ISSN(Online): 2319-8753 ISSN (Print) : 2347-6710 International Journal of Innovative Research in Science, Engineering and Technology (An ISO 3297: 2007 Certified Organization) Vol. 4, Issue 9, September 2015 Copyright to IJIRSET DOI:10.15680/IJIRSET.2015.0409100 9002 Some generalization on Fixed Point Theorems Related to Fuzzy Metric Spaces Mahesh Kumar Verma, Manoj Kumar Jha, Vikas Kumar Mishra Dept. of Mathematics, Govt GB College Hardibazar, Korba, India Dept. of Mathematics, Rungta Engineering College, Raipur, India Dept. of Mathematics, Rungta College of Engineering & Technology, Raipur, India ABSTRACT: In the present paper some generalization on fixed point and common fixed point theorems in complete Fuzzy 4-metric spaces are established which are motivated by Gahler [13-15], Sharma, Sharma and Isekey [30],Sharma , S.[31], Shrivastava R., Singhal J.[36]. KEYWORDS: Fuzzy metric spaces, fuzzy 2- metric spaces, fuzzy 3- metric spaces, fuzzy 4- metric spaces fixed point, Common fixed point. I. INTRODUCTION In 1965, the concept of fuzzy sets was introduced by Zadeh [36]. After that many authors have expansively developed the theory of fuzzy sets and applications. Especially, Deng [8], Erceg [10], Kaleva and Seikhala [23], Karamosil and Michalek [25], have introduced the concept of fuzzy metric spaces in different ways. Recently, many authors [1,6,11,17,20,21,22,27,28,31,32 ] have also studied the fixed point theory in the fuzzy metric spaces and [2,3,4,5,19,26,33] have studied for fuzzy mappings which opened an avenue for further development of analysis in such spaces and such mappings. Consequently in due course of time some metric fixed point results were generalized to fuzzy metric spaces by various authors. Gahler in a series of papers [13, 14, and 15] investigated 2-metric spaces. Sharma, Sharma and Iseki [30] studied for the first time contraction type mappings in 2-metic space. We [34, 35] have also worked on 2-Metric spaces and 2- Banach spaces for rational expressions. We know that 2 and 3-metric space is a real valued function of a point triples on a set X, which abstract properties were suggested by the area function in Euclidean spaces. Now it is natural to expect 4-Metric space, which is suggested by the volume function. II. SOME FIXED POINT THEOREMS IN FUZZY 2-METRIC SPACE Definition (3 A): A binary operation *: [0, 1] x [0,1] x [0,1] → [0,1] is called a continuous t-norm if ([0,1],*) is an abelian topological monodies with unit 1 such that  ଵ ∗ ଵ ∗ ଵ ≥ ଶ ∗ ଶ ∗ ଶ whenever  ଵ ≥ ଶ ,  ଵ ≥ ଶ ,  ଵ ≥ ଶ , for all  ଵ ,  ଶ ,  ଵ ,  ଶ ,and  ଵ ,  ଶ are in [0,1]. Definition (3 B): The 3-tuple (X, M, *) is called a fuzzy 2-metric space if X is an arbitrary set, * is continuous t-norm and M is fuzzy set in  ଷ × [0, ∞) satisfying the followings ( ܯܨ ′ − 1): ܯ( ݔ, ݕ, ݖ,0) = 0 ( ܯܨ ′ − 2): ܯ( ݔ, ݕ, ݖ, ݐ) = 1, ∀ ݐ≻ 0, ݔ⇔= ݕ ( ܯܨ ′ − 3): ܯ( ݔ, ݕ, ݐ)= ܯ( ݔ, ݖ, ݕ, ݐ)= ܯ( ݕ, ݖ, ݔ, ݐ), Summary about three variable ( ܯܨ ′ − 4): ܯ( ݔ, ݕ, ݖ, ݐ ଵ , ݐ ଶ , ݐ ଷ ) ≥ ܯ( ݔ, ݕ, ݑ, ݐ ଵ ) ܯ∗( ݔ, ݑ, ݖ, ݐ ଶ ) ܯ∗( ݔ, ݕ, ݖ, ݐ ଷ ) ( ܯܨ ′ − 5): ܯ( ݔ, ݕ, ݖ):[0,1) → [0,1]  ݏ ݐ݋ݐݑ݋ݏݑ, ∀ ݔ, ݕ, ݖ, ݑ ߝ, ݐ ଵ , ݐ ଶ , ݐ ଷ ≻ 0