Some Common Fixed Point Theorems using Faintly Compatible Maps in Intuitionistic Fuzzy Metricspace

Transcription

1 Research Article Some Common Fixed Point Theorems using Faintly Compatible Maps in Intuitionistic Fuzzy Metricspace IJAESTR International Journal ISSN(Online): Vol. 4I(1) The Author(s) Kamal Wadhwa Govt. Narmada P.G. College Hoshangabad, M.P. India. Ashlekha Dubey Sant Hirdaram Girls College Bhopal M.P. India. Abstract In present paper we proved Some Common Fixed Point theorems for four self-mappings by using the general contractive condition given by Khichiet al. [7] improvises the result by replacing the occasionally weakly compatible (owc) mappings by the faintly compatible pair of mapping in an Intuitionistic Fuzzy Metric Space. Keywords Intuitionistic Fuzzy Metric Space, Common Fixed Point, Property (E.A),Sub Sequentially Continuity, faintly Compatible maps. Mathematics Subject Classification: 52H25, 47H INTRODUCTION The development of fuzzy sets by Zadeh leads to develop a lot of literature regarding the theory of fuzzy sets its application. A large number of renowned Mathematicians worked with fuzzy sets in different branches of Mathematics. One such is the Fuzzy Metric Space. In 1986, Atanassov [4] introduced studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets which is introduced by Zadeh [13]. In 2004, Park [10] defined the concept of intuitionistic fuzzy metric space with the help of continuous t norms continuous t-conorms. In 2006, Alaca et al. [2] using the notion of intuitionistic fuzzy sets defined the concept of intuitionistic fuzzy metric space with the help of continuous t norms continuous t-co norms as a generalization of fuzzy metric space which is introduced by Kramosilet al. [8]. In 1998, Jungcket al. [6] introduced the notion of weakly compatible mappings showed that compatible mappings are weakly compatible but not conversely. Al-Thagafiet al.[3] introduced the concept of occasionally weakly compatible (owc) mappings which is more general than the concept of weakly compatible mappings. Aamri et al. [1] generalized the concepts of non-compatibility by defining the notion of (E.A) property in metric space. Pant et al. [9] introduced the concept of conditional compatible maps.bishtet al. [5] criticize the concept of occasionally weakly compatible (owc) as follows Under contractive conditions the existence of a common fixed point occasional weak compatibility are equivalent conditions, consequently, proving existence of fixed points by assuming occasional weak compatibility is equivalent to proving the existence of fixed points by assuming the existence of fixed points. Therefore use of occasional weak compatibility is a redundancy for fixed point theoremsunder contractive conditions to removes this redundancy we used faintly compatiblemapping in our paper which is weaker than weak compatibility or semi compatibility. Faintly compatible maps introduced by Bishtet al. [5 ] is an improvement of conditionally compatible maps.using these concepts Wadhwa et al. [11,12] proved some common fixed point theorems. In this paper we prove some common fixed point for four mappings using the concept of faintly compatible pair of mappings in an Intuitionistic fuzzy metric space. Preliminary Notes Definition 2.1 [7].A mapping : [0, 1] [0, 1] [0, 1] is called a continuous t-norm if * is satisfying the following conditions: (i) is commutative associative; Corresponding Author Prof. Kamal Wadhwa, Govt. Narmada P.G. College Hoshangabad, M.P. India wadhwakamal68@gmail.com

2 434 I n t e r n a t i o n a l J o u r n a l V o l. 4 I ( 1 ) (ii) is continuous; (iii) a 1= a for all a [0, 1]; (iv) a b c d whenever a c b c b d for all a,b,c,d [0, 1]. Definition 2.2 [7]. A mapping : [0, 1] [0, 1] [0, 1] is called a continuous t- co norm.if is satisfying the following conditions: (i) is commutative associative; (ii) is continuous; (iii) a 0=a for all a [0, 1]; (iv) a b c d whenever a c b c b d for all a,b,c,d [0, 1] Definition 2.3 [7]: A 5-tuple (X, M, N,, ) is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, * is a continuous t- norm, is a continuous t co norm M, N are fuzzy sets on X 2 (0, ) satisfying the following conditions: (i) M(x, y, t) + N(x, y, t) 1 for all x, y X t >0; (ii) M(x, y, 0) =0 for all x, y X; (iii) M(x, y, t) =1 for all x, y X t > 0 if only if x=y; (iv) M(x, y, t) =M(y, x, t) for all x, y X t > 0; (v) M(x, y, t) M(y, z, s) M(x, z, t + s) for all x, y, z X s, t > 0; (vi) For all x, y X, M(x, y, ): [0, ) [0, 1] is continuous; (vii) lim M(x, y, t)=1 for all x, y X t > 0; (viii) N(x, y, 0) =1 for all x, y X; (ix) N(x, y, t) =0 for all x, y X t > 0 if only if x=y; (x) N(x, y, t) =N(y, x, t) for all x, y X t > 0; (xi) N(x, y, t) N(y, z, s) N(x, z, t + s) for all x, y, z X s, t > 0; (xii) For all x, y X, N(x, y, ): [0, ) [0, 1] is continuous; (xiii) lim N(x, y, t)=0 for all x, y X t > 0; Then (X, M, N,, )is called an intuitionistic fuzzy metric on X. The functions M(x, y, t) N(x, y, t) denote the degree of nearness non-nearness between x y with respect to t respectively. Example 1[7] Let (X, d) is a metric space. Define t-norm a b = ab or a b = Min(a, b) t= co norm a b =Max (a, b) for all x, y X t > 0, M (x, y, t) =,N (, ) (x, y, t) = (, ) (, ) Then (X, M, N,, ) is an intuitionistic fuzzy is a fuzzy metric space.we call this intuitionistic fuzzy metric (M, N) induced by the metric d the stard intuitionistic fuzzy metric. Definition 2.4[7]: Let (X, M, N,, ) be an intuitionistic fuzzy metric space then a) Sequence {x n } in X is said to be Cauchy sequence if, for all t > 0 p > 0, lim M(x, x, t) =1, lim N(x, x, t) =0. b) A sequence {x n } in X is said to be convergent to a point x X if, for all t > 0, lim M(x, x, t) =1, lim M(x, x, t) =0 since are continuous, the limit is uniquely determined from M(x, y, t) * M(y, z, s) M(x, z, t + s) for all x, y, z X s, t > 0 of N(x, y, t) N(y, z, s) N(x, z, t + s) for all x, y, z X s, t > 0. Definition 2.5[7]: An intuitionistic fuzzy metric space (X, M, N, *, ) is said to be complete if only if every Cauchy sequence in X is convergent.

3 435 I n t e r n a t i o n a l J o u r n a l V o l. 4 I ( 1 ) Definition 2.6[7]: Let A B be mappings from an intuitionistic fuzzy metric space (X, M, N,, ) into itself. The maps A B are said to be compatible if, for all t>0, lim M(ABx, BAx, t) =1, lim N(ABx, BAx, t) = 0 whenever {x n } is a sequence in X such that lim n Ax n =lim n Bx n =xfor some x X. Definition 2.7 Let A B be mappings from an intuitionistic fuzzy metric Space (X, M, N,, ) into itself. The maps A B are said to be Conditionally compatible:iff whenever the set of sequence {x n } in X such that lim Ax = lim Bx is non-empty, there exists a sequence {z n }in X Such that lim Az = lim Bz = t,for some t X lim M(ABz, BAz, t) =1, lim N(ABz, BAz, t) = 0 for all t > 0. Faintly compatible[11]iff (A, B) is conditionally compatible A B commute on a non-empty subset of the set of coincidence points, whenever the set of coincidence points is nonempty. Satisfy the property (E.A.)[11]if there exists a sequence {x n } in X such that lim Ax = lim Bx =t for some t X. Sub sequentially continuous[11] if there exists a sequence {x n } in X such that lim Ax = lim Bx = x X satisfy lim ABx =Ax, lim BAx =Bx. Semi-compatible:lim M(ABx, Bx, t) = 1,lim N(ABx, Bx, t) = 0 whenever {x n } is a sequence such that lim n Ax n = lim n Bx n = x, for some x X. Definition 2.8 [7]: Two self mappings A B of an intuitionistic fuzzy metric space (X, M, N,, ) is said to be non-compatible if there exists at list one sequence{x n } Such that lim n Ax n= lim n Bx n =z for some z in X but neither lim n M (ABx n,bax n,t) 1, lim n N (ABx n,bax n,t) 0 or the limit does not exists. Definition 2.9[7]: Let (X, M, N,, ) be an intuitionistic fuzzy metric space. Let A B be self-maps on X. Then a point x in X is called a coincidence point of A B iff Ax=Bx. In this case, w=ax=bx is called a point of coincidence of A B. Definition 2.10 [7]: A pair of self mappings (A, B) of an intuitionistic fuzzy metric Space (X,M,N,, ) is said to be weakly compatible if they commute at their Coincidence points i.e Ax=Bx for some x in X, then ABx=BAx. Lemma 1.[7]: Let (X, M, N,, ) be an intuitionistic fuzzy metric space for all x,y α (0, 1) M(x, y, αt) (M(x,y,t) N(x,y,αt) (N(x,y, t) then x=y. in X, t >0 if there exists a number 2. MAIN RESULTS Theorem (3.1) Let (X, M, N,, ) be a complete intuitionistic fuzzy metric space let P, Q, R S be self mappings of X. If there exist α (0, 1) such that M (Px,Qy, αt) φ Min M(Px, Rx, t), N (Px,Qy, αt) ψ Max N(Px, Rx, t),, M(Qy, Sy, t), N(Qy, Sy, t) (3.1.1)

4 436 I n t e r n a t i o n a l J o u r n a l V o l. 4 I ( 1 ) for all x, y X, t >0, a,b,c >0 where φ, ψ: [0 1] [0, 1] such that φ (t) >t ψ (t) <t for all t [0, 1) respectively. Ifpairs (P, R) (Q, S) satisfies E.A. property with sub sequentially Continuous, faintly compatible map then P, Q, R S have a Unique common fixed point in X. Proof :(P, R) (Q, S) satisfy E.A property which implies that there exist sequences {x n } {y n } in X such that lim Px = lim Rx = t 1 for some t 1 X also lim Qx = lim Sx = t 2 for some t 2 X. Since pairs (P, R) (Q, S) are faintly compatible therefore conditionally compatibility of (P, R) (Q, S) implies that there exist sequences {z n } {z n' } in X satisfying lim Pz = lim Rz = u for some u X, such that M (PRz n, RPz n, t) =1, N (PRz n, RPz n, t) =0 also lim Qz = lim Sz = v for some v X, such that M (QSz n ', SQz n ', t) =1, N (QSz n ', SQz n ', t) =0.As the pairs (P, R) (Q, S) are sub sequentially continuous, we get lim PRz = Pu, lim RPz = Ru so Pu = Ru,also lim QSz = Qv, lim SQz = Sv so Qv = Sv. Since pairs (P, R) (Q, S) are faintly compatible, we get PRu = RPu& So PPu = PRu =RPu = RRu also QSv = SQv& So QQv=QSv=SQv=SSv. Now we show that Pu=Qv. Let x=u y=v in (3.1.1) we have M (Pu,Qv, αt) φ Min M(Pu, Ru, t), φ Min M(Pu, Pu, t), φ Min M(Pu, Pu, t), φ(min{1, M(Pu, Qv, t), 1, }) φ (M(Pu, Qv, t) (M(Pu, Qv, t) N(Pu,Qv, αt) ψ Max N(Pu, Ru, t), ψ Max N(Pu, Pu, t), ψ Max N(Pu, Pu, t), M(Pu, Qv, t), M(Qv, Sv, t) N(Pu, Qv, t), N(Qv, Sv, t), N(Qv, Sv, t) ψ(max{0, N(Pu, Qv, t), 0}) ψ (N(Pu, Qv, t) N (Pu, Qv, t) From the condition of φ ψ we get M (Pu,Qv, αt) (M(Pu,Qv,t) N (Pu,Qv, αt) (N(Pu,Qv, t))., M(Qv, Sv, t), M(Qv, Sv, t), N(Qv, Sv, t)

5 437 I n t e r n a t i o n a l J o u r n a l V o l. 4 I ( 1 ) Then by the Lemma 1 it is clear that Pu=Qv. Now we show that PPu= Pu QQv =Pu. Let x=pu y=v in (3.1.1) We have, M(PPu,Qv,αt) φ Min M(PPu, RPu, t), φ Min M(PPu, RPu, t), φ Min M(PPu, PPu, t), φ(min{1, M(PPu, Qv, t), 1, }) φ (M(PPu, Pu, t) M (PPu,Pu,t) (,, ) (,, ) (,, ) (,, ) (,, ) (,, ) N(PPu,Qv,αt) ψ Max N(PPu, RPu, t), ψ Max N(PPu, RPu, t), ψ(max{0, N(PPu, Qv, t), 0}) ψ (N(PPu, Pu, t) (N(PPu, Pu, t) From the condition of φ ψ we get M(PPu, Qv, t), M(Qv, Sv, t), M(Qv, Sv, t) (,, ) (,, ) (,, ) (,, ) (,, ) (,, ), N(Qv, Sv, t), M(Qv, Sv, t), N(Qv, Sv, t) ψ Max N(PPu, PPu, t), a + b + c N(PPu, Qv, t), N(Qv, Sv, t) a + b + c M (PPu,Pu, αt) M(PPu,Pu,t) N (PPu,Pu, αt) N(PPu,Pu, t)) Then by the Lemma 1 it is clear that PPu=Pu. Now we have to show that Pu=QQv. Let x=u y=qv in (3.1.1). We have, M(Pu,QQv,αt) φ Min M(Pu, Ru, t), φ Min M(Pu, Pu, t), φ Min M(Pu, Pu, t), (,, ) (,, ) (,, ) (,, ) (,, ) (,, ) φ(min{1, M(Pu, QQv, t), 1, }) M(Pu, QQv, t), M(QQv, SQv, t), M(QQv, SQv, t), M(QQv, SQv, t)

6 438 I n t e r n a t i o n a l J o u r n a l V o l. 4 I ( 1 ) φ (M(Pu, QQv, t) M (Pu,QQv,t) N(Pu,QQv,αt) ψ Max N(Pu, Ru, t), (,, ) (,, ) (,, ), N(QQv, SQv, t) ψ Max N(Pu, Pu, t), ψ Max N(Pu, Pu, t), (,, ) (,, ) (,, ) N(Pu, QQv, t), N(QQv, SQv, t) ψ(max{0, N(Pu, QQv, t), 0}) ψ (N(Pu, QQv, t) N (Pu,QQv, t) From the condition of φ ψ we get M (Pu,QQv, αt) (M(Pu,QQv,t) N (Pu,QQv, αt) (N(Pu,QQv, t)) Then by the Lemma 1 it is clear that Pu=QQv. Now we haveppu=rpu=pu,pu=qqv=qpupu=qqv=sqv=spu since Qv=Pu. Hence we havep (Pu) =R (Pu) =Q (Pu) =S (Pu). Let Pu=z thenp (z) =R (z) =Q (z) =S (z) Where z is a common fixed point of P, Q, R, S. Hence the uniqueness of the fixed point holds from (3.1.1)., N(QQv, SQv, t) Corollary (3.2): Let (X, M, N, *, ) be a complete intuitionistic fuzzy metric space let P, Q, there exist α (0, 1) such that R S be self mappings of X. If M (Px,Qy, αt) φ Min M(Px, Rx, t), N (Px,Qy, αt) ψ Max N(Px, Rx, t),, M(Qy, Ry, t), N(Qy, Ry, t) (3.2.1) for all x, y X, t >0, a,b,c >0 where φ, ψ: [0 1] [0, 1] Such that φ (t) >t ψ (t) <t for all t [0, 1) respectively. If pairs (P, R) (Q, R) satisfies E.A. property with sub Sequentially continuous,faintly compatible maps then P, Q R S have a unique common fixed point in X. Theorem (3.3): Let (X, M, N, *, ) be a complete intuitionistic fuzzy metric space let

7 439 I n t e r n a t i o n a l J o u r n a l V o l. 4 I ( 1 ) P, Q, R S be self mappings of X. If there exist α (0, 1) such that M (Px,Qy, αt) φ Min M(Rx, Sy, t), M(Rx, Qy, t), N (Px,Qy, αt) ) ψ Max N(Rx, Sy, t), N(Rx, Qy, t), (3.3.1) (,, ) (,, ) (,, ) (,, ), M (Px, Sy, t), N (Px, Sy, t) for all x, y X, t >0, a,b,c>0 where φ, ψ: [0 1] 4 [0, 1] Such that φ (t, t, 1, t) >t ψ (t, t, 0, t) <t for all t [0, 1) respectively. If pairs (P, R) (Q, S) satisfies E.A. property with sub sequentially continuous, faintly compatible maps then P, Q, R S have a unique common fixed point in X. Proof:(P, R) (Q, S) satisfy E.A property which implies that there exist sequences {x n } {y n } in X such that lim Px = lim Rx = t 1 for some t 1 Xalso lim Qx = lim Sx = t 2 for Rome t 2 X. Since pairs (P, R) (Q, S) are faintly compatible therefore conditionally compatibility of (P, R) (Q, S) implies that there exist sequences {z n } {z n' } in X satisfying lim Pz =lim Rz = u for some u X, Such that M (PRz n, RPz n, t) =1, N (PRz n, RPz n, t) =0 also lim Qz = lim Sz = v for some v X, Such that M (QSz n ', SQz n ', t) =1,N (QSz n ', SQz n ', t) =0. As the pairs (P, R) (Q, S) are sub sequentially continuous, we get lim PRz = Pu, lim RPz = Ru so Pu = Ru also lim QSz = Qv, lim SQz = Sv so Qv = Sv. Since pairs (P, R) (Q, S) are faintly compatible, we get PRu = RPu& so PPu=PRu=RPu=RRu alsoqsv=sqv& so QQv=QSv=SQv=SSv. Now we show that Pu=Qv. Let x=u y=v in (3.3.1) we have M (Pu,Qv, αt) φ Min M(Ru, Sv, t), M(Ru, Qv, t), φ Min M(Pu, Qv, t), M(Pu, Qv, t),, M (Pu, Sv, t) φ(min{m(pu, Qv, t), M(Pu, Qv, t), 1, M (Pu, Sv, t)}) φ(m(pu, Qv, t)) M(Pu, Qv, t) N (Pu,Qv, αt) ψ Max N(Ru, Sv, t), N(Ru, Qv, t), ψ Max N(Pu, Qv, t), N(Pu, Qv, t),, N (Pu, Sv, t) ψ(max{n(pu, Qv, t), N(Pu, Qv, t), 1, N (Pu, Sv, t)}) ψ(n(pu, Qv, t)) (N(Pu, Qv, t) From the condition of φ ψwe get (,, ) (,, ) (,, ) (,, ), M (Pu, Sv, t), N (Pu, Sv, t)

8 440 I n t e r n a t i o n a l J o u r n a l V o l. 4 I ( 1 ) M (Pu,Qv, αt) (M(Pu,Qv,t) N (Pu,Qv, αt) (N(Pu,Qv, t)) Then by the Lemma 1 it is clear that Pu=Qv. Now we show that PPu=Pu Let x=pu y=v in (3.3.1) we have, M(PPu,Qv,αt) φ Min M(RPu, Sv, t), M(RPu, Qv, t), (,, ) (,, ), M (PPu, Sv, t) φ Min M(PPu, Qv, t), M(PPu, Qv, t), φ(min{m(ppu, Qv, t), M(PPu, Qv, t), 1, M (PPu, Sv, t)}) φ(m(ppu, Qv, t)) M(PPu, Qv, t) N(PPu,Qv,αt) ψ Max N(RPu, Sv, t), N(RPu, Qv, t), ψ Max N(PPu, Qv, t), N(PPu, Qv, t),, M (PPu, Sv, t) (,, ) (,, ) ψ(max{n(ppu, Qv, t), N(PPu, Qv, t), 1, N (PPu, Sv, t)}) ψ(n(ppu, Qv, t)) N (PPu,Qv, t) From the condition of ψ φ we get M (PPu,Pu, αt) (M(PPu,Pu,t) N (PPu,Pu, αt) (N(PPu,Pu,t)) Then by the Lemma 1 it is clear that PPu=Pu. Now we have to show that Pu=QQv Let x=u y=qv in (3.3.1) we have, M(Pu,QQv,αt) φ Min M(Ru, SQv, t), M(Ru, QQv, t), φ Min M(Pu, QQv, t), M(Pu, QQv, t), φ(min{m(pu, QQv, t), M(Pu, QQv, t), 1, M (Pu, SQv, t)}) φ(m(pu, QQv, t)) M (Pu,QQv, t) N(Pu,QQv,αt) ψ Max N(Ru, SQv, t), N(Ru, QQv, t),, N (PPu, Sv, t) (,, ) (,, ), M (Pu, SQv, t) (,, ) (,, ), N (PPu, Sv, t), M (Pu, SQv, t), N (Pu, SQv, t)

9 441 I n t e r n a t i o n a l J o u r n a l V o l. 4 I ( 1 ) ψ Max N(Pu, QQv, t), N(Pu, QQv, t), ψ(max{n(pu, QQv, t), N(Pu, QQv, t), 1, N (Pu, SQv, t)}) ψ(n(pu, QQv, t)) N (Pu,QQv, t) From the condition of φ ψ we get, N (Pu, SQv, t) M (Pu,QQv, αt) (M(Pu,QQv,t) N (Pu,QQv, αt) (N(Pu,QQv, t)) Then by the Lemma 1 it is clear that Pu=QQv. Now we have PPu=RPu=Pu,Pu=QQv=QPu Pu=QQv=SQv=SPu since Qv=Pu. Hence we havep (Pu) =R (Pu) =Q (Pu) =S (Pu). Let Pu=z thenp (z) =R (z) =Q (z) =S (z). Where z is a common fixed point of P, Q, R, S. Hence the uniqueness of the fixed point holds from (3.3.1) Reference [1] Aamri,M. Moutawakil, D. El. Some new common fixed point theoremsunder stric contractive conditions, J. Math. Anal. Appl., 270 (2002), [2] Alaca, C.,Turkoglu, D Yildiz, C. Fixed points in intuitionistic fuzzymetric spaces, SmalleritChoas, Solitons& Fractals, 29(5) (2006), [3] Al-Thagafi, M. A Shahzad, N. Generalized I-nonexpansiveselfmaps invariant Approximations,Acta. Math. Sinica, 24(5) (2008), [4] Atanassov, K., Intuitionistic Fuzzy sets, Fuzzy sets system, 20(1986) [5] Bisht, R.K Shahzad, N. Faintly compatible mappings common fixed points, Fixed point theory applications, 2013, 2013:156. [6] Jungck, G. Rhodes, B.E Fixed Point for occasionally weakly compatible mappings, Fixed Point Theory, 7(2006) [7] Khichi,Surendra Singh Singh, Amardeep Intuitionistic Fuzzy Metric spaces Common Fixed Point Theorems Using Occasionally Weakly Compatible (OWC) self-mappings International Journal of Fuzzy Mathematics systems Volume 4, Number 2(2014), pp [8] Kramosil.I Michalek,J. Fuzzy metric statistical metric spaces kybernetica,11 (1975) [9] Pant, R.P Bisht, R.K. Occasionally weakly compatible mappings fixed points. Bull. Belg. Math. Soc. Simon Stevin, 19 (2012), [10] Park, J. H., Intuitionistic fuzzy metric spaces, chaos, Solutions & Fractals 22(2004), [11] Wadhwa, Kamal Bharadwaj, VedPrakash. Some common Fixed Point Theorems Using Faintly Compatible Maps in Fuzzy Metric space. International Journal of Fuzzy Mathematics systems Volume 4, Number 2(2014), pp [12] Wadhwa, Kamal Bharadwaj, VedPrakash. Common Fixed Point Theorems Using Faintly Compatible Mappings in Fuzzy Metric Spaces. Computer Engineering Intelligent Systems Vol.5, No.7, [13] Zadeh, L.A. Fuzzy sets, Inform. Control 8(1965),