On the fuzzy stability problem of generalized cubic mappings

Abstract

We introduce a fuzzy anti-β-norm and generalized cubic mapping and then investigate the Hyers-Ulam-Rassias stability in quasi β-Banach space and the fuzzy stability by using a fixed point in fuzzy anti-β Banach space for the generalized cubic function.

Keywords

Hyers-Ulam-Rassias stability cubic mapping quasi-β-mormed space fixed point fuzzy anti-normed space fuzzy anti-β-normed space

1 Introduction

In 1965, the notion of fuzzy sets has been introduced by Zadeh [45]. The fuzzy theory has become a very active area of research and a lot of developments have been made in the theory of fuzzy sets in various problems arising in the field of science such as control problems, information, physics, statistics, engineering, economics, finance and even social sciences; see [1 , 30]. Hence there is no doubt that the notion of fuzzy set theory is half century old, which has got popularity among the researcher from different parts of the globe and has been applied in almost all the branches of science and technology since its inception; see [13 , 46].

The concept of stability problem of a functional equation was first posed by Ulam [41] concerning the stability of group homomorphisms as follows: Let G₁ be a group and let G₂ be a metric group with the metric d (· , ·). Given ɛ > 0, does there exist a δ > 0 such that if a function h : G₁ → G₂ satisfies the inequality d (h (xy) , h (x) h (y)) < δ for all x, y ∈ G₁ then there is a homomorphism H : G₁ → G₂ with d (h (x) , H (x)) < ɛ for all x ∈ G₁? Hyers [16] gave a partial answer to the question of Ulam and Hyers’ theorem was generalized in various directions. In particular, the result of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [33] for linear mappings by considering a unbounded Cauchy difference. This result of Rassias lead mathematicians working in stability of functional equations to establish what is known today as Hyers-Ulam-Rassias stability or Cauchy-Rassias stability as well as to introduce new definitions of stability concepts. During the last three decades, several stability problems of a large variety of functional equations have been extensively studied and generalized by a number of authors see [23,. 36]. Isac and Rassias [17] were first to provide applications of new fixed point theorem for the stability theory of functional equations. By using fixed point methods the stability problems of several functional equations have been extensively investigated by a number of authors; see [7 , 31].

Jun and Kim [20] introduced the following cubic functional equation:

$\begin{matrix} f (2 x + y) + f (2 x - y) \\ = 2 f (x + y) + 2 f (x - y) + 12 f (x) \end{matrix}$ and established a general solution. Najati [28] investigated the following generalized cubic functional equation:

$\begin{matrix} f (sx + y) + f (sx - y) \\ = sf (x + y) + sf (x - y) + 2 (s^{3} - s) f (x) \end{matrix}$ (1.1) for a positive integer s ≥ 2 . Also, Jun and Kim [19] proved the Hyers-Ulam-Rassias stability of a Euler-Lagrange type cubic mapping as follows:

$\begin{matrix} f (sx + y) + f (x + sy) \\ = (s + 1) (s - 1)^{2} [f (x) + f (y)] \\ + s (s + 1) f (x + y), \end{matrix}$ (1.2) where $s \in ℤ (s \neq 0, \pm 1) .$

In this paper, we deal with the following the functional equation:

$\begin{matrix} f (ax + y) - f (x - ay) - af (x + y) + a^{2} f (x - y) \\ = (a^{2} - 1) (a + 1) f (x) + (a^{2} - 1) (a - 1) f (y) \end{matrix}$ (1.3) for all $a \in ℤ (a \neq 0, \pm 1) .$

We will use the following definition to prove Hyers-Ulam-Rassias stability for the generalized cubic functional equation in the quasi β-normed space. Let β be a real number with 0 < β ≤ 1 and $𝕂$ be either $ℝ$ or $ℂ .$

Definition 1.1. Let X be a linear space over a field $𝕂 .$ A quasi β-norm || · || is a real-valued function on X satisfying the following statements:

||x||≥0 for all x ∈ X and ||x||=0 if and only if x = 0 .

||λx|| = |λ|^β · ||x|| for all $λ \in 𝕂$ and all x ∈ X .

There is a constant K ≥ 1 such that ||x + y|| ≤ K (||x|| + ||y||) for all x, y ∈ X .

The pair (X, || · ||) is called a quasi β-normed space if || · || is a quasi β-norm on X . The smallest possible K is called the modulus of concavity of || · || . A quasi β-Banach space is a complete quasi-β-normed space.

A quasi β-norm || · || is called a (β, p)-norm (0 < p ≤ 1) if (3) takes the form ||x + y||^p ≤ ||x||^p + ||y||^p for all x, y ∈ X . In this case, a quasi β-Banach space is called a (β, p)-Banach space; see [6 , 34].

In 1984. Katsaras [21] and Wu and Fang [43] independently introduced the notion of fuzzy norm. Since then some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view; see [4 , 44]. In 2003, Bag and Samanta [4] modified the definition of Cheng and Mordeson [9]. Bag and Samanta [4] introduced the following definition of fuzzy normed spaces. The notion of fuzzy stability of functional equations was given in the paper [27]. Introducing a fuzzy anti-norm linear space which depends on the idea of fuzzy norm from Bag and Samanta [5], Jebril and Samanta investigated its important properties [8].

Now, we will use the definition of fuzzy anti-normed spaces to investigate a fuzzy version of Hyers-Ulam-Rassias stability in the fuzzy anti-normed algebra setting.

Definition 1.2. [18] Let X be a real vector space. A function $N : X \times ℝ \to [0, 1]$ is called a fuzzy anti-norm on X if for all x, y ∈ X and all $s, t \in ℝ,$

N (x, t) =1 for t ≤ 0

N (x, t) =0 if and only if x = 0 for all t > 0

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

N (x + y, s + t) ≤ max {N (x, s) , N (y, t)}

N (x, t) is a non-increasing function of $t \in ℝ$ and $lim_{t \to \infty} N (x, t) = 0,$

for x ≠ 0, N (x, ·) is continuous on $ℝ .$

The pair (X, N) is called a fuzzy anti-normed space.

The property (aN3) implies that N (- x, t) = N (x, t) for all x ∈ X and t > 0 . It is easy to show that (aN4) is equivalent the following condition: $\begin{matrix} N (x + y, t) & \leq & \max {N (x, t), N (y, t)}, \\ for all x, y \in X and t \in ℝ . \end{matrix}$

Definition 1.3. Let X be a real vector space. A fuzzy anti-norm $N : X \times ℝ \to [0, 1]$ is called a fuzzy anti-β-norm on X if (aN3) in Definition 1.2 takes the form $\begin{array}{l} (a N^{'} 3) N (c x, t) \\ = N (x, \frac{t}{{| c |}^{β}}) (c \neq 0, 0 < β \leq 1) . \end{array}$

Example 1.4. Let (X, || · ||) be a β-normed space. Define $N (x, t) = {\begin{matrix} \frac{| | x | |}{t + | | x | |} & when t > 0, t \in ℝ \\ 1 & when t \leq 0, \end{matrix}$ where x ∈ X . We note that $N (cx, t) = \frac{| | cx | |}{t + | | cx | |} = \frac{| | x | |}{\frac{t}{| c |^{β}} + | | x | |} = N (x, \frac{t}{| c |^{β}}),$ for all x ∈ X and $c \in ℝ (c \neq 0, 0 < β \leq 1) .$ Then (X, N) is a fuzzy anti-β-normed space induced by the β-norm || · || .

Definition 1.5. Let (X, N) be a fuzzy anti-β-normed vector space. A sequence {x_n} in X is said to be convergent or converge if there exists an x ∈ X such that $lim_{n \to \infty} N (x_{n} - x, t) = 0$ for all t > 0 . In this case, x is called the limit of the sequence {x_n} and we denote it by $N - lim_{n \to \infty} x_{n} = x .$

Definition 1.6. Let (X, N) be a fuzzy anti-β-normed vector space. A sequence {x_n} in X is called Cauchy if for each ɛ > 0 and each t > 0 there exists an $n_{0} \in ℕ$ such that for all n ≥ n₀ and all integer d > 0, we have N (x_n+d - x_n, t) < ɛ .

It is well-known that every convergent sequence in a fuzzy anti-β-normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy anti-β-normed space is said to be fuzzy anti-β complete and the fuzzy anti-β-normed vector space is called a fuzzy anti-β Banach space.

Now, we will state the theorem, the alternative of fixed point in a generalized metric space.

Definition 1.7. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies

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

d (x, y) = d (y, x) for all x, y∈ X ;

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

Das was to provide applications of fuzzy metric for fuzzy sequence spaces; see [11, 12]. Also, Lot of research work has been done on fuzzy fixed point theory by using the notion of generalized metric, see for instance Tripathy et al. [37 –39]. The following is the alternative of fixed point results due to Margolis and Diaz [24] and Rus [35].

Theorem 1.8. (The alternative of fixed point) fixed Suppose that we are given a complete generalized metric space (X, d) and a strictly contractive mapping J : X → X with Lipschitz constant 0 < L < 1 . Then for each given x ∈ X, either $d (J^{n} x, J^{n + 1} x) = \infty all n \geq 0,$ or there exists a natural number n₀ such that

d (Jⁿx, Jⁿ⁺¹x)< ∞ for all n≥ n₀ ;

The sequence {Jⁿx} is convergent to a fixed point y^* of J ;

y^* is the unique fixed point of J in the set $Y = {y \in X | d (J^{n_{0}} x, y) < \infty};$

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

In this paper, we will investigate the Hyers-Ulam-Rassias stability in quasi β-normed space and then the fuzzy stability by using a fixed point in fuzzy anti-β Banach space for the generalized cubic function f : X → Y satisfying the Equation (1.3). Let us fix some notations which will be used throughout this paper. Let $a \in ℤ (a \neq 0, \pm 1) .$

2 A cubic functional equation

In this section let X and Y be real vector spaces and we investigate the general solution of the functional Equation (1.3). Before we proceed, we would like to introduce some basic definitions concerning n-additive symmetric mappings and key concepts which are found in [40, 42]. Let n be a positive integer. A function A_n : Xⁿ → Y is called n-additive if it is additive in each of its variables. A function A_n is said to be symmetric if A_n (x₁, ⋯ , x_n) = A_n (x_σ(1), ⋯ , x_σ(n)) for every permutation {σ (1) , ⋯ , σ (n)} of {1, 2, ⋯ , n} . If A_n (x₁, x₂, ⋯ , x_n) is an n-additive symmetric map, then Aⁿ (x) will denote the diagonal A_n (x, x, ⋯ , x) and Aⁿ (rx) = rⁿAⁿ (x) for all x ∈ X and all $r \in ℚ .$ Such a function Aⁿ (x) will be called a monomial function of degree n (assuming Aⁿnotequiv0). Furthermore the resulting function after substitution x₁ = x₂ = ⋯ = x_s = x and x_s+1 = x_s+2 = ⋯ = x_n = y in A_n (x₁, x₂, ⋯ , x_n) will be denoted by A^s,n-s (x, y) .

Theorem 2.1. A function f : X → Y is a solution of the functional Equation (1.3) if and only if f is of the form f (x) = A³ (x) for all x ∈ X, where A³ (x) is the diagonal of the 3-additive symmetric mapping A₃ : X³ → Y .

Proof. Assume that f satisfies the functional Equation (1.3). Letting x = y = 0 in the Equation (1.3), we have $a (a - 1) (2 a + 1) f (0) = 0,$ that is, f (0) =0 . Let y = 0 in the Equation (1.3). Then we get $f (ax) = a^{3} f (x)$ (2.1) for all x ∈ X . Putting x = 0 in the Equation (1.3), we get $(- a^{3} + a^{2}) (f (y) + f (- y)) = 0$ (2.2) for all y ∈ X . Hence we have f (- y) = - f (y) , for all y ∈ X . That is, f is odd. We can rewrite the functional Equation (1.3) in the form $\begin{matrix} f (x) - \frac{1}{(a^{2} - 1) (a + 1)} f (ax + y) \\ + \frac{1}{(a^{2} - 1) (a + 1)} f (x - ay) \\ + \frac{a}{2 (a^{2} - 1) (a + 1)} f (x + y) \\ - \frac{a^{2}}{(a^{2} - 1) (a + 1)} f (x - y) + \frac{a - 1}{a + 1} f (y) = 0 \end{matrix}$ for all x, y ∈ X and an integer a (a ≠ 0, ± 1) . By Theorems 3.5 and 3.6 in [42], f is a generalized polynomial function of degree at most 3, that is, f is of the form $f (x) = A^{3} (x) + A^{2} (x) + A^{1} (x) + A^{0} (x)$ (2.3) for all x ∈ X, where A⁰ (x) = A⁰ is an arbitrary element of Y, and Aⁱ (x) is the diagonal of the i-additive symmetric mapping A_i : Xⁱ → Y for i = 1, 2, 3 . By f (0) =0 and f (- x) = - f (x) for all x ∈ X, we get A⁰ (x) = A⁰ = 0 and A² (x) =0 . It follows that $f (x) = A^{3} (x) + A^{1} (x)$ for all x ∈ X . By (2.1) and Aⁿ (rx) = rⁿAⁿ (x) for all x ∈ X and $r \in ℚ,$ we obtain that A¹ (x) =0 . Hence we get f (x) = A³ (x) for all x ∈ X .

Conversely, assume that f (x) = A³ (x) for all x ∈ X, where A³ (x) is the diagonal of a 3-additive symmetric mapping A₃ : X³ → Y . Note that $\begin{matrix} A^{3} (qx + ry) & = & q^{3} A^{3} (x) + 3 q^{2} {rA}^{2, 1} (x, y) \\ + 3 {qr}^{2} A^{1, 2} (x, y) + r^{3} A^{3} (y) \\ c^{s} A^{s, t} (x, y) & = & A^{s, t} (cx, y), \\ c^{t} A^{s, t} (x, y) & = & A^{s, t} (x, cy) \end{matrix}$ where 1 ≤ s, t ≤ 2 and $c \in ℚ .$ Thus we may conclude that f satisfies the Equation (1.3). □

3 Hyers-Ulam-Rassias stability over a quasi β-Banach space

Throughout this section, let X be a real linear space and let Y be a quasi β-Banach space with a quasi β-norm || · ||_Y . Let K be the modulus of concavity of || · ||_Y . We will investigate the Hyers-Ulam-Rassias stability for the functional Equation (1.3); see also the paper [10].

For a given mapping f : X → Y and all fixed integer a (a ≠ 0, ± 1), let

$\begin{matrix} D_{a} f (x, y) & : = & f (ax + y) - f (x - ay) \\ - af (x + y) + a^{2} f (x - y) \\ - (a^{2} - 1) (a + 1) f (x) \\ - (a^{2} - 1) (a - 1) f (y) \end{matrix}$ (3.1) for all x, y ∈ X .

Theorem 3.1. Suppose that there exists a mapping φ : X² → [0, ∞) for which a mapping f : X → Y satisfies f (0) =0, $| | D_{a} f (x, y) | |_{Y} \leq φ (x, y)$ (3.2) and the series $\sum_{j = 0}^{\infty} (\frac{K}{| a |^{3 β}})^{j} φ (a^{j} x, a^{j} y)$ converges for all x, y ∈ X . Then there exists a unique generalized cubic mapping C : X → Y satisfying the Equation (1.3) and the inequality $| | f (x) - C (x) | |_{Y} \leq \frac{K}{| a |^{3 β}} \sum_{j = 0}^{\infty} (\frac{K}{| a |^{3 β}})^{j} φ (a^{j} x, 0),$ (3.3) for all x ∈ X .

Proof. On letting y = 0 in inequality (3.4), since f (0) =0 we have $| | D_{a} f (x, 0) | |_{Y} = | | f (ax) - a^{3} f (x) | |_{Y} \leq φ (x, 0)$ that is, $| | f (x) - \frac{1}{a^{3}} f (ax) | |_{Y} \leq \frac{1}{| a |^{3 β}} φ (x, 0),$ (3.4) for all x ∈ X .

We note that putting x = ax and multiplying $\frac{1}{| a |^{3 β}}$ in the inequality (3.4), we get $\frac{1}{| a |^{3 β}} | | f (ax) - \frac{1}{a^{3}} f (a^{2} x) | |_{Y} \leq \frac{1}{| a |^{3 β}} \frac{1}{| a |^{3 β}} φ (ax, 0),$ (3.5) for all x ∈ X .

Combining two inequalities (3.4) and (3.5), we have

$\begin{matrix} | | f (x) - (\frac{1}{a^{3}})^{2} f (a^{2} x) | |_{Y} \\ \leq \frac{K}{| a |^{3 β}} (φ (x, 0) + \frac{1}{| a |^{3 β}} φ (ax, 0)), \end{matrix}$ (3.6) for all x ∈ X .

Since K ≥ 1, inductively using the previous note we have the following inequalities

$\begin{matrix} | | f (x) - (\frac{1}{a^{3}})^{k} f (a^{k} x) | |_{Y} \\ \leq \frac{K}{| a |^{3 β}} \sum_{j = 0}^{k - 1} (\frac{K}{| a |^{3 β}})^{j} φ (a^{j} x, 0), \end{matrix}$ (3.7) for all x ∈ X, $k \in ℕ$ and also

$\begin{matrix} | | (\frac{1}{a^{3}})^{k} f (a^{k} x) - (\frac{1}{a^{3}})^{t} f (a^{t} x) | |_{Y} \\ \leq \frac{K}{| a |^{3 β}} \sum_{j = k}^{t - 1} (\frac{K}{| a |^{3 β}})^{j} φ (a^{j} x, 0), \end{matrix}$ (3.8) for all x ∈ X and k, $t \in ℕ (k < t) .$

Since the right-hand side of the previous inequality (3.8) tends to 0 as t → ∞ , hence ${(\frac{1}{a^{3}})^{n} f (a^{n} x)}$ is a Cauchy sequence in the quasi β-Banach space Y . Thus we may define $C (x) = lim_{n \to \infty} (\frac{1}{a^{3}})^{n} f (a^{n} x),$ for all x ∈ X . Since K ≥ 1, replacing x and y by aⁿx and aⁿy respectively and dividing by |a|^3βn in the inequality (3.4), we have $\begin{matrix} {(\frac{1}{| a |^{3 β}})}^{n} | | D_{a} f (a^{n} x, a^{n} y) | |_{Y} \\ = {(\frac{1}{| a |^{3 β}})}^{n} | | f (a^{n} (ax + y)) - f (a^{n} (x - ay)) \\ - af (a^{n} (x + y)) + a^{2} f (a^{n} (x - y)) \\ - (a^{2} - 1) (a + 1) f (a^{n} x) \\ - (a^{2} - 1) (a - 1) f (a^{n} y) | |_{Y} \\ \leq {(\frac{K}{| a |^{3 β}})}^{n} φ (a^{n} x, a^{n} y) \end{matrix}$ for all x, y ∈ X .

On taking n → ∞ , the definition of C implies that C satisfies (1.3) for all x, y ∈ X, that is, C is the generalized cubic mapping. Also, the inequality (3.7) implies the inequality (3.3).

Now, it remains to show the uniqueness. Assume that there exists T : X → Y satisfying (1.3) and (3.3). Then $\begin{matrix} | | T (x) - C (x) | |_{Y} \\ = {(\frac{1}{| a |^{3 β}})}^{n} | | T (a^{n} x) - C (a^{n} x) | |_{Y} \\ \leq {(\frac{1}{| a |^{3 β}})}^{n} K (| | T (a^{n} x) - f (a^{n} x) | |_{Y} \\ + | | f (a^{n} x) - C (a^{n} x) | |_{Y}) \\ \leq \frac{2 K^{2}}{| a |^{3 β} K^{n}} \sum_{j = n}^{\infty} {(\frac{K}{| a |^{3 β}})}^{j} φ (a^{j} x, 0) \end{matrix}$ for all x ∈ X . On letting n → ∞ , we immediately have the uniqueness of C . □

Corollary 3.2. Let θ ≥ 0, p < 3 be a real number and X be a normed linear space with norm || · || . Suppose f : X → Y is a mapping satisfying f (0) =0 and $| | D_{a} f (x, y) | |_{Y} \leq θ (| | x | |^{p} + | | y | |^{p})$ (3.9) for all x, y ∈ X and all t > 0 . Then $C (x) : = N - lim_{n \to \infty} \frac{1}{a^{3 n}} f (a^{n} x)$ exists for each x ∈ X and defines a generalized cubic mapping C : X → Y such that $| | f (x) - C (x) | |_{Y} \leq \frac{θ K}{(| a |^{3 β} - K | a |^{p β})} | | x | |^{p}$ for all x ∈ X and all t > 0 .

Proof. The proof follows from Theorem 3.1 by taking φ (x, y) = θ (||x||^p + ||y||^p) for all x, y ∈ X . □

Theorem 3.3. Suppose that there exists a mapping φ : X² → [0, ∞) for which a mapping f : X → Y satisfies f (0) =0, $| | D_{a} f (x, y) | |_{Y} \leq φ (x, y)$ (3.10) and the series $\sum_{j = 1}^{\infty} (| a |^{3 β} K)^{j} φ (a^{- j} x, a^{- j} y)$ converges for all x, y ∈ X . Then there exists a unique generalized cubic mapping C : X → Y which satisfies the Equation (1.3) and the inequality $| | f (x) - C (x) | |_{Y} \leq K \sum_{j = 1}^{\infty} (| a |^{3 β} K)^{j} φ (a^{- j} x, 0),$ (3.11) for all x ∈ X .

Proof. If x is replaced by $\frac{1}{a} x$ in the inequality (3.4), then the proof follows from the proof of Theorem 3.1. □

4 Fuzzy fixed point stability over a fuzzy anti-β Banach space

Let us fix some notations which will be used throughout this section. We assume X is a vector space and (Y, N) is a fuzzy anti-β Banach space. Using fixed point method, we will prove the Hyers-Ulam stability of the functional equation satisfying Equation (1.3) in fuzzy anti-β Banach space.

Theorem 4.1. Let φ : X² → [0, ∞) be a function such that there exists an 0 < L < 1 with $φ (x, y) \leq \frac{L}{| a |^{3 β}} φ (ax, ay)$ (4.1) for all x, y ∈ X . Let f : X → Y be a mapping satisfying f (0) =0 and $N (D_{a} f (x, y), t) \leq \frac{φ (x, y)}{t + φ (x, y)}$ (4.2) for all x, y ∈ X and all t > 0 . Then $C (x) : = N - lim_{n \to \infty} a^{3 n} f (\frac{x}{a^{n}})$ exists for each x ∈ X and defines a generalized cubic mapping C : X → Y such that $N (f (x) - C (x), t) \leq \frac{L φ (x, 0)}{| a |^{3 β} (1 - L) t + L φ (x, 0)}$ (4.3) for all x ∈ X and all t > 0 .

Proof. On putting y = 0 in the inequality (4.2), we have $N (f (ax) - a^{3} f (x), t) \leq \frac{φ (x, 0)}{t + φ (x, 0)}$ (4.4) for all x ∈ X and all t > 0 .

We note that on letting $x = \frac{x}{a}$ in the inequality (4.4) we have $N (f (x) - a^{3} f (\frac{x}{a}), t) \leq \frac{φ (\frac{x}{a}, 0)}{t + φ (\frac{x}{a}, 0)} .$

The inequality (4.1) implies that $N (f (x) - a^{3} f (\frac{x}{a}), t) \leq \frac{\frac{L}{| a |^{3 β}} φ (x, 0)}{t + \frac{L}{| a |^{3 β}} φ (x, 0)} .$ On putting $t = \frac{L}{| a |^{3 β}} t,$ we have $N (f (x) - a^{3} f (\frac{x}{a}), \frac{L}{| a |^{3 β}} t) \leq \frac{φ (x, 0)}{t + φ (x, 0)},$ (4.5) for all x ∈ X and all t > 0 .

We consider the set $S : = {g : X \to X}$ and the mapping d defined on S × S by $\begin{matrix} d (g, h) = \inf {μ \in ℝ^{+} | N (g (x) - h (x), μ t) \\ \leq \frac{φ (x, 0)}{t + φ (x, 0)}, \forall x \in X and t > 0} \end{matrix}$ where inf ∅ = + ∞ , as usual. Then (S, d) is a complete generalized metric space; see [25, Lemma 2.1]. Now let’s consider the linear mapping J : S → S such that $Jg (x) : = a^{3} g (\frac{x}{a})$ for all x ∈ X . Let g, h ∈ S be given such that d (g, h) = ɛ . Then $N (g (x) - h (x), ɛ t) \leq \frac{φ (x, 0)}{t + φ (x, 0)}$ for all x ∈ X and all t > 0 . $\begin{matrix} N (Jg (x) - Jh (x), L ɛ t) \\ = N (a^{3} g (\frac{x}{a}) - a^{3} h (\frac{x}{a}), L ɛ t) \\ = N (g (\frac{x}{a}) - h (\frac{x}{a}), \frac{L}{| a |^{3 β}} ɛ t) \\ \leq \frac{φ (\frac{x}{a}, 0)}{\frac{L}{| a |^{3 β}} t + φ (\frac{x}{a}, 0)} \\ \leq \frac{φ (x, 0)}{t + φ (x, 0)} \end{matrix}$ for all x ∈ X and all t > 0 . d (g, h) = ɛ implies that d (Jg, Jh) ≤ Lɛ . Hence we get $d (Jg, Jh) \leq L d (g, h)$ for all g, h ∈ S . The inequality (4.5) implies that $d (f, Jf) \leq \frac{L}{| a |^{3 β}} .$ By Theorem 1.8, there exists a mapping C : X → Y such that

C is a fixed point of J, that is, $C (\frac{x}{a}) = \frac{1}{a^{3}} C (x)$ (4.6) for all x ∈ X . The mapping C is a unique fixed point of J in the set M = {g ∈ S | d (f, g) < ∞} . This means that C is a unique mapping satisfying the Equation (4.6) such that there exists a μ ∈ (0, ∞) satisfying $N (f (x) - C (x), μ t) \leq \frac{φ (x, 0)}{t + φ (x, 0)}$ for all x ∈ X and all t> 0 ;

d (Jⁿf, C) →0 as n → ∞ . This implies the following equality $N - lim_{n \to \infty} a^{3 n} f (\frac{x}{a^{n}}) = C (x)$ for all x∈ X ;

$d (f, C) \leq \frac{1}{1 - L} d (f, Jf),$ which implies the inequality $d (f, C) \leq \frac{1}{1 - L} \cdot \frac{L}{| a |^{3 β}} = \frac{L}{| a |^{3 β} (1 - L)} .$

It implies that $N (f (x) - C (x), \frac{L}{| a |^{3 β} (1 - L)} t) \leq \frac{φ (x, 0)}{t + φ (x, 0)}$ for all x ∈ X and all t > 0 . On replacing t by $\frac{| a |^{3 β} (1 - L)}{L} t,$ we have $N (f (x) - C (x), t) \leq \frac{L φ (x, 0)}{| a |^{3 β} (1 - L) t + L φ (x, 0)}$ for all x ∈ X and all t > 0 . That is, the inequality (4.3) holds. On letting $x = \frac{x}{a^{n}}$ and $y = \frac{y}{a^{n}}$ in the inequality (4.2), we have $N (a^{3 n} D_{a} f (\frac{x}{a^{n}}, \frac{y}{a^{n}}), | a |^{3 β n} t) \leq \frac{φ (\frac{x}{a^{n}}, \frac{y}{a^{n}})}{t + φ (\frac{x}{a^{n}}, \frac{y}{a^{n}})}$ for all x, y ∈ X, all t > 0 and all $n \in ℕ .$ Replacing t by $\frac{t}{| a |^{3 β n}},$ $\begin{matrix} N (a^{3 n} D_{a} f (\frac{x}{a^{n}}, \frac{y}{a^{n}}), t) \\ \leq \frac{φ (\frac{x}{a^{n}}, \frac{y}{a^{n}})}{\frac{t}{| a |^{3 β n}} + φ (\frac{x}{a^{n}}, \frac{y}{a^{n}})} \leq \frac{L^{n} φ (x, y)}{t + L^{n} φ (x, y)} \end{matrix}$ for all x, y ∈ X, all t > 0 and all $n \in ℕ .$ Since $lim_{n \to \infty} \frac{L^{n} φ (x, y)}{t + L^{n} φ (x, y)} = 0$ for all x, y ∈ X and all t > 0, we may conclude that $N (D_{a} C (x, y), t) = 0$ for all x, y ∈ X and all t > 0 . Thus the mapping C : X → Y is the generalized cubic mapping. □

Corollary 4.2. Let θ ≥ 0, p > 3 be a real number and X be a normed linear space with norm || · || . Suppose f : X → Y is a mapping satisfying f (0) =0 and $N (D_{a} f (x, y), t) \leq \frac{θ (| | x | |^{p} + | | y | |^{p})}{t + θ (| | x | |^{p} + | | y | |^{p})}$ (4.7) for all x, y ∈ X and all t > 0 . Then $C (x) : = N - lim_{n \to \infty} a^{3 n} f (\frac{x}{a^{n}})$ exists for each x ∈ X and defines a generalized cubic mapping C : X → Y such that $\begin{matrix} N (f (x) - C (x), t) \\ \leq \frac{θ | a |^{3 β} | | x | |^{p}}{| a |^{3 β} (| a |^{p β} - | a |^{3 β}) t + θ | a |^{3 β} | | x | |^{p}} \end{matrix}$ for all x ∈ X and all t > 0 .

Proof. The proof follows from Theorem 4.1 on taking φ (x, y) = θ (||x||^p + ||y||^p) for all x, y ∈ X and L = |a|^(3-p)β . □

Footnotes

Acknowledgment

The authors thank to the reviewers for the careful and detailed reading of the manuscript and suggesting valuable comments and some related references.

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