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Common random fixed point theorems of contractions in

  1. 1. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.8, 2014 78 Common Random Fixed Point Theorems of Contractions in Partial Cone Metric Spaces over Non normal Cones PIYUSH M. PATEL(1) Research Scholar of Sainath University,Ranchi(Jarkhand). pmpatel551986@gmail.com RAMAKANT BHARDWAJ(2) Associate Professor in Truba Institute of Engineering & Information Technology, Bhopal (M.P). drrkbhardwaj100@gmail.com SABHAKANT DWIVEDI(3) Department of Mathematics, Institute for Excellence in higher education Bhopal 1. ABSTRACT: The purpose of this paper is to prove existence of common random fixed point in the setting of partial cone metric space over the non-normal cones. Key wards: common fixed point,cone metric space, random variable 2. INTRODUCTION AND PRELIMINARIES Random nonlinear analysis is an important mathematical discipline which is mainly concerned with the study of random nonlinear operators and their properties and is needed for the study of various classes of random equations. The study of random fixed point theory was initiated by the Prague school of Probabilities in the 1950s [4, 13, and 14]. Common random fixed point theorems are stochastic generalization of classical common fixed point theorems. The machinery of random fixed point theory provides a convenient way of modeling many problems arising from economic theory and references mentioned therein. Random methods have revolutionized the financial markets. The survey article by Bharucha-Reid [1] attracted the attention of several mathematicians and gave wings to the theory. Itoh [18] extended Spacek's and Hans's theorem to multivalued contraction mappings. Now this theory has become the full edged research area and various ideas associated with random fixed point theory are used to obtain the solution of nonlinear random system (see [2,3,7,8,9 ]). Papageorgiou [11, 12], Beg [5,6] studied common random fixed points and random coincidence points of a pair of compatible random and proved fixed point theorems for contractive random operators in Polish spaces. In 2007, Huang and Zhang [9] introduced the concept of cone metric space and establish some fixed point theorems for contractive mappings in normal cone metric spaces. Subsequently, several other authors [10, 17, ] studied the existence of fixed points and common fixed points of pings satisfying contractive type condition on a normal cone metric space. In 2008, Rezapour and Hamlbarani [17] omitted the assumption of normality in cone metric space, which is a milestone in developing fixed point theory in cone metric space. In this paper we prove existence of common random fixed point in the setting of cone random metric spaces under weak contractive condition. Recently, Dhagat et al. [19] introduced the concept of cone random metric space and proved an existence of random fixed point under weak contraction condition in the setting of cone random metric spaces. The purpose of this paper to find common random fixed point theorems of contractions in partial cone metric spaces over non normal cones. Definition 2.1. Let X be a nonempty set and let 𝑃 be a cone of a topological vector space E . A partial cone metric on X is a mapping PXX:p  such that, for each Xthtgtf )(),(),( , t ,
  2. 2. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.8, 2014 79      t ththptgthpthtfptgtfpp tgtfptftfpp tftgptgtfpp tgtftgtgptftfptgtfpp ))(),(())(),(())(),(())(),(()3( ))(),(())(),(()3( ))(),(())(),(()2( ),()())(),(())(),(())(),(()1( The pair ),( pX is called a partial cone metric space over 𝑃. Definition 2.2. A function f: Ω → C is said to be measurable if 1 ( )f B C   for every Borel subset B of H. Definition 2.3. A function :F C C  is said to be a random operator if (., ):F x C  is measurable for every x C Definition 2.4. A measurable :g C  is said to be a random fixed point of the random operator :F C C  if ( , ( )) ( )F t g t g t for all t  Definition 2.5. A random operator F: Ω×C → C is said to be continuous if for fixed t , ( ,.):F t C C is continuous. Lemma:2.6 Let P be solid cone of a topological vector space E and     .)(,)(,)( Ethtgtf nnn  if ntgthtf nnn  )()()( , and there exists some Et )( Such that )()()()()()( tththenttgandttf nnn    Lemma:2.7 Let P be solid cone of a normed vector space  .,E then for each sequence   .)( Etfn  )()()()( . ttfimpliesttf nn    moreover if P is normal, then )()( ttfn   implies )()( . ttfn  Lemma:2.8 Let P be solid cone of a normed vector space    andKE n ,.,   .)( Ptfn     )(,sup)( tfKthenKandtf nnnnn . Theorem 2.9 Let ),( pX  be partial cone metric space. The mapping :,ST XX  are called contractions restricted with variable positive linear bounded mappings if there exist )4,3,2,1(:  iXXLi  such that ))),((),(()),((),(())(),(( )),((),(())(),(( ))),((),(())(),(())(),(())(),(())),((( 4 3 21 ttfTtgpttgStfptgtfL ttgStgptgtfL ttfTtfptgtfLtgtfptgtfLttfTp    (*))(),(  Xtgtf In particular if (*) is holds with then T and S are called contractions restricted with positive linear bounded mappings 3. Main Result: )4,3,2,1())(),((  iAandAtgtfL iii 
  3. 3. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.8, 2014 80 Theorem 3.1. Let ( , )X p be a 𝜃-complete partial cone metric space over a solid cone 𝑃 of a normed vector space ( , . )E and let , :T S X X  be contractions restricted with variable positive linear bounded random mappings. If 3 4( ( ( ), ( )) ( ( ), ( ))) 1p L f t g t L f t g t  and 2 4( ( ( ), ( )) ( ( ), ( ))) 1p L f t g t L f t g t  ( ), ( ) (1)f t g t X   1 2 1l l  and 3l   , where (.)p denotes the spectral radius of linear bounded mappings, 1 1 (t),g(t) X 2 2 (t),g(t) X 3 3 (t),g(t) X l sup ( ( ), ( )) l sup ( ( ), ( )) l sup ( ( ), ( )) , (2) f f f K f t g t K f t g t K f t g t                                      1 1 1 2 4 2 2 1 3 4 3 2 3 4 ( ( ), ( )) ( ( ), ( ))[L ( ( ), ( )) ( ( ), ( )) ( ( ), ( ))] ( ( ), ( )) ( ( ), ( ))[L ( ( ), ( )) ( ( ), ( )) ( ( ), ( ))] ( ( ), ( )) ( ( ), ( ))[I ( ( ), ( )) ( ( ), ( ))] K f t g t L f t g t f t g t L f t g t L f t g t K f t g t L f t g t f t g t L f t g t L f t g t K f t g t L f t g t L f t g t L f t g t f           ( ), ( ) (3)t g t X                                       where 1( ( ), ( ))L f t g t and 2 ( ( ), ( ))L f t g t denote the inverse of 3 4I ( ( ), ( )) ( ( ), ( ))L f t g t L f t g t  and 2 4I ( ( ), ( )) ( ( ), ( ))L f t g t L f t g t  respectively, then andT S have common random fixed point in X Moreover if 1 2 3 4( ( ( ), ( )) ( ( ), ( )) ( ( ), ( )) 2 ( ( ), ( ))) 1 ( ), ( ) (4) p L f t g t L f t g t L f t g t L f t g t f t g t X                then andT S have unique common random fixed point (t)h X such that, for each (t)nh X , (t) (t)p nh h   where (t)nh is defined by 1 ( (t),t) ;n is even number (t) (5) S( (t),t) ;n is odd number n n n T h h h                            Proof. For each Xyx , by (1), the inverse of ))(),(())(),((1 43 tgtfLtgtfL  and ))(),(())(),((1 42 tgtfLtgtfL  exist. then, it is clear that 1L and 2L are meaningful and 321 ,, KKK are well defined.      0 431 ))(),(())(),(())(),(( i i tgtfLtgtfLtgtfL Xyx , , which is together with )4,3,2(:  iXXLi  implies that )2,1(:  iXXLi  and hence ).3,2,1(:  iXXKi  by(*),(5) , (p4)  XXL :4   )6())(),(())(),(())(),(( 0 422    i i tgtfLtgtfLtgtfL
  4. 4. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.8, 2014 81 )7( ))](),(())(),(())[(),(( ))(),(())(),(( ))(),(())(),(( ))(),(())(),(( ))](),(())(),(())[(),(( ))(),(())(),(( ))(),(())(),(( ))(),(())(),(( )),((),),((())(),(( 22121221224 22121223 1221222 1221221 12122221224 22121223 1221222 1221221 1222212                    NktftfptftfptftfL tftfptftfL tftfptftfL tftfptftfL tftfptftfptftfL tftfptftfL tftfptftfL tftfptftfL ttfSttfTptftfp kkkkkk kkkk kkkk kkkk kkkkkk kkkk kkkk kkkk kkkk And so   )8( ))](),(( ))(),(( ))(),(())(),(( ))(),(())(),(())(),(( 122 1224 12221221 221212241222                Nktftfp tftfL tftfLtftfL tftfptftfLtftfLI kk kk kkkk kkkkkk Act the above inequality with ))(),(( 1221 tftfL kk  ; then,  XXL :1 , )9()),(),(())(),(())(),(( 12212212212   NktftfptftfKtftfp kkkkkk Similarly ,by (*),(p3), (p4) and  XXL :4 )10( ))](),(())(),(())[(),(( ))(),(())(),(( ))(),(())(),(( ))(),(())(),(( ))](),(())(),(())[(),(( ))(),(())(),(( ))(),(())(),(( ))(),(())(),(( )),((),),((( ))(),(())(),(( 3222122212224 221212223 322212222 122212221 3212222212224 221212223 322212222 122212221 1222 22323222                      NktftfptftfptftfL tftfptftfL tftfptftfL tftfptftfL tftfptftfptftfL tftfptftfL tftfptftfL tftfptftfL ttfSttfTp tftfptftfp kkkkkk kkkk kkkk kkkk kkkkkk kkkk kkkk kkkk kk kkkk And so
  5. 5. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.8, 2014 82   )11( ))](),(( ))(),(( ))(),(())(),(( ))(),(())(),(())(),(( 1222 12224 1222312221 32221222412222                Nktftfp tftfL tftfLtftfL tftfptftfLtftfLI kk kk kkkk kkkkkk Act the above inequality with ))(),(( 12222 tftfL kk  ; then, by: ,:2  XXL )12(.,))(),(())(),(()(),(( 2212122221222   NktftfptftfKtftfp kkkkkk Moreover ,:, 21  XXKK )13()),(),(())(),((........ ))........(),(())(),(())(),(( 10101 122212212212    NktftfptftfK tftfKtftfKtftfp kkkkkk In the following, we will prove that )14())(),((   tftfp mn For m>n, we have four cases 2qn2p,m(4)12qn2p,(3)m2qn1,+2p=m(2)1;+2q=n1,+2p=m(1)  Where p and q are two non negative integers such that qp  . We only show that (14) holds for case (1) the proof of three cases is similar. It follows for (p4) and (13) that )15(. ))(),(())(),((.....))(),(())(),(( ))(),(())(),(()).....(),(())(),(( .....))(),(())(),(()).....(),(())(),(())(),(( ))(),(())(),(())........(),(())(),(( ))(),(( ))(),(())(),((.........))(),(())(),(( ))(),(( ))(),(( 10101122211222 101013222212221 101011222122112222 1010112221221 1021 12221232222212 1212                qptftfptftfKtftfKtftfK tftfptftfKtftfKtftfK tftfptftfKtftfKtftfKtftfK tftfptftfKtftfKtftfK tftfKpK tftfptftfptftfptftfp tftfp tftfp ppqp ppqq qqqqqq qqqq ppppqqqq pq mn By ,121 ll        )16(, 1 ))(),(( ))(),(( ))(),((..... ))(),(( 21 1021211 10 1 211211 10212 1 1 1 2 1 12 1 1 1021                   qp ll tftfplllll tftfpllllll tftfpllllllll tftfKpK q p qi p qi ii ppppqqqq Which implies that . ))(),(( tftfp mn ,and hence  ))(),(( tftfp mn by lemma(2.7) Thus by (15) and lemma (2.6)  ))(),(( tftfp mn ;that is (14) holds. It is prove that  )(tfn is a Cauchy sequence in  pX, ,and so by the  completeness of  pX, , there exits Xth )( such that )()( thtf p n   and ))(),(( ththp ; that is,
  6. 6. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.8, 2014 83 )17())(),((   thtfp n . For all ,Nk  by (*) and (p4),               ))(),(( )),((),())(),(())(),(( ))(),(())(),(( )),((),())(),(( ))(),(())(),(( ))(),((),(,),( ))(),(()(),(),),(()(),),(( 2 122124 212123 122 12121 212 22 thtfp tthTtfptfthptfthL tftfptfthL tthTthptfthL tfthptfthL thtfpttfStthTp thtfptfthtthTpthtthTp k kkk kkk k kk kk kk                       )18())(),(()),((),( ))(),(()),((),( )),((),())(),(( ))(),(( ))(),(()))(),(())(),(( )),((),())(),(( ))(),(())(),(( 2 2 122 124 212123 122 12121                   thtfptthTthp thtfptthTthp tthtfptfthp tfthL tfthpthtfptfthL tthTthptfthL tfthptfthL k k kk k kkk k kk And so       )19( ))(),(())(),(())(),(( ))(),(())(),(())(),(())(),(( ))(),),((())(),(())(),(( 2124123 12124123121 124122        NkthtfptfthLtfthLI tfthptfthLtfthLtfthL thtthTptfthLtfthLI kkk kkkk kk Act the above inequality with ))(),(( 122 tfthL k ;then,  XXL :1 , )20())(),((,)(),((,))(,),((( 212312122   NkthtfpKtfthpKthtthTp kkkk Where ))(),(( 12212,2 tfthpKK kk   and ))(),(( 12212,3 tfthpKK kk   . It is clear that { 12,2 kK },{ 12,3 kK } subsets of  and   12,312,2 sup,sup kkkk KK by 21ll <1and 3l .then its follows from the lemma (3) and (17)that )21(,)(),(()(),(( 1212,31212,2    tfthpKtfthpK kkkk Which together with lemma (2.6)and (20) implies that   )),((),( tthTthp Therefore )()),(( thtthT  by (p1) and( p3).similarly we can show that )()),(( thtthS  . Hence )(th is common random fixed point of T and
  7. 7. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.8, 2014 84 Now we show the uniqueness of fixed point. Let )(tf and )(th be two common random fixed point of T and S then by (*) and (p3) )3,2(:  iXXLi        )22()),(),(())(),((2))(),(())(),(())(),(( ))(),(())(),(())(),(())(),(( ))(),(())(),((2))(),(( ))),((),(())),((),(())(),(( ))),((),(())(),(()),((),(())(),(()),(),(())(),(( ))),(()),),((())(),(( 4321 32 41 4 321       tfthptfthLtfthLtfthLtfthL tftfptfthLththptfthL tfthptfthLtfthL tthTtfpttfSthptfthL ttfStfptfthLtthTthptfthLtfthptfthL ttfStthTptfthp And so   )23()),(),(())(),((2))(),(())(),(())(),(( 4321  tfthptfthLtfthLtfthLtfthLI It follow from (3) that the inverse of  ))(),((2))(),(())(),(())(),(( 4321 tfthLtfthLtfthLtfthLI  exists(denoted by)   1 4321 ))(),((2))(),(())(),(())(),((   tfthLtfthLtfthLtfthLI and    1 4321 ))(),((2))(),(())(),(())(),(( tfthLtfthLtfthLtfthLI by Neumann’s formula with  1 4321 ))(),((2))(),(())(),(())(),((   tfthLtfthLtfthLtfthLI ;then ))(),(( tfthp and hence )()( tfth  by (p1) and (p3)the proof is completed. REFERENCES: 1. A.T. Bharucha-Reid, Random Integral equations, Mathematics in Science and Engineer- ing, vol. 96, Academic Press, New York(1972). 2. A.T. Bharucha-Reid, Fixed point theorems in Probabilistic analysis Bulletin of the American Mathematical Society, 82(5) (1976), 641-657. 3. C.J. Himmelberg, Measurable relations, Fundamenta Mathematicae, 87 (1975), 53-72. 4. D.H. Wagner, Survey of measurable selection theorem, SIAM Journal on Control and Optimization, 15(5) (1977), 859-903 5. I. Beg, Random fixed points of random operators satisfying semi contractivity conditions Mathematics Japonica, 46(1) (1997), 151-155. 6. I. Beg, Approximation of random fixed points in normed spaces Nonlinear Analysis 51(8) (2002), 1363-1372. 7. I. Beg and M. Abbas, Equivalence and stability of random fixed point iterative procedures Journal of Applied Mathematics and Stochastic Analysis, (2006), 19 pages 8. I. Beg and M. Abbas, Iterative procedures for solutions of random operator equations in Banach spaces, Journal of Mathematical Analysis and Applications, 315(1) (2006), 181-201. 9. L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332(2) (2007), 1468-1476 10. M. Abbas and G. Jungck, Common fixed point results for non commuting mappings continuity in cone metric spaces, J. Math. Anal. Appl., 341 (2008), 416-420. 11. N.S. Papageorgiou, Random fixed point theorems for measurable multifunctions in Banach spaces, Proceedings of the American Mathematical Society, 97(3) (1986),507- 514. 12. N.S. Papageorgiou, On measurable multifunctions with stochastic domain, Journal of Australian Mathematical Society. Series A, 45(2) (1988), 204-216. 13. O. Hans, Reduzierende zufallige transformationen, Czechoslovak Mathematical Journal (82) (1957), 154-1587. 14. O. Hans, Random operator equations, Proceeding of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, University of California Press, California, (1961), 185-202. 15. R. Penaloza, A characterization of renegotiation proof contracts via random fixed points in Banach spaces, working paper 269, Department of Economics, University of Brasilia, Brasilia, December (2002). . 16 Sh. Rezapour, A review on topological properties of cone metric spaces, in Proceedings of the International Conference on Analysis, Topology and Appl. (ATA 08), Vrinjacka
  8. 8. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.8, 2014 85 Banja, Serbia, May-June (2008). 17. Sh. Rezapour and R. Hamlbarani, Some notes on the paper "Cone metric spaces and fixed point theorems of contractive mappings", J. Math. Anal. Appl., 345(2) (2008), 719-724. 18. S. Itoh, Random fixed point theorems with an application to random differential equations in Banach spaces, Journal of Mathematical Analysis and Applications, 67(2) (1979), 261-273. 19. V.B. Dhagat, R. Shrivastav and V. Patel, Fixed point theorems in cone random metric spaces, Advances in Fixed Point Theory, 2(3) (2012), 357-363. 20.V.M. Sehgal and S.P. Singh, On random approximations and a random fixed point theorem for set valued mappings, Proceedings of the American Mathematical Society, 95(1) (1985), 91-94.
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