R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec ...
R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec ...
R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec ...
R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec ...
R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec ...
R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec ...
R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec ...
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International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.

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  1. 1. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145 RESEARCH ARTICLE www.ijera.com OPEN ACCESS Fixed Point Theorem for Selfmaps On A Fuzzy 2-Metric Space Under Occasionally Subcompatibility Condition K.P.R. Sastry 1, M.V.R. Kameswari 2, Ch. Srinivasa Rao 3, and R. Venkata Bhaskar 4 1 8-28-8/1, Tamil Street, Chinna Waltair, Visakhapatnam -530 017, India Department of Engineering Mathematics, GIT, Gitam University, Visakhapatnam -530 045, India 3 Department of Mathematics, Mrs. A.V.N. College, Visakhapatnam -530 001, India 4 Department of Mathematics, Raghu Engineering College, Visakhapatnam -531 162, India. 2 Abstract In this paper, we prove a common fixed point theorem for three selfmaps and extend it to four and six continuous selfmaps on a fuzzy 2-metric space, using the notion of occasionally sub compatible map with respect to another map. We show that the result of Surjeet Singh Chauhan and Kiran Utreja [7] follows as a corollary. Mathematical subject classification (2010): 47H10, 54H25 Key words:Fuzzy 2-metric space, coincidence point, weakly compatible maps, occasionally sub compatiblemap w.r.t another map, sub compatible of type A. For 1.Introduction The concept of fuzzy sets was introduced by (i) is commutative and associative L.A.Zadeh [9] in 1965 which became active field (ii) is continuous of research for many researchers. In 1975, (iii) Karmosil and Michalek [5] introduced the concept (iv) of a fuzzy metric space based on fuzzy sets; this notion was further modified by George and Veermani [2] with the help of t-norms. Many are examples of authors made use of the definition of a fuzzy metric continuous t-norms. space due to George and Veermani [2] in proving fixed point theorems in fuzzy metric spaces. Gahler Definition 2.3: (Kramosil. I and Michelek. J [5]) [1] introduced and studied 2-metric spaces in a A triple is said to be a fuzzy metric space series of his papers. Iseki, Sharma, and Sharma [3] (FM space, briefly) if X is a nonempty set, is a investigated, for the first time, contraction type continuous t-norm and M is a fuzzy set on mappings in 2-metric spaces. In this paper we which satisfies the following introduced the notion of occasionally sub conditions: compatible map with respect to another map. Using this notion we prove a common fixed point theorem for three selfmaps and extend it to four and six For and continuous selfmaps on a fuzzy 2-metric space also we show that the result of Surjeet Singh Chauhan (i) and Kiran Utreja [8] follows as a corollary. (ii) 2. Preliminaries We begin with some known definitions and results. (iii) (iv) Definition 2.1 :( Zadeh. L.A [9]) A fuzzy set A in a nonempty set X is a function with domain X and values in . (v) Definition 2.2: (Schweizer.B and Sklar. A [6] ) A function is said to be a continuous t-norm if satisfies the following conditions: www.ijera.com is left continuous. Then is called a fuzzy metric space on X. The function nearness between and denotes the degree of with respect to t. 1139 | P a g e
  2. 2. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145 Example 2.4: (George and Veeramani [2]) Let be a metric space. Define and for all all www.ijera.com (This corresponds to tetrahedron inequality in 2-metric space. The function value may be interpreted as the probability that the area of triangle is less than t.) is left continuous. Example 2.8: Let be a metric space. Define and and , Then is a Fuzzy metric space. It is called the Fuzzy metric space induced by for all Definition 2.5: (Gahler [1]) Let X be a nonempty set. A real valued function on is said to be a 2-metric on X if given distinct elements exists an element such that of and all Then , there Definition 2.9: (Jinkyu Han[4]) Let fuzzy 2-metric space. Then, , . is a 2-metric space. Example 2.6: Let and let the area of the triangle spanned by x, y and z which may be given explicitly by the formula      d  x, y , z   x1 y2 z3  y3z2  x2 y1z3  y3z1  x3 y1z2  y2 z1 , where      x  x1 , x2 , x3 , y  y1 , y 2 , y3 , z  z1 , z 2 , z3 pair  Then is called a 2-metric space. Definition 2.7: ( S.Sharma [7] ) A triple is said to be a fuzzy metric 2-space, if X is a nonempty set, is a continuous t-norm and M is a fuzzy set on which satisfies the following conditions: for and t1 , t2 , t3 , for all if and only if at least two of the three points are equal, , ( symmetry about first three variables) be a A sequence in a fuzzy 2- metric space is said to be convergent to a point if . A sequence in a fuzzy 2- metric space is called a Cauchy sequence, if for any and there exists such that, for all and , when at least two of are equal, The pair is a fuzzy 2- metric space.  A fuzzy 2- metric space in which every Cauchy sequence is convergent is said to be complete. Definition 2.10: Let and be maps from a fuzzy 2-metric space into itself. A point x in X is called a coincidence point of A and B if . Definition 2.11: Two selfmaps and of a fuzzy 2-metric space are said to be weakly compatible if they commute at their coincidence points, that is if for some , then . Definition 2.12: Two selfmaps and of a fuzzy 2-metric space are said to be sub compatible if there exists a sequence in such that Definition 2.13: (Surjeet singh Chauhan and Kiran Utreja [8]) Two selfmaps and of a fuzzy 2-metric space are said to be sub compatible of type A, if there exists a sequence in such that , and t1  t2  t3 , www.ijera.com The following two results are proved in [4]. 1140 | P a g e
  3. 3. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145 Lemma 2.14: For all non-decreasing function. is a Lemma 2.15: Let be a fuzzy 2-metric If there k   0,1 such with exists and Definition 3.3: Let and be selfmaps of a fuzzy 2-metric space . is said to be sub compatible with respect to if there exists a sequence such that that for all space. www.ijera.com then . and further Surjeet Singh Chauhan and Kiran Utreja [6] proved the following Theorem Theorem 2.16: (Surjeet Singh Chauhan and Kiran Utreja [8]) Let and be six self mappings of a fuzzy 2-metric space with continuous t-norm satisfying . Suppose the pairs and are sub compatible of type A having the same coincidence point and ,     M Sx, PQy, z , t   M Sx, ABx, z , t   M PQy , Ty , z , t  M  ABx, PQy , z , t   M  ABx, Ty , z , t   M Sx , Ty , z , kt  Now we state and prove our main results. Theorem 3.4: Let and be three continuous self mappings of a fuzzy 2-metric space with continuous t-norm satisfying . Suppose are occasionally sub compatible with respect to and     Then fixed point in and k   0,1 , t  0 . have a unique common . for all x, y, z  X and some Then and If further then and in . 3. Main Results In this section we first introduce the notions of occasionally sub compatible map weakly sub compatible map and sub compatible map with respect to another map. We use these notions to prove some fixed point theorems. Further we show that the result of Surjeet Singh Chauhan and Kiran Utreja [7] follows as a corollary. Definition 3.1: Let and be selfmaps of a fuzzy 2-metric space . is said to be occasionally sub compatible with respect to , if there exists a sequence such that   (3.4.1) (2.16.1) for all x, y, z  X and some   M Ax, By , z , kt  M By , Sx , z , t  M Ax , Sy , z , t  M Ax , By , z , t k   0,1 , t  0 . have a coincidence point in . and are weakly compatible have a unique common fixed point Proof: We may suppose, without loss of generality, that has at least three elements. Since is occasionally sub compatible with respect to , there exists a sequence in such that and (since A and S are continuous) Thus a is coincidence point of and Also since is occasionally sub compatible with respect to , there exists a sequence in such that Definition 3.2: Let and be selfmaps of a fuzzy 2-metric space . is said to be weakly sub compatible with respect to , if and . Thus b is coincidence point of www.ijera.com and 1141 | P a g e 
  4. 4. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145 Take in  (3.4.1). Then www.ijera.com We have  M  Aa , Bb, z , kt   M  Bb, Sa, z ,t  M  Aa, Sb, z, t  M  Aa, Bb, z, t     M  Aa , Bb, z , kt   M  Bb, Aa, z , t  M  Aa, Bb, z, t M  Aa, Bb, z, t  M  Aa, Bb, z, kt   M  Aa, Bb, z, t  t  0 (since, by given condition,  is minimum t-norm) Therefore  M  Aa, Bb, z, t   1 t  0 and For uniqueness, suppose fixed point of and  Thus for all Aa  Bb Since is fixed point of . is common fixed point of and is another common Then and  Bb, w, z , kt   M  ABb, Bw, z , kt  M   M  w, Bb , z ,t M  Bb , w, z ,t M  Bb , w , z ,t    M  Bw, SBb , z ,t M  ABb , Sw, z ,t M  ABb , Bw, z ,t  Take in  (3.4.1). Then  M  Bb, w, z , kt   M  Bb, w, z , kt  t  0  M  Aa , Bb , z ,kt   M  Bb , Sa , z ,t M  Aa ,Sb , z ,t M  Aa , Bb , z ,t   M  Bb, w, z , kt   1 t  0  M  Aa, Ba, z , kt   M  Ba, Aa, z , t   M  Aa, Aa, z , t   M  Aa, Ba, z , t   M  Aa, Ba, z , kt   M  Ba, Aa, z , t  1  M  Aa, Ba, z , t    M  Aa, Ba, z , kt   M  Aa, Ba, z , t  t  0 Bb  w  M  Aa, Ba, z , t   1 t  0  This completes the proof of Theorem. Aa  Ba Since , Therefore is coincidence point of Take show that Theorem 3.5: Let and mappings of a fuzzy 2-metric space and . in (3.4.1), then, as above we can is coincidence point of and . with continuous t-norm satisfying . Suppose and are continuous and and are occasionally sub compatible with respect to and respectively and Since and are weakly compatible, they commute at their coincidence points. Thus we have, M  Sx, PQy , z , t   M  Sx , ABx , z , t      M  Sx , Ty , z , kt   M  PQy , Ty , z , t   M  ABx , PQy , z , t  M  ABx, Ty , z , t     and Take in (3.4.1). Then   M  Aa , BBb , z ,kt   M  BBb ,Sa , z ,t M  Aa ,SBb , z ,t M  Aa ,BBb , z ,t  M M M  BBb , Aa , z , t   M  Aa , BSb , z , t      M  Aa , BBb , z , t      Aa , BBb, z , kt    M  BBb , Aa , z , t   M  Aa , BBb , z , t      M  Aa , BBb , z , t      Aa , BBb, z , kt     Aa , BBb, z , kt   M  Aa , BBb, z , t  t  0  M  Aa , BBb , z , t   1 t  0 M  Aa  BBb  Bb  BBb Therefore is fixed point of be six self (3.5.1) for all x, y, z  X and some Then in . and k   0,1 , t  0 . have a coincidence point If further and compatible then and common fixed point in . are weakly have a unique Proof: We may suppose, without loss of generality, that has at least three elements. Since is occasionally sub compatible with respect to , there exists a sequence in such that . We have. and Therefore is fixed point of www.ijera.com . 1142 | P a g e
  5. 5. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145 Therefore www.ijera.com is fixed point of . Thus a is coincidence point of Take Also since is occasionally sub compatible with respect to , there exists a sequence in such that M  Sa , PQTb, z , t   M  Sa , ABa , z , t  M  PQTb, TTb, z , t   M  Sa , TTb , z , kt     M  ABa , PQTb, z , t   M  ABa , TTb, z , t   M  Sa , TPQb , z , t   M  Sa , Sa , z , t      M  Sa , TTb , z , kt    M TPQb , TTb , z , t   M  Sa , TPQb , z , t   M  Sa , TPQb , z , t     M  Sa , TTb , z , t   M  Sa , Sa , z , t      M  Sa , TTb , z , kt    M TTb , TTb , z , t   M  Sa , TTb , z , t   M  Sa , TTb , z , t       M  Sa , TTb , z , t   1  1    M  Sa , TTb , z , kt     M  Sa , TTb , z , t   M  Sa , TTb , z , t    and Thus b is coincidence point of Take and in (3.5.1). Then   M Sa , Tb , z , kt  M  Sa , PQb, z , t   M  Sa, ABa, z , t     M  PQb, Tb, z , t   M  ABa, PQb, z , t  M  ABa , Tb, z , t      M  Sa, Tb, z, t   M  Sa, Sa, z, t   M Tb, Tb, z , t      M  Sa, Tb, z, kt     M  Sa, Tb, z, t   M  Sa, Tb, z, t     in (3.5.1). Then  Sa , TTb, z , kt   M  Sa , TTb, z , t  t  0  M  Sa , TTb , z , kt   1 t  0 M  Sa  TTb  M  Sa, Tb, z, kt   M  Sa, Tb, z, t  t  0  Sa  TSa  M  Sa, Tb, z, kt   1 t  0  Sa  Tb Therefore Since and are weakly compatible, they commute at their coincidence points. We have  M  Sa, Tb, z, kt   M  Sa, Tb, z, t  11 M  Sa, Tb, z, t   M  Sa, Tb, z, t  is fixed point of . Thus we have, Therefore is fixed point of . and Thus Take is common fixed point M  SSa , PQb, z , t   M  SSa , ABSa , z , t     M  SSa , Tb , z , kt    M  PQb , Tb , z , t   M  ABSa , PQb , z , t  M  ABSa , Tb, z , t     M  SSa , Tb, z , t   M  SSa , SABa , z , t     M  SSa , Tb , z , kt    M Tb , Tb , z , t   M  SABa , Tb , z , t   M  SABa , Tb, z , t     M  SSa , Tb, z , t   M  SSa , SSa , z , t     M  SSa , Tb , z , kt    M Tb , Tb , z , t   M  SSa , Tb , z , t   M  SSa , Tb, z , t     For uniqueness, suppose fixed point of is another common Then  M  SSa, Tb, z, kt   M  SSa, Tb, z, t  1 1  M  SSa, Tb, z, t   M  SSa, Tb, z, t   M  SSa, Tb, z, kt   M  SSa, Tb, z, t  t  0  M  SSa, Tb, z, kt   1 t  0   of in (3.5.1). Then SSa  Tb SSa  Sa Therefore is fixed point of . We have. www.ijera.com 1143 | P a g e
  6. 6. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145    M Sa , w, z , kt  M SSa , Tw, z , kt  www.ijera.com Also since and commute, the pair is weakly compatible. M  SSa , PQw, z , t   M  SSa , ABSa , z , t     Therefore , from Theorem 3.5, is fixed point of   M  PQw, Tw, z , t   M  ABSa , PQw, z , t  . M  ABSa , Tw, z , t     Now we prove that is fixed point of and  M  Sa, w, z , t   M  Sa, Sa, z , t   M  w, w, z , t       M  Sa, w, z , t   M  Sa, w, z , t   Take   in (3.6.1). Then  M  Sa, w, z , t  1  M  SAa , PQb, z , t   M  SAa , ABAa , z , t        1 M  Sa, w, z , t   M  SAa , Tb , z , kt   M  PQb , Tb , z , t   M  ABAa , PQb , z , t    M  ABAa , Tb, z , t     M  Sa, w, z , t   M  Sa, w, z , kt   M  Sa, w, z , t  t  0 Since M  Sa, w, z , kt   1t  0 Sa  w Theorem 3.6: Let and be six continuous self mappings of a fuzzy 2-metric space with continuous t-norm satisfying . Suppose and are occasionally sub compatible with respect to and respectively  M  SAa , Tb, z , t   M  SAa , SAa , z , t    M Tb, Tb, z , t   M  SAa , Tb, z , t   M  SAa , Tb, z , t     M  SAa , Tb, z , t   1  1   M  SAa , Tb , z , kt     M  SAa , Tb, z , t   M  SAa , Tb, z , t   M  SAa, Tb, z, kt   M  SAa, Tb, z, t  t  0  Now we extend Theorem 3.5 to six continuous selfmaps.  M  Sx , PQy , z , t   M  Sx , ABx , z , t     M  PQy , Ty , z , t   M  ABx , PQy , z , t  M  ABx , Ty , z , t     (3.6.1) for all x, y, z  X and some  k   0,1 , t  0 .  M  SAa, Tb, z, kt   1t  0   SAa  Tb ASa  Sa Therefore is fixed point of If and have a coincidence further , then have a unique common fixed point in and Since and commute, the pair weakly compatible. Therefore, from Theorem 3.5, . www.ijera.com is fixed point of Therefore Therefore is fixed point of . in (3.6.1). Then M  Sa , PQPb, z , t   M  Sa , ABa , z , t     M  Sa , TPb , z , kt   M  PQPb , TPb , z , t   M  ABa , PQPb , z , t  M  ABa , TPb, z , t     . Proof: From Theorem 3.5, we can prove the following a is coincidence point of coincidence point of and and . From Theorem 3.5, we have Take Then point in . commute,  M SAa , Tb , z , kt  This completes the proof of Theorem. M Sx , Ty , z , kt  and Since and commute, so b is Sa  Tb . is is fixed point of 1144 | P a g e
  7. 7. R. Venkata Bhaskar et al Int. Journal of Engineering Research and Applications ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.1139-1145 M  Sa , TPb, z , t   M  Sa , Sa , z , t       M  Sa , TPb , z , kt   M TPb , TPb , z , t   M  Sa , TPb , z , t  M  Sa , TPb, z , t        M Sa , TPb , z , kt  M  Sa , TPb, z , t   1  1     M  Sa , TPb, z , t   M  Sa , TPb, z , t      M  Sa , TPb , z , kt   1t  0   M Sa , TPb , z , kt  M Sa , TPb , z , t t  0  [1] Sa  PTb  ACKNOWLEDGMENT The fourth author (R.Venkata Bhaskar) is grateful to 1. The authorities of Raghu Engineering College for granting necessary permissions to carry on this research and, 2. GITAM University for providing the necessary facilities to carry on this research. References Sa  TPb  Sa  PSa Therefore [2] is fixed point of . [3] From Theorem 3.5, we have is fixed point of [4] Therefore Therefore is fixed point of . and [5] For uniqueness, suppose is another common fixed point of and in . [6] Then [7] Thus is common fixed point of in .    M Sa , w, z , kt  M SSa , Tw, z , kt  M  SSa , PQw, z , t   M  SSa , ABSa , z , t       M  PQw, Tw, z , t   M  ABSa , PQw, z , t  M  ABSa , Tw, z , t      M  Sa, w, z, t   M  Sa, Sa, z, t       M  w, w, z, t   M  Sa, w, z, t       M  Sa, w, z, t    M  Sa, w, z, t  11 M  Sa, w, z, t        M  Sa, w, z, t      M  Sa, w, z, kt   M  Sa, w, z, t  t  0 www.ijera.com [8] [9] Gahler.S., 2-metrische Raume and ihre topologische structure, Math. Nachr. 26(1983),115-148 George,A. and Veeramani, P., On some results in Fuzzy metric spaces, Fuzzy sets and system, 64 (1994), 395-399. Iseki.K, P.L. Sharma, B.K. Sharma, Contrative type mappings on 2-metric space, Math. Japonica 21 (1976), 67-70. Jinkyu Han., A common fixed point theorem on fuzzy 2-metric spaces, Journal of the Chungcheong Mathematical Society, Volume 23, No.4(2010),645-656 Kramosil,I. and Michelek,J., Fuzzy metric and statistical metric, kybernetica 11 (1975), 336-344. Schweizer.B. and Sklar. A, Probabilistic Metric spaces, Pacific J. Math.10 (1960), 313-334 S.Sharma, On fuzzy metric spaces, Southeast Asian Bull. Of Math, 26(2002), no.1.133.145. Surjeet singh Chauhan and Kiran Utreja., A Common Fixed point theorem in fuzzy 2metirc space, Int.J.Contemp.Math.Sciences, Vol.8, 2013, no2, 85-91. Zadeh. L.A, Fuzzy sets, Inform and control, 189 (1965), 338-353.  M  Sa, w, z, kt   1t  0  Sa  w This completes the proof of Theorem. Note 3.7: It is evident that Theorem 2.16 becomes a corollary of Theorem 3.6, because the pairs and are sub compatible of type A implies that and are occasionally sub compatible with respect to and respectively. www.ijera.com 1145 | P a g e

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