the K/98/213 abdus salam international centre for theoretical physics

PICARD ITERATIONS FOR SOLUTION OF NONLINEAR EQUATIONS

IN CERTAIN BANACH SPACES

Chika Moore

Available at: http://uw.ictp.trieste.it/-pub-off IC/98/213

United Nations Educational Scientific and Cultural Organization and

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

PICARD ITERATIONS FOR SOLUTION OF NONLINEAR EQUATIONS

IN CERTAIN BANACH SPACES

Chika Moore’ Department of Mathematics and Computer Science, Nnamdi Azikiwe University,

P.M.B. 5025, Awka, Anambm State, Nigeria2 and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

Let E be a real puniformly smooth Banach space and let A : D(A) c E H E be locally Lipschitz and strongly quasi-accretive. It is proved that a Picard recursion process converges strongly to the unique solution of the equation As = f, f E R(A) with the convergence being at least as fast as a geometric progression with ratio w E (0,l). Related results deal with the convergence of Picard iterations to the fixed point of locally Lipschitz strong hemicontractions and to the solutions of nonlinear equations of the forms z + TX = f and x - XTx = f where T is an accretivetype operator.

M IRAMARE - TRIESTE

November 1998

‘Regular Associate of the Abdus Salam ICTP. This research was partly supported by Research Grant RG/MATHS/AF/AC No. 97-210 from the Third World Academy of Sciences (TWAS).

‘Permanent address.

1 Introduction

Several iteration processes have been established for the constructive approximation of solutions -to several classes of (nonlinear) operator equations and several convergence results established using these iterative processes (see e.g., [l-19] and the references cited therein). The most classical of these processes seems to be the Picard process. It is a trite fact that if the Picard process converges then other processes are virtually uninteresting for that same problem, except possibly for the consideration of errors. For not only is the Picard process simpler, but also whenever it converges, it normally does so at least as fast as a geometric progression with a ratio less than 1. In fact, other iterative procedures (for instance, the averaging processes of Krasnoselskii, Schaeffer, later Mann and much later Ishikawa) were developed specifically to tackle situations where the Picard iteration process fails or seems to have failed. Thus, the recent result of C. E. Chidume [8] that if E is an arbitrary real normed linear space and A : E H E is strongly quasi-accretive and Lipschitz, then a Picard-like iteration process converges strongly to the solution of the equation Ax = f is really interesting.

Let E be a real Banach space and E* its dual. For 1 < p < 03, the duality mapping JP : E I-+ 2E* is defined by

JPx := {f * E E* : (x, f’) = [lx/*, 11 f *II = I~xII*-~}

where (., .) denotes the generalized duality pairing between E and E*. Recall that a map A : E I+ E is said to be accretive if ‘dz, y E D(A) 3j,(z - y) E Jp(x - y) such that

(Ax - 4, jp(x - Y>> L 0 (1)

and is said to be strongly accretive if A - ICI is accretive where k E (0,l) is a constant and I denotes the identity operator on E. Let S(T) = {x* E D(A) : Ax’ = f} # 8 denote the solution set of the equation Ax = f. If (1) holds for all x E D(A) and y = x* E S(T), then A is said to be quasi-accretive. The notion of strongly quasi-accretive is similarly defined. A is said to be m-accretive if the operator (I + A) is surjective. Related to this class of operators is the class of dissipative operators where an operator A is said to be dissipative if and only the operator (-A) is accretive. The notions of quasi-dissipative and m-dissipative are similarly defined. Closely, related to the class of accretive operators is the class of pseudocontractive maps. A map T : E H E is said to be pseudocontractive if Vx, y E D(T), 3jp(x-y) E JP(x- y) such that

((I- T>x - (I - Tb,jp(x - Y>> L 0 (2)

Observe that T is pseudocontractive if and only if A = (I - T) is accretive. Thus, the mapping theory of accretive operators is intimately tied to the fixed point theory of pseudocontractions. A map T is called hemicontractive if and only if A = (I - T) is quasi-accretive.

2 Preliminaries

Let E be a real normed linear space of dimension dim.E 2 2. The modulus of smoothness of E is defined by

p&) := sup 1 IIX + Y II + IIX - Y II 2

- 1 : 11x11 = 1, llyll = T} ; 7 > 0

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If p&r) > 0 VT > 0 then E is said to be smooth. If these exist a constant c > 0 and a real number 1 < p < 00 such that

P&) 5 c7-p

then E is said to be p-uniformly smooth. Typical examples of such spaces are the Lebesgue Lp, the sequence ep and the Sobolev WT (1 < p < co) spaces. In fact, for 1 < p 5 2 these spaces are p-uniformly smooth and for 2 5 p < oo they are 2-uniformly smooth.

If E is a real p-uniformly smooth Banach space, then (see e.g., [7]) the following geometric inequality holds:

lb + YIP I I141P + p(y&(x)) + CPllYllP

for all Z, y E E and some real positive constant C, > 1.

3 Main Theorems

3.1 Iterative solution of the equation Ax = f

Theorem 1 Let E be a real p-uniformly smooth Banach space and let A : D(A) c E I-+ E be a locally Lipschitz and strongly quasi-accretive operator with open domain D(A) in E such that the equation Ax = f has a solution x* E D(A) for f E R(A) arbitrary but fixed. Define Ax : D(A) H E by

Axx := x - X(Ax - f); kfx E D(A)

Then, there exist a neighbourhood B of x* and a real number X E> (0,l) such that starting with an arbitrary x0 E B the Picard sequence {xcn} generated by

xn+l = &x,x; 7220 (3)

remains in B and converges strongly to x* with convergence being at least as fast as a geometric progression.

Proof. Since A is locally Lipschitz, there is an r > 0 such that A is Lipschitz on B = Br(x*) = {x E E : 112 - x*1/ _< r} c D(A). Let k E (0,l) and L > 1 denote the strong accretivity and Lipschitz constants of A respectively. Observe that f = Ax*. Pick an arbitrary so E B, choose

( ) 1

A= -L (p-1) LPC,

and generate the sequence {xn}+s as in (3). We now prove that x,, E B, V n > 0. Suppose that xn E B. Then,

jlx,z+~ - x*llP = llxn - x* - X(Ax, - Ax*)j/P

L llxn- x*IIp - pX(Ax, - Ax*&(xn - x*)) + XpCpIIAxn - Ax*IIP

I (1 - pkX + Lr’CpXP) /lxn - x*Ijp = [l - (pk - LpC,xP-‘) X] 1(x, - x*IIP

= [I-(p-1)x(&)&] llxn-x*IIp (4)

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Hence, since 20 E B by choice of the initial guess, it follows by the inductive hypothesis that the sequence {zn} remains in B. Set

and observe that w E (0,l) since

k< LCj’P

(p - 1)V ; Vl<p<oo

Hence, iterating further from (4), we obtain

II% - x*lIp I (wPy 11x0 - x*llp

or, equivalently,

Since w” --t 0 as n + 00 the assertions of the theorem follow and the proof is complete. 0

3.2 Iteration of fixed points of strong hemicontractions

Theorem 2 Let E be a p-uniformly smooth Banach space and let T : D(T) c E I+ E be a locally Lipschitz strong hemicontraction with open domain D(T) and a jixed point x* E D(T). For some real number X define the mapping TA : D(T) I--+ E by

TAx := x - X(1 - T)x; Vx E D(T).

Then, there exist a neighbourhood B of x* and a real number X E (0,l) such that the Picard sequence {x~}~~o generated from an arbitrary x0 E B by

remains in B and converges strongly to x* with convergence being at least as fast as a geometric progression.

Proof. Proceeding as in the proof of Theorem 1, we choose B = I&.(x*) c D(T) such that T is Lipschitz on B. Let k E (0,l) and L > 1 denote respectively the strong hemicontractive and Lipschitz constants of T. Let L, = (1 + L) denote the Lipschitz constant of (I - T). Choose

A= pk 2cp L?:

-I- p-1

and with an arbitrary xo E B generate the sequence as in (5). Observe that (I-T) is strongly quasi-accretive. Thus, the rest follow as in the proof of Theorem 1 and the proof is complete. 0

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3.3 Iterative solution of the equation x + Ax = f

Theorem 3 In Theorem 1, let A be a locally Lipschitz quasi-accretive operator with open domain D(A) such that the equation x + Ax = f has a solution x* E D(A). Let the sequence {xn} be generated by (1) where AA : D(A) C E I+ E is defined by

Axx := x - X(x + Ax - f); Vx E D(A)

Then, there exist a neighbourhood B of x* and a real number X E (0,l) such that starting with any initial guess x0 E B the sequence {xcn} remains in B and converges strongly to x1 with convergence being at least as fast a geometric progression with ratio w E (0,l).

Proof. Observe that the operator (I + A) is strongly quasi-accretive with constant k E (0,l) and also locally Lipschitz with constant L, = (1 + L). Theorem 1, therefore, applies and the proof is complete. 0

3.4 Iterative solution of the equation x - yAx = f

Theorem 4 Let E be a real p-uniformly smooth Banach space and let A : D(A) C E H E be a locally Lipschitz quasi-dissipative operator such that the equation x - yAx = f has a solution x* E D(A) for all y > 0 and for f E R(I - ?A) arbitrary but fixed. For some real number X define the operator AA : D(A) c) E by

Axx:=x-X(x-yAx- f); VXE D(A)

Then, there exist a neighbourhood B of x* and a real number X E (0,l) such that starting from an arbitrary initial guess x0 E B the sequence {xn} generated by (1) remains in B and converges strongly to x* with convergence being at least as fast as a geometric progression with ratio w E (0,l).

Proof. Since A is quasi-dissipative and locally Lipschitz with constant L > 1, we have that the operator I - yA is strongly quasi-accretive with constant k E (0,l) and locally Lipschitz with constant L, = (1 + yL). The rest now follow as in the proof of Theorem 1. 0

Remarks 1 1. It is known that the spaces L,, p, f? Wr for 1 < p < 00 and m 2 0 are p-uniformly smooth. Thus, our theorems hold for large classes of spaces.

2. It is a noteworthy consequence of Theorem 1 that for locally Lipschitz strongly quasi- accretive operators with open domains in p-uniformly smooth Banach spaces, other it- eration processes are virtually uninsteresting for the construction of the solution to the equation Ax = f. The iteration process used is simpler and the convergence rate at least as fast as a geometric progression. This rate of convergence seems to provide a least upper bound for iteration processes within this setting: the choice of X is optimal.

3. The present technique coould not readily extend the theorems to uniformly smooth spaces as it seemed impossible to choose a fixed X0 E (0,l) for which convergence would be guaranteed.

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4. The techniques of our theorems easily extend to the case where the operators are multi- valued once a single-valued selection cam be made; the rest is a mere repetition of the argument above. Moreover, it is known that locally Lipschitz maps need not be continuous.

5. If the domain D(A) of the operator A is closed but with a nonempty interior, that is, D(A)’ # 8, then our theorems still hold provided that x* E D(A)O.

6. Our theorems still hold if in, say, Theorem 1 the operator A is locally strongly quasi- accretive and locally Lipschitz. Hence, we have the following theorem.

Theorem 5 Let E be a real p-uniformly smooth Banach space and let A : D(A) c E ++ E be locally strongly quasi-accretive and locally Lipschitz. Suppose that the domain D(A) is open and that the equation Ax = f has a solution x* E D(A), V f E R(A) arbitrary but fixed. For some real number X define AA by Axx := x - X(Ax - f). Then, there exist a neighbourhood B of x* and a real number X such that starting with an arbitrary x0 E B the iterative sequence {x~} generated by (3) converges strongly to x* with convergence being at least as fast as a geometric progression with ratio w E (0,l).

Proof. Since A is locally strongly quasi-accretive, there is a real number rl > 0 such that A is strongly quasi-accretive in Brl(x*). Also, since it is locally Lipschitz, there is a real number r2 > 0 such that A is Lipschitz in BTZ(x*). Let r = min{r1, r-2) and set B = Br(x*). Then, A is both strongly quasi-accretive and Lipschitz in B. Let k E (0,l) and L > 1 denote the strong accretivity and Lipschitz constants of A in B respectively. The rest of the argument now follows as in the proof of Theorem 1 and the proof is complete. 0

Obviously, similar theorems can be stated for the other theorems in this paper.

4 Conclusion

In this concluding section, we establish the strong convergence of the usual Mann and Ishikawa iteration processes to the solution of the equation Ax = f where A is $-strongly quasi-accretive in real Banach spaces which are both uniformly smooth and uniformly convex.

Recall that an operator A is said to be $-strongly quasi-accretive if there exists a continuous strictly increasing function $J : [0, 02) H [0, oo) with q(O) = 0 such that for all x E D(A), X* E S(A) there exists j(x - x*) E J(x - x*) such that

(Ax - f,j(x - x*)> L $(11x - x*11)11x - x*1/

If we now define the function C#J : [0, oo) I-+ (0,l) by

Tw> @) := 1 + t + t)(t)

then, it easily follows that the operator A satisfies the following inequality:

(Ax - f,j(x - x*>> 2 4(//x - x*ii)b - x*ii2

In the sequel, the following results of shall be needed.

Lemma R (Reich, [IS]): Let E be a Banach space which is both uniformly convex and uniformly smooth. Let A : D(A) c E H E be m-accretive and let Jr = (I + rA)-‘. Then for x E E the

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strong limit Qx := lim,,o Jrx exists and Q : E H cl(D(A)) is a nonexpansive retraction of E onto cl(D(A)), the closure of the domain D(A) of A.

Lemma XR (Xu & Roach, [19]): Let E be a real uniformly smooth Banach space. Then, there exist positive constants D and C such that for all x,y E E the following inequality holds:

lb + pII2 5 11412 + 2(~7j(x)) + D max { Ilxll + lIdI T} Pdllvll) (6)

For the rest of this paper, Q denotes the nonexpansive retraction in Lemma R. We have the following theorems.

Theorem 6 Let K be a closed subset of a real Banach space E which is both uniformly smooth and uniformly convex. Let A : K H E be $-strongly quasi-accretive with (I - A)(K) bounded. Let {an), &J, k-d, {4J, {%I, (4 C IO, 11 b e real sequences satisfying the following conditions:

(4 (ii)

(iii)

a, + b, + c, = 1 = u; + b:, + c;; n>O lim b, = ,,leW b; = ikW ck = 0 n-+cc

1 b, = 00 and c, = o( b,) n>O

and let {un}, {vn} b e b ounded sequences in E. Let f E R(A) be arbitrary but fied and define T:KHE byTx:= f +x- Ax, V’s E K. Then, the sequence {x,} generated from an arbitrary xo E K by

%+I = Qzn; Zn = Unxn + bnTyn + Gxvn; n20 (7) yn=Qwn; wn = akx, + b;Tx, + ckun; n>O (8)

converges strongly to the solution x* E K of the equation Ax = f.

Proof. Let

d = max i ~~~{llTx - x*lli7 ~{{llvn - 5*lI}7 ~{{\I% - X*/l) - I and observe that

IIY~ - x*II = IlQwn - &x*11

I llwn -x*(1 5 &llxn --*II + b~I[Txc, - x*II + c~IIu~ - X*/I

I d(a:, + b:, + c;) = d

llxn+l -x*ll = IlQzn - Qx*II

5 11-h - X*11 i anllXn - X* II + bnIITyn --*II + CnIIvn - x*(( I d(an + bn + in) = d

SO that the sequences {x,}, {yn} are bounded. Furthermore, let

Ml = 4d2; M3 = 2M; M4 = max{Ml, M2, M3)

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Then, we have the following estimates:

+M(b:, + c;> pE[@:, +&)I b:, + c',

so that

IhI+ - x*l12 < a;llzn - 2*112 + 2 [l - 9uvn - bill>] d4lYn - x*l12 fMo%-L~nll~(~n - x*) - j(yn - x*)11 + Mlc, +M(bn + c-n> pE[@n + %>I

bn + c, >

+M@n + G-J PEkGz + 41 bn + c, > + Mlc, + Mzb,c:,

+Moanbn llj(xn - x*> - j(yn - x*>ll

5 [l - c$(llyn - ~*ll)~n1211~n - Z*ll2 + On (9) where

072 PEW:, + 41

= M4 C, + bnc; + bn(bL + CL) t b:, + dn >

fhz llj(Gx - X*) - j(yn - X*)/l + (bn + Cn)

Observe that

so that since E is uniformly smooth and so j is uniformly continuous on bounded sets, then we have that

llj(xn - x*) - j(yn - x*)ll -+ 0 as n -+ m

Moreover, we have that

IIY~ - x*ll = IlQwn - &a* (1 <_ IIwn - X* (1 < Ilxn - X* [I + d(bk + CL)

and

IlYn - x*(1 = IlQwn - X* 11 = IJxn - X* - (Qxn - Qwn>ll

1 11% - X*11 - 11% - %I1

L 11X7x - x*11 - d(b:, + CL)

so that

Hence, II’, - 5*ll - d(bk + cb> I llY7I - x*1) 5 JIzn - 5*ll + d(bk + Ck)

liy$ I( Yn - x* 11 = lirrizf JIx, - x* II

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Claim 1 lim inf,,, JJx, - x* II = 0:

Suppose not. Then, liminf jlvn - x*11 = 6 > 0. Thus, there is an integer N sufficiently large such that V n >_ N, $(llyn - X*/I) 2 %$. Then, V n 2 N we have that

where <p, = l(xn - x*jj2. Ob serve that a, = o(b,) thus by the Lemma, @p, -+ 0 as n -+ 0~) contradicting the assumption that liminf 11~~ - z*II # 0. Thus, the claim follows. Now,

II%+1 - x*/I2 = ((G - X* - bn(xn - yn) - bn(Ayn - AZ*) - %(x, - ~n)((~

i II% - x*112 - 2bn(Xn - Yn,j(Xn - X*)) + M(bn + in> PE[M2(bn + %)I

bn + C, > -2bn(Ayn - AZ*, j(xn - x*)) - 2C,(Xn - Vn,j(Xn - X*))

L IIxn - x*I12 + Mlbn(bL + ($1 - 2bn$(IIyn - x*II)IIY~ - x*II +M(bn + in>

PE[MP(bn -I- %>I

bn + Gz > + Mlc,

+M2bIJj(xn - xc’> - j(yn - x*)/l

Also,

IIYn -x*11 1 lIxn+l - x*11 - llyn - xn+lll = j/xn+l - x*II - IIQWn - QznII

1 llxn+l - x*ll - 11% - Xnll - II&z - &Jj

2 IIxn+l - x*/J - Mo(bn + bk + C, + CL)

Since liminf (lx, - x*11 = 0 there exists a subsequence (11~~~ - x*/l} such that ((xn3 - x*/j -+ 0 as j -+ co. Thus, given any E > 0 there is an integer ju > 0 such that

(I&j - X*II 5 5, ‘Jj L jo

Let

and observe that p -+ 0 as n + 00. Hence, there are integers Nr > 0 such that n

and N2 > 0 such that Mo(b,+b~+c,+c~) 5 % Vn> Nz

Choose j, sufficiently large such that nj, > max{nj,, Nr, Nz}. Let yn = Ijyn - x*/l and an = IIxn - x*11?

Claim 2

lIxnje +k -z*$, Vk>O

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Suppose that the claim is true for I; = m but not true for I; = m + 1. Then,

IIxn,.+m+l - x*ll > i and $ (ynj,+m) > $ (:)

Then,

E2 ,<@ n3. +m+l 2 @nj. +m - 2bnj* +mYnj. +m $ (Tnj. +m) + bn3* +monj, +m

+bnj.+mi+ i 0

EL <-z which is absurd. Since the claim holds trivially for k = 0 , it holds holds for all k > 0 by the inductive hypothesis. Since, E > 0 is arbitrary, the assertion of the theorem follows immediately. This completes the proof. 0

Theorem 7 Let K, E, A, T, f, x* be as in Theorem 6 and let the real sequences {an}, {bn}, {GI} C [0, l] satisfy the following conditions:

(9 Un+bn+Cn=l; Vn>O (ii) lim b,=O=J$mc, 71-00

(iii) c c, = cc n>O

and let {un} be a bounded sequence in E. Then, the sequence {xn} generated from an arbitrary x0 E K by

xn+l = anxn + bnTxn + Gun, n>O (10) converges strongly to x*.

Proof. Set bk E 0 and c’, E 0 in the proof of Theorem 6 and the assertion follows immediately. II

Remarks 2 1. By setting c, G 0 G CL in Theorems 6 and 7, we obtain that the usual Mann and Ishikawa iteration processes converge strongly under the hypothesis of our theorems.

2. The requirement that E be uniformly convex as imposed in Theorems 6 and 7 is merely to assure the existence of the nonexpansive retraction Q; otherwise, it is sufficient that E be uniformly smooth.

3. The conclusion of Theorems 6 and 7 easily extend to the approximation of fixed points of $-strong hemicontractions and to the iterative solution of operator equations of the forms x + Ax = f (for a quasi-accretive A) and x - XAx (for a dissipative A). Furthermore, these theorems easily extend to set-valued maps. 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.

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