### Common fixed point theorems for multi-valued contractions

S. Sedghi, N. Shobe
2007 International Mathematical Forum
In this paper, a common fixed point theorem for weakly compatible maps in fuzzy metric spaces is proved. Mathematics Subject Classification: 54E40; 54E35; 54H25 Keywords: Common fixed points;weakly compatible mappings; Complete fuzzy metric space 2. * is continuous, 3. a * 1 = a for all a ∈ [0, 1], 4. a * b ≤ c * d whenever a ≤ c and b ≤ d, for each a, b, c, d ∈ [0, 1]. Two typical examples of continuous t-norm are a * b = ab and a * b = min(a, b). Definition 1.2. A 3-tuple (X, M, * ) is called
more » ... X, M, * ) is called a fuzzy metric space if X is an arbitrary (non-empty) set, * is a continuous t-norm, and M is a fuzzy set on X 2 × (0, ∞), satisfying the following conditions for each x, y, z ∈ X and t, s > 0, Let (X, M, * ) be a fuzzy metric space . For t > 0, the open ball B(x, r, t) with center x ∈ X and radius 0 < r < 1 is defined by Let (X, M, * ) be a fuzzy metric space. Let τ be the set of all A ⊂ X with x ∈ A if and only if there exist t > 0 and 0 < r < 1 such that B(x, r, t) ⊂ A. Then τ is a topology on X (induced by the fuzzy metric M). This topology is Hausdorff and first countable. A sequence {x n } in X converges to x if and only if M(x n , x, t) → 1 as n → ∞, for each t > 0. It is called a Cauchy sequence if for each 0 < ε < 1 and t > 0, there exits n 0 ∈ N such that M(x n , x m , t) > 1 − ε for each n, m ≥ n 0 . The fuzzy metric space (X, M, * ) is said to be complete if every Cauchy sequence is convergent. A subset A of X is said to be F-bounded if there exists t > 0 and 0 < r < 1 such that M(x, y, t) > 1 − r for all x, y ∈ A. Lemma 1.3.  Let (X, M, * ) be a fuzzy metric space. Then M(x, y, t) is non-decreasing with respect to t, for all x, y in X. Example 1.4. Let X = R. Denote a * b = a.b for all a, b ∈ [0, 1]. For each t ∈]0, ∞[, define M(x, y, t) = t t + |x − y| for all x, y ∈ X.