LetE be a real Banach Space andK a nonempty closed convex (not necessarily bounded) subset ofE

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

Volume 3, Issue 1, Article 3, 2002

APPROXIMATION OF FIXED POINTS OF ASYMPTOTICALLY DEMICONTRACTIVE MAPPINGS IN ARBITRARY BANACH SPACES

D.I. IGBOKWE

DEPARTMENT OFMATHEMATICS

UNIVERSITY OFUYO

UYO, NIGERIA.

epseelon@aol.com

Received 14 May, 2001; accepted 17 July, 2001.

Communicated by S.S. Dragomir

ABSTRACT. LetE be a real Banach Space andK a nonempty closed convex (not necessarily 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. Furthermore, the boundedness assumption imposed on the subsetK in ([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.

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

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

1. INTRODUCTION

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

LetKbe a nonempty subset ofE. A mappingT :K →Kis calledk-strictly asymptotically pseudocontractive mapping, with sequence{kn} ⊆[1,∞),lim

n kn = 1(see for example [3, 4]), if for allx, y ∈K there existsj(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,

ISSN (electronic): 1443-5756

c 2002 Victoria University. All rights reserved.

043-01

for all n ∈ N. T is called an asymptotically demicontractive mapping with sequence k_n ⊆ [0,∞), lim

n k_n = 1 (see for example [3, 4]) if F(T) = {x ∈ K : T x = x} 6= ∅ and for all x∈K andx^∗ ∈F(T), there existsk∈[0,1)andj(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 alln ∈ N. Furthermore, T is uniformly L-Lipschitzian, if there exists a constant L > 0, such that

(1.3) kTⁿx−Tⁿyk ≤Lkx−yk,

for allx , y ∈Kandn∈N.

It is clear that ak-strictly asymptotically pseudocontrative mapping with a nonempty fixed point setF(T)is asymptotically demicontrative. The classes ofk-strictly asymptotically pseu- docontractive and asymptotically demicontractive maps were first introduced in Hilbert spaces by Qihou [4]. In a Hilbert space,j is 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 con- vergence theorem for the iterative approximation of fixed points of k-strictly asymptotically pseudocontractive mappings and asymptotically demicontractive mappings in Hilbert spaces.

Recently, Osilike [3], extended the theorems of Qihou [4] concerning the iterative approxima- tion of fixed points ofk- strictly asymptotically demicontractive mappings from Hilbert spaces to much more general realq-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. Let K be a closed convex and bounded subset of E and T : K → K a completely continuous uniformlyL-Lipschitzian asymptotically demicontractive mapping with a sequence k_n ⊆ [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≥1and 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 Theorem 1.1,c_q is a constant appearing in an inequality which characterizesq-uniformly smooth Banach spaces. In Hilbert spaces,q = 2, c_q = 1and withβ_n = 0∀n, Theorems 1 and 2 of Qihou [4] follow from Theorem 1.1 (see Remark 2 of [3]).

It is our purpose in this paper to extend Theorem 1.1 from real q-uniformly smooth Ba- nach 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)

xn+1 =a⁰_nxn+b⁰_nTⁿyn+c⁰_nvn, n≥1,

where{a_n}, {b_n}, {c_n}, {a⁰_n}, {b⁰_n}, {c⁰_n}are real sequences in[0,1]. a_n+b_n+c_n = 1 = a⁰_n+b⁰_n+c⁰_n,{u_n}and{v_n}are bounded sequences inK. 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.

2. MAINRESULTS

In the sequel, we shall need the following:

Lemma 2.1. Let E be a normed space, andK a nonempty convex subset of E. LetT : K → K be uniformly L-Lipschitzian mapping and let {a_n}, {b_n}, {c_n}, {a⁰_n}, {b⁰_n} and {c⁰_n} be sequences in[0,1]witha_n+b_n+c_n =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.

(2.1)

Proof. Setλ_n =kx_n−Tⁿx_nk. Then

kx_n−T x_nk ≤ kx_n−Tⁿx_nk+LkTⁿ⁻¹x_n−x_nk

≤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(xn−1−Tⁿ⁻¹xn−1) +b⁰_n−1(Tⁿ⁻¹yn−1−Tⁿ⁻¹xn−1) +c⁰_n−1(v_n−1−Tⁿ⁻¹x_n−1)k

≤λ_n+L²kTⁿ⁻¹y_n−1−x_n−1k+L²c⁰_n−1kv_n−1−x_n−1k

+Lkxn−1−Tⁿ⁻¹xn−1k+L²kyn−1−xn−1k+Lc⁰_n−1kvn−1−xn−1k +Lc⁰_n−1kxn−1−Tⁿ⁻¹xn−1k

=λ_n+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+ 2Lλ_n−1+L(1 +L)c⁰_n−1kv_n−1−x_n−1k+L²(1 +L)ky_n−1 −x_n−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−1kxn−1 −Tⁿ⁻¹xn−1k

≤λn+L(L²+ 2L+ 1)λn−1+L(1 +L)c⁰_n−1kvn−1−xn−1k +L²(1 +L)cn−1ku_n−1−xn−1k+Lc⁰_n−1kx_n−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 numbers 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.

Proof. Observe that

a_n+1 ≤ (1 +δ_n)a_n+b_n

≤ (1 +δ_n)[(1 +δn−1)an−1+bn−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 then lim

n a_n = 0

completing the proof of Lemma 2.2.

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

n k_n= 1, andP∞

n=1(k_n² −1)<∞. Let{a_n}, {b_n}, {c_n}, {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 =∞, (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 allx, y ∈E, there existsj(x+y)∈J(x+y) such that

(2.3) kx+yk² ≤ kxk²+ 2hy, j(x+y)i.

Letx^∗ ∈ F(T)and letM > 0be such thatkun−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(x_n−x^∗)k²−2b⁰_nhx_n+1−Tⁿx_n+1, j(x_n+1−x^∗)i+

+ 2b⁰_nhxn+1−Tⁿxn+1, j(xn+1−x^∗)i+ 2b⁰_nhTⁿyn−xn, j(xn+1−x^∗)i + 2c⁰_nhv_n−x_n, j(x_n+1−x^∗)i

=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_nhv_n−x_n, j(x_n+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

kxn+1−x^∗k² ≤ k(xn−x^∗)k²−b⁰_n(1−k)kxn+1−Tⁿxn+1k²

+b⁰_n(k_n² −1)kx_n+1−x^∗k²+ 2(b⁰_n)²hTⁿy_n−x_n, j(x_n+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_nkv_n−x_nkkx_n+1−x^∗k

=k(xn−x^∗)k²−b⁰_n(1−k)kxn+1−Tⁿxn+1k²+ (k²_n−1)kxn+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)

Observe that

ky_n−x^∗k = ka_n(x_n−x^∗) +b_n(Tⁿx_n−x^∗) +c_n(u_n−x^∗)k

≤ kx_n−x^∗k+Lkx_n−x^∗k+M

= (1 +L)kx_n−x^∗k+M, (2.6)

so that

kTⁿyn−xnk ≤ Lkyn−x^∗k+kxn−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

≤ kxn−x^∗k+L[(1 +L)kxn−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]ky_n−x^∗k+c⁰_nM

≤[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

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

(2.9)

Substituting (2.7)-(2.9) in (2.5) we obtain,

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

+ (k_n²−1){[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[[bn(1 +L) +cn] + [b⁰_n(1 +L+c⁰_n](1 +L)]kxn−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 + (k_n²−1){[1 +L(1 +L)]²kx_n+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 +kxn−x^∗k][[1 +L(1 +L)]kxn−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+c_n]}{[1 +L(1 +L)]kx_n−x^∗k+M(1 +L)}.

Sincekx_n−x^∗k ≤1 +kx_n−x^∗k², we have

(2.10) kx_n+1−x^∗k² ≤[1 +δ_n]kx_n−x^∗k²+σ_n−b⁰_n(1−k)kx_n+1−Tⁿx_n+1k², where

δn= (k_n²−1){[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+cn][[1 +L(1 +L)] +M(1 +L)]}.

SinceP∞

n=1(k_n²−1)<∞, condition (iii) implies thatP∞

n=1δ_n<∞andP∞

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 +δj]kx1−x^∗k²+

∞

Y

j=1

[1 +δj]

∞

X

j=1

σj <∞, sinceP∞

n=1δ_n<∞andP∞

n=1σ_n <∞. Hence{kx_n−x^∗k}^∞_n=1 is bounded. Letkx_n−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, whereµn=M²δn+σnso 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(vn−xn)k².

(2.12)

For arbitraryu, v ∈E, setx=u+vandy=−vin (2.3) to obtain

(2.13) kv+uk² ≥ kuk²+ 2hv, j(u)i.

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

≥ kxn−Tⁿxnk²+ 2hb⁰_n(Tⁿyn−xn) +Tⁿxn−Tⁿxn+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_nkv_n−x_nk}kx_n−Tⁿx_nk

≤ 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

≤ kxn+1−Tⁿxn+1k²+ 2(1 +L)kxn−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 +kx_n−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)

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

(2.14) Since lim

n→∞b⁰_n = 0, lim

n→∞c⁰_n = 0 and lim inf

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

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

n→∞ kxn −T xnk = 0,

completing the proof of Lemma 2.3.

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

n k_n = 1, P∞

n=1(k_n² −1)<∞. Let{a_n}, {b_n}, {c_n},{a⁰_n}, {b⁰_n}, {c⁰_n}, {un}, and {vn} be as in Lemma 2.3 and let {xn} 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. 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]

≥ 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)(k²_n−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, where

L= 1 + 2 +p

{(1−k)(D²−1)}

1−k .

HenceT is uniformlyL-Lipschitzian. SinceF(T)6=∅, thenT is uniformly L-Lipschitzian and asymptotically demicontractive and hence the result follows from Lemma 2.3.

Remark 2.5. It is shown in [3] that ifEis a Hilbert space andT :K →K isk-asymptotically pseudocontractive with sequence{k_n}then

kTⁿx−Tⁿyk ≤ D+√ k 1−√

kkx−yk ∀x, y ∈K, where k_n ≤D, ∀n ≥1.

Theorem 2.6. LetEbe a real Banach space andK a nonempty closed convex subset ofE. Let T :K →Kbe a completely continuous uniformlyL-Lipschitzian asymptotically demicontrac- tive 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 Lemma 2.3. Then the sequence {x_n}generated from an arbitraryx₁ ∈Kby

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 Lemma 2.3,lim inf

n kxn−Tⁿxnk= 0, hence there exists a subsequence{xnj}of {x_n}such thatlim

n kx_nj−T x_njk= 0.

Since {xnj} is bounded and T is completely continuous, then {T xnj} has a subsequence {T x_jk} which converges strongly. Hence {x_njk} converges strongly. Suppose lim

k→∞x_njk = p.

Then lim

k→∞T x_njk = T p. lim

k→∞kx_njk −T x_njkk = kp−T pk = 0so thatp ∈ F(T). It follows from (2.11) that

kx_n+1−pk² ≤ kx_n−pk²+µ_n

Lemma 2.2 now implies lim

n→∞kx_n−pk= 0completing the proof of Theorem 2.6.

Corollary 2.7. Let E be an arbitrary real Banach space and K a nonempty closed convex subset ofE. let T : K → K be ak-strictly asymptotically pseudocontractive mapping with F(T) 6= ∅and sequence {k_n} ⊂ [1,∞)such thatlim

n k_n = 1, andP∞

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 Lemma 2.3. Then the sequence {x_n}generated from an arbitraryx₁ ∈Kby

yn = anxn+bnTⁿxn+cnun, n ≥1, x_n+1 = a⁰_nx_n+b⁰_nTⁿy_n+c⁰_nv_n, n≥1, converges strongly to a fixed point ofT.

Proof. As shown in Corollary 2.4,T is uniformlyL-Lipschitzian and sinceF(T)6=∅thenT is asymptotically demicontractive and the result follows from Theorem 2.6.

Remark 2.8. If we setb_n =c_n = 0, ∀n ≥ 1in Lemma 2.3, Theorem 2.6 and Corollaries 2.4 and 2.7, we obtain the corresponding results for the modified Mann iteration method with errors in the sense of Xu [7].

Remark 2.9. Theorem 2.6 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 2.6 is proved without the boundedness condition imposed on the subsetK 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 2.10. 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 modifi- cations of the proofs of the corresponding results for the Ishikawa iteration method with errors in the sense of Xu [7].

Theorem 2.11. LetEbe a real Banach space and letT :E →Ebe a uniformlyL-Lipschitzian asymptotically demicontractive mapping with sequence{k_n} ⊂ [1,∞) such thatlim

n k_n = 1, andP∞

n=1(k_n² −1) < ∞. Let{u_n}and {v_n} be sequences inE such 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≤α_n, β_n ≤1,n ≥1;

(ii) P∞

n=1α_n=∞ (iii) P∞

n=1α²_n<∞andP∞

n=1β_n <∞.

Let{xn}be the sequence generated from an arbitraryx1 ∈Eby y_n = (1−β_n)x_n+β_nTⁿx_n+u_n, n ≥1, xn+1 = (1−αn)xn+αnTⁿyn+vn, n≥1, Thenlim inf

n→∞ kx_n−T x_nk= 0.

Corollary 2.12. LetEbe a real Banach space and letT :E →Ebe ak-strictly asymptotically pseudocontractive map withF(T)6=∅and sequence{k_n} ⊂[1,∞)such thatlim

n k_n = 1, and P∞

n=1(k²_n−1)< ∞. Let{u_n}, {v_n}, {α_n}and{β_n}be as in Theorem 2.11 and let{x_n}be the sequence generated from an arbitraryx₁ ∈E by

yn = (1−βn)xn+βnTⁿxn+un, n ≥1, x_n+1 = (1−α_n)x_n+α_nTⁿy_n+v_n, n≥1, Thenlim inf

n→∞ kx_n−T x_nk= 0.

Theorem 2.13. LetE, T, {u_n}, {v_n}, {α_n}and{β_n}be as in Theorem 2.11. If in addition T : E → E is completely continuous then the sequence {x_n} generated from an arbitrary x₁ ∈E by

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

Corollary 2.14. LetE,T,{u_n}, {v_n}, {α_n}and{β_n}be as in Corollary 2.12. If in addition T is completely continuous, then the sequence{x_n}generated from an arbitraryx, y ∈Eby

y_n = (1−β_n)x_n+β_nTⁿx_n+u_n, n ≥1, xn+1 = (1−αn)xn+αnTⁿyn+vn, n≥1, converges strongly to a fixed point ofT.

Remark 2.15. (a) If K is a nonempty closed convex subset of E andT : K → K, then Theorems 2.11 and 2.13 and Corollaries 2.12 and 2.14 also hold provided that in each case the sequence{x_n}lives inK.

(b) If we setβ_n= 0, ∀n≥1in Theorems 2.11 and 2.13 and Corollaries 2.12 and 2.14, we obtain the corresponding results for modified Mann iteration method with errors in the sense of Liu [2].

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