Existence of common fixed points of weak** commuting mappings which satisfies the contractive condition involving pair of mappings in a complete metric space under certain is shown. Key words: commuting mappings, weak ** commuting mapping

Gagrani, Shweta

English

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Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications  43 Common Fixed Point and Weak** Commuting Mappings Shweta Gagrani e-mail: shweta_gagrani@yahoo.com  Abstract  Existence of common fixed points of weak** commuting mappings which satisfies the contractive condition involving pair of mappings in a complete metric space under certain is shown. Key words: commuting mappings, weak ** commuting mapping.  1. Introduction    A study of the common fixed points and weak** commuting mappings is fascinating field of research lying at the intersection of non-linear analysis. A wide spread interest in the domain and vast amount of mathematical activity have led to many remarkable new results.  In 1976, Jungck [4] investigated and found interdependence between commuting mappings and common fixed points and proved the followings:  Let T be a continuous mapping of a complete metric space (X, d) into itself. Then T has a fixed point in X, if and only if there exists an α ∈ (0, 1) and a mapping S : X → X which commutes with T and satisfies:  (1) S(X) ⊂ T(X) and d(Sx, Sy) ≤ d(Tx, Ty) For all x, y in X. Indeed, S and T have a unique common fixed point if and only if (1) holds for some α ∈ (0, 1).  Further, in 1977, Singh [10] generalized the above result and proved that two continuous and commuting mappings from a complete metric space into itself satisfies some conditions, then two commuting mappings have a unique common fixed point.  Das and Vishwanathana Naik [1] have proved a theorem for two commuting mappings. Fisher [2] proved a common fixed point of commuting mappings, Rhoades and Seesa [8] established some fixed point theorems for three pair wise weakly commuting self maps satisfying a very general contractive definitions. Khan and Imdad [5], considering a pair of self maps {A, T} of metric space (X, d) satisfying a weaker condition the commutativity: namely weak* commuting pair of mappings, that is   d(ATx, TAx) ≤ d(A2x, T2x) For each x in X. B. Fisher [2] has been proved following theorem for two commuting mappings T and S. If S is a mapping and T is a continuous mapping of the complete metric space into itself and satisfying the inequality :  (2) d(STx, TSy) ≤ k {d(Tx, TSy) + d(Sy, STx)} for all x, y in X, where 0 ≤ k ≤ 1/2, then S and T have a unique common fixed point.  In 1986, Pathak [7] has been further generalized a result of Khan and Imdad [5] by considering a pair of self maps {A, T} of a metric space (X, d) satisfying a weaker condition, then commutativity: namely, weak* commuting pair of mappings, that is     d(ATx, TAx)  ≤ d(A2x, T2x) for each x ∈ X. In 1995, Lohani and Badshah [6] further generalized the result of B. Fisher[2, 3]  The purpose of this note is to prove some results concerning fixed points of weak** commuting mappings defined on complete metric spaces and satisfying some new functional inequality. Definition 1.1.   According to Seesa [9] two self maps S and T defined on metric space (X, d) are said to be weakly commuting maps iff   d(STx, TSx) ≤ d(Sx, Tx) for all x in X.  Defintion 1. 2.  Two self mappings S and T of metric space (X, d) is called weak** commuted, if S(X) ⊂ T(X) and for any x ∈ X, d(S2T2x, T2S2x) ≤ d(S2Tx, TS2x) ≤ d(ST2x, T2Sx) ≤ d(STx, TSx) ≤ d(S2x,                                                                                                     T2x) Definition 1.3.  A map S : X → X, X being a metric space, is called an idempotent, if S2 = S.  We further generalize the result of Fisher [2, 3], Pathak [7] and Lohani & Badshah [6] by using another type of rational expression.  Theorem 1.1.  If S is a mapping and T is a continuous mapping of complete metric space {S, T} is weak** commuting pair and the following condition : Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications  44 ),(),(),(),(),(),(),d(T),(222222222222222222222222ySxTdxTSySdySTxTdxTSySdySTySdySTxTdxTSxySTxTSdβα+++≤  for all x, y in X, where 0 ≤ α + β < 1, then S and T have a unique common fixed point. Proof.   Let x be an arbitrary point in X. Define  (S2T2)nx = x2n or T2(S2T2)nx = x2n+1 Where n =0, 1, 2, 3…., by contractive condition (A), ( ) ( )( ) ( )( )( )xTSTSxTSTdxTSTSxTSTSdxTSTSTxTSTdxTSTSxTSTSdxTSTSTxTSTSdTSTSTxTSTdxTSTSxTSTdxTSTSTxTSTSdxxdnnnnnnnnnnxnnnnnnn1222212221222212222122222122212222122221222221222212222212221222212221222222222122)(,)())(,)(())((,)(()(,)()((,)()((,)()(,)()))((,)((),(−−−−−−−−−−−−−−−++++=≤βα ))(,)(())(,)((),())(,)(())((,)(())((,)(())(,)((122221222122221222122122221222122222122122221222 2xTSTSxTSTdxTSTSxTSTdxxdxTSTSxTSTdxSTTTSxSTTdxTSTSTxTSTdxTSTSxTSTdnnnnnnnnnnnn−−−−+−−−−−−+≤+≤βαβα                 ( )( ) ).,())(,(2122212nnnnxxdxTSxd−−+≤+≤βαβα Proceeding in the same manner d(x2n, x2n+1) < (α + β)2n-1d(x1, x2). Also d(xn, xm) ≤ )( 1, +=∑ imnii xxd   for m > n. Since k < 1, it follows that the sequence {xn} is Cauchy sequence in the complete metric space X and so it has a limit in X, that is    limn→∞x2n = u = limn→∞x2n+1 and since T is continuous, we have   u = limn→∞x2n+1 = limn→∞T2(x2n) = T2u. Further,   d(x2n+1, S2u) = d(T2(S2T2)n+1x, S2u)     = d(T2(S2T2)n+1x, S2(T2u)) for u = T2u Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications  45      ),(),(),()],(),(),(),([),)())(,(),)(()))(,(),())(,)())(,)((223222232222232222122122222212212222222221221222122uxdxuduSxdxuduSuduSxdxxduTxTSdxTSTuTduTSxTSdxTSTuTduTSuTduTSxTSdxTSTxTSdnnnnnnnnnnnnnn+++++++++++++++++=+++≤βαβαtaking limit as n → ∞, it follows that   d(u, S2u) = 0. which implies  d(u, S2u) = 0 and so u = S2u = T2u. Now consider weak** commutativity of pair {S, T} implies that S2T2u = T2S2u, S2Tu = TS2u, ST2u = T2Su and so S2Tu = Tu and T2Su = Su. Now  d(u, Su) = d(S2T2u, T2S2(Su)) ( )0),(0),(10),(),(),(),(),(),(),(),(),())(,(),(),())(,(),())(,()),(())(,()]),(())(),(())(,(),([222222222222222222222222≤⇒≤−⇒=+++=+++=+++≤SuudSuudSuuduSudSuuduSudSuSudSuuduudSuuduSudSuSuduSudSuSudSuTSuduSudSuSuTduTSSuSdSuSTuTduTSSuSduSSTuSSduSSTuTduTSuTdββαβαβα                             Hence Su = u, similarly we can show that Tu = u. Hence u is a common fixed point of S and T.  Now suppose that x is an another common fixed point of S and T. Then            ),(),( 2222 vSTuTSdvud =        ),(),(),(),(),(),(),( 22222222222222222222vSuTduTSvSdvSTuTduTSvSdvSTvSdvSTuTduTSuTdβα +++≤                         ( )0),(0),(1),(),(),(),(),(),(),(≤≤−+++≤vudvudvuduvdvudvudvudvudvudββα Then it follows that u = v. Hence S and T have a unique common fixed point. Theorem 1.2.   If S is mapping and T is a continuous mapping of a complete metric space X into itself and satisfying {S, T} is weak** commuting pair and the following condition : ),(),(),()],([)],([),( 222222222222222222 ySxTdySTySdxTSxTdySTySdxTSxTdySTxTSd βα +++≤      (B) for all x, y in X, where α , β ≥ 0 with 2α + β < 1, then S and T have a unique common fixed point. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications  46 Proof.   Let x be an arbitrary point in X. Define  (S2T2)nx = x2n or T2(S2T2)nx = x2n+1 Where n =0, 1, 2, 3…., by contractive condition (B),  ( )xTSTSxTSTdxTSTSTxTSTSdxTSTSxTSTdTSTSTxTSTSdTSTSxTSTdxTSTSTxTSTSdxTSTxTSdxxdnnnnnnnnnnnnnnnn1222212221222221222212222122221222221222221222212221222221222222222122)(,)()))((,)(())(,)((,)}])((,)({([}])(,)({([))((,.)(()))(,)((),(−−−−−−−−−−−−++++==β ),()],(),([),(),(),()],[()],([21212221221212221221222212nnnnnnnnnnnnnnnnxxdxxdxxdxxdxxdxxdxxdxxd−+−−+−+−++≤+++≤βαβα ( ) ),()(),(1),(),()(212122122212nnnnnnnnxxdxxdxxdxxd−++−+≤−++≤βαααβα ),(1),( 212122 nnnn xxdxxd −+−+≤αβα                   ),( 212 nn xxkd −≤  where k = αβα−+1. Proceeding in the same manner, we have d(x2n, x2n+1) ≤ k2n-1d(x1,x2). Also d(xn, xm) ≤ ∑=+mniii xxd ),( 1   for m > n. Since k < 1, it follows that the sequence {xn} is Cauchy sequence in the complete metric space X and so it has a limit in X, that is   limn→∞x2n = u = limn→∞x2n+1 and since T is continuous, we have  u = limn→∞x2n+1 = limn→∞T2(x2n)m = T2u. Further  d(x2n+3, S2u) = d(T2(S2T2)n+1x, S2u)                               = d(T2(S2T2)n+1x, S2(T2u))    (since u = T2u)                                          ))(,(})(,){(),(])(,)[()],([ 12221222122222212221222222xTSuTdxTSTxTSduTSuTdxTSTxTSduTSuTd nnnnn++++++++≤ βα),()],())(,([ 2223222222 uTxdxxduTSuTd nnn +++ ++≤ βα  ),()),(),(( 2232222 uxdxxduSud nnn +++ ++≤ βα  Taking limit as n → ∞, it follows that   d(u, S2u) ≤ 0, which implies that d(u, S2u) = 0 and so that u = S2u = T2u. Now consider weak** commutativity of pair {S, T}, implies that S2T2u = T2S2u , S2Tu, TS2u, ST2u  = T2Su  and so S2Tu = Tu and T2Su= Su. Now d( u, Su) = d(S2T2u, T2S2(Su))                        ))(,())(),((),()](),([)],([ 2222222222222222SuSuTdSuSTSuSduTSuTdSuSTSuSduTSuTdβα +++≤  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications  47                        ),(),(),()],([)],([SuudSuSuduudSuSuduudβα +++≤                    (1- β) d(u, Su) ≤ 0     this implies that (1-β) ≠ 0. Hence d(u, Su) = 0 or Su = u. Similarly we can show that Tu = u. Hence u is a common fixed point of S and T. Now suppose that v is another common fixed point of S and T, then  d(u, v) = d(S2T2u, T2S2v)                      ),(),(),()],([)],([ 2222222222222222vSuTdvSTvSduTSuTdvSTvSduTSuTdβα +++≤                       ≤ ),()],[()],([ vudvvduud βα ++  (1- β) d(u, v) ≤0. Since (1-β) ≠ 0, then d(u, v) = 0. Thus it follows that u = v. Hence S and T have a unique common fixed point. Example 1.1. Let X = [0,1] with Euclidean metric space and define S and T by 2,2xTxxxSx =+=  for all x ∈ X, then [0, 1/5] ⊂ [0,1/4], where Sx = [0, 1/5] and Tx = [0, 1/4]      168163),( 2222+−+=xxxxxSTxTSd                          )168()163(5 2+−+=xxx                     ≤ )84)(82(2++ xxx                    = 8482 +−+ xxxx                   = d (S2Tx,TS2x) ⇒ d(S2T2x, T2S2x) ≤ d(S2Tx, TS2x) d(S2Tx, TS2x) = 8482 +−+ xxxx                       = )84)(82(2 2++ xxx                      ≤ )84)(8(3 2++ xxx                      = 848 +−+ xxxx                     = d(ST2x , T2Sx) ⇒ d(S2Tx, TS2x) ≤ d(ST2 x, T2 Sx) d(ST2 x, T2 Sx) = 848 +−+ xxxx                         = )8)((8(3 2++ xxx                        ≤ )42)(4(2++ xxx                        = 424 +−+ xxxx                        = d(STx, TSx) Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications  48 ⇒ d(ST2 x, T2 Sx) ≤ d(STx, TSx) d(STx, TSx) = 424 +−+ xxxx                     =  )42)(4(2++ xxx                     ≤ )43(43 2+xx                    = 434 +−xxx                   = d(T2x, S2x) ⇒  d(STx, TSx) ≤ d(T2x, S2x) using [0, 1] for x ∈X, we conclude that definition (1.2) as follows :  d(S2T2x, T2S2x) ≤ d(S2Tx, TS2x) ≤ d(ST2 x, T2 Sx) ≤ d(STx, TSx) ≤ d(T2x, S2x) for any x∈ X.  References [1] Dass K.M. and Vishwanathan, Naik, K. Common fixed point theorem  for commuting maps on a metric space, Proc. Amer. Math. Soc. 77(1979), 369. [2] Fisher, B. Common fixed point theorem, Indian Jour. Pure Appl. Math. 20(1978), 136.  [3] Fisher, B. Common fixed point of commuting mappings, Bull. Inst. Math. Acad. Sincia 9(1981), 399.  [4] Jungck, G. Commuting maps and fixed points, Amer. Math. Monthly  83(1976), 261.  [5] Khan, M.S. and Imdad, M. Fixed point theorems for a class  of  mappings, Indian Jour. Pure Apll. Math.14(1983), 1220.  [6] Lohani P.C. and Badshah, V.H. Common fixed point and weak** commuting mappings, Bull. Cal. Math. Soc. 87(1995), 289.  [7] Pathak, H.K. Weak* commuting mappings and fixed points, Indian Jour. Pure Appl. Math. 17(1986), 201.  [8] Rhoades, B.E. and Seesa, S. Common fixed point theorems for three mappings under a weak commutativity condition, Indian Jour. Pure Appl. Math. 17 (1986), 47.  [9] Seesa, S. On a weak commutativity condition of mappings in a fixed point consideration, Publ. Inst. Math. 32(1982), 149.  [10] Singh, S.L. On common fixed points of commuting mappings, Math. Sem. Notes,  Kobe Univ. 5(1977), 131.   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.  More information about the publisher can be found in the IISTE’s homepage:  http://www.iiste.org  CALL FOR JOURNAL PAPERS The IISTE is currently hosting more than 30 peer-reviewed academic journals and collaborating with academic institutions around the world.  There’s no deadline for submission.  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Common Fixed Point and Weak** Commuting Mappings

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Common Fixed Point and Weak** Commuting Mappings

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