Common fixed point theorems in L-fuzzy metric spaces

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Abstract

In this paper, at first we prove a common fixed point theorem in L-fuzzy metric space. Secondly, to introduce the concept of compatible mappings of type (P) in L-fuzzy metric space, which is equivalent to the concept of compatible and compatible mappings of type (A) under some appropriate conditions. In the sequel, we derive some relations between these mappings. Then, we prove a coincidence point theorem and a fixed point theorem for compatible mappings of type (P) in L-fuzzy metric space, for four mappings.

Section snippets

Introduction and preliminaries

The notion of fuzzy sets was introduced by Zadeh [18]. Various concepts of fuzzy metric spaces were considered in [6], [7], [12], [13]. Many authors have studied fixed theory in fuzzy metric spaces. The most interesting references are [2], [3], [10], [11], [15], [16].

In the sequel, we shall adopt usual terminology, notation and conventions of L-fuzzy metric spaces introduced by Saadati et al. [17].

Definition 1.1

[10]

Let L=(L,⩽L) be a complete lattice, and U a non-empty set called universe. An L-fuzzy set A on U

The main results

Theorem 2.1

Let {An} be a sequence of mappings Ai of a complete L-fuzzy metric space (X,M,T) which has the property (C), into itself such that, for any two mappings Ai, Aj,M(Aim(x),Ajm(y),αi,jt)⩾LM(x,y,t)for some m; here 0 < αi,j < k < 1 for i, j = 1, 2, … , x, y ∈ X and t > 0. Then the sequence {An} has a unique common fixed point in X.

Proof

Let x0 be an arbitrary point in X and define a sequence {xn} in X by x1=A1m(x0),x2=A2m(x1),…. Then we haveM(x1,x2,t)=M(A1m(x0),A2m(x1),t)⩾LM(x0,x1,t/α1,2),M(x2,x3,t)=M(A2m(x1),A3m(x2),t)⩾LM(x

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