Fixed point results for contractive mappings in fuzzy metric spaces and its applications

Abstract

In this paper, we introduce a new fuzzy contraction mapping and prove that such mappings have fixed point in τ-complete fuzzy metric spaces. As an application, we shall utilize the results obtained to show the existence and uniqueness of random solution for the following random linear random operator equation. Moreover, we shall show the existence and uniqueness of the solutions for nonlinear Volterra integral equations on a kind of particular fuzzy metric space.

Keywords

Random fixed point random operator random operator equation contractive mapping fixed point fuzzy metric space non-archimedean fuzzy metric space

1 Introduction

The concept of fuzzy metric spaces was obtained in different ways [5 , 29–32]. To obtain a Hausdorff topology of fuzzy metric spaces, [9 , 18] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [19] and showed that every ordinary metric induces a fuzzy metric in the sense of George and Veeramani [9]

Fixed point theorems are crucial for proving the existence of solutions to a wide range of issues in mathematics, physics, engineering, medicine, and the social sciences. The study of fixed point theorems in fuzzy mathematics was instigated by Weiss [39], Butnariu [4], Singh and Talwar [37], Estruch and Vidal [6], Wang et al. [40], Mihet [27], Qiu et al. [28] and Beg and Abbas [3]. Heilpern [14] developed the fuzzy Banach contraction principle on a complete metric linear space with the d_∞-metric for fuzzy sets and proposed the idea of fuzzy contraction mappings (see, [11 , 42]).

Azam and Beg [2] established common fixed point theorems for a pair of fuzzy mappings in metric linear space that meets Edelstein, Alber, and Guerra-Delabriere type contractive requirements. In fuzzy metric spaces, Rashid [30] provide a generalization of Hicks-type and SB-type contractions. For these new type contraction mappings on fuzzy metric spaces, he proved several fixed point theorems.

We introduce a new fuzzy contraction mapping and show that it has a fixed point in complete fuzzy metric spaces in this study. The results will be used to demonstrate the existence and uniqueness of a random solution for the following random linear random operator problem. We’ll also demonstrate the existence and uniqueness of solutions for nonlinear Volterra integral equations on a certain fuzzy metric space.

This paper’s structure is as follows: We review some concepts and the uniform structure of fuzzy metric spaces in Section 2. We will use the results to establish the existence and uniqueness of a random solution for the following random linear random operator problem in Section 3. The fourth section explains several popular fixed point theorems for Contraction type mappings. Our results expand and extend several existing results in fuzzy metric spaces and probabilistic metric spaces, see [18 , 39].

In the sequel, respectively, N, I, ^R +, R denote natural numbers, integer numbers, positive real numbers and real numbers.

2 Preliminaries

First, let’s review some basic concepts used in the sequel.

Definition 1. [17, 34] Let Δ : [0, 1] × [0, 1] → [0, 1]. Then Δ is said to be t-norm if and only if for all x, y, z ∈ [0, 1], we have

Δ (x, y) = Δ (y, x) (commutativity),

Δ (x, y) ≤ Δ (x, z), if y ≤ z (monotonicity),

Δ (x, Δ (y, z)) = Δ (Δ (x, y) , z) (associativity),

Δ (x, 1) = x.

Definition 2. [34] A binary operation Δ : [0, 1] × [0, 1] → [0, 1] is said to be a continuous t-norm if ([0, 1] , T) is a topological monoid with unit 1 such that Δ (a, b) ≤ Δ (c, d) whenever a ≤ c, b ≤ d for all a, b, c, d ∈ [0, 1].

Definition 3. [19] A triplet (X, M, Δ) is called a fuzzy metric space (briefly, a FM-space)(in the sense of Kramosil and Michalek) if X is an arbitrary (non-empty) set, Δ is a continuous t-norm and M is a fuzzy set on X × X × [0, ∞) such that the following axioms hold:

M (x, y, 0) =0 for all x, y ∈ X,

M (x, y, t) =1 for every t > 0 if and only if x = y,

M (x, y, t) = M (y, x, t) for all x, y ∈ X and t > 0,

M (x, y, .) : [0, ∞) → [0, 1] is left continuous for all x, y ∈ X,

M (x, z, t + s) ≥ Δ (M (x, y, t) , M (y, z, s)) for all x, y, z ∈ X and for all t, s ∈ [0, ∞).

Definition 4. [9] A triplet (X, M, Δ) is called a fuzzy metric space (briefly, a FM-space) if X is an arbitrary (non-empty) set, Δ is 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:

M (x, y, t) >0,

M (x, y, t) =1 if and only if x = y,

M (x, y, t) = M (y, x, t),

M (x, z, t + s) ≥ Δ (M (x, y, t) , M (y, z, s)),

M (x, y, .) : [0, ∞) → [0, 1] is continuous.

If (X, M, Δ) is a fuzzy metric space, we will say that (M, Δ) is a fuzzy metric on X. If we replace (FM-4) by

M (x, z, max {s, t}) ≥ Δ (M (x, y, t) , M (y, z, s)),

then a triplet (X, M, Δ) is called a non-Archimedean fuzzy metric space. Since (FM-4A) implies (FM-4) then each non-Archimedean fuzzy metric space is a fuzzy metric space. We will refer to the fuzzy metric spaces in the sense of George and Veeramani as GV-fuzzy metric spaces.

Example 1. Let (X, d) be an ordinary metric space and let β be a nondecreasing and continuous function from (0, ∞) into (0, 1) such that $lim_{t \to \infty} β (t) = 1$ . Some examples of these functions are $β (t) = \frac{t}{t + 1}, β (t) = 1 - e^{- t}$ , and $β (t) = e^{- \frac{1}{t}}$ . Let Δ (a, b) ≤ ab for all a, b ∈ [0, 1]. For each t ∈ (0, ∞), define M (x, y, t) = [β (t)] ^d(x,y) for all x, y ∈ X. It is easy to see that it is non-Archimedean fuzzy metric space.

Example 2. Let (X, d) be a metric space. Denote by a . b the usual multiplication for all a, b ∈ [0, 1], and let M_d be the function defined on X × X × (0, ∞) by $M_{d} (x, y, t) = \frac{t}{t + d (x, y)}$

Then (X, M_d, .) is a fuzzy metric space called standard fuzzy metric space, and (M_d, .) is called the standard fuzzy metric of d (see [9]).

Let (X, M, Δ) be a non-Archimedean fuzzy metric space. The open ball B_M (x, r, t) for t > 0 with centre x ∈ X and radius r, 0 < r < 1 is defined by: $B_{M} (x, r, t) = {y \in X : M (x, y, t) > 1 - r}$ A subset A of a fuzzy metric space (X, M, Δ) is said to be open if given any point x ∈ A, there exists 0 < r < 1, and t > 0 such that B (x, r, t) ⊆ A.

The family: $\begin{matrix} τ_{M} = {A \subset X : x \in A if only if there exists t > 0 \\ and 0 < r < 1 such that B (x, r, t) \subset A} \end{matrix}$ is a topology on X, τ_M is called the topology on X induced by the fuzzy metric M. (X, τ_M) is a Hausdorff first countable topological space (see [9]).

Definition 5. ([34]) Let (X, M, Δ) be a Menger fuzzy metric space. Then

A sequence {x_n} in X is said to be τ-convergent to x ∈ X, denoted it by x_nxrightarrowτx, if for any ɛ > 0 and λ ∈ (0, 1), there exists a positive integer N₀ = N₀ (ɛ, λ) such that M (x_n, x, ɛ) >1 - λ, whenever n ≥ N₀.

A sequence {x_n} in X is said to be τ-Cauchy if for any ɛ > 0 and λ ∈ (0, 1), there exists a positive integer N₀ = N₀ (ɛ, λ) such that M (x_n, x_m, ɛ) >1 - λ, whenever n, m ≥ N₀.

(X, M) is said to be τ-complete if every τ-Cauchy sequence in X is τ-convergent to some point in X.

Definition 6. [[34]] A mapping T : X → X is said to be τ-continuous if, for any sequence {x_n} in X such that x_nxrightarrowτx, Tx_nxrightarrowτTx.

Definition 7. [[30]] Let (X, M, Δ) be a fuzzy metric space with continuous t-norm Δ. A mapping T : X → X is a contraction mapping (or a SB-contraction mapping) if and only if there is an μ ∈ (0, 1) such that $M (Tx, Ty, t) \geq M (x, y, \frac{t}{μ})$ (1) for all x, y ∈ X and t > 0.

3 Fixed point theorems for contraction type mappings

We present some fixed point theorems for numerous contraction type mappings in FM-spaces in this section.

Definition 8. A mapping ψ : R → ^R + is called a distribution function if it is nondecreasing and left continuous with inf ψ = 0 and sup ψ = 1. We will denote $D$ by the set of all distribution functions. Let H denote the specific distribution function defined by $H (t) = {\begin{matrix} 0, & if t \leq 0; \\ 1, & if t > 0 . \end{matrix}$

Proposition 1. [30] Let(X, M, Δ) be a fuzzy metric space with continuous t-norm Δ. Let T : X → X be a contraction mapping satisfying condition (1). Then either

T has a unique fixed point, or for every

p₀ ∈ X,sup {G_p
₀ (t) : t ≥ 0} <1, where $\begin{matrix} G_{p_{0}} (t) & = & inf {M (p_{0}, p_{m}, t) : p_{m} = \\ {Tp}_{m - 1}, m \in N} . \end{matrix}$

Definition 9. [[34]] A set A in FM-space (X, M, Δ) is said to be fuzzy bounded if $sup_{s > 0} sup_{t < s} inf_{p, q \in A} M (p, q, t) = 1 .$ (2)

Clearly, it follows that the condition (1) is equivalent to the following condition: $sup_{s > 0} inf_{p, q \in A} M (p, q, t) = 1 .$ In what follows, we shall adopt the following notations:

Definition 10. Let (X, M, Δ) be a fuzzy metric space and T be a self-mapping on (X, M, Δ). The set $O_{T} (x; 0, \infty) = {T^{n} x}_{n = i = 0}^{\infty} = {x, Tx, T^{2} x, \dots}$ is called the orbit of x. If for an x ∈ X, every τ-Cauchy sequence in O_T (x ; 0, ∞) is τ-converges to a point in X, then the fuzzy metric space (X, M, Δ) is said to be (x, T)-orbitally τ-complete. Moreover, for any x, y ∈ X and positive integer i, we denote

$O_{T} (x; i, \infty) = {T^{n} x}_{n = i}^{\infty}$ ,

O_T (x, y ; 0, ∞) = O_T (x ; 0, ∞) ∪ O_T (y ; 0, ∞),

O_T (x, y ; i, ∞) = O_T (x ; i, ∞) ∪ O_T (y ; i, ∞), respectively.

Example 3. Let X = [0, ∞) , Δ (a, b) = min {a, b} for all a, b ∈ [0, 1] and let $M (x, y, t) = \frac{t}{t + | x - y |}$ for all x, y ∈ X, t > 0. Define T : X → X by $Tx = {\begin{matrix} \frac{x}{3}, & if 0 \leq x < 1; \\ x + 1, & if 1 \leq x < \infty . \end{matrix}$ for all x ∈ X. If we take 0 ≤ x₀ < 1, then $O_{T} (x_{0}; 0, \infty) = {x_{0}, \frac{x_{0}}{3}, \frac{x_{0}}{3^{2}}, \dots, \frac{x_{0}}{3^{n - 1}}, \dots}$ for n = 1, 2, ⋯ and so (X, M, Δ) is (x₀ ; 0, T)-orbitally τ-complete.

Example 4. Let X = (0, ∞) , Δ (a, b) = min {a, b} for all a, b ∈ [0, 1] and let $M (x, y, t) = \frac{t}{t + | x - y |}$ for all x, y ∈ X, t > 0. Define T : X → X by $Tx = {\begin{matrix} \frac{x}{3}, & if 0 < x < 1; \\ 1, & if 1 \leq x < \infty . \end{matrix}$ for all x ∈ X. If we take 0 < x₀ < 1, then $O_{T} (x_{0}; 0, \infty) = {x_{0}, \frac{x_{0}}{3}, \frac{x_{0}}{3^{2}}, \dots, \frac{x_{0}}{3^{n - 1}}, \dots}$ for n = 1, 2, ⋯. But (X, M, Δ) is not (x₀ ; 0, T)-orbitally τ-complete. Now if we take 1≤ x₀ < ∞, then O_T (x₀ ; 0, ∞) = {x₀, 1, 1, ⋯} and so (X, M, Δ) is (x₀ ; 0, T)-orbitally τ-complete.

Example 5. Let X = [0, ∞) , Δ (a, b) = min {a, b} for all a, b ∈ [0, 1] and let $M (x, y, t) = {\begin{matrix} \frac{1}{1 + max {x, y}}, & if x \neq y; \\ 1, & if x = y . \end{matrix}$ for all x, y ∈ X, t > 0. (X, M, Δ) is non-Archimedean fuzzy metric space. Define T : X → X by T (x) = x/3 for all x ∈ X. If we take x₀ = 1, then $O_{T} (1; 0, \infty) = {1, \frac{1}{3}, \frac{1}{3^{2}}, \dots, \frac{1}{3^{n - 1}}, \dots}$ for n ∈ N and so (X, M, Δ) is (1 ;0, T)- orbitally τ-complete.

Definition 11. [[34]] A function φ : ^R + → ^R + is said to satisfy the condition (Φ) if it is strictly increasing, φ (0) =0, and $lim_{n \to \infty} φ^{n} (t) = \infty$ for all t > 0, where φⁿ (t) denotes the n-th iteration of φ (t).

Lemma 1. [[34]]Let φ (t) satisfy the condition (Φ). Then

φ (t) > t for all t > 0,

for any positive integer n = 1, 2, ⋯, $φ^{n} (t) > φ^{n - 1} (t) > \dots > φ (t) > t, t > 0 .$

Throughout this section, we always assume that (X, M, Δ) is a τ-complete non-Archimedean FM-space, Δ is a continuous t-norm, and φ (t) is a function satisfying the condition (Φ).

Theorem 1.Let T be a τ-continuous self-mapping on (X, M, Δ). Suppose that for each x ∈ X, the orbit O_T (x ; 0, ∞) is fuzzy bounded, and that for any x ∈ X, there exists a positive integer m (x) such that, for all t ≥ 0, $\begin{matrix} inf_{p, q \in O_{T} (x; m (x), \infty)} M (p, q, t) \\ \geq & inf_{p, q \in O_{T} (x; 0, \infty)} M (p, q, φ (t)) . \end{matrix}$ (3) Then T has a fixed point in X, and for any x₀ ∈ X, the sequence {x_n} defined by x_n = Tⁿx₀, n ∈ N, τ-converges to a fixed point of T.

Proof. For any x₀ ∈ X, by assumptions, there exists a positive integer m (x₀) such that $\begin{matrix} inf_{p, q \in O_{T} (x; m (x_{0}), \infty)} M (p, q, t) \geq \\ inf_{p, q \in O_{T} (x_{0}; 0, \infty)} M (p, q, φ (t)), \end{matrix}$ t ≥ 0. Now we choose a sequence {m (j)} of positive integers defined as follows: $m (0) = m (x_{0}), m (j + 1) = m (j) + m (x_{m (j)}),$ j = 0, 1, ⋯, where x_m(j) = T^m(j)x₀ .

By the induction, we can prove the following inequality holds: $\begin{matrix} inf_{p, q \in O_{T} (x_{0}; m (j + 1), \infty)} M (p, q, t) \\ \geq inf_{p, q \in O_{T} (x_{0}; m (j), \infty)} M (p, q, φ (t)) \\ \geq \dots \end{matrix}$ (4) $\begin{matrix} \geq inf_{p, q \in O_{T} (x_{0}; m (0), \infty)} M (p, q, φ^{j + 1} (t)) \\ \geq inf_{p, q \in O_{T} (x_{0}; 0, \infty)} M (p, q, φ^{j + 2} (t)) \\ \geq sup_{s < φ^{j + 2} (t)} inf_{p, q \in O_{T} (x_{0}; 0, \infty)} M (p, q, s) . \end{matrix}$ By using Lemma 1 $lim_{j \to \infty} inf_{p, q \in O_{T} (x_{0}; m (j + 1), \infty)} M (p, q, t) = 1, t \geq 0,$ which implies that {x_n} is a τ-Cauchy sequence in X. By the τ-completeness of (X, M, Δ), we can suppose x_nxrightarrowτx_* ∈ X. By the τ-continuity of T, it is easy to see that x_* is a fixed point of T. This completes the proof. □ Theorem 2. Let T be a τ-continuous self-mapping on (X, M, Δ). Suppose that for each x ∈ X, the orbit O_T (x ; 0, ∞) is fuzzy bounded, and suppose that there exists a positive integer m such that, for any x ∈ X, the following holds: $inf_{p, q \in O_{T} (x; m, \infty)} M (p, q, t)$ (5) $\geq inf_{p, q \in O_{T} (x; 0, \infty)} M (p, q, φ (t)), t \geq 0 .$ Then the conclusion of Theorem 1 still holds.

Lemma 2. Let T be a τ-continuous self-mapping on (X, M, Δ). Then the following conditions are equivalent:

There exists a positive integer m such that, for any x ∈ X, $\begin{matrix} inf_{p, q \in O_{T} (x; m, \infty)} M (p, q, t) \geq \\ inf_{p, q \in O_{T} (x; 0, \infty)} M (p, q, φ (t)), t \geq 0 . \end{matrix}$

There exists a positive integer m such that, for any x ∈ X and any non-negative integer k, $\begin{matrix} M (T^{m} x, T^{m + k} x, t) \geq \\ inf_{p, q \in O_{T} (x; 0, \infty)} M (p, q, φ (t)), t \geq 0 . \end{matrix}$

Proof. The proof is straightforward and so is omitted.

□ Theorem 3. Let T be a τ-continuous self-mapping on (X, M, Δ) Suppose that for each x ∈ X, the orbit O_T (x ; 0, ∞) is fuzzy bounded, and that the condition (i) or (ii) in Lemma 2 is satisfied. Then the conclusion of Theorem 1 still holds.

Theorem 4.Let T be a τ-continuous self-mapping on (X, M, Δ) Suppose that there exist positive integers m, n such that for any x, y ∈ X and any t ≥ 0, $M (T^{m} x, T^{n} x, t) \geq$ (6) $inf_{p, q \in O_{T} (x, y; 0, \infty)} M (p, q, φ (t)), t \geq 0 .$ Then T has a unique fixed point in X, and for any x₀ ∈ X, the sequence {x_n} defined by x_n = Tⁿx₀, n = 1, 2, ⋯, τ-converges to this fixed point.

Proof. Without loss of generality, we can assume that m ≥ n. For any x ∈ X and a non-negative integer k, put y = T^m-n+kx. It follows from (5) that, for all t ≥ 0, we have $\begin{matrix} M (T^{m} x, T^{m + k} x, t) \\ \geq inf_{p, q \in O_{T} (x, T^{m - n + k} x; 0, \infty)} M (p, q, φ (t)) \\ = inf_{p, q \in O_{T} (x; 0, \infty)} M (p, q, φ (t)) . \end{matrix}$ By Theorem 4, T has a fixed point in X, and for any x₀ ∈ X, the sequence {x_n = Tⁿx₀} τ-converges to a fixed point x_* of T.

Suppose that y_* is another fixed point of T. Then for all t ≥ 0, we have $\begin{matrix} M (x_{*}, y_{*}, t) & = M (T^{m} x_{*}, T^{n} y_{*}, t) \\ \geq inf_{p, q \in O_{T} (x_{*}, y_{*}; 0, \infty)} M (p, q, φ (t)) \\ = M (x_{*}, y_{*}, φ (t)) . \end{matrix}$ Repeating the above procedure by the condition (Φ), we obtain $M (x_{*}, y_{*}, t) \geq M (x_{*}, y_{*}, φ^{n} (t)) \to H (t), n \to \infty$ which implies that x_* = y_*. This completes the proof.

□ Theorem 5. Let (X, M, Δ) be a τ-complete non-Archimedean FM-space, Δ be a continuous t-norm satisfying the condition Δ (t, t) ≥ t for all t ∈ [0, 1]. Suppose T : X → X is a contraction mapping, i.e., there exists a constant k ∈ (0, 1) such that, for all t ≥ 0 and x, y ∈ X, $M (Tp, Tq, t) \geq M (p, q, \frac{t}{k}) .$ (7) Then T has a unique fixed point x_*, and for each x₀ ∈ X, the sequence {x_n} defined by x_n = Tⁿx₀, n = 1, 2, ⋯, τ-converges to the point x_* in X.

Proof. Take $φ (t) = \frac{t}{k}, t \geq 0$ , and then it is easy to see that φ (t) satisfies the condition (Φ). Since T is a contraction mapping, obviously, T is τ-continuous. We now prove that under the condition (6) and Δ (t, t) ≥ t, t ∈ [0, 1], for each x ∈ X, O_T (x ; 0, ∞) is fuzzy bounded.

In fact, for each x₀ ∈ X and any λ ∈ (0, 1), by the property of distribution function, there exists a number t₁ = t₁ (λ) >0 such that for all t ≥ t₁, $M (x_{0}, {Tx}_{0}, t) > 1 - λ .$ (8) If k₁ ∈ (k, 1), for any positive integer n > 1, it follows from (6) that $M (x_{0}, T^{n} x_{0}, k_{1} t)$ (9) $\begin{matrix} \geq Δ (M (x_{0}, {Tx}_{0}, (k_{1} - k) t), \\ M ({Tx}_{0}, T^{n} x_{0}, kt)) \\ \geq Δ (M (x_{0}, {Tx}_{0}, (k_{1} - k) t), \\ M (x_{0}, T^{n - 1} x_{0}, t)) \\ \geq min {M (x_{0}, {Tx}_{0}, (k_{1} - k) t), \\ M (x_{0}, T^{n - 1} x_{0}, t)} . \end{matrix}$ From (8), we have $M (x_{0}, T^{n - 1} x_{0}, t) \geq$ (10) $\begin{matrix} min {M (x_{0}, {Tx}_{0}, (k_{1} - k) k_{1}^{- 1} t), \\ M (x_{0}, T^{n - 2} x_{0}, k_{1}^{- 1} t)} . \end{matrix}$ Substituting (9) into (8), simplifying the expression and noting $k_{1}^{- 1} > 1$ , we have $M (x_{0}, T^{n} x_{0}, k_{1} t) \geq$ (11) $min {M (x_{0}, {Tx}_{0}, (k_{1} - k) t), M (x_{0}, T^{n - 2} x_{0}, t)} .$ Repeating the above procedures, we have $M (x_{0}, T^{n} x_{0}, k_{1} t) \geq$ (12) $\begin{matrix} min {M (x_{0}, {Tx}_{0}, (k_{1} - k) t), M (x_{0}, x_{0}, t)} \\ = & M (x_{0}, {Tx}_{0}, (k_{1} - k) t) . \end{matrix}$ Therefore, we have $M (x_{0}, T^{n} x_{0}, t) \geq$ (13) $M (x_{0}, {Tx}_{0}, (k_{1} - k) k_{1}^{- 1} t), n > 1 .$ Combining (7) and (12), since $t \geq max {t_{1}, \frac{k_{1} t_{1}}{k_{1} - k}} = \frac{k_{1} t_{1}}{k_{1} - k}$ , we have M (x₀, Tⁿx₀, t) >1 - λ for all n = 0, 1, 2, ⋯. Hence we have $inf_{q \in O_{T} (x_{0}; 0, \infty)} M (x_{0}, q, t) > 1 - λ$ (14) for all $t \geq \frac{k_{1} t_{1}}{k_{1} - k}$ . In view of arbitrariness of λ ∈ (0, 1), we have $sup_{t > 0} inf_{q \in O_{T} (x_{0}; 0, \infty)} M (x_{0}, q, t) = 1 .$ (15) However, for any given p₀, q₀ ∈ O_T (x ; 0, ∞), we have $\begin{matrix} M (p_{0}, q_{0}, t) \geq Δ (M (p_{0}, x_{0}, \frac{t}{2}), \\ M (x_{0}, q_{0}, \frac{t}{2})) \end{matrix}$ (16) $\begin{matrix} \geq Δ (inf_{q \in O_{T} (x_{0}; 0, \infty)} M (p_{0}, x_{0}, \frac{t}{2}), \\ inf_{q \in O_{T} (x_{0}; 0, \infty)} M (x_{0}, q_{0}, \frac{t}{2})) \\ \geq inf_{q \in O_{T} (x_{0}; 0, \infty)} M (x_{0}, q, \frac{t}{2}) . \end{matrix}$ By the arbitrariness of p₀, q₀ ∈ O_T (x₀ ; 0, ∞), it follows from (14) and (15) that $\begin{matrix} sup_{t > 0} inf_{p, q \in O_{T} (x_{0}; 0, \infty)} M (p, q, t) \\ \geq & sup_{t > 0} inf_{q \in O_{T} (x_{0}; 0, \infty)} M (x_{0}, q, \frac{t}{2}) = 1 . \end{matrix}$ This shows that, for any x₀ ∈ X, the orbit O_T (x₀ ; 0, ∞) is fuzzy bounded. Thus the conclusion of Theorem 5 can be obtained from Theorem 4. This completes the proof. □ Theorem 6.Let (X, M, Δ) be a τ-complete non-Archimedean FM-space with a t-norm Δ satisfying $sup_{0 \leq t \leq 1} Δ (t, t) = 1$ . Let T : X → X be a mapping satisfying the following conditions:

for each x ∈ X, $sup_{t \in R} G_{x} (t) = 1$ , where $G_{x} (t) = inf_{n \in I^{+}} M (x, T^{n} x, t),$

for each x ∈ X, there exists a positive integer m (x) ∈ ^I + such that for any y ∈ X and t ∈ ^R +, $\begin{matrix} M (T^{m (x)} x, T^{m (x)} y, t) \geq min {M (x, y, φ (t)), \\ M (x, T^{m (x)} x, φ (t)), \\ M (x, T^{m (x)} y, φ (t))}, \end{matrix}$ (17) where φ (t) is a left-continuous function satisfying the condition (Φ). Then

for any x₀ ∈ X, the sequence {x_n} defined by x_n = T^m(x_n-1)x_n-1, n = 1, 2, ⋯, τ-converges to some point x_* in X,

In addition, if there exists t^* ∈ ^R + such that M (x_*, T^{m
_*}x_*, t^*) =1, then x_* is the unique fixed point of T in X, and the iterative sequence {Tⁿx₀} τ-converges to x_*.

Proof. (a) We prove that {x_n} is a τ-Cauchy sequence in X. Letting m (x_i) = m_i, i = 1, 2, ⋯, for any n ∈ ^I +, by (16), we have $\begin{matrix} M (x_{i}, T^{n} x_{i}, t) \geq min {M (x_{i - 1}, T^{n} x_{i - 1}, \\ φ (t)), M (x_{i - 1}, T^{m_{i - 1} + n} x_{i - 1}, φ (t)), \\ M (x_{i - 1}, T^{m_{i - 1} + n} x_{i - 1}, φ (t))} \\ \geq & G_{x_{i - 1}} (φ (t)) . \end{matrix}$ Therefore, G_{x
_i} (t) ≥ G_{x
_i-1} (φ (t)) ≥ ⋯ ≥ G_{x
₀} (φ (t)) and so we have

$\begin{matrix} M (x_{i}, x_{i + j}, t) & = & M (x_{i}, T^{m_{i + j - 1} + \dots + m_{i}} x_{i}, t) \\ \geq & G_{x_{i}} (t) \geq G_{x_{0}} (φ (t)) . \end{matrix}$ (18) Since the distribution function is nondecreasing, $sup_{t \in R} G_{x_{0}} (t) = 1$ and $x_{*}^{i} (t) \to \infty$ as i→ ∞, for all t > 0, we know that $G_{x_{0}} (x_{*}^{i} (t)) \to 1$ as i→ ∞. This implies that {x_n} is a τ-Cauchy sequence in X. Since Xis τ-complete, hence x_n → x_* in X.

(b) First we prove that x_* is a fixed point of T^m(x_*). Letting m (x_*) = m_*, by the assumption, there exist t_* ∈ R such that

M (x_*, T^{m
_*}x_*, t_*) =1. Denote

$t_{0} = inf {t : M (x_{*}, T^{m_{*}} x_{*}, t) = 1} .$ (19) It is obvious that t₀ ≤ t_*. Next we prove that t₀ = 0. In fact, if t₀ > 0, by the left-continuity of x_*, there exist t₁, t₂ ∈ ^R +, 0 < t₂ < t₁ < t₀, such that x_* (t₂) > t₀. From (18), it follows that $\begin{matrix} M (x_{*}, T^{m_{*}} x_{*}, t_{1}) < 1, and \\ M (x_{*}, T^{m_{*}} x_{*}, φ (t_{2})) = 1 . \end{matrix}$ On the other hand, it follows from $\begin{matrix} M (x_{*}, T^{m_{*}} x_{i}, φ (t_{2})) \geq \\ Δ (M (x_{*}, x_{i}, φ (t_{2}) - t_{0}) M (x_{i}, T^{m_{*}} x_{i}, t_{0}) \\ \geq & Δ (M (x_{*}, x_{i}, φ (t_{2}) - t_{0}), G_{x_{0}} (φ^{i} (t_{0}))) \\ \to & 1 (i \to \infty) \end{matrix}$ that $\begin{matrix} M (T^{m_{*}} x_{i}, T^{m_{*}} x_{*}, t_{2}) \geq \\ min {M (x_{*}, x_{i}, φ (t_{2})), M (x_{*}, T^{m_{*}} x_{*}, φ (t_{2})), \\ M (x_{*}, T^{m_{*}} x_{i}, φ (t_{2}))} \to 1 (i \to \infty) . \end{matrix}$

Since $\begin{matrix} M (T^{m_{*}} x_{i}, T^{m_{*}} x_{*}, t_{1} - t_{2}) \geq \\ Δ (M (x_{*}, x_{i}, \frac{t_{1} - t_{2}}{2}), \\ G_{x_{0}} (φ^{i} (\frac{t_{1} - t_{2}}{2}))) \to 1 (i \to \infty), \end{matrix}$ we have $\begin{matrix} M (T^{m_{*}} x_{i}, T^{m_{*}} x_{*}, t_{1}) \geq \\ Δ (M (x_{*}, T^{m_{*}} x_{*}, φ (t_{2})), M (T^{m_{*}} x_{i}, T^{m_{*}} x_{*}, t_{2})) \\ \to 1 (i \to \infty), \end{matrix}$ which contradicts M (x_*, T^{m
_*}x_*, t₁) <1. Hence we have t₀ = 0 and so M (x_*, T^{m
_*}x_*, t) =1 for all t > 0, i.e., T^{m
_*}x_* = x_*.

Next we prove that x_* is the unique fixed point of T^{m
_*}. Then, for any t > 0, we have

$\begin{matrix} M (x_{*}, y_{*}, t) & = & M (T^{m_{*}} x_{*}, T^{m_{*}} y_{*}, t) \\ \geq & min {M (x_{*}, y_{*}, φ (t)), M (x_{*}, x_{*}, φ (t)), \\ M (x_{*}, y_{*}, φ (t))} \\ = & M (x_{*}, y_{*}, φ (t)) \\ \geq & \dots \\ \geq & M (x_{*}, y_{*}, φ^{i} (t)) \\ \to & 1 (i \to \infty), \end{matrix}$

i.e., x_* = y_*. Thus, the fixed point of T^{m
_*} is unique.

Finally, we prove that x_*is also the unique fixed point of T and Tⁿx₀xrightarrowτx_*. In fact, since T^{m
_*}x_* = x_*, T^{m
_*}Tx_* = Tx_*. Noting that x_* is the unique fixed point of T^{m
_*}, thus we have x_* = Tx_*. The uniqueness of the fixed point x_* is obvious.

For any n ∈ ^I +, n > m_*, we may write it by n = km_* + s, 0 ≤ s < m_*. For any t > 0, by (16), we have, for all t > 0,

$\begin{matrix} M (x_{*}, T^{n} x_{0}, t) & = & M (T^{m_{*}} x_{*}, T^{m_{*} k + s} x_{0}, t) \\ (since x_{*} is a fixed point of T^{m_{*}}) \\ \geq & min {M (x_{*}, T^{(k - 1) m_{*} + s} x_{*}, φ (t)), \\ M (x_{*}, T^{n} x_{0}, φ (t))} \\ \geq & \dots \\ \geq & min {M (x_{*}, T^{(k - 1) m_{*} + s} x_{0}, φ (t)), \\ M (x_{*}, T^{n} x_{0}, φ^{i} (t))} \\ \to & 1 (k \to \infty) . \end{matrix}$

Letting i→ ∞, we obtain, for all t > 0, $\begin{matrix} M (x_{*}, T^{n} x_{0}, t) & \geq & M (x_{*}, T^{(k - 1) m_{*} + s} x_{0}, φ (t)) \\ \geq & \dots \\ \geq & M (x_{*}, T^{s} x_{0}, φ^{k} (t)) \\ \to & 1 (k \to \infty) . \end{matrix}$ This implies that Tⁿx₀ → x_* as k→ ∞. This completes the proof. □ Theorem 7. Let (X, M, Δ) be a τ-complete non-Archimedean FM-space and the t-norm satisfy the condition that for any t₀ ∈ (0, 1] is continuous at t = 1. Let T : X → X be a mapping satisfying the following conditions:

for each x ∈ X, $sup_{t \in R} G_{x} (t) = 1$ ,

for each x ∈ X, there exists m (x) ∈ ^I + such that for all y ∈ X and t ∈ ^R +, $\begin{matrix} M (T^{m (x)} x, T^{m (x)} y, t) \geq \\ min {M (x, y, \frac{t}{k}), \\ M (x, T^{m (x)} x, \frac{t}{k}), M (x, T^{m (x)} y, \frac{t}{k})}, \end{matrix}$ where k ∈ (0, 1) is a constant.

Then T has a unique fixed point x_* ∈ X, and for any x₀ ∈ X, the iterative sequence {Tⁿx₀} τ-converges to the fixed point x_*.

Proof. First we note that, under the assumption on Δ, $sup_{0 < t < 1} Δ (t, t) = 1$ . Letting Φ (t) = t/k, then it satisfies the conditions in Theorem 6.

Next, we prove that, for all t > 0, $M (x_{*}, T^{m_{*}} x_{*}, t) = 1 .$ In fact, for any t > 0, we have $\begin{matrix} M (x_{*}, T^{m_{*}} x_{i}, t) \geq \\ Δ (M (x_{*}, x_{i}, \frac{t}{2}), G_{x_{0}} (\frac{t}{2 k^{i}})) \\ \to 1 (i \to \infty) . \end{matrix}$ Taking k₁ ∈ (k, 1), we have $0 < \frac{k}{k_{1}} < 1$ and $\begin{matrix} M (T^{m_{*}} x_{i}, T^{m_{*}} x_{*}, \frac{kt}{k_{1}}) \geq \\ min {M (x_{*}, x_{i}, \frac{t}{k_{1}}), M (x_{*}, T^{m_{*}} x_{*}, \frac{t}{k_{1}}), \\ M (x_{*}, T^{m_{*}} x_{i}, \frac{t}{k_{1}})} \end{matrix}$ for all t > 0. Since M (x_*, x_i, t/k₁) →1 and M (x_*, T^{m
_*}x_i, t/k₁) →1 for all t > 0 (i→ ∞), there exists n ∈ I⁺ such that, for i > N₀, we have $\begin{matrix} M (T^{m_{*}} x_{i}, T^{m_{*}} x_{*}, \frac{kt}{k_{1}}) \geq \\ M (x_{*}, T^{m_{*}} x_{*}, \frac{t}{k_{1}}), t > 0 . \end{matrix}$ On the other hand, we have $\begin{matrix} M (x_{*}, T^{m_{*}} x_{*}, t) \\ \geq Δ (M (x_{*}, T^{m_{*}} x_{i}, (1 - \frac{k}{k_{1}}) t), \\ M (T^{m_{*}} x_{i}, T^{m_{*}} x_{*}, \frac{kt}{k_{1}})) \\ \geq & Δ (M (x_{*}, T^{m_{*}} x_{i}, (1 - \frac{k}{k_{1}}) t), \\ M (x_{*}, T^{m_{*}} x_{*}, \frac{t}{k_{1}})) . \end{matrix}$ Letting i→ ∞ in the preceding expression, we have, for all t > 0, $\begin{matrix} M (x_{*}, T^{m_{*}} x_{*}, t) & \geq & M (x_{*}, T^{m_{*}} x_{*}, \frac{t}{k_{1}}) \\ \geq & \dots \\ \geq & M (x_{*}, T^{m_{*}} x_{*}, \frac{t}{k_{1}^{j}}) \\ \to & 1 (j \to \infty) . \end{matrix}$ Hence M (x_*, T^{m
_*}x_*, t) =1 for all t > 0. By Theorem 6, the conclusion is obtained. This completes the proof. □ By the same way as stated above, we can prove the following:

Theorem 8.Let T be a self-mapping on a τ-complete non-Archimedean FM-space (X, M, Δ). Suppose that for each x ∈ X, the orbit O_T (x ; 0, ∞) is fuzzy bounded, and that for each x ∈ X there exists an integer m (x) ≥1 such that for all y ∈ X and t ≥ 0, $\begin{matrix} M (T^{m (x)} x, T^{m (x)} y, t) \geq min {M (x, y, \frac{t}{k}), \\ M (x, T^{m (x)} x, \frac{t}{k}), M (x, T^{m (x)} y, \frac{t}{k})}, \end{matrix}$ where k is a constant with k ∈ (0, 1). Then T has a unique fixed point x_* in X and Tⁿx₀ → x_* for each x₀ ∈ X.

As an application, in the following, we shall utilize the results obtained above to show the existence and uniqueness of random solution for the following random linear random operator equation $x (ω) = T (ω, x (ω)) + y (ω) .$ (20) In the sequel, we shall always assume that $(Ω, A, P)$ is a complete probability measure space, (X, d) is a separable linear metric space, and $(X, B)$ is a measurable space, where $B$ is the σ-algebra of all Borel subsets of X

Let T be a mapping from Ω × X into X. For any positive integer i and x, y ∈ X, we denote $\begin{matrix} O_{T} (ω, x; 0, \infty) = {x = x_{0} (ω), x_{1} (ω) \\ = T (ω, x_{0} (ω)), \dots, \\ x_{n} (ω) = T (ω, x_{n - 1} (ω)), \dots} \\ O_{T} (ω, x, y; 0, \infty) = O_{T} (ω, x; 0, \infty) \cup \\ O_{T} (ω, y; 0, \infty) \\ O_{T} (ω, x; i, \infty) = {x_{i} (ω) = T (ω, x_{i - 1} (ω)), \\ \dots, x_{n} (ω) = T (ω, x_{n - 1} (ω)), \dots} \\ O_{T} (ω, x, y; i, \infty) = O_{T} (ω, x; i, \infty) \\ \cup O_{T} (ω, y; i, \infty) \end{matrix}$ Let A ⊂ X and denote $δ (A) = sup_{x, y \in A} d (x, y)$ .

Definition 12. [7] A (real-valued) random variable, often denoted by X (or some other capital letter), is a function mapping a probability space (S, P) into the real line R. Associated with each point s in the domain S the function X assigns one and only one value X (s) in the range R. (The set of possible values of X (s) is usually a proper subset of the real line; i.e., not all real numbers need to occur. If S is a finite set with m elements, then X (s) can assume at most m different values as s varies in S.)

Example 6. (roll of two dice) Consider a random roll of two dice. The natural sample space is $S = {(i, j) : 1 \leq i \leq 6, 1 \leq j \leq 6},$ where each of the 36 points in S is assigned equal probability p (s) =1/36 . The random variable X might record the sum of the values on the two dice, i.e., $X (s) = X ((i, j)) = i + j .$

The proof of the following lemmas are straightforward and are omitted here. Lemma 3. Let x (ω) be an X-valued random variable. Let T (. , .) be a d-continuous random operator. Then T (ω, x (ω)) is also an X-valued random variable.

Definition 13. Let (X, d) be a metric space. The space (X, M, Δ) is called an induced non-Archimedean FM-space if Δ = min and M is defined by $\begin{matrix} M (x, y, t) & = & H (t - d (x, y)) = {\begin{matrix} 0, & if d (x, y) \geq t; \\ 1, & if d (x, y) < t . \end{matrix} \end{matrix}$ It is known that if (X, d) is a complete metric space, then the induced non-Archimedean FM-space (X, M, Δ) is a τ-complete non-Archimedean FM-space and that a sequence {x_n} in X τ-converges to a point x_* ∈ X if and only if {x_n} in X converges in the metric d to x_*.

Lemma 4. Let (X, d) be a complete linear metric space and (X, M, Δ) be an induced non-Archimedean FM-space from (X, d). Let {x_n} be a sequence in X. Then we have $x_{n} \overset{τ}{\to} x i f a n d o n l y i f x_{n} \overset{d}{\to} x .$

Theorem 9. Let (X, d) be a complete separable linear metric space and (X, M, Δ) the induced τ-complete non-Archimedean FM-space. Suppose that T (. , .) : Ω × X → X is a d-continuous random operator, and that for any x ∈ X and any ω ∈ Ω, O_s (ω, x ; 0, ∞) is d-bounded. Suppose further that there exist positive integers m, n satisfying the following: $P {ω \in Ω : d (S^{n} (ω, x), S^{m} (ω, y))$ (21) $= φ (ω, δ (O_{s} (ω, x, y; 0, \infty)))} = P (E_{x, y}) = 1$ for all x, y∈, where the random operator S : Ω × X → X is defined as follows: $S (ω, x) = T (ω, x) + y (ω),$ and φ : Ω × [0, ∞) → [0, ∞) is a random function satisfying the following conditions (Φ₁) and (Φ₂):

for any ω ∈ Ω, φ (ω, 0) =0 and φ (ω, .) is strictly increasing respect to t ∈ [0, ∞).

for any ω ∈ Ω and any t > 0, $lim_{n \to \infty} φ^{- n} (ω, t) = \infty$ , where φ^-1 (ω, .) denotes the inverse function of φ (ω, .), and φ^-n (ω, .) is the n-th iterative of φ^-1 (ω, .).

Then for any given X-valued random variable

\tilde{y} (ω)

, there exists a unique random solution x_* (ω) of the following nonlinear random operator equation:

x (ω) = T (ω, x (ω)) + \tilde{y} (ω),

(22) and for any X-valued random variable x₀ (ω), the sequence

x_{n} (ω) = T (ω, x_{n - 1} (ω)) + \tilde{y} (ω), n = 1, 2, \dots

of X-valued random variable converges almost surely to x_* (ω).

Proof. By the separability of X and the continuity of T (ω, .), it is easy to see that $G = ⋂_{x \in X} ⋂_{y \in X} E_{x, y} \in A and P (G) = 1,$ where the set E_x,y is defined by (20). For any fixed $\tilde{ω} \in G$ , it follows from (20) that $d (S^{n} (\tilde{ω}, x), S^{m} (\tilde{ω}, y)) \leq φ (\tilde{ω}, δ (\tilde{ω}, x, y; 0, \infty)),$ x, y ∈ X. Hence for all x, y ∈ X and any t ≥ 0, we have, by condition (Φ₁), $\begin{matrix} M (S^{n} (\tilde{ω}, x), S^{m} (\tilde{ω}, y), t) = \\ H (t - d (S^{n} (\tilde{ω}, x), S^{m} (\tilde{ω}, y))) \\ \geq & H (t - φ (\tilde{ω}, δ (\tilde{ω}, x, y; 0, \infty))) \\ = & H (φ (\tilde{ω}, φ^{- 1} (\tilde{ω}, t)) \\ - φ (\tilde{ω}, δ (\tilde{ω}, x, y; 0, \infty))) \\ = & H (φ^{- 1} (\tilde{ω}, t) - δ (\tilde{ω}, x, y; 0, \infty)) \\ = & inf_{p, q \in O_{s} (\tilde{ω}, x, y; 0, \infty)} M (p, q, φ^{- 1} (\tilde{ω}, t)) . \end{matrix}$ Next since $φ (\tilde{ω}, t)$ satisfies the conditions (Φ₁) and (Φ₂), we have that $φ^{- 1} (\tilde{ω}, t)$ satisfies the condition (Φ). From Lemma (4), it follows that $S (\tilde{ω}, .)$ is τ-continuous. Since each d-bounded set in X is fuzzy bounded, it follows that for any x ∈ X, the orbit $O_{s} (\tilde{ω}, x; 0, \infty)$ is fuzzy bounded. Hence all the conditions of Theorem 4 are satisfied. By using Theorem 4, there exist an unique fixed point $x_{*} (\tilde{ω}) \in X$ and for any X-valued random variable x₀ (ω), when $ω = \tilde{ω}$ the sequence ${x_{n} (\tilde{ω})}_{n = 0}^{\infty}$ converges in τ (hence in d) to $x_{*} (\tilde{ω})$ . By the arbitrariness of $\tilde{ω} \in G$ , this implies that x_n (ω) → x_* (ω) a . s . By Lemma 3, it is easy to know that {x_n (ω)} is a sequence of X-valued random variables, so that its strong limit x_* (ω) is also an X-valued random variable, and x_* (ω) is the unique random fixed point of S (. , .). Accordingly, x_* (ω) is the unique random solution of the equation (21). This completes the proof. □

4 Common fixed point theorems for contraction type mappings

Throughout this section, we always assume that (X, M, Δ) is a τ-complete non-Archimedean FM-space, Δ is a continuous t-norm which is stronger than Δ_m = max {sum - 1, 0}, i.e., Δ (a, b) ≥ Δ_m (a, b) for all a, b ∈ [0, 1], and the function φ (t) satisfies the condition (Φ).

Theorem 10. Let (X, M, Δ), Δ, and φ be the same as above. Let ${T_{n}}_{n = 1}^{\infty}$ be a sequence of self-adjoint on (X, M, Δ). Suppose that there exists a functional sequence ${m_{n} (x)}_{n = 1}^{\infty} : (X, M, Δ) \to I^{+}$ such that for each n ∈ ^I + and each x ∈ X, m_n (x) |m_b (T_nx) and that, for any i, j ∈ ^I +, i ≠ j and x, y ∈ X, the following holds: $M (T_{i}^{m_{i} (x)} x, T_{j}^{m_{j} (y)} y, t) \geq$ (23) $min_{p, q \in {x, y, T_{i}^{m_{i} (x)} x, T_{j}^{m_{j} (x)} y}} M (p, q, φ (t))$ for all t ≥ 0. Suppose further that there exists some x₀ ∈ X such that the set ${x_{n}}_{n = 0}^{\infty} \subset X$ defined by $x_{n} = T_{n}^{m_{n} (x_{n - 1})} x_{n - 1}$ (24) for all n = 1, 2, ⋯ is fuzzy bounded. Then there exists an unique common fixed point x^* ∈ X and x_nxrightarrowτx^*.

Proof. For given x₀ ∈ X, we prove that the sequence {x_n} defined by (2) is a τ-Cauchy sequence in X. In fact, for any i, j ∈ ^I +, it follows from (1) that, for all t ≥ 0, $\begin{matrix} M (x_{i}, x_{j}, t) = M (T_{i}^{m_{i} (x_{i - 1})} x_{i - 1}, \\ T_{j}^{m_{j} (x_{j - 1})} x_{j - 1}, t) \end{matrix}$ (25) $\geq min_{p, q \in {x_{i - 1}, x_{j - 1}, x_{i}, x_{j}}} M (p, q, φ (t)) .$ Therefore, for any m, n ∈ ^I + (m < n), from (3) we have $inf_{m \leq i, j \leq n} M (x_{i}, x_{j}, t) \geq inf_{m - 1 \leq i, j \leq n} M (x_{i}, x_{j}, φ (t)) .$ By the arbitrariness of n ∈ ^I + (n > m), we have $\begin{matrix} inf_{m \leq i, j} M (x_{i}, x_{j}, t) \geq inf_{m - 1 \leq i, j} M (x_{i}, x_{j}, φ (t)) \\ \geq sup_{u < φ (t)} inf_{i, j \geq m - 1} M (x_{i}, x_{j}, φ (t)) . \end{matrix}$ (26) By the induction, it is easy to prove that $\begin{matrix} inf_{m \leq i, j} M (x_{i}, x_{j}, t) \geq inf_{0 \leq i, j} M (x_{i}, x_{j}, φ (t)) \\ \geq sup_{u < φ^{m} (t)} inf_{i, j \geq 0} M (x_{i}, x_{j}, u) . \end{matrix}$ (27) for all m = 1, 2, ⋯ and t ≥ 0. In view of condition (Φ) and the fuzzy boundedness of the set {x_n} defined by (2), it follows that for all t ≥ 0, $\begin{matrix} lim_{m \to \infty} inf_{m \leq i, j} M (x_{i}, x_{j}, t) \geq \\ lim_{m \to \infty} (sup_{u < φ^{m} (t)} inf_{i, j \geq 0} M (x_{i}, x_{j}, u)) \\ = & {\begin{matrix} 0, & if t = 0; \\ sup_{u > 0} inf_{i, j \geq 0} M (x_{i}, x_{j}, u), & if t > 0 . \end{matrix} \\ = & {\begin{matrix} 0, & if t = 0; \\ 1, & if t > 0 . \end{matrix} \end{matrix}$ (28) Consequently, for any given ɛ > 0 and 0 < λ < 1, there exists N = N (ɛ, λ) ∈ ^I + such that for all m ≥ N, $inf_{m \leq i, j} M (x_{i}, x_{j}, ɛ) > 1 - λ .$ Therefore, we have $M (x_{i}, x_{j}, ɛ) > 1 - λ, i, j \geq N .$ (29) This implies that the sequence {x_n} defined by (2) is a τ-Cauchy sequence in X. By the τ-completeness of X, we can suppose that x_nxrightarrowτx^* ∈ X.

Now we prove that x^* is a common periodic point of {T_n}, i.e., $T_{n}^{m_{n} (x^{*})} x^{*} = x^{*}, n = 1, 2, \dots .$ Indeed, for any i ∈ ^I +, it follows from (1) that, for any n > i, we have $M (x_{n}, T_{i}^{m_{i} (x^{*})} x^{*}, t) =$ (30) $\begin{matrix} M (T_{n}^{m_{n} (x_{n - 1})} x_{n - 1}, T_{i}^{m_{i} (x^{*})} x^{*}, t) \\ \geq & min_{p, q \in {x_{n - 1}, x^{*}, x_{n}, T_{i}^{m_{i} (x^{*})} x^{*}}} M (p, q, φ (t)) \end{matrix}$ for all t ≥ 0. Let G₀ be the set of all discontinuity points of $M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, t)$ . Since φ^m is strictly increasing, we know that φ^-m (G₀) is the set of discontinuity points of $M (x_{n}, T_{i}^{m_{i} (x^{*})} x^{*}, φ^{m} (t))$ , m = 1, 2, ⋯. Moreover, G₀, φ^-m (G₀), m = 1, 2, ⋯ are the countable sets and so $G = G_{0} \cup (\sum_{m = 1}^{\infty} φ^{- m} (G_{0}))$ is also countable. Let $\tilde{G} = R^{+} \ G$ . When t = 0 or $t \in \tilde{G}$ (i.e., t is a common continuity point of $M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, t)$ and $M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, φ^{m} (t))$ , m = 1, 2, ⋯, it follows from (8) that $\begin{matrix} M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, t) = lim_{n \to \infty} M (x_{n}, T_{i}^{m_{i} (x^{*})} x^{*}, t) \\ \geq M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, φ (t)) . \end{matrix}$ Repeating this procedure, we can prove that $\begin{matrix} M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, t) \geq M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, φ (t)) \geq \\ \dots \geq M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, φ^{n} (t)) . \end{matrix}$ Letting n→ ∞ and noting the condition (Φ), for all $t \in \tilde{G}$ or t = 0, we have $M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, t) = H (t)$ (31) $= {\begin{matrix} 0, & if t = 0; \\ 1, & if t > 0 . \end{matrix}$ When t ∈ G with t > 0, by the density of real numbers, there exist $t_{1}, t_{2} \in \tilde{G}$ such that 0 < t₁ < t < t₂. Since the distribution function is nondecreasing, we have, from (9), $\begin{matrix} 1 = H (t_{1}) = M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, t_{1}) \\ \leq M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, t) \\ \leq M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, t_{2}) = 1 . \end{matrix}$ This shows that,for all t ∈ G with t > 0, $M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, t) = H (t) .$ (32) Combining (9) and (10), we have $M (x^{*}, T_{i}^{m_{i} (x^{*})} x^{*}, t) = H (t), t \geq 0,$ i.e., $x^{*} = T_{i}^{m_{i} (x^{*})} x^{*}, i = 1, 2, \dots$ .

To prove that x^* is the unique periodic point of ${T_{n}}_{n = 1}^{\infty}$ , we proceed as follows:

Suppose that y^* ∈ X is another periodic point of some T_j, i.e., $y^{*} = T_{j}^{m_{j} (y^{*})} y^{*} .$ Then for any i ∈ ^I +, i ≠ j, we have $\begin{matrix} M (x^{*}, y^{*}, t) & = & M (T_{i}^{m_{i} (x^{*})} x^{*}, T_{j}^{m_{j} (y^{*})} y^{*}, t) \\ \geq & min_{p, q \in {x^{*}, y^{*}}} M (p, q, φ (t)) \\ = & M (x^{*}, y^{*}, φ (t)), t \geq 0 . \end{matrix}$ Repeating this procedure, for all n = 1, 2, ⋯ and t ≥ 0, we can prove $M (x^{*}, y^{*}, t) \geq M (x^{*}, y^{*}, φ^{n} (t)) .$ Letting n→ ∞ and noting the condition (Φ), we have $M (x^{*}, y^{*}, t) = H (t), t \geq 0, i . e ., x^{*} = y^{*} .$ Furthermore, by the assumption, since

m_n (x) |m_n (T_nx) for all x ∈ X and for all n ∈ ^I +, there exists, for each i = 1, 2, ⋯ some k_i ∈ ^I + such that $m_{i} (T_{i} x^{*}) = k_{i} m_{i} (x^{*}) .$ (33) Hence we have, for each i = 1, 2, ⋯, $\begin{matrix} T_{i} x^{*} & = & T_{i} T_{i}^{m_{i} (x^{*})} x^{*} = T_{i} T_{i}^{2 m_{i} (x^{*})} x^{*} \\ = & \dots = T_{i} T_{i}^{k_{i} m_{i} (x^{*})} x^{*} \\ = & T_{i} T_{i}^{m_{i} (T_{i} x^{*})} x^{*} = T_{i}^{m_{i} (T_{i} x^{*})} T_{i} x^{*} . \end{matrix}$ This implies that T_ix^* is also a periodic point of T_i. Since x^* is the unique periodic point of T_i, we have x^* = T_ix^*, i = 1, 2, ⋯, which means that x^* is the desired unique common fixed point of ${T_{n}}_{n = 1}^{\infty}$ . This completes the proof. □

Corollary 1. Let ${T_{n}}_{n = 1}^{\infty}$ , (X, M, Δ) and ${m_{n} (x)}_{n = 1}^{\infty}$ be the same as in Theorem 10. Suppose that for any i, j ∈ ^I +, and any x, y ∈ X, the following holds: $\begin{matrix} M (T_{i}^{m_{i} (x)} x, T_{j}^{m_{j} (y)} y, t) \geq \\ min_{p, q \in {x, y, T_{i}^{m_{i} (x)} x, T_{j}^{m_{j} (y)} y}} M (p, q, \frac{t}{h}) \end{matrix}$ for all t ≥ 0. Suppose further that there exists x₀ ∈ X such that the set ${x_{n}}_{n = 1}^{\infty}$ defined by (2) is fuzzy bounded. Then there exists an unique common fixed point x^* ∈ X and x_nxrightarrowτx^*.

Proof. Taking φ (t) = t/h, h ∈ (0, 1), it is easy to see that φ (t) satisfies the condition (Φ). So the result follows now from Theorem 10. □

Corollary 2. Let ${T_{n}}_{n = 1}^{\infty}$ be a sequence of self-mappings on (X, M, Δ). Suppose that there exists a sequence ${m_{n}}_{n = 1}^{\infty}$ in ^I + such that for any x, y ∈ X and any i, j ∈ ^I +, i ≠ j, the following holds: $M (T_{i}^{m_{i}} x, T_{j}^{m_{j}} y, t) \geq$ (34) $min_{p, q \in {x, y, T_{i}^{m_{i}} x, T_{j}^{m_{j}} y}} M (p, q, \frac{t}{h})$ for all t ≥ 0. Suppose further that there exists x₀ ∈ X such that the set ${x_{n}}_{n = 1}^{\infty}$ defined by (2) is fuzzy bounded. Then there exists an unique common fixed point x^* ∈ X and x_nxrightarrowτx^*.

Throughout this section, we always assume that (X, M, Δ) is a τ-complete non-Archimedean FM-space, Δ is a continuous t-norm, and the function φ (t) satisfies the condition (Φ). Suppose further that S, T are self-mappings on (X, M, Δ) and that they are commutative and τ-continuous. Furthermore, we denote $O_{S, T} (x; 0, \infty) = {S^{i} T^{j} x}_{i, j = 0}^{\infty}, x \in X .$

Theorem 11. Let (X, M, Δ), S, T and φ be the same as above. Suppose that, for each x ∈ X, the set O_S,T (x ; 0, ∞) is fuzzy bounded. Suppose further that there exist m, m′ ∈ ^I +, m + m′ ≥ 1, such that for each x ∈ X and t ≥ 0, the following holds: $\begin{matrix} inf_{p, q \in O_{S, T} (S^{m} T^{m^{'}} x; 0, \infty)} M (p, q, t) \geq \\ inf_{p, q \in O_{S, T} (x; 0, \infty)} M (p, q, φ (t)) . \end{matrix}$ (35) Then for each x₀ ∈ X, the sequence ${S^{n} T^{n}}_{n = 0}^{\infty}$ converges in τ to some common fixed point x^* ∈ X of S and T.

Proof. Letting h = max {m, m′}, for any given x₀ ∈ X, from (B2), we have $\begin{matrix} inf_{p, q \in O_{S, T} (S^{h} T^{h} x_{0}; 0, \infty)} M (p, q, t) \geq \\ inf_{p, q \in O_{S, T} (S^{m} T^{m^{'}} x_{0}; 0, \infty)} M (p, q, t) \\ \geq & inf_{p, q \in O_{S, T} (x_{0}; 0, \infty)} M (p, q, φ (t)) \\ \geq & sup_{u < Φ (t)} inf_{p, q \in O_{S, T} (x_{0}; 0, \infty)} M (p, q, u) \end{matrix}$ for all t ≥ 0. By the induction, we can prove the following inequality holds: $\begin{matrix} inf_{p, q \in O_{S, T} (S^{nh} T^{nh} x_{0}; 0, \infty)} M (p, q, t) \geq \\ inf_{p, q \in O_{S, T} (S^{(n - 1) h} T^{(n - 1) h} x_{0}; 0, \infty)} M (p, q, φ (t)) \\ \geq & \dots \end{matrix}$ (36) $\begin{matrix} \geq & inf_{p, q \in O_{S, T} (x_{0}; 0, \infty)} M (p, q, φ^{n} (t)) \\ \geq & sup_{u < φ^{n} (t)} inf_{p, q \in O_{S, T} (x_{0}; 0, \infty)} M (p, q, u) \end{matrix}$ for all t ≥ 0. By the condition (Φ), we have $\begin{matrix} lim_{n \to \infty} inf_{p, q \in O_{S, T} (S^{nh} T^{nh} x_{0}; 0, \infty)} M (p, q, t) \geq \\ lim_{n \to \infty} sup_{u < φ^{n} (t)} inf_{p, q \in O_{S, T} (x_{0}; 0, \infty)} M (p, q, u) \\ = & {\begin{matrix} sup_{u > 0} inf_{p, q \in O_{S, T} (x_{0}; 0, \infty)} M (p, q, u), & if t > 0; \\ 0, & if t = 0 . \end{matrix} \\ = & H (t), t \geq 0 . \end{matrix}$ Therefore, for any given ɛ > 0 and 0 < λ < 1, there exists a positive integer N = N (ɛ, λ) such that $\begin{matrix} inf_{p, q \in O_{S, T} (S^{nh} T^{nh} x_{0}; 0, \infty)} M (p, q, ɛ) \\ > 1 - λ, n \geq N . \end{matrix}$ This implies that the subsequences of ${S^{i} T^{j} x_{0}}_{i, j = 0}^{\infty}$ in which the indexes i and j are both convergent to ∞, are τ-Cauchy sequences in X. By the τ-completeness of X, they all converge in τ to the same limit x^* ∈ X. In particular, the following three subsequences converge in τ to the limit x^*: $\begin{matrix} {S^{n} T^{n} x_{0}}_{n = 0}^{\infty}, {S (S^{n} T^{n}) x_{0}}_{n = 0}^{\infty}, \\ {T (S^{n} T^{n} x_{0})}_{n = 0}^{\infty} . \end{matrix}$ By the τ-continuity of S and T, we have $x^{*} = {Sx}^{*} = {Tx}^{*},$ that is, x^* is a common fixed point of S and T. This completes the proof. □ Taking S = I_X (the identity mapping on X),we have

Corollary 3. Let (X, M, Δ), and φ be the same in Theorem 11. Suppose that T is a τ-continuous self-mapping on (X, M, Δ) and that for each x ∈ X the set $O_{T} (x; 0, \infty) = {T^{n} x}_{n = 0}^{\infty}$ is fuzzy bounded. If there exists m ∈ ^I + such that for all t ≥ 0 and x ∈ X, $\begin{matrix} inf_{p, q \in O_{T} (T^{m} x; 0, \infty)} M (p, q, t) \geq \\ inf_{p, q \in O_{T} (x; 0, \infty)} M (p, q, φ (t)) . \end{matrix}$ Then for any x₀ ∈ X, the sequence ${T^{n} x_{0}}_{n = 0}^{\infty}$ τ-converges to some fixed point of T in X.

Theorem 12.Let (X, M, Δ), S, T and φ be the same in Theorem 11. Suppose that, for each x ∈ X, the set O_S,T (x ; 0, ∞) is fuzzy bounded. Suppose further that there exist m, m′, n, n′ ∈ ^I +, m + m′ ≥ 1, n + n′ ≥ 1, such that for each x, y ∈ X and t ≥ 0, the following holds: $\inf_{p, q \in O_{S, T} (S^{m} T^{m^{'}} x; 0, \infty) \cup O_{S, T} (S^{n} T^{n^{'}} y; 0, \infty)} M (p, q, t) \geq \inf_{p, q \in O_{S, T} (x; 0, \infty) \cup O_{S, T} (y; 0, \infty)} M (p, q, φ (t)) .$ (37) Then there exists a unique common fixed point x^* of S and T in X, and for any x₀ ∈ X, the sequence ${S^{n} T^{n} x_{0}}_{n = 0}^{\infty}$ converges in τ to x^*.

Proof. Letting h = max {m, m′, n, n′} and taking x = y in (B4), we have $\begin{matrix} inf_{p, q \in O_{S, T} (S^{h} T^{h} x; 0, \infty)} M (p, q, t) \geq \\ inf_{p, q \in O_{S, T} (x; 0, \infty)} M (p, q, φ (t)) \end{matrix}$ for all x ∈ X and t ≥ 0. By Theorem 11, we know that, for each x₀ ∈ X, the sequence ${S^{n} T^{n} x_{0}}_{n = 0}^{\infty}$ converges in τ to some common fixed point x^* ∈ X of S and T. Now we prove that x^* is the unique common fixed point of S and T in X. Suppose that this is not the case. Then there exists another common fixed point y^* of S and T in X. From (37), we have $M (x^{*}, y^{*}, t) = \inf_{p, q \in O_{S, T} (S^{m} T^{m^{'}} x^{*}; 0, \infty) \cup O_{S, T} (S^{n} T^{n^{'}} y^{*}; 0, \infty)} M (p, q, t) \geq \inf_{p, q \in O_{S, T} (x^{*}; 0, \infty) \cup O_{S, T} (y^{*}; 0, \infty)} M (p, q, φ (t)) = M (x^{*}, y^{*}, φ (t)), t > 0.$ Repeating the procedure, for all t ≥ 0 and n = 1, 2, ⋯, we can prove $M (x^{*}, y^{*}, t) \geq M (x^{*}, y^{*}, φ^{n} (t)) .$ Letting n→ ∞ and using the condition (Φ), we have $M (x^{*}, y^{*}, t) = H (t), t \geq 0, i . e ., x^{*} = y^{*} .$ This completes the proof. □ As a consequence of Theorem 12 with S = I_X, we have

Corollary 4. Let (X, M, Δ), and φ be the same in Theorem 12. Suppose that T is a τ-continuous self-mapping on (X, M, Δ) and that for each x ∈ X the set $O_{T} (x; 0, \infty) = {T^{n} x}_{n = 0}^{\infty}$ is fuzzy bounded. If there exist m, n ∈ ^I + such that for all x, y ∈ X and t ≥ 0, $\inf_{p, q \in O_{T} (T^{m} x; 0, \infty) \cup O_{T} (T^{n} y; 0, \infty)} M (p, q, t) \geq \inf_{p, q \in O_{T} (x; 0, \infty) \cup O_{T} (y; 0, \infty)} M (p, q, φ (t))$ Then for any x₀ ∈ X, the sequence ${T^{n} x_{0}}_{n = 0}^{\infty}$ τ-converges to some fixed point of T in X.

Lemma 5. [34]Let φ : [0, ∞) → [0, ∞) satisfies the condition (Φ) and let ψ : [0, ∞) → [0, ∞) defined by $ψ (t) = {\begin{matrix} 0, & t = 0; \\ inf {s > 0 : φ (s) > t}, & t > 0 . \end{matrix}$ (40) Then we have the following:

φ (t) < t for all t > 0,

φ (ψ (t)) ≤ t and ψ (φ (t)) = t for all t ≥ 0,

ψ (t) ≥ t for all t ≥ 0,

$lim_{n \to \infty} ψ^{n} (t) = \infty$ for all t > 0,

$\sum_{n = 0}^{\infty} φ^{n} (t) < \infty$ for all t > 0.

Definition 14. [35] Let Δ be a t-norm satisfying condition $sup_{0 < t < 1} Δ (t, t) = 1$ . Then Δ is said to be of h-type if the family of functions {Δ^m (t)} is equi-continuous at t = 1, where Δ¹ (t) = Δ (t, t) , Δ^m (t) = Δ (t, Δ^m-1 (t)) , t ∈ [0, 1], m = 2, 3, ⋯.

Theorem 13. Let (X, M, Δ) be τ-complete non-Archimedean FM-space, where Δ is a left-continuous t-norm of h-type. Let T : X → X satisfying the following condition: for any x, y ∈ X and u ∈ Tx, there exists a point v ∈ Ty such that

$\begin{matrix} M (u, v, φ (t)) & \geq & min {M (x, y, t), M (x, Tx, t), \\ M (y, Ty, t)} \end{matrix}$ (41) for all t > 0, where the function φ satisfies the condition (Φ). Then T has a fixed point in X.

To prove Theorem 13, we need the following lemma.

Lemma 6. Let (X, M, Δ) be τ-complete non-Archimedean FM-space, where Δ is a left-continuous t-norm of h-type. If a sequence {x_n} in X satisfies the following condition: for any n ∈ N and t > 0, $M (x_{n}, x_{n + 1}, t) \geq M (x_{0}, x_{1}, ψ^{n} (t)),$ (42) where φ is a function satisfying the condition (Φ) and ψ is defined by (38). Then {x_n} is a τ-Cauchy sequence in X.

Proof. Since Δ is a t-norm of h-type, for any λ ∈ (0, 1), there exists a δ (λ) ∈ (0, 1) such that for any t > δ (λ), Δ^m (t) >1 - λ for all m ∈ N. Since φ satisfies the condition (Φ), $\sum_{n = 0}^{\infty} φ^{n} (t_{0}) < \infty$ . Hence for any t > 0, there exists N₀ ∈ N such that $\sum_{n = k}^{\infty} φ^{n} (t_{0}) < t$ for any k ≥ N₀. By (40) and Lemma 5, for any k ≥ N₀, we have $\begin{matrix} M (x_{k}, x_{k + m}, t) \geq M (x_{k}, x_{k + m}, \sum_{n = k}^{\infty} φ^{n} (t_{0})) \\ \geq Δ (M (x_{k}, x_{k + 1}, φ^{k} (t_{0})), Δ (M (x_{k + 1}, x_{k + 2}, \\ φ^{k + 1} (t_{0}))), \dots Δ (M (x_{k + m - 2}, x_{k + m - 1}, \\ φ^{k + m - 2} (t_{0}))), Δ (M (x_{k + m - 1}, x_{k + m}, \\ φ^{k + m - 1} (t_{0})), \dots) \\ \geq Δ^{m} (M (x_{0}, x_{1}, t_{0})) > 1 - λ \end{matrix}$ for all m ∈ N, which implies that {x_n} is a τ-Cauchy sequence in X. This completes the proof. □ Proof. [Proof of Theorem 13] Let x₀ ∈ X and x₁ ∈ Tx₀. By Lemma 5 and the condition (1), there exists x₂ ∈ Tx₁ such that $\begin{matrix} M (x_{1}, x_{2}, t) \geq M (x_{1}, x_{2}, φ (ψ (t))) \\ \geq min {M (x_{0}, x_{1}, ψ (t)), M (x_{0}, {Tx}_{0}, ψ (t)) \\ M (x_{1}, {Tx}_{1}, ψ (t))} \end{matrix}$ (43) $min {M (x_{0}, x_{1}, ψ (t)), M (x_{1}, x_{2}, ψ (t))}$ for all t > 0, where ψ is defined by (38). Using (48) repeatedly, we have $\begin{matrix} M (x_{1}, x_{2}, t) \geq min {M (x_{0}, x_{1}, ψ (t)), \\ M (x_{0}, x_{1}, ψ^{2} (t)), M (x_{1}, x_{2}, ψ^{2} (t))} \\ = min {M (x_{0}, x_{1}, ψ (t)), M (x_{1}, x_{2}, ψ^{2} (t))} \\ \geq \dots \\ min {M (x_{0}, x_{1}, ψ (t)), M (x_{1}, x_{2}, ψ^{n} (t))} . \end{matrix}$ Letting n→ ∞, we have $M (x_{1}, x_{2}, t) \geq M (x_{0}, x_{1}, ψ (t))$ for all t > 0. Taking this procedure repeatedly, we define a sequence {x_n} in X satisfying $x_{n + 1} \in {Tx}_{n} and$ (44) $M (x_{n}, x_{n + 1}, t) \geq M (x_{n - 1}, x_{n}, ψ (t))$ for all t > 0. Thus, for any n ∈ N and t > 0, we have $M (x_{n}, x_{n + 1}, t) \geq M (x_{n - 1}, x_{n}, ψ (t))$ (45) $\geq \dots \geq M (x_{0}, x_{1}, ψ^{n} (t)) .$ Therefore, by Lemma 6, {x_n} is a τ-Cauchy sequence in X. Since (X, M, Δ) is a τ-complete, we assume that x_n → x^* ∈ X.

Next, we prove that x^* is a fixed point of T. In fact, for any t > 0 and ɛ > 0, from (14), we have $\begin{matrix} M (x_{n + 1}, {Tx}^{*}, t - ɛ) \geq M (x_{n + 1}, {Tx}^{*}, \\ φ (ψ (t - ɛ))) \\ = sup_{y \in Tx} M (x_{n + 1}, Tx, φ (ψ (t - ɛ))) \\ \geq min {M (x_{n}, x^{*}, ψ (t - ɛ)), \end{matrix}$ (46) $\begin{matrix} M (x_{n}, x_{n + 1}, ψ (t - ɛ)), \\ M (x^{*}, {Tx}^{*}, ψ (t - ɛ))} \\ \geq min {M (x_{n}, x^{*}, ψ (t - ɛ)), M (x_{0}, x_{1}, \\ ψ^{n + 1} (t - ɛ)), M (x^{*}, {Tx}^{*}, ψ (t - ɛ))} \end{matrix}$ If M (x^*, Tx^*, ψ (t - ɛ)) =1, then we have $\begin{matrix} M (x_{n + 1}, {Tx}^{*}, t - ɛ) \geq min { \\ M (x_{n}, x^{*}, ψ (t - ɛ)), M (x_{0}, x_{1}, ψ^{n + 1} (t - ɛ))} . \end{matrix}$ Letting n→ ∞, we have M (x^*, Tx^*, t - ɛ) ≥1. By the arbitrariness of ɛ ∈ (0, 1), we have M (x^*, Tx^*, t) =1 for all t > 0, i.e., x^* ∈ Tx^*. Thus, the conclusion is proved.

If M (x^*, Tx^*, ψ (t - ɛ)) <1, then letting n→ ∞ in (44), we have $M (x^{*}, {Tx}^{*}, t - ɛ) \geq M (x^{*}, {Tx}^{*}, ψ (t)) .$ (47) Hence, we have

$\begin{matrix} M (x^{*}, {Tx}^{*}, t) \geq Δ (M (x^{*}, x_{n + 1}, ɛ), \\ M (x_{n + 1}, {Tx}^{*}, t - ɛ)) . \end{matrix}$ (48) Letting n→ ∞ in (46), from (45) and the left continuity of Δ, we have $M (x^{*}, {Tx}^{*}, t) \geq M (x^{*}, {Tx}^{*}, t - ɛ) .$ Thus, as t → ɛ, by the continuity of ψ and the left continuity of distribution function, it follows that $M (x^{*}, {Tx}^{*}, t) \geq M (x^{*}, {Tx}^{*}, ψ (t)) .$ Taking this procedure repeatedly, we obtain $\begin{matrix} M (x^{*}, {Tx}^{*}, t) \geq M (x^{*}, {Tx}^{*}, ψ (t)) \\ \geq \dots \geq M (x^{*}, {Tx}^{*}, ψ^{n} (t)) . \end{matrix}$ Therefore, as n→ ∞, M (x^*, Tx^*, t) =1 for all t > 0, i.e., x^* ∈ Tx^*. This completes the proof. □

5 Applications

As an application, in the sequel we use some result stated above to show the existence and uniqueness of the solutions for nonlinear Volterra integral equations on a kind of particular fuzzy metric space.

In what follows, let [0, a] be a fixed real interval (0< a < ∞) and (X, norm . _X) a real Banach space. We denote by C ([0, a] ; X) the Banach space of all X-valued continuous functions defined on [0, a] with norm defined by $‖ x_{C} = sup_{0 \leq t \leq a} ‖ {x (t)}_{X},$ (49)x (t) ∈ C ([0, a] ; X). As well as the norm norm . _C, the space C ([0, a] ; X) can be endowed with another norm norm . _* which is defined as follows: $‖ x_{*} = sup_{0 \leq t \leq a} (e^{- Lt} ‖ x (t)),$ (50)x (t) ∈ C ([0, a] ; X), where L is any positive number. It is clear that the norm norm . _* is equivalent to the norm norm . _C.

We also denote by (C ([0, a] ; X) , M, min) the induced non-archimedean FM-space, where M is the mapping from C ([0, a] ; X) × C ([0, a] ; X) into [0, ∞) defined by $M (x, y, t) = {\begin{matrix} 0, & if t < ‖ {x - y}_{*}; \\ 1, & if t \geq ‖ {x - y}_{*} . \end{matrix}$ x (s) , y (s) ∈ C ([0, a] ; X) , t ∈ R .

Now we study the existence and uniqueness of the solutions of the following kind of nonlinear Volterra integral equations: $x (t) = y (t) + \int_{0}^{t} K (t, s, x (s)) ds,$ (51) 0 ≤ t ≤ a, where y (t) ∈ C ([0, a] , X) is any given function.

Theorem 14. Let (X, norm . _X), C ([0, a] ; X) and (C ([0, a] ; X) , M, min) be the same stated above. Suppose that the following conditions are satisfied:

K (t, s, x (s))∈

C ([0, a] × [0, a] × C ([0, a] ; X) ; X) and $‖ K_{C} = sup_{t, s \in [0, a], x \in X} ‖ {K (t, s, x)}_{X} < \infty,$

there exist m ∈ ^I + and a constant β ∈ (0, 1) such that $\begin{matrix} M (T^{m} x, T^{m} y, t) = \\ min_{p, q \in {x, y, T^{m} x, T^{m} y}} M (p, q, \frac{t}{β}) \end{matrix}$ for all x, y ∈ C ([0, a] ; X) and t ∈ ^R +, where the mappings T and T^m are defined as follows: $\begin{matrix} (Tx) (t) & = & y (t) + \int_{0}^{t} K (t, s, x (s)) ds, \\ (T^{m} x) (t) & = & y (t) + \int_{0}^{t} K (t, s, T^{m - 1} x (s)) ds, \end{matrix}$

for any x (t) ∈ C ([0, a] ; X), the set

$O_{T} (x; 0, \infty) = {T^{n} x (t)}_{n = 0}^{\infty}$ is bounded.

Then for any x₀ (t) ∈ C ([0, a] ; X), the sequence

{T^{n} x_{0} (t)}_{n = 0}^{\infty}

converges in the ‖ . _C to a solution x^* (t) ∈ C ([0, a] ; X) of the equation (45).

Proof. The conclusion follows immediately from Corollary 2. □ Next, we apply our main results to study the existence of a solution of boundary value problems for fractional differential equations involving the Caputo fractional derivative.

Let X = C ([0, 1] , R) be the Banach space of all continuous functions from [0, 1] into R with the norm $‖ x_{\infty} = sup_{t \in [0, 1]} | x (t) | .$ Define M : X² × (0, ∞) → [0, 1] by $M (x, y, t) = e^{- \frac{‖ x - y}{t}}$ for all x, y ∈ X and t ∈ (0, ∞).

It is well known that (X, M, Δ) is a τ-complete non-Archimedean fuzzy metric space with

Δ (a, b) = a . b with a, b ∈ [0, 1].

Now, let us recall the following basic notions which will be needed subsequently.

Definition 15. [16] For a function u given on the interval [a, b] the Caputo fractional derivative of function u order β > 0 is defined by

$\begin{matrix} (^{c} D_{a^{+}}^{β}) u (t) = \\ \frac{1}{Γ (n - β)} \int_{a}^{t} (t - s)^{n - β - 1} u^{(n)} (s) ds \\ (n - 1 \leq β < n, n = [β] + 1), \end{matrix}$ (52) where [β] denotes the integer part of the positive real number β and Γ is a gamma function.

Consider the boundary value problem for fractional order differential equation given by: ${c_{D_{a^{+}}^{β}} (y (t)) = h (t, y (t)), t \in [0, 1], 1 < β \leq 2; y (0) = c_{0}, y^{'} (1) - c_{1},$ (53) where $c_{D_{a^{+}}^{β}}$ denotes the Caputo fractional derivative of order β, h : [0, 1] → R is a continuous function and c₀, c₁ are real constants.

Definition 16. [1] A function y ∈ C² ([0, 1] , R), with its β-derivative existing on [0, 1] is said to be a solution of (60) if y satisfies the equation $c_{D_{a^{+}}^{β}} (y (t)) = h (t, y (t))$ on [0, 1] and the conditions y (0) = c₀, y′ (1) = c₁. The following lemma is required in what follows.

Lemma 7. [1] Let 1 < β ≤ 2 and u : [0, 1] → R be continuous. A function y is a solution of the fractional integral equation $\begin{matrix} y (t) & = & \frac{1}{Γ (β)} \int_{0}^{t} (t - s)^{β - 1} u (s) ds \\ - & \frac{t}{Γ (β - 1)} \int_{0}^{1} (t - s)^{β - 2} u (s) ds \\ + & c_{0} + c_{1} t \end{matrix}$ if and only if y is a solution of the fractional boundary value problem ${c_{D_{a^{+}}^{β}} (y (t)) = u (t), y (0) = c_{0}, y^{'} (1) - c_{1,}$ $\begin{matrix} y^{'} (1) & = & c_{2} + \frac{t}{Γ (β - 1)} \int_{0}^{1} (t - s)^{β - 2} u (s) ds \\ = c_{1}, where c_{0}, c_{1}, c_{2} \in R . \end{matrix}$

Theorem 15. Suppose that

for all x, y ∈ X, t ∈ [0, 1] such that $| h (t, x (t)) - h (t, y (t)) | \leq$ (54) $\begin{matrix} λ | x (t) - y (t) |, where \\ 0 < \frac{1}{k} = λ (\frac{1}{Γ (β + 1)} + \frac{1}{Γ (β)}) < 1; \end{matrix}$

there exists y₀ ∈ X such that $\begin{matrix} y_{0} (t) & \leq & \frac{1}{Γ (β)} \int_{0}^{t} (t - s)^{β - 1} h (s, y_{0} (s)) ds \\ - & \frac{t}{Γ (β - 1)} \int_{0}^{1} (t - s)^{β - 2} h (s, y_{0} (s)) ds \\ + & c_{0} + c_{1} t; \end{matrix}$

h is nondecreasing in the second variable.

Then, the equation (61) has a unique solution in X.

Proof. Define Q : X → X by $\begin{matrix} Qy (t) & = & \frac{1}{Γ (β)} \int_{0}^{t} (t - s)^{β - 1} h (s, y (s)) ds \\ - & \frac{t}{Γ (β - 1)} \int_{0}^{1} (t - s)^{β - 2} h (s, y (s)) ds \\ + & c_{0} + c_{1} t; c_{0}, c_{1} \in R . \end{matrix}$ First, we prove that Q is continuous. Let {y_n} be a sequence such that y_n → y in X. Then for each t ∈ [0, 1]

$\begin{matrix} | {Qy}_{n} (t) - Qy (t) | \leq \\ \frac{1}{Γ (β)} \int_{0}^{t} (t - s)^{β - 1} | h (s, y_{n} (s)) - h (s, y (s)) | ds \\ + \frac{1}{Γ (β - 1)} \int_{0}^{1} (t - s)^{β - 2} | h (s, y_{n} (s)) - h (s, y (s)) | ds . \end{matrix}$

As h is a continuous function, we have $\begin{matrix} lim_{n \to \infty} ‖ {h (t, y_{n} (t)) - h (t, y (t))}_{\infty} = 0 \\ \Leftrightarrow lim_{n \to \infty} ‖ {{Qy}_{n} - Qy}_{\infty} = 0 \\ \Leftrightarrow lim_{n \to \infty} e^{- \frac{‖ {{Qy}_{n} - Qy}_{\infty}}{t}} = 1 \\ \Leftrightarrow lim_{n \to \infty} {Qy}_{n} = Qy . \end{matrix}$ Hence, Q is continuous.

Clearly, the fixed points of the operator Q are solutions of the Equation (60). We will use Theorem 13 to prove that Q has a fixed point.

Let x, y ∈ X, then for all t ∈ [0, 1]. Observe that

$\begin{matrix} | Qy (t) - Qx (t) | \leq \\ \frac{1}{Γ (β)} \int_{0}^{t} (t - s)^{β - 1} | h (s, y (s)) - h (s, x (s)) | ds \\ + \frac{t}{Γ (β - 1)} \int_{0}^{1} (t - s)^{β - 2} | h (s, y (s)) - h (s, x (s)) | ds \\ \leq \frac{1}{Γ (β)} \int_{0}^{t} (t - s)^{β - 1} | h (s, y (s)) - h (s, x (s)) | ds \\ + \frac{1}{Γ (β - 1)} \int_{0}^{1} (1 - s)^{β - 2} | h (s, y (s)) - h (s, x (s)) | ds \\ \leq \frac{1}{Γ (β)} \int_{0}^{t} (t - s)^{β - 1} λ | y (s) - x (s) | ds \\ + \frac{1}{Γ (β - 1)} \int_{0}^{1} (1 - s)^{β - 2} λ | y (s) - x (s) | ds \\ \leq \frac{1}{Γ (β)} \int_{0}^{t} (t - s)^{β - 1} λ ‖ {y - x}_{\infty} ds \\ + \frac{1}{Γ (β - 1)} \int_{0}^{1} (1 - s)^{β - 2} λ ‖ {y - x}_{\infty} ds \\ \leq \frac{λ ‖ {y - x}_{\infty}}{Γ (β)} \int_{0}^{t} (t - s)^{β - 1} ds \\ + \frac{λ ‖ {y - x}_{\infty}}{Γ (β - 1)} \int_{0}^{1} (1 - s)^{β - 2} ds \\ \leq λ (\frac{1}{Γ (β + 1)} + \frac{1}{Γ (β)}) ‖ {y - x}_{\infty} \end{matrix}$

$\begin{matrix} = \frac{1}{k} ‖ {y - x}_{\infty} . \end{matrix}$ Hence, $k ‖ {Qy - Qx}_{\infty} \leq ‖ {y - x}_{\infty} .$ This gives $e^{- k \frac{‖ {Qy - Qx}_{\infty}}{t}} \geq e^{- \frac{‖ {y - x}_{\infty}}{t}} .$ Therefore, $\begin{matrix} M (Qx, Qy, φ (t)) \geq M (x, y, t) \\ \geq min {M (x, y, t), M (Qx, x, t), M (y, Qy, t)}, \end{matrix}$ with $φ (t) = \frac{t}{k}$ and k > 1.

Therefore, all conditions of Theorem 13 are satisfied. Hence, Q has a fixed point which is a solution for the equation (51) in X. Finally, observe that if x, y ∈ X are two fixed point of Q in X, then x ≤ max {x, y} , y ≤ max {y, x}, and z = max {x, y} ∈ X. Additionally, M (x, z, t) >0 and M (x, y, t) >0 for all t > 0. Therefore, it follows from Theorem 13 that the fixed point of Q is unique and thus the solution of (51) is also unique in X. This completes the proof.

□ Finally, we provide the following example which supports Theorem 15.

Example 7. Consider the boundary value problem of the fractional differential equation ${c_{D_{0^{+}}^{\frac{3}{2}}} (y (t)) = \frac{y (t)}{3 (1 + y (t))}, t \in [0, 1]; y (0) = 0, y^{'} (1) - 1$ (55) Take $h (t, y (t)) = \frac{y (t)}{3 (1 + y (t))}, (t, y (t)) \in [0, 1] \times [0, \infty) .$ Let x (t) , y (t) ∈ [0, ∞) and t ∈ [0, 1]. Then $\begin{matrix} | h (t, y (t)) - h (t, x (t)) | & = & \frac{1}{3} | \frac{y (t)}{1 + y (t)} - \frac{y (t)}{1 + y (t)} | \\ = & \frac{1}{3} | \frac{y (t) - x (t)}{(1 + y (t)) (1 + x (t))} | \end{matrix}$ $\begin{matrix} \leq & \frac{1}{3} | y (t) - x (t) | . \end{matrix}$ Hence, condition(i) of Theorem 15 is satisfied for $λ = \frac{1}{3}$ . Also, we have $\frac{1}{3} [\frac{1}{Γ (5 / 2)} + \frac{1}{Γ (3 / 2)}] = \frac{7}{9 \sqrt{π}} < 1 .$ Thus, (52) holds. Taking y₀ = 0, then $\begin{matrix} 0 \leq \frac{1}{Γ (3 / 2)} \int_{0}^{t} (t - s)^{1 / 2} h (s, 0) ds \\ - \frac{t}{Γ (1 / 2)} \int_{0}^{1} (1 - s)^{- 1 / 2} h (s, 0) ds \\ + t = t, t \in [0, 1] . \end{matrix}$ This shows that condition (ii) of Theorem 15 is also fulfilled. Therefore, Equation (53) has a unique solution on [0, 1].

6 Conclusion

In this article, a new fuzzy contraction mapping is presented, and it is demonstrated that such mappings have a fixed point in τ-complete fuzzy metric spaces. For the following random linear random operator problem, we applied the results to demonstrate the existence and uniqueness of the random solution. On a specific fuzzy metric space, we also demonstrated the existence and uniqueness of the solutions for the nonlinear Volterra integral equations. The presence and existence of a solution to boundary value problems for fractional differential equations using the Caputo fractional derivative were also investigated using our primary results.

Footnotes

Acknowledgements

The authors take this opportunity to thank the referees for their very helpful comments on the submitted version of the paper.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.

Funding

This study is not funded.

Declaration

This manuscript does not contain any studies with human participants or animals performed by the author.

Consent

The manuscript is submitted with the consent of the author.

Contribution

This manuscript is a work of the author in all aspects.

References

Agarwal

R.P.

, Benchohra

and Hamani

, Boundary value problems for fractional differential equations, Georgian Math J 16 (2009), 401–411.

Azam

and Beg

, Common fixed points of fuzzy maps,, Math Comput Model 49 (2009), 1331–1336.

Beg

and Abbas

, Invariant approximation for fuzzy nonexpansive mappings,, Math Bohem 136(1) (2011), 51–59.

Butnariu

, Fixed point for fuzzy mapping,, Fuzzy Sets Syst 7 (1982), 191–207.

Chang

S.S.

, Probabilistic contractor and the solutions for nonlinear equations in Probabilistic normed spaces, Chines Science Bull 19 (1990), 1451–1454.

Estruch

V.D.

and Vidal

, A note on fixed fuzzy points for fuzzy mappings, Proceedings of the II Italian-Spanish Congress on General Topology and Its Applications (Italian) (Trieste, 1999), Rend Ist Mat Univ Trieste 32(suppl. 2) (2001), 39–45.

Fristedt

, Gray

A modern approach to probability theory, Boston: BirkhÃd’user (1996).

Fang

J.X.

, A note on fixed point theorems of Hadzic, Fuzzy Sets and Systems 48(3) (1992), 391–395.

George

and Veeramani

, On some results in fuzzy metric spaces,, J Fuzzy Sets Syst 64 (1994), 395–399.

10.

George

and Veeramani

, On some results of analysis for fuzzy metric spaces,, Fuzzy Sets and Systems 90(3) (1997), 365–368.

11.

Gupta

and Verma

, Common fixed points using (ψ, φ)-type contractive maps in fuzzy metric spaces, , Journal of Applied Research and Technology 18 (2020), 27–33.

12.

Hadzić

, Fixed point theorems for multivalued mappings in some classes of fuzzy metric spaces, Fuzzy Sets and Systems 29 (1989), 115–125.

13.

Hadzić

, Pap

Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications, vol. 536, Kluwer Academic Publishers, Dordrecht, 2001.

14.

Heilpern

, Fuzzy mappings and fixed point theorems,, J Math Anal Appl 83(1981), 566–569.

15.

Huang

, Carić.

, Dosenović

, Rakić

, Brdar

, Fixed-Point Theorems in Fuzzy Metric Spaces via Fuzzy F-Contraction, Mathematics 9 (2021), 641. https://doi.org/10.3390/math9060641.

16.

Kilbas

A.A.

and Marzan

S.A.

, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, 82–86, 142;, English Translation Differ Equ 41 (2005), 84–89.

17.

Klement

E.P.

, Mesiar

, Pap

Triangular norms, Kluwer Academic Publishers, Dordrecht, 2000.

18.

Kaleva

and Seikkala

, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984), 215–229.

19.

Kramosil

and Michalek

, Fuzzy metric and Statistical metric spaces,, Kybernetika 11 (1975), 326–334.

20.

Lee

A.C.

and Padgett

W.J.

, Random contractors and the solution of random nonlinear equations, Nonlinear Anal TMA 1 (1977), 173–185.

21.

Lee

A.C.

and Padgett

W.J.

, Random contractors with random nonlinear majorant functions, Nonlinear Anal TMA 3 (1979), 707–715.

22.

Lee

A.C.

and Padgett

W.J.

, Solutions of random operator equations by random step-contractors, Nonlinear Anal TMA 4 (1980), 145–151.

23.

Mihet

, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems 144(3) (2004), 431–439.

24.

Mihet

, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems 158(8) (2007), 915–921.

25.

Mihet

, Fuzzy Θ-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets and Systems 159(6) (2008), 739–744.

26.

Mihet

, A note on fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems 251 (2014), 83–91.

27.

Mihet

On fuzzy ɛ-contractive mappings in fuzzy metric spaces, Fixed Point Theory Appl 2007, Article ID 87471 (2007).

28.

Qiu

, Shu

and Guan

, Common fixed point theorems for fuzzy mappings under φ-contraction condition,, Chaos Solitons Fractals 41(1) (2009), 360–367.

29.

Rashid

M.H.M.

, Some Results on Fuzzy Metric Spaces,, Int J Open Problems Compt Math 9(4) (2016), 1–23.

30.

Rashid

M.H.M.

, Hicks Type Contractions and SB-type Contractions in Fuzzy Metric Spaces,, Mayfeb J Math 3 (2016), 16–23.

31.

Rashid

M.H.M.

, Minimax theorems in fuzzy metric spaces, Comp Appl Math 37 (2018), https://doi.org/10.1016/j.jsurg.2016.07.008.

32.

Rashid

M.H.M.

Fuzzy Contractor and Nonlinear Operator Equation in fuzzy normed spaces, Novi Sad Journal of Mathematics to appear.

33.

Samreen

, Hussain

, Aydi

and Gaba

Y.U.

, κ _ G-Contractive Mappings in Fuzzy Metric Spaces, Journal of Mathematics 2022, Article ID 5920124, 10 pages.

34.

Schweizer

, Sklar

Probabilistical Metric Spaces, Dover Publications, New York, 2005.

35.

Serstnev

A.N.

, The notion of the Random norm spaces, Dolk Akad Nauk USSR 149 (1963), 280–283.

36.

Stevens

, Metrically generated probabilistic metric spaces, , {Fundamenta Mathematicae 61 (1967), 259–269 10.4064/fm-61-3-259-269.

37.

Singh

S.L.

and Talwar

, Fixed points of fuzzy mappings,, Soochow J Math 19(1) (1993), 95–102.

38.

Wardowski

, Fuzzy contractive mappings and fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 222 (2013), 108–114.

39.

Weiss

M.D.

, Fixed points and induced fuzzy topologies for fuzzy sets,, J Math Anal Appl 50 (1975), 142–150.

40.

Wang

and Wu

, Fuzzy α-almost convex mappings and fuzzy fixed point theorems for fuzzy mappings,, Ital J Pure Appl Math 17 (2005), 137–150.

41.

Xiao

J.Z.

and Zhu

X.H.

, On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets and Systems 125(2) (2002), 153–161.

42.

Xiao

J.Z.

and Zhu

X.H.

, Fuzzy normed operators space and its completeness, Fuzzy Sets and Systems 133(3) (2003), 389–399.