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:epseelon@aol.com
c
2000Victoria University ISSN (electronic): 1443-5756 043-01
Approximation of Fixed Points of Asymptotically Demicontractive Mappings in
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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 gen- eralization 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.
Contents
1 Introduction. . . 3 2 Main Results . . . 7
References
Approximation of Fixed Points of Asymptotically Demicontractive Mappings in
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1. Introduction
LetE be an arbitrary real Banach space and letJ denote the normalized duality mapping fromE into2E∗ 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 {kn} ⊆ [1,∞), lim
n kn = 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(kn2 −1)kx−yk2, for all n ∈ N. T is called an asymptotically demicontractive mapping with sequencekn⊆[0,∞),lim
n kn= 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(kn2 −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 demicontrac- tive 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 ≤kn2kx−yk2+kk(I−Tn)x−(I−Tn)yk2 and
(1.5) kTnx−Tnyk2 ≤k2nkx−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 demi- contractive mappings in Hilbert spaces. Recently, Osilike [3], extended the the- orems 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(kn2 − 1)<∞. Let{αn}and{βn}be real sequences satisfying the conditions.
(i) 0≤αn, βn≤1,n≥1;
(ii) 0<∈≤cqαq−1n (1 +Lβn)q ≤ 12{q(1−k)(1 +L)−(q−2)}− ∈, for alln ≥1 and for some∈>0; and
(iii) P∞
n=0βn <∞.
Then the sequence{xn}generated from an arbitraryx1 ∈K by yn= (1−βn)xn+βnTnxn, n ≥1,
xn+1 = (1−αn)xn+αnTnyn, n≥1 converges strongly to a fixed point ofT.
In Theorem1.1,cqis a constant appearing in an inequality which character- izesq-uniformly smooth Banach spaces. In Hilbert spaces,q = 2, cq = 1 and withβn = 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
yn =anxn+bnTnxn+cnun, n≥1, (1.6)
xn+1 =a0nxn+b0nTnyn+c0nvn, n≥1,
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where {an}, {bn}, {cn}, {a0n}, {b0n}, {c0n}are real sequences in [0,1]. an+ bn+cn = 1 = a0n+b0n+c0n, {un}and{vn}are bounded sequences in K. If we setbn=cn = 0in (1.6) we obtain the modified Mann iteration method with errors in the sense of Xu [7] given by
(1.7) xn+1 =a0nxn+b0nTnxn+c0nvn, 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{an},{bn},{cn}, {a0n},{b0n}and{c0n}be sequences in[0,1]withan+bn+cn =a0n+b0n+c0n= 1.
Let {un}, {vn}be bounded sequences inK. For arbitraryx1 ∈ K, generate the sequence{xn}by
yn=anxn+bnTnxn+cnun, n ≥1 xn+1 =a0nxn+b0nTnyn+c0nvn, n ≥0.
Then
kxn−T xnk ≤ kxn−Tnxnk+L(1 +L)2kxn+1−Tn−1xn−1k
+L(1 +L)c0n−1kvn−1−xn−1k+L2(1 +L)cn−1kun−1−xnk +Lc0n−1kxn−1−Tn−1xn−1k.
(2.1)
Proof. Setλn=kxn−Tnxnk. Then
kxn−T xnk ≤ kxn−Tnxnk+LkTn−1xn−xnk
≤λn+L2kxn−xn−1k+LkTn−1xn−1−xnk
=λn+L2ka0n−1xn+b0n−1Tn−1yn−1 +c0n−1vn−1−xn−1k +Lka0n−1xn−1+b0n−1Tn−1yn−1+c0n−1vn−1−Tn−1xn−1k
=λn+L2kb0n−1(Tn−1yn−1−xn−1) +c0n−1(vn−1−xn−1)k
+Lka0n−1(xn−1−Tn−1xn−1) +b0n−1(Tn−1yn−1−Tn−1xn−1) +c0n−1(vn−1 −Tn−1xn−1)k
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≤λn+L2kTn−1yn−1−xn−1k+L2c0n−1kvn−1−xn−1k
+Lkxn−1−Tn−1xn−1k+L2kyn−1−xn−1k+Lc0n−1kvn−1−xn−1k +Lc0n−1kxn−1−Tn−1xn−1k
=λn+Lλn−1+L(1 +L)c0n−1kvn−1−xn−1k+L2kTn−1yn−1−xn−1k +L2kyn−1−xn−1k+Lc0n−1kxn−1−Tn−1xn−1)k
≤λn+ 2Lλn−1+L(1 +L)c0n−1kvn−1−xn−1k+L2(1 +L)kyn−1−xn−1k +L2kTn−1xn−1 −xn−1k+Lc0n−1kxn−1 −Tn−1xn−1)k
=λn+L(1 +L)λn−1+L(1 +L)c0n−1kvn−1−xn−1k
+L2(1 +L)kbn−1(Tn−1xn−1 −xn−1) +cn−1(un−1−xn−1)k +Lc0n−1kxn−1−Tn−1xn−1k
≤λn+L(L2+ 2L+ 1)λn−1 +L(1 +L)c0n−1kvn−1−xn−1k +L2(1 +L)cn−1kun−1−xn−1k+Lc0n−1kxn−1−Tn−1xn−1k, completing the proof of Lemma 1.
Lemma 2.2. Let{an}, {bn}and{δn}be sequences of nonnegative real num- bers satisfying
(2.2) an+1 ≤(1 +δn)an+bn, n≥1.
IfP∞
n=1δn<∞andP∞
n=1bn <∞then lim
n→∞anexists. If in addition{an}has a subsequence which converges strongly to zero then lim
n→∞an= 0.
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Proof. Observe that
an+1 ≤ (1 +δn)an+bn
≤ (1 +δn)[(1 +δn−1)an−1+bn−1] +bn
≤ ...≤
n
Y
j=1
(1 +δj)a1+
n
Y
j=1
(1 +δj)
n
X
j=1
bj
≤ ...≤
∞
Y
j=1
(1 +δj)a1+
∞
Y
j=1
(1 +δj)
∞
X
j=1
bj <∞.
Hence{an}is bounded. LetM >0be such thatan≤M, n≥1. Then an+1 ≤(1 +δn)an+bn≤an+M δn+bn =an+σn
where σn = M δn + bn. It now follows from Lemma 2.1 of ([6, p. 303]) that lim
n an exists. Consequently, if {an} 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{kn} ⊆[1,∞), such thatlim
n kn = 1, andP∞
n=1(kn2− 1)<∞. Let{an}, {bn}, {cn},
{a0n},{b0n},{c0n} be real sequences in[0,1]satisfying:
(i) an+bn+cn = 1 =a0n+b0n+c0n, (ii) P∞
n=1b0n=∞,
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(iii) P∞
n=1(b0n)2 <∞,P∞
n=1c0n<∞,P∞
n=1bn<∞, andP∞
n=1cn <∞.
Let{un}and{vn}be bounded sequences inK and let{xn}be the sequence generated from an arbitraryx1 ∈K by
yn =anxn+bnTnxn+cnun, n≥1, xn+1 =a0n+b0nTnyn+c0nvn, n ≥1, thenlim inf
n kxn−T xnk= 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
kxn+1−x∗k2
=k(1−b0n−c0n)xn+b0nTnyn+c0nvn−x∗k2
=k(xn−x∗) +b0n(Tnyn−xn) +c0n(vn−xn)k2
≤ k(xn−x∗)k2+ 2hb0n(Tnyn−xn) +c0n(vn−xn), j(xn+1−x∗)i
=k(xn−x∗)k2−2b0nhxn+1−Tnxn+1, j(xn+1−x∗)i + 2b0nhxn+1−Tnxn+1, j(xn+1−x∗)i
+ 2b0nhTnyn−xn, j(xn+1−x∗)i+ 2c0nhvn−xn, j(xn+1−x∗)i
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=k(xn−x∗)k2 −2b0nhxn+1−Tnxn+1, j(xn+1−x∗)i + 2b0nhxn+1−xn, j(xn+1−x∗)i
+ 2b0nhTnyn−Tnxn+1, j(xn+1−x∗)i + 2c0nhvn−xn, j(xn+1−x∗)i.
(2.4)
Observe that
xn+1−xn =b0n(Tnyn−xn) +c0n(vn−xn).
Using this and (1.2) in (2.4) we have kxn+1−x∗k2
≤ k(xn−x∗)k2 −b0n(1−k)kxn+1−Tnxn+1k2 +b0n(k2n−1)kxn+1−x∗k2
+ 2(b0n)2hTnyn−xn, j(xn+1−x∗)i + 2b0nhTnyn−Tnxn+1, j(xn+1−x∗)i + 3c0nhvn−xn, j(xn+1−x∗)i
≤ k(xn−x∗)k2 −b0n(1−k)kxn+1−Tnxn+1k2 + (kn2−1)kxn+1−x∗k2
+ 2(b0n)2kTnyn−xnkkxn+1−x∗k + 2b0nLkxn+1−ynkkxn+1−x∗k + 3c0nkvn−xnkkxn+1−x∗k
=k(xn−x∗)k2−b0n(1−k)kxn+1−Tnxn+1k2 + (kn2−1)kxn+1−x∗k2+ [2(b0n)2kTnyn−xnk + 2b0nLkxn+1−ynk+ 3c0nkvn−xnk]kxn+1−x∗k.
(2.5)
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Observe that
kyn−x∗k = kan(xn−x∗) +bn(Tnxn−x∗) +cn(un−x∗)k
≤ kxn−x∗k+Lkxn−x∗k+M
= (1 +L)kxn−x∗k+M, (2.6)
so that
kTnyn−xnk ≤ Lkyn−x∗k+kxn−x∗)k
≤ L[(1 +L)kxn−x∗k+M] +kxn−x∗k
≤ [1 +L(1 +L)]kxn−x∗k+M L, (2.7)
kxn+1−x∗k = ka0n(xn−x∗) +b0n(Tnyn−x∗) +c0n(vn−x∗)k
≤ kxn−x∗k+Lkyn−x∗k+M
≤ kxn−x∗k+L[(1 +L)kxn−x∗k+M] +M
= [1 +L(1 +L)]kxn−x∗k+ (1 +L)M, (2.8)
and
kxn+1−ynk=ka0n(xn−yn) +b0n(Tnyn−yn) +c0n(vn−yn)k
≤ kxn−ynk+b0n[kTnyn−x∗k+kyn−x∗k]
+c0n[kvn−x∗k+kyn−x∗k]
=kbn(Tnxn−xn) +cn(un−xn)k+b0n[Lkyn−xnk+kyn−x∗k]
+c0nM +c0nkyn−x∗k
≤bn(1 +L)kxn−x∗k+cnM +cnkxn−x∗k + [b0n(1 +L) +c0n]kyn−x∗k+c0nM
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≤[bn(1 +L) +cn]kxn−x∗k+cnM
+ [b0n(1 +L) +c0n][(1 +L)kxn−x∗k+M] +c0nM
≤ {[bn(1 +L) +cn] + [b0n(1 +L) +c0n](1 +L)}kxn−x∗k +M[b0n(1 +L) + 2c0n+cn].
(2.9)
Substituting (2.7)-(2.9) in (2.5) we obtain, kxn+1−x∗k2
≤ k(xn−x∗)k2−b0n(1−k)kxn+1−Tnxn+1k2
+ (k2n−1){[1 +L(1 +L)]kxn+1−x∗k+M(1 +L)}2
+{(b0n[[1 +L(1 +L)]kxn−x∗k+M L] + 3c0n[M+kxn−x∗k]
+ 2b0nL[[bn(1 +L) +cn] + [b0n(1 +L+c0n](1 +L)]kxn−x∗k
+M[b0n(1 +L) + 2c0n+cn]}{[1 +L(1 +L)]kxn−x∗k+M(1 +L)}
≤ k(xn−x∗)k2−b0n(1−k)kxn+1−Tnxn+1k2 + (k2n−1){[1 +L(1 +L)]2kxn+1−x∗k2
+ 2M(1 +L)[1 +L(1 +L)]kxn−x∗k+M2(1 +L)2}
+ 2(b0n)2[[1 +L(1 +L)]kxn−x∗k+M L][[1 +L(1 +L)]kxn−x∗k +M(1 +L)] + 3c0n[M +kxn−x∗k][[1 +L(1 +L)]kxn−x∗k +M(1 +L)] + 2b0nL{[[bn(1 +L) +cn]
+ [b0n(1 +L) +c0n](1 +L)]kxn−x∗k
+M[b0n(1 +L) + 2c0n+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−b0n(1−k)kxn+1−Tnxn+1k2,
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where
δn= (k2n−1){[1 +L(1 +L)]2+ 2M(1 +L)[1 +L(1 +L)]}
+ 2(b0n)2{[1 +L(1 +L)]2+M(1 +L)[1 +L(1 +L)]
+M L[1 +L(1 +L)]}
+ 3c0n{[1 +L(1 +L)] +M[1 +L(1 +L)] +M(1 +L)}
+ 2b0nL{{[bn(1 +L) +cn] + [b0n(1 +L) +c0n](1 +L)}
× {[1 +L(1 +L)] +M(1 +L)}
+M[b0n(1 +L) + 2c0n+cn][1 +L(1 +L)]}
and
σn = (kn2 −1){2M(1 +L)[1 +L(1 +L)] +M2(1 +L)2}
+ 2(b0n)2{[1 +L(1 +L)]M(1 +L) +M L[1 +L(1 +L)] +M2L(1 +L)}
+ 3c0n{M[1 +L(1 +L)] +M2(1 +L) +M(1 +L)
+ 2b0nL{[[bn(1 +L) +cn] + [b0n(1 +L) +c0n](1 +L)][M(1 +L)]
+M[b0n(1 +L) + 2c0n+cn][[1 +L(1 +L)] +M(1 +L)]}.
Since P∞
n=1(k2n − 1) < ∞, condition (iii) implies that P∞
n=1δn < ∞ and P∞
n=1σn<∞. From (2.10) we obtain
kxn+1−x∗k2 ≤ [1 +δn]kxn−x∗k+σn
≤ ...≤
n
Y
j=1
[1 +δj]kx1−x∗k2+
n
Y
j=1
[1 +δj]
n
X
j=1
σj
≤
∞
Y
j=1
[1 +δj]kx1−x∗k2+
∞
Y
j=1
[1 +δj]
∞
X
j=1
σj <∞,
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since P∞
n=1δn < ∞andP∞
n=1σn < ∞. Hence{kxn−x∗k}∞n=1 is bounded.
Letkxn−x∗k ≤M, n≥1. Then it follows from (2.10) that (2.11) kxn+1−x∗k2 ≤ kxn−x∗k2+M2δn+σn
−b0n(1−k)kxn+1−Tnxn+1k2, n ≥1 Hence,
b0n(1−k)kxn+1−Tnxn+1k2 ≤ kxn−x∗k2− kxn+1−x∗k2+µn, whereµn =M2δn+σnso that,
(1−k)
n
X
j=1
b0jkxj+1−Tjxj+1k2 ≤ kx1−x∗k2+
n
X
j=1
µj <∞, Hence,
∞
X
n=1
b0nkxn+1−Tnxn+1k2 <∞, and condition (ii) implies thatlim inf
n→∞ kxn+1−Tnxn+1k= 0. Observe that kxn+1−Tnxn+1k2 = k(1−b0n−c0n)xn+b0nTnyn+c0nvn−Tnxn+1k2
= kxn−Tnxn+b0n(Tnyn−xn) +Tnxn−Tnxn+1 +c0n(vn−xn)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 kxn+1−Tnxn+1k2
=kxn−Tnxn+b0n(Tnyn−xn) +Tnxn−Tnxn+1+c0n(vn−xn)k2
≥ kxn−Tnxnk2+ 2hb0n(Tnyn−xn) +Tnxn−Tnxn+1 +c0n(vn−xn), j(xn−Tnxn)i.
Hence
kxn−Tnxnk2
≤ kxn+1−Tnxn+1k2+ 2kb0n(Tnyn−xn)
+Tnxn−Tnxn+1+c0n(vn−xn)kkxn−Tnxnk
≤ kxn+1−Tnxn+1k2+ 2{b0nkTnyn−xnk+Lkxn+1−xnk +c0nkvn−xnk}kxn−Tnxnk
≤ kxn+1−Tnxn+1k2+ 2{b0nkTnyn−xnk+Lb0nkTnyn−xnk +Lc0nkvn−xnk+c0nkvn−xnk}kxn−Tnxnk
≤ kxn+1−Tnxn+1k2+ 2(1 +L)kxn−x∗k
× {(1 +L)b0nkTnyn−xnk+ (1 +L)c0nkvn−xnk}
≤ kxn+1−Tnxn+1k2+ 2(1 +L)kxn−x∗k
× {(1 +L)b0n[[1 +L(1 +L)]kxn−x∗k+M L]
+ (1 +L)c0n[M +kxn−x∗k], (using (2.6))
≤ kxn+1−Tnxn+1k2
+ 2(1 +L)M{(1 +L)b0n[[1 +L(1 +L)]M +M L]
+ (1 +L)c0n[M +M]}, (since kxn−x∗k ≤M)
Approximation of Fixed Points of Asymptotically Demicontractive Mappings in
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(2.14) =kxn+1−Tnxn+1k2+ 2b0n(1 +L)4M2+ 4c0n(1 +L)2M.
Since lim
n→∞b0n = 0, lim
n→∞c0n = 0 andlim inf
n→∞ kxn+1 −Tnxn+1k = 0, it follows from (2.14) thatlim inf
n→∞ kxn−Tnxnk = 0. It then follows from Lemma 1 that lim inf
n→∞ kxn−T xnk= 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(k2n− 1) < ∞. Let {an}, {bn}, {cn},{a0n}, {b0n}, {c0n}, {un}, and {vn} be as in Lemma2.3 and let{xn}be the sequence generated from an arbitraryx1 ∈ K by
yn =anxn+bnTnxn+cnun, n≥1, xn+1 =a0n+b0nTnyn+c0nvn, 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−(k2n−1)kx−yk2}
= 1 2[√
1−kk(I −Tn)x−(I−Tn)yk +p
kn2−1kx−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)(k2n−1)}
1−k ]kx−yk.
Furthermore,
kTnx−Tnyk − kx−yk ≤ k(I−Tn)x−(I−Tn)yk
≤ [2 +p
{(1−k)(kn2 −1)}
1−k ]kx−yk, from which it follows that
kTnx−Tnyk ≤[1 + 2 +p
{(1−k)(kn2−1)}
1−k ]kx−yk.
Since{kn}is bounded, letkn ≤D, ∀n≥1. Then kTnx−Tnyk ≤ [1 + 2 +p
{(1−k)(D2−1)}
1−k ]kx−yk
≤ Lkx−yk,
Approximation of Fixed Points of Asymptotically Demicontractive Mappings in
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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{kn}then
kTnx−Tnyk ≤ D+√ k 1−√
kkx−yk ∀x, y ∈K, where kn≤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{kn} ⊂ [1,∞)such thatlim
n kn= 1andP∞
n=1(kn2−1)<∞. Let{an},{bn},{cn},{a0n}, {b0n}, {c0n}, {un}, and{vn}be as in Lemma2.3. Then the sequence{xn}gen- erated from an arbitraryx1 ∈K by
yn = anxn+bnTnxn+cnun, n ≥1, xn+1 = a0nxn+b0nTnyn+c0nvn, n ≥1, converges strongly to a fixed point ofT.
Proof. From Lemma2.3,lim inf
n kxn−Tnxnk= 0, hence there exists a subse- quence{xnj}of{xn}such thatlim
n kxnj −T xnjk= 0.
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Since{xnj}is bounded andT is completely continuous, then{T xnj}has a subsequence{T xjk}which converges strongly. Hence{xnjk}converges strongly.
Suppose lim
k→∞xnjk =p. Then lim
k→∞T xnjk =T p. lim
k→∞kxnjk −T xnjkk
=kp−T pk= 0so thatp∈F(T). It follows from (2.11) that kxn+1−pk2 ≤ kxn−pk2+µn 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{kn} ⊂ [1,∞)such that limn kn= 1, andP∞
n=1(k2n−1)<∞. Let{an}, {bn}, {cn},{a0n}, {b0n}, {c0n}, {un}, and{vn} be as in Lemma2.3. Then the sequence {xn} generated from an arbitraryx1 ∈Kby
yn = anxn+bnTnxn+cnun, n ≥1, xn+1 = a0nxn+b0nTnyn+c0nvn, 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 setbn = cn = 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 condi- tion 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{an},{bn},{cn},{a0n},{b0n},{c0n},{un}, and{vn}are completely independent of any geometric properties of underlying Banach space.
Remark 2.4. Prototypes for our iteration parameters are:
b0n = 1
3(n+ 1), c0n = 1
3(n+ 1)2, a0n = 1−(b0n+c0n), bn = cn= 1
3(n+ 1)2, an= 1− 1
3(n+ 1)2, n ≥1.
The proofs of the following theorems and corollaries for the Ishikawa itera- tion 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 kn = 1, andP∞
n=1(k2n−1) < ∞. Let {un}and{vn}be sequences inEsuch thatP∞
n=1kunk<∞andP∞
n=1kvnk<∞, and let{αn} and{βn}be sequences in[0,1]satisfying the conditions:
(i) 0≤αn, βn≤1,n≥1;