11.common fixed point theorems in non archimedean normed space
Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.1, No.2, 2011Common Fixed Point Theorems in Non-Archimedean Normed Space Vishal Gupta1*, Ramandeep Kaur2 1. Department of Mathematics, Maharishi Markandeshwar University, Mullana, Ambala, Haryana, (India) 2. Department of Mathematics, Karnal Institutes of Technology and Management Karnal, Haryana, (India) * E-mail of the corresponding author: firstname.lastname@example.org, email@example.comAbstractThe purpose of this paper is to prove some common fixed point theorem for single valuedand multi-valued contractive mapping having a pair of maps on a spherically completenon-Archimedean normed space.Key-Words: Fixed point, Contractive mapping, Non-Archimedean normed space,spherically complete metric space.1. IntroductionC.Petals et al. (1993) proved a fixed point theorem on non-Archimedean normed spaceusing a contractive condition. This result is extended by Kubiaczyk(1996) from singlevalued to multi valued contractive mapping. Also for non expansive multi valuedmapping, some fixed point theorems are proved. In 2008, K.P.R Rao(2008) proved somecommon fixed point theorems for a pair of maps on a spherically complete metric space.2. PreliminariesDefinition 2.1. A non-Archimedean normed space ( X , ) is said to be sphericallycomplete if every shrinking collection of balls in X has a non empty intersection.Definition 2.2. Let ( X , ) be a normed space and T : X X , then T is said to becontractive iff whenever x & y are distinct points in X, Tx Ty x yDefinition 2.3. Let ( X , ) be a normed space let T : X Comp( X ) (The space of allcompact subsets of X with Hausdroff distance H), then T is said to be a multivaluedcontractive mapping if H (Tx, Ty) x y for any distinct points in X.3. Main ResultsTheorem 3.1. Let X be a non-Archimedean spherically complete normed space. If f andT are self maps on X satisfying T ( X ) f ( X ) Tx Ty f ( x) f ( y) x, y X , x yThen there exist z X such that fz Tz .Further if f and T are coincidentally commuting at z then z is unique common fixed pointof f and T.1|P agewww.iiste.org
Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.1, No.2, 2011Proof: Let Ba B f (a), f (a) Ta denote the closed spheres centered at fa with radii f (a) Ta , and let F be the collection of these spheres for all a X . The relation Ba Bb iff Bb Bais a partial order. Consider a totally ordered subfamily F1 of F . From the sphericallycompleteness of X, we have Ba B Let fb B and Ba FBsince fb Ba implies f (b) f (a) f (a) T (a) ………….. a F1(3.1)Let x Bb , then x fb fb Tb max fb fa , fa Ta , Ta Tb fa Ta , Ta Tb fa Ta Ta Tb fa fb Now x fa max x fb , fb fa m a fa TaImplies x Ba . Hence Bb Ba for any Ba F1 .Thus Bb is an upper bound in F for the family F1 and hence by Zorn’s Lemma, F has amaximal element (say) Bz , z X .Suppose fz TzSince Tz T ( X ) f ( X ), w X such that Tz fw , clearly z w .Now fw Tw Tz Tw fz fwThus fz Bw and hence Bz Bw , It is a contradiction to maximality of Bz .Hence fz TzFurther assume that f and T are coincidently commuting at z.Then f 2 z f ( fz ) f (Tz) Tf ( z) T (Tz) T 2 zSuppose fz zNow T ( fz ) T ( z ) f ( fz ) f ( z ) f 2 z fz T ( fz ) TzHence fz z . Thus z fz TzLet v be a different fixed point, for v z , we have z v Tz Tv fz fv z vThis is a contradiction. The proof is completed.Theorem 3.2 Let X be a non-Archimedean spherically complete normed space. Let f : X X and T : X C ( X ) (the space of all compact subsets of X with the Hausdroffdistance H) be satisfying T(X ) f (X ) (3.2) H (Tx, Ty) fx fy (3.3)For any distinct points x and y in X. then there exist z X such that fz Tz .Further assume that2|P agewww.iiste.org
Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.1, No.2, 2011 fx fw H T ( fx), Tw for all x, y, w X with fx Ty (3.4)And f and T are coincidentally commuting at z. (3.5)Then fz is the unique common fixed point of f and T.Proof: Let Ba fa, d ( fa, Ta) denote the closed sphere centered at fa with radius d ( fa, Ta) and F be the collection of these spheres for all a X . Then the relation Ba Bb iff Bb Bais a partial order on F . Let F1 be a totally ordered subfamily of F .From the sphericallycompleteness of X, we have Ba B Let fb B and Ba aF1 1, then fb Ba B FHence fb fa d ( fa, Ta) (3.6)If a b , then Ba Bb .Assume that a bSince Ta is complete, w Ta such that fa w d ( fa, Ta) (3.7)Consider x Bb , then x fb d fb, Tb inf fb c c Tb max fb fa , fa w ,inf w c max d ( fa, Ta), H (Ta, TbTb c d ( fa, Ta) (3.8)Also x fa max x fb , fb fa d ( fa, Ta)Thus x Ba and Bb Ba for any Ba F1 .Thus Bb is an upper bound in F for the familyF1 and hence by Zorn’ Lemma F has maximal element say Bz , for some z XSuppose f z Tz , since Tz is compact, there exist k Tz such that d ( fz, Tz ) fz kSince T ( X ) f ( X ), u X Such that k fuThus d ( fz, Tu) fz fu (3.9)Clearly z u .Now d ( fu, Tu) H (Tz, Tu) fz fuHence fz Bu , thus Bz BuIt is a contradiction to the maximality of Bz ,hence fz Tz .Further assume (3.4) and (3.5)Write fz p , then p Tz ,from (3.6) p fp fz fp H (Tfz, Tp) H (Tp, Tp) 0Implies fp p . From (3.7) p fp fTz Tfz TpThus fz p is a common fixed point of f and T .Suppose q X , q p is such that q fq Tq3|P agewww.iiste.org
Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.1, No.2, 2011From (3.6) and (3.7) p q fp fq H (Tfp, Tq) H (Tp, Tq) fp fq p qImplies p q .Thus p fz is the unique common fixed point of f and T.ReferencesKubiaczyk, N.Mostafa Ali(1996), A multivalued fixed point theorems in Non-Archimedean vector spaces, Nov Sad J. Math, 26(1996).K.P.R Rao, G.N.V Kishore(2008), Common fixed point theorems in Ultra Metric Spaces,Punjab Uni. J.M ,40,31-35 (2008).Petals.C., Vidalis.T.(1993), A fixed point theorem in Non-Archimedean vector spaces,Proc. Amer. Math. Soc.118(1993).4|P agewww.iiste.org
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