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

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


Introduction
In 1940, Ulam 1 raised a question concerning the stability of homomorphisms as follow.Given a group G 1 , a metric group G 2 with the metric d •, • , and a positive number ε, does there exist a δ > 0 such that if a mapping f : G 1 → G 2 satisfies the inequality d f xy , f x f y < δ, 1.1 for all x, y ∈ G 1 then there exists a homomorphism F : for all x ∈ G 1 ?When this problem has a solution, we say that the homomorphisms from G 1 to G 2 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 G 1 and G 2 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.
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 f 1 , f 2 : R → R defined by f 1 x ax and f 2 x ax 2 , where a is a real constant, then f 1 satisfies 1.3 , and f 2 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 ax 2 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 Cȃdariu 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.

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 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.

2.3
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 .
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 8f 0 lim 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 12 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 d Jg, Jh ≤ Ld g, h , 2.14 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 2.20 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 a 1 , a 2 , b 1 , b 2 with a 1 , b 1 / 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.
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 J 0 g 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 any g, h ∈ S. That is, J is a strictly contractive self-mapping of S with the Lipschitz constant L. Also we see that

2.30
for all x ∈ V \ {0}, which implies that d f, Jf ≤ L/8 < ∞.Therefore, according to Theorem 2.1, the sequence {J n f} 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.

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.

Journal of Applied Mathematics
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 for all x ∈ V \ {0}.In particular, the mappings F i , i 1, 2, 3, are represented by 12 for all x, y ∈ V and i 1, 2, 3. Put for all x, y ∈ V and i 1, 2, 3, then ϕ 1 satisfies 2.5 , ϕ 2 satisfies 2.18 , and ϕ 3 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 F 1 : V → Y satisfying 3.6 , which is represented by 2.7 .Observe that, by 3.2 and 3.3 , 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 F 1 0 0, we get AF 1 x, y 0, 3.17 for all x, y ∈ V .According to Theorem 2.4, there exists a unique mapping F 2 : V → Y satisfying 3.7 , which is represented by 2.7 .By using the similar method to prove 3.9 , we can show that F 2 is represented by 3.10 .In particular, if φ 2 x, y is continuous, then ϕ 2 is continuous on V \ {0} 2 , and we can say that f 2 is an additive map by Theorem 2.4.On the other hand, according to Theorem 2.5, there exists a unique mapping F 3 : V → Y satisfying 3.8 which is represented by 2.25 .Observe that, by 3.2 and 3.5 , 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 F 3 0 0, we get AF 3 x, y 0, 3.21 for all x, y ∈ V .

Journal of Applied Mathematics
Next, by Theorem 2.4, there exists a unique mapping F 2 : V → Y satisfying 3.24 , which is represented by 2.7 .By using the similar method to prove 3.26 , we can show that F 2 is represented by 3.27 .In particular, φ 2 x, y is continuous, then ϕ 2 is continuous on V \ {0} 2 , and we can say that f 2 is a quadratic map by Theorem 2.4.On the other hand, according to Theorem 2.5, there exists a unique mapping F 3 : V → Y satisfying 3.25 which is represented by 2.25 .Observe that for all x, y ∈ V \ {0}.Taking the limit as n → ∞ in the above inequality and using F 3 0 0, we get QF 3 x, y 0, 3.39 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.for all x ∈ X \ {0}.Moreover if p < 0, then f is itself a quadratic-additive mapping.

Corollary 3 . 3 . 2 −
Let X be a normed space, and let Y be a Banach space.Suppose that the mapping f : X → Y satisfies the inequality Df x, y ≤ θ x p y p 3.40 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 f x − F x ≤ 2 p x p if 0 < p < 1 3.41 3.28for all x ∈ V .Moreover, if φ 2 x, y is continuous, then f 2 itself is a quadratic mapping.