Stability of an n-Dimensional Mixed-Type Additive and Quadratic Functional Equation in Random Normed Spaces

In 1940, Ulam 1 gave a wide-ranging talk before a mathematical colloquium at the University of Wisconsin, in which he discussed a number of important unsolved problems. Among those was the following question concerning the stability of homomorphisms. Let G1 be a group, and let G2 be a metric group with a 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 is a homomorphism H : G1 → G2 with d h x ,H x < ε for all x ∈ G1? If the answer is affirmative, we say that the functional equation for homomorphisms is stable. Hyers 2 was the first mathematician to present the result concerning the stability of functional equations. He answered the question of Ulam for the case where G1 and G2 are assumed to be Banach spaces. This result of Hyers is stated as follows. Let f : E1 → E2 be a function between Banach spaces such that


Introduction
In 1940, Ulam 1 gave a wide-ranging talk before a mathematical colloquium at the University of Wisconsin, in which he discussed a number of important unsolved problems.Among those was the following question concerning the stability of homomorphisms.
Let G 1 be a group, and let G 2 be a metric group with a metric d •, • .Given ε > 0, does there exist a δ > 0 such that if a function h : G 1 → G 2 satisfies the inequality d h xy , h x h y < δ for all x, y ∈ G 1 , then there is a homomorphism H : If the answer is affirmative, we say that the functional equation for homomorphisms is stable.Hyers 2 was the first mathematician to present the result concerning the stability of functional equations.He answered the question of Ulam for the case where G 1 and G 2 are assumed to be Banach spaces.This result of Hyers is stated as follows.
Let f : E 1 → E 2 be a function between Banach spaces such that for some δ > 0 and for all x, y ∈ E 1 .Then the limit A x lim n → ∞ 2 −n f 2 n x exists for each x ∈ E 1 , and A : E 1 → E 2 is the unique additive function such that f x − A x ≤ δ for every x ∈ E 1 .Moreover, if f tx is continuous in t for each fixed x ∈ E 1 , then function A is linear.
We remark that the additive function A is directly constructed from the given function f, and this method is called the direct method.The direct method is a very powerful method for studying the stability problems of various functional equations.Taking this famous result into consideration, the additive Cauchy equation f x y f x f y is said to have the Hyers-Ulam stability on E 1 , E 2 if for every function f : E 1 → E 2 satisfying the inequality 1.1 for some δ ≥ 0 and for all x, y ∈ E 1 , there exists an additive function A : In 1950, Aoki 3 generalized the theorem of Hyers for additive functions, and in the following year, Bourgin 4 extended the theorem without proof.Unfortunately, it seems that their results failed to receive attention from mathematicians at that time.No one has made use of these results for a long time.
In 1978, Rassias 5 addressed the Hyers's stability theorem and attempted to weaken the condition for the bound of the norm of Cauchy difference and generalized the theorem of Hyers for linear functions.
Let f : E 1 → E 2 be a function between Banach spaces.If f satisfies the functional inequality for some θ ≥ 0, p with 0 ≤ p < 1 and for all x, y ∈ E 1 , then there exists a unique additive function If, in addition, f tx is continuous in t for each fixed x ∈ E 1 , then the function A is linear.This result of Rassias attracted a number of mathematicians who began to be stimulated to investigate the stability problems of functional equations.By regarding a large influence of Ulam, Hyers, and Rassias on the study of stability problems of functional equations, the stability phenomenon proved by Rassias is called the Hyers-Ulam-Rassias stability.For the last thirty years, many results concerning the Hyers-Ulam-Rassias stability of various functional equations have been obtained see 6-17 .In this paper, applying the fixed point method, we prove the Hyers-Ulam-Rassias stability of the n-dimensional mixed-type additive and quadratic functional equation in random normed spaces.Every solution of 1.3 is called a quadratic-additive function.
Throughout this paper, let n be an integer larger than 1.

Preliminaries
We introduce some terminologies, notations, and conventions usually used in the theory of random normed spaces see 18, 19 .The set of all probability distribution functions is denoted by Δ : F : 0, ∞ → 0, 1 | F is left-continuous and nondecreasing on 0, ∞ , F 0 0, and F ∞ 1}.

2.1
Let us define The set Δ is partially ordered by the usual pointwise ordering of functions, that is, F ≤ G if and only if F t ≤ G t for all t ≥ 0. The maximal element for Δ in this order is the distribution function ε 0 : 0, ∞ → 0, 1 given by Definition 2.2 See 19 .Let X be a vector space, τ a continuous t-norm, and let Λ : X → D be a function satisfying the following conditions: R 1 Λ x t ε 0 t for all t > 0 if and only if x 0; R 2 Λ αx t Λ x t/|α| for all x ∈ X, α / 0, and for all t ≥ 0; R 3 Λ x y t s ≥ τ Λ x t , Λ y s for all x, y ∈ X and all t, s ≥ 0.
A triple X, Λ, τ is called a random normed space briefly, RN-space .If X, • is a normed space, we can define a function Λ : X → D by for all x ∈ X and t > 0. Then X, Λ, τ M is a random normed space, which is called the induced random normed space.
i A sequence {x n } in X is said to be convergent to a point x ∈ X if, for every t > 0 and ε > 0, there exists a positive integer N such that ii A sequence {x n } in X is called a Cauchy sequence if, for every t > 0 and ε > 0, there exists a positive integer N such that iii An RN-space X, Λ, τ is called complete if and only if every Cauchy sequence in X converges to a point in X.
Definition 2.4.Let X be a nonempty set.A function d : We now introduce one of the fundamental results of the fixed point theory.For the proof, we refer to 20 or 21 .
Theorem 2.5 See 20, 21 .Let X, d be a complete generalized metric space.Assume that Λ : X → X is a strict contraction with the Lipschitz constant L < 1.If there exists a nonnegative integer n 0 such that d Λ n 0 1 x, Λ n 0 x < ∞ for some x ∈ X, then the following statements are true: In 2003, Radu 22 noticed that many theorems concerning the Hyers-Ulam stability of various functional equations follow from the fixed point alternative Theorem 2.5 .Indeed, he applied the fixed point method to prove the existence of a solution of the inequality 1.1 and investigated the Hyers-Ulam stability of the additive Cauchy equation see also [23][24][25][26] .Furthermore, Mihet ¸and Radu 27 applied the fixed point method to prove the stability theorems of the additive Cauchy equation in random normed spaces.
In 2009, Towanlong and Nakmahachalasint 28 established the general solution and the stability of the n-dimensional mixed-type additive and quadratic functional equation 1.3 by using the direct method.According to 28 , a function f : E 1 → E 2 is a quadraticadditive function, where E 1 and E 2 are vector spaces, if and only if there exist an additive function a : E 1 → E 2 and a quadratic function q : E 1 → E 2 such that f x a x q x for all x ∈ E 1 .

Hyers-Ulam-Rassias Stability
Throughout this paper, let X be a real vector space and let Y, Λ, τ M be a complete RN-space.For a given function f : X → Y , we use the following abbreviation: We will now prove the stability of the functional equation 1.3 in random normed spaces by using fixed point method.
Proof.We will first treat the case where ϕ satisfies the condition i .Let S be the set of all functions g : X → Y with g 0 0, and let us define a generalized metric on S by It is not difficult to show that S, d is a complete generalized metric space see 29 or 30, 31 .
Consider the operator J : S → S defined by Jf x : Then we can apply induction on m to prove for all x ∈ X and m ∈ N.
Let f, g ∈ S and let u ∈ 0, ∞ be an arbitrary constant with d g, f ≤ u.For some 0 < α < n satisfying the condition i , it follows from the definition of d, R 2 , R 3 , and i that for all x ∈ X \ {0} and t > 0, which implies that That is, J is a strict contraction with the Lipschitz constant 0 < α/n < 1.Moreover, by R 2 , R 3 , and 3.2 , we see that for all x ∈ X \ {0} and t > 0. Hence, it follows from the definition of d that Now, in view of Theorem 2.5, the sequence {J m f} converges to the unique "fixed point" 3.12 for all x ∈ X.
By Theorem 2.5, 3.11 , and the definition of d, we have that is, the first inequality in 3.4 holds true.
We will now show that F is a quadratic-additive function.It follows from R 3 and the definition of τ M that 3.14 for all x 1 , x 2 , . . ., x n ∈ X \ {0}, t > 0, and m ∈ N. Due to the definition of F, the first four terms on the right-hand side of the above inequality tend to 1 as m → ∞.By a somewhat tedious manipulation, we have

3.15
Hence, it follows from R 2 , R 3 , definition of τ M , 3.2 , and i that which tends to 1 as m → ∞ for all x 1 , x 2 , . . ., x n ∈ X \ {0} and t > 0. Therefore, 3.14 implies that Λ DF x 1 ,x 2 ,...,x n t 1 3.17 for any x 1 , . . ., x n ∈ X \ {0} and t > 0. By R 1 , this implies that DF x 1 , . . ., x n 0 for all x 1 , . . ., x n ∈ X \ {0}, which ends the proof of the first part.Now, assume that ϕ satisfies the condition ii .Let S, d be the same as given in the first part.We now consider the operator J : S → S defined by for all g ∈ S and x ∈ X.Notice that for all x ∈ X and m ∈ N. Let f, g ∈ S and let u ∈ 0, ∞ be an arbitrary constant with d g, f ≤ u.From R 2 , R 3 , the definition of d, and ii , we have for all x ∈ X \ {0}, t > 0, and for some α > n 2 satisfying ii , which implies that That is, J is a strict contraction with the Lipschitz constant 0 < n 2 /α < 1.Moreover, by R 2 , 3.2 , and ii , we see that for all x ∈ X \ {0} and t > 0. This implies that d f, Jf ≤ 1/ 2α < ∞ by the definition of d.Therefore, according to Theorem 2.5, the sequence {J m f} converges to the unique "fixed point" F : X → Y of J in the set T {g ∈ S | d f, g < ∞} and F is represented by 3.24 the second inequality in 3.4 holds true.
Next, we will show that F is a quadratic-additive function.As we did in the first part, we obtain the inequality 3.14 .In view of the definition of F, the first four terms on the right-hand side of the inequality 3.14 tend to 1 as m → ∞.Furthermore, a long manipulation yields By a similar way presented in the proof of Theorem 3.1, we can also prove the preceding theorem if the domains of relevant functions include 0. Theorem 3.2.Let X be a real vector space, Z, Λ , τ M an RN-space, Y, Λ, τ M a complete RNspace, and let ϕ : X n → Z be a function.Assume that ϕ satisfies one of the conditions i and ii in Theorem 3.1 for all x 1 , x 2 , . . ., x n ∈ X and t > 0. If a function f : X → Y satisfies f 0 0 and 3.2 for all x 1 , x 2 , . . ., x n ∈ X and t > 0, then there exists a unique quadratic-additive function F : X → Y satisfying 3.4 for all x ∈ X and t > 0. Now, we obtain general Hyers-Ulam stability results of 1.3 in normed spaces.If X is a normed space, then X, Λ, τ M is an induced random normed space.We get the following result.
Corollary 3.3.Let X be a real vector space, Y a complete normed space, and let ϕ : X \ {0} n → 0, ∞ be a function.Assume that ϕ satisfies one of the following conditions: iii ϕ nx

2 . 2 Definition 2 . 1
See 18  .A function τ : 0, 1 × 0, 1 → 0, 1 is called a continuous triangular norm briefly, continuous t-norm if τ satisfies the following conditions: a τ is commutative and associative; b τ is continuous; c τ a, 1 a for all a ∈ 0, 1 ; d τ a, b ≤ τ c, d for all a, b, c, d ∈ 0, 1 with a ≤ c and b ≤ d.Typical examples of continuous t-norms are τ P a, b ab, τ M a, b min{a, b}, and τ L a, b max{a b − 1, 0}.