Aitken delta-squared generalized Juncgk-type iterative procedure

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1 Atke delta-squared geeralzed Jucgk-type teratve procedure M. De la Se Isttute of Research ad Developmet of Processes. Uversty of Basque Coutry Campus of Leoa (Bzkaa) PO Box Blbao, 488- Blbao. SPAIN emal: Abstract. Ths paper dscusses a geeral Atke delta-squared geeralzed Jugck-modfed S -teratve scheme. The study apples geeralzed versos of Atke delta-squared procedure ad Veter s theorem to dscuss postvty ad global stablty of the geeralzed Jugck teratve scheme whch s of terest umercal methods ad ts accelerato of covergece. eywords. Atke delta squared, Jugck terato, Veter theorem, stablty.. Itroducto Iteratve schemes are of terest umercal computatos ad ts propertes related to accelerato of covergece to solve scetfc ad egeerg problems. A very commo teratve scheme s the so called Jugck teratve scheme, whch volves the use of two coupled mappgs, ad ts varous extesos, [-3]. Such a scheme s useful also fxed pot theory to fd commo fxed pots of both mappgs. I ths paper, we exted such a teratve scheme ad study ts stablty ad postvty uder certa parametrcal restrctos. Also, we study the accelerato of covergece by proposg a Atke type delta squared procedure for accelerato of covergece ad combg the results wth the geeralzato of a backgroud Veter s stablty theorem result, whose basc form s well - kow dscrete parametrcal recursve detfcato, [4].. Notato The real sequeces y ad x are equvalet, deoted byy, f both have the same lmt. x. The teratve sequece ad some prelmares The followg Atke delta-squared geeralzed Jugck- modfed S -teratve scheme s of terest order to accelerate the covergece of the modfed sequece to the same lmt as the umodfed oe provded that such a lmt exsts: A Sy ASy a T z at y ; z, z C (..) b Sz bt z (..) (..3) S y S y (..4) S y ; N, where S T : C C, are two mappgs o a oempty subset C of a Baach space X, subject to T C SC, a ad b are real sequeces,, ad are bary sequeces

2 takg values or ; ad s a fte dfferece operator defg the correctg Atke-type terms of the above teratve procedure as follows: ; S y S y S y S x 3 (..) ; S y S y S y S y (..) ; N. The bary sequeces ad have two fuctos, amely, a) To remove the Atke correcto f t would mply dvso by zero at some terato (.e. f or S y s zero) ; b) To decde f mplemetg the Atke correcto or ot at ay partcular terato whch does o mply dvso by zero. The subsequet two techcal prelmary results reflect the features that the sequeces S S y ca have dfferet lmts f such lmts exst ad that uder mld codtos the Atke correcto leads to the same lmts as ts stadard Jugck teratve process provded that such lmts exst. x ad Lemma.. The followg propertes hold: () Assume that a, ad S coverges to Sz z. The, the followg lmt exsts: lm a b Sz a S y a b T y () Assume that b, ad S coverges to Sy lm a Sz b a b Sz a b T y y. The, the followg lmt exsts: () Assume that a,, b,, S coverges to Sz ad S coverges to The, the followg lmt exsts: lm a b Sz a S y a b T y If, addto, Sx Sy the lm a b Sz T y If Sy Sz ad f ab lm the lm T y Sz. z z y Sy. Proof: Oe gets obtag explctly T from (..), sce a, ad after ts replacemet to (..3), oe gets: a b Sz a b S abt y ; N Property () follows sce z Sz S. To prove Property (), oe obtas explctly T from (..3), sce b, ad oe gets by replacg t to (..) that z S y b a b Sz a b T y a, N

3 Property () follows sce S y Sy. Property () follows from Propertes () -(). Lemma.. The subsequet propertes follow: () If S x coverges to a fte lmt Sz the a T z at y at x at y a T x a T y Furthermore, f the () If y ASz ad Sz S coverges to a fte lmt ASz Sz provded that for ay N N ASz Sy the ASy Sz faster tha Sz Sz. Sy provded that for ay N b Sz bt x b Sx bt x S y S y S y S y Sz b T x b Furthermore, f the ASy Sy ASy Sy faster tha Sy Sy. Proof: Note from drect computatos that j ; N S x Sz A I Sz j j Sz 3 ad, sce S coverges to a fte lmt Sy y Sz A I Sz, oe gets: j j j lm Sz Sz lm lm lm A I A I A I Sz ASz Sz A I S z A Sz A I Sz A I ASz 3 S z sce Sz from (..), A I Sz Sz ad Sz Sz Sz Sz S z The, ASz Sz ad A A Sz yeld: ASz Sz Sz Sz 3 3. O the other had, smple calculatos wth (..) 3

4 sce ASz Sz so that f the ASz Sz faster tha Sz Sz sce Sz as so that the lmt below exsts: lm A Sz Sz lm Sz Property () has bee proved. Property () s proved mutats-mutads from the evaluato of A I S y Sz j j j ; N so that Sy A I S y Sy lm j j j Sy Sy lm AS y Sy A I S y A I S y ASy S y3 S y The followg result proves propertes of stablty ad covergece for the case whe T,S : X X are lear ad the sequeces a ad b are bouded whle o-ecessarly oegatve. Lemma.3. Assume that a ad b are real bouded sequeces ad that T, S : X X are lear ad S : X X s, furthermore, oe-to-oe ad of closed rage. The, the followg propertes hold: () If, for some k lm sup a, R T, oe has sup a T k k, lm sup b T k lm, lm sup b k S ; S k k k k S k k k k k k S wth the equalty S S k k beg guarateed f ; S k k The, z ad codtos of Lemma.. () If T ad y are bouded. Furthermore, Az ad S b S b M ; S a S a b S b x ad y are bouded. Furthermore, Az ad The, codtos of Lemma.. () If T ad S b S b M ; a a M S Ay are also bouded uder the ; N ; N Ay are also bouded uder the 4

5 The, x ad codtos of Lemma.. (v) If y are bouded. Furthermore, Az ad T ad a the zad Az ad the codtos of Lemma., z ad y. (v) If Lemma., Ay are also bouded uder the y are bouded, equvalet ad coverge to zero. Uder Ay are bouded, equvalet ad coverge to zero faster tha T ad b the y Ax ad Ay are bouded, equvalet ad coverge to zero faster tha x ad z are bouded ad coverge to zero. Uder the codtos of Proof: The varous propertes for the Atke-modfed sequeces Az ad Ay whch follow from the correspodg oes for z ad y are drect uder Lemma. so that o specfc proofs are eeded. Thus, oly proofs for the geeralzed Jugck-modfed S -teratve scheme are ow gve. Oe gets from (..) ad (..3) S z a T z a T y (..5) S y b S b T z (..6) where the mmum modulus of S s postve,.e. S closed rage the z ; N y S a S a b S b T T z y., sce S : X X s lear, oe-to-oe ad of ad oe gets uder the gve assumptos ad recursve calculatos that k S k z z ; N S k kk k S z z z z ; N Ths proves the boudedess of Property () for the umodfed sequeces ad also for the Atke deltasquared corrected sequeces uder the codtos of Lemma.. Property () follows from the subsequet relatos whch follow from the gve assumptos: z z z y S b S b z M z M z ; N ; Property () follows uder suffcet codtos to prove the boudedess of the sequeces uder Property () holds, by combg both gve assumptos. Propertes (v)-(v) follow from (..5), (..6) T as follows: wth z, sup y S b S z lm so that y f a ; ad z f y ad b. 3. Exteded Veter s Theorem ad ts use the modfed Jucgk teratve scheme 5

6 The subsequet prelmary result s the used: Theorem 3.. Cosder the followg teratve scheme parameterzed by real sequeces ad costats: x x ; x (3.) where,,,, s bouded, ; N N (3..),, as (3..) ad defe the real costat, as lm ).The, the followg propertes hold: () If ; N N the (whch exsts sce as x (Veter s theorem [ ] ). Furthermore, x ; lm x () If ; N () If ; N the x (3.3). x s bouded, x ad x the x s bouded ad x. (v) If lm sup the x s bouded. If, furthermore, sup N the sup N x f N Proof: If ; N x sup the x from Veter s theorem. The secod part of Property () follows by rewrtg (3.) for ; N the equvalet form x x x ; x leadg to x x x ; x ; N so that oe gets va recursve calculatos: x x x wth whch leads to (3.3) ad Property () s prove. O the other had, oe gets drectly from (3.) for lm whch yelds x x x x 6

7 x x ad hece Property (). If, addto, ; N the x, f x x, x N ad Property (). Note that, sce, (3.) may be equvaletly rewrtte as follows: x x x x x x bouded. Hece, x sup x (3.4.) x sup x (3.4.) ; N wth x what mples sup x x sup or x sup sup x x sup sup (3.6) The lm sup m (3.4.) that x s bouded f lm sup x whch s guarateed from (3.6) f m k what s tur guarateed f lm sup lm sup x lm sup sup, If, furthermore, sup f N x the, t follows from (3.4.) that sup x x sup N N (3.5). The, oe gets from wth ts frst equalty beg strct f x. Thus, Property (v) has bee prove. Theorem 3.. Cosder the teratve scheme (..)-(.4) uder the subsequet costrats: ) b,, b ) a,, a, T o y or, a, a, T o z 3) T, S : X X are lear ad S : X X s oe-to-oe wth closed rage wth the sequece S T cossts of composte postve operators. 7

8 4) Defe oegatve real sequeces ad, b b, such that., of geeral terms satsfyg The, the teratve scheme (..)-(..4) s globally stable for ay z, y whle t fulfls S, ad t s a oegatve sequece, z S, T,, respectvely, AS faster tha S, respectvely, y z y y Sy. z T ; ad AS z Proof: The teratve scheme (..) (..4) verfes Theorem 3. wth the deftos x ; N N uder the above costrats -4 wth b of geeral term Sy at y z b b Sz at y z T z b Sz a T S Sy a T S Sz N b ; beg oegatve for z ad y, summable, ad coverget to zero. The, the result follows from Theorem 3. usg Lemmas.-.3. ACNOWLEDGEMENTS The author s very grateful to the Spash Govermet by ts support of ths research trough Grat DPI-365, ad to the Basque Govermet by ts support of ths research trough Grats IT378- ad SAIOTE S-PEUN5. He s also grateful to the Uversty of Basque Coutry by ts facal support through Grat UFI /7. REFERENCES [] I. Icha, Vscosty terato method for geeralzed equlbrum pots ad fxed pot problems of fte famly of oexpasve mappgs, Appled Mathematcs ad Computato, Vol. 9, Issue 6, pp ,. [] M. De la Se, Stable terato procedures metrc spaces whch geeralze a Pcard-type terato, Fxed Pot Theory ad Applcatos, Vol., Artcle ID 5757, 5 pages,.do:.55//9539. [3] M. De la Se, Stablty ad covergece results based o fxed pot theory for a geeralzed vscosty teratve scheme, Fxed Pot Theory ad Applcatos, Artcle IDF 3458, Vol. 9, do:.55/ 9/3458, 9. [4] M. Delase, Ole optmzato of the free parameters dscrete adaptve cotrol systems, IEE Proceedgs- D Cotrol Theory ad Applcatos, Vol. 3, Issue 4, pp , 984. [5] J.J. Mambres ad M. De la Se, Applcato of umercal - methods to the accelerato of the covergece of the adaptve cotrol algorthms- The oe-dmesoal case, Computers & Mathematcs wth Applcatos-Part A, Vol., Issue, pp , 986. [6] M. Abbas, T. Nazr ad S. Romaguera, Fxed pot results for geeralzad cyclc cotracto mappgs partal metrc spaces, Revsta de la Real Academa de Cecas Exactas, Fscas y Naturales, Sere A: Matematcas,Vol. 6, No., pp ,. [7] M. De la Se, Lkg cotractve self-mappgs ad cyclc Mer-eeler cotractos wth aa selfmappgs, Fxed Pot Theory ad Applcatos, Vol., Artcle ID 5757, 3 pages,.do:.55//5757. [8] W.S. Du, New coe fxed pot theorems for olear multvalued maps wth ther applcatos, Appled Mathematcs Letters, Vol. 4, No., pp. 7-78,. 8

9 [9] W. Laowag ad B. Payaak, Commo fxed pots for some geeralzed multvalued oexpasve mappgs uformly covex metrc spaces, Fxed Pot Theory ad Applcatos, Artcle umber, do:.86/ ,. [] H.. Nashe ad M.S. ha, A applcato of fxed pot theorem to best approxmato locally covex space, Appled Mathematcs Letters, Vol. 3, Issue, pp. -7,. [] A. Latf ad M.A. utb, Fxed pots for w-cotractve multmaps, Iteratoal Joural of Mathematcs ad Mathematcal Sceces, Vol. 9, Artcle ID , 7 pages, 9, do:.55/9/ [] T. Husa ad A. Latf, Fxed pots of multvalued oexpasve maps, Iteratoal Joural of Mathematcs ad Mathematcal Sceces, Vol. 4, No.3, pp. 4-43, 99. [3] M. S. ha, Commo fxed pot theorems for multvalued mappgs, Pacfc Joural of Mathematcs, Vol. 95, No., pp , 98. [4] J.M. Medel, Dscrete Techques of Parameter Estmato: The Equato Error Formulato, Marcel Dekker Ic.,