Theoretical Mathematics & Applications, vol. 4, no. 4, 04, 9-8 ISSN: 79-9687 (print), 79-9709 (online) Scienpress Ltd, 04 A fixed point theorem on compact metric space using hybrid generalized ϕ - weak contraction L. Shambhu Singh,*, Th. Chhatrajit Singh and M. Krishnanando Singh 3 Abstract In this paper we obtain common fixed points of two self mappings defined on a compact metric space using the hybrid generalized ϕ -weak contractions. Mathematics Subject Classification: 54E45 Keywords: Common fixed point; compact metric space; contractive mapping; hybrid generalized ϕ - weak contraction. Introduction Many mathematical researcher obtained fixed points and fixed point results on contractive mappings viz, Song [, 3, 4], Al-Thagafi and Shahzad [0], Reich [4], Kalinke [5], Paliwal [7] Rhoades [8], Hussain and Jungck [] and D.M.College of Science Imphal, Manipur, India. * Corresponding author. E-mail: lshambhu008@rediffmail.com National Institute of Technology, Manipur, India. 3 D.M.College of Science Imphal, Manipur, India. Article Info: Received: July, 04. Revised: August 30, 04. Published online : November 5, 04.
0 A fixed point theorem on compact metric space using hybrid generalized others. The concept of weak contraction in fixed point theory was introduced by Alber and Guerre Delabriere [6] in 997 for single value mappings in Hilbert space and proved existence of fixed points using ϕ - weak contraction on a complete metric space. He also highlighted the relation between ϕ - weak contraction with that of Boyd and Wong type [] and the Reich type contraction [4]. Qingnian Zhang, Yisheng Song [6] proved the existence of fixed point in complete metric space for generalized ϕ - weak contraction. The aim of this paper is to prove the existence of a unique common fixed point of a hybrid generalized ϕ - weak contraction mappings for a pair a pair of self-mappings in compact metric space. Before proving the main results we need the following definitions for our main results. Definition. Let ( X, d ) be a metric space and f : X X be a mapping f is said to be contractive if for each xy, X there exists k ( 0,) such that d( xy, ) k d( xy, ). Definition. A self mapping g from a metric space ( X, d ) into itself is called a ϕ - weak contraction if for each xy, X there exists a function ϕ: [ 0, ) ( 0, ) such that ϕ ( t) > 0 for all t [ 0, ) and d( gx, gy) d( x, y) ϕ ( d( x, y) ). ϕ 0 = 0 and Definition.3 Let ( X, d ) be a metric space and f : X X be a self-mapping f is said to have a fixed point on X if there exists x X such that fx = x.
L.S. Singh, T.C. Singh and M. K. Singh Example. Let ( X, d), X = R be the usual metric space. For f : R R 3 defined by fx = x x R the fixed points of R are 0, and. This shows that the fixed point, if exists may not be unique. Definition.4 Let S and T be two self-mappings from a metric space ( X, d ) into itself A point x X will be a coincident point of S and T if Sx = Tx. Example. Consider Sx = x, Tx 3 = x defined on (, ) Rd. The points x = 0 and x = are coincident points of the mappings S and T. = = ( 3 ) = 6 and STx S Tx S x x Since = = = 6, therefore STx = TSx TSx T Sx T x x x R and hence S and T are also commutative on R.. Main results Theorem. Let ( X, d ) be a compact metric space and ST, : X X be two mappings such that for all xy, X : (i) S( X ) or T( X ) is a closed subspace of X and.. (..) (ii) d( SxTy, ) C( xy, ) C( xy, ) ϕ (..) where ϕ: [ 0, ) [ 0, ) is a lower semi-continuous with ϕ ( t) > 0 and and, (, ) + (, ) (, ) + (, ) d Sx x d y Ty d x Ty d y S x C( xy, ) = max d( xy, ),, ϕ 0 = 0 (..3) then S and T have coincident points z X which are also the unique common fixed points of the mappings S and T.
A fixed point theorem on compact metric space using hybrid generalized Proof: Let x X be any arbitrary point. There exists point x, x, x 3,... in X such that Inductively, we have, x = Tx, x = Sx, x = Tx,..., 0 3 x = Sx, x = Tx, for all n 0. n+ n+ n+ n Choosing x, n+ xn for x and y when n is odd (, ) = (, ) d x x d Sx Tx n+ n n n ( ) (, ) (, ) C x x ϕ C x x (using (ii)) C x If d( x, x ) > d( x, x ) n+ n n n then C( x, x ) = d( x, x ) n n n+ n Also, d ( x, x ) = d ( Sx, Tx ) n+ n n n n n n n (, ) n xn (, ) (, ) d Sx x + d Tx x d( x x ) n n n n max n, n,, (, ) + (, ) d x Tx d x Sx n n n n (, ) (, ) d x x + d x x ( x ) n+ n n n max d xn, n,, ( ) (, ) (, ) C x x ϕ C x x n n n n = d( x, x ) C( x, x ) n+ n n n (, ) + (, ) d x x d x x n n n n+ ϕ (..4) which is a contradiction. Similarly, we have another contradiction if n is taken an even number. Thus, (, ) (, ) (, ) d x x C x x d x x n 0 (..5) n+ n n n n n
L.S. Singh, T.C. Singh and M. K. Singh 3 Since d( xn, xn ) d( xn, xn ) therefore, the sequence { (, )} d x x n n n 0 is non-increasing on R +, the set of all positive real which is bounded from below i.e., there exists a positive real number r 0 such that Now, lim d x, x = lim C x, x = r (..6) n From (..4), we have, mn, n+ n n n n ( ) n n lim (, ) ϕ r =ϕ C x x n ( C( xn xn ) ) liminf ϕ,. n ϕ lim d x, x lim C x, x liminf C x, x n+ n n n n n n n n i.e., r r ϕ ( r) i.e., ϕ( r) 0 i.e., ϕ ( r) = 0 Thus, r = 0 lim d( x, x ) = 0 (..7) n n+ n To prove that { x n } is a Cauchy sequence. If not, there exists some N a large number for a given > 0 such that (, x ) d x n+ m+ > Now, < d( x, x ) n+ m+ (, ) = d Sx Tx n m (, ) (, ) C x x ϕ C x x n m n m Letting mn,, we have 0 < which is a contradiction and hence { x } is a Cauchy sequence. As S( X ) or T( X ) is a closed subset of (, ) X d which is n
4 A fixed point theorem on compact metric space using hybrid generalized bounded the sequence n converging to some point z X as ( X, d ) { } x has a convergent subsequence x nk as k T X of lim x z. or lim Sx nk+ nk z Sz = z k We shall prove that Tz = z. If not, d z, Tz d Sz, Tx n + C zx +, n k S X is a compact subspace of (, ) + (, ) (, ) + (, ) d z Sz d x Tx d z Tx d x Sz n+ n+ n+ n+ max. d zx, n+,, (, ) + (, ) (, ) + (, ) d z z d z Tz d z Tz d z Sz max. d( zz, ),, (, ) d ( z, Tz) d z Tz max d( zz, ),, (, ) d z Tz which is a contradiction and hence, Tz = z. Thus, Sz = Tz = z z is a common fixed point of the mappings S and T. For uniqueness, let there be another fixed point of the mappings S and T and z such that Sz = Tz = z. To prove z = z d z, z = d Sz, Tz C( zz, ) (, ) + ( ', ') (, ) + (, ) d Sz z d z Tz d z Tz d z Tz max. d( zz, ),, (, ) + (, ) (, ) + (, ) d z z d z z d z z d z z = max. d( zz, ),,
L.S. Singh, T.C. Singh and M. K. Singh 5 = d( zz, ) d( zz, ) < d( zz, ) a contradiction z z = Thus, the fixed point is unique. Theorem. Let ( X, d ) be a compact metric space and ST, : X Xbe two mappings such that for all xy, X. (i) S( X ) or T( X ) is a closed subspace of X and (ii) d( SxTy, ) C( xy, ) ϕ C( xy, ) where ϕ: [ 0, ) [ 0, ) is a lower semi-continuous with ϕ ( t) > 0 and ϕ 0 = 0 and (, ) +ϕ( y, Sx) d x Ty C( xy, ) = max d( xy, ), d( xsx, ), d( yty, ), then S and T have a unique common fixed point z X such that Sz = Tz = z. Theorem.3 Let ( X, d ) be a complete metric space and ST, : X X be two mappings such that for all xy, X d( SxTy, ) C( xy, ) ϕ C( xy, ) where ϕ: ( 0, ) [ 0, ) is a lower semi-continuous with ϕ ( t) > 0 and d ( x, Sx) + d ( y, Ty) d ( x, Ty) + d ( ysx) and C( xy, ) max d( xy, ),, = they S and T have a unique common fixed point z X such that Sz = Tz = z. ϕ 0 = 0 Theorem.4 Let ( X, d ) be a compact metric space and S: X defined on X such that for xy, X. X a mapping
6 A fixed point theorem on compact metric space using hybrid generalized (, ) (, ) ϕ (, ) d Sx Sy C x y C x y where ϕ: [ 0, ) [ 0, ) is a lower semi-continuous function with ( t) ϕ > 0, ϕ 0 = 0 and (, ) + (, ) (, ) + (, ) d s Sx d y Sy d x Sy d y Sx C( xy, ) = max d( xy, ), then there exists a unique point z X such that Sz = z. Proof: The details of the proofs of the Theorems.,.3 and also.4 are omitted. Example. Let X = [ 0,] =, a closed and bounded subset of equipped with the usual metric d( xy, ) = x y consider the mappings S: X T : X X defined by X and For xy, X, Sx = x Tx =, 0 (, ) + (, ) (, ) + (, ) d x Sx d y Ty d x Ty d y Sx C( xy, ) = max d( xy, ),, x x + y x + y x = max x y,, = x y, 0 y x = x+ y x, x y x x = + x y x, x x y
L.S. Singh, T.C. Singh and M. K. Singh 7 ϕ t = t, t > 0 6 Taking the mappings S and T satisfy all the condition of Theorem of. and hence x = 0 is the unique common fixed points of S and T. References [] Kannan R., Some results on fixed points, Bull. Calcutta Math. Soc., 60, (968), 7-76. [] Boyd D.W., Wong T.S.W., on non linear contractions, Proc. Amer. Soc., 0, (969), 458-464. [3] Reich S., Some fixed point problems, Atti. Accad. Naz. Lincei, 57, (974), 94-98. [4] Wong C.S., on Kannan type, Proc. Amer. Math. Sci., 05, III MR 50# 099 Zbl 9-47004, (975). [5] Kalinke A.K., on a fixed point theorem for Kannan type mappings, Math. Taponic.,33(5), (988), 7-73. [6] Alber Y.J., Guerre-Delabriere S., Principles of weakly contractive maps in Hilbert Spaces, in: I Gohberg, Yu lyubich (Eds). New Results in operator Theory, in : Advance and Appl., 98, Birkhauser, Basel (997), 7-. [7] Paliwal V.C., Fixed point theorems in metric spaces, Bull Cal. Math. Soc., 80, (998), 005-008. [8] Rhoades B.E., Some theorems on weakly contractive maps, Non linear Anal., 47, (00), 683-693. [9] Shahzad. M., Invariant approximation, generalised I-contractions and R-sub weakly commuting maps, Fixed Point Theory Appl.,, (005), 79-86. [0] Al. Thagafi M.A., Shahzad M., Non commuting self maps and invariant approximations, Non linear Anal., 64, (006), 778-786.
8 A fixed point theorem on compact metric space using hybrid generalized [] Hussain N., Jungck G., Common fixed point and in variant approximation results for non commuting generalized ( f, g ) - non expansive maps., Journal of Math. Anal. Appl., 3, (006), 85-86. [] Song Y., Coincidence points for non commuting f - weakly contractive mappings, Int. Compute. Appl. Math. (IJCAM), (), (007), 5-57. [3] Song Y., Common fixed points and invariant approximations for generalized (f-g) non-expansion mappings, Common. Math. Anal.,, (007), 7-6. [4] Song Y., Xu S., A note on common fixed points for Banach operator pairs, Int. J. Contemp. Math. Sci.,, (007), 63-66. [5] Shambhu L., Mahendra Y., Singh M.R., Some common fixed point theorem in complete in complete metric spaces IJMSEA, ISSH : 0973-944, (IV), (008), 63-7. [6] Zhang Q., Song Y., Fixed point Theory for generalized ϕ - weak contraction, Applied Mathematics letters,, (009), 75-78.