A fixed point approach to approximate quadratic functional equation in non-Archimedean L * -fuzzy normed spaces

Abstract

We establish some stability results concerning the 2-dimensional vector variable quadratic functional equation $f (x + y, z + w) + f (x - y, z - w) = 2 f (x, z) + 2 f (y, w)$ in non-Archimedean $L^{*}$ -fuzzy normed spaces.

Keywords

ℒ*-fuzzy metric and normed space Hyers-Ulam stability quadratic functional equation non-Archimedean ℒ*-fuzzy normed space

1 Introduction

The fuzzy set theory is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. It has also very useful applications in various field, e.g., population dynamic [5], chaos control [14, 15], computer programming [16], nonlinear dynamical systems [21], nonlinear operators [30], statistical coverage [29], etc. The fuzzy topology proves to be a very useful tool to deal with such situations where the use of classical theories breaks down. In 1984, Katsaras [24] defined fuzzy norms on linear spaces and Wu and Fang [43] also introduced a notion of fuzzy normed spaces and gave the generalization of the Kolmogoroff normalized theorem for fuzzy topological linear spaces. In [6], Biswas defined and studied fuzzy inner products on linear spaces. Since then some mathematicians have defined fuzzy metrics and norms on linear spaces from various points of view [4 , 44]. In 1994, Cheng and Mordeson [9] introduced a definition of fuzzy norm on linear spaces in such a manner that the corresponding induced fuzzy metrics are of Kramosil and Michalek type. In 2003, Bag and Samanta [3] modified the definition of Cheng and Mordeson by removing a regular condition.

Let X be a real linear space. A function $N : X \times ℝ \to [0, 1]$ (the so-called fuzzy subset) is said to be a fuzzy norm on X if for all x, y ∈ X and all $s, t \in ℝ$ ,

(N1) N (x, c) =0 for c ⩽ 0;

(N2) x = 0 if and only if N (x, c) =1 for all c > 0;

(N3) $N (cx, t) = N (x, \frac{t}{| c |})$ if c ≠ 0;

(N4) N (x + y, s + t) ⩾ min {N (x, s) , N (y, t)};

(N5) N (x, .) is a non-decreasing function on $ℝ$ and $lim_{t \to \infty} N (x, t) = 1$ ;

(N6) For x ≠ 0, N (x, .) is left-continuous on $ℝ$ .

The pair (X, N) is called a fuzzy normed linear space. One may regard N (x, t) as the truth value of the statement ‘the norm of x is less than or equal to the real number t’.

Definition 1.1. (1) The triple $(X, M, T)$ is said to be an $L$ -fuzzy metric space if X is an arbitrary (non-empty) set, $T$ is a continuous t-norm on $L$ and $M$ is an $L$ -fuzzy set on X² × (0, + ∞) satisfying the following conditions: for all x, y, z ∈ X and t, s ∈ (0, + ∞),

(a) $M (x, y, t) \geq_{L} 0_{L}$ ;

(b) $M (x, y, t) = 1_{L}$ for all t > 0 if and only if x = y;

(d) $T (M (x, y, t), M (y, z, s) \leq_{L} M (x, z, t + s)$ ;

(e) $M (y, z, .) : (0, + \infty) \to L$ is left-continuous.

In this case, $M$ is called an $L$ -fuzzy metric.

(2) If $M = M_{M, N}$ is an intuitionistic fuzzy set, then the 3-tuple $(X, M_{M, N}, T)$ is said to be an intuitionistic fuzzy metric space.

Example 1.2. Let (X, d) be a matric space. Denote $T (a, b) = (a_{1} b_{1}, min (a_{2} + b_{2}, 1))$ for all a = (a₁, a₂) , b = (b₁, b₂) ∈ L^* and let $M_{M, N}$ be the intuitionistic fuzzy set on X × (0, + ∞) defined as follows: $M_{M, N} (x, t) = (\frac{{ht}^{n}}{{ht}^{n} + md (x, y)}, \frac{md (x, y)}{{ht}^{n} + md (x, y)})$ for all $t, h, m, n \in ℝ^{+}$ . Then $(X, M_{M, N}, T)$ is an intuitionistic fuzzy metric space.

Example 1.3. Let $X = ℕ$ . Define $T (a, b) = (max (0, a_{1} + b_{1} - 1), a_{2} + b_{2} - a_{2} b_{2})$ for all a = (a₁, a₂) , (b₁, b₂) ∈ L^* and let $M_{M, N}$ be the intuitionistic fuzzy set on X × (0, + ∞) defined as follows. $M_{M, N} (x, t) = {\begin{matrix} (\frac{x}{y}, \frac{y - x}{y}) & if x \leq y \\ (\frac{y}{x}, \frac{x - y}{x}) & if y \leq x \end{matrix}$ for all x, y ∈ X and t > 0. Then $(X, M_{M, N}, T)$ is an intuitionistic fuzzy metric space.

Stability problem of a functional equation was first posed by Ulam [41] which was answered by Hyers [22] and then generalized by Aoki [1] and Rassias [32] for additive mappings and linear mappings, respectively. Since then several stability problems for various functional equations have been investigated in [10–12 , 33]; various fuzzy stability results concerning Cauchy, Jensen, quadratic and cubic functional equations were discussed in [26–28 , 42]; and some random stability results concerning Jensen and cubic functional equations were discussed in [8, 38]. The stability problem for the 2-dimensional vector variable quadratic functional equation was proved by Bae and Park [2] for mappings f : X × X → Y, where X is a real normed space and Y is a Banach space.

In this paper, we determine the stability of the following 2-dimensional vector variable quadratic functional equation $\begin{matrix} f (x + y, z + w) + f (x - y, z - w) \\ = 2 f (x, z) + 2 f (y, w) \end{matrix}$ (1) in non-Archimedean $L^{*}$ -fuzzy normed spaces. Functional equation (1) has a solution like f (x, y) = ax² + bxy + cy².

2 Preliminaries

2.1 $L^{*}$ -fuzzy normed spaces

In this subsection, we recall some terminology, notation and definitions used in [17 , 36].

Consider the set L^* and the order relation ⩽_{L
^*} defined by $\begin{matrix} L^{*} = {(x_{1}, x_{2}) : (x_{1}, x_{2}) \in [0, 1]^{2}, x_{1} + x_{2} ⩽ 1}, \\ (x_{1}, x_{2}) ⩽_{L^{*}} (y_{1}, y_{2}) \Leftrightarrow x_{1} ⩽ y_{1}, x_{2} ⩾ y_{2}, \\ (x_{1} . x_{2}), (y_{1}, y_{2}) \in L^{*} . \end{matrix}$ Then (L^*, ⩽ _{L
^*}) is a complete lattice ([34]). We denote its unit by 0_{L
^*} = (0, 1) and 1_{L
^*} = (1, 0).

Definition 2.1. Let U be a nonempty set called the universe. An L^*-fuzzy set in U is defined as a mapping $A : U \to L^{*}$ . For each u in U, $A (u)$ represents the degree (in L^*). An intuitionstic fuzzy set $A_{ζ, η}$ in a universal set U is an object $A_{ζ, η} = {(ζ_{A} (u), η_{A} (u)) : u \in U}$ , where $ζ_{A} (u) \in [0, 1]$ and $η_{A} (u) \in [0, 1]$ for all u ∈ U are called the membership degree and the non-membership degree, respectively, of $u \in A_{ζ, η}$ and, furthermore, satisfy $ζ_{A} (u) + η_{A} (u) ⩽ 1$ .

Definition 2.2. A triangular norm (t-norm) on L^* is a mapping $T : (L^{*})^{2} : \to L^{*}$ satisfying the following conditions:

$(a) (\forall x \in L^{*}) (T (x, 1_{L^{*}}) = x)$ (boundary condition);

$(b) (\forall (x, y) \in (L^{*})^{2}) (T (x, y) = T (y, x))$ (commutativity);

$(c) (\forall (x, y, z) \in (L^{*})^{3}) (T (x, T (y, z)) = T (T (x, y), z))$ (associativity);

(d) (∀ (x, x′, y, y′) ∈ (L^*) ⁴) (x ⩽ _{L
^*}x′ and $y ⩽_{L^{*}} y^{'} \Rightarrow T (x, y) ⩽_{L^{*}} T (x^{'}, y^{'}))$ (monotonicity).

A t-norm $T$ on L^* is said to be continuous if, for any x, y ∈ L^* and any sequences {x_n} and {y_n} which converge to x and y, respectively, $lim_{n \to \infty} T (x_{n}, y_{n}) = T (x, y) .$ For examples, let a = (a₁, a₂) , b = (b₁, b₂) ∈ L^*, consider $T (a, b) = (a_{1} b_{1}, min {a_{2} + b_{2}, 1})$ and $M (a, b) = (min {a_{1}, b_{1}}, max {a_{2}, b_{2}})$ . Then $T (a, b)$ and $M (a, b)$ are continuous t-norms.

Now, we define a sequence $T^{n}$ recursively by $T^{1} = T$ and $\begin{matrix} T^{n} (x^{(1)}, \dots, x^{(n + 1)}) \\ = T (T^{n - 1} (x^{(1)}, \dots, x^{(n)}), x^{(n + 1)}) \end{matrix}$ for all n ⩾ 2 and x⁽ⁱ⁾ ∈ L^*.

Definition 2.3. A negator on L^* is any decreasing mapping $N : L^{*} : \to L^{*}$ satisfying $N (0_{L^{*}}) = 1_{L^{*}}$ and $N (1_{L^{*}}) = 0_{L^{*}}$ . If $N (N (x)) = x$ for all x ∈ L^*, then $N$ is called an involutive negator. A negator on [0, 1] is a decreasing mapping N : [0, 1] → [0, 1] satisfying N (0) =1 and N (1) =0. N_s denotes the standard negator on [0, 1] defined by N_s (x) =1 - x for all x ∈ [0, 1].

The definition of an intuitionistic fuzzy normed space is given below (see [34]).

Definition 2.4. (1) Let $L = (L^{*}, ⩽_{L^{*}})$ . The triple $(X, P, T)$ is said to be an $L$ -fuzzy normed space if X is a vector space, $T$ is a continuous t-norm on L^* and $P$ is an L^*-fuzzy set on X × (0, + ∞) satisfying the following conditions for all x, y ∈ X and t, s > 0,

$(a) P (x, t) >_{L^{*}} 0_{L^{*}}$ ;

$(b) P (x, t) = 1_{L^{*}}$ if and only if x = 0;

$(c) P (α x, t) = P (x, \frac{x}{| α |})$ for all α ≠ 0;

$(d) P (x + y, t + s) ⩾_{L^{*}} T (P (x, t), P (y, s))$ ;

$(e) P (x, .) : (0, \infty) \to L^{*}$ is continuous;

$(f) lim_{t \to 0} P (x, t) = 0_{L^{*}}$ and $lim_{t \to \infty} P (x, t) = 1_{L^{*}}$ .

In this case $P$ is called an $L$ -fuzzy norm (briefly, L^*-fuzzy norm).

(2) If $P = P_{μ, ν}$ is an intuitionistic fuzzy set (see Definition 2.1), then the triple $(X, P_{μ, ν}, T)$ is said to be an intuitionstic fuzzy normed space (briefly, IFN-space). In this case $P = P_{μ, ν}$ is said an intuitionistic fuzzy norm on X.

Note that, if $P$ is an L^*-fuzzy norm on X, then the following are satisfied:

$(i) P (x, t)$ is nondecreasing with respect to t for all x ∈ X.

$(ii) P (x - y, t) = P (y - x, t)$ for all x, y ∈ X and t > 0(see [18 , 34]).

Example 2.5. Let (X, ∥ . ∥) be a normed space. Let $T (a, b) = (a_{1} b_{1}, min (a_{2} + b_{2}, 1))$ for all a = (a₁, a₂) , b = (b₁, b₂) ∈ L^* and μ, ν be membership and non-membership degree of an intuitionistic fuzzy set defined by $P_{μ, ν} (x, t) = (μ_{x} (t), ν_{x} (t)) = (\frac{t}{t + m ∥ x ∥}, \frac{∥ x ∥}{t + ∥ x ∥})$ for all $t \in ℝ^{+}$ in which m > 1. Then $(X, P, T)$ is an IFN-space. Here, μ (x, t) + ν (x, t) =1 for x = 0 and μ (x, t) + ν (x, t) <1 for x ≠ 0.

Let $M (a, b) = (min {a_{1}, b_{1}}, max {a_{2}, b_{2}})$ for all a = (a₁, a₂) , b = (b₁, b₂) ∈ L^* and μ, ν be membership and non-membership degree of an intuitionistic fuzzy set defined by $\begin{matrix} P (x, t) \\ = (μ_{x} (t), ν_{x} (t)) = (e^{- ∥ x ∥ / t}, e^{- ∥ x ∥ / t} (e^{∥ x ∥ / t} - 1)) \end{matrix}$ for all x ∈ X and $t \in ℝ^{+}$ . Then $(X, P, M)$ is an IFN-space.

Definition 2.6. (1) A sequence {x_n} in an $L^{*}$ -fuzzy normed spaces $(X, P, T)$ is said to be convergent to a point x ∈ X (denoted by $x_{n} \overset{P}{\to} x$ ) if $P (x_{n} - x, t) \to 1_{L^{*}}$ as n→ ∞ for every t > 0.

(2) A sequence {x_n} in an $L^{*}$ -fuzzy normed space $(X, P, T)$ is said to be Cauchy sequence if, for any 0 < ɛ < 1 and t > 0, there exists $n_{0} \in ℕ$ such that $P (x_{n} - x_{m}, t) >_{L^{*}} (N_{s} (ɛ), t), \forall n, m ⩾ n_{0},$ where N_s is the standard negator.

(3) An $L^{*}$ -fuzzy normed space $(X, P, T)$ is said to be complete if every Cauchy sequence in X is convergent

The similar concepts of convergence and Cauchy sequences in an intuitionistic fuzzy normed space were studied in [36].

Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies

(1) d (x, y) =0 if and only if x = y;

(2) d (x, y) = d (y, x) for all x, y ∈ X;

(3) d (x, y) ⩽ d (x, z) + d (y, z) for all x, y, z ∈ X.

For explicitly later use, we recall a fundamental result in fixed point theory.

Theorem 2.7. (The fixed point alternative theorem [7]) Let (Ω, d) be a complete generalized metric space and J : Ω → Ω be a strictly contractive mapping with Lipschitz constant 0 < L < 1, that is, $d (Jx, Jy) ⩽ Ld (x, y), \forall x, y \in Ω .$ Then, for each given x ∈ Ω, either $d (J^{n} x, J^{n + 1} x) = \infty, \forall n ⩾ 0,$ or $d (J^{n} x, J^{n + 1} x) < \infty, \forall n ⩾ n_{0},$ for some nonnegative integer n₀. Actually, if second alternative holds, then the sequence {Jⁿx} converges to a fixed point y^* of J and

(1) y^* is the unique fixed point of J in the set Δ = {y ∈ Ω : d (J^{n
₀}x, y) < ∞};

(2) $d (y, y^{*}) ⩽ \frac{1}{1 - L} d (y, Jy)$ for all y ∈ Δ.

2.2 Non-Archimedean $L^{*}$ -fuzzy normed spaces

In 1896, Hensel [20] introduced a field with a valuation in which does not have the Archimedean property. And authors [35, 39] have had some studies on it.

Definition 2.8. Let $K$ be a field. A non-Archimedean absolute value on $K$ is a function $| . | : K \to [0, + \infty)$ such that, for any $a, b \in K$ ,

(i) |a|⩾0 and equality holds if and only if a = 0,

(ii) |ab| = |a||b|,

(iii) |a + b| ⩽ max {|a|, |b|} (the strict triangle inequality).

Note that |n|⩽1 for each integer n. we always assume, in addition, that | . | is non-trivial, i.e., there exists an a₀ ≠ 0, 1.

Definition 2.9. A non-Archimedean $L^{*}$ -fuzzy normed spaces is a triple $(V, P, T)$ , where V is a vector space, $T$ is a continuous t-norm on L^* and $P$ is an $L^{*}$ -fuzzy set on V × [0, ∞) satisfying the following conditions: for all x, y ∈ V and t, s ∈ (0, ∞),

$(a) 0_{L^{*}} <_{L^{*}} P (x, t)$ ;

$(b) P (x, t) = 1_{L^{*}}$ if and only if x = 0;

$(c) P (α x, t) = P (x, \frac{t}{| α |})$ for all α ≠ 0;

$(d) T (P (x, t), P (y, s)) ⩽_{L^{*}} P (x + y, max {t, s})$ ;

$(e) P (x, .) : (0, \infty) \to L^{*}$ is continuous;

$(f) lim_{t \to 0} P (x, t) = 0_{L^{*}}$ and $lim_{t \to \infty} P (x, t) = 1_{L^{*}}$ .

Example 2.10. Let (X, ∥ . ∥) be a non-Archimedean normed linear space. Then the triple $(X, P, min)$ , where $P (x, t) = {\begin{matrix} 0, & if t ⩽ ∥ x ∥; \\ 1, & if t > ∥ x ∥, \end{matrix}$ is a non-Archimedean normed space in which L = [0, 1].

Example 2.11. Let (X, ∥ . ∥) be a non-Archimedean normed linear space.

Denote $T_{M} (a, b) = (min {a_{1}, b_{1}}, max {a_{2}, b_{2}})$ for all a = (a₁, a₂) , b = (b₁, b₂) ∈ L^* and $P_{μ, ν}$ be the intuitionistic fuzzy set on X × (0, + ∞) defined as follows: $P_{μ, ν} (x, t) = (\frac{t}{t + ∥ x ∥}, \frac{∥ x ∥}{t + ∥ x ∥})$ for all $t \in ℝ^{+}$ . Then $(X, P_{μ, ν}, T_{M})$ is a non-Archimedean intuitionistic fuzzy normed space.

3 Stability in non-Archimedean $L^{*}$ -fuzzy normed spaces

In this section, we prove the Hyers-Ulam stability of quadratic functional equation in non-Archimedean $L^{*}$ -fuzzy normed spaces, based on the fixed point method. For notational convenience, given a function f : X × X → Y, we define the difference operator $\begin{matrix} D_{q} f (x, y, z, w) = & f (x + y, z + w) + f (x - y, z - w) \\ - 2 f (x, z) - 2 f (y, w) \end{matrix}$ We begin with a Hyers-Ulam type theorem in non-Archimedean $L^{*}$ -fuzzy normed spaces for the functional equation (1).

Theorem 3.1. Let X be a linear space and let $(Z, P^{'}, T^{'})$ be a non-Archimedean $L^{*}$ -fuzzy normed space. Let φ : X⁴ → Z be a function, for some real number with α < 4, such that $P^{'} (φ (2 x, 2 y, 2 z, 2 w), t) ⩾_{L^{*}} P^{'} (α φ (x, y, z, w), t)$ for all x, y, z, w ∈ X and all t > 0. Let $(Y, P, T)$ be a non-Archimedean $L^{*}$ -fuzzy Banach space and let f : X × X → Y be a mapping, with f (0, 0) =0, such that $P (D_{q} f (x, y, z, w), t) ⩾_{L^{*}} P^{'} (φ (x, y, z, w), t)$ (2) for all x, y, z, w ∈ X and all t > 0. Then there exists a unique quadratic mapping F : X × X → Y satisfying (1) such that

$\begin{matrix} P (F (x, y) - f (x, y), t) \\ ⩾_{L^{*}} P^{'} (φ (x, x, y, y), (4 - α) t), \end{matrix}$ (3) for all x, y ∈ X and all t > 0.

Proof. Putting y = x and w = z in (2), we obtain $P (f (2 x, 2 z) - 4 f (x, z), t) ⩾_{L^{*}} P^{'} (φ (x, x, z, z), t)$ for all x, z ∈ X and all t > 0. Replacing y by z in the above inequality, we get $P (f (2 x, 2 y) - 4 f (x, y), t) ⩾_{L^{*}} P^{'} (φ (x, x, y, y), t)$ (4) for all x, y ∈ X and all t > 0.

Consider the set Ω : = {g : X × X → Y} and introduce a complete generalized metric on Ω (see [7]) (as usual, inf ∅ = ∞): $\begin{matrix} d (g, h) & = inf {K \in ℝ^{+} : P (g (x, y) - h (x, y), Kt) \\ ⩾_{L^{*}} P^{'} (φ (x, x, y, y), t), \forall x, y \in X, t > 0} . \end{matrix}$ Now, we consider the mapping J : Ω → Ω such that $J (g (x, y)) = \frac{1}{4} g (2 x, 2 y)$ for all x, y ∈ X and we prove that J is a strictly contractive mapping of Ω with the Lipschitz constant $\frac{α}{4}$ .

Let g, h ∈ Ω be given such that d (g, h) = ɛ. Then $P (g (x, y) - h (x, y), ɛ t) ⩾_{L^{*}} P^{'} (φ (x, x, y, y), t)$ (5) for all x, y ∈ X, t > 0. Hence $\begin{matrix} P (Jg (x, y) - Jh (x, y)), ɛ t) \\ ⩾_{L^{*}} P (g (2 x, 2 y) - h (2 x, 2 y)), 4 ɛ t) \\ ⩾_{L^{*}} P^{'} (φ (x, x, y, y), \frac{4 t}{α}) \end{matrix}$ for all x, y ∈ X and t > 0. By definition, $d (Jg, Jh) ⩽ \frac{α}{4} ɛ$ . Therefore, $d (Jg, Jh) ⩽ \frac{α}{4} d (g, h) for all g, h \in Ω .$

It follows from (4) that $d (f, Jf) ⩽ \frac{1}{4}$ . Therefore, by Theorem 2.1, there exists a mapping F : X → Y satisfying:

(1) F is a fixed point of J, that is, $F (2 x, 2 y) = 4 F (x, y) for all x, y \in X .$ The mapping F is a unique fixed point of J in the set Δ = {g ∈ Ω : d (g, f) < ∞}. This implies that F is a unique mapping satisfying (3) such that there exists a K > 0 satisfying $P (f (x, y) - F (x, y), Kt) ⩾_{L^{*}} P^{'} (φ (x, x, y, y), t)$ for all x, y ∈ X and t > 0;

(2) d (Jⁿf, F) →0 as n→ ∞. This implies the equality $F (x, y) : = lim_{n \to \infty} J^{n} f (x, y) = lim_{n \to \infty} \frac{1}{4^{n}} f (2^{n} x, 2^{n} y)$ for all x, y ∈ X;

(3) $d (f, F) ⩽ \frac{1}{1 - L} d (f, Jf)$ with f ∈ Δ, which implies the inequality $d (f, F) ⩽ \frac{1}{4 - α},$ from which it follows $P (F (x, y) - f (x, y), \frac{t}{4 - α}) ⩾_{L^{*}} P^{'} (φ (x, x, y, y), t)$ for all x, y ∈ X and t > 0. This implies that the inequality (3) holds.

It remains to show that F is a quadratic mapping. Replacing x and y by 2ⁿx and 2ⁿy in (2), respectively, we get $\begin{matrix} P (4^{- n} D_{q} f (2^{n} x, 2^{n} y, 2^{n} z, 2^{n} w), t) \\ ⩾_{L^{*}} P^{'} (φ (2^{n} x, 2^{n} y, 2^{n} z, 2^{n} w), 4^{n} t) \\ ⩾_{L^{*}} P^{'} (φ (x, y, z, w), \frac{4^{n} t}{α^{n}}) . \end{matrix}$ Taking the limit as n→ ∞, we find that F fulfills (1).

To prove the uniqueness of the mapping F, assume that there exists another quadratic mapping G : X × X → Y which satisfies (1). For fix x, y ∈ X, we know that F (2ⁿx, 2ⁿy) =4ⁿH (x, y) and G (2ⁿx, 2ⁿy) =4ⁿG (x, y) for all $n \in ℕ$ . It follows from (3) that $\begin{matrix} P (F (x, y) - G (x, y), t) \\ = P (\frac{F (2^{n} x, 2^{n} y)}{4^{n}} - \frac{G (2^{n} x, 2^{n} y)}{4^{n}}, t) \\ ⩾_{L^{*}} T (P (\frac{F (2^{n} x, 2^{n} y)}{4^{n}} - \frac{f (2^{n} x, 2^{n} y)}{4^{n}}, t), \\ P (- \frac{G (2^{n} x, 2^{n} y)}{4^{n}} + \frac{f (2^{n} x, 2^{n} y)}{4^{n}}, t)) \\ ⩾_{L^{*}} T^{2} (P^{'} (φ (2^{n} x, 2^{n} x, 2^{n} y, 2^{n} y), 4^{n} (4 - α) t)) \\ ⩾_{L^{*}} T^{2} (P^{'} (φ (x, x, y, y), \frac{4^{n} (4 - α) t}{α^{n}})) \end{matrix}$ for all x, y ∈ X, all $n \in ℕ$ and all t > 0, where $T^{2} (a) = T (a, a)$ for all a ∈ [0, 1]. Since $lim_{n \to \infty} \frac{4^{n} (4 - α) t}{α^{n}} = \infty$ for all t > 0, we get $lim_{n \to \infty} P^{'} (φ (x, x, y, y), \frac{4^{n} (4 - α) t}{α^{n}}) = 1$ for all x, y ∈ X and all t > 0. Therefore $P (F (x, y) - G (x, y), t) = 1$ . Thus it is concluded that F (x, y) = G (x, y).

Corollary 3.2. Let p be a nonnegative real number with p < 2, X be a normed space with norm $∥ . ∥, (Z, P^{'}, T)$ be a non-Archimedean $L^{*}$ -fuzzy normed space, $(Y, P, T)$ be complete non-Archimedean $L^{*}$ -fuzzy normed space, and let z₀ ∈ Z. If f : X × X → Y is a mapping such that $\begin{matrix} P (D_{q} f (x, y, z, w), t) \\ ⩾_{L^{*}} P^{'} ((∥ x ∥^{p} + ∥ y ∥^{p} + ∥ z ∥^{p} + ∥ w ∥^{p}) z_{0}, t) \end{matrix}$ for all x, y, z, w ∈ X and t > 0, then there exists a unique quadratic mapping F : X × X → Y such that $\begin{matrix} P (F (x, y) - f (x, y), t) \\ ⩾_{L^{*}} P^{'} ((∥ x ∥^{p} + ∥ y ∥^{p}) z_{0}, \frac{(4 - 2^{p}) t}{2}) \end{matrix}$ for all x, y ∈ X and t > 0.

Theorem 3.3. Let X be a linear space and let $(Z, P^{'}, T^{'})$ be a non-Archimedean $L^{*}$ -fuzzy normed space. Let φ : X⁴ → Z be a function, for some real number with α > 4, such that $P^{'} (φ (\frac{x}{2}, \frac{y}{2}, \frac{z}{2}, \frac{w}{2}), t) ⩾_{L^{*}} P^{'} (φ (x, y, z, w), α t)$ (6) for all x, y, z, w ∈ X and all t > 0. Let $(Y, P, T)$ be a non-Archimedean $L^{*}$ -fuzzy Banach space and let f : X × X → Y, with f (0, 0) =0, be a mapping such that satisfies (2). Then there exists a unique quadratic mapping F : X × X → Y satisfying (1) such that

$\begin{matrix} P (F (x, y) - f (x, y), t) \\ ⩾_{L^{*}} P^{'} (φ (x, x, y, y), (α - 4) t) \end{matrix}$ (7) for all x, y ∈ X and all t > 0.

Proof. It follows from (4) that $P (f (x, y) - 4 f (\frac{x}{2}, \frac{y}{2}), t) ⩾_{L^{*}} P^{'} ((φ (x, x, y, y), t)$ for all x, y ∈ X and all t > 0.

Consider the set Ω : = {g : X × X → Y} and introduce the generalized metric on X, $\begin{matrix} d (g, h) & = inf {K \in ℝ^{+} : P (g (x, y) - h (x, y), Kt) \\ ⩾_{L^{*}} P^{'} (φ (x, x, y, y), t), \forall x, y \in X, t > 0} . \end{matrix}$

Now we consider the linear mapping J : X → X, such that $Jg (x, y) : = 4 g (\frac{x}{2}, \frac{y}{2})$ for all x, y ∈ X. We prove that J is a strictly contractive mapping of Ω with the Lipschitz constant $\frac{4}{α}$ .

Let g, h ∈ Ω be given such that d (g, h) = ɛ. Hence, from (5) and (6), we have $\begin{matrix} P (Jg (x, y) - Jh (x, y), ɛ t) \\ = P (g (\frac{x}{2}, \frac{y}{2}), h (\frac{x}{2}, \frac{y}{2}), \frac{ɛ}{4} t) \\ ⩾_{L^{*}} P^{'} (φ (\frac{x}{2}, \frac{x}{2}, \frac{y}{2}, \frac{y}{2}), \frac{1}{4} t) \\ ⩾_{L^{*}} P^{'} (φ (x, x, y, y), \frac{α}{4} t) \end{matrix}$ for all x, y ∈ X and all t > 0. By definition, $d (Jg, Jh) ⩽ \frac{α}{4} ɛ$ . Therefore, $d (Jg, Jh) ⩽ \frac{4}{α} d (g, h)$ for all x, y ∈ X and all t > 0. It follows from (4) that $d (f, Jf) = \frac{1}{α}$ . Therefore, by Theorem 2.1, J has a unique fixed point in the set Δ = {g ∈ Ω : d (g, f) < ∞}. Let F be the fixed point of J, that is, $4 F (\frac{x}{2}, \frac{y}{2}) = F (x, y)$ for all x, y ∈ X, which satisfies that there exists K > 0 such that $P (f (x, y) - F (x, y), Kt) ⩾_{L^{*}} P^{'} (φ (x, x, y, y), t)$ for all x, y ∈ X and all t > 0. On the other hand, we have $lim_{n \to \infty} d (J^{n} f, F) = 0 .$ This implies the following quality $F (x, y) : = lim_{n \to \infty} J^{n} f (x, y) = lim_{n \to \infty} 4^{n} f (\frac{x}{2^{n}}, \frac{y}{2^{n}})$ for all x, y ∈ X and all t > 0. It follows from $d (f, F) ⩽ \frac{1}{1 - L} d (f, Jf)$ that $d (f, F) ⩽ \frac{1}{α (1 - L)} .$ This implies that (7) holds. The remainder of proof is similar to the proof of Theorem 3.

Corollary 3.4. Let p be a nonnegative real number with p > 2, X be a normed space with norm $∥ . ∥, (Z, P^{'}, T)$ be a non-Archimedean $L^{*}$ -fuzzy normed space, $(Y, P, T)$ be complete non-Archimedean $L^{*}$ -fuzzy normed space, and let z₀ ∈ Z. If f : X × X → Y is a mapping such that $\begin{matrix} P (D_{q} f (x, y, z, w), t) \\ ⩾_{L^{*}} P^{'} ((∥ x ∥^{p} + ∥ y ∥^{p} + ∥ z ∥^{p} + ∥ w ∥^{p}) z_{0}, t) \end{matrix}$ for all x, y, z, w ∈ X and t > 0, then there exists a unique quadratic mapping F : X × X → Y such that $\begin{matrix} P (F (x, y) - f (x, y), t) \\ ⩾_{L^{*}} P^{'} ((∥ x ∥^{p} + ∥ y ∥^{p}) z_{0}, \frac{(2^{p} - 4) t}{2}) \end{matrix}$ for all x, y ∈ X and t > 0.

Example 3.5. Let (Y, ∥ . ∥) be a non-Archimedean Banach space, $(Y, P, T)$ a non-Archimedean $L^{*}$ -fuxzzy normed space, in which

$P (y, t) = (\frac{t}{t + ∥ y ∥}, \frac{∥ y ∥}{t + ∥ y ∥}), \forall y \in Y, t > 0,$ and $(Z, P^{'}, T^{'})$ be a complete non-Archimedean $L^{*}$ -fuxzzy normed space. Define $\begin{matrix} φ (x, y, z, w, t) & = (\frac{t}{1 + t}, \frac{1}{1 + t}), \\ P^{'} (z, t) & = (\frac{t}{1 + t}, \frac{1}{1 + t}) . \end{matrix}$ It is easy to see that $\begin{matrix} P (D_{q} f (x, y, z, w), t) & = (\frac{t}{t + ∥ y ∥}, \frac{∥ y ∥}{t + ∥ y ∥}) \\ ⩾_{L^{*}} (\frac{1}{2 + t}, \frac{1 + t}{2 + t}) \\ = P^{'} (φ (x, y, z, w), t) \end{matrix}$ for some fixed y ∈ Y and t > 0. Thus all the conditions of Theorem 3 hold and so there exists a unique quadratic mapping F : X × X → Y such that $\begin{matrix} P (F (x, y) - f (x, y), t) \\ ⩾_{L^{*}} (\frac{(4 - α) t}{1 + (4 - α) t}, \frac{1}{1 + (4 - α) t}) . \end{matrix}$

4 Conclusion

Fuzzy set theory offers a strict mathematical basis in which vague conceptual phenomenon can be exactly and thoroughly studied. The information of intuitionistic fuzzy set become more meaningful and applicable since it comprises the degree of belongingness, degree of non-belongingness and the hesitation margin. Distance measure between intuitionistic fuzzy sets is an important perception in fuzzy mathematics because of its wide applications in real world, such as pattern recognition, machine learning, decision making and market prediction. In this paper, we make a contribution to the intuitionistic fuzzy by using fixed point results to study the stability a two dimensional quadratic functional equation in non-Archimedean $L^{*}$ -fuzzy normed spaces.

Competing interests

The authors declare that they have no competing interests.

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A fixed point approach to approximate quadratic functional equation in non-Archimedean L * -fuzzy normed spaces

Abstract

Keywords

1 Introduction

2.1 L * -fuzzy normed spaces

2.2 Non-Archimedean L * -fuzzy normed spaces

3 Stability in non-Archimedean L * -fuzzy normed spaces

Competing interests

References

2.1 $L^{*}$ -fuzzy normed spaces

2.2 Non-Archimedean $L^{*}$ -fuzzy normed spaces

3 Stability in non-Archimedean $L^{*}$ -fuzzy normed spaces