In this paper we study a unique common fixed point theorem. The existence of fixed point for two weakly compatible maps is established under new contractive condition of integral type by using another functions.Keywords: Fixed point, Complete metric space, Weakly Compatible maps

Gupta, Vishal

Mani, Naveen

English

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Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.1, No.1, 2011  1 | P a g e  www.iiste.org A Common Fixed Point Theorem for Two Weakly Compatible Mappings Satisfying a New Contractive Condition of Integral Type Vishal Gupta1*, Naveen Mani2 1. Department Of Mathematics, Maharishi Markandeshwar University, Mullana, Ambala, Haryana, (India) 2. Department Of Mathematics, Maharishi Markandeshwar University, Mullana, Ambala, Haryana, (India)    * E-mail of the corresponding author: vishal.gmn@gmail.com, vkgupta09@rediffmail.com  Abstract In this paper we study a unique common fixed point theorem. The existence of fixed point for two weakly compatible maps is established under new contractive condition of integral type by using another functions   and  . Keywords: Fixed point, Complete metric space, Weakly Compatible maps.  1. Introduction The concept of commutativity has been used and generalized by many authors in several ways. For this Jungck (1976) proved a common fixed point theorem for commuting maps generalizing the Banach’s fixed point theorem. On the other hand Sessa (1982) has introduced the concept of weakly commuting .It is further generalized by Jungck (1988), so called compatibility. It can be easily verified that when two mappings are commuting then they are compatible. Clearly commuting, weakly commuting mappings are compatible but conversely need not be true.  The study of fixed point theorems satisfying various types of contractive inequalities has been a very active field of research during the last few decades. Such condition involves rational, irrational and general type expressions. To study more about this matter we recommended going deep into the survey articles by Rhodes (1977), (1983).  In (2002) Branciari obtained a fixed point result for a single mapping satisfying an analogue of a Banach contraction principle for integral type inequality as below: Theorem 1.1(Branciari 2002) Let (X, d) be a complete metric space,   [   )       a mapping such that for each         ∫  ( )   (    )  ≤   ∫  ( )   (   )  Where   [    )  [    )  is a “Lebesgue-integrable function” which is summable on each compact subset of   , non-negative, and such that for each    ,  ∫  ( )        then   has a unique fixed point such that for each                  Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.1, No.1, 2011  2 | P a g e  www.iiste.org After this result, there are many theorems dealing with mappings satisfying a general contractive condition of integral type.  Vishal Gupta et al. (2008) prove a common fixed point theorem for R-weakly commuting fuzzy maps satisfying a general contractive condition of integral type. Recently Vishal Gupta (2008) proves a Common Fixed Point Theorem for Compatible Mapping. Some other is noted in (Abbas et al. 2007; Altun 2007; Branciari 2002; Rhoades 1987; Rhoades 2007; Kumar S et al.  2007; Vijayaraju et al. 2005) 2. Preliminaries We recall the definitions of complete metric space and other results that will be needed in the sequel.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (    )                                                    (     )                                                      (    )                                           (      )                                                   (    )                                                                      3. Main Result Theorem3.1. Let S and T be self compatible maps of a complete metric space (X, d) satisfying the following conditions  (i) S(X)   T(X)                                     (3.1) (ii)  ∫  ( )   (     )  ≤  ∫  ( )   (     )  -  ∫  ( )   (     )      (3.2) Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.1, No.1, 2011  3 | P a g e  www.iiste.org for each  x,y   X where   [    )  [    ) is a continuous and non decreasing function and   [    )  [    ) is a lower semi continuous and non decreasing function such that         ( ) =  ( ) = 0 if and only if     also  [    )  [    )  is a “Lebesgue-integrable function” which is summable on each compact subset of   , non-negative, and such that for each    ,  ∫  ( )        Then S and T have a unique common fixed point. Proof : Let    be an arbitrary point of X. Since S(X)   T(X).Choose a point   in X such that S  =T  . Continuing this process, in general, choose      such that                                      For each integer     from (3.2)  ∫  ( )   (       )     ∫  ( )   (       )  -  ∫  ( )    (       )    (3.3)    ∫  ( )   (       )  Since   is continuous and has a monotone property, Therefore   ∫  ( )   (       )  ∫  ( )   (       )  Let us take     = ∫  ( )   (       )   then it follows that    is monotone decreasing and lower bounded sequence of numbers. Therefore there exist     such that       as    . Suppose that       Taking limit as     on both sides of (3.3) and using that   is lower semi continuous, we get,     ( )   ( )   ( )   ( )               This is a contradiction. Therefore   = 0. This implies                                             as                                                  ∫  ( )   (       )     as                                          (3.5) Now we prove that      is a Cauchy sequence. Suppose it is not. Therefore there exist an     and subsequence  {  ( )} &  {  ( )} such that for each positive integer  ,  ( )is minimal in the sense that     (  ( )   ( ))         (  ( )     ( ))            (3.6) Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.1, No.1, 2011  4 | P a g e  www.iiste.org Now,     (  ( )   ( ))    (  ( )   ( )  )    (  ( )     ( ))         (  ( )   ( )  )         (3.7) Now              ∫  ( )      ∫  ( )   (  ( )   ( ))       ∫  ( )     (  ( )   ( )  )   Letting      and from (3.5)                                            ∫  ( )   (  ( )   ( ))                                 (3.8) Now consider the triangle inequality,  (  ( )   ( ))    (  ( )   ( )  )    (  ( )     ( )  )    (  ( )     ( ))  (  ( )     ( )  )    (  ( )     ( ))    (  ( )   ( ))    (  ( )   ( )  ) and therefore, ∫  ( )   (  ( )   ( ))  ∫  ( )   (  ( )   ( )  )  (  ( )     ( )  )   (  ( )     ( ))      ∫  ( )   (  ( )     ( )  )  ∫  ( )   (  ( )     ( ))   (  ( )   ( ))   (  ( )   ( )  )  Taking     and using (3.5) and (3.8) in above inequalities, we get               ∫  ( )   (  ( )     ( )  )     ∫  ( )   (  ( )     ( )  )  This implies,                                              ∫  ( )   (  ( )     ( )  )                            (3.9) Now from (3.2), we have             ∫  ( )   (  ( )   ( ))  ≤  ∫  ( )   (  ( )     ( )  )  -  ∫  ( )   (  ( )     ( )  )  Taking limit as     and using (3.8) and (3.9), we get  ( ) ≤  ( ) -  ( ) This is a contradiction. Hence       is a Cauchy sequence. Since (X, d) is complete metric space, therefore there exist a point   such that Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.1, No.1, 2011  5 | P a g e  www.iiste.org                      Consequently, we can find   in X such that  ( )    . Now,   ∫  ( )   (        )  ≤  ∫  ( )   (       )  -  ∫  ( )   (       )  On taking limit as     implies   (∫  ( )   (     ) ) ≤  ( ) -  ( ) And so  (∫  ( )   (     ) )     implies that  ( )    . Hence   is the point of coincidence of   and  . Now we prove that   is the unique point of coincidence of   and  . Suppose not, therefore there exist  (    ) and there exist α in X such that  (α)    (α)   .  Using (3.2) we have               ∫  ( )   (     α)   ∫  ( )   (     α)  ≤  ∫  ( )   (     α)  -  ∫  ( )    (     α)                  ∫  ( )   (     α)   ∫  ( )   (     α)  This is a contradiction which implies    .  This proves uniqueness of point of coincidence of   and  . Therefore by using lemma (2.4), the result is proved.  References  Abbas M. and Rhoades B.E. (2007), Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized contractive condition of integral type, Fixed Point Theory and Applications, Volume 2007, article ID 54101, 9 pages Altun.I, Turkoglu.D, Rhoades B.E. (2007), Fixed points of weakly compatible maps satisfying a general contractive condition of integral type, Fixed Point Theory and Applications, Volume 2007,Article ID 17301, 9 pages  Branciari A. (2002), A fixed point theorem for mapping satisfying a general contractive condition of integral type, International journal of Mathematics and Mathematical Sciences, 29:9 (2002) 531-536. Gairola U.C. and Rawat A.S. (2008), A fixed point theorem for integral type inequality, International Journal of Math. Analysis, Volume 2, 2008, no. 15,709 – 712.   Mathematical Theory and Modeling  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.1, No.1, 2011  6 | P a g e  www.iiste.org Gupta Vishal et.al (2008), Common fixed point theorem for R-weakly commuting fuzzy maps satisfying a general contractive condition of integral type, International jounnal of Mathematical Science Engg. Appls, Vol. 2 no.II (2008) pp.193-203. Gupta Vishal et.al (2008), A Common Fixed Point Theorem for Compatible Mapping Accepted to publish in Innovative Systems Design and engineering (U.S.A), ISSN-2222, 1727(Paper), ISSN-2222-2871(Online)  Jungck G. (1976), Commuting mappings and fixed point, Amer. Math. Monthly, (1976), 261-263 Jungck G. (1986), Compatible mappings and common fixed points, International Journal of Mathematics and Mathematical Sciences, 9 (1986), 771 – 779. Jungck G. (1988), Compatible mappings and common fixed points (2), International Journal of Mathematics and Mathematical Sciences, 11 (1988), 285 - 288 Kumar S., Chugh R. and Kumar R. (2007), Fixed point theorem for compatible mapping satisfying a contractive condition of integral , Soochow Journal of Mathematics, Vol. 33, No. 2, pp. 181 – 185, April 2007. Pathak H.K., Tiwari R., Khan M.S. (2007), A common fixed point theorem satisfying integral type implicit relations, Applied Mathematics E – Notes, 7(2007), 222 – 228.  Rhoades.B.E. (1977), A comparison of various definitions of contractive mapping, Trans.Amer.Math.Soc.226 (1977), 257 – 290. Rhoades.B.E. (1983), Contractive definitions revisited, Topological Methods in Nonlinear Functional Analysis, Contemp. Math., Vol. 21, American Mathematical Society, Rhode Island, 1983, pp. 189 – 203. Rhoades.B.E. (1987), Contractive definitions, Nonlinear Analysis, World Science Publishing, Singapore, 1987, pp. 513 – 526. Rhoades.B.E. (2003), Two fixed point theorems for mapping satisfying a general contractive condition of integral type, International Journal of Mathematics and Mathematical Sciences, 2003:63, 4007 – 4013.  Sessa S. (1982), On a weak commutativity conditions of mappings in fixed point consideration, Publ. Math. Beograd, 32:46(1982), 146-153.  Vijayaraju P., Rhoades B.E. and Mohanraj R. (2005), A fixed point theorem for a pairs of maps satisfying a general contractive condition of integral type, International Journal of Mathematics and Mathematical Sciences, 2005:15, 2359 - 2364. This academic article was published by The International Institute for Science, Technology and Education (IISTE).  The IISTE is a pioneer in the Open Access Publishing service based in the U.S. and Europe.  The aim of the institute is Accelerating Global Knowledge Sharing.  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A Common Fixed Point Theorem for Two Weakly Compatible Mappings Satisfying a New Contractive Condition of Integral Type

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A Common Fixed Point Theorem for Two Weakly Compatible Mappings Satisfying a New Contractive Condition of Integral Type

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