Commentationes Mathematicae Universitatis Carolinae
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1 Commentationes Mathematicae Universitatis Carolinae Josef Kolomý Note on the paper of I. Fenyö: On the solution of non-linear equations in Banach space Commentationes Mathematicae Universitatis Carolinae, Vol. 1 (1960), No. 4, Persistent URL: Terms of use: Charles University in Prague, Faculty of Mathematics and Physics, 1960 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library
2 Commentationes Mathemaiicae Universitatis Carolinae 1, t (1960) NOTĚ ON THE PAPEH OF I.FENYO: ON THE SOLUTION OP NON-INEAR EQUATIONS IN BANACH SPACE Josef KOLOMÍ, Praha It is shown that in tbe well known article of I.Fenyo [l] the proofs of both theorems are erroneous. Theorem 3 is a cor-sction of the tht -rem 2 ( xj ) 0 Let the equation <D fcx) ~ y be given, where a non-linear operátor f(x) maps Banach space A in, Banach space Y * Wě áenote by F(*) the ex prese ion f(x)-~y «I.Fenyo proves [lj the f ollowing theo rems : THEOREM 1«Let fcx) maps B -space X ín S -space Y and has in fl neighbourhood of the element A e X the following properties; 1) In the point X e it has the Frecheťs derivát ion, therq 4 exists tthj' and II/VA0)~'Í// = B 2 5 Wfm -fix Ji á C ÍU ~X. I for 11A -*o 11 á * < T&í flr Then the equation (l) has a unique solution X in <t-nei jíhbourhood of X for every % / which yields the inequality The iterative process defineal by the equality, is conyergent in the norm of the space x* of (1). - X to the solution THEOREM 2«Let fcx) maps B -' space X in B -space / and has in /fc-neigh a bourhood of the element X0eX continuous Frecheťs deri«yation, If there the operátor / Cx0) exists and the inequality >] á «Í4-«)
3 nolds for every x for which / x-x 0 /( K 9 then the equation (l) has a unique solution x in ft -neighbourhooá of the element X 0 éx for every y*y which yields the inedit, i f, h f M ] f á K (1_ 0 _ The modified Newton-Kantorowitch iterative process definea by the equality r c r / -\-4crc where F(x) ='/(x)- $, is convergent in the norm of the space X to the solution A of (1). The proof of theorem 1 is establiáed on the validity of the inequalities IlFíx) 1) á 7 $F'(x) íl = 7 yielding in l x-xoh~ fc. We show that these estimati* ons are not held* Above all the estimation does not follow from the assumptions of theorem 1 for every A from H X -X 0 / á K«does not Further the inequality / p (x) íl = nť hold for every X for which is /fx~x c íí ú t valid* and PROOF* There exists the operátor is the operátor T(x)=(j-f(x 0 r*[f(xj-f(xy)' HTCx) lř á j _. < ~Y~ for every X from Hx-xJJá /í,. It follows from the inequalities \\rm-\fu 0 >f(x))tů íf'cx.r'1 \\ťcx.)-rtx)i z á&cllx-xj) ú BCa< -f and Banach theorem. Further we obtain F'6< 0 )ry*)= FVx 0 ) (J-fCx.T* &'(*.)-f(*)])= - Poo * Yx.r* /Vx)-FVx) FVx)" 4 -{FVxjTYxrt-Too FVxo) M. Therefore the operátor i 0 O ~ exists in 1 -neighbourhood of the element x 0 and \\ F l Cx)" i /I < -J~B
4 ; Let us now assume that I F'(x) li a -Jfrom // x - x a // ú K. Then for every X Therefore the above mentioned estimation does not hold. Further the application of theorem 1 is not advantageous, because in practical problems it is yielding when //x*-.x<, // < 6 where is rather small such that X, can be considered as a good approximate solution of the equation (1). The proof of theorem 2 (see [l] ) is erroneous because: a) the equalities (25), (26) (see j] ) do not hold beoause Lagrange formula has no validity for operators in generál (see [2J, 3) b) from the condition of theorem 2 does not foliow the uni city of the solution x* of (1) in -neighbourhood of x.ex THEOREM 3. Let f(x) maps B -space A in E -space y and has the following propertiesi 1) For every A from Hx~X 0 ljá# it has the derivát ion of Prechet, there exists the operátor 7 (**) and 2)\\ťM-f'Cx0)l\áC ' forlu-xjlf/t and0<8c<y. Then the equation (1) has a unique solution x* in -t-neigh- bourhood of x X for every element e / which yields the inequality \\<y -/<Tx0)/Iá - rt. (4-%). The modified Newton-Kantorowitch iterative process defined by the equauty s - p ' F( is convergent in the norm of the space X X of the equation (1)..J to the solution REFERENCES, lj I.FENYO, Ueber die Losung der im Hanaehschen Saume defih nierten nichtlinearen Gleichungen. Acta t
5 Mathematicae Ac.Scientiarum Hungariae, No 5, (1954), z} M.M»Vajnberg, Variacionnyje metody issledovania nelinejnych operátorov, Moskva 1956 Í3j L.V.Kantorowitch - G.P,Akilov f Funkcionalnyj analiz v normirovannych prostranstvech* Moskva 1959 c \ * m\.\x- «"
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