Introduction. Common Fixed Point Theorems For Six Self Mappings Complying Common (E.A.) Property in Fuzzy Metric Space

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1 Inernaional Journal of Mahemaics Trends and Technology (IJMTT) - Volume 56 Number-6 February 08 Common Fixed Poin Theorems For Six Self Mappings Complying Common (E.A.) Propery in Fuzzy Meric Space Rachana Soni and Sanjay Sharma Deparmen of Applied Mahemaics Bhilai Insiue of Technology, Durg (C.G)India. Absrac In his paper, we prove common xed poin heorems for six self mappings in a fuzzy meric spaces complying common (E. A.) propery. In order o prove he resuls, we uilize implici relaions. We also give an example o validae main resuls. Subjec: 00 AMS Classi caion: 47H0, 54H5. Keywords: relaion. Fuzzy Meric Space, Commuing mappings, Common (E.A.) propery, Implici Inroducion The idea of fuzzy mahemaics is iniiaed by Zadeh [] in 965. Las few decades were very dynamic for fuzzy mahemaics and he recen researches winessed he fuzzi caion in almos every area of mahemaics e.g. arihmeic, opology, graph heory, logic, di erenial equaions ec. Fuzzy se heory has pracical applicaions in applied sciences such as image processing, medical sciences, conrol heory, neural work heory, mahemaical modeling. Fuzzy meric space is inroduced by Kramosil and Michalek [0] by generalizing he probabilisic meric space o fuzzy background. George and Veeramani [7] vaguely modi ed he above concep o ge a Hausedor opology on his space. On he oher side, Fixed Poin Theory is one of mos valuable research branches in Nonlinear Analysis. I can be uilized o various di eren absrac meric spaces. In las wo decades xed poin heorems have been largely explored in he seings of fuzzy meric spaces. Jungck [9] launched he noion of compaible maps for a pair of self mappings and Aamri e al [8] generalized he concep of non-compaibiliy by inroducing he noion of propery (E.A.) for self mappings which conained he class of non-compaible mappings in meric spaces, many resuls showed conracion maps saisfying propery(e.a.) in seings of fuzzy meric spaces for insance Kumar e al in [4], Sedghi e al in [] and Imdad e al in [6]. Popa and Turkoglu [] proved some xed poin heorems for hybrid mappings saisfying implici relaions. Popa [9] uilized he family of implici real funcions o prove he exisence of xed poins. rachanasoni007@gmail.com ssharma_bi@yahoo.co.in hp:// Page 48

2 Inernaional Journal of Mahemaics Trends and Technology (IJMTT) - Volume 56 Number-6 February 08 In his paper we prove common xed poin heorems for six self mappings complying common (E.A.) propery on fuzzy meric space employing implici relaion. Our resul exends he several exising resuls in lieraure. Preliminaries De niion..([0]) A binary operaion : [0, ] [0, ] [0, ] is coninuous -norm if sais es he following condiions: (i) is commuaive and associaive, (ii) is coninuous, (iii) a = a for every a [0,], (iv) a b c d whenever a b and c d for all a, b, c, d [0,]. De niion..([]) The -uple (X,M, ) is called a fuzzy meric space(fm-space) if X is an arbirary se, is a coninuous -norm and M is a fuzzy se in X (0, ) saisfying, for every x, y, z X and, s > 0, he following condiions: (FM-)M (x, y, 0) = 0, (FM-) M (x, y, ) = for all > 0 if and only if x = y, (FM-) M (x, y, ) = M (y, x, ), (FM-4)M (x, y, ) M (y, z, s) M (x, z, + s), (FM-5)M (x, y, ) : (0, ) [0, ] is coninuous. Remark ([8]). Le (X, M, ) be a fuzzy meric space. Then M (x, y, ) is nondecreasing on (0, ) for all x, y X. Remark ([]) Le (X, M, ) be a fuzzy meric space. ion on X (0, ) for all x, y X. Then M (x, y, ) is coninuous func- De niion.. ( []) A pair of self mappings (A,B) of a fuzzy meric space (X,M, ) is said o be commuing if M(ABx, BAx, ) = for all x X and > 0. De niion.4. ([0]) A pair of self mappings (A, B) of a fuzzy meric space (X, M, ) is said o be weakly commuing if M (ABx, BAx, ) M (Ax, Bx, ) for all x X and > 0, Remark ([6]) Le (X, M, ) be a fuzzy meric space. If here exiss r (0, ) such ha M (x, y, r) M (x, y, ) for all x, y X and > 0, hen x = y De niion.5. ([9]) A pair of self mappings (A, B) of a fuzzy meric space (X, M, ) is said o be compaible (or asympoically commuing) if for all > 0 limn M (ABxn, BAxn, ) =, De niion.6. ([4]) A pair of self mappings (A, B) of a fuzzy meric space (X, M, ) is said o be weakly compaible if hey commue a he coincidence poins i.e., if Au = Bu for some u = X, hen ABu = BAu. De niion.7. ([5]) A pair of self mappings (A, B) of a fuzzy meric space (X, M, ) is said o have he propery (E.A.) if here exiss a sequence {xn } in X such ha limn Axn = hp:// Page 49

3 Inernaional Journal of Mahemaics Trends and Technology (IJMTT) - Volume 56 Number-6 February 08 limn Bxn = y for some y X. We can see ha compaible as well as noncompaible pairs saisfy he propery (E.A.). De niion.8. ([5]) Two pairs of self mappings (A, P ) and (B, Q) de ned on fuzzy meric space (X, M, ) are said o share common propery (E.A.) if here exis sequences {xn } and {yn } in X such ha, limn Axn = limn P xn = limn Byn = limn Qyn = z for some z X. De niions.9. Le A and B be self mappings of fuzzy meric space (X, M, ), hen a poin x X is said o be a (i) coincidence poin of A and B if Bx = Ax, (ii) xed poin of B if Bx = x. Implici Relaion We uilize implici relaions o prove common xed poin resuls. Le M be he se of all coninuous funcions ψ : [0, ]4 R non-decreasing 4 coordinae variables in rs argumen and saisfying he following condiions: (a) ψ(v,, v, ) 0 v, (b) ψ(v,,, v) 0 v, (c) ψ(v, v,, ) 0 v. Example.: De ne ψ : [0, ] R as ψ(,,, 4 ) = We can see clearly ψ sais es all condiions (a), (b) and (c). Thus ψ M. Main Resuls Theorem.. Le A, B, P, Q, S and T be self mappings of a fuzzy meric space (X, M, ) saisfying he following condiions: (i) The pairs (AP, S) and (BQ, T ) share he common E.A. propery, (ii) For any x, y X and > 0, ψ in M such ha, ψ M (AP x, BQy, ), M (Sx, T y, ), M (Sx, AP x, ), M (T y, BQy, ) 0, (iii) AP = P A and eiher AS = SA or P S = SA, (iv) BQ = QB and eiher BT = T B or QT = T Q, If he range of one of S and T is closed subspace of X hen A, B, P, Q, S and T have unique common xed poin. Proof: The pairs (AP, S) and (BQ, T ) share he common (E.A.) propery i.e. here exis wo sequences {xn } and {yn } in X such ha, hp:// Page 40

4 Inernaional Journal of Mahemaics Trends and Technology (IJMTT) - Volume 56 Number-6 February 08 4 limn AP xn = limn Sxn = limn BQyn = limn T yn = Sv = BQv = z(say) f or z X. Suppose S(X) is closed space of X, hus here exiss a poin v X such ha, z = Sv = BQv Now, we claim ha AP v = z, ake x = v and y = yn, by (ii), ψ M (AP v, BQyn, ), M (Sv, T yn, ), M (Sv, AP v, ), M (T yn, BQyn, ) 0 as n ψ M (AP v, z, ), M (z, z, ), M (z, AP v, ), M (z, z, ) 0, ψ M (AP v, z, ),, M (z, AP v, ) 0, as ψ is non decreasing in he rs argumen, ψ M (AP v, z, ),, M (AP v, z, ), 0 using condiion (a) of implici relaions M (AP v, z, ), hence AP v = z = Sv = BQv Implies he pair (AP, S) has a coinciden poin v similarly by aking y = v and x = xn in (ii), we ge BQv = T v = z Thus AP v = Sv = BQv = T v = z, N ow, AP Sv = SAP v and BQT v = T BQv i.e. AP z = Sz and BQz = T z we now show ha AP z = z, Taking x = z and y = v in (ii) we ge, ψ M (AP z, BQv, ), M (Sz, T v, ), M (Sz, AP z, ), M (T v, BQv, ) 0 ψ M (AP z, z, ), M (AP z, z, ), M (AP z, AP z, ), M (BQv, BQv, ) 0 ψ M (AP z, z, ), M (AP z, z, ),, ) 0 From he condiion (c) of implici relaions, M (AP z, z, ) hence we ge, AP z = z Thus AP z = z = Sz Now since AP = P A, hence z = AP z = P Az f inally P Az = AP z = Sz = z Suppose A commues wih S so AS = SA hus SAz = ASz = z Since AP = P A, we have AP Az = A(P Az) = Az hp:// Page 4

5 Inernaional Journal of Mahemaics Trends and Technology (IJMTT) - Volume 56 Number-6 February 08 5 Now we show ha Az = z, T aking x = Az and y = v in (ii) we ge, ψ M (AP Az, BQv, ), M (SAz, T v, ), M (SAz, AP Az, ), M (T v, BQv, ) 0 ψ M (Az, z, ), M (z, z, ), M (z, Az, ), M (z, z, ) 0 as ψ is non decreasing in he rs argumen, M (Az, z, ),, M (Az, z, ), 0 from he condiion (a) of implici relaion M (Az, z, ) Hence, M (Az, z, ) = hus we have Az = z i.e. Az = Sz = z Similarly if P commues wih S hen by aking, x = P z and y = v we ge, P z = Az = Sz = z we now show ha BQz = z by aking x = v and y = z we ge BQz = z, since B commues wih Q z = BQz = QBz i.e. z = QBz = BQz = T z Now suppose B commues wih T i.e. BT = T B hus we have T Bz = BT z = Bz And if BQ = QB, we have BQBz = B(BQz) = Bz Taking x = v and y = Bz in (ii), we ge, Bz = z Since BQz = z i.e. BQz = QBz = Qz = z Thus, Bz = Qz = z Similarly if Q commues wih T, hen by aking x = v and y = T z, we ge T z = z Therefore, we have proven ha, Az = Bz = P z = Qz = Sz = T z = z Hence z is a common xed poin of six self mappings A, B, P, Q, S and T in X. Uniqueness: Now we'll prove uniqueness of he xed poin, If l is also a common xed poin of A, B, P, Q, S and T hen by aking x = z and y = l in (ii) ψ M (AP z, BQl, ), M (Sz, T l, ), M (Sz, AP z, ), M (T l, BQl, ) 0 Since z and l are xed poins of mappings A, B, P, Q, S andt we have Az = Bz = P z = Qz = Sz = T z = z and Al = Bl = P l = Ql = Sl = T l = l ψ M (z, l, ), M (z, l, ), M (z, z, ), M (l, l, ) 0 hp:// Page 4

6 Inernaional Journal of Mahemaics Trends and Technology (IJMTT) - Volume 56 Number-6 February 08 6 ψ M (z, l, ), M (z, l, ),, 0 From he condiion (c) of implici relaions, M (z, l, ) z = l. Theorem.. Le A, B, P and Q be self mappings of a fuzzy meric space (X, M, ) saisfying he following: (v) The pair (AP, BQ) sais es he common E.A. propery, (vi) Wih anoher pair of self mapping S and T, AB and BQ sais es he condiion(ii). If he range of one of AP and BQ is closed subspace of X hen AB and P Q have common xed poin. Proof: All he condiions of he Theorem. are sais ed ensuring he resuls. Now one needs o prove ha AB and P Q have common xed poin, he pair (AP, BQ) sais es E.A. propery, Hence, limn AP xn = limn BQxn Now ake x = z, y = xn ands = AP, T = BQ in he condiion (ii) we ge, ψ M (AP z, BQxn, ), M (AP z, BQxn, ), M (AP z, AP z, ), M (BQxn, BQxn, ) 0 as n ψ M (AP z, z, ), M (AP z, z, ),, 0 From he condiion (c) of implici relaions M (AP z, z, ) = AP z = z Now we claim ha BQz = z, ake x = xn and y = z in (ii), we ge, ψ M (AP xn, BQz, ), M (AP xn, BQz, ), M (AP xn, AP xn, ), M (BQz, BQz, ) 0 ψ M (z, BQz, ), M (z, BQz, ),, ) 0 From he condiion (c) of implici relaion, we ge, z = BQz AP z = BQz = z, hus AP and BQ have common xed poin z. Example.. Le (X, M, ) is a fuzzy meric space where X = [0, 0] and M (x, y, ) =, f or > 0, def ine ψ : [0, ]4 R, + x y hp:// Page 4

7 Inernaional Journal of Mahemaics Trends and Technology (IJMTT) - Volume 56 Number-6 February 08 7 ψ(,,, 4 ) = (A) Clearly ψ sais es all he condiions (a), (b) and (c) of implici relaions. Now de ne, Ax =, Bx = P x = x, Qx = x x 0 Sx = x > 0 4 x 0 Tx = x > 0 A he coincidence poin, he pairs (AP, S) and (BQ, T ) share he common E.A. propery, Condiion (): x, y 0 LHS of inequaliy (ii) ψ (M (,, ), M (,, ), M (,, ), M (,, ) = ψ,,, ), now from (A) = 0 which sais es (ii) Condiion (): x > 0, y 0 From he LHS of (ii) ψ M (,, ), M ( 4,, ), M ( 4,, ), M (,, ) = ψ,,, Now from (A) + = = which sais es (ii) Condiion (): x 0, y > 0, From he LHS of (ii) = ψ M (,, ), M (,, ), M (,, ), M (,, ) = ψ, M (,, ),, M (,, ) Now from (A) hp:// Page 44

8 Inernaional Journal of Mahemaics Trends and Technology (IJMTT) - Volume 56 Number-6 February = + = which sais es (ii) Condiion (4): x > 0, y > 0 Now de ned mappings and LHS of (ii) give us, = ψ M (,, ), M ( 4,, ), M ( 4,, ), M (,, ) = ψ,,, ( For > 0 ) Hence all four condiions of variables saisfy (ii). Clearly, A, B, P, Q, S and T saisfy he hypohesis of Theorem. and have a unique common xed poin in X. References [] Kang Shin Min e al, Common xed poin heorems of R-weakly commuing mappings in fuzzy meric spaces, Inernaional Journal of Mahemaical Analysis, Vol.9, No. (05) [] Manro Saurabh, A common xed poin heorem for R-weakly commuing maps saisfying propery(e.a.) in fuzzy meric spaces using implici relaion, I.J. Modern Educaion and Compuer Science, (0) [] Sedghi e al, On xed poins of weakly commuing mappings wih propery (E.A.), Journal of Advanced Sudies in Topology, Vol., No. (0). [4] Kumar S. and Fisher B., A common xed poin heorems in fuzzy meric space using propery E.A. and implici relaion, Thai Journal of Mahemaics, 8, No., (00) [5] Abbas M., Alun I. and Gopal D., Common xed poin heorems for noncompaible mappings in fuzzy meric spaces, Bullein of Mahemaical Analysis and Applicaions, vol., no. (009) [6] Imdad M. and Ali J., Jungck's common xed poin heorem and E. A. propery, Aca Mah. Sin.(Engl. Ser.), 4, No. (008) [7] Imdad M. and Ali Javed, Some common xed poin heorems in fuzzy meric spaces, Mahemaical Communicaions (006) 5-6. [8] Aamri M. and Mouawakil, Some new common xed poin heorems under sric conracive condiions, J. Mah. Anal. Appl., 70(00)8-88. [9] Popa V., some xed poin heorems for compaible mappings saisfying an implici relaion, Demonsraio Mah., (999) hp:// Page 45

9 Inernaional Journal of Mahemaics Trends and Technology (IJMTT) - Volume 56 Number-6 February 08 9 [0] Vasuki R., Common xed poins of R-weakly commuing mappings in fuzzy meric spaces, Indian J. Pure. Appl. Mah. 0 (999), [] Vasuki R., A common xed poin heorem in a fuzzy meric space, Fuzzy Ses and Sysems 97 (998), [] Popa V. and D. Turkoglu, Some xed poin heorems for hybrid conracions saisfying an implici relaion, Sud. Cerce Sin. Ser. Mah. Univ. Bacau., 8(998) [] George A and Veeramani P., On some resuls of analysis for fuzzy meric spaces, Fuzzy ses and sysems, vol. 90, no. (997) [4] Pahak H. K., Fixed poin heorems for weak compaible muli-valued and single-valued mappings, Aca Mah. Hungar. 67 (995) [5] Pan R. P., Common xed poin for noncommuing mapping, J. Mah. Anal. Appl., 88(994) [6] Mishra S. N., Sharma N. and Singh S. L., Common xed poins of maps on fuzzy meric spaces, In. J. Mah. Sci. 7 (994)5-58. [7] George A. and Veeramani P., On some resul in fuzzy meric space, Fuzzy Ses and Sysems, Vol. 64, No.. (994), [8] Grabiec M., Fixed poins in fuzzy meric spaces, Fuzzy Ses and Sysems, 7 (989), [9] Jungck G., Compaible mappings and common xed poins, Inerna. J. Mah. Sci.9(986) [0] Kramosil I. and Michalak J., Fuzzy Meric and saisical meric spaces, Kyberneica vol. No. 5 (975) [] Jungck G, Commuing Mappings and Fixed Poins, The American Mahemaical Monhly, Vol. 8, No. 4 (976), pp [] Zadeh L. A., Fuzzy Ses, Inform. Conrol Vol 89 (965) 8-5. hp:// Page 46