In this paper we give prove some fixed point theorems for contractive conditions in vector metric space, our results are generalized form of some well known results. Our results are motivated by Ishak Altun and Cuneyt Cevik [10]   Keywords: - fixed point, vector metric space

Choudhary, Sanjay

Patkar, Shyam

Shrivastava, Rakesh

English

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Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.4, No.22, 2014  88 Fixed Point Theorem Result in Vector Metric Spaces.  Rakesh Shrivastava1, Shyam Patkar2  and Sanjay Choudhary3 1. JNCT Bhopal (M.P) 2. Truba Institute of Engineering & Information Bhopal (M.P) 3. Professor and Head Govt. NMV Hoshangabad (M.P) Abstract  In this paper we give prove some fixed point theorems for contractive conditions in vector metric space, our results are generalized form of some well known results. Our results are motivated by Ishak Altun and Cuneyt Cevik [10]   Keywords: - fixed point, vector metric space.   Theorem 1.1Let X be an vector metric space with E is Archimedes, suppose the mappings  satisfies the following conditions  for all       Where  is a constant and     is complete subspace of X ,  Then G and F have a unique point of coincidence in X, moreover of F and G are weakly compatible, then they have a unique common fixed point in X. Proof:-Let  define the sequence  by   We have that             Where  Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.4, No.22, 2014  89     Now since E is Archimedean then  is an E-cauchy sequence since the range of F contains the range of T and the range of at least one is E-complete. There exists  such that  Hence there exists a sequence  in E such that an and  on the other hand, we can find  such that  Let us show that  we have       )  Case-I   when      For all n, since.  At least one of the following four cases hold for all n, Case-II     Case-III          Case-IV       Case-V          Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.4, No.22, 2014  90    Since the infimum of sequence on the right side of last inequality are zero then  Therefore z is a point of coincidence of F and G if  is another point of coincidence then there is   ]     Where   Hence  If F and G are weakly compatible then it is obvious, that Z is unique common fixed point of F and G. Theorem 1.2 :- Let X be an vector metric space with E is Archimedean, suppose the mapping  satisfies the following conditions      1. ,  2. F(X) or G(X) is E-complete subspace of X then F and G have a unique point of coincidence in X moreover if F and G are weakly compatible then they have a unique common fixed point is X  Proof:- Let us define the sequence  and  as in the proof of theorem we show that    We have to consider three case   Only second case if      Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.4, No.22, 2014  91     Where   In the proof of theorem, that  is an E-cauchy sequence then there exist  is E   such that  Now we have to show that  we have       Case-I when        At least one of three cases holds for all n, consider only the case of  because the other two cases have shown that the proof of theorem it is satisfied      This is  Since  hence z is a point of coincidence of F and G the uniqueness of z as in the proof of theorem 1.1 also if F and G are weakly compatible then it is obvious that z is unique common fixed point of G and F by [1.1] Theorem (1.3) Let X be an vector metric space with E is Archimedean suppose the mapping  satisfies the following conditions  1. For all    Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.4, No.22, 2014  92   Where b,c,d,e,f,g,h,i,j and R are nonnegative and b+c+d+e+g+h+i+j+R<1 2.  3. F(x) or G(x) is E-complete subspace of X then F and G have a unique point of coincidence in X moreover if F and G are weakly compatible then have a  unique common fixed point in X  Proof:-Let us define the sequence  and  as in the proof of theorem (1.1) we have to show that  for some  and all n consider  for all n then.  and  Hence    If we choose  Then   In the proof of theorem (1.1) we illustrate that  is an E couchy sequence, then there exist  and  in E such that  Let us show that  we have       For all n. then   Hence Z is a point of coincidence of F and G. Then uniqueness of z is easily seen. Also if F and G are weakly compatible, then if is obvious that z is unique common fixed point of G and F by [1.1].  Acknowledgement: Author [Rks] , thankful to Ishak Altun and Cuneyt Cevik for motivation by their results for extension.      Reference  [1] M. Abbas , G. Jungck, “common fixed point result for non commuting mappings without continuity metric space” J, Math Anal Appl. 341(2008) 416-420 [2] C.D Alprantis K.C Border, “Infinite, Dimensional Analysis”  Sprinager- Verlag Berlin 1999 [3] I, Altun. D.Turkoglu, “Some fixed point theorem for weakly compatible mappings satisfying an implicit relation”, Taiwanese J, Math 13(4) (2009) 1291-1304 [4] I. Altan D. Turkoglu. BE. Rhoades, “Fixed points of weakly compatible mapps satisfying a general contractive condition of integral type” fixed point theorem. Appl. 2007 Art ID 17301 9 App Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.4, No.22, 2014  93 [5] C. Covik. I ,Altam, “ vector metric spaces and some properties”  Topal. Met Nonlin Anal 34(2) (2009), 375-382  [6] L.G. Huong X Zhong, “Cone metric spaces and fixed point theorems of contractive mappings” J math. Anal 332,(2007) 1468-1476. [7] G. Jangck S. Radenovic, S. Radojevic , “common fixed point theorem for weakly compatible pairs on cone metric spaces”, fixed- point theory. Appl. 2009,Art ID 643840 13 P.P. [8] Z. Kadelburg’sRadenovic. V.  Rakocevic, “topological, vector space valued cone metric spaces and fixed point theorem”, fixed point theory Appl. 2010 Art ID 170253. [9] I. ShakAltan and ciineytcevik, “some common fixed point theorems in vector metric spaces”. (2011), 105-113. [10] Ishak Altun and Cuneyt Cevik, “SOME COMMON FIXED POINT THEOREMS IN VECTOR METRIC SPACES” Filomat 25:1 (2011), 105-113.       The IISTE is a pioneer in the Open-Access hosting service and academic event management.  The aim of the firm is Accelerating Global Knowledge Sharing.  More information about the firm can be found on the homepage:  http://www.iiste.org  CALL FOR JOURNAL PAPERS There are more than 30 peer-reviewed academic journals hosted under the hosting platform.   Prospective authors of journals can find the submission instruction on the following page: http://www.iiste.org/journals/  All the journals articles are available online to the readers all over the world without financial, legal, or technical barriers other than those inseparable from gaining access to the internet itself.  Paper version of the journals is also available upon request of readers and authors.   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Fixed Point Theorem Result in Vector Metric Spaces.

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Fixed Point Theorem Result in Vector Metric Spaces.

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