Mathematical Theory and Modeling                                              www.iiste.orgISSN 2224-5804 (Paper) ISSN 222...
Mathematical Theory and Modeling                                                         www.iiste.orgISSN 2224-5804 (Pape...
Mathematical Theory and Modeling                                                                                     www.i...
Mathematical Theory and Modeling                                                                                          ...
Mathematical Theory and Modeling                                                                                          ...
Mathematical Theory and Modeling                                              www.iiste.orgISSN 2224-5804 (Paper) ISSN 222...
International Journals Call for PaperThe IISTE, a U.S. publisher, is currently hosting the academic journals listed below....
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11.a common fixed point theorem for two weakly compatible mappings satisfying a new contractive condition of integral type

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11.a common fixed point theorem for two weakly compatible mappings satisfying a new contractive condition of integral type

  1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.1, 2011A Common Fixed Point Theorem for Two Weakly CompatibleMappings 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.comAbstractIn this paper we study a unique common fixed point theorem. The existence of fixedpoint for two weakly compatible maps is established under new contractive condition ofintegral type by using another functions and .Keywords: Fixed point, Complete metric space, Weakly Compatible maps.1. IntroductionThe concept of commutativity has been used and generalized by many authors in severalways. For this Jungck (1976) proved a common fixed point theorem for commuting mapsgeneralizing the Banach’s fixed point theorem. On the other hand Sessa (1982) hasintroduced 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 commutingthen they are compatible. Clearly commuting, weakly commuting mappings arecompatible but conversely need not be true.The study of fixed point theorems satisfying various types of contractive inequalities hasbeen a very active field of research during the last few decades. Such condition involvesrational, irrational and general type expressions. To study more about this matter werecommended going deep into the survey articles by Rhodes (1977), (1983).In (2002) Branciari obtained a fixed point result for a single mapping satisfying ananalogue 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 oneach compact subset of , non-negative, and such that for each , ∫ ( )then has a unique fixed point such that for each1|Pagewww.iiste.org
  2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.1, 2011After this result, there are many theorems dealing with mappings satisfying a generalcontractive condition of integral type.Vishal Gupta et al. (2008) prove a common fixed point theorem for R-weakly commutingfuzzy maps satisfying a general contractive condition of integral type. Recently VishalGupta (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. PreliminariesWe recall the definitions of complete metric space and other results that will be needed inthe sequel. ( ) ( ) ( ) ( ) ( )3. Main ResultTheorem3.1. 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)2|Pagewww.iiste.org
  3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.1, 2011for each x,y X where [ ) [ ) is a continuous and non decreasingfunction and [ ) [ ) is a lower semi continuous and non decreasingfunction such that ( ) = ( ) = 0 if and only if also [ ) [ ) isa “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 suchthat S =T . Continuing this process, in general, choose suchthat 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 andlower bounded sequence of numbers. Therefore there exist such thatas . Suppose that Taking limit as on both sides of (3.3) and using that is lower semicontinuous, 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 , ( )isminimal in the sense that ( ( ) ( )) ( ( ) ( ))(3.6)3|Pagewww.iiste.org
  4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.1, 2011Now, ( ( ) ( )) ( ( ) ( ) ) ( ( ) ( )) ( ( ) ( ) )(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 completemetric space, therefore there exist a point such that4|Pagewww.iiste.org
  5. 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.1, 2011Consequently, we can find in X such that ( ) .Now, ( ) ( ) ( ) ∫ ( ) ≤ ∫ ( ) - ∫ ( )On taking limit as implies ( ) (∫ ( ) )≤ ( )- ( ) ( )And so (∫ ( ) ) implies that ( ) . Hence is the point ofcoincidence 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 ofcoincidence 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 – 712.5|Pagewww.iiste.org
  6. 6. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.1, 2011 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 - 2364.6|Pagewww.iiste.org
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