the m abdus salam international centre for theoretical physics

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1 the m abdus salam international centre for theoretical physics K/98/211 ITERATIVE SOLUTION OF EQUATIONS INVOLVING K-p.d. OPERATORS C.E. Chidume Chika Moore

2 Available at: http : //WV. ictp. trieste. it/-pub- off IC/98/211 United Nations Educational Scientific Cultural Organization International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ITERATIVE SOLUTION OF EQUATIONS INVOLVING K-p.d. OPERATORS The Abdus Salam International C.E. Chidume Centre for Theoretical Physics, Trieste, Italy Chika Moore2 Department of Mathematics Computer Science, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambm State, Nigeria3 The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract Let E be an arbitrary real Banach space let A : D(A) 2 E H E be a K-positive definite operator such that the equation AZ = f has a solution for f E E arbitrary but fixed. It is proved that a steepest-descent-like iteration method with errors converges strongly to the solution. MIRAMARE - TRIESTE November chidumeoictp.trieste.it 2Regular Associate of the Abdus Salam ICTP. Research of the author was supported in part by Research Grant RG/MATHS/AF/AC No from the Third World Academy of Sciences (TWAS). 3Permanent address.

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6 contradicting the assumption that IIKmII > B. Thus (16) holds Vn 2 0. Let us now prove that IIK~LnII 5 B * llk~n+1ll 5 B; v?-t 2 0 (17) From (13) (16), we obtain that [I+ 2a(bn + Cn)]llK~n+l II2 2 B2 + 2P2BM(bn + G) (bn + 2%) + WBMG~ 2 B2 + 6P2BM(bn + ~1% + 2/?BM(bn + k)* 2Mi3 1 5 l+:(b,+k) B2 so that (17) follows. Since, by the choice of B, IlK~oII 2 B it follows by induction on n that IIKPnII I: B IlK~nlI 5 B; vn 2 0 Now, from (12) it follows that there exists a constant MO > 0 such that lik7nl12 I llkhl12 + Mobil Then, from (13) we have that there exists a constant d > 0 such that = IIK~nl12+ II+ Wbn + ~n>]llk~+lll~ 5 IIK~nl12 + 2adk; Vn > 0 Then (18) becomes (since b, + c, > bn) (1 + 2abn)@n+l < an + 2adG; Vn > 0 Observe that for 0 < x < 1, the following estimate holds Hence, (18) becomes (1+x)- < n+l 5 (1-2abn)Qn + dam (19) Since c,, = bi b, + 0 as n + 00, a straight forward application of Lemma HY readily yields ---f 0 as n -+ oo. Thus, IIK,u~II + 0 as n That is, 0 = iii~ IIAxn - f II = die lla(xn - x*)11 2 0, $im llk(xn - X*)11 2 f dew IlXn - X*II (18) (where 13 is the boundedness constant of K-l) proof. 0 so that xn + x* as n This completes the Corollary 1 Let E, A, K f be as in Theorem 1 let {un} be a sequence in D(A) such that {Ku,} is bounded. Then there exists a positive real number X such that if the real sequences {a,}, {bn}, {cn} G (0, l] satisfy the following conditions (9 on+bn+h=l; Vn/O (ii) b, 5 X;,llW b, = 0 (iii) c, = b; c b, = 00 then starting with any initial guess x0 E D(A), the sequence iteratively generated by n>o X72+1 = anxn+bn(xn-c1n)+cn(un+xn-cln); Vn>O Pn = K- (Ax, -f); Vn > 0 converges strongly to the unique solutiuon to the equation Ax = f. (20) (21)

7 Proof. The corollary easily follows from Theorem 1 by setting b; = 0. 0 Theorem 2 In Theorem 1, let xn+l = anxn + bn(x, - ^In)+Gz(Vn+Xn-~n); Vn>Cl Yn = a~xn+b~(x,-~n)+c~(un+xn--); vn>o 7n = K- (Ay, - f) Pn=Kvl(Axn-f); Vn>o (22) (23) (24 Then {xn} converges strongly to x*. Proof.Observe that Kyn = Kpn - (b; + ck)&-n + c;aun; Vn > 0 K/-h+1 = Kpn - (bn + G)Am + &AU,; Vn 2 0 (25) (26) so that llk~ IIKPnI12-2a(bn + ~)llkpn+1112 W2(bn + h){(bn + ~)lik~nll + (bl + ck>llk~ii + CnIIKvnII +c~iik~nii)~iik~+lii + WnIIKvnII-IIKPn+lII IIKYnI12 5 IIKPnl12-2a(bk + c~)llkynl12 + W2(% + ck) {(% + ck>iik~ll + c~llkunii} -IIKmII +2~~~IIK~nII~IlK~nll The rest now follows as in Theorem 1 the proof is complete. 0 Remarks 1. Our theorems are actually valid in arbitrary real normed linear spaces; the convergence of the sequence of iterates does not depend on the completeness of the underlying linear space. 2. The assumption that {Ku,} {Kvn} are bounded sequences can be replaced by the assumption that {un} {vn} are bounded sequences in E since the boundedness of the former set of sequences follows from the boundedness of the later set of sequences. Acknowledgments The authors are most grateful to the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. This research was carried out while the second author was vising the Abdus Salam ICTP as an Associate; a generous contribution from the Swedish International Development Cooperation Agency (SIDA) made the visit possible.

8 References 1. C. E. Chidume S. J. Aneke; Existence, uniqueness approximation of a solution for a K-positive definite opemtor equation, Applicable Analysis 50 (1993), C. E. Chidume M. 0. Osilike; Approximation of a solution for a K-positive definite operator equation, J. Math. Anal. Appl. 210 (1997), l Bai Chuanzhi; Approximation of a solution for a K-positive definite operator equation in uniformly smooth separable Banach spaces, J. Math. Anal. Appl. accepted, to appear (1998/99). 4. Chika Moore; Iterative approximation of the solution to a K-accretive operator equation in certain Banach spaces, Indian J. Pure Appl. Math. 21(12) (1990), W. M. Patterson, 3rd; Iterative methods for the solution of a linear operator equation in Hilbert space - a survey, Lect. Notes Math. No. 394 Springer-Verlag, Berlin. Heidelberg. New York, W. V. Petryshyn; Direct iterative methods for the solution of linear operator equations in Hilbeti space, Trans. Amer. Math. Sot. 105 (1962), W. V. Petryshyn; On the generalized overrelaxa:cation method for operation equations, Proc. Amer. Math. Sot. 14 (1963), W. V. Petryshyn; On a class of K-p.d non K-p.d operators operator equations, J. Math. Anal. Appl. 10 (1965), l Haiyun Zhou Jia Yuting; Approximation of fixed points of strongly pseudowntractive maps without Lipschitz assumption, Proc. Amer. Math. Sot. 125(6) (1997),