Some new fixed point results in non-Archimedean fuzzy metric spaces

Abstract. In this paper, we introduce the notions of fuzzy (α, β, φ)-contractive mapping, fuzzy α-φ-ψ-contractive mapping and fuzzy α-β-contractive mapping and establish some results of fixed point for this class of mappings in the setting of non-Archimedean fuzzy metric spaces. The results presented in this paper generalize and extend some recent results in fuzzy metric spaces. Also, some examples are given to support the usability of our results.


Introduction
The concept of fuzzy metric space was introduced in different ways by some authors (see i.e. [1,2]) and further to this, the fixed point theory in this kind of spaces has been intensively studied (see [3][4][5][6][7][8][9][10][11]).Here, we underline as the notion of fuzzy metric space, introduced by Kramosil and Michalek [2] was modified by George and Veeramani [12,13] that obtained a Hausdorff topology for this class of fuzzy metric spaces.Recently, Miheţ [14] enlarged the class of fuzzy contractive mappings of Gregori and Sapena [7] and proved a fuzzy Banach contraction result for complete non-Archimedean fuzzy metric spaces, see also Vetro [15].Now, we briefly describe our reasons for being interested in results of this kind.The applications of fixed point theorems are remarkable in different disciplines of mathematics, engineering and economics in dealing with problems arising in approximation theory, game theory and many others (see [16] and references therein).Consequently, many researchers, following the Banach contraction principle, investigated the existence of weaker contractive conditions or extended previous results under relatively weak hypotheses on the metric space.Motivated by Samet et al. [17], we introduce the class of fuzzy (α, β, ϕ)-contractive mappings, fuzzy α-φ-ψ-contractive mappings and fuzzy α-β-contractive mappings.The reader is referred to [18][19][20] for some discussions and applications on a non-Archimedean metric space and its induced topology.For example, let X be a non-Archimedean metric space, some assumptions on X can allow to extend a group of isometries of X to the group of Mobius transformations on X.Additionally, this result applies when the metric space is a field, that is, the p-adic numbers Q p , and it is known that many metrics arise from valuations on a ring.Also for this, our results can be of interest in such areas of mathematics as algebra, geometry, group theory, functional analysis and topology.In this paper, we give fixed point results for some new classes of fuzzy contractive mappings.Our results substantially generalize and extend several comparable results in the existing literature, in particular we consider a recent result of Shen et al. [21].

Preliminaries
For the sake of completeness, we briefly recall some basic concepts used in the following.Definition 2. A fuzzy metric space in the sense of George and Veeramani is an ordered triple (X, M, ) such that X is a nonempty set, a continuous t-norm and M is a fuzzy set on X × X × (0, +∞) satisfying the following conditions for all x, y, z ∈ X and t, s > 0: Then the triple (X, M, ) is called a fuzzy metric space.If we replace (F4) by (F6) M (x, y, t) M (y, z, s) M (x, z, max{t, s}), then the triple (X, M, ) is called a non-Archimedean fuzzy metric space.Since, (F6) implies (F4) then each non-Archimedean fuzzy metric space is a fuzzy metric space.Definition 3. Let (X, M, ) be a fuzzy metric space (or a non-Archimedean fuzzy metric space).Then (i) a sequence {x n } converges to x ∈ X, if and only if lim n→+∞ M (x n , x, t) = 1 for all t > 0; (ii) a sequence {x n } in X is a Cauchy sequence if and only if for all ∈ (0, 1) and t > 0, there exists n 0 such that M (x n , x m , t) > 1 − for all m, n n 0 ; (iii) the fuzzy metric space (or the non-Archimedean fuzzy metric space) is called complete if every Cauchy sequence converges to some x ∈ X.

Main results
The following theorem is our first result on the existence of fixed points for fuzzy (α, β, ϕ)-contractive mappings.
x n for some n ∈ N, then x = x n is a fixed point for f and the result is proved.Hence, we suppose that x n+1 = x n for all n ∈ N. Since f is an α-admissible mapping and α(x 0 , f x 0 ) = α(x 0 , x 1 ) 1, we deduce that By continuing this process, we get α(x n , x n+1 ) 1 for all n ∈ N ∪ {0}.Similarly, we deduce that β(x n , t) k(t) for all n ∈ N ∪ {0} and all t > 0. Also define τ n (t) = M (x n , x n+1 , t) for all n ∈ N ∪ {0} and all t > 0. From (1) with x = x n−1 and y = x n we get Since ϕ is decreasing, then τ n−1 (t) < τ n (t), that is, the sequence {τ n (t)} is an increasing sequence for all t > 0. Take lim n→+∞ τ n (t) = τ (t).We will show that τ (t) = 1 for all t > 0. Suppose, to the contrary, that 0 < τ (t 0 ) < 1 for some t 0 > 0. Since τ n (t 0 ) τ (t 0 ) and ϕ is continuous, by taking the limit as n → +∞ in (2) with t = t 0 , we obtain which is a contradiction.Hence, τ (t) = 1 for all t > 0. Now, we want show that {x n } is a Cauchy sequence.Assuming it is not true, then there exists ∈ (0, 1) and t 0 > 0 such that for all k ∈ N there exist Assume that m(k) is the least integer exceeding n(k) satisfying the above inequality.Equivalently, M (x m(k)−1 , x n(k) , t 0 ) > 1 − (4) and so, for all k, we get By taking limit as n → +∞ in (5), we deduce that Using the continuity of the function ϕ, by taking the limit as k → +∞ in the above inequality, we get Now, if ϕ(1 − ) = 0 then by (ϕ2) we have = 0, which is a contradiction.Otherwise, we assume that ϕ(1 − ) > 0. Then 1 k(t 0 ), which is a contradiction, since 0 < k(t 0 ) < 1.Thus {x n } is a Cauchy sequence.The completeness of (X, M, ) ensures that the sequence {x n } converges to some z ∈ X, that is, for all t > 0, Since, x n = x n+1 for all n ∈ N ∪ {0}, by (F2), we get 0 < τ n (t) = M (x n , x n+1 , t) < 1 for all t > 0. Hence, there exists a subsequence {x n(r) } of {x n } such that x n(r) = z for all n ∈ N. From (1) with x = x n(r) and y = z, we have Taking the limit as n → +∞ in the above inequality, we have by taking the limit as n → +∞, we obtain www.mii.lt/NAHence, M (z, f z, t) = 1 and so z = f z.Now, we assume that y = f y implies α(y, f y) 1 and β(x, t) < 1 for all x ∈ X and all t > 0. We show that z is the unique fixed point of f .Assume that w = z is another fixed point of f and M (z, w, t) < 1 for all t > 0, then we have which is a contradiction and hence M (z, w, t) = 1 for t > 0, that is, w = z.Definition 7. Let (X, M, ) be a non-Archimedean fuzzy metric space and f : X → X be an α-admissible mapping.Also, suppose that ψ, φ : [0, 1] → [0, 1] are two continuous functions such that ψ is decreasing, ψ(t) > ψ(1) − φ(1) and φ(t) > 0 for all t ∈ (0, 1).We say, f is a fuzzy α-φ-ψ-contractive mapping if holds for all x, y ∈ X and all t > 0.
For this class of mappings we have the following result of existence and uniqueness of fixed point.Theorem 2. Let (X, M, ) be a complete non-Archimedean fuzzy metric space, α : X × X → [0, +∞), ψ, φ : [0, 1] → [0, 1] as in Definition 7 and f be a fuzzy α-φ-ψcontractive mapping such that the following assertions hold: (i) there exists x 0 ∈ X such that α(x 0 , f x 0 ) 1; (ii) if {x n } is a sequence such that α(x n , x n+1 ) 1 for all n ∈ N and x n → x as n → +∞, then α(x, f x) 1.Then f has a fixed point.Moreover, if y = f y implies α(y, f y) 1, then f has a unique fixed point.
Proof.Define a sequence {x n } in X by x n = f n x 0 = f x n−1 for all n ∈ N. If x n+1 = x n for some n ∈ N, then x = x n is a fixed point for f and the result is proved.Hence, we suppose that x n+1 = x n for all n ∈ N.Then, 0 < M (x n , x n+1 , t) < 1.Since f is an α-admissible mapping and α(x 0 , f x 0 ) = α(x 0 , x 1 ) 1, we deduce that α(x 1 , x 2 ) = α(f x 0 , f x 1 ) 1.By continuing this process, we get α(x n , x n+1 ) 1 for all n ∈ N ∪ {0}.From (6) with x = x n−1 and y = x n , we obtain Since ψ is decreasing, then M (x n−1 , x n , t) < M (x n , x n+1 , t).It follows that {M (x n , x n+1 , t)} is an increasing sequence in (0, 1] and hence there exists l(t) ∈ (0, 1] such that for all t > 0. Let us prove that l(t) = 1 for all t > 0. Suppose that there exists t 0 > 0 such that 0 < l(t 0 ) < 1.By taking the limit as n → +∞ in (7), we have Then φ(l(t 0 )) = 0, which is a contradiction and so l(t) = 1 for all t > 0. We will show that {x n } is a Cauchy sequence.Again, assuming it is not true and proceeding as in the proof of Theorem 1, there exist ∈ (0, 1) and t 0 > 0 such that for all k ∈ N there exist From ( 6) with x = x m(k) and y = x n(k) , we deduce Applying the continuity of the functions φ and ψ, by taking the limit as k → +∞ in the above inequality, we get and so φ(1 − ) = 0, which is a contradiction.Then {x n } is a Cauchy sequence.Since (X, M, ) is a complete non-Archimedean fuzzy metric space, then the sequence {x n } converges to some z ∈ X, that is, for all t > 0, we have Assume that there exists t 0 > 0 such that 0 < M (z, f z, t 0 ) < 1.Then by ( 6) and (ii) we get, www.mii.lt/NA and so Then, {M (x n , x n+1 , t)} is an increasing sequence in (0, 1].Thus there exists l(t) ∈ (0, 1] such that lim n→+∞ M (x n , x n+1 , t) = l(t) for all t > 0. We will prove that l(t) = 1 for all t > 0. By ( 9) we deduce Regarding the property of the function β, we conclude that Next, we will prove that {x n } is a Cauchy sequence.Suppose, to the contrary, that {x n } is not a Cauchy sequence.Proceeding as in the proof of Theorem 1, there exist ∈ (0, 1) and t 0 > 0 such that, for all k ∈ N, there exist From (8) with x = x m(k) and y = x n(k) we deduce Taking the limit as k → +∞ in the above inequality we get and so = 0, which is a contradiction.Then {x n } is a Cauchy sequence.Since (X, M, ) is a complete space, then the sequence {x n } converges to some z ∈ X such that, for all t > 0, we have lim n→+∞ M (x n , z, t) = 1.
Taking the limit as n → +∞ in the above inequality, we have for all t > 0 and then that is, z = f z.Now, we assume that y = f y implies α(y, f y) 1.We show that z is the unique fixed point of f .Suppose that y, z are two fixed points of f and there exists t 0 > 0 such that 0 < M (y, z, t 0 ) < 1.Using (8), we deduce and hence which implies M (y, z, t 0 ) = 1 that is a contradiction.Therefore, M (y, z, t) = 1 for all t > 0 and so y = z.
Finally, we briefly discuss a recent result of Shen et al. [21].Precisely, we consider the following theorem.
Theorem 4. (See [21].)Let (X, M, ) be a complete fuzzy metric space and f be a selfmapping on X. Assume that k : (0, +∞) → (0, 1) is a function and ϕ ∈ Φ.Also, suppose that ϕ M (f x, f y, t) k(t)ϕ M (x, y, t) holds for all x, y ∈ X with x = y and all t > 0. Then f has a unique fixed point.
Shen et al. [21] claimed that if {x n } is not a Cauchy sequence, then there exist 0 < < 1 and two sequences {p(n)} and {q(n)} such that, for all t > 0, we have Here, we note that if (X, d) is a complete metric space, then (X, M, ) is a complete fuzzy metric space if M (x, y, t) = t t + d(x, y) for all x, y ∈ X and t > 0.
Now, if M (x p(n) , x q(n) , t) 1 − holds for all t > 0, clearly, we have which is a contradiction with respect to (12) that implies Thus, the proof is wrong.
On the other hand, by taking α(x, y) = 1 and β(x, t) 2 = k(t) in Theorem 1, we deduce the following correct version of Theorem 4.
Theorem 5. Let (X, M, ) be a complete non-Archimedean fuzzy metric space and f be a self-mapping on X. Assume that k : (0, +∞) → (0, 1) is a function and ϕ ∈ Φ.Also, suppose that ϕ M (f x, f y, t) k(t)ϕ M (x, y, t) holds for all x, y ∈ X with x = y and all t > 0. Then f has a unique fixed point.

Examples
In this section, we will present some examples to illustrate the usefulness of the proposed theoretical results.
Proof.Clearly, (X, M, ) is a non-Archimedean fuzzy metric space.Without loss of generality we assume that x > y.Since that is, α(x, f x)α(y, f y)ϕ M (f x, f y, t) β(x, t)β(y, t)ϕ M (x, y, t) for all x, y ∈ X with x = y and hence f is a fuzzy (α, β, ϕ)-contractive mapping.Then all the conditions of Theorem 1 hold and f has a fixed point (here x = 0 is a fixed point of f ).Moreover, for all x ∈ X, we have α(x, f x) 1 and so the fixed point of f is unique.Also, f x = x/(2(x + 2)) and α(x, y) = 1.By the similar method in the proof of Example 1, we can show that α(x, f x)α(y, f y)ϕ M (f x, f y, t) β(x, t)β(y, t)ϕ M (x, y, t) .
By the similar proof as in Example 2 the conditions (i) and (ii) of Theorem 3 hold.Then by Theorem 3, f has a unique fixed point.