Stabilities of Cubic Mappings in Various Normed Spaces: Direct and Fixed Point Methods

Theorem 1.1 (Th. M. Rassias). Let f : E → E^′ be a mapping from a normed vector space E into a Banach space E^′ subject to the inequality

for all x, y ∈ E, where ϵ and p are constants with ϵ > 0 and p < 1. Then, the limit exists for all x ∈ E and L : E → E′ is the unique additive mapping which satisfies for all x ∈ E. If p < 0, then inequality (1.1) holds for x, y ≠ 0 and (1.3) for x ≠ 0. Also, if for each x ∈ E the mapping t ↦ f(tx) is continuous in t ∈ ℝ, then L is ℝ-linear.

Theorem 1.2 (J. M. Rassias). Let f : E → E^′ be a mapping from a real normed vector space E into a Banach space E^′ subject to the inequality

for all x, y ∈ E, where ϵ and r = p + q are constants with ϵ > 0 and r ≠ 1. Then, L : E → E^′ is the unique additive mapping which satisfies for all x ∈ E.

Definition 2.1. A function  T : [0,1] ² → [0,1]  is a continuous triangular norm (briefly  a t-norm) if Tsatisfies the following conditions:

• (i)

  T is commutative and associative;

• (ii)

  T is continuous;

• (iii)

  T(x, 1) = x for all x ∈ [0,1];

• (iv)

  T(x, y) ≤ T(z, w) whenever x ≤ z and y ≤ w for all x, y, z, w ∈ [0,1].

Definition 2.2. A random normed space (briefly  RN-space) is a triple (X, μ^′, T), where  X  is a vector space,  T  is a continuous t-norm and  μ^′ : X → D⁺  is a mapping such that the following conditions hold:

•  

  (i)  $$ for all t > 0 if and only if x = 0;

•  

  (ii) $$ for all α ∈ ℝ, α ≠ 0, x ∈ X and t ≥ 0;

•  

  (iii) $$, for all x, y ∈ X and t, s ≥ 0.

Definition 2.3. Let  (X, μ^′, T)   be an RN-space.

•  

  (i) A sequence {x_n} in X is said to be convergent to x ∈ X in X if, for all t > 0, $$.

•  

  (ii) A sequence {x_n} in X is said to be Cauchy sequence in X if, for all t > 0, $$.

•  

  (iii) The RN-space (X, μ^′, T) is said to be complete if every Cauchy sequence in X is convergent.

Theorem 2.4. If (X, μ^′, T) is RN-space and {x_n} is a sequence such that x_n → x, then $$.

Definition 2.5. Let  X  be a vector space over a field  𝕂  with a non-Archimedean valuation  |·|. A function  ∥·∥:X → [0, ∞) is called a non-Archimedean norm if the following conditions hold:

• (a)

  ∥x∥ = 0 if and only if x = 0 for all x ∈ X;

• (b)

  ∥rx∥ = |r | ∥x∥ for all r ∈ 𝕂 and x ∈ X;

• (c)

  the strong triangle inequality holds:

  $$ ()
  for all x, y ∈ X. Then (X, ∥·∥) is called a non-Archimedean normed space.

Definition 2.6. Let  {x_n} be a sequence in a non-Archimedean normed space  X.

• (a)

  A sequence $$ in a non-Archimedean space is a Cauchy sequence if and only if, the sequence $$ converges to zero.

• (b)

  The sequence {x_n} is said to be convergent if, for any ε > 0, there are a positive integer N and x ∈ X such that

  $$ ()
  for all n ≥ N. Then, the point x ∈ X is called the limit of the sequence {x_n}, which is denoted by lim _n→∞x_n = x.

• (c)

  If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space.

Definition 2.7. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies the following conditions:

• (a)

  d(x, y) = 0 if and only if x = y for all x, y ∈ X;

• (b)

  d(x, y) = d(y, x) for all x, y ∈ X;

• (c)

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

Theorem 2.8. Let (X, d) be a complete generalized metric space and J : X → X a strictly contractive mapping with Lipschitz constant L < 1. Then, for all x ∈ X, either

for all nonnegative integers n or there exists a positive integer n₀ such that

• (a)

  d(Jⁿx, Jⁿ⁺¹x) < ∞ for all n₀ ≥ n₀;

• (b)

  the sequence {Jⁿx} converges to a fixed point y^* of J;

• (c)

  y^* is the unique fixed point of J in the set $$;

• (d)

  d(y, y^*) ≤ (1/(1 − L))d(y, Jy) for all y ∈ Y.

Lemma 3.1. Let E₁ and E₂ be real vector spaces. A function f : E₁ → E₂ satisfies the functional equation (1.7) if and only if f : E₁ → E₂ satisfies the functional equation (1.9). Therefore, every solution of functional equation (1.9) is also cubic function.

Proof. Let f : E₁ → E₂ satisfy the equation (1.7). Putting x = y = 0 in (1.7), we get f(0) = 0. Set y = 0 in (1.7) to get f(−y) = −f(y). By induction, we lead to f(kx) = k³f(x) for all positive integer k. Replacing y by x + y in (1.7), we have

for all x, y ∈ E₁. Once again replacing y by y − x in (1.7), we have for all x, y ∈ E₁. Adding (3.1) to (3.2) and using (1.7), we obtain for all x, y ∈ E₁. By using the previous method, by induction, we infer that for all x, y ∈ E₁ and each positive integer m ≥ 3.

Let f : E₁ → E₂ satisfy the functional equation (1.9) with the positive integer m ≥ 3. Putting x = y = 0 in (1.9), we get f(0) = 0. Setting x = 0, we get f(−y) = −f(y). Let k be a positive integer. Replacing y by kx + y in (1.9), we have

for all x, y ∈ E₁. Replacing y by y − kx in (1.9), we have for all x, y ∈ E₁. Adding (3.5) to (3.6), we obtain for all x, y ∈ E₁ and for all integer k ≥ 1. Let ψ_m(x, y) = f(mx + y) + f(mx − y) for each integer m ≥ 0. Then, (3.7) means that for all x, y ∈ E₁ and for all integer k ≥ 1. For k = 1 and k = m in (3.8), we obtain for all x, y ∈ E₁. By the proof of the first part, since f : E₁ → E₂ satisfies the functional equation (1.9) with the positive integer m ≥ 3, then f satisfies the functional equation (1.9) with the positive integer k ≥ m. It follows from (3.9) that f satisfies the functional equation (1.7) and

Theorem 3.2. Let X be a real linear space, (Z, μ^′, min ) an RN-space, and ψ : X² → Z a function such that, for some 0 < α < m³,

and, for all x, y ∈ X and t > 0, $$. Let (Y, μ, min ) be a complete RN-space. If f : X → Y is a mapping with f(0) = 0 such that for all x ∈ X and t > 0 then the limit C(x) = lim _n→∞(f(mⁿx)/m³ⁿ) exists for all x ∈ X and defines a unique cubic mapping C : X → Y such that

Proof. Putting y = 0 in (3.12) we see that, for all x ∈ X,

Replacing x by mⁿx in (3.14) and using (3.11), we obtain So This implies that Replacing x by $$ in (3.17), we obtain As {f(mⁿx)/m³ⁿ} is a Cauchy sequence in complete RN-space (Y, μ, min ), so there exists some point C(x) ∈ Y such that lim _n→∞(f(mⁿx)/m³ⁿ) = C(x). Fix x ∈ X and put p = 0 in (3.18). Then, we obtain and so, for every ϵ > 0, we have Taking the limit as n → ∞ and using (3.21), we get Since ϵ was arbitrary by taking ϵ → 0 in (3.22), we get Replacing x and y by mⁿx and mⁿy in (3.12), respectively, we get, for all x, y ∈ X and for all t > 0, Since $$, we conclude that C(mx ± y) = C(x ± y) + 2(m³ − m)C(x). To prove the uniqueness of the cubic mapping C, assume that there exists another cubic mapping L : X → Y which satisfies (3.13).

By induction one can easily see that, since f is a cubic functional equation, so, for all n ∈ ℕ and every x ∈ X, C(mⁿx) = m³ⁿC(x), and L(mⁿx) = m³ⁿL(x), we have

so Since $$, it follows that, for all t > 0, μ_C(x)−L(x)(t) = 1 and so C(x) = L(x). This completes the proof.

Corollary 3.3. Let X be a real linear space, (Z, μ^′, min ) an RN-space, and (Y, μ, min ) a complete RN-space. Let 0 < r < 1 and z₀ ∈ Z, and let f : X → Y be a mapping with f(0) = 0 and satisfying

for all x, y ∈ X and t > 0. Then, the limit C(x) = lim _n→∞(f(mⁿx)/m³ⁿ) exists for all x ∈ X and defines a unique cubic mapping C : X → Y such that for all x ∈ X and t > 0.

Proof. Let α = m^3r, and let ψ : X² → Z be defined as ψ(x, y) = (| | x | |^r+| | y | |^r)z₀.

Remark 3.4. In Corollary 3.3, if we assume that ψ(x, y) = (∥x∥^r.∥y∥^r)z₀ or ψ(x, y) = (∥x∥^r+s + ∥y∥^r+s + ∥x∥^r∥y∥^s)z₀, then we get Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability, respectively. But, since we put  y = 0  in this functional equation, the Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability corollaries will be obvious. Meanwhile, the JMRassias mixed product-sum stability when r + s = 3  is an open question.

Corollary 3.5. Let X be a real linear space, (Z, μ^′, min ) an RN-space, and (Y, μ, min ) a complete RN-space. Let z₀ ∈ Z, and let f : X → Y be a mapping with f(0) = 0 and satisfying

for all x, y ∈ X and t > 0. Then, the limit C(x) = lim _n→∞(f(mⁿx)/m³ⁿ) exists for all x ∈ X and defines a unique cubic mapping C : X → Y such that for all x ∈ X and t > 0.

Proof. Let α = 1, and let ψ : X² → Z be defined by ψ(x, y) = δz₀.

Theorem 3.6. Let X be a real linear space, (Z, μ^′, min ) an RN-space, and ψ : X² → Z a function such that for some 0 < α < 1/m³

and, for all x, y ∈ X and t > 0, Let (Y, μ, min ) be a complete RN-space. If f : X → Y is a mapping with f(0) = 0 and satisfying (3.12), then the limit C(x) = lim _n→∞m³ⁿf(x/mⁿ) exists for all x ∈ X and defines a unique cubic mapping C : X → Y such that

Proof. Putting y = 0 in (3.12) and replacing x by x/m, we obtain that for all x ∈ X

Replacing x by x/mⁿ in (3.34) and using (3.31), we obtain So This implies that The rest of the proof is similar to the proof of Theorem 3.2.

Corollary 3.7. Let X be a real linear space, (Z, μ^′, min ) an RN-space, and (Y, μ, min ) a complete RN-space. Let r > 1 and z₀ ∈ Z, and let f : X → Y be a mapping with f(0) = 0 and satisfying (3.27). Then, the limit C(x) = lim _n→∞m³ⁿf(x/mⁿ) exists for all x ∈ X and defines a unique cubic mapping C : X → Y such that

for all x ∈ X and t > 0.

Proof. Let α = m^−3r, and let ψ : X² → Z be defined as ψ(x, y) = (| | x | |^r+| | y | |^r)z₀.

Corollary 3.8. Let X be a real linear space, (Z, μ^′, min ) be an RN-space, and (Y, μ, min ) a complete RN-space. Let z₀ ∈ Z, and let f : X → Y be a mapping with f(0) = 0 and satisfying (3.29). Then, the limit C(x) = lim _n→∞m³ⁿf(x/mⁿ) exists for all x ∈ X and defines a unique cubic mapping C : X → Y such that

for all x ∈ X and t > 0.

Proof. Let α = 1/m⁴, and let ψ : X² → Z be defined by ψ(x, y) = δz₀.

Theorem 4.1. Let X be a linear space, (Y, μ, T_M) a complete RN-space, and Φ a mapping from X² to D⁺(Φ(x, y) is denoted by Φ_x,y) such that there exists 0 < α < 1/m³ such that

for all x, y ∈ X and t > 0. Let f : X → Y be a mapping with f(0) = 0 and satisfying for all x, y ∈ X and t > 0. Then, for all x ∈ X exists and C : X → Y is a unique cubic mapping such that for all x ∈ X and t > 0.

Proof. Putting y = 0 in (4.2) and replacing x by x/m, we have

for all x ∈ X and t > 0. Consider the set and the generalized metric d in S defined by where inf   ∅ = +∞. It is easy to show that (S, d) is complete (see [26], Lemma 2.1). Now, we consider a linear mapping J : S → S such that for all x ∈ X. First, we prove that J is a strictly contractive mapping with the Lipschitz constant m³α.

In fact, let g, h ∈ S be such that d(g, h) < ϵ. Then, we have

for all x ∈ X and t > 0, and so for all x ∈ X and t > 0. Thus, d(g, h) < ϵ implies that This means that for all g, h ∈ S. It follows from (4.5) that By Theorem 2.8, there exists a mapping C : X → Y satisfying the following.

(1) C is a fixed point of J, that is,

for all x ∈ X.

The mapping C is a unique fixed point of J in the set

This implies that C is a unique mapping satisfying (4.14) such that there exists u ∈ (0, ∞) satisfying for all x ∈ X and t > 0.

(2) d(Jⁿf, C) → 0 as n → ∞. This implies the equality

for all x ∈ X.

(3) d(f, C) ≤ d(f, Jf)/(1 − m³α) with f ∈ Ω, which implies the inequality d(f, C) ≤ α/(2 − 2m³α)  and so

for all x ∈ X and t > 0. This implies that inequality (4.4) holds. Now, we have for all x, y ∈ X, t > 0, and n ≥ 1, and so, from (4.1), it follows that Since for all x, y ∈ X and t > 0, we have for all x, y ∈ X and t > 0. Thus, the mapping C : X → Y is cubic. This completes the proof.

Corollary 4.2. Let X be a real normed space, θ ≥ 0, and p a real number with p ∈ (1, +∞). Let f : X → Y be a mapping with f(0) = 0 and satisfying

for all x, y ∈ X and t > 0. Then, for all x ∈ X, the limit C(x) = lim _n→∞m³ⁿf(x/mⁿ) exists and C : X → Y is a unique cubic mapping such that for all x ∈ X and t > 0.

Proof. The proof follows from Theorem 4.1 if we take

for all x, y ∈ X and t > 0. In fact, if we choose α = m^−3p, then we get the desired result.

Theorem 4.3. Let X be a linear space, (Y, μ, T_M) a complete RN-space, and Φ a mapping from X² to D⁺ ( Φ(x, y) is denoted by Φ_x,y) such that for some 0 < α < m³

for all x, y ∈ X and t > 0. Let f : X → Y be a mapping with f(0) = 0 and satisfying (4.2). Then, for all x ∈ X, the limit C(x)∶ = lim _n→∞f(mⁿx)/m³ⁿ exists and C : X → Y is a unique cubic mapping such that for all x ∈ X and t > 0.

Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 4.1. Now, we consider a linear mapping J : S → S such that

for all x ∈ X.

Let g, h ∈ S be such that d(g, h) < ϵ. Then, we have

for all x ∈ X and t > 0 and so for all x ∈ X and t > 0. Thus, d(g, h) < ϵ implies that This means that for all g, h ∈ S.

Putting y = 0 in (4.2), we see that, for all x ∈ X,

It follows from (4.33) that By Theorem 2.8, there exists a mapping C : X → Y satisfying the following.

(1) C is a fixed point of J, that is,

for all x ∈ X.

The mapping C is a unique fixed point of J in the set

This implies that C is a unique mapping satisfying (4.35) such that there exists u ∈ (0, ∞) satisfying for all x ∈ X and t > 0.

(2) d(Jⁿf, C) → 0 as n → ∞. This implies the equality

for all x ∈ X.

(3) d(f, C) ≤ d(f, Jf)/(1 − α/m³) with f ∈ Ω, which implies the inequality d(f, C) ≤ 1/(2m³ − 2α), and so

for all x ∈ X and t > 0. The rest of the proof is similar to the proof of Theorem 4.1.

Corollary 4.4. Let X be a real normed space, θ ≥ 0, and p a real number with p ∈ (0,1). Let f : X → Y be a mapping with f(0) = 0 and satisfying (4.23). Then, for all x ∈ X, the limit C(x) = lim _n→∞f(mⁿx)/m³ⁿ exists and C : X → Y is a unique cubic mapping such that

for all x ∈ X and t > 0.

Proof. The proof follows from Theorem 6.3 if we take

for all x, y ∈ X and t > 0. In fact, if we choose α = m^3p, then we get the desired result.

Remark 4.5. In Corollaries 4.2 and 4.4, if we assume that Φ_x,y(t) = t/(t + θ(∥x∥^p · ∥y∥^p)) or Φ_x,y(t) = t/(t + θ(∥x∥^p+q + ∥y∥^p+q + ∥x∥^p∥y∥^q)), then we get Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability, respectively. But, since we put y = 0 in this functional equation, the Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability corollaries will be obvious. Meanwhile, the JMRassias mixed product-sum stability when p + q = 3 is an open question.

Theorem 5.1. Let ζ : G² → [0, +∞) be a function such that

for all x, y ∈ G and let for each x ∈ G the limit exist. Suppose that f : G → X a mapping with f(0) = 0 and satisfying the following inequality: Then, the limit C(x)∶ = lim _n→∞m³ⁿQ(x/mⁿ) exists for all x ∈ G and defines a cubic mapping C : G → X such that Moreover, if then C is the unique cubic mapping satisfying (5.4).

Proof. Putting y = 0 in (5.3), we get

for all x ∈ G. Replacing x by x/mⁿ⁺¹ in (5.6), we obtain It follows from (5.1) and (5.7) that the sequence {m³ⁿf(x/mⁿ)} _n≥1 is a Cauchy sequence. Since X is complete, {m³ⁿf(x/mⁿ)} _n≥1 is convergent. Set C(x)∶ = lim _n→∞m³ⁿf(x/mⁿ).

Using induction, one can show that

for all n ∈ ℕ and all x ∈ G. By taking n to approach infinity in (5.8) and using (5.2), one obtains (5.4). By (5.1) and (5.3), we get for all x, y ∈ G. Therefore, the function C : G → X satisfies (1.9). To prove the uniqueness property of C, let L : G → X be another function satisfying (5.4). Then, for all x ∈ G. Therefore, C = L, and the proof is complete.

Corollary 5.2. Let ξ : [0, ∞)→[0, ∞) be a function satisfying

Let κ > 0, and let f : G → X be a mapping with f(0) = 0 and satisfying the following inequality: for all x, y ∈ G. Then there exists a unique cubic mapping C : G → X such that

Proof. Defining ζ : G² → [0, ∞) by ζ(x, y): = κ(ξ(|x|) + ξ(|y|)), we have

for all x, y ∈ G. The last equality comes from the fact that |m|³ξ(1/|m|) < 1. On the other hand, for all x ∈ G, exists. Also, Applying Theorem 5.1, we get the desired result.

Theorem 5.3. Let ζ : G³ → [0, +∞) be a function such that

for all x, y ∈ G, and let for each x ∈ G the limit exist. Suppose that f : G → X a mapping with f(0) = 0 and satisfying (5.3). Then, the limit C(x)∶ = lim _n→∞f(mⁿx)/m³ⁿ exists for all x ∈ G and defines a cubic mapping C : G → X such that Moreover, if then C is the unique cubic mapping satisfying (5.19).

Proof. Putting y = 0 in (5.3), we get

for all x ∈ G. Replacing x by mⁿx in (5.21), we obtain It follows from (5.17) and (5.22) that the sequence {f(mⁿx)/m³ⁿ} _n≥1 is convergent. Set C(x): = lim _n→∞f(mⁿx)/m³ⁿ. On the other hand, it follows from (5.22) that for all x ∈ G and all nonnegative integers p, q with q > p ≥ 0. Letting p = 0, passing the limit q → ∞ in the last inequality, and using (5.18), we obtain (5.19). The rest of the proof is similar to the proof of Theorem 5.1.

Theorem 6.1. Let ζ : X² → [0, ∞) be a function such that there exists an L < 1 with

for all x, y ∈ X. Let f : X → Y be a mapping with f(0) = 0 and satisfying the following inequality: for all x, y ∈ X. Then, there is a unique cubic mapping C : X → Y such that for all x ∈ X.

Proof. Putting y = 0 in (6.2) and replacing x by x/m, we have

for all x ∈ X. Consider the set and the generalized metric d in S defined by where inf  ∅ = +∞. It is easy to show that (S, d) is complete (see [26], Lemma 2.1).

Now, we consider a linear mapping J : S → S such that

for all x ∈ X. Let g, h ∈ S be such that d(g, h) = ϵ. Then, for all x ∈ X, and so for all x ∈ X. Thus, d(g, h) = ϵ implies that d(Jg, Jh) ≤ Lϵ. This means that for all g, h ∈ S. It follows from (6.4) that By Theorem 2.8, there exists a mapping C : X → Y satisfying the following.

(1) C is a fixed point of J, that is,

for all x ∈ X. The mapping C is a unique fixed point of J in the set This implies that C is a unique mapping satisfying (6.12) such that there exists μ ∈ (0, ∞) satisfying for all x ∈ X.

(2) d(Jⁿf, C) → 0 as n → ∞. This implies the equality

for all x ∈ X.

(3) d(f, C) ≤ d(f, Jf)/(1 − L) with f ∈ Ω, which implies the inequality

This implies that inequality (6.3) holds.

By (6.1) and (6.2), we obtain

for all x, y ∈ X and n ∈ ℕ. So, for all x, y ∈ X. Thus, the mapping C : X → Y is cubic, as desired.

Corollary 6.2. Let θ ≥ 0, and let r be a real number with 0 < r < 1. Let f : X → Y be a mapping with f(0) = 0 and satisfying inequality

for all x, y ∈ X. Then, the limit C(x) = lim _n→∞m³ⁿf(x/mⁿ) exists for all x ∈ X and C : X → Y is a unique cubic mapping such that for all x ∈ X.

Proof. The proof follows from Theorem 6.1 by taking

for all x, y ∈ X. In fact, if we choose L = |2m³|^1−r, then we get the desired result.

Theorem 6.3. Let ζ : X² → [0, ∞) be a function such that there exists an L < 1 with

for all x, y ∈ X. Let f : X → Y be a mapping with f(0) = 0 and satisfying the inequality (6.2). Then, there is a unique cubic mapping C : X → Y such that

Proof. By (5.21), we know that

for all x ∈ X.

Let (S, d) be the generalized metric space defined in the proof of Theorem 6.1. Now, we consider a linear mapping J : S → S such that

for all x ∈ X. Let g, h ∈ S be such that d(g, h) = ϵ. Then, ∥g(x) − h(x)∥≤ϵζ(x, 0) for all x ∈ X, and so for all x ∈ X. Thus, d(g, h) = ϵ implies that d(Jg, Jh) ≤ Lϵ. This means that for all g, h ∈ S. It follows from (6.24) that By Theorem 2.8, there exists a mapping C : X → Y satisfying the following.

(1) C is a fixed point of J, that is,

for all x ∈ X. The mapping C is a unique fixed point of J in the set This implies that C is a unique mapping satisfying (6.29) such that there exists μ ∈ (0, ∞) satisfying for all x ∈ X.

(2) d(Jⁿf, C) → 0 as n → ∞. This implies the equality

for all x ∈ X.

(3) d(f, C) ≤ d(f, Jf)/(1 − L) with f ∈ Ω, which implies the inequality

This implies that inequality (6.23) holds.

The rest of the proof is similar to the proof of Theorem 6.1.

Corollary 6.4. Let θ ≥ 0, and let r be a real number with r > 1. Let f : X → Y be a mapping with f(0) = 0 and satisfying (6.19). Then, the limit C(x) = lim _n→∞(f(mⁿx)/m³ⁿ) exists for all x ∈ X and C : X → Y is a unique cubic mapping such that

for all x ∈ X.

Proof. The proof follows from Theorem 6.3 by taking

for all x, y ∈ X. In fact, if we choose L = |2m³|^r−1, then we get the desired result.

Remark 6.5. In Corollaries 6.2 and 6.4, if we assume that ζ(x, y) = θ(∥x∥^r · ∥y∥^r) or ζ(x, y) = θ(∥x∥^r+s + ∥y∥^r+s + ∥x∥^r∥y∥^s), then we get Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability, respectively. But, since we put y = 0 in this functional equation, the Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability corollaries will be obvious. Meanwhile, the JMRassias mixed product-sum stability when r + s = 3 is an open question.