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Tii-p chi Tin hoc va Dieu khien uoc, T 17, S.l A NOTE ON REGULARIZATION BY LINEAR NGUYEN (2001), 17-20 OPERATORS BUONG Abstract The aim of this note is to give an improvement in our results of convergence rates of the regularized solutions for ill-posed operator equations involving monotone operators and in their convergence rates in combination with finite-dimensional approximations of reflexive Banach spaces Tom tJi t Bai trrnh bay mot di tien tot ho'n cho toc d E properties X, 0, g(O) = 0, g(t) -> +00, Xhb; if hi a, 151 a -> 0, as as t -> +00, and Ah a -> 0, the sequence are also {xhD} E So, xi:/l of (1.2) can be approximated An( iL x ) with the following - A(x)]] ~ hg(]]x]]), ]]/b converges with by solution + a B" x - Inb = P,: AiL Pn, B" = P~ B Pn, I~'= Pr: I~, under • This work was supported by the National of the finite-dimensional Fundamental (1.3) the conditions Research Program problem that in Natural Sciences 18 NGUYEN N BUONG The convergence rates of the sequences {Xh'b} and {xh'n, are given by (see [1]) 'I'h eor ern 1.1 Assume that the following conditions (i) A is Frchet differentiable in some neighbourhood (ii) There exists a constant L IIA'(x) (iii) There exists an element >0 such Lilvll (iv) U (So) of So - A'(y)11 ::; Lllx - v E D(B) such yll, V x E So, Y E U(So) that = BX1' < 2mD· if a is chosen as a ~ (h + 0)1", < J-L < I, we obtain IIXhh- xIII = Remark: hold: that A'(Xl)'V Then, where x;~~' denotes the solution of (1.3), (JlIIax = O((h + o)u), = min{l- (J J-L, J-L/2} 1/3, when J-L = 2/3 Set /3" = IIP~BPnxl In = 11(1 - Pn)X& - BX111, T'heor ern 1.2 Let the following conditions hold: (i) Conditions (i) - (iv) of Theorem 2.1 are fulfilled (ii) a t s chosen + + In)l" + f3n as a ~ (h Then IIX;:~'- xIII = where (J = O((h +0 In)U + /3r~/2) , min{1 - J-L, J-L/2} in [2) we can prove that the sequences {X;~D} and {Xh6'} In this note, by using the approach converge with faster rates RESULTS 'I'heorern 2.1 Suppose that the following conditions hold: (i) A tS tunce-Frectiei differentiable with IIA"II ::; M, M is a positive (ii) There exists an element v E D(B), Bv # 0, such that A'(Xl)V (iii) Then, Mllvll such + 0)1", that a ~ (h < J-L < I, we have + o)u), O((h (J = min{1- J-L,J-L}' From (1.1) and (1.2) it follows that A(X;:b) - A(xd ,L,;:.c Set·; •-' :J ";f< - ,;J~' ~:' ~~i = BX1' < 2mD' if a is chosen IIX~b- xIII = Proof constant + aB(x~b P/;b = - xd = fb - fo 11 + A(X;:6) - Aj,(Xh6) - aBxl' + t(X;;'b - xl))dt + «B A'(XI o l it~s._e~:y."t9 see that P'~b has the inversion P:t ) with IIP,~tl)ll::; 1/(mDa) And, we have A NOTE ON REGULARIZATION BY LINEAR OPERATORS 19 On the other hand, aIIP;,'b(-l) EX111= a [llp:}-l) + A'(Xl) (P,';b - PI~b)vll] :S; allvll + aIIPI~}-l)(Ph'b :S; a(llvll + IIEvll) + II( :S; a(llvll + IIEvll) + Mllvllllxf,o - X111/(2mn) mn - A'(X1)VII r l; A'(XI + t(Xf,o - xIJ)dt - A'(X1))v/mnll mn Therefore, Consequently, Hence, Remark: With /-l Set (3n 'I'heor ern 2.2 Suppose a ~ (h + + I,,)" + (3n, First, we estimate where A" = P,:APn, and max{.Bn, IIEnv - O((h Ph't 11 o P*A'( n is linear, bounded Ilpl:~'(-l)p,:(Jb IIp