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Fixed point theorem for compatible mappings of type (R) in fuzzy metric spaces
1.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 03 Issue: 01 | Jan-2016 www.irjet.net p-ISSN: 2395-0072 _______________________________________________________________________________________ © 2016, IRJET ISO 9001:2008 Certified Journal Page 248 FIXED POINT THEOREM OF COMPATIBLE OF TYPE (R) USING IMPLICIT RELATION IN FUZZY METRIC SPACE Dr Mahendra Singh Bhadauriya1 , Ruchi Gangil2 1 Assistant Professor, ITM Group of Institutions, Gwalior (M.P), India 2 Assistant Professor, ITM Group of Institutions, Gwalior (M.P), India ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstract In this paper authors prove a Common fixed print theorem for compatible mapping of type (R) in Fuzzy metric space by using implicit relation. Our result modifies as well as generalize the results of M.Koireng et al [10]. In [1] Cho et al introduced the concept of semi compatibility in the abstract D-metric space. Recently B. Singh et al [15] introduced the concept of semi compatible mapping in the context of a fuzzy metric space. The earliest important result of compatible mapping was obtained by jungck [7]. Pathak, Chang and Cho [11] introduced the concept of compatible mapping of type (P). In 2004, R.Singh et al [12] introduced the concept by idea of compatible mapping of type (R) by combing the definition of compatible mapping and compatible mapping of type (P). Our aspire in this paper is to define compatible of type R in fuzzy metric spaces and prove some common fixed point theorem of compatible map of type (R) by generalized some interesting result [9]. Key Words: 54H25, 54E50. 1. INTRODUCTION The concept of fuzzy sets was introduced at first by Zadeh [17] which laid the foundation of fuzzy mathematics. George and Veeramani In [5] modified the notion of fuzzy metric space introduced by Kramosil and Michalek [8]. They also obtained that every metric space induces a fuzzy metric spaces. Sessa [16] proved a generalization of commutatively which called weak commutatively. Further Jungek [7] more generalized commutatively which is called compatibility in metric space. Sessa introduced the concept of weakly commuting mappings and Jungck defined the notions of compatible mappings in order to generalize the concept of weak commutativity and showed that weak commuting mappings are compatible but the converse is not true. In recent years, a number of a fixed point theorems and coincidence theorems have been obtained by various authors utilizing this notion. Jungck further weakened the notion of weak compatibility and Jungck and Rhoades further extended weak compatibility. 2. PRELIMINARIES AND DEFINATIONS: Definition 2.1 [5] The 3-tuple (X, M, *) is called a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm and M is a fuzzy set on ),0(2 X which satisfying the following conditions: (1) 0),,( tyxM (2) 1),,( tyxM If and only if yx (3) ),,(),,( txyMtyxM (4) ),,(),,(*),,( stzxMszyMtyxM (5) ]1,0[),0(:,.),( yxM is continuous, for all Xzyx ,, and 0, st Example 2.1.1 : Let ),( dX be a metric space and abba * or },min{* baba . Let ),( ),,( yxdt t tyxM for all Xyx , and 0t then ,*),( MX is a fuzzy metric space. Definition 2.2 [13] A sequence }{ nx in a fuzzy metric space ,*),( MX is said to be a Cauchy sequence if and only if for each , 0t there exists Nx such that 1),,( txxM mn , For all 0, xmn The sequence }{ nx is said to converge to a point x in X if for each 0 , 0t there exists Nx 0 such that 1),,( txxM n for all 0xn A fuzzy metric space ,*),( MX is said to be complete if every Cauchy sequence in it converges to a point in it.
2.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 03 Issue: 01 | Jan-2016 www.irjet.net p-ISSN: 2395-0072 _______________________________________________________________________________________ © 2016, IRJET ISO 9001:2008 Certified Journal Page 249 Definition 2.3 [15] A pair of self mappings (A, S) of fuzzy metric space ,*),( MX is said to be compatible if 1),,( tSAxASxMLim nn n Whenever }{ nx is a sequence in X such that xAxLimSxLim n n n n , for some Xx Definition 2.4 [14] A pair (A, S) of self mappings of a fuzzy metric space is said to be semi compatible if SxASxLim n n whenever }{ nx is a sequence in X such that xSxLimASxLim n n n n So (A, S) is semi compatible and SyAy implies SAyASy by taking yxn }{ and SyAyx . Proposition 2.1 [2] In a fuzzy metric space ,*),( MX limit of a sequence is unique. Proof: Let xxn }{ and yxn }{ then ),,(1),,( tyxMLimtxxMLim n n n n Now )2/,,(*)2/,,(),,( txyMtxxMtyxM nn 1*1),,( tyxM (By taking limit n ) 1),,( tyxM for all 0t Thus yx and hence the limit is unique. Proposition 2.2 [15] If (A, S) is a semi-compatible pair of self maps of a fuzzy metric space ,*),( MX and S in continuous then ),( SA is compatible. Definition 2.5 [9] Self mappings T and S of a fuzzy metric space (X, M, ∗) is said to be compatible of type (R) if 1),,( tTxSTxMLim nn n For all t > 0 1),,( tTTxSSxMLim nn n For all t > 0 Whenever { nx } is a sequence in X such that tSxLimAxLim n n n n For some z ∈ X. Lemma 2.1 [15] let (X, M, *) be a fuzzy metric space If there exists k ϵ X such that M(x, y, k t) ≥ M(x, y, t/kn) for positive integer n taking limit as n → ∞, M(x, y, k/t) ≥ 1 and hence x = y. Lemma 2.2 [14] The only t-norm * satisfying r*r ≥ r for all r∈[0,1] is the minimum t-norm, that is, a* b = min {a , b} for all a, b ∈ [0,1] Lemma 2.3 [13] :- Let (X, M, *) be a fuzzy metric space, if Proposition 2.3 [9] Let (X, M, *) be a fuzzy metric space and let T and S be Compatible mappings of type(R) and T z = S z for some z ∈ X then TTz = TSz = STz = SSz. Proposition 2.4 [9] Let (X , M , * ) be a fuzzy metric space and let A and S be Compatible mappings of type(R) and let T nx , S nx → z as n →∞ for some z∈ X. Then (I) 1),,( zTzSTxMLim n n for if S is continuous at t (ii) 1),,( zSzSTxMLim n n for if T is continuous at t (iii) TSz = STz and Tz=Sz if T and S are continuous at z. A Class of Implicit Relation – Let be the set of all real and continuous from R5 ]1,0[: satisfying the following conditions: (A-1) Is non-increasing in second, third, fourth and fifth argument (A-2) 0),,,,( vvuvu vu 0),,,,( vvvvu vu Example:- },,,.{),,,,( 5432154321 ttttMaxtttttt 3. MAIN RESULT THEOREM 3.1 Let A, B, S and T is self mappings of a complete fuzzy metric space ,*),( MX with continuous t-norm defined by ]1,0[},{}*min{* bababa satisfying the following conditions: (i) )(XA )(XT , )(XB )(XS (ii) One of A, B, S and T are continuous. (iii) Pairs (A, S) and (B, T) are compatible of type (R) (iv) Some )1,0(k such that for all 0,, tXyx ),,,(),,,(),,,( tAxSxMtTySxMktByAxM ),,(),,,( tAxTyMktByTyM ) 0 (v) 1),,(,, tyxMXyx as t Then A, B, S and T have a unique common fixed point. Proof: Let Xx 0 be any point as )()( XTXA and )()( XBXS , Xx 1 and Xx 2 such that 10 TxAx and 21 SxBx . Inductively we construct a sequence }{ ny in X such that 12212 nnn TxAxy And 221222 nnn SxBxy ; )( 22 nn Sxy n = 0, 1 With 122 , nn xyxx using contractive condition, we get
3.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 03 Issue: 01 | Jan-2016 www.irjet.net p-ISSN: 2395-0072 _______________________________________________________________________________________ © 2016, IRJET ISO 9001:2008 Certified Journal Page 250 ),,(),,,(),,,(( 22122122 tAxSxMtTxSxMktBxAxM nnnnnn 0),,(),,,(, 2121212 tAxTxMktBxTxM nnnn ),,(),,,(),,,(( 1221222212 tyyMtyyMktyyM nnnnnn ),,,( 2212 ktyyM nn 0)),,( 1212 tyyM nn And ),,(),,,(),,,(( 1221222212 tyyMtyyMktyyM nnnnnn 1),,,( 2212 ktyyM nn ) 0 Since is non-increasing and ψ is non decreasing in fifth argument therefore, 0).,(),.,( ),.,(),.,(),.,( 12122212 1221222212 tyyMktyyM tyyMtyyMktyyM nnnn nnnnnn Therefore by (A-2), ),,(),,( 212212 tyyMktyyM nnnn and similarly ),,(),,( 122212 tyyMktyyM nnnn Hence ),,(),,( 11 tyyMktyyM nnnn for all n. Now we show that 1),,( tyyMLim npn n For all p and t > 0 Now ),,( 1 tyyM nn )/,,( 1 ktyyM nn )/,,( 2 2 ktyyM nn .............. 1)/,,( 01 n ktyyM As n kt / as n Thus the result holds for 1p . By induction hypothesis suppose that the result hotels for p = r, now 11*1)2/,,(*)2/,,( ),,( 1 1 tyyMtyyM tyyM rnrnrnn rnn Thus the result holds for 1 rp Hence }{ ny is a Cauchy sequence in X and as X is complete we get Xzyn }{ . Hence zSxzAx nn 22 , ... (I) zBxzTx nn 1212 , .... (II) Let S be continuous and pairs (A, S) are compatible of type(R) we get AS Szx n 2 , SzSSx n 2 From contractive condition we get ),,(),,,(( 122122 tTxSSxMktBxASxM nnnn , ),,(),,,( 121222 ktBxTxMtASxSSxM nnnn 0)),,( 212 tAxTxM nn Taking limit as n we get 0)),,(),,,( ),,,(),,,(),,,(( tzzMktzzM tSzSzMtzSzMktzSzM 0)1,1,1),,,(),,,(( tzSzMktzSzM Since is non increasing and ψ is non decreasing in third, fourth and fifth argument therefore we can write 0)),,(),,,( ),,,(),,,(),,,(( tzSzMtzSzM tzSzMtzSzMktzSzM ),,(),,( tzSzMktzSzM and ),,(),,( tzSzNktzSzN zSz Step II: Put 12, nxyzx in condition (IV), ),,(),,,(),,,(( 1212 tAzSzMtTxSzMktBxAzM nn , 0),,(),,,( 121212 tAzTxMktBxTxM nnn As n 0)),,(, ),,(),,,(),,,(),,,(( tAzzM ktzzMtAzSzMtzSzMktzAzM zSz 0)),,(),,,( ),,,(),,,(),,,(( tAzzMktzzM tAzzMtzzMktzAzM 0)),,(,1),,,(,1),,,(( tzAzMtzAzMktzAzM Since is non-increasing and ψ is non –decreasing in second and fourth argument 0),,(),,,( ),,,(),,,(),,,(( tzAzMtzAzM tzAzMtzAzMktzAzM ),,(),,( tzAzMktzAzM And ),,(),,( tzAzNktzAzN SzzAz Step III: As XuXTXA ),()( zTuAz Put nxx 2 , uy is condition (IV), we have ),,(),,,(),,,(( 2222 tAxSxMtTuSxMktBuAxM nnnn , 0),,(),,,( 2 tAxTuMktBuTuM n Taking Limit n 0),,(),,,( ),,,(),,,(),,,(( tzTuMktBuTuM tzzMtTuzMktBuzM zTuAz 0),,(),,,( ),,,(),,,(),,,(( tzzMktBuzM tzzMtzzMktBuzM 0)1,,,( ,1,1),,,(( ktBuzM ktBuzM Since is non increasing and ψ is non decreasing in second, third and fifth argument then
4.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 03 Issue: 01 | Jan-2016 www.irjet.net p-ISSN: 2395-0072 _______________________________________________________________________________________ © 2016, IRJET ISO 9001:2008 Certified Journal Page 251 0),,(),,,( ),,,(),,,(),,,(( tBuzMktBuzM tBuzMtBuzMktBuzM ),,(),,(( tBuzMktBuzM And ),,(),,(( tBuzNktBuzN zBu And we get zBuTu and as (B, T) is compatible of type R so we get TBuBTu i.e. TzBz Step IV: Take zx and zy in condition (IV), we get ),,,(),,,(),,,(( tAzSzMtTzSzMktBzAzM 0)),,(),,,( tAzTzMktBzTzM 0),,(),,,( ),,,(),,,(),,,(( tzBzMktBzBzM tzzMtBzzMktBzzM 0),,(,1,1),,,(),,,(( tzBzMktBzzMktBzzM Since is non increasing in second, third and fifth argument 0),,(),,,( ),,,(),,,(),,,(( tzBzMtzBzM tzBzMtBzzMktBzzM ),,(),,( tzBzMktBzzM Bz = z Therefore zTzBz And so we get zTzSzBzAz Hence z is a common fixed point A, B, S, and T Uniqueness: Let z and 'z be two common fixed points of the maps A, B, S and T. Then zSzTzBzAz And ''''' zSzTzBzAz Using condition (IV), we get ),,,(),,',(),,',(( tAzSzMtTzSzMktBzAzM 0)),,'(),,','( tAzTzMktBzTzM 0)),,'( ),,','(),,,(),,',(),,',(( tzzM ktzzMtzzMtzzMktzzM 0),,'(,1,1),,',(),,',(( tzzMtzzMktzzM Since is none increasing in third and fourth argument so? 0)),',( ),,',(),,,(),,',(),,',(( ' tzzM tzzMtzzMtzzMktzzM 0)),',( ),,',(),,,(),,',(),,',(( ' tzzM tzzMtzzMtzzMktzzM ),',(),',( tzzMktzzM and 'zz . Hence z is a unique common fixed point maps A, B, S, T. COROLLARY 3.1 Let A, B, S and T be self mappings of a complete fuzzy metric space ,*),( MX with continuous t-norm and co t- norm defined by ]1,0[},{}*min{* bababa Satisfying I to III 0),,(),,,( ),,,(),,,( tAyTxMtAxSxM tTySxMktByAxM Then A, B, S and T have a unique common fixed point. COROLLARY 3.2 Let A, B, S and T be self mappings of a complete fuzzy metric space ,*),( MX with Continuous t-norm defined by ]1,0[},{}*min{* bawherebaba Satisfying i, ii, iii, v of theorem3.1 and there exist some )1,0(k such that for all 0,, tXyx 0),,(),2,,( ),,,(),,,(),,,( tAxTyMtByTyM tAxSxMtTySxMktByAxM Then A, B, S and T have a unique common fixed point. ACKNOWLEDGEMENT Authors are thankful to Dr. P. S. Bisen for their valuable suggestion towards the improvement of this paper. REFERENCES . [1] Y.J Cho, B.K. Sharma, and D.R. Sahu, Semi- compatibility and fixed points, Math. Japonica 42(1), 91-98, 1995. [2] Cho, on common fixed points in fuzzy metric spaces, Inter. Math. Forum 1 (10), 471-479, 2006. [3] Zi- Ke Deng, Fuzzy pseudo metric spaces, J. Math. Anal. Appl.86, 74-95, 1982. [4] M.A, Erceg., Metric spaces in Fuzzy set theory, J. Math. Anal. Appl., 69, 205-230, 1979. [5] A. George and P. Veeramani, On same results of analysis for fuzzy metric spaces, Fuzzy sets and systems 90, 365-368, 1997. [6] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27, 385-389, 1988.
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International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 03 Issue: 01 | Jan-2016 www.irjet.net p-ISSN: 2395-0072 _______________________________________________________________________________________ © 2016, IRJET ISO 9001:2008 Certified Journal Page 252 [7] G. Jungck, Compatible mappings and common fixed points. Internat. J.Math. and Math. Sci., 9(4), 771-779, 1986. [8] Kramosil and J. Machalek, Fuzzy metric and statistical metric spaces, Kybernetika 11, 336- 344, 1975.. [9] M. Koireng et.al. , Common fixed point theorem of Compatibility of type R in fuzzy metric space, Gen.math.Notes, pp 58 –62, May2012. [10] O. Kaleva and S. Seikkala, on fuzzy metric spaces, Fuzzy Sets and Systems 12, 215-229, 1984. [11] Pathak, Chang, Cho, Fixed point theorems for compatible mappings of type (P), Indian journal of Math. 36(2), 151-166, 1994. [12] Y Rohan ,M.R.Singh And Shambhu , Common Fixed Point Of Compatible Mapping Of Type (R) In Banach spaces,Proc.Of ath.Soci.Bhu,20, 77-87, 2004. [13] B.Singh and M.S.Bhadauriya, Common Fixed Point Theorems for Pair of Self Mapping Satisfying Common (E. A) like Property in Fuzzy Metric Spaces, International Journal of Computer Applications 75(16) , 29-32, August 2013. [14] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math.,10 , 314- 334. 1960. [15] B. Singh and M.S Chouhan, Common fixed points of compatible maps in fuzzy metric spaces, fuzzy sets and systems, 115, 471-475, 2000. [16] S. Sessa On weak commutativity condition of mappings in fixed point considerations, publ.inst. Math. Besgrad, 32 (46) 149-153. 1982. [17] L .A Zadeh, Fuzzy sets, Inform and Control, 8, 338-353, 1965.
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