&nbsp;In this paper we prove a common fixed point theorem for weakly compatible mappings satisfies certain&nbsp; contractive condition in non- empty closed subset of a separable Hilbert Space. Our results generalize and extend the result Chauhan [7]

Gupta, R.N

Modi, Geeta

English

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ISSN 2347-1921                                                           2882 | P a g e                                                       J a n u a r y  1 1 ,  2 0 1 5  COMMON FIXED POINT THEOREM FOR WEAKLY COMPATIBLE MAPPINGS IN HILBERT SPACE  Geeta Modi1,  1. Prof. and Head Department of Mathematics, Govt. M.V.M. Bhopal (M.P.) India. modi.geeta@gmail.com R.N.Gupta2 2.Asst. Prof., Department of Mathematics, Jai Narain College of Technology, Bhopal (M. P.), India.  ramnareshgupta@gmail. com  ABSTRACT In this paper we prove a common fixed point theorem for weakly compatible mappings satisfies certain contractive condition in non- empty closed subset of a separable Hilbert Space. Our results generalize and extend the result Chauhan [7]. Keywords Common fixed point, random operators, weakly compatible. Hilbert Space. .Academic Discipline And Sub-Disciplines Mathematics. AMS Subject classification (2000): 47H10, 54H25.                 Council for Innovative Research Peer Review Research Publishing System Journal: JOURNAL OF ADVANCES IN MATHEMATICS Vol.9, No 8 www.cirjam.com , editorjam@gmail.com ISSN 2347-1921                                                           2883 | P a g e                                                       J a n u a r y  1 1 ,  2 0 1 5    1.INTRODUCTION The study of random fixed point theory is started by Prague school of Probabilists in 1950 [8, 11]. Bharucha-Reid [5] has attracted much attention of many mathematicians by his survey article in this literature. Bharucha-Reid and Reagan [5, 10] obtain the solution of non linear random system by using random fixed point theory. The structure of common random fixed point and random coincidence points for a pair of compatible random operators in Polish space studied by Beg [1, 2] and Beg and Shahzad [3, 4].Chouhan [7] has proved a fixed point theorem for four random operators in Separable Hilbert space. In this paper we will prove a common fixed point theorem for weakly compatible random operators by using contractive condition in separable Hilbert spaces. For this we construct a sequence of measurable function of random fixed point to the four random operators. 2.PRELIMINARY NOTES Let  be a closed subset of Separable Hilbert space  and  a measurable space. Definition 2.1: A function is called measurable if  for each Borel subset  of . Definition 2.2: A function is called random operator if  is measurable for all  Definition 2.3: A measurable function is called a random fixed point to the random operator if  for all  Definition 2.4: A random operator is called continuous if for fixed  if   is continuous  Definition 2.5: Two random operators are called compatible if  and are compatible for all  Definition 2.6: Two random operators are called weakly compatible if  for some measurable mapping compatible    For all  3. MAIN RESULTS Theorem 3.1: Let  be a non-empty closed subset of a Separable Hilbert space . Let  and  be four continuous random operators defined on  such that for satisfy the following Conditions   Where   (2) The pair  and  are weakly compatible. Then  and  have unique common random fixed point in  Proof: Let  be an arbitrary measurable mapping for all . We construct a sequence of mappings {  Suppose that { {  are two sequences such that ISSN 2347-1921                                                           2884 | P a g e                                                       J a n u a r y  1 1 ,  2 0 1 5  = ,  = .  Firstly we show that {  is a Cauchy sequence.    If    and  Then = . Therefore  is a common random fixed point of   and  Now let the sequence {  and {  have no two consecutive terms equal at the same order. For all  and    =           Where  So in general for all we have,   Taking  we get  Thus for all , {  is a Cauchy sequence.  Hence {  is convergent in Separable Hilbert space. Suppose that  as  for  Since  is closed and  is a function from  to  Now we shall show that  is a common random fixed point of   and  ISSN 2347-1921                                                           2885 | P a g e                                                       J a n u a r y  1 1 ,  2 0 1 5  For                Letting we have because are subsequence of     Since  and  are weakly compatible. Therefore      For all  since  Similarly we can prove that for all   ISSN 2347-1921                                                           2886 | P a g e                                                       J a n u a r y  1 1 ,  2 0 1 5  Himmelberg [9] had proved if  is a continuous random operator on closed subset then for any measurable function  the function  is also measurable function.  Thus  is a sequence of measurable function. And hence  is also a measurable function. This implies that  is a common random fixed point of and  Uniqueness  Suppose that be the another common random fixed point of and  Therefore for all ,                                                                                                                                 ,  Now            Since   .  Hence  and  have a common unique random fixed point in  Example Suppose that  define as  and . Let  and  are two sequence such that  and  Now since  and    And  Then clearly ) and  are weakly compatible. Now          ISSN 2347-1921                                                           2887 | P a g e                                                       J a n u a r y  1 1 ,  2 0 1 5   Hence theorem is verified with condition (1) and (2). REFERENCES [1] Beg, I. 1997. Random fixed point of random operators satisfying semi-contractive conditions, Mahtmatica Japonica, 46, No.1, 151-155. [2] Beg, I. 2002. Approximation of random fixed point in normed space, Nonlinear Analysis 51, No.8, 1363-1372. [3] Beg, I. and Shahzad, N.1993. Random fixed points of random multivalued operators on Polish spaces, Non-linear Analysis Theory Method and Applications, 20, No.7, 835-847. [4] Beg, I. and Shahzad, N. 1995. Common random fixed points of random multivalued operators on Metric spaces, Bollet-tino Della Unione Mathematica Italiana, Series VII.A 9, No.3, 493-503. [5] Bharucha-Reid, A.T. 1976. Fixed point theorems in Probabilistic analysis, Bulletin of the American Mathematical Society, 82, No.5, 641-657. [6] Bharucha-Reid, A.T. 1972. Random Integral Equation, Academic press, New York. [7] Chauhan, S.S. 2013. Common fixed point theorem for four random operators in Hilbert space, Advance in Applied Science Research, 4(2), 219-222. [8] Han’s, O.1957.  Reduzierende Zufllige transformation, Czechoslovak Mathematical Journal, 7 (82),154-158. [9] Himmelberg, C. J. 1975. Measurable relations, Fund. Math., 87, 53-72. [10] Regan, D.O. 1998.  Fixed points and random fixed points for weakly inward approximable maps, Proceedings of the American Mathematical Society, 126, No.10, 3045-3053. [11] Spacek, A. 1955.  Zufallige Gleichungen, Czechoshvak Mathematical Journal, 5(80), 462-466. Author’ biography with Photo Dr. Geeta Modi has been awarded Ph.D., in 1990. She has 31 years of teaching experience. She is currently working as a Professor & Head of Department of Mathematics, Govt. MVM Bhopal. She is presently The Chairman of Board of Studies (Mathematics) Barkatullah University Bhopal, and associated with the Member of central Board of Studies (Mathematics) Government of Madhya Pradesh. She is V.C. nominee member of Board of Studies IEHE Bhopal Madhya Pradesh, India. She has published more than 45 articles in national and international journals. 08 Research scholar awarded Ph.D. under her supervision and 06 research scholar registered.       R. N. Gupta is a research scholar in the Department of Mathematics, Govt. MVM   Bhopal, Madhya Pradesh,India. His field of research in Fixed Point theorey and has 15 years of teaching experience.   

COMMON FIXED POINT THEOREM FOR WEAKLY COMPATIBLE MAPPINGS IN HILBERT SPACE

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COMMON FIXED POINT THEOREM FOR WEAKLY COMPATIBLE MAPPINGS IN HILBERT SPACE

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