Fixed Point Approximation of Weakly Commuting Mappings in Banach Space

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1 BULLETIN of the Bull. Malaysia Math. S. So. (Seod Series) 3 (000) 8-85 MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Fied Poit Approimatio of Weakly Commutig Mappigs i Baah Spae ZAHEER AHMAD AND ABDALLA J. ASAD Departmet of Applied Mathematis, Z.H. College of Egg. ad Teh., Aligarh Muslim Uiversity, Aligarh-000 (U.P.), Idia Abstrat. A fied poit theorem for a pair of weakly ommutig mappigs usig Ma iterative proess is preseted. I the proess the results due to Rhoades, Kha Imdad ad hosh are improved.. Itrodutio The Ma iterative proess (f.[8) assoiated with a self mappig T of a Baah spae X is desribed as follows: For 0 ε X we defie = ( ) for > 0, where { } satisfies (i) 0 = (ii) 0 < for > 0 (iii) diverges. The above sheme has bee used by may authors e.g. Bose ad Mukherji [, Das et al. [, Dotso ad Seter [3, Emmauelle [4, Ishikawa [6, Massa [9 ad Rhoades [ et. Usig Ma iterative proess desribed above Rhoades [ has proved the followig geeralizatio of a theorem of Pal ad Maiti [0. Theorem. Let T be a self mappig of a Baah spae X suh that y Ty α y holds for all,y i X where < α <. Let { } be a sequee of Ma iterates assoiated with T ad with { } satifyig (i) ad (ii) ad lim = h > 0 istead of (iii). If { } overges the it overges to the fied poit of T.

2 8 Z. Ahmad ad A.J. Asad The itet of this ote is to preset yet aother etesio of Theorem whih i tur geeralizes earlier results due to hosh [5 ad Kha-Imdad [7.. Mai result Before presetig our result, we reall that a pair of self-mappig {, T} of a ormed liear spae X is said to be weakly ommutig (f. [) if T for all ε X. Clearly a ommutig pair is weakly ommutig but the overse eed ot be true i geeral. Now we prove the followig: Theorem. Let ad T be two self mappigs of a Baah spae X suh that the iequality y Ty α y β [ Ty y holds for all,y i X, where β < h, ( β / h) < α < ( β / h), lim = h > 0. The the sequee { } defied by = ( ) (.) overges to a poit p ε X where { } is a dereasig sequee ad ejoys the properties of Theorem. Also if {,T} is a weakly ommutig pair ad is otiuous at {p} the p is a oiidee poit of ad T. Further if p = p the ad T have a ommo fied poit. Proof. I order to prove that { } is overget, we show that it is a Cauhy sequee. Cosider for > 0 (.3 ) = α [ β

3 Now usig (.) we have Fied Poit Approimatio of Weakly Commutig Mappigs i Baah Spae 83 ad = / { ( ) } (.4) h ( ) = h = / { ( ) } (.5) = { ) ( h ( ) ( ) ) Substitutig these values from (.4) ad (.5) i (.3) it follows that k L k 0 where k = ( α β / h) /( β / h) < whih implies that { } is a Cauhy sequee ad so overges to some poit p i X Now osider p = { )) ( p) O lettig, it yields that { } also overges to p. Sie is otiuous at {p}, the sequees { } Also sie ad T are weakly ommutig ad so p ad { } overge to p. T whih o lettig, yields T p 0. This implies that { T} also overges to p.

4 84 Z. Ahmad ad A.J. Asad To prove { T} overges to Tp, osider T Tp T p p Tp β p α p [ T p Tp O lettig, it yields p Tp β p Tp whih is a otraditio. This implies that p = Tp = p' ( say) Thus, we have show that p is the oiidee poit ad T. Further if we assume p = p ad sie ad T weakly ommute, Tp ' p' = Tp Tp p Tp = 0 ad so Tp ' = p' = p = p = p' so that p ' ( = p = Tp) is a ommo fied poit of ad T. This ompletes the proof. Remark. As remarked i hosh [5, we do ot assume the overgee of the sequee } but rather it is a osequee of oditio (.) { Remark. If we set β = 0 i Theorem, the we get a sharpeed form of Theorem of Kha et al. [7 as it ivolves weak ommutativity istead of ommutativity. Remark 3. By settig β = 0 ad = I (the idetity mappig) we get a Theorem of hosh [5 whih refies the ited Theorem due to Rhoades [. Settig α = 0 we get yet aother result.

5 Fied Poit Approimatio of Weakly Commutig Mappigs i Baah Spae 85 Referees. R.K. Bose ad R.N. Mukherkjee, Approimatig fied poits of some mappig, Pro. Amer. Math. So. 8 (98), K.M. Das, S.P. Sigh ad B. Watso, A ote o Ma iteratio for quasi-oepasive mappigs, No-Liear Aalysis 6 (98), W.. Dotso, Jr. ad H.F. Seter, Approimatig fied poits of o-epasive mappig, Pro. Amer. Math. So. 44 (974), Emmauell, A remark o overgee of Ma-Ishikawa iterative proess for oepasive mappigs, No-Liear Aalysis 7 (983), K.M. hosh, A ote o a theorem of Rhoades, Math. Sem. Notes 7 (980), S. Ishikawa, Fied poits by a ew iteratio method, Pro. Amer. Math. So. (974), M.S. Kha ad M. Imdad, A ote o ommo fied poits i Baah spaes, Math. Notae 3 (984), W.R. Ma, Mea value methods i iteratios, Pro. Amer. Math. So. 4 (953), S. Massa, Fied poit approimatios for quasi-oepasive-mappig, Le Mathematishe 37 (98), T.K. Pal ad M. Maiti, O orbital otiuity ad Ciri geeralized otratios, lasik Math. 3 (33) (978), B.E. Rhoades, Etesio of some fied poit theorems of Ciri, Maiti ad Pal, Math. Sem. Notes 6 (978), S. Sessa, O a weak ommutativity oditio i fied poit osideratio, Pub. Ist. Math. 3 (46) (98), Keywords: oiidee poit, fied poit, weakly ommutig mappig. 980 AMS Mathematis Subjet Classifiatio: Primary 54H5, Seodary 47H0