Hyers-Ulam-Rassias Fuzzy Stability of Bi-additive θ -random operator inequalities: A fixed point technique

Abstract

We attempt to solve some bi-additive θ-random operator inequalities and use the fixed point technique to prove the fuzzy version of Hyers-Ulam-Rassias stability of them.

Keywords

Hyers-Ulam-Rassias stability fuzzy sets 54H20 46L05 39B62 43A22

1 Introduction and preliminaries

Inspired by the basic notions of Katsaras’ paper [1], using the concept of Minkowski functionals of L-fuzzy sets introduced by Höhle [2], and fuzzy metric space by Kaleva and Seikkala [3], in 1988, Morsi [4] introduced a notion of fuzzy (pseudo) normed spaces. Next, by using the notion of random normed spaces introduced by S̆erstnev [5], and studied by MuS̆tari [6], Radu [7], Cheng and Mordeson [8], Rano and Bag [9], Shi [10, 11] defined the fuzzy normed spaces. Next, some authors studied fuzzy functional analysis and applications, in particular, Hyers-Ulam-Rassias stability problems in fuzzy normed spaces [12 –16].

Let (Γ, Σ, ξ) be a probability measure space. Assume that $(T, B_{T})$ and $(S, B_{S})$ are Borel measureable spaces, in which T and S are fuzzy Banach spaces and F : Γ × T² → S is a random operator. In fuzzy Banach spaces, first we solve the additive θ-random operator inequalities $ν (F (γ, t + s, p - r) + F (γ, t - s, p + r) - 2 F (γ, t, p) + 2 F (ω, s, r), τ)$ (1.1) $\begin{matrix} \geq ν (θ (2 F (γ, \frac{t + s}{2}, p - r) + 2 F (γ, \frac{t - s}{2}, p + r) - 2 F (γ, t, p) + 2 F (γ, s, r)), τ), \\ ν (2 F (γ, \frac{t + s}{2}, p - r) + 2 F (γ, \frac{t - s}{2}, p + r) - 2 F (γ, t, p) + 2 F (γ, s, r), τ) \end{matrix}$ (1.2) $\geq ν (θ (F (γ, t + s, p - r) + F (γ, t - s, p + r) - 2 F (γ, t, p) + 2 F (γ, s, r)), τ)$ where $0 \neq θ \in ℂ$ is fixed and |θ|<1.

By the fixed point technique, we study the Hyers-Ulam-Rassias stability of the above additive θ-random operator inequalities (1.1) and (1.2) in fuzzy Banach spaces.

2 Preliminaries

In this paper, we let I = [0, 1] and J = (0, 1].

Definition 2.1. ([17, 18]) A continuous triangular norm (shortly, a ct-norm) is a continuous mapping Δ from I² to I such that

(a) Δ (τ, υ) = Δ (υ, τ) and Δ (τ, κ (υ, ϑ)) = Δ (κ (τ, υ) , ϑ) for all τ, υ, ϑ ∈ I;

(b) Δ (τ, 1) = τ for all τ ∈ I;

Some examples of the t-norms are:

(1) Δ_P (τ, υ) = τυ;

(2) Δ_M (τ, υ) = min {τ, υ} = ⋀ {τ, υ};

(3) Δ_L (τ, υ) = max {τ + υ - 1, 0} (: the Lukasie-wicz t-norm).

Definition 2.2. ([19, 20]) Suppose that Δ is a ct-norm, S is a vector space and ν is a fuzzy set from S × (0, ∞) to J. In this case, the ordered tuple (S, ν, Δ) is called a fuzzy normed space (in short, FN-space) if the following conditions are satisfied:

(FN1) ν (s, τ) =1 for all τ > 0 if and only if s = 0;

(FN2) $ν (α s, τ) = ν (s, \frac{τ}{| α |})$ for all s ∈ S and $α \in ℝ$ with α ≠ 0;

(FN3) ν (t + s, τ + ς) ≥ Δ (ν (t, τ) , ν (s, ς)) for all t, s ∈ S and τ, ς ≥ 0.

(FN4) ν (s, .) : (0, ∞) → J is continuous.

A complete FN-space is said to be fuzzy Banach space (in short FB-space).

Let (S, ∥ · ∥) be a linear normed space. Then $ν (s, ς) = exp (- \frac{∥ s ∥}{ς})$ for all ς > 0 defines a fuzzy norm and the ordered tuple (S, ν, Δ_M) is a FN-space.

Let (Γ, Σ, ξ) be a probability measure space. Assume that $(T, B_{T})$ and $(S, B_{S})$ are Borel measureable spaces, in which T and S are complete FN spaces. A mapping F : Γ × T → S is said to be a random operator if {γ : F (γ, t) ∈ B} ∈ Σ for all t in T and $B \in B_{S}$ . Also, F is random operator, if F (γ, t) = s (γ) be a S-valued random variable for every t in T. A random operator F : Γ × T → S is called linear if F (γ, αt₁ + βt₂) = αF (γ, t₁) + βF (γ, t₂) almost every where for each t₁, t₂ in T and α, β are scalers, and bounded if there exists a nonnegative real-valued random variable M (γ) such that $ν (F (γ, t_{1}) - F (γ, t_{2}), M (γ) τ) \geq ν (t_{1} - t_{2}, τ),$ almost every where for each t₁, t₂ in T and τ > 0.

Recently, some authors have published some papers on approximation of functional equations in several spaces by the direct technique and the fixed point technique, for example, fuzzy Menger normed algebras [23], fuzzy metric spaces [24, 25], fuzzy normed spaces [26], non-Archimedian random Lie C^*-algebras [27], non-Archimedean random normed spaces [28], random multi-normed space [29], see also [30, 31] and [32, 33].

Note that, a [0, ∞]-valued metric is called a generalized metric.

Theorem 2.3. ([21, 22]) Consider a complete generalized metric space (T, δ) and a strictly contractive function Λ : T → T with Lipschitz constant β < 1. So, for every given element t ∈ T, either $δ (Λ^{n} t, Λ^{n + 1} t) = \infty$ for each $n \in ℕ$ or there is $n_{0} \in ℕ$ such that

(1) δ (Λⁿt, Λⁿ⁺¹t) < ∞ , ∀ n ≥ n₀;

(2) the fixed point s^* of Λ is the convergent point of sequence {Λⁿt};

(3) in the set V = {s ∈ T ∣ δ (Λ^{n
₀}t, s) < ∞}, s^* is the unique fixed point of Λ;

(4) (1 - β) δ (s, s^∗) ≤ δ (s, Λs) for every s ∈ V.

3 Bi-additive θ-random operator inequality (1.1)

Now we generalize a result of Park [34, 35] (see also, [36 –41]).

Lemma 3.1. Let the random operator F : Γ × T² → S hold in (1.1) and F (γ, 0, p) = F (γ, t, 0) =0 almost every where for each t, s, p, r ∈ T and γ ∈ Γ, so F : Γ × T² → S is bi-additive.

Proof. Assume that F : Γ × T² → S satisfies (1.1).

Putting t = s and r = 0 in (1.1), imply that F (γ, 2t, p) =2F (γ, t, p) almost every where for all t, p ∈ T and γ ∈ Γ. Thus $\begin{matrix} ν (F (γ, t + s, p - r) + F (γ, t - s, p + r) - 2 F (γ, t, p) + 2 F (γ, s, r), τ) \\ \geq ν (θ (2 F (γ, \frac{t + s}{2}, p - r) + 2 F (γ, \frac{t - s}{2}, p + r) - 2 F (γ, t, p) + 2 F (γ, s, r)), τ) \\ = ν (θ (F (γ, t + s, p - r) + F (γ, t - s, p + r) - 2 F (γ, t, p) + 2 F (γ, s, r)), τ) \end{matrix}$

and so

$\begin{matrix} F & (γ, t + s, p - r) + F (γ, t - s, p + r) \\ - 2 F (γ, t, p) + 2 F (γ, s, r) = 0 \end{matrix}$ (3.1) for all t, s, p, r ∈ T . γ ∈ Γ, τ > 0.

Putting r = 0 in (3.1), implies that F (γ, t + s, p) + F (γ, t - s, p) =2F (γ, t, p) and so $F (γ, t_{1}, p) + F (γ, s_{1}, p) = 2 F (γ, \frac{t_{1} + s_{1}}{2}, p) = F (γ, t_{1} + s_{1}, p)$ for all t₁ : = t + s, s₁ : = t - s, p ∈ X, since |θ|≤1 and F (γ, 0, p) =0 for all p ∈ T. So F : Γ × T² → S is additive in the second variable.

Similarly, one can show that F : Γ × T² → S is additive in the third variable. Hence F : Γ × T² → S is a random operator bi-additive.□

Theorem 3.2. Let (Γ, Σ, ξ) be a probability measure space, let φ : T⁴ × (0, ∞) → J be a fuzzy set such that there is an L < 1 with $φ (\frac{t}{2}, \frac{s}{2}, p, 0, τ) \geq φ (t, s, p, 0, \frac{2 τ}{L})$ for all t, s, p, r ∈ T, τ > 0,

$lim_{n \to \infty} φ (\frac{t}{2^{n}}, \frac{s}{2^{n}}, p, 0, \frac{τ}{2^{n}}) = 1$ (3.2) for all t, s ∈ T, τ > 0. Let F : Γ × T² → S be a random operator satisfying F (γ, t, 0) = F (γ, 0, p) =0 for all t, p ∈ T, γ ∈ Γ and

$\begin{matrix} ν (F (γ, t + s, p - r) + F (γ, t - s, p + r) - 2 F (γ, t, p) + 2 F (γ, s, r), τ) \\ \geq min {ν (θ (2 F (γ, \frac{t + s}{2}, p - r) + 2 F (γ, \frac{t - s}{2}, p + r) - 2 F (γ, t, p) + 2 F (γ, s, r)), τ), \\ φ (t, s, p, r, τ)} \end{matrix}$ (3.3) for all t, s, p, r ∈ T, γ ∈ Γ, τ > 0. So, there is a unique bi-additive random operator Q : Γ × T² → S in which

$\begin{matrix} ν (F (γ, t, p) - Q (γ, t, p), τ) \\ \geq φ (t, t, p, 0, \frac{2 (1 - L) τ}{L}) \end{matrix}$ (3.4) for all t, p ∈ T, γ ∈ Γ, τ > 0.

Proof. Putting r = 0 and s = t in (3.3), we get

$ν (F (γ, 2 t, p) - 2 F (γ, t, p), τ) \geq φ (t, t, p, 0, τ)$ (3.5) for each t, p ∈ X, γ ∈ Γ, τ > 0. So $\begin{matrix} ν (F (γ, t, p) - 2 F (γ, \frac{t}{2}, p), τ) \\ \geq φ (\frac{t}{2}, \frac{t}{2}, p, 0, τ) \end{matrix}$ for all t, p ∈ X, γ ∈ Γ, τ > 0. On the set $\begin{matrix} B : = {K : Γ \times T^{2} \to S; \\ K (γ, t, 0) = K (γ, 0, p) = 0 \forall t, p \in T, γ \in Γ} \end{matrix}$ (3.6) we define the following generalized metric:

$\begin{matrix} δ (F, K) = \\ inf {μ \in ℝ_{+} : ν (F (γ, t, p) - K (γ, t, p), τ) \\ \geq φ (t, t, p, 0, \frac{τ}{μ}), \forall t, p \in T, γ \in Γ, τ > 0}, \end{matrix}$ (3.7) In [42], Mihet and Radu proved that (B, δ) is complete (see also [43]).

Now we consider the linear mapping Λ : B → B such that $Λ F (γ, t, p) : = 2 F (γ, \frac{t}{2}, p)$ (3.8) for all t, p ∈ T,γ ∈ Γ . Consider F, k ∈ B such that δ (F, K) = ɛ . So, $ν (F (γ, t, p) - K (γ, t, p), t) \geq φ (t, t, p, 0, \frac{τ}{ɛ})$ for each t, p ∈ T, γ ∈ Γ, τ > 0. Hence $\begin{matrix} ν (Λ F (γ, t, p) - Λ K (γ, t, p), τ) \\ = ν (F (γ, \frac{t}{2}, p) - K (γ, \frac{t}{2}, p), \frac{τ}{2}) \\ \geq φ (\frac{t}{2}, \frac{t}{2}, p, 0, \frac{τ}{2 ɛ}) \\ \geq φ (t, t, p, 0, \frac{τ}{L ɛ}) \end{matrix}$ for all t, p ∈ T, γ ∈ Γ, τ > 0 . Then, δ (F, K) = ɛ we conclude that δ (ΛF, Λk) ≤ Lɛ . This means that $δ (Λ F, Λ K) \leq L δ (F, K)$ for all F, K ∈ B. By (3.5) we have that $ν (F (γ, t, p) - 2 F (γ, \frac{t}{2}, p), τ) \geq φ (\frac{t}{2}, \frac{t}{2}, p, 0, τ)$

for all t, p ∈ T, γ ∈ Γ, τ > 0 . So $ν (F (γ, t, p) - Λ F (γ, t, p)), τ) \geq φ (t, t, p, 0, \frac{2 τ}{L})$ for all t, p ∈ T, γ ∈ Γ, τ > 0. Hence $δ (F, Λ F) < \frac{L}{2} < 1 < \infty$ . Theorem 2.3, implies that, there exists a random operator Q : Γ × T² → S such that

(1) A fixed point for function Λ, is Q, $Q (γ, t, p) = 2 Q (γ, \frac{t}{2}, p)$ (3.9) for all t, p ∈ T, γ ∈ Γ, which is unique in the set $M = {K \in B : δ (F, K) < \infty} .$

(2) δ (ΛⁿF, Q) →0 as n→ ∞, which implies that $lim_{n \to \infty} 2^{n} F (γ, \frac{t}{2^{n}}, p) = Q (γ, t, p)$ for each t, p ∈ T, γ ∈ Γ . (3) $δ (F, Q) \leq \frac{1}{1 - L} δ (F, Λ F)$ , which implies that $δ (F, Q) \leq \frac{1}{1 - L} .$ By use of definition δ we get $ν (F (γ, t, p) - Q (γ, t, p), τ) \geq φ (t, t, p, 0, \frac{2 (1 - L) τ}{L})$ for all t, p ∈ T, γ ∈ Γ, τ > 0 . It follows from (3.3) that $\begin{matrix} ν (Q (γ, t + s, p - r) + Q (γ, t - s, p + r) - 2 Q (γ, t, p) + 2 Q (γ, s, r), τ) \\ = lim_{n \to \infty} ν (2^{n} (F (γ, \frac{t + s}{2^{n}}, p - r) + F (γ, \frac{t - s}{2^{n}}, p + r) - 2 F (γ, \frac{t}{2^{n}}, p) + 2 Fg (γ, \frac{s}{2^{n}}, r)), τ) \\ \geq min {lim_{n \to \infty} ν (2^{n} θ (2 F (γ, \frac{t + s}{2^{n + 1}}, p - r) + 2 F (γ, \frac{t - s}{2^{n + 1}}, p + r) - 2 F (γ, \frac{t}{2^{n}}, p) + 2 F (γ, \frac{s}{2^{n}}, r)), τ), \\ lim_{n \to \infty} φ (\frac{t}{2^{n}}, \frac{t}{2^{n}}, p, 0, \frac{τ}{2^{n}})} \\ \geq ν (θ (2 Q (γ, \frac{t + s}{2}, p - r) + 2 Q (γ, \frac{t - s}{2}, p + r) - 2 Q (γ, t, p) + 2 Q (γ, s, r)), τ) \end{matrix}$ for all t, s, p, r ∈ T, γ ∈ Γ, τ > 0, since $lim_{n \to \infty} φ (\frac{t}{2^{n}}, \frac{t}{2^{n}}, p, o, \frac{τ}{2^{n}}) = 1$ . So

$\begin{matrix} ν (Q (γ, t + s, p - r) + Q (γ, t - s, p + r) - 2 Q (γ, t, p) + 2 Q (γ, s, r), τ) \\ \geq ν (θ (2 Q (γ, \frac{t + s}{2}, p - r) + 2 Q (γ, \frac{t - s}{2}, p + r) - 2 Q (γ, t, p) + 2 Q (γ, s, r)), τ) \end{matrix}$

for all t, s, p, r ∈ T, γ ∈ Γ, τ > 0. Now, Lemma 3.1, implies that the random operator Q : Γ × T² → S is bi-additive.□

Corollary 3.3. Let (Γ, Σ, ξ) be a probability measure space. Assume that q > 1, σ ≥ 0 and F : Γ × T² → S be a random operator satisfying F (γ, t, 0) = F (γ, 0, p) =0 and $\begin{matrix} ν (F (γ, t + s, p - r) + F (γ, t - s, p + r) \\ - 2 F (γ, t, p) + 2 F (γ, s, r), τ) \\ \geq min {ν (θ (2 F (γ, \frac{t + s}{2}, p - r) \\ + 2 F (γ, \frac{t - s}{2}, p + r) \\ - 2 F (γ, t, p) + 2 F (γ, s, r)), τ), \\ exp (- \frac{σ (∥ t ∥^{q} + ∥ s ∥^{q}) (∥ p ∥^{q} + ∥ r ∥^{q})}{τ})} \end{matrix}$ (3.10)for all t, s, p, r ∈ T, γ ∈ Γ, τ > 0. So, there is a unique bi-additive random operator Q : Γ × T² → S such that $\begin{matrix} ν (F (γ, t, p) - Q (γ, t, p), τ) \\ \geq exp (- \frac{2 σ ∥ t ∥^{q} ∥ p ∥^{q}}{(2^{q} - 2) τ}) \end{matrix}$ for all t, p ∈ T, γ ∈ Γ, τ > 0.

Proof. In Theorem 3.2 put $φ (t, s, p, r, τ) = exp (- \frac{σ (∥ t ∥^{q} + ∥ s ∥^{q}) (∥ p ∥^{q} + ∥ r ∥^{q})}{τ})$ for all t, s, p, r ∈ T, τ > 0 with L = 2^1-q. □

Theorem 3.4. Let (Γ, Σ, ξ) be a probability measure space, let φ : T⁴ × (0, ∞) → J be a fuzzy set such that there is an L < 1 with $φ (t, s, p, 0, τ) \geq φ (\frac{t}{2}, \frac{s}{2}, p, 0, \frac{τ}{2 L})$ for all t, s, p, r ∈ T, τ > 0, $lim_{n \to \infty} φ (2^{n} t, 2^{n} s, p, o, 2^{n} τ) = 1$ for all t, s, p ∈ T, τ > 0. Let F : Γ × T² → S be a random operator satisfying (3.3) and F (γ, t, 0) = F (γ, 0, p) =0 for all t, s ∈ T, γ ∈ Γ . Then there exists a unique bi-additive random operator Q : Γ × T² → S such that

$\begin{matrix} ν (F (γ, t, p) - Q (γ, t, p), τ) \\ \geq φ (t, t, p, 0, 2 (1 - L) τ) \end{matrix}$ (3.11) for all t, p ∈ T, γ ∈ Γ, τ > 0.

Proof. Putting r = 0 and s = t in (3.3), we get $\begin{matrix} ν (F (γ, 2 t, p) - 2 F (γ, t, p), τ) \geq φ (t, t, p, 0, τ) \end{matrix}$ for all t, p ∈ T, γ ∈ Γ, τ > 0. So $\begin{matrix} ν (F (γ, t, p) - \frac{1}{2} F (γ, 2 t, p), τ) \\ \geq φ (t, t, p, 0, 2 τ) \end{matrix}$ for all t, p ∈ T, γ ∈ Γ, τ > 0. On the set $\begin{matrix} B : = {K : Γ \times T^{2} \to S; \\ K (γ, t, 0) = K (γ, 0, p) = 0 \forall t, p \in T, γ \in Γ} \end{matrix}$ (3.12) we define the following generalized metric:

$\begin{matrix} δ (F, K) = inf {μ \in ℝ_{+} : ν (F (γ, t, p) \\ - K (γ, t, p), τ) \geq φ (t, t, p, 0, \frac{τ}{μ}), \\ \forall t, p \in T, γ \in Γ, τ > 0}, \end{matrix}$ (3.13)

In [42], Mihet and Radu proved that (B, δ) is complete (see also [43]). Now, we consider the linear mapping Λ : B → B such that $Λ F (γ, t, p) : = \frac{1}{2} F (γ, 2 t, p)$ (3.14) for all t, p ∈ T,γ ∈ Γ .

Consider F, K ∈ B such that δ (F, K) = ɛ . Then $ν (F (γ, t, p) - K (γ, t, p), t) \geq φ (t, t, p, 0, \frac{2 τ}{ɛ})$ for all t, p ∈ T, γ ∈ Γ, τ > 0. Hence $\begin{matrix} ν (Λ F (γ, t, p) - Λ K (γ, t, p), τ) \\ = ν (F (γ, 2 t, p) - K (γ, 2 t, p), 2 t) \\ \geq φ (2 t, 2 t, p, 0, \frac{4 τ}{ɛ}) \geq φ (t, t, p, 0, \frac{2 τ}{L ɛ}) . \end{matrix}$ for all t, p ∈ T, γ ∈ Γ, τ > 0 . Then, δ (F, K) = ɛ we conclude that δ (ΛF, ΛK) ≤ Lɛ . This means that $δ (Λ F, Λ K) \leq L δ (F, K)$ for all F, K ∈ B. By (3.5) we have that $ν (F (γ, t, p) - \frac{1}{2} F (γ, 2 t, p), τ) \geq φ (t, t, p, 0, 2 τ)$ for all t, p ∈ T, γ ∈ Γ, τ > 0 . So $ν (F (γ, t, p) - Λ F (γ, t, z)), τ) \geq φ (t, t, p, 0, 2 τ)$ for all t, p ∈ T, γ ∈ Γ, τ > 0. Hence δ (F, ΛF)<1 < ∞. Theorem 2.3, implies that, there exists a random operator Q : Γ × T² → S such that (1) A fixed point for function Λ, is Q, $Q (γ, t, p) = \frac{1}{2} Q (γ, 2 t, p)$ (3.15) for all t, p ∈ T, γ ∈ Γ, which is unique in the set $M = {K \in B : δ (F, K) < \infty} .$ (2) δ (ΛⁿF, Q) →0 as n→ ∞. This implies the equality $lim_{n \to \infty} \frac{1}{2^{n}} F (γ, 2^{n} t, p) = Q (γ, t, p)$ for all t, p ∈ T, γ ∈ Γ, τ > 0 . (3) $δ (F, Q) \leq \frac{1}{1 - L} δ (F, Λ F)$ , which implies the inequality $δ (F, Q) \leq \frac{1}{1 - L} .$ By use of definition δ we get $ν (F (γ, t, p) - Q (γ, t, p), τ) \geq φ (t, t, p, 0, 2 (1 - L) τ)$ for all t, p ∈ T, γ ∈ Γ, τ > 0 . The rest of the proof is similar to the proof of the Theorem 3.2 □

Corollary 3.5. Let (Γ, Σ, ξ) be a probability measure space. Assume that q < 1 and σ ≥ 0 and F : Γ × T² → S be a random operator satisfying (3.3) and F (γ, t, 0) = F (γ, 0, p) =0 for all t, p ∈ T, γ ∈ Γ. Then there exists a unique bi-additive random operator Q : Γ × T² → S such that

$\begin{matrix} ν (F (γ, t, p) - Q (γ, t, p), τ) \\ \geq exp (- \frac{2 σ ∥ t ∥^{q} ∥ p ∥^{q}}{(2 - 2^{q}) τ}) \end{matrix}$ (3.16) for all t, p ∈ T, γ ∈ Γ, τ > 0.

Proof. In Theorem 3.4 put $\begin{matrix} φ (t, s, p, r, τ) \\ = exp (- \frac{σ (∥ t ∥^{q} + ∥ s ∥^{q}) (∥ p ∥^{q} + ∥ r ∥^{q})}{τ}) \end{matrix}$ for all t, s, p, r ∈ T, ττ > 0, with L = 2^q-1. □

4 Bi-additive θ-random operator inequality (1.2)

We solve and investigate the bi-additive θ-random operator inequality (1.2) in complex FN spaces.

Lemma 4.1. If a random operator F : Γ × T² → S satisfies F (γ, 0, p) = F (γ, t, 0) =0 and

$\begin{matrix} ν (2 F (γ, \frac{t + s}{2}, p - r) + 2 F (γ, \frac{t - s}{2}, p + r) \\ - 2 F (γ, t, p) + 2 F (γ, s, r), τ) \\ \geq ν (θ (F (γ, t + s, p - r) + F (γ, t - s, p + r) \\ - 2 F (γ, t, p) + 2 F (γ, s, r)), τ) \end{matrix}$ (4.1) for all t, s, p, r ∈ T, γ ∈ Γ, τ > 0, then F : Γ × T² → S is bi-additive.

Proof. Assume that F : Γ × T² → S satisfies (4.1).

Letting s = r = 0 in (4.1), we get $4 F (γ, \frac{t}{2}, p) = 2 F (γ, t, p)$ for all t, p ∈ T, γ ∈ Γ. Thus $\begin{matrix} ν (F (γ, t + s, p - r) + F (γ, t - s, p + r) \\ - 2 F (γ, t, p) + 2 F (γ, s, r), τ) \\ = ν (2 F (γ, \frac{t + s}{2}, p - r) + 2 F \\ (γ, \frac{t - s}{2}, p + r) - 2 F (γ, t, p) + 2 F (γ, s, r), τ) \\ \geq ν (θ (F (γ, t + s, p - r) + F (γ, t - s, p + r) \\ - 2 F (γ, t, p) + 2 F (γ, s, r)), τ) \end{matrix}$ and so $\begin{matrix} F (γ, t + s, p - r) + F (γ, t - s, p + r) \\ - 2 F (γ, t, p) + 2 F (γ, s, r) = 0 \end{matrix}$ for all t, s, p, r ∈ T, γ ∈ Γ.

Using similar method of Lemma 3.1, the proof will be complete. □

Theorem 4.2. Let (Γ, Σ, ξ) be a probability measure space, let φ : T⁴ × (0, ∞) → J be a fuzzy set such that there is an L < 1 with $φ (\frac{t}{2}, \frac{s}{2}, p, 0, τ) \geq φ (t, s, p, 0, \frac{2 τ}{L})$ for all t, s, p, r ∈ T. Let F : Γ × T² → S be a random operator satisfying F (γ, t, 0) = F (γ, 0, p) =0 for all t, p ∈ T, γ ∈ Γ and $\begin{matrix} ν (2 F (γ, \frac{t + s}{2}, p - r) + 2 F (γ, \frac{t - s}{2}, p + r) \\ - 2 F (γ, t, z) + 2 F (γ, s, r), τ) \\ \geq min {ν (θ (F (γ, t + s, p - r) + F (γ, t - s, p + r) \\ - 2 F (γ, t, p) + 2 F (γ, s, r)), τ), \\ φ (t, s, p, r, τ)} \end{matrix}$ (4.2) for all t, s, p, r ∈ T, γ ∈ Γ, τ > 0 . Then there exists a unique bi-additive random operator Q : Γ × T² → S such that

$\begin{matrix} ν (F (γ, t, p) - Q (γ, t, p), τ) \\ \geq φ (t, 0, p, 0, 2 (1 - L) τ) \end{matrix}$ (4.3) for all t, p ∈ T, γ ∈ Γ, τ > 0.

Proof. Letting s = r = 0 in (4.2), we get

$\begin{matrix} ν (4 F (γ, \frac{t}{2}, p) - 2 F (γ, t, p), τ) \\ \geq φ (t, 0, p, 0, τ) \end{matrix}$ (4.4) for all t, p ∈ T, γ ∈ Γ, τ > 0. So $\begin{matrix} ν (F (γ, t, p) - 2 F (γ, \frac{t}{2}, p), τ) \\ \geq φ (t, 0, p, 0, 2 τ) \end{matrix}$ for all t, p ∈ T, γ ∈ Γ, τ > 0. On the set $\begin{matrix} B : = {K : Γ \times T^{2} \to S; \\ K (γ, t, 0) = K (γ, 0, p) = 0 \forall t, p \in T, γ \in Γ} \end{matrix}$ we define the following generalized metric: $\begin{matrix} δ (F, K) = \inf {μ \in ℝ_{+} : ν (F (γ, t, p) - K (γ, t, p), τ) \\ \geq φ (t, t, p, 0, \frac{τ}{μ}), \forall t, p \in T, γ \in Γ, τ > 0}, \end{matrix}$ In [42], Mihet and Radu proved that (B, δ) is complete (see also [43]). Now, we consider the linear mapping Λ : B → B such that $Λ F (γ, t, p) : = 2 F (γ, \frac{t}{2}, p)$ for all t, p ∈ T,γ ∈ Γ . Consider F, K ∈ B such that δ (F, K) = ɛ . Then $ν (F (γ, t, p) - K (γ, t, p), t) \geq φ (t, t, p, 0, \frac{τ}{ɛ})$ for all t, p ∈ T, γ ∈ Γ, τ > 0. Hence $\begin{matrix} ν (Λ F (γ, t, p) - Λ K (γ, t, p), τ) \\ = ν (F (γ, \frac{t}{2}, p) - K (γ, \frac{t}{2}, p), \frac{τ}{2}) \\ \geq φ (\frac{t}{2}, \frac{t}{2}, p, 0, \frac{τ}{2 ɛ}) \\ \geq φ (t, t, p, 0, \frac{τ}{L ɛ}) \end{matrix}$ for all t, p ∈ T, γ ∈ Γ, τ > 0 . Then, δ (F, K) = ɛ implies that δ (ΛF, ΛK) ≤ Lɛ . This means that $δ (Λ F, Λ K) \leq L δ (F, K)$ for all F, K ∈ B. By (4.4) we have that $ν (F (γ, t, p) - 2 F (γ, \frac{t}{2}, p), t) \geq φ (t, t, p, 0, 2 τ)$ for all t, p ∈ T, γ ∈ Γ, τ > 0 . So $ν (F (γ, t, p) - Λ F (γ, t, p), τ) \geq φ (t, t, p, 0, 2 τ)$ for all t, p ∈ T, γ ∈ Γ, τ > 0. Hence $δ (F, Λ F) < \frac{1}{2} < 1 < \infty$ . Theorem 2.3, implies that, there exists a random operator Q : Γ × T² → S such that (1) A fixed point for function Λ, is Q, $Q (γ, t, p) = 2 Q (γ, \frac{t}{2}, p)$ (4.5) for all t, p ∈ T, γ ∈ Γ, which is unique in the set $M = {K \in B : δ (F, K) < \infty} .$ (2) δ (ΛⁿF, Q) →0 as n→ ∞, which implies that $lim_{n \to \infty} 2^{n} F (γ, \frac{t}{2^{n}}, p) = Q (γ, t, p)$ for all t, p ∈ T, γ ∈ Γ . (3) $δ (F, Q) \leq \frac{1}{1 - L} δ (F, Λ F)$ , which implies that $δ (F, Q) \leq \frac{1}{1 - L} .$ By use of definition δ we get $ν (F (γ, t, p) - Q (γ, t, p), t) \geq φ (t, t, p, 0, 2 (1 - L) τ)$ for all t, p ∈ T, γ ∈ Γ, τ > 0 . Using similar method of the Theorem 3.2, the proof will be complete. □

Corollary 4.3. Let (Γ, Σ, ξ) be a probability measure space. Assume that q > 1 and σ ≥ 0 and F : Γ × T² → S be a random operator satisfying F (γ, t, 0) = F (γ, 0, p) =0 and $\begin{matrix} ν (2 F (γ, \frac{t + s}{2}, p - r) + 2 F (γ, \frac{t - s}{2}, p + r) \\ - 2 F (γ, t, p) + 2 F (γ, s, r), τ) \\ \geq min {ν (θ (F (γ, t + s, p - r) \\ + F (γ, t - s, p + r) - 2 F (γ, t, p) + 2 F (γ, s, r)), τ), \\ exp (- \frac{σ (∥ t ∥^{q} + ∥ s ∥^{q}) (∥ p ∥^{q} + ∥ r ∥^{q})}{τ})} \end{matrix}$ (4.6) for all t, s, p, r ∈ T, γ ∈ Γ, τ > 0. So, there is a unique bi-additive random operator Q : Γ × T² → S such that

$\begin{matrix} ν (F (γ, t, p) - Q (γ, t, p), τ) \\ \geq exp (- \frac{2^{q - 1} σ ∥ t ∥^{q} ∥ p ∥^{q}}{(2^{q} - 2) τ}) \end{matrix}$ (4.7) for all t, p ∈ T, γ ∈ Γ, τ > 0.

Proof. In Theorem 4.2 put $φ (t, s, p, r, τ) = \frac{τ}{τ + σ (∥ t ∥^{q} + ∥ s ∥^{q}) (∥ p ∥^{q} + ∥ r ∥^{q}}$ for all t, s, p, r ∈ T, τ > 0, with L = 2^1-q.

□

Theorem 4.4. Let (Γ, Σ, ξ) be a probability measure space. Assume that φ : T⁴ × (0, ∞) → J be a fuzzy set such that there exists an L < 1 with $φ (t, s, p, 0, τ) \geq φ (\frac{t}{2}, \frac{s}{2}, p, 0, \frac{τ}{2 L})$ for all t, s, p, r ∈ T. Let F : Γ × T² → S be a random operator satisfying (4.2) and F (γ, t, 0) = F (γ, 0, p) =0 for all t, p ∈ T, γ ∈ Γ. Then there is a unique bi-additive random operator Q : Γ × T² → S such that

$\begin{matrix} ν (F (γ, t, p) - Q (γ, t, p), τ) \\ \geq φ (u, 0, z, 0, \frac{2 (1 - L)}{L} τ) \end{matrix}$ (4.8)for all t, p ∈ T, γ ∈ Γ, τ > 0.

Proof. Letting s = r = 0 in (4.2), we get

$\begin{matrix} ν (4 F (γ, \frac{t}{2}, p) - 2 F (γ, t, p), τ) \\ \geq φ (t, 0, p, 0, τ) \end{matrix}$ (4.9) for all t, p ∈ T, γ ∈ Γ, τ > 0. So $\begin{matrix} ν (F (γ, t, p) - \frac{1}{2} F (γ, 2 t, p), τ) \\ \geq φ (2 t, 0, p, 0, 4 τ) \geq φ (t, 0, p, 0, \frac{2 τ}{L}) \end{matrix}$

for all t, p ∈ T, γ ∈ Γ, τ > 0. On the set $\begin{matrix} B : = {K : Γ \times T^{2} \to S; K (γ, t, 0) = K (γ, 0, p) = 0 \\ \forall t, p \in T, γ \in Γ} \end{matrix}$ we define the following generalized metric: $\begin{matrix} δ (F, K) = \inf {μ \in ℝ_{+} : ν (F (γ, t, p) - K (γ, t, p), τ) \\ \geq φ (t, t, p, 0, \frac{τ}{μ}), \forall t, p \in T, γ \in Γ, τ > 0}, \end{matrix}$ In [42], Mihet and Radu proved that (B, δ) is complete (see also [43]). Now, we consider the linear mapping Λ : B → B such that $Λ F (γ, t, p) : = \frac{1}{2} F (γ, 2 t, p)$ for all t, p ∈ T, γ ∈ Γ . Let F, k ∈ B be given such that δ (F, K) = ɛ . Then $ν (F (γ, t, p) - K (γ, t, p), t) \geq φ (t, t, p, 0, \frac{τ}{ɛ})$ for all t, p ∈ T, γ ∈ Γ, τ > 0. Hence $\begin{matrix} ν (Λ F (γ, t, p) - Λ K (γ, t, p), τ) = ν (F (γ, 2 t, p) \\ - K (γ, 2 t, p), 2 t) \\ \geq φ (2 t, 2 t, p, 0, \frac{τ}{2 ɛ}) \geq φ (t, t, p, 0, \frac{τ}{L ɛ}) \end{matrix}$ for all t, p ∈ T, γ ∈ Γ, τ > 0 . Then δ (F, K) = ɛ we conclude that δ (ΛF, ΛK) ≤ Lɛ . This means that $δ (Λ F, Λ K) \leq L δ (F, K)$ for all F, K ∈ B. It follows from (4.9) that $ν (F (γ, t, p) - \frac{1}{2} F (γ, 2 t, p), t) \geq φ (2 t, 0, p, 0, 4 τ)$ for all t, p ∈ T, γ ∈ Γ, τ > 0 . So $\begin{matrix} ν (F (γ, t, p) - Λ F (γ, t, p), τ) \geq φ (2 t, 0, p, 0, 4 τ) \\ \geq φ (t, t, p, 0, \frac{2 τ}{L}) \end{matrix}$ for all t, p ∈ T, γ ∈ Γ, τ > 0. Hence $δ (F, Λ F) < \frac{L}{2} < 1 < \infty$ . Theorem 3.2, implies that, there exists a random operator Q : Γ × T² → S such that (1) A fixed point for Λ, is Q, $Q (γ, t, p) = \frac{1}{2} Q (γ, 2 t, p)$ (4.10) for all t, p ∈ T, γ ∈ Γ, which is unique in the set $M = {k \in B : δ (F, k) < \infty} .$

(2) δ (ΛⁿF, Q) →0 as n→ ∞, which implies that $lim_{n \to \infty} \frac{1}{2^{n}} F (γ, 2^{n} t, p) = Q (γ, t, p)$ for all t, p ∈ T, γ ∈ Γ . (3) $δ (F, Q) \leq \frac{1}{1 - L} δ (F, Λ F)$ , which implies that $δ (F, Q) \leq \frac{1}{1 - L} .$ (4.11) By use of definition δ we get $\begin{matrix} ν (F (γ, t, p) - Q (γ, u, z), τ) \\ \geq φ (t, t, p, 0, \frac{2 (1 - L) τ}{L}) \end{matrix}$ for all t, p ∈ T, γ ∈ Γ, τ > 0 . Using similar method of the Theorem 3.4, the proof will be complete. □

Corollary 4.5. 0et (Γ, Σ, ξ) be a probability measure space. Assume that q < 1 and σ ≥ 0 and F : Γ × T² → S be a random operator satisfying (4.3) and F (γ, t, 0) = F (γ, 0, p) =0 for all t, p ∈ T, γ ∈ Γ. So, there is a unique bi-additive random operator Q : Γ × T² → S such that

$\begin{matrix} ν (F (γ, t, p) - Q (γ, t, p), τ) \\ \geq exp (- \frac{σ ∥ t ∥^{q} ∥ p ∥^{q}}{2 (2 - 2^{q}) τ}) \end{matrix}$ (4.12)for all t, p ∈ T, γ ∈ Γ, τ > 0.

Proof. In Theorem 4.4 put $\begin{matrix} φ (t, s, p, r, τ) = exp (- \frac{σ (∥ t ∥^{q} + ∥ s ∥^{q}) (∥ p ∥^{q} + ∥ r ∥^{q})}{τ}) \end{matrix}$ for all t, s, p, r ∈ T, with L = 2^q-1. □

5 Stability of Stochastic Differential Equations

In this section, we prove the Hyers-Ulam-Rassias stability of the stochastic differential equations of the form

$y^{'} (γ, x) = F (γ, x, y (γ, x)) .$ (5.1)

Theorem 5.1. For given real numbers a and b with a < b, let I = [a, b] be a closed interval. Let β be positive constants with 0 < β < 1. Assume that $F : Γ \times I \times ℝ \to ℝ$ continuous random operator which satisfies a Lipschitz condition

$\begin{matrix} ν (F (γ, x, y) - F (γ, x, z), β t) \\ \geq ν (y - z, t) \end{matrix}$ (5.2) for any x ∈ I, γ ∈ Γ, $y, z \in ℝ$ and t > 0. If a continuous random operator $y : Γ \times I \to ℝ$ satisfies

$\begin{matrix} ν (y (γ, x) - y (γ, c) \\ - \int_{c}^{x} F (γ, τ, y (γ, τ)) d τ, t) \geq φ (x, t) \end{matrix}$ (5.3) for all x ∈ I, γ ∈ Γ and t > 0, where φ : I × ∞ → (0, 1] be a fuzzy set with

$inf_{z \in [c, x]} φ (z, t) \geq φ (x, t)$ (5.4) for all x ∈ I and t > 0, then there exists a unique continuous random operator $y_{0} : Γ \times I \to ℝ$ such that

$\begin{matrix} y_{0} (γ, x) \\ = y_{0} (γ, c) + \int_{c}^{x} F (γ, τ, y_{0} (γ, τ)) d τ \end{matrix}$ (5.5) (consequently, y₀ is a solution to (5.1)) and

$ν (y (γ, x) - y_{0} (γ, x), t) \geq φ (x, (1 - β) t)$ (5.6) for all x ∈ I, γ ∈ Γ and t > 0.

Proof. Define a set X of all continuous random operator $f : Γ \times I \to ℝ$ by

$X = {f : Γ \times I \to ℝ}$ (5.7) and introduce a generalized metric on X as follows:

$\begin{matrix} δ (f, g) = inf {C > 0 : ν (f (γ, x) - g (γ, x), Ct) \\ \geq φ (x, t), \forall x \in I, γ \in Γ, t > 0} \end{matrix}$ (5.8) In [42], Mihet and Radu proved that (B, δ) is complete (see also [43]). Now, we consider the linear map Λ : X → X is defined by

$\begin{matrix} (Λ y) (γ, x) = y (γ, c) \\ + \int_{c}^{x} F (γ, τ, y (γ, τ)) d τ (x \in I, γ \in Γ) \end{matrix}$ (5.9) for all f ∈ X. We show that Λ is strictly contractive on X. For any f, g ∈ X, let ɛ > 0 be an arbitrary constant with δ (f, g) ≤ ɛ, so, we have

$ν (f (γ, x) - g (γ, x), ɛ t) \geq φ (x, t),$ (5.10) for any x ∈ I, γ ∈ Γ and t > 0. Let, c = ξ₁ < ξ₂ < . . . < ξ_k = x, τ_i ∈ [ξ_i, ξ_i+1] and △s_i = ξ_i - ξ_i-1, i = 1, 2, . . . , k. By using, (5.2), (5.4), (5.8) and (5.10), we have $\begin{matrix} ν ((Λ f) (γ, x) - (Λ g) (γ, x), ɛ β t) \\ = ν (\int_{c}^{x} {F (γ, τ, f (γ, τ)) - F (γ, τ, g (γ, τ))} d τ, ɛ β t) \\ = ν (lim_{‖ Δ s ‖ \to 0} \sum_{i = 1}^{k} {F (γ, τ_{i}, f (γ, τ_{i})) - F (γ, τ_{i}, g (γ, τ_{i}))} Δ s_{i}, ɛ β t) \\ = lim_{‖ Δ s ‖ \to 0} ν (\sum_{i = 1}^{k} {F (γ, τ_{i}, f (γ, τ_{i})) - F (γ, τ_{i}, g (γ, τ_{i}))} Δ s_{i}, ɛ β t) \\ \geq lim_{‖ Δ s ‖ \to 0} ⋀ ν (F (γ, τ_{i}, f (γ, τ_{i})) - F (γ, τ_{i}, g (γ, τ_{i})), \frac{ɛ β t}{| Δ s_{i} | k}) \\ \geq lim_{‖ Δ s ‖ \to 0} ⋀ ν (f (γ, τ_{i}) - g (γ, τ_{i}), \frac{ɛ t}{| Δ s_{i} | k}) \\ \geq inf_{τ \in [c, x]} ν (f (γ, τ) - g (γ, τ), ɛ t) \\ \geq inf_{τ \in [c, x]} φ (τ, t) \\ \geq φ (x, t) \end{matrix}$ for all x ∈ I, γ ∈ Γ and t > 0. So, we have δ (Λf, Λg) ≤ ɛβ. Hence, we can conclude that δ (Λf, Λg) ≤ βδ (f, g) for any f, g ∈ X, this shows, Λ is a strictly contractive mapping on X with Lipschitz constant β ∈ (0, 1). By using (5.3) and (5.9), we conclude that δ (Λy, y) ≤1 and so, δ (Λⁿ⁺¹y, Λⁿy)≤ βⁿ < ∞.

Theorem 3.2, implies that, then there exists a unique continuous random operator $y_{0} : Γ \times I \to ℝ$ such that such that

(1) A fixed point for Λ, is y₀, i.e., $Λ (y_{0}) = y_{0} .$ (5.11)

(2) δ (Λⁿy, y₀) →0 as n→ ∞.

(3) $δ (y, y_{0}) \leq \frac{1}{1 - β} δ (y, Λ y)$ , which implies that $\begin{matrix} ν (y (γ, x) - y_{0} (γ, x), t) \geq φ (x, (1 - β) t) \end{matrix}$ for all x ∈ I, γ ∈ Γ and t > 0. □

6 Conclusion

In this paper, we considered two bi-additive θ-random operator inequalities in fuzzy normed spaces and proved the Hyers-Ulam-Rassias stability of them by using fixed point method. As an application, we proved the Hyers-Ulam-Rassias stability of the stochastic differential equations of the form $\begin{matrix} y^{'} (γ, x) = F (γ, x, y (γ, x)) . \end{matrix}$

Footnotes

Acknowledgments

The authors are thankful to the three anonymous referees for giving valuable comments and suggestions which helped to improve the final version of this paper.

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