Abstract
In this article, we propose to determine some stability results for the functional equation of cubic in random 2-normed spaces which seems to be a quite new and interesting idea. Also, we define the notion of continuity, approximately and conditional cubic mapping in random 2-normed spaces and prove some interesting results.
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1 Introduction and preliminaries
In 1940, Ulam [1] proposed the following question concerning the stability of group homomorphisms:
Let G1 be a group and let G2 be a metric group with the metric d(., .). Given ϵ > 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 exists a homomorphism H : G1 → G1 with d(h(x), H(x)) < ϵ for all x ∈ G1?
In next year, Hyers [2] answers the problem of Ulam under the assumption that the groups are Banach spaces and then generalized by Aoki [3] and Rassias [4] for additive mappings and linear mappings, respectively. Since then several stability problems for various functional equations have been investigated in [5–12].
The stability problem for the cubic functional equation was proved by Jun and Kim [5] for mappings f: X → Y, where X is a real normed space and Y is a Banach space. Later on, the problem of stability of cubic functional equation were discussed by many mathematician.
An interesting and important generalization of the notion of a metric space was introduced by Menger [13] under the name of statistical metric space, which is now called a probabilistic metric space. An important family of probabilistic metric spaces is that of probabilistic normed spaces. The theory of probabilistic normed spaces is important as a generalization of deterministic results of linear normed spaces. The theory of probabilistic normed spaces was initiated and developed in [14, 15] and further it was extended to random 2-normed spaces by Goleţ [16] using the concept of 2-norm of Gahler [17]. For more details of probabilistic and random/fuzzy 2-normed space, we refer to [18–22] and references therein.
In this article, we establish Hyers-Ulam stability concerning the cubic functional equations in random 2-normed spaces which is quite a new and interesting idea to study with.
In this section, we recall some notations and basic definitions used in this article.
A distribution function is an element of Δ+, where Δ+ = {f : ℝ → [0, 1]; f is left-continuous, nondecreasing, f(0) = 0 and f(+∞) = 1} and the subset D+ ⊆ Δ+ is the set D+ = {f ∈ Δ+; l-f(+∞) = 1}. Here l-f(+∞) denotes the left limit of the function f at the point x. The space Δ+ is partially ordered by the usual point-wise ordering of functions, i.e., f ≤ g if and only if f(x) ≤ g(x) for all x ∈ ℝ. For any a ∈ ℝ, H a is a distribution function defined by
The set Δ, as well as its subsets, can be partially ordered by the usual pointwise order: in this order, H0 is the maximal element in Δ+.
A triangle function is a binary operation on Δ+, namely a function τ : Δ+ × Δ+ → Δ+ that is associative, commutative nondecreasing and which has ε0 as unit, that is, for all f, g, h ∈ Δ+, we have:
-
(i)
τ(τ(f, g), h) = τ(f, τ(g, h)),
-
(ii)
τ(f, g) = τ(g, f),
-
(iii)
τ(f, g) = τ(g, f) whenever f ≤ g,
-
(iv)
τ(f, H 0) = f.
A t-norm is a continuous mapping * : [0, 1] × [0, 1] → [0, 1] such that ([0, 1], *) is abelian monoid with unit one and c * d ≥ a * b if c ≥ a and d ≥ b for all a, b, c, d ∈ [0, 1].
The concept of 2-normed space was first introduced in [17] and further studied in [23–25].
Let X is a linear space of a dimension d, where 2 ≤ d < ∞. A 2-normed on X is a function ∥., .∥ : X × X → ℝ satisfying the following conditions, for every x, y ∈ X (i) ∥x, y∥ = 0 if and only if x and y are linearly dependent; (ii) ∥x, y∥ = ∥y, x∥; (iii) ∥αx, y∥ = |α|∥x, y∥, α ∈ ℝ; (iv) ∥x + y, z∥ ≤ ∥x, z∥ + ∥y, z∥. In this case (X, ∥., . ∥) is called a 2-norm space.
Example 1.1. Take X = ℝ2 being equipped with the 2-norm ∥x, y∥ = the area of the parallelogram spanned by the vectors x and y, which may be given explicitly by the formula
Recently, Goleţ [16] introduced the notion of random 2-normed space and further studied by Mursaleen [26].
Let X be a linear space of a dimension greater than one, τ is a triangle function, and . Then ℱ is called a probabilistic 2-norm on X and a probabilistic 2-normed space if the following conditions are satisfied:
-
(i)
if x and y are linearly dependent, where denotes the value of at t ∈ ℝ,
-
(ii)
if x and y are linearly independent,
-
(iii)
for every x, y in X,
-
(iv)
for every t > 0, α ≠ 0 and x, y ∈ X,
-
(v)
whenever x, y, z ∈ X.
If (v) is replaced by
(v') , for all x, y, z ∈ X and , then triple is called a random 2-normed space (for short, RTN-space).
Example 1.2. Let (X, ∥., .∥) be a 2-normed space with ∥x, z∥ = ∥x1z2 - x2z1∥, x = (x1, x2), z = (z1, z2) and a * b = ab for a, b ∈ [0, 1]. For all x ∈ X, t > 0 and nonzero z ∈ X, consider
Then is a random 2-normed space.
Remark 1.3. Note that every 2-normed space (X, ∥., .∥) can be made a random 2-normed space in a natural way, by setting , for every x, y ∈ X, t > 0 and a * b = min{a, b}, a, b ∈ [0, 1].
2 Stability of cubic functional equation
In the present section, we define the notion of convergence, Cauchy sequence and completeness in RTN-space and determine some stability results of the cubic functional equation in RTN-space.
The functional equation
is called the cubic functional equation, since the function f(x) = cx3 is its solution. Every solution of the cubic functional equation is said to be a cubic mapping.
We shall assume throughout this article that X and Y are linear spaces; and are random 2-normed spaces; and is a random 2-Banach space.
Let φ be a function from X × X to Z. A mapping f : X → Y is said to be φ-approximately cubic function if
for all x, y ∈ X, t > 0 and nonzero z ∈ X, where
We define:
We say that a sequence x = (x k ) is convergent in or simply ℱ-convergent to ℓ if for every ϵ > 0 and θ ∈ (0, 1) there exists k0 ∈ ℕ such that whenever k ≥ k0 and nonzero z ∈ X. In this case we write and ℓ is called the ℱ-limit of x = (x k ).
A sequence x = (x k ) is said to be Cauchy sequence in or simply ℱ-Cauchy if for every ϵ > 0, θ > 0 and nonzero z ∈ X there exists a number N = N(ϵ, z) such that for all n, m ≥ N. RTN-space is said to be complete if every ℱ-Cauchy is ℱ-convergent. In this case is called random 2-Banach space.
Theorem 2.1. Suppose that a function φ : X × X → Z satisfies φ(2x, 2y) = αφ(x, y) for all x, y ∈ X and α ≠ 0. Let f : X → Y be a φ-approximately cubic function. If for some 0 < α < 8,
and for all x, y ∈ X, t > 0 and nonzero z ∈ X. Then there exists a unique cubic mapping C : X → Y such that
for all x ∈ X, t > 0 and nonzero z ∈ X.
Proof. For convenience, let us fix y = 0 in (2). Then for all x ∈ X, t > 0 and nonzero z ∈ X
Replacing x by 2nx in (5) and using (3), we obtain
for all x ∈ X, t > 0 and nonzero z ∈ X; and for all n ≥ 0. By replacing t by αnt, we get
It follows from and (6) that
for all x ∈ X, t > 0 and n > 0 where . By replacing x with 2mx in (7), we have
Thus
for all x ∈ X, t > 0, m > 0, n ≥ 0 and nonzero z ∈ X. Hence
for all x ∈ X, t > 0 m ≥ 0, n ≥ 0 and nonzero z ∈ X. Since 0 < α < 8 and , the Cauchy criterion for convergence shows that is a Cauchy sequence in . Since is complete, this sequence converges to some point C(x) ∈ Y. Fix x ∈ X and put m = 0 in (8) to obtain
for all t > 0, n > 0 and nonzero z ∈ X. Thus we obtain
for large n. Taking the limit as n → ∞ and using the definition of RTN-space, we get
Replace x and y by 2nx and 2ny, respectively, in (2), we have
for all x, y ∈ X, t > 0 and nonzero z ∈ X. Since
we observe that C fulfills (1). To Prove the uniqueness of the cubic function C, assume that there exists a cubic function D : X → Y which satisfies (4). For fix x ∈ X, clearly C(2nx) = 8nC(x) and D(2nx) = 8nD(x) for all n ∈ ℕ. It follows from (4) that
Therefore
Thus for all x ∈ X, t > 0 and nonzero z ∈ X. Hence C(x) = D(x).
Example 2.2. Let X be a Hilbert space and Z be a normed space. By ℱ and , we denote the random 2-norms given as in Example 1.1 on X and Z, respectively. Let φ : X × X → Z be defined by φ(x, y) = 8(∥x∥2 + ∥y∥2)zο, where zο is a fixed unit vector in Z. Define f : X → X by f(x) = ∥x∥2x + ∥x∥2xο for some unit vector xο ∈ X. Then
Also
Thus,
Hence, conditions of Theorem 2.1 for α = 4 are fulfilled. Therefore, there is a unique cubic mapping C : X → X such that for all x ∈ X, t > 0 and nonzero z ∈ X.
By a modification in the proof of Theorem 2.1, one can easily prove the following:
Theorem 2.3. Suppose that a function φ : X × X → Z satisfies for all x, y ∈ X and α ≠ 0. Let f : X → Y be a φ-approximately cubic function. If for some α > 8
and for all x, y ∈ X, t > 0 and nonzero z ∈ X. Then there exists a unique cubic mapping C : X → Y such that
for all x ∈ X, t > 0 and nonzero z ∈ X.
3 Continuity in random 2-normed spaces
In this section, we establish some interesting results of continuous approximately cubic mappings.
Let f : ℝ → X be a function, where ℝ is endowed with the Euclidean topology and X is an random 2-normed space equipped with random 2-norm ℱ. Then, f is said to be random 2-continuous or simply ℱ-continuous at a point sο ∈ ℝ if for all ϵ > 0 and all 0 < α < 1 there exists δ > 0 such that
for each s with 0 < |s - sο| < δ and nonzero z ∈ X.
A mapping f : X → Y is said to be (p, q)-approximately cubic function if, for some p, q and some zο ∈ Z,
for all x, y ∈ X, t > 0 and nonzero z ∈ X.
Theorem 3.2. Let X be a normed space and let f : X → Y be a (p, q)-approximately cubic function. If p, q < 3, there exists a unique cubic mapping C : X → Y such that
for all x ∈ X, t > 0 and nonzero z ∈ X. Furthermore, if for some x ∈ X and all n ∈ ℕ, the mapping g : ℝ → Y defined by g(s) = f(2nsx) is ℱ-continuous. Then the mapping s ↦ C(sx) from ℝ to Y is ℱ-continuous; in this case, C(rx) = r3C(x) for all r ∈ ℝ.
Proof. Suppose that a function φ : X × X → Z satisfies φ(x, y) = (∥x∥p+∥y∥q)zο. Existence and uniqueness of the cubic mapping C satisfying (9) are deduced from Theorem 2.1. Note that for each x ∈ X, t ∈ ℝ and n ∈ ℕ, we have
Fix x ∈ X and sο ∈ ℝ. Given ϵ > 0 and 0 < α < 1. From (10) follows that
for all |s - sο| < 1 and s ∈ ℝ. Since , there exists nο ∈ ℕ such that
for all |s - sο| < 1 and s ∈ ℝ. By the ℱ-continuity of the mapping , there exists δ < 1 such that for each s with 0 < |s - sο| < δ, we have
It follows that
for each s with 0 < |s - sο| < δ. Hence, the mapping s ↦ C(sx) is ℱ-continuous.
Now, we use the ℱ-continuity of s ↦ C(sx) to establish that for all rο ∈ ℝ. For each r, ℚ is a dense subset of ℝ, we have C(rx) = r3C(x). Fix rο ∈ ℝ and t > 0. Then, for 0 < α < 1 there exists δ > 0 such that
for each r ∈ ℝ and 0 < |r - rο| < δ. Choose a rational number r with 0 < |r - rο| < δ and . Then
Thus . Hence, we conclude that .
Remark 3.2. We can also prove Theorem 3.1 for the case when p, q > 3. In this case, there exists a unique cubic mapping C : X → Y such that for all x ∈ X, t > 0 and nonzero z ∈ X.
4 Approximately and conditional cubic mapping in random 2-normed spaces
In this section, we obtain completeness in RTN-space through the existence of some solution of a stability problem for cubic functional equation.
A mapping f : ℕ∪{0} → X is said to be approximately cubic if for each α ∈ (0, 1) there exists some n α ∈ ℕ such that , for all n ≥ 2m ≥ n α and nonzero z ∈ X.
By a conditional cubic mapping, we mean a mapping f : ℕ ∪ {0} → X such that (1) holds whenever x ≥ 2y.
It can be easily verified that for each conditional cubic mapping f : ℕ ∪ {0} → X, we have f(2n) = 23nf(1).
Theorem 4.1. Let be a RTN-space such that for each approximately cubic mapping f : ℕ ∪ {0} → X, there exists a conditional cubic mapping C : ℕ ∪ {0} → X, such that
for nonzero z ∈ X. Then is a random 2-Banach space.
Proof. Let (x n ) be a Cauchy sequence in a RTN-space. By induction on k, we can find a strictly increasing sequence (n k ) of natural numbers such that
for each n, m ≥ n k and nonzero z ∈ X. Let and define f : ℕ ∪ {0} → X by f(k) = k3y k . Let α ∈ (0, 1). and find some nο ∈ ℕ such that . One can easily verified that
for each k ≥ 2j, and nonzero z ∈ X. Then
for j > nο and nonzero z ∈ X. Since and k - j ≥ j, we have
Clearly
The inequalities 2k - j ≥ j and 2k - j > k imply
Therefore . This shows that f is approximately cubic type mapping. By our assumption, there exists a conditional cubic mapping C : ℕ ∪ {0} → X, such that . In particular, . This means that
Hence the subsequence converges to y = C(1). Therefore, the Cauchy sequence (x n ) also converges to y.
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Alotaibi, A., Mohiuddine, S.A. On the stability of a cubic functional equation in random 2-normed spaces. Adv Differ Equ 2012, 39 (2012). https://doi.org/10.1186/1687-1847-2012-39
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DOI: https://doi.org/10.1186/1687-1847-2012-39