the international centre for theoretical physics K/98/212 STEEPEST DESCENT METHOD FOR A CLASS OF NONLINEAR EQUATIONS C.E. Chidume and Chika Moore

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1 umted nations educanonal, scientific and cultural orgamzation the abdus salam international centre for theoretical physics K/98/212 STEEPEST DESCENT METHOD FOR A CLASS OF NONLINEAR EQUATIONS C.E. Chidume and Chika Moore

2 Available at: pub-off IC/98/212 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS STEEPEST DESCENT METHOD FOR A CLASS OF NONLINEAR EQUATIONS The Abdus Salam International C.E. Chidume Centre for Theoretical Physics, Trieste, Italy and Chika Moore2 Department of Mathematics and Computer Science, Nnamdi Arikiwe University, P.M.B , Awka, Anambra State, Nigeria3 and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract Let E be a real uniformly smooth Banach space and let A : E ++ E be a $-strongly quasiaccretive operator such that the equation Aa: = f has a solution Z* E D(A). We introduce a generalized steepest descent method and prove that it converges strongly to z*. Our iteration parameters are independent of the geometry of E. We also obtain explicit convergence rates. Furthermore, we provide an affirmative answer to a question raised by Li-Shan Liu (see Math. Review 96E47107). Related results deal with the iterative approximation of fixed points of $-strong pseudocontractions. MIRAMARE - TRIESTE November chidumeoictp.trieste.it 2Regular Associate of the Abdus Salam ICTP. 3Permanent address.

3 1 Introduction Let E be a real normed linear space. A mapping U with domain D(U) and range R(U) in E is said to be accretive if V Z, y E D(U) and V s > 0 the following inequality holds YII i 11x - Y + s(ux - UY>Il Let E* denote the dual space of E and let J : E I--+ 2E* denote the normalized duality mapping on E defined by Jx := {f E E* : (z, f*) = 11412, Ilf*ll = IMI} where (.,.) denotes the generalized duality pairing between elements of E and E*. Using a result of Kato [19], it turns out that U is accretive if and only if Vx, y E D(U) 3j(x - y) E J(x - y) such that (Ux - UYJ(a: - Y)) 2 0 The mapping U is said to be strongly accretive if there exists a positive constant Q such that U - 01 is accretive where I is the identity map of E. The map U is said to be $-strongly accretive if there exists a continuous, strictly increasing function 7+!~ : [0, oo) I+ [O,oo) with g(o) = 0 such that V x, y E D(U) the following inequality holds (Ux - Uy,j(x - Y>> 1 em - YII>IIX - YII (1) Let N(U) := {x E E : Ux = 0). If IV(U) # 0, and (1) holds V y E N(U) and x E E then U is said to be $-strongly quasi-accretive. In the special case in which G(t) := kt for some k E (0,l) and (1) holds Vy E N(U) called quasi-accretive. and Vx E E, then U is called strongly quasi-accretive. If k = 0, U is Closely related to the accretive maps is the class of pseudocontractive operators where an operator T is said to be pseudocontractive,(strongly pseudocontractive, +-strongly pseudocontrac- tive, or hemicontractive) if the operator U = I - T is accretive (strongly accretive, $-strongly accretive, or quasi-accretive) respectively. Thus, the mapping theory of accretive operators is closely related to the fixed point theory of pseudocontractions. These classes of operators have been studied extensively by various authors (see e.g., [l-15, ). A mapping T is said to be demicontinuous if it is continuous from the strong to the weak topologies on E; that is, if {xn} c D(T) and x, -+ x* as n ---t 03 then TX, - TX* as n -+ cw where - denotes weak convergence. The accretive operators were introduced by Browder [l] and Kato [18] and are of interest mainly because of the fact that many physically significant problems can be modelled in terms of an initial-value problem of the following form; g+ux=o; x(0) = x0 where U is of the accretive-type in an appropriate Banach space. Such evolution equations arise typically in models involving heat, wave or Schrodinger equations. Observe that elements of N(U), the kernel of U, are precisely the equilibrium points of the system (2). Consequently, considerable efforts have been addressed to developing techniques for constructing the zeros of accretive-type operators (or equivalently, the fixed points of 2

4 pseudocontractive-type space, Vainberg [31] and Zarantonello defined by and proved that if: operators) (see e.g., [3-14, 16, ). In this connection, but in Hilbert [37] independently introduced the steepest descent method %+1 = Ga - cnt%, x0 E H, n 2 0 (3) (i) T = I + U, where U is accretive and Lipschitzean and I is the identity map on H; (ii) k=xe (0,l);nLO then the sequence {zn} defined by (3) converges strongly to an element of N(T). This result has been extended in several directions by various authors (see e.g., [3-14, 19, 22-29, 34-35, ). Very recently, Morales and Chidume [22] obtained the following result which generalizes and extends several important known results in this connection (see, e.g., the introduction and remarks in [22]): Theorem MC ([22])Let E be a uniformly smooth Bunach space and let A : E H E be a bounded demicontinuous mapping which is also ti-strongly uccretive on E. Let f E E and let x0 E E be un arbitrary initial value. Then there exists a real constant rg > 0 such that the approximating scheme x,+1 = x7-i -cn(ax,--f);nzo (4 converges strongly to the unique solution of the equation Ax = f, provided that the real sequence {G} satisfies the following conditions: (i) 0 < c, 5 TO, n 2 0; (ii) 2 c, = cm; (iii) 2 hb(cn) < 03 n=o n=o In Theorem MC, the b in condition (iii) is a function which depends on the geometry of the underlying Banach space E (see e.g. [22]). C onsequently, condition (iii) is, in general, not convenient in applications. Nevanlinna and Reich (Israel J. Math. 32 (1979), 4458), however, have shown that for any given continuous nondecreasing function b with b(0) = 0, sequences {A,} always exist such that (i) 0 < A, < 1,n 2 0; (ii) 1 A, = 00; (iii) CX,b(X,) < 03. If E = L,(l < p < oo), we can choose any sequence {A,} c [,/!I, with s = p if 1 < p I 2 and s=2 ifpl2. Observe that this condition implies lim c, = 0. It is our purpose in this paper to first extend Theorem MC to a generalized steepest descent method: an Ishikawa-type iteration process (see e.g., [11-131, [19], [38]) and then to weaken condition (iii) to lime, = 0, which is independent of the geometry of E. Furthermore, we obtain explicit convergence rates. Our method, which is of independent interest, also provides an affirmative answer to a question raised by Li-Shan Liu (see Math. Review 96E47107) in a setting even much more general than it is posed. 2 Preliminaries Let E be a real Banach space. The modulus of smoothness of E is defined by PE(7 -> := {f (II x + YII + 11x - Yll> - 1 : llxll = 1; IIYII = 7) ; T>O 3

5 The Banach space E is said to be smooth if PE(T) > 0 and is said to be uniformly smooth if PE (7)!!?I - =o. r It is known that if E is smooth then the duality map J is single-valued and if E is uniformly smooth then J is uniformly continuous on bounded sets. For our next result, we need the notion of a subdifferential. The subdifferential of a functional f on a Banach space E with dual space E* is a map defined by c3f (x) = {x* E E* : f(y) 1 f(x) + (Y - x,x*), VY E El Lemma 1 Let E be un arbitrary real Bunuch space. Then for arbitrary x, y E E there exists j(x - y) E J(x - y) such that lb + Yl ~ (YdX + Y)> (5) Proof. It is easy to see that j is the subdifferential 64,~~ of the convex functional cp : E I-+ %z+u{o} defined by Then, for any u, w E E we have that from which we obtain P(W) - P(U) 1 (w - u,.w) 11~112 L llwl12 -I- 2b - w, j(u)> Set u := x + y and w := x to obtain the inequality (5) and the proof is complete. 0 We shall also need the following results. Lemma X-R ([35]) Let E be a uniformly smooth Bunuch space. Then there exist positive constants D and C such that V x, y E E Yl (YA4) + D mtxx { lkdl + Il?h ;} PE(ll?d> (6) Lemma W ([32]) Let {@.n}n,o be a sequence of positive reals, and let {6~}n20 C [0, 11, {~n},+.o be real sequences such that C- n,o - 6, = 03 and a, = o(&). Suppose that the following inequality holds Then Qn --f 0 us n Q, n+l 2 (1 - &)@n + an; n 2 0 (7) 3 Main Results We prove the following theorems. 4

6 3.1 Iterative solution of the equation Ax = f Theorem 1 Let E be a un$ormly smooth Banach space and let A : E H E be a bounded $- strongly quasi-accretive operator such that the equation Ax = f has a unique solution x* E D(A) for any jixed f E E. Then there exists a real number 70 > 0 such that if the real sequences {(Y,},Lo, {Pn},,>o satisfy the following conditions: then the sequence {xn}~20 iteratively generated from an arbitrary x0 E D(A) by Xn+l = Xn -an(ayn-f); nlo Yn = Xn -,&(&t-f); n ) (9) converges strongly to x*. Proof. Without loss of generality, we may set f = 0. Then $~(llxo - x*~~)~~xo - x*ii 5 (Axe, j(xo - x*1> I IIAxoll-lh - x*11 so that $(11x0 -x*11> 5 IIAxoll. W e now make the following definitions: a0 := sup {Y : NY) 5 IIAxoll~ m0 := sup { IlAull : IIu - x011 I 5ao) do := Dmax 5as, g -t 1 b. := a0 z where D and C are the constants appearing in (6). Ob serve that if IIAxoll = 0 then we are done since the Theorem is trivially true in this case. So, we assume that llaxoii > 0. Since E is uniformly smooth, r- p~(~) + 0 as r Hence, we can choose 70 > 0 such that ~ 0 < 7 I 70, ao@(bo) ao~(ao) 4do do (10) Since E is uniformly smooth, given any E = %$$$ > 0 there exists 6, > 0 such that YII I 6* * IW - j(y>ii F E Choose 0 < 6 < 6,. Define (11) With this 70 generate the sequence {xcn}n20 as given by (8) and (9). Observe that 11x0 - x*11 i a0; IIAxoII 5 mo Also 11~0 -x*11 < 11~0 - x x0 - x*11 = PoWOII + 11x0 - x*ll I 2ao so that, IlAyoll 5 m. 5

7 Claim 1 The sequence (xcn} is bounded. Suppose the contrary, for contradiction, and let no > 0 be the first natural number such that Then, I\s,,-~ - x*11 < a0 so that ]lx,,-1 using (11) we have the following estimates. llxna - x*1] > a0 (12) - x0\) 5 2~~ and so I\Azn,-lIl 5 mo. firthermw SO that [[AynD-~I1 5 mo. Also, IIYno-1 - x*ii 2 &--1IIA~no-lll + Ilxno--1 - x*ti 5 2aa. ttym-1 - ZOll I &-ltthzo-l/t + ttx?lo-1 - zoll 5 3a0 Thus, 11xno - ~011 < 3ao and so ~~Azn,,~~ 5 mg. Furthermore, using (6), (9) to (13), we have that 5 IIxno - x*tt2-2pno$~(jl~no - X*tt)tI%~ - x*it + d&w 5 Iho - $*\I2 - Pn, I II%, - x*u2 2a0ffG0) C - do i m(mopno) A0 II PE(mOPno) p Thus, ))yn, - x*11 2 ljx,, - x* 11 < 2ao. Hence, l}yno - x0\] 5 3~0 and so \~AY,~ that so that Moreover, tlvno - xn,ii = PnoIIAxcnoli I +mm 5 fi IMJno - It *) - j&lo - x*)ll 2 ao?wo) = 8mo IlYno - x*ii 1 IIxno -x*il-&ollaxno(i Lao-bo=bo Using (6), (lo), (11) and (14) we now have llxno+l - x*l12 I bn, - x*l12 - %&4~,,,j(~,, - xc*)) X*/I + ~~ttaynoi[, z} CU,, [pe a~~~ ] L ho - Ic* II2-2Qno@(IlYno - Z* II) l/y720 - CC* \I +2~~,llAynoIl.llj(Y~, - z*) - j(xno - s*)tt +dgan, [ PE(z y)] 11 I mo. Observe 5 llxn, - x*l12 - %o aow0) - { ao$@o) aold(@ I IIGl.o - x* 112 6

8 so that ll~~+l- z*ll i lizno IlA~o+lll L mo. Also, - z*ll. Thus, llz~,~+r - z*ll 5 2ae and I/z,~+~ - zoll 5 3ao so that bm+l - x*11 2 ll%o - x* II - ano II&no II 1 bo (15) Thus, using (6), (8), (lo), (11) and (15) we have PE(~OPno+l) IlYnofl - x*/l2 I ll~no+l - x*l12 - &Lo+1 aow0) - do { [ P no+1 I} ~0 that IIYno+l - z*ii 5 2~0; IIYno+l - ~011 < 3~ and liayno+rii 5 me. Suppose I/Go+1 - x*11 > ao. Using the same arguments as above we have that ll%0+2 - x*11 I IlGo+l - %*I] 5 I/ho - z*ll 2 2aO If, on the other hand, we assume that IIzno+r - CC* II 5 ue, then there are two possibilities: either 11~~ - z*ii I us v n 1 no + 1, in which case we have that the sequence is bounded, or there exists an integer k 1 no + 1 such that ~~~~ - z*/l > a0 and llzk-l - z*ij 5 a~, in which case we are back to (12) and the argument given above applies provided that IIAy~-lI/ 5 mo and IIAz~-~II _< mo. However, and so IIAs~-~II < mo. Moreover, IIQC-1 - ~011 I IlWl - x* x* - zoll 5 2ao IlYk-I - zoll 5 bc-~ - a-l/l + IIn+ - ~011 5 &liiasrc-lil + 2uo 5 3uo so that (IAy~+lll 5 mu, as required. Therefore, llzn - ~*~~ < 2uo Vn > 0 contradicting the assumption that the sequence is unbounded. Thus, (5,) is bounded (and {yn} is also bounded). This establishes claim 1. Now, observe that 11% - Z* 11 - mopn I IlYn - Z* 115 II& - 5* 11 + mopn Vn 10 (16) so that, in particular, lim~f llzn - z*ll = lirmrigf llyn - 5*11. Define the function 4 : [0, oo) H [0, 1) by ( ) = 1 + r + g(r) where + is the function appearing in (1). It is easy to see that #(r) = 0 _ (Az,j(z- z*>> 2 $(Ib - ~*ll)b - x*11?u~ - x*11> > 1+llz-z*ll+~(l/s-z*l/)II - *~~2 = 4(ll~ - ~*ll)ll~ - x*l12 r = 0 and that Claim 2 lim inf,,, llyn - LC*/ = 0 (= lim inf,,, 11~~ - z*[[) 7

9 Suppose not. Then, lim inf 4(llyn - X*/I) = X > 0 for some positive number A. So, there exists an integer NO > 0 such that Furthermore, 4(llun - x*11) 2 S, \J n 2 NO Hence, Vn 1 NO > 0, we have IlYn - x*l12 L IlZn - x*l12 - ~PnllA~nII~ll~n - x*1/ + P;llA~ll~ 2 11~ - z*l12-2pnljaxnii.jlxn - z*ll II%+1-2* II2 I IIG - z* II2-2%4( llyn - XT* II) II?./n - z* II2 + +2&n lla~nil~llj(yn - z*) - j(xn - z*> II + +Dmax { 11 x, - x* IJ + cr,iiay,lj, ;} cr, [ pe ;~yn ] I 11% - x*11* - anj+ [llxn - x*l12-2pnlla~n~~~~~~n - x*11] + +2~nPnIIAynI)-Ilj(yn - x*) -j(~n - x*)11 + doan [pe(~mo)l I [l - A%] 11% - x*l12 + ~l%llj(yn - xc > - j(xn - x*)11 +M2an [ pe ~~an)] (17) where and Define MI = ~sup{aiiax~i~.~~x~ -x*/j + IIAYnll : n L 0) MZ = an= E; (11~ - x*1( + an/laynll), i - Then (17) now I= IIXn - X*l12; 6, := xcr, (7n := a, Mlljj(yn - X*) - j(xn - x*)11 + MZ Qi n+l 5 (1 - b)@n + Qn which is exactly the same as (7). n t0 asn--+m sincec&=m anda,=o(&). Thus, llz, - x* II + 0 and so llyn - x*11 + o as n By continuity, 4( llyn - X* II) -+ 4(O) = 0 as n + oo; a contradiction. Hence, the claim holds. Claim 3 lim,+, 11% - X*11 = 0 Using (5), we have that IIXn+l -x*i12 i Ilxn -x*l12-2an(ayn,j(xn+l -z*)> = IlXn - x*i12 -hx(ayn,j(yn -Xc*>> +2Qn(Ayn, j(yn - xc*) - j(sn+l - x*)) 5 IlXn - X*)j2-2%z+(IIYn - X*II).llYn - X*/l +2~nII&nII*Ilj(~n - z*) - j(xn+l - x*)11 I 11% -x*[12 + Ml~nIlj(yn - X*) -~(GI+I -x*)11 8

10 where Now set Ml = 2w3IA~d - Pn = II% - X* 11 Y un = Ml Ilj(yn - xc*) - j( xn+l - x*)11 and 3;2 = llyn - x*11 to obtain the inequality Observe that so that Pi+1 5 Pi - bz$(%)~n + Qn~n J$ IIY~ -xn+lii 5 n!$m(anliaynii +&II&t/l) = 0 )I& Ilj(Yn - X*> - j(%+l - X*)11 = 0 and hence un -+ 0 as n -+ co. By claim 2, there is a subsequence {pnj} of {pn} such that Pnj + 0 as J -+ co. Thus, given any E > 0 there exist a ju > 0 sufficiently large such that an integer Nr > 0 sufficiently large such that and also an integer N2 > 0 sufficiently large such that mo(% + A) < ii Vn>Np Observe that ^fn 2 P71+1- mo(an + Pn) nj. 2 m={njo, Nl, N2) Pnj, < E Let Clearly, We claim that hj,i-k < &; t/k>0 Suppose that for some k = m, pnj,+ n < E but that pnje+m+l > E. Then, so that we now have Ynj,+m 1 Pnj,+m+l - mo(%j,+m + Pnj,+m) > E - i = 4 E2 < PiI,+m+l 5 PZj* +nz - 2ffnj, +m ti((m,, +m)%j, +m + Wzj, fmcnj, +m which is absurd. Thus, = E2 - cynj*+,lc) Pnj, +k < E * Pnj, +k+l < E Since Pnj. < E (that is, for k = 0), it follows by induction on k that pnj,+k 2 E, Vk 1 0. Since & is arbitrary, it follows that pnj,+k + 0 as k -+ w that is, pn --t 0 as n --f 03. This completes the proof. 0 9

11 Corollary 1 In Theorem 1, let the sequence be generated by x,+1 =xc,-%(&n-f); n 2 0 (18) where the real sequence {c,i}n,o c - (0, yoyo] satisfies the following conditions: Then, the conclusion of Theorem 1 still holds. (9 c Gl = 00 and (ii) $-nwh = o n>o Corollary 2 Let E be a uniformly smooth Banach space and let A : E H E be a bounded demicontinuous $-strongly quasi-accretive mapping. Let {cy~} and {p,} satisfy the conditions in Theorem 1. Then the sequence {x~}~~o generated by (8) and (9) converges strongly to the unique solution x* of the equation Ax = f. Proof. By a result of Kartsatos [17], A(E) is an open set, but since A(E) is known to be closed, then A is surjective and hence the equation Ax = f is solvable for any fixed f E E. The rest of the argument now follows as in Theorem Iterative solution of the equation x + Ax = f Observe that the identity mapping on E satisfies all the conditions on $. Thus, if A is quasiaccretive then the operator G = I+A is $-strongly quasi-accretive with -z)(t) = t. Consequently, the iterative solution of the equation x + Ax = f readily follows from Theorem 1 above. Recall that an operator U is called dissipative if (-U) is accretive. The other notions of dissipativity are similarly defined. Suppose that the operator A is quasi-dissipative. Then, it easily follows that the operator T = I - XA is again $-strongly quasi-accretive with G(t) = t. This implies that the iterative solution of the equation x - XAx = f readily follows ifrom Theorem 1. Hence, we have the following results. Theorem 2 Let E be a uniformly smooth Banach space and let A : E I-+ E be a bounded quasi-accretive operator such that the equation x + Ax = f has a solution x* E D(A) where f E E is an arbitrary but fixed vector. Let {cy,}, {,&} be as given in Theorem 1. Then the sequence {xn}n>o iteratively generated from an arbitrary x0 E D(A) by converges strongly to x*. Xn+l = Xn -an(yn+ayn-f); v ln.20 (19) Yn = Xn - Pn(xn + Axn -f); VnLO (20) Proof. Observe that the operator (I + A) is $-strongly quasi-accretive. Thus, Theorem 1 applies. Cl Corollary 3 Let E be a uniformly smooth Banach space and let A be a bounded dissipative operator such that the equation x - XAx = f has a solution x* E D(A) where f E E is an arbitrary but fixed vector. Let {cr,}, {p,} b e as in Theorem 1. Then the sequence {xn} iteratively generated by converges strongly to x*. Xn+l = Xn - %(Yn - XAy, - f); Vn 2 0 (21) Yn = Xn - pn(xn - XAX, - f); Vn 2 0 (22) 10

12 3.3 Fixed Points Hemicontractions Theorem 3 Let E be a uniformly smooth Banach space and let T : E H E be a +-strong hemicontraction such that (I - T) is bounded and T has a fixed point x* E D(T). Then there exists a positive real number ^/o such that the sequence {x~},,~ - iteratively generated from un arbitrary x0 E D(T) by Xn+l := Xn - Qn(Yn - Tyn); n L 0 (23) yn I= Xn - Pn(xn - TX,); n > 0 (24 where the real sequences {cy~}~>~; {&}n,o - strongly to x*. satisfy conditions (i) - (iii) of Theorem 1, converges Proof. If we set A = (I - T), then A is a bounded $-strongly quasi-accretive mapping. The hypotheses of Theorem 1 are then satisfied since x* is a zero of A if and only if it is a fixed point of T. The proof is complete. 0 Corollary 4 In Theorem 3 let the sequence {xn} be iteratively generated by %+l = X72 -k(xn-txn); n>o (25) where the real sequence {G} satisfies the conditions in Corollary 3. Then the conclusion of Theorem 3 still holds. 3.4 Fixed Points of Quasi-nonexpansive maps In the special case where A = I - T and T is quasi-nonexpansive (i.e., the fixed point set F(T) := {x E E : TX = x} # 0 and [[Ta: - Tx*II 5 llx - x*11 for all x E D(T) and x* E F(T) ), our next corollaries prove that, in a certain sense, the condition 1)x - Txll > f (d(x, F(T))), where f : [0, co) I+ [0, co) is strictly increasing and f (0) = 0 and d(x, F(T)) = inf{ /lx - x*/l : x* E F(T)}, is sufficient for the strong convergence of the iteration processes (8), (9) and (18) to a fixed point of T. Corollary 5 Let E be a uniformly smooth Banach space, let K c E be a nonempty closed and convex subset of E and let T : K H K be a quasi- nonexpansive mapping. Then, for any initial guess x0 E K, the iteration process (8) and (9) converges strongly to a fixed point x* of T provided there exists a strictly increasing function f : [0, m) H [0, co) with f (0) = 0 such that Ilxn - TxnII 1 f (d(xn, F(T))), Vn L 0 Corollary 6 In Corollary 6, let the iteration process be generated by (18), then the conclusions of Corollary 6 holds. 3.5 Conclusion Remarks 1 1. The methods of our proof easily carry over to the case of set-valued maps since we can make suitable single-valued selections under the hypotheses of our theorems so that the rest of the arguments then become identical with those of this paper. 2. If we make the choice cr, = (n++yg )- = pn in Theorem 1 and the choice c, = (n+$ )- in Corollary 3, we obtain that the rate of convergence is of the form llxn - x*11 = O(n-i) in either case. 11

13 3. In [19] (see also, Math. Review 96E47107), Liu proved that if the sequence {xn} is bounded and A is strongly accretive and Lipschitzian, then iteration processes of the Mann and Ishikawa types converge strongly to a solution of the equation Az = f in uniformly smooth Banach spaces. The question of whether or not the assumption that the sequence {xn} be bounded can be weakened was asked by Liu (see e.g., Math. Review 96E47107). Claim 1 of our Theorem 1 proves that even in the more general setting of our theorems, the sequence {xcn} is always bounded. 4. Several results on the strong convergence of the steepest descent method to a solution of the equation Az = $, where A is Q-strongly quasi-accretive, in uniformly smooth Banach spaces have been established (see e.g., [9, 22, 24, 351). The results in [35] contain a gap. More precisely, while the authors in [35] are trying to prove the boundedness of the sequence {z~}, it can be noticed that formula (2.11) (p.347 of [35]) holds for n = nj, but not necessarily for its successor. This gap is inherited in [9]. The main result in [22] fills this gap but uses an iteration parameter that is dependent on the geometry of E. The results in [24] are very special cases of our theorems since they are established in q-uniformly smooth Banach spaces and for Lipschitz maps. Moreover, the iteration methods considered in [9], [22] and [35] are special cases of our iteration method in which,& G 0 Vn > 0. Our theorems, therefore, generalize and unify most of the important known results in this connection. Acknowledgments The authors are most grateful to the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The second author is most grateful to the Committee on Development and Exchange of the International Mathematical Union, Cedex Prance for a travel support that enabled him take up a visiting fellowship during which this research was carried out. This work was done within the framework of the Associateship Scheme of the Abdus Salam ICTP. Financial support from the Swedish International Development Agency is acknowledged. References 1. F. E. Browder; Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. Amer. Math. Sot. 73 (1967), F. E. Browder; Nonlinear operators and nonlinear equations of evolution in Banach spaces, in Proc. Symp. Pure Math. 18(2), Amer. Math. Sot., Providence, RI, F. E. Browder and W. V. Petryshyn; Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20, (1967), R. E. Bruck; The iterative solution of the equation y E x+tx for a monotone operator T in Hibert space, Bull. Amer. Math. Sot. 79 (1973), C. E. Chidume; The iterative solution of the equation f E x + TX for a monotone operator T in LP spaces, J. Math. Anal. Appl. 116 (1986), C. E. Chidume; Iterative approximation of fixed points of Lipschitzean strictly pseudocontractive mappings, Proc. Amer. Math. Sot. 99 (1987), C. E. Chidume; Approximation of fixed points of strongly pseudo-contractive mappings; Proc. Amer. Math. Sot. lzo(2) (1994), C. E. Chidume; Iterative solution of nonlinear equations with strongly accretive operators, J. Math. Anal. Appl. 192 (1995), C. E. Chidume; Steepest descent approximations for accretive operator equations, Nonlinear Analysis, TMA 26(2) (1996),

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