Available at: http://www.ictp.trieste.it/~pub-off IC/98/214
United Nations Educational Scientific and Cultural Organizationand
International Atomic Energy Agency
THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
ITERATIVE APPROXIMATIONOF FIXED POINTS OF DEMICONTRACTIVE MAPS
Chika Moore1
Department of Mathematics and Computer Science, Nnamdi Azikiwe University,P.M.B. 5025, Awka, Anambra State, Nigeria2
andThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
Abstract
Let K be a closed convex nonempty subset of a real q-uniformly smooth Banach space and
let T : K i—> K be demicontractive with (I — T) demiclosed at 0 G K. It is proved that the
Mann and Ishikawa iteration processes with errors (in the sense of Xu [24]) converge weakly to
a fixed point of T. If K is compact and convex, then the convergence is strong. Related results
deal with other sufficient conditions for strong or weak convergence of the iteration processes
used.
MIRAMARE - TRIESTE
November 1998
1 Regular Associate of the Abdus Salam ICTP.This research was supported in part by Research Grant RG/MATHS/AF/AC No.97-210 from the Third worldAcademy of Sciences (TWAS).
2Permanent address.
1 Introduction:
Let H be a Hilbert space. In [1,7] the following class of maps was introduced. A map T with
domain D(T) and range R(T) in H is said to be strictly pseudocontractive (in the sense of
[1,7,9,13]) if there exists a constant k G [0,1) such that Vs,j/e D(T) we have that
2 < ||a; - y| |2 + k\\x - y - (Tx - Ty)\\2 (1)
Let F(T) = {x G D(T) : Tx = x}. If F(T) / 0 and there exists a constant k G [0,1) such that
\fx G D(T) and x* G F(T) the following inequality holds
| |Ta;-a;* | | 2 < \\x - x*\\2 + k\\x-Txf (2)
then, T is said to be demicontractive. It is easy to see that a strictly pseudocontractive map
with a fixed point is demicontractive. Thus, the class of demicontractive maps properly includes
the class of strictly pseudocontractive maps with fixed points (see e.g., [1,7,17]). The important
class of quasi-nonexpansive maps (where a map T is said to be quasi-nonexpansive if F(T) = 0
and 11To; - x*\\ < \\x - x*\\; V s e D(T) and Va;* G F(T)) is also a subclass of this class of
maps. Moreover, if the fixed point set F(T) is nonempty, then the classes of maps studied by
such authors as Goebel et al [6], Kannan [9] and Wong [21] are special subclasses of the class of
demicontractive maps. Furthermore, If we set λ = 2k) , then it is routine to see that, in Hilbert
spaces, (??) is equivalent to the condition: there exists a constant λ > 0 such that M x G D(T)
and x* G F(T) we have that
-Tx, x-x*)>X\\x-Tx\\2 (3)
which is the condition satisfied by the class of maps introduced by Maruster [12]. Thus, the class
of nonlinear maps introduced in 1977 by Hicks and Kubicek [7] and Maruster [12] independently
coincide in Hilbert spaces. It is easy to observe from (??) that
||Ta?-a?*|| < ||aj-aj*|| + \/fc||a;-ra;||
< (l + \/fc)||a;-a;*|| + \/fc||ra;-a;*||
(l-\/fc)||Ta;-a;*|| < (1 + Vk)\\x - x*\\
\\Tx-x*\\
Wth ^ ( l ± | ) (4,
and from (??) that
\\x-x*\\ > A||a:-Ta:| | > A(||Ta: - x*\\ - | |a;-a;*| |)
so that
| |Ta:-a:*| | < L | | a : - a : * | | ; with L = l + A"1 (5)
Several authors have studied this class of nonlinear maps (see e.g., [1-9, 10, 12-20]) and conver-
gence theorems established for iteration processes of the Mann-type (see e.g., [11]).
It is our purpose in this paper to establish strong and weak convergence of both the Mann
and Ishikawa iteration processes (with errors) to a fixed point of a demicontractive map in real
q-uniformly smooth Banach spaces.
2
2 Preliminaries
Let E be a normed linear space of dimension dim.E > 2. The modulus of smoothness of E is
defined by
1 : ||x|| l , | W
The linear space E is said to be smooth if ΡE(Τ) > 0 Vr > 0. If there exists a constant c > 0
and a real number 1 < q < oo such that
then E is said to be q-uniformly smooth. Typical examples of such spaces are the Lesbesgue Lp,
the sequence £p and the Sobolev Wpm spaces for 1 < p < oo. In fact, these spaces are p-uniformly
smooth if 1 < p < 2 and 2-uniformly smooth for p > 2.
Let E* denote the dual of E and let Jp : E i-> 2E* denote the generalized duality mapping
defined by
/ r — { f* (z w* • I T f*\ — \\r\\P- II f*ll — H r P " 1 !,Jp.L •— \J t -̂ • \-L> J I — II-HI ) IIJ II — II-HI J
where (.,.) denotes the generalized duality pairing. It is known (see e.g., [22,23]) that Jp is
single-valued (and denoted by jp) and 'Lipschitz' Holder-continuous with constant L* > 0 HE
is p-uniformly smooth. That is,
WjpX-jpVW^LtWx-yW"-1, Vx,yeE
Moreover, Mo / x G E, Jpx = \\x\\p~2J2X where J = J2 is the normalized duality map.
If E is a real q-uniformly smooth Banach space, then (see e.g., [22]) the following inequality
holds: V x,y G E and some positive constant Cq we have that
\\x + y\\q<\\xr + q(y,jp{x))+Cq\\y\\q (6)
In [2], Chidume extended the condition (??) to arbitrary real Banach spaces thus: Vx G
D(T), x* G F(T) and j(x — x*) G J(x — x*) there exists a positive constant λ such that
(x-Tx,j(x-x*))>X\\x-Tx\\2 (7)
If E is q-uniformly smooth, then condition (??) becomes
(x-Tx,jq(x-x*)) > \\x-x*\r2(x-Tx,j(x-x*))
> \\x-x*\\q-2.X\\x-Tx\\2
> \q-l\\x-Tx\\q
So that we now have the following inequality
(Tx - x*, jq(x - x*)) < \\x-x*\\q -Xq-l\\x-Tx\\q (8)
We shall need the following definitions in the sequel: Let K be a subset of a Banach space
E. A map T : K i-> E is said to be demiclosed at z G E if the weak convergence of {xn} C K
to some point p G K and the strong convergence of {Txn} to z implies that Tp = z. A map
T : K <-^ E is said to be demicompact at z G K if for any bounded sequence {xn} C K such
that (I — T)xn -^ z as n —> 00 then there exist a subsequence {xn j} and a point p e K such
that x n j —> p as j —>• 00 and (I — T)p = z.
We now prove the following theorems.
3
3 Main Theorems
Theorem 1 Let K be a closed convex nonempty subset of a real q-uniformly smooth Banach
space and let T : K i—> K be demicontractive with (I — T) demiclosed at o e K. Let {un}, {vn}
be bounded sequences in K and let the real sequences {an}, {bn}, {cn}, {a'n}, {b'n}, {c'n} C [0,1]
satisfy the following conditions:
(i) an + bn + cn = 1 = a'n + b'n + c'n; n > 0
(ii) ^ bn = oo and ^ cn < oon>0 n>0
(iii) ^ bnb^ < oo and ^ bncn < s<n>0 n>0
where s = min{1, q — 1}. Then, the sequence {xn} generated from an arbitrary X0 G K by
xn+1 = anxn + bnTyn + cnvn; n > 0 (9)
yn = <4a:ra + ft^T^ + c'nun; n>0 (10)
converges weakly to a fixed point ofT.
Proof. Let x* G K be a fixed point of T. Then, we have the following estimates:
\\h'(Tr — r*) 4- r'(v — r*)\\q < b'q\\Tr — r*\\q 4- C rqhi — r*\\q\\unyi xn x ) -|- cn\u,n x ) \ \ ^ wn ii-i xn x || -|- OgCra \\an x \\
-\-ab'(q~l^r' \\v — r*\\ WTr — r*\\q~l
i<lun cn\\u'n -L W-W1-Ln X ||
< Tqh'q\\<r — <r*\\q -i- C r'q\\ii — r*\\q
— T*) — i (v — T*)\\ < L \W —II I I 9 " 1x ) Jq\yn X ) \ \ L ±j*\\£n yn\\
< L , {[(L + l)b'n + dn\ \\xn -x*\\+ dn\\un - x*
\Vn-x*\\q
r' (v — r* i (r — r*)\ 4- C, \\b' (Tr — r*\ -\- r' (11 — r*)cn\u"n. -L i Jq\-Ln X )) i~ ̂ q\\Un[-L Xn X ) -\- Cn[U,n X )
— rx\ an \\xn x || q+qan un' (\\r — r* \\q — \q~l I IT —Trn i ||xra x \\ A \\xn J-xn
ln Cn\\Un' Hi/ — o™* II 11 o™ — T * \\q~l A- C \\h (Tv _ T * U ^ (11 — <r*\\\q
n\\Un -h \\-\\-hn X || \ ^qW^ny1-^n X ) \ (^n\U>n X ) \ \
~l) (n' 4- / I / )MHT T * I | 9 n\q~ln'(q~l)h' I IT T T \\q
^ / ) ' ( < ? - ! ) • / II , , _ ™ * | | | | T T — T * ! ! 9 " 1 -I- I ^ V ^ H i / — T * l l g
-^ q n n 11 n I r l l ^ i II ' q n 11 ^ 11
-Vi ) ll^ra X \\ qA (l,n Vnv\Xn J- %n\\
qc'n ( a ^ 1 ) + CqLilb^) \\un - x*\\.\\xn - x*
+qc'n I a ra« ^ + G ' g ^ ^ ^ ^ I \\Un - x'-\\.\\xn - x"
The last inequality follows from the fact that for x, y G [0,1] the following estimate holds
f(x, y) = il-x-y)q + q i l - x - y)q~lx + CqLqxq < (1 - x)q + q(1 - a;)"-1^ + C q L q x q
Now,
3J 11 S : Q"n 11 ^"ra ^ II ~T~ ^ q 11 ^ n \-L y n X ) -\- Cn (Vn X )\\
+qaql~
1bn{Tyn - x*,jqixn - x*)) + qa^lCn{vn - x*,jqixn - x*))
< aqjxn - x*\\q + qatlbn (\\yn ~ x*\\q - \q-l\\yn - Tyn
+qaql~
1bn(Tyn - x*,jq(xn - x*) - jq(yn - x*))
+qaqr1cn{vn - x*,jqixn - x*)) + Cqbqn\\Tyn - x*\\q
+C2qc
qn\\vn - x*\\q + qCqb
qn-
lcn\\vn - x*\\.\\Tyn - x*\\q~l
< Inq A- nnq~lh A- C Tqhq\ \\v — T* 11̂ — n\q~lnq~lh Hi/ — Tii II— X^n'q^n " n T W » J J Un I \\Xn X || qA U,n Un\\yn -1 yn\\
— n \ q ~ l n ' ^ q ~ ^ > f n n q ~ 1 A - C T q b q ~ 1 ^ b ft' I I T — T T II9
qA Un \qun -\- K^/qLi Un I UnUn\\Xn J- Xn\\
\ n (n'(q-l) I n r<?-l?,/(<?-l)A\ (nnq~l 4- H Tqhq~l\h o1 Wti T * I I I I T-\-q \Un -\-lsqL/ Un I \qtln ~\- isqLz Un I UnCn\\Un — X | | . | |Xn — .
C2q ( g a r 1 + CqL%tl) bnc'«\\un - x*\\q + C&\\vn - x*\\q
tlbn ([(L + l)b'n + c'n]\\xn -x*\\ + c'Jun - x*\\)q~l x
\\xn-x*\\+c'n\\un-x*
1bt1Cn\\Vn ~X*\\. ([(L + l)b'n + C'n + l]\\xn -X*\\+ C'jun -
-nnq~lr Wit —r*\\ \\r — v*\\q~lqan tn\\vn x \\.\\xn x 11
Let
M = max<sup{||-ura - a ;* | | } , sup{||vra - x*n>0 n>0
We consider the following two cases:
case 1: all n > 0 such that \\xn — x*\\ > M: In this case, we have that
\\xn+i-x*\\q <(l+-fn)\\xn-x*\\q, with ^ 7 r a < o o
case 2: all n > 0 such that \\xn — x*\\ < M In this case, we have that for some constant
M1 > Mq the following inequality holds
Hi- , 1 — 1-*II* < Hi- — 1-*II* -I- M . ' vH-^ri+l X || \ \\Xn X || -f- IVIl /n
Thus, limra^oo \\xn — x*\\ exists and so the sequences {xn}, {yn}, {Txn}, {Tyn} are bounded.
Furthermore, for some positive constant M0 we then have
q\q-laq-lbn\\yn - Tyn\\q < \\xn - x*\\q - \\xn+l - x*\\q + M0(bn^ + bnc',° + cn)
Iterating downwards, we have
q\q~l Y, <~lbn\\yn - Tyn\\q < \\x0 - x*\\q + Mo ^ ( W l s + bj° + cn) < oo
ra>0 ra>0
so that from the hypothesis, we deduce that
liminf \\yn -Tyn\\ = 0
and so there exists a subsequence {ynj} C {yn} such that
ynj -^ V and ynj - Tynj -^ o as j - • oo
Since (I — T) is demiclosed at o e i f and K is weakly closed, we have that p G F(T), the fixed
point set of T. Thus, for any /* e i£* we have that
fiVnj -p) ->0 as j ^ oo
Now,
f*(xnj -P) = f*(xnj - ynj) + f*(yn. -p) = b'nJ*{xnj - Txnj) + dnj*(xn] - unj) + f*(yn. -p)
so that
I™ \f*{xni -p)\ < lim {b'n]\f{xn]-Txnj)\+d \f{xn]-unj)\ + \f*(yn. -p)\\ = 0j ^ o o J 3—>oo L J J J j J J j j
and hence xnj -^ p as j —>• . Observe that for some constant M0 > 0,
a'Jxn-Tyn\\ < \\yn-Tyn\\+b'n\\Txn-Tyn\\+c'n\\un-Tyn\\
< \\yn-Tyn\\+M0(b'n + c'n)
Hence,lim \\xnj -Tyn\\ = 0
We now claim thatf*(xnj+k-p)^0 as j ^oo ; Vfc>0
Suppose that the claim is true for some k = m. Then, from
f*(ynj+m -p) = b'n.+mf*(xnj+m - Txnj+m)
+c'nj+mf*(Unj+m ~ xnj+m) + f*(xnj+m-p)
we see that
Vrij+m -^P as j -^ oo
so that, additionally, from
.TOcrv+m+i ~p) = bnj+mf*(Tynj+m-xnj+m)
+Cnj+mf*(vnj+m - xnj+m) + f*(xnj+m - p)
we have that
\f*(xnj+m+i-p)\ < Mcnj+m + \f(xnj+m-p)\
+ \\f*\\-\\xnj+m-Tynj+m\\ -^ 0 as j -• oo
Since, the claim is trivially true for k = 0, it follows by the inductive hypothesis that the claimholds for all k > 0. Hence, {xn} converges weakly to a fixed point of T. This completes theproof. •
Theorem 2 Let E and the sequences be as in Theorem 1. Suppose that K is compact convexand that T : K i-> K is demicontractive. Then, the sequence {xn} converges strongly to a fixedpoint ofT.
Proof. Proceeding as in the proof of Theorem 1, we obtain that liminf \\yn — Tyn\\ = 0. Thisimmediately implies that some subsequence {ynj} of {yn} converges strongly to a fixed point,say p, of T. From
\\xn P\\ 2i \\xn yn\\ i^ \\yn p\\
<" yj ll/v. T1'?1 II 4 - r1 \\T ii— "rall"t/ra 1 -^nW \ l-n\vhn "-
II 4 - Wii-rill \ \\yn
it follows that {xnj} also converges strongly top. This implies that liminf \\xn— p\\ = 0. Observe
that from hypotheses of the theorem, σn = M0(bnb'n + bnc^ + cn) is summable and that with
Φn = \\xn -p\\q we have
Now, given any ε > 0 there exist an integer j0 sufficiently large such that
and another integer N 1 sufficiently large such that
since the tail of a summable series is arbitrarily small. Choose j * sufficiently large such that
njt > max{n j 0, N1}
Then, for any k > 0 we have that
k+1
r=0
< ε
+
ε = ε
- 4 4 2
Since, ε > 0 is arbitrary, it follows that xn —x a;* strongly as n —>• . This completes the proof.
n
Theorem 3 Let T,K,E be as in Theorem 1 and let {un} be a bounded sequence in K. Let
{an}, {bn}, {cn} ^ [0,1] be real sequences satisfying the following conditions:
(i) an + bn + cn = 1; n>0 (ii) J ^ bn = oo (iii) J ^ cn < oora>0 ra>0
Then, the sequence {xn} generated from an arbitrary X0 G K by
xn+1 = anxn + bnTxn + cnun; n > 0 (1 1)
converges weakly to a fixed point ofT.
Proof. Let
<Tn •= [q^ + CgL^bt^llXn-xT^+C^llUn-xT1] \K ~ X*
δn := q\q-laq
n-lbn
Then, proceeding as in the proof of Theorem 1, we easily obtain
A I I T _ T V I I * < I I T — T * I I * — I I T M _ / J - * | | 9 - | _ / TO r a l l - t r a -t • t r i l l ^ \\Xn X \\ W-Ln+l X \\ -\- (Tn cn
so that
5 3 Sn\\Xn ~ Txn\\q < \\XO - X*\\q + ^ σncn < OOra>0 ra>0
Since J2n>oδn = oc then liminfra^oo \\xn — Txn\\q = 0. So, there exists a subsequence {xnj} of
{xn} such that
xnj — Txnj —> o and xnj -^ p as j —> oo
Since K is weakly closed and (I — T) is demiclosed at o G K then p £ K and (I — T)p = o.
We now claim that
lim \f*{x nj+k-p)\=0; Vfc>0 and V f* e E*
Suppose that the claim is true for k = m. Then,
lim \f(xnj+m+i-p)\ < lim \f*(xnj+m - p)| + lim 6 r a .+ m |/*(a; r a .+ m - Txnj+m)|J —>OO J—>OO J—>OO
+ lim cnj+m\f*(xnj+m-unj+m)\
= 0
Since the claim is trivially true for k = 0 the claim follows by the inductive hypothesis. Thus,
the assertion of the theorem holds and the proof is complete. •
Theorem 4 Let E and the sequences be as in Theorem 3 and let K be compact and convex.
Suppose that T : K i—> K is demicontractive. Then, the sequence {xn} converges strongly to a
fixed point of T.
Proof. Follows as in the proof of Theorem 2 with minimal changes. •
Corollary 1 In Theorem 1, let T : K i—> K be demicontractive and continuous at a point p, a
cluster point of {xn}. Let
(i) 0 < α < bn < 1, n > 0 (ii) J ^ cn < oo, (iii) J ^ b^ < oo, (iv) ^ cn < s<n>0 n>0 n>0
where s = min{1, q — 1}. Then, p e F(T) and {xn} converges strongly to p.
Proof. Let δn = q\q~1aq~1bn. Then, for some positive constant d we have that δn > d > 0, so
that from
5n\\Vn ~ T y n \ \ q < \\Xn - X*\\q - \\Xn+1 - X*\\q + M i ( & ; s + 4 S + Cn)
and, consequently,
d E WVn ~ TVnW < \\XO ~ X*\\q + M i J^ibn + Cn + Cra) < OOn>0 n>0
we conclude that
and also
lim \\xn -Tyn\\ = 0n—>oo
Since p is a cluster point of {xn}, there is a subsequence {xn j} of {xn} which converges strongly
to p. Then,
'Jnj -p\\ < an.\\xnj -p\\ + b'n.\\Txnj -p\\ + c'n.\\unj -p\\ -»• 0 as j oo
Hence, ynj —>• p as j —>• oo and so Tynj —>• Tp as j —>• . Since l i m ^ o o \\ynj — Tynj\\ =
\\p -Tp\\ = 0, it follows that p G F(T). As in the proof of Theorem 1, we obtain
So that since M1 J2n>o(cn + &« + cn) < °°> it follows that lim \\xn — p\\ exists. Hence, since
Hindoo \\xnj — p\\ = 0, we have that lim^^oo \\xn —p\\ = 0. This completes the proof. •
Corollary 2 In Theorem 3, let T : K i—> K be demicontr active and continuous at a point p, a
cluster point of {xn}. Let
(i) an + bn + cn = 1; n > 0 (II) 0 < α < bn < 1; (iii) J ^ c n < oora>0
Then, {xn} converges strongly to p.
Proof. Follows easily from Corollary 1 on setting b'n = 0 = c'n. •
Corollary 3 In Corollary 1, letT : K i-> K be a demicontractive map which is also demicom-
pact at o G K. Then, {xn} converges strongly to a fixed point ofT.
Proof. Since {yn} is bounded and the sequence {yn — Tyn} converges strongly to o, then by
the demicompactness of T there is a subsequence {ynj} of {yn} which converges strongly to
some point p G F(T). From
\xnj - p\\ < \\ynj - p\\ + b'n. \\xnj - Txnj || + c'n. \\xnj - unj |nj \\ \\
< \\ynj -P\\ + M*Q/n. + c'n.) ^ 0 as j - . oo
we have that {xnj} converges strongly to p. The rest now follows as in the proof of Corollary
1. •
Corollary 4 In Corolary 2, letT : K i—> K be a demicontractive map which is also demicompact
at o G K. Then, {xn} converges strongly to a fixed point of T.
Proof. Follows easily from Corollary 3. •
4 Remarks
1. By setting cn = 0 = dn in our theorems, we obtain that the usual Mann and Ishikawa
iteration processes converge (strongly and weakly, according to conditions) to a fixed point
of T.
2. The convergence theorems in this paper do not depend on the Opial condition and as such
our theorems hold, in particualr, in the Lebesgue Lp (p = 2) spaces.
3. Suppose that A : E i-> E is a continuous linear operator, with zero as an eigenvalue,
satisfying the condition that for all z G D(A) and some λ a positive constant,
10
Suppose further that f G R(A). Then we may apply, say, Corollary 1 to prove that the
Mann and Ishikawa iteration methods (with errors) converge strongly to a solution of the
equation Ax = f. The details are routine.
Acknowledgments
This research was carried out while the author was visiting the Abdus Salam International
Centre for Theoretical Physics, Trieste, Italy as an Associate; a generous grant from the Swedish
International Development Cooperation Agency (SIDA) made the visit possible. The author is
most grateful to both.
References
1. F. E. Browder and W. V. Petryshyn; Construction of fixed points of nonlinear mappings
in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-228.
2. C. E. Chidume; The solution by iteration of nonlinear equations in certain Banach spaces,
J. Nig. Math. Soc. 3 (1984), 57-62.
3. C. E. Chidume; An iterative method for nonlinear demiclosed monotone-type operators,
Dynamical Systems & Appl. 3 (1994), 349-356.
4. C. E. Chidume and Chika Moore; Fixed point iteration for pseudocontractive maps, Proc.
Amer. Math. Soc. (1998/99), accepted, to appear.
5. J. B. Diaz and F. T. Metcalf; On the set of sub sequential limit points of successive approx-
imations, Trans. Amer. Math. Soc. 135 (1969), 459-485.
6. K. Goe'bel, W. A. Kirk and T. N. Shimi; A fixed point theorem in uniformly convex spaces,
Bol. U.M.I. 7 (1973), 67-75.
7. T. L. Hicks and J. D. Kubicek; On the Mann iteration process in Hilbert spaces, J. Math.
Anal. Appl. 59 (1977), 498-504.
8. S. Ishikawa; Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974),
147-150.
9. R. Kannan; Fixed point theorems in reflexive Banach spaces, Proc. Amer. Math. Soc. 38
(1974), 111-118.
10. L. S. Liu; Ishikawa and Mann iterative process with errors for nonlinear strongly accretive
operator equations, J. Math. Anal. Appl. 194 (1995), 114-125.
11. W. R. Mann; Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
12. S. Maruster; The solution by iteration of nonlinear equations in Hilbert spaces, Proc. Amer.
Math. Soc. 63(1) (1977), 69-73.
11
13. S. A. Naimpally and K. L. Singh; Extensions of some fixed point theorems of Rhoades, J.
Math. Anal. Appl. 96 (1983), 437-446.
14. M. O. Osilike; Iterative method for nonlinear monotone-type operators in uniformly smooth
Banach spaces, J. Nig. Math. Soc. 12 (1993), 73-79.
15. W. V. Petryshyn and T. E. Williamson jr.; Strong and weak convergence of the sequence
of successive approximations for nonexpansive mappings, J. Math. Anal. Appl. 43 (1973),
459-750.
16. L. Qihou; The convergence theorems of the sequence of Ishikawa iterates for hemicontrac-
tive mappings, J. Math. Anal. Appl. 148 (1990), 55-62.
17. L. Qihou; Convergence theorems for the sequence of iterates for asymptotically demicon-
tractive and hemicontractive mappings, Nonlinear Anal. TMA 26(11) (1996), 1835-1842.
18. B. E. Rhoades; comments on two fixed point iteration methods, J. Math. Anal. Appl. 56(1976), 741-750.
19. H. F. Senter and W. G. Dotson; Approximating fixed points of nonexpansive mappings,
Proc. Amer. Math. Soc. 44 (1974), 375-380.
20. X. L. Weng; The iterative solution of nonlinear equations in certain Banach spaces, J. Nig.
Math. Soc. 11(1) (1992), 1-7.
21. C.S. Wong; Fixed points and characterization of certain maps, Pacific J. Math. 54 (1974),
305-312.
22. H. K. Xu; Inequalities in Banach spaces with applications, Nonlinear Anal. TMA 16(2)
(1991), 1127-1138.
23. Z. B. Xu and G. F. Roach; Characteristic inequalities for uniformly convex and uniformly
smooth Banach spaces, J. Math. Anal. Appl. 157 (1991), 189-210.
24. Y. Xu; Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive
operator equations, J. Math. Anal. Appl. 224 (1998), 91-101.
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