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Common fixed point theorems for self maps on a fuzzy metric space involving integral type in equalities

K.P.R.Sastry1, G.A.Naidu2, N.Umadevi3
 8-28-8/1, Tamil Street, Chinna Waltair, Visakhapatnam-530017, India1
Department of Mathematics, Andhra university, Visakhapatnam-530003, India2
Department of Mathematics, Raghu Engineering College, Visakhapatnam-531 162, India3
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Abstract

Malhotra [7] proved a common fixed point theorem in fuzzy metric spaces for occasionally weakly compatible mappings with integral type in equality by reducing its minimum value. In this paper we prove a common fixed point theorem in fuzzy metric spaces for occasionally weakly compatible mappings with integral type inequality involving some special type of Lebesgue integrable functions.

Keywords
Fuzzy metric space, occasionally weakly compatible (owc) mappings, common fixed point.
Mathematical Subject Classification (2010): 47H10, 54H25

INTRODUCTION
Fuzzy set was introduced by Zadeh [15].Kramosil and Michalek [6] introduced the notion of a fuzzy metric space, George and Veermani [4]modified the notion of fuzzy metric spaces with the help of continuous t-norms. Grabiec[3],Subramanyam[13], Vasuki[14], Pant and Jha [9]obtained some analogous results proved by Balasubramaniam [1] et al. Subsequently, it was developed extensively by many authors and used in various fields. Jungck [5] introduced the notion of compatible maps for a pair of self maps.George and Veermani [4], Sessa [11] initiated the tradition of improving commutative condition in fixed point theorems by introducing the notion of weak commuting property. Further Jungck and Rhoades [5] gave a more generalized condition defined as compatibility in metric spaces. Jungck and Rhoades [5] also introduced the concept of weakly compatible maps.Malhotra [7] proved a common fixed point theorem in fuzzy metric spaces for occasionally weakly compatible mappings with integral type in equality by reducing its minimum value. In this paper we prove a common fixed point theorem for selfmaps on fuzzy metric spaces for occasionally weakly compatible mappings with integral type inequality involving some special type of Lebesgue integrable functions (Definition [2.1]).
Definition 1.1 :( Zadeh.L.A. [14]) A fuzzy set A in a nonempty set X is a function with domain X and values in 0,1 .
Definition 1.2: ( Schweizer.B. and Sklar. A. [9]) A function ∗ ∶ 0,1 × 0,1 → 0,1 is said to be a continuous t-norm if ∗ satisfies the following conditions:
 
Definition 1.3: (Kramosil. I. and Michalek. J. [5]) A triple (X, M,*) is said to be a fuzzy metric space (FM space, briefly) if X is a nonempty set, * is a continuous t-norm and M is a fuzzy set on X2 × [0, ∞) satisfying the following conditions:
Then M is called a fuzzy metric space on X.
The function M(x, y, t) denotes the degree of nearness between x and y with respect to t.
Definition 1.6: (George.A. and Veeramani.P. [3]) Let (��, ��,∗) be a fuzzy metric space.
Then,
Definition 1.7: (Malhotra.S.K.Naveen Verma and Ravindra Sen [6]) Two self maps f and g of a set ��are occasionally weakly compatible (owc) iff there is a point �� in �� which is a coincidence point of �� and �� at which �� and �� commute.
Malhotra et.al.[7] proved a common fixed point theorem for four self maps on a fuzzy metric space which satisfy an inequality involving a function ∅ ∈ Φ.
Theorem 1.11: (Malhotra et.al.[7]) Let (��, ��,∗) be a complete fuzzy metric space and let F, G, S and T are self–mapping of X. Let the pairs {F,S} and {G ,T} be owc . If there exists �� ∈ (0,1) such that

II. MAIN RESULT
In this section we introduce a special class of Lebesgue integrable function and use this notion to prove a fixed point theorem.
Hence �� is a common fixed point of �� and ��. From (2.2.2) we have ���� = ���� = ��,followingthe above argument follows that ���� = ���� = ��. Thus �� is a common fixed point of ��, ��, �� and ��.
Hence common fixed point of F, G, S and T is unique.
Proof : By (2.3.1)
Hence result follows from theorem (2.2)
Hence result follows from theorem (2.2)

ACKNOWLEDGEMENT
The third author (N.U) is grateful to Raghu Engineering College authorities for giving permission and the management of SITAM for giving facilities to carry on this research.
 

References
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