inequalities considered by Qihou [4].

In [4], Qihou using the modified Mann iteration method introduced by Schu [5], proved convergence theorem for the iterative approximation of fixed points ofk-strictly asymptotically pseudocontractive mappings and asymptotically demicontractive mappings in Hilbert spaces. Recently, Osilike [3], extended the theorems of Qihou [4] concerning the iterative approximation of fixed points ofk- strictly asymptotically demicontractive mappings from Hilbert spaces to much more general real q-uniformly smooth Banach spaces, 1 < q < ∞, and to the much more general modified Ishikawa iteration method. More precisely, he proved the following:

Theorem 1.1. (Osilike [3, p. 1296]): Letq >1and letEbe a realq-uniformly smooth Banach space. LetK be a closed convex and bounded subset ofE and

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T : K → K a completely continuous uniformlyL-Lipschitzian asymptotically demicontractive mapping with a sequence kn ⊆ [1,∞)satisfyingP∞

n=1(k_n² − 1)<∞. Let{α_n}and{β_n}be real sequences satisfying the conditions.

(i) 0≤α_n, β_n≤1,n≥1;

(ii) 0<∈≤c_qα^q−1_n (1 +Lβ_n)^q ≤ ¹₂{q(1−k)(1 +L)^−(q−2)}− ∈, for alln ≥1 and for some∈>0; and

(iii) P∞

n=0β_n <∞.

Then the sequence{x_n}generated from an arbitraryx₁ ∈K by y_n= (1−β_n)x_n+β_nTⁿx_n, n ≥1,

x_n+1 = (1−α_n)x_n+α_nTⁿy_n, n≥1 converges strongly to a fixed point ofT.

In Theorem1.1,c_qis a constant appearing in an inequality which character- izesq-uniformly smooth Banach spaces. In Hilbert spaces,q = 2, cq = 1 and withβ_n = 0∀n, Theorems 1 and 2 of Qihou [4] follow from Theorem1.1 (see Remark 2 of [3]).

It is our purpose in this paper to extend Theorem1.1 from realq-uniformly smooth Banach spaces to arbitrary real Banach spaces using the more general modified Ishikawa iteration method with errors in the sense of Xu [7] given by

y_n =a_nx_n+b_nTⁿx_n+c_nu_n, n≥1, (1.6)

x_n+1 =a⁰_nx_n+b⁰_nTⁿy_n+c⁰_nv_n, n≥1,

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where {a_n}, {b_n}, {c_n}, {a⁰_n}, {b⁰_n}, {c⁰_n}are real sequences in [0,1]. a_n+ bn+cn = 1 = a⁰_n+b⁰_n+c⁰_n, {un}and{vn}are bounded sequences in K. If we setb_n=c_n = 0in (1.6) we obtain the modified Mann iteration method with errors in the sense of Xu [7] given by

(1.7) x_n+1 =a⁰_nx_n+b⁰_nTⁿx_n+c⁰_nv_n, n≥1.

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2. Main Results

In the sequel, we shall need the following:

Lemma 2.1. LetEbe a normed space, andK a nonempty convex subset ofE.

LetT :K →Kbe uniformly L-Lipschitzian mapping and let{a_n},{b_n},{c_n}, {a⁰_n},{b⁰_n}and{c⁰_n}be sequences in[0,1]withan+bn+cn =a⁰_n+b⁰_n+c⁰_n= 1.

Let {u_n}, {v_n}be bounded sequences inK. For arbitraryx₁ ∈ K, generate the sequence{x_n}by

y_n=a_nx_n+b_nTⁿx_n+c_nu_n, n ≥1 x_n+1 =a⁰_nx_n+b⁰_nTⁿy_n+c⁰_nv_n, n ≥0.

Then

kx_n−T x_nk ≤ kx_n−Tⁿx_nk+L(1 +L)²kx_n+1−Tⁿ⁻¹xn−1k

+L(1 +L)c⁰_n−1kvn−1−xn−1k+L²(1 +L)cn−1kun−1−x_nk +Lc⁰_n−1kxn−1−Tⁿ⁻¹xn−1k.

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Proof. Setλ_n=kx_n−Tⁿx_nk. Then

kxn−T xnk ≤ kxn−Tⁿxnk+LkTⁿ⁻¹xn−xnk

≤λ_n+L²kx_n−xn−1k+LkTⁿ⁻¹xn−1−x_nk

=λ_n+L²ka⁰_n−1x_n+b⁰_n−1Tⁿ⁻¹yn−1 +c⁰_n−1vn−1−xn−1k +Lka⁰_n−1xn−1+b⁰_n−1Tⁿ⁻¹yn−1+c⁰_n−1vn−1−Tⁿ⁻¹xn−1k

=λ_n+L²kb⁰_n−1(Tⁿ⁻¹yn−1−xn−1) +c⁰_n−1(vn−1−xn−1)k

+Lka⁰_n−1(x_n−1−Tⁿ⁻¹x_n−1) +b⁰_n−1(Tⁿ⁻¹y_n−1−Tⁿ⁻¹x_n−1) +c⁰_n−1(vn−1 −Tⁿ⁻¹xn−1)k

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≤λ_n+L²kTⁿ⁻¹yn−1−xn−1k+L²c⁰_n−1kvn−1−xn−1k

+Lkxn−1−Tⁿ⁻¹xn−1k+L²kyn−1−xn−1k+Lc⁰_n−1kvn−1−xn−1k +Lc⁰_n−1kx_n−1−Tⁿ⁻¹x_n−1k

=λn+Lλn−1+L(1 +L)c⁰_n−1kvn−1−xn−1k+L²kTⁿ⁻¹yn−1−xn−1k +L²kyn−1−xn−1k+Lc⁰_n−1kxn−1−Tⁿ⁻¹xn−1)k

≤λ_n+ 2Lλn−1+L(1 +L)c⁰_n−1kvn−1−xn−1k+L²(1 +L)kyn−1−xn−1k +L²kTⁿ⁻¹xn−1 −xn−1k+Lc⁰_n−1kxn−1 −Tⁿ⁻¹xn−1)k

=λ_n+L(1 +L)λn−1+L(1 +L)c⁰_n−1kvn−1−xn−1k

+L²(1 +L)kbn−1(Tⁿ⁻¹xn−1 −xn−1) +cn−1(un−1−xn−1)k +Lc⁰_n−1kx_n−1−Tⁿ⁻¹x_n−1k

≤λn+L(L²+ 2L+ 1)λn−1 +L(1 +L)c⁰_n−1kvn−1−xn−1k +L²(1 +L)cn−1kun−1−xn−1k+Lc⁰_n−1kxn−1−Tⁿ⁻¹xn−1k, completing the proof of Lemma 1.

Lemma 2.2. Let{a_n}, {b_n}and{δ_n}be sequences of nonnegative real num- bers satisfying

(2.2) a_n+1 ≤(1 +δ_n)a_n+b_n, n≥1.

IfP∞

n=1δ_n<∞andP∞

n=1b_n <∞then lim

n→∞a_nexists. If in addition{a_n}has a subsequence which converges strongly to zero then lim

n→∞a_n= 0.

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Proof. Observe that

a_n+1 ≤ (1 +δ_n)a_n+b_n

≤ (1 +δ_n)[(1 +δ_n−1)a_n−1+b_n−1] +b_n

≤ ...≤

n

Y

j=1

(1 +δ_j)a₁+

n

Y

j=1

(1 +δ_j)

n

X

j=1

b_j

≤ ...≤

∞

Y

j=1

(1 +δ_j)a₁+

∞

Y

j=1

(1 +δ_j)

∞

X

j=1

b_j <∞.

Hence{a_n}is bounded. LetM >0be such thata_n≤M, n≥1. Then a_n+1 ≤(1 +δ_n)a_n+b_n≤a_n+M δ_n+b_n =a_n+σ_n

where σ_n = M δ_n + b_n. It now follows from Lemma 2.1 of ([6, p. 303]) that lim

n a_n exists. Consequently, if {a_n} has a subsequence which converges strongly to zero thenlim

n an = 0completing the proof of Lemma2.2.

Lemma 2.3. LetE be a real Banach space andKa nonempty convex subset of E. LetT :K →Kbe uniformlyL-Lipschitzian asymptotically demicontractive mapping with a sequence{k_n} ⊆[1,∞), such thatlim

n k_n = 1, andP∞

n=1(k_n²− 1)<∞. Let{an}, {bn}, {cn},

{a⁰_n},{b⁰_n},{c⁰_n} be real sequences in[0,1]satisfying:

(i) a_n+b_n+c_n = 1 =a⁰_n+b⁰_n+c⁰_n, (ii) P∞

n=1b⁰_n=∞,

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(iii) P∞

n=1(b⁰_n)² <∞,P∞

n=1c⁰_n<∞,P∞

n=1b_n<∞, andP∞

n=1c_n <∞.

Let{u_n}and{v_n}be bounded sequences inK and let{x_n}be the sequence generated from an arbitraryx₁ ∈K by

y_n =a_nx_n+b_nTⁿx_n+c_nu_n, n≥1, x_n+1 =a⁰_n+b⁰_nTⁿy_n+c⁰_nv_n, n ≥1, thenlim inf

n kx_n−T x_nk= 0.

Proof. It is now well-known (see e.g. [1]) that for all x, y ∈ E, there exists j(x+y)∈J(x+y)such that

(2.3) kx+yk² ≤ kxk²+ 2hy, j(x+y)i.

Let x^∗ ∈ F(T)and let M > 0be such that kun−x^∗k ≤ M, kvn −x^∗k ≤ M, n ≥1. Using (1.6) and (2.3) we obtain

kx_n+1−x^∗k²

=k(1−b⁰_n−c⁰_n)x_n+b⁰_nTⁿy_n+c⁰_nv_n−x^∗k²

=k(x_n−x^∗) +b⁰_n(Tⁿy_n−x_n) +c⁰_n(v_n−x_n)k²

≤ k(x_n−x^∗)k²+ 2hb⁰_n(Tⁿy_n−x_n) +c⁰_n(v_n−x_n), j(x_n+1−x^∗)i

=k(xn−x^∗)k²−2b⁰_nhxn+1−Tⁿxn+1, j(xn+1−x^∗)i + 2b⁰_nhx_n+1−Tⁿx_n+1, j(x_n+1−x^∗)i

+ 2b⁰_nhTⁿy_n−x_n, j(x_n+1−x^∗)i+ 2c⁰_nhv_n−x_n, j(x_n+1−x^∗)i

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=k(x_n−x^∗)k² −2b⁰_nhx_n+1−Tⁿx_n+1, j(x_n+1−x^∗)i + 2b⁰_nhx_n+1−x_n, j(x_n+1−x^∗)i

+ 2b⁰_nhTⁿy_n−Tⁿx_n+1, j(x_n+1−x^∗)i + 2c⁰_nhvn−xn, j(xn+1−x^∗)i.

(2.4)

Observe that

x_n+1−x_n =b⁰_n(Tⁿy_n−x_n) +c⁰_n(v_n−x_n).

Using this and (1.2) in (2.4) we have kx_n+1−x^∗k²

≤ k(x_n−x^∗)k² −b⁰_n(1−k)kx_n+1−Tⁿx_n+1k² +b⁰_n(k²_n−1)kx_n+1−x^∗k²

+ 2(b⁰_n)²hTⁿyn−xn, j(xn+1−x^∗)i + 2b⁰_nhTⁿy_n−Tⁿx_n+1, j(x_n+1−x^∗)i + 3c⁰_nhv_n−x_n, j(x_n+1−x^∗)i

≤ k(x_n−x^∗)k² −b⁰_n(1−k)kx_n+1−Tⁿx_n+1k² + (k_n²−1)kx_n+1−x^∗k²

+ 2(b⁰_n)²kTⁿy_n−x_nkkx_n+1−x^∗k + 2b⁰_nLkx_n+1−y_nkkx_n+1−x^∗k + 3c⁰_nkvn−xnkkxn+1−x^∗k

=k(x_n−x^∗)k²−b⁰_n(1−k)kx_n+1−Tⁿx_n+1k² + (k_n²−1)kx_n+1−x^∗k²+ [2(b⁰_n)²kTⁿy_n−x_nk + 2b⁰_nLkx_n+1−y_nk+ 3c⁰_nkv_n−x_nk]kx_n+1−x^∗k.

(2.5)

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Observe that

kyn−x^∗k = kan(xn−x^∗) +bn(Tⁿxn−x^∗) +cn(un−x^∗)k

≤ kx_n−x^∗k+Lkx_n−x^∗k+M

= (1 +L)kx_n−x^∗k+M, (2.6)

so that

kTⁿy_n−x_nk ≤ Lky_n−x^∗k+kx_n−x^∗)k

≤ L[(1 +L)kx_n−x^∗k+M] +kx_n−x^∗k

≤ [1 +L(1 +L)]kx_n−x^∗k+M L, (2.7)

kx_n+1−x^∗k = ka⁰_n(x_n−x^∗) +b⁰_n(Tⁿy_n−x^∗) +c⁰_n(v_n−x^∗)k

≤ kx_n−x^∗k+Lky_n−x^∗k+M

≤ kx_n−x^∗k+L[(1 +L)kx_n−x^∗k+M] +M

= [1 +L(1 +L)]kx_n−x^∗k+ (1 +L)M, (2.8)

and

kxn+1−ynk=ka⁰_n(xn−yn) +b⁰_n(Tⁿyn−yn) +c⁰_n(vn−yn)k

≤ kx_n−y_nk+b⁰_n[kTⁿy_n−x^∗k+ky_n−x^∗k]

+c⁰_n[kv_n−x^∗k+ky_n−x^∗k]

=kb_n(Tⁿx_n−x_n) +c_n(u_n−x_n)k+b⁰_n[Lky_n−x_nk+ky_n−x^∗k]

+c⁰_nM +c⁰_nky_n−x^∗k

≤b_n(1 +L)kx_n−x^∗k+c_nM +c_nkx_n−x^∗k + [b⁰_n(1 +L) +c⁰_n]kyn−x^∗k+c⁰_nM

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≤[b_n(1 +L) +c_n]kx_n−x^∗k+c_nM

+ [b⁰_n(1 +L) +c⁰_n][(1 +L)kx_n−x^∗k+M] +c⁰_nM

≤ {[b_n(1 +L) +c_n] + [b⁰_n(1 +L) +c⁰_n](1 +L)}kx_n−x^∗k +M[b⁰_n(1 +L) + 2c⁰_n+cn].

(2.9)

Substituting (2.7)-(2.9) in (2.5) we obtain, kxn+1−x^∗k²

≤ k(x_n−x^∗)k²−b⁰_n(1−k)kx_n+1−Tⁿx_n+1k²

+ (k²_n−1){[1 +L(1 +L)]kx_n+1−x^∗k+M(1 +L)}²

+{(b⁰_n[[1 +L(1 +L)]kx_n−x^∗k+M L] + 3c⁰_n[M+kx_n−x^∗k]

+ 2b⁰_nL[[b_n(1 +L) +c_n] + [b⁰_n(1 +L+c⁰_n](1 +L)]kx_n−x^∗k

+M[b⁰_n(1 +L) + 2c⁰_n+c_n]}{[1 +L(1 +L)]kx_n−x^∗k+M(1 +L)}

≤ k(x_n−x^∗)k²−b⁰_n(1−k)kx_n+1−Tⁿx_n+1k² + (k²_n−1){[1 +L(1 +L)]²kxn+1−x^∗k²

+ 2M(1 +L)[1 +L(1 +L)]kx_n−x^∗k+M²(1 +L)²}

+ 2(b⁰_n)²[[1 +L(1 +L)]kx_n−x^∗k+M L][[1 +L(1 +L)]kx_n−x^∗k +M(1 +L)] + 3c⁰_n[M +kx_n−x^∗k][[1 +L(1 +L)]kx_n−x^∗k +M(1 +L)] + 2b⁰_nL{[[b_n(1 +L) +c_n]

+ [b⁰_n(1 +L) +c⁰_n](1 +L)]kx_n−x^∗k

+M[b⁰_n(1 +L) + 2c⁰_n+cn]}{[1 +L(1 +L)]kxn−x^∗k+M(1 +L)}.

Sincekxn−x^∗k ≤1 +kxn−x^∗k², we have

(2.10) kxn+1−x^∗k² ≤[1 +δn]kxn−x^∗k²+σn−b⁰_n(1−k)kxn+1−Tⁿxn+1k²,

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where

δ_n= (k²_n−1){[1 +L(1 +L)]²+ 2M(1 +L)[1 +L(1 +L)]}

+ 2(b⁰_n)²{[1 +L(1 +L)]²+M(1 +L)[1 +L(1 +L)]

+M L[1 +L(1 +L)]}

+ 3c⁰_n{[1 +L(1 +L)] +M[1 +L(1 +L)] +M(1 +L)}

+ 2b⁰_nL{{[b_n(1 +L) +c_n] + [b⁰_n(1 +L) +c⁰_n](1 +L)}

× {[1 +L(1 +L)] +M(1 +L)}

+M[b⁰_n(1 +L) + 2c⁰_n+c_n][1 +L(1 +L)]}

and

σ_n = (k_n² −1){2M(1 +L)[1 +L(1 +L)] +M²(1 +L)²}

+ 2(b⁰_n)²{[1 +L(1 +L)]M(1 +L) +M L[1 +L(1 +L)] +M²L(1 +L)}

+ 3c⁰_n{M[1 +L(1 +L)] +M²(1 +L) +M(1 +L)

+ 2b⁰_nL{[[b_n(1 +L) +c_n] + [b⁰_n(1 +L) +c⁰_n](1 +L)][M(1 +L)]

+M[b⁰_n(1 +L) + 2c⁰_n+c_n][[1 +L(1 +L)] +M(1 +L)]}.

Since P∞

n=1(k²_n − 1) < ∞, condition (iii) implies that P∞

n=1δ_n < ∞ and P∞

n=1σ_n<∞. From (2.10) we obtain

kx_n+1−x^∗k² ≤ [1 +δ_n]kx_n−x^∗k+σ_n

≤ ...≤

n

Y

j=1

[1 +δ_j]kx₁−x^∗k²+

n

Y

j=1

[1 +δ_j]

n

X

j=1

σ_j

≤

∞

Y

j=1

[1 +δ_j]kx₁−x^∗k²+

∞

Y

j=1

[1 +δ_j]

∞

X

j=1

σ_j <∞,

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since P∞

n=1δ_n < ∞andP∞

n=1σ_n < ∞. Hence{kx_n−x^∗k}^∞_n=1 is bounded.

Letkxn−x^∗k ≤M, n≥1. Then it follows from (2.10) that (2.11) kx_n+1−x^∗k² ≤ kx_n−x^∗k²+M²δ_n+σ_n

−b⁰_n(1−k)kx_n+1−Tⁿx_n+1k², n ≥1 Hence,

b⁰_n(1−k)kx_n+1−Tⁿx_n+1k² ≤ kx_n−x^∗k²− kx_n+1−x^∗k²+µ_n, whereµ_n =M²δ_n+σ_nso that,

(1−k)

n

X

j=1

b⁰_jkx_j+1−T^jx_j+1k² ≤ kx₁−x^∗k²+

n

X

j=1

µ_j <∞, Hence,

∞

X

n=1

b⁰_nkx_n+1−Tⁿx_n+1k² <∞, and condition (ii) implies thatlim inf

n→∞ kx_n+1−Tⁿx_n+1k= 0. Observe that kx_n+1−Tⁿx_n+1k² = k(1−b⁰_n−c⁰_n)x_n+b⁰_nTⁿy_n+c⁰_nv_n−Tⁿx_n+1k²

= kx_n−Tⁿx_n+b⁰_n(Tⁿy_n−x_n) +Tⁿx_n−Tⁿx_n+1 +c⁰_n(v_n−x_n)k².

(2.12)

For arbitraryu, v ∈E, setx=u+v andy=−v in (2.3) to obtain (2.13) kv +uk² ≥ kuk²+ 2hv, j(u)i.

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From (2.12) and (2.13), we have kx_n+1−Tⁿx_n+1k²

=kx_n−Tⁿx_n+b⁰_n(Tⁿy_n−x_n) +Tⁿx_n−Tⁿx_n+1+c⁰_n(v_n−x_n)k²

≥ kx_n−Tⁿx_nk²+ 2hb⁰_n(Tⁿy_n−x_n) +Tⁿx_n−Tⁿx_n+1 +c⁰_n(v_n−x_n), j(x_n−Tⁿx_n)i.

Hence

kx_n−Tⁿx_nk²

≤ kx_n+1−Tⁿx_n+1k²+ 2kb⁰_n(Tⁿy_n−x_n)

+Tⁿx_n−Tⁿx_n+1+c⁰_n(v_n−x_n)kkx_n−Tⁿx_nk

≤ kx_n+1−Tⁿx_n+1k²+ 2{b⁰_nkTⁿy_n−x_nk+Lkx_n+1−x_nk +c⁰_nkvn−xnk}kxn−Tⁿxnk

≤ kx_n+1−Tⁿx_n+1k²+ 2{b⁰_nkTⁿy_n−x_nk+Lb⁰_nkTⁿy_n−x_nk +Lc⁰_nkv_n−x_nk+c⁰_nkv_n−x_nk}kx_n−Tⁿx_nk

≤ kx_n+1−Tⁿx_n+1k²+ 2(1 +L)kx_n−x^∗k

× {(1 +L)b⁰_nkTⁿy_n−x_nk+ (1 +L)c⁰_nkv_n−x_nk}

≤ kx_n+1−Tⁿx_n+1k²+ 2(1 +L)kx_n−x^∗k

× {(1 +L)b⁰_n[[1 +L(1 +L)]kx_n−x^∗k+M L]

+ (1 +L)c⁰_n[M +kxn−x^∗k], (using (2.6))

≤ kx_n+1−Tⁿx_n+1k²

+ 2(1 +L)M{(1 +L)b⁰_n[[1 +L(1 +L)]M +M L]

+ (1 +L)c⁰_n[M +M]}, (since kx_n−x^∗k ≤M)

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(2.14) =kx_n+1−Tⁿx_n+1k²+ 2b⁰_n(1 +L)⁴M²+ 4c⁰_n(1 +L)²M.

Since lim

n→∞b⁰_n = 0, lim

n→∞c⁰_n = 0 andlim inf

n→∞ kx_n+1 −Tⁿx_n+1k = 0, it follows from (2.14) thatlim inf

n→∞ kx_n−Tⁿx_nk = 0. It then follows from Lemma 1 that lim inf

n→∞ kx_n−T x_nk= 0, completing the proof of Lemma2.3.

Corollary 2.4. LetEbe a real Banach space andKa nonempty convex subset of E. Let T : K → K be a k-strictly asymptotically pseudocontractive map withF(T)6=∅and sequence{kn} ⊂[1,∞)such thatlim

n kn= 1, P∞

n=1(k²_n− 1) < ∞. Let {a_n}, {b_n}, {c_n},{a⁰_n}, {b⁰_n}, {c⁰_n}, {u_n}, and {v_n} be as in Lemma2.3 and let{x_n}be the sequence generated from an arbitraryx₁ ∈ K by

y_n =a_nx_n+b_nTⁿx_n+c_nu_n, n≥1, x_n+1 =a⁰_n+b⁰_nTⁿy_n+c⁰_nv_n, n ≥1, Thenlim inf

n→∞ kxn−T xnk= 0.

Proof. From (1.1) we obtain k(I−Tⁿ)x−(I−Tⁿ)ykkx−yk

≥ 1

2{(1−k)k(I−Tⁿ)x−(I−Tⁿ)yk²−(k²_n−1)kx−yk²}

= 1 2[√

1−kk(I −Tⁿ)x−(I−Tⁿ)yk +p

k_n²−1kx−yk][√

1−kk(I−Tⁿ)x−(I−Tⁿ)yk −√

k²−1kx−yk]

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≥ 1 2[√

1−kk(I−Tⁿ)x−(I −Tⁿ)yk]

[√

1−kk(I−Tⁿ)x−(I−Tⁿ)yk −√

k²−1kx−yk]

so that 1

2

√

1−k[√

1−kk(I−Tⁿ)x−(I−Tⁿ)yk]−√

k²−1kx−yk ≤ kx−yk.

Hence

k(I−Tⁿ)x−(I−Tⁿ)yk ≤[2 +p

{(1−k)(k²_n−1)}

1−k ]kx−yk.

Furthermore,

kTⁿx−Tⁿyk − kx−yk ≤ k(I−Tⁿ)x−(I−Tⁿ)yk

≤ [2 +p

{(1−k)(k_n² −1)}

1−k ]kx−yk, from which it follows that

kTⁿx−Tⁿyk ≤[1 + 2 +p

{(1−k)(k_n²−1)}

1−k ]kx−yk.

Since{k_n}is bounded, letk_n ≤D, ∀n≥1. Then kTⁿx−Tⁿyk ≤ [1 + 2 +p

{(1−k)(D²−1)}

1−k ]kx−yk

≤ Lkx−yk,

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where

L= 1 + 2 +p

{(1−k)(D²−1)}

1−k .

Hence T is uniformlyL-Lipschitzian. Since F(T) 6= ∅, then T is uniformly L-Lipschitzian and asymptotically demicontractive and hence the result follows from Lemma2.3.

Remark 2.1. It is shown in [3] that ifEis a Hilbert space andT : K → Kis k-asymptotically pseudocontractive with sequence{k_n}then

kTⁿx−Tⁿyk ≤ D+√ k 1−√

kkx−yk ∀x, y ∈K, where k_n≤D, ∀n≥1.

Theorem 2.5. Let E be a real Banach space andK a nonempty closed con- vex subset of E. Let T : K → K be a completely continuous uniformly L-Lipschitzian asymptotically demicontractive mapping with sequence{k_n} ⊂ [1,∞)such thatlim

n k_n= 1andP∞

n=1(k_n²−1)<∞. Let{a_n},{b_n},{c_n},{a⁰_n}, {b⁰_n}, {c⁰_n}, {u_n}, and{v_n}be as in Lemma2.3. Then the sequence{x_n}gen- erated from an arbitraryx₁ ∈K by

y_n = a_nx_n+b_nTⁿx_n+c_nu_n, n ≥1, x_n+1 = a⁰_nx_n+b⁰_nTⁿy_n+c⁰_nv_n, n ≥1, converges strongly to a fixed point ofT.

Proof. From Lemma2.3,lim inf

n kx_n−Tⁿx_nk= 0, hence there exists a subsequence{x_n_j}of{x_n}such thatlim

n kx_n_j −T x_n_jk= 0.

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Since{x_n_j}is bounded andT is completely continuous, then{T x_n_j}has a subsequence{T xj_k}which converges strongly. Hence{xn_jk}converges strongly.

Suppose lim

k→∞x_n_jk =p. Then lim

k→∞T x_n_jk =T p. lim

k→∞kx_n_jk −T x_n_jkk

=kp−T pk= 0so thatp∈F(T). It follows from (2.11) that kx_n+1−pk² ≤ kx_n−pk²+µ_n Lemma2.2 now implies lim

n→∞kxn−pk = 0completing the proof of Theorem 2.5.

Corollary 2.6. Let E be an arbitrary real Banach space and K a nonempty closed convex subset ofE. letT :K →K be ak-strictly asymptotically pseu- docontractive mapping withF(T)6= ∅and sequence{k_n} ⊂ [1,∞)such that limn k_n= 1, andP∞

n=1(k²_n−1)<∞. Let{a_n}, {b_n}, {c_n},{a⁰_n}, {b⁰_n}, {c⁰_n}, {un}, and{vn} be as in Lemma2.3. Then the sequence {xn} generated from an arbitraryx₁ ∈Kby

y_n = a_nx_n+b_nTⁿx_n+c_nu_n, n ≥1, xn+1 = a⁰_nxn+b⁰_nTⁿyn+c⁰_nvn, n ≥1, converges strongly to a fixed point ofT.

Proof. As shown in Corollary 2.4, T is uniformly L-Lipschitzian and since F(T)6=∅thenT is asymptotically demicontractive and the result follows from Theorem2.5.

Remark 2.2. If we setb_n = c_n = 0, ∀n ≥ 1in Lemma2.3, Theorem2.5 and Corollaries 2.4 and2.6, we obtain the corresponding results for the modified Mann iteration method with errors in the sense of Xu [7].

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Remark 2.3. Theorem 2.5 extends the results of Osilike [3] (which is itself a generalization of a theorem of Qihou [4]) from realq-uniformly smooth Banach space to arbitrary real Banach space.

Furthermore, our Theorem 2.5 is proved without the boundedness condition imposed on the subset K in ([3, 4]) and using the more general modified Ishikawa Iteration method with errors in the sense of Xu [7]. Also our iteration parameters{a_n},{b_n},{c_n},{a⁰_n},{b⁰_n},{c⁰_n},{u_n}, and{v_n}are completely independent of any geometric properties of underlying Banach space.

Remark 2.4. Prototypes for our iteration parameters are:

b⁰_n = 1

3(n+ 1), c⁰_n = 1

3(n+ 1)², a⁰_n = 1−(b⁰_n+c⁰_n), b_n = c_n= 1

3(n+ 1)², a_n= 1− 1

3(n+ 1)², n ≥1.

The proofs of the following theorems and corollaries for the Ishikawa iteration method with errors in the sense of Liu [2] are omitted because the proofs follow by a straightforward modifications of the proofs of the corresponding results for the Ishikawa iteration method with errors in the sense of Xu [7].

Theorem 2.7. LetEbe a real Banach space and letT :E →E be a uniformly L-Lipschitzian asymptotically demicontractive mapping with sequence{kn} ⊂ [1,∞)such thatlim

n k_n = 1, andP∞

n=1(k²_n−1) < ∞. Let {u_n}and{v_n}be sequences inEsuch thatP∞

n=1ku_nk<∞andP∞

n=1kv_nk<∞, and let{α_n} and{β_n}be sequences in[0,1]satisfying the conditions:

(i) 0≤αn, βn≤1,n≥1;