We investigate the stability of the functional equation 2f(x + y) + f(x − y) + f(y − x) − 3f(x) − f(−x) − 3f(y) − f(−y) = 0 by using the fixed point theory in the sense of Cădariu and Radu.

1. Introduction

In 1940, Ulam [1] raised a question concerning the stability of homomorphisms as follow. Given a group G₁, a metric group G₂ with the metric d(·, ·), and a positive number ε, does there exist a δ > 0 such that if a mapping f : G₁ → G₂ satisfies the inequality

()

for all x, y ∈ G₁ then there exists a homomorphism F : G₁ → G₂ with

()

for all x ∈ G₁? When this problem has a solution, we say that the homomorphisms from G₁ to G₂ are stable. In the next year, Hyers [2] gave a partial solution of Ulam′s problem for the case of approximate additive mappings under the assumption that G₁ and G₂ are Banach spaces. Hyers′ result was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering the stability problem with unbounded Cauchy’s differences. The paper of Rassias had much influence in the development of stability problems. The terminology Hyers-Ulam-Rassias stability originated from this historical background. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see [5–12].

Almost all subsequent proofs, in this very active area, have used Hyers′ method of [2]. Namely, the mapping F, which is the solution of a functional equation, is explicitly constructed, starting from the given mapping f, by the formulae F(x) = lim _n→∞ (1/2ⁿ)f(2ⁿx) or F(x) = lim _n→∞ 2ⁿf(x/2ⁿ). We call it a direct method. In 2003, Cădariu and Radu [13] observed that the existence of the solution F for a functional equation and the estimation of the difference with the given mapping f can be obtained from the fixed point theory alternative. This method is called a fixed point method. In 2004, they applied this method [14] to prove stability theorems of the Cauchy functional equation:

()

In 2003, they [15] obtained the stability of the quadratic functional equation:

()

by using the fixed point method. Notice that if we consider the functions f₁, f₂ : ℝ → ℝ defined by f₁(x) = ax and f₂(x) = ax², where a is a real constant, then f₁ satisfies (1.3), and f₂ holds (1.4), respectively. We call a solution of (1.3) an additive map, and a mapping satisfying (1.4) is called a quadratic map. Now we consider the functional equation:

()

which is called the Cauchy additive and quadratic-type functional equation. The function f : ℝ → ℝ defined by f(x) = ax² + bx satisfies this functional equation, where a, b are real constants. We call a solution of (1.5) a quadratic-additive mapping.

In this paper, we will prove the stability of the functional equation (1.5) by using the fixed point theory. Precisely, we introduce a strictly contractive mapping with the Lipschitz constant 0 < L < 1. Using the fixed point theory in the sense of Cădariu and Radu, together with suitable conditions, we can show that the contractive mapping has the fixed point. Actually the fixed point F becomes the precise solution of (1.5). In Section 2, we prove several stability results of the functional equation (1.5) using the fixed point theory, see Theorems 2.3, 2.4, and 2.5. In Section 3, we use the results in the previous sections to get a stability of the Cauchy functional equation (1.3) and that of the quadratic functional equation (1.4), respectively.

2. Main Results

We recall the following result of the fixed point theory by Margolis and Diaz.

Theorem 2.1 (see [16] or [17].)Suppose that a complete generalized metric space (X, d), which means that the metric d may assume infinite values, and a strictly contractive mapping J : X → X with the Lipschitz constant 0 < L < 1 are given. Then, for each given element x ∈ X, either

()

or there exists a nonnegative integer k such that

(1)
d(Jⁿx, Jⁿ⁺¹x)<+∞ for all n ≥ k,
(2)
the sequence {Jⁿx} is convergent to a fixed point y^* of J,
(3)
y^* is the unique fixed point of J in Y∶ = {y ∈ X, d(J^kx, y)<+∞},
(4)
d(y, y^*)≤(1/(1 − L))d(y, Jy) for all y ∈ Y.

Throughout this paper, let V be a (real or complex) linear space, and let Y be a Banach space. For a given mapping f : V → Y, we use the following abbreviation:

()

for all x, y ∈ V. If f is a solution of the functional equation Df ≡ 0, see (1.5), we call it a quadratic-additive mapping. We first prove the following lemma.

Lemma 2.2. If f : V → Y is a mapping such that Df(x, y) = 0 for all x, y ∈ V∖{0}, then f is a quadratic-additive mapping.

Proof. By choosing x ∈ V∖{0}, we get

()

Since f(0) = 0, we easily obtain Df(x, y) = 0 for all x, y ∈ V.

Now we can prove some stability results of the functional equation (1.5).

Theorem 2.3. Let φ : (V∖{0}) ² → [0, ∞) be a given function. Suppose that the mapping f : V → Y satisfies

()

for all x, y ∈ V∖{0}. If there exist constants 0 < L, L^′ < 1 such that φ has the property

()

for all x, y ∈ V∖{0}, then there exists a unique quadratic-additive mapping F : V → Y such that

()

for all x ∈ V∖{0}. In particular, F is represented by

()

for all x ∈ V.

Proof. It follows from (2.5) that

()

for all x, y ∈ V∖{0}, and

()

for all x ∈ V∖{0}. From this, we know that f(0) = 0. Let S be the set of all mappings g : V → Y with g(0) = 0. We introduce a generalized metric on S by

()

It is easy to show that (S, d) is a generalized complete metric space. Now we consider the mapping J : S → S, which is defined by

()

for all x ∈ V. Notice

()

for all n ∈ ℕ and x ∈ V. Let g, h ∈ S, and let K ∈ [0, ∞] be an arbitrary constant with d(g, h) ≤ K. From the definition of d, we have

()

for all x ∈ V∖{0}, which implies that

()

for any g, h ∈ S. That is, J is a strictly contractive self-mapping of S with the Lipschitz constant L. Moreover, by (2.4), we see that

()

for all x ∈ V∖{0}. It means that d(f, Jf) ≤ 3/16 < ∞ by the definition of d. Therefore, according to Theorem 2.1, the sequence {Jⁿf} converges to the unique fixed point F : V → Y of J in the set T = {g ∈ S∣d(f, g) < ∞}, which is represented by (2.7) for all x ∈ V. By the definition of F, together with (2.4) and (2.7), it follows that

()

for all x, y ∈ V∖{0}. By Lemma 2.2, we have proved that

()

for all x, y ∈ V.

Theorem 2.4. Let φ : (V∖{0}) ² → [0, ∞) be a given function. Suppose that the mapping f : V → Y satisfies (2.5) for all x, y ∈ V∖{0}. If there exists a constant 0 < L < 1/2 such that φ has the property

()

for all x, y ∈ V∖{0}, then there exists a unique quadratic-additive mapping F : V → Y satisfying (2.6) for all x ∈ V∖{0}. Moreover, if φ(x, y) is continuous, then f itself is a quadratic-additive mapping.

Proof. It follows from (2.18) that

()

for all x, y ∈ V∖{0}, and

()

for all x ∈ V∖{0}. By the same method used in Theorem 2.3, we know that there exists a unique quadratic-additive mapping F : V → Y satisfying (2.6) for all x ∈ V∖{0}. Since φ is continuous, we get

()

for all x, y ∈ V∖{0} and for any fixed integers a₁, a₂, b₁, b₂ with a₁, b₁ ≠ 0. Therefore, we obtain

()

for all x ∈ V∖{0}, where ψ(x) is defined by ψ(x) = φ(x, x) + φ(−x, −x). Since f(0) = 0 = F(0), we have shown that f ≡ F. This completes the proof of this theorem.

We continue our investigation with the next result.

Theorem 2.5. Let φ : (V∖{0}) ² → [0, ∞). Suppose that f : V → Y satisfies the inequality ∥Df(x, y)∥ ≤ φ(x, y) for all x, y ∈ V∖{0}. If there exists 0 < L < 1 such that the mapping φ has the property

()

for all x, y ∈ V∖{0}, then there exists a unique quadratic-additive mapping F : V → Y such that

()

for all x ∈ V∖{0}. In particular, F is represented by

()

for all x ∈ V.

Proof. By the similar method used to prove f(0) = 0 in the proof of Theorem 2.3, we can easily show that f(0) = 0. Let the set (S, d) be as in the proof of Theorem 2.3. Now we consider the mapping J : S → S defined by

()

for all g ∈ S and x ∈ V. Notice that

()

and J⁰g(x) = g(x), for all x ∈ V. Let g, h ∈ S, and let K ∈ [0, ∞] be an arbitrary constant with d(g, h) ≤ K. From the definition of d, we have

()

for all x ∈ V∖{0}. So

()

for any g, h ∈ S. That is, J is a strictly contractive self-mapping of S with the Lipschitz constant L. Also we see that

()

for all x ∈ V∖{0}, which implies that d(f, Jf) ≤ L/8 < ∞. Therefore, according to Theorem 2.1, the sequence {Jⁿf} converges to the unique fixed point F of J in the set T∶ = {g ∈ S∣d(f, g) < ∞}, which is represented by (2.25). Since

()

the inequality (2.24) holds. From the definition of F(x), (2.4), and (2.23), we have

()

for all x, y ∈ V∖{0}. By Lemma 2.2, F is quadratic additive.

Remark 2.6. If φ satisfies the equality φ(x, y) = φ(−x, −y) for all x, y ∈ V∖{0} in Theorems 2.3, 2.4, and 2.5, then the inequalities (2.6) and (2.24) can be replaced by

()

for all x ∈ V∖{0}, respectively.

3. Applications

For a given mapping f : V → Y, we use the following abbreviations:

()

for all x, y ∈ V. Using Theorems 2.3, 2.4, and 2.5 we will show the stability results of the additive functional equation Af ≡ 0 and the quadratic functional equation Qf ≡ 0 in the following corollaries.

Corollary 3.1. Let f_i : V → Y, i = 1,2, 3, be mappings for which there exist functions ϕ_i : (V∖{0}) ² → [0, ∞), i = 1,2, 3, such that

()

for all x, y ∈ V∖{0}. If there exists 0 < L < 1 such that

()

for all x, y ∈ V∖{0}, then there exist unique additive mappings F_i : V → Y, i = 1,2, 3, such that

()

for all x ∈ V∖{0}. In particular, the mappings F_i, i = 1,2, 3, are represented by

()

for all x ∈ V. Moreover, if ϕ₂(x, y) is continuous, then f₂ is itself an additive mapping.

Proof. Notice that

()

for all x, y ∈ V and i = 1,2, 3. Put

()

for all x, y ∈ V and i = 1,2, 3, then φ₁ satisfies (2.5), φ₂ satisfies (2.18), and φ₃ satisfies (2.23). Therefore, ∥Df_i(x, y)∥ ≤ φ_i(x, y), for all x, y ∈ V∖{0} and i = 1,2, 3. According to Theorem 2.3, there exists a unique mapping F₁ : V → Y satisfying (3.6), which is represented by (2.7). Observe that, by (3.2) and (3.3),

()

as well as

()

for all x ∈ V∖{0}. From this and (2.7), we get (3.9). Moreover, we have

()

for all x, y ∈ V∖{0}. Taking the limit as n → ∞ in the above inequality and using F₁(0) = 0, we get

()

for all x, y ∈ V. According to Theorem 2.4, there exists a unique mapping F₂ : V → Y satisfying (3.7), which is represented by (2.7). By using the similar method to prove (3.9), we can show that F₂ is represented by (3.10). In particular, if ϕ₂(x, y) is continuous, then φ₂ is continuous on (V∖{0})², and we can say that f₂ is an additive map by Theorem 2.4. On the other hand, according to Theorem 2.5, there exists a unique mapping F₃ : V → Y satisfying (3.8) which is represented by (2.25). Observe that, by (3.2) and (3.5),

()

as well as

()

for all x ∈ V∖{0}. From these and (2.25), we get (3.11). Moreover, we have

()

for all x, y ∈ V∖{0}. Taking the limit as n → ∞ in the above inequality and using F₃(0) = 0, we get

()

for all x, y ∈ V.

Corollary 3.2. Let f_i : V → Y, i = 1,2, 3, be mappings for which there exist functions ϕ_i : (V∖{0}) ² → [0, ∞), i = 1,2, 3, such that

()

for all x, y ∈ V∖{0}. If there exists 0 < L < 1 such that the mapping ϕ₁ satisfies (3.3), ϕ₂ satisfies (3.4), and ϕ₃ satisfies (3.5) for all x, y ∈ V∖{0}, then there exist unique quadratic mappings F_i : V → Y, i = 1,2, 3, such that

()

for all x ∈ V∖{0}. In particular, the mappings F_i, i = 1,2, 3, are represented by

()

for all x ∈ V. Moreover, if ϕ₂(x, y) is continuous, then f₂ itself is a quadratic mapping.

Proof. Notice that

()

for all x, y ∈ V and i = 1,2, 3. Put φ_i(x, y)∶ = ϕ_i(x, y) + (1/2)(ϕ_i(x, −y) + ϕ_i(y, −x)), for all x, y ∈ V and i = 1,2, 3, then φ₁ satisfies (2.5), φ₂ satisfies (2.18), and φ₃ satisfies (2.23). Moreover,

()

for all x, y ∈ V∖{0} and i = 1,2, 3. According to Theorem 2.3, there exists a unique mapping F₁ : V → Y satisfying (3.23) which is represented by (2.7). Observe that

()

as well as

()

for all x ∈ V∖{0}. From this and (2.7), we get (3.26) for all x ∈ V. Moreover, we have

()

for all x, y ∈ V∖{0}. Taking the limit as n → ∞ in the above inequality, we get

()

for all x, y ∈ V∖{0}. Using F₁(0) = 0, we have

()

for all x, y ∈ V∖{0}. Therefore, QF₁(x, y) = 0 for all x, y ∈ V.

Next, by Theorem 2.4, there exists a unique mapping F₂ : V → Y satisfying (3.24), which is represented by (2.7). By using the similar method to prove (3.26), we can show that F₂ is represented by (3.27). In particular, ϕ₂(x, y) is continuous, then φ₂ is continuous on (V∖{0}) ², and we can say that f₂ is a quadratic map by Theorem 2.4. On the other hand, according to Theorem 2.5, there exists a unique mapping F₃ : V → Y satisfying (3.25) which is represented by (2.25). Observe that

()

for all x ∈ V∖{0}. It leads us to get

()

for all x ∈ V∖{0}. From these and (2.25), we obtain (3.28). Moreover, we have

()

for all x, y ∈ V∖{0}. Taking the limit as n → ∞ in the above inequality and using F₃(0) = 0, we get

()

for all x, y ∈ V.

Now, we obtain Hyers-Ulam-Rassias stability results in the framework of normed spaces using Theorems 2.3 and 2.4.

Corollary 3.3. Let X be a normed space, and let Y be a Banach space. Suppose that the mapping f : X → Y satisfies the inequality

()

for all x, y ∈ X∖{0}, where θ ≥ 0 and p ∈ (−∞, 0)∪(0,1)∪(2, ∞). Then there exists a unique quadratic-additive mapping F : X → Y such that

()

for all x ∈ X∖{0}. Moreover if p < 0, then f is itself a quadratic-additive mapping.

Proof. This corollary follows from Theorems 2.3, 2.4, and 2.5, and Remark 2.6, by putting

()

for all x, y ∈ X∖{0} with L = 2^p−1 < 1 if p < 1, L = 2^2−p < 1 if p > 2, and L^′ = 2^−p < 1 if p > 0.

Corollary 3.4. Let X be a normal space let and Y be a Banach space. Suppose that the mapping f : X → Y satisfies the inequality

()

for all x, y ∈ X∖{0}, where θ ≥ 0 and p + q ∈ (−∞, 0)∪(0,1)∪(2, ∞). Then there exists a unique quadratic-additive mapping F : X → Y such that

()

for all x ∈ X∖{0}. Moreover, if p + q < 0, then f is itself a quadratic-additive mapping.

Proof. This corollary follows from Theorems 2.3, 2.4, 2.5, and Remark 2.6, by putting

()

for all x, y ∈ X∖{0} with L = 2^p+q−1 < 1 if p + q < 1, L = 2^2−p−q < 1 if p + q > 2, and L^′ = 2^−p−q < 1 if p + q > 0.

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A Fixed Point Approach to the Stability of the Cauchy Additive and Quadratic Type Functional Equation

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3. Applications

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A Fixed Point Approach to the Stability of the Cauchy Additive and Quadratic Type Functional Equation

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