Random Stability of Quadratic Functional Equations

In this paper, we investigate the generalized Hyers-Ulam stability on random p-normed spaces associated with the following generalized quadratic functional equation f(kx + y) + f(kx − y) = 2f(x +y) + 2f(x − y) + 2(k − 2)f(x) − 2f(y) ,where k is a fixed positive integer via two methods


Introduction
The theor y of random normed spaces is important as a generalization of the deterministic results of normed spaces and also in the study of random operator equations. The notion of a random normed space corresponds to the situations when we do not know exactly the norm of the point and we know only probabilities of p ossible values of this norm. The random normed spaces may provide us the appropriate tools to study the geometr y of nuclear physics and have useful applications in quantum particle physics.
In 1940, Ulam [1] suggested the stability problem of functional equations concerning the stability of group homomorphisms as follows: When is it true that a function that approximately satisfies a functional equation must be close to an exact solution of the equation? If the problem accepts a solution, we say the equation is stable. The famous Ulam stability problem was partially solved by Hyers [2] for the linear functional equation of Banach spaces. Hyers' s theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias approach. The paper of Rassias has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. Cădariu and Radu [6] applied the fixed-point method to investigation of the Jensen functional equation. They could present a short and simple proof (different from the direct method initiated by Hyers in 1941) for the generalized Hyers-Ulam stability of Jensen functiona l equation and for quadratic functional equation.
The quadratic function f ( x ) = c 2 satisfies the functional equation and therefore, the functional equation (1.1) is called the quadratic functional equation. The generalized Hyers-Ulam-Rassias stability theorem for the quadratic functional equation (1.1) was proved by [7,8,9] and the references therein. The stability problems of several functional equations have been extensively investigated by a number of authorss, and there are many interesting results concerning this paper [10,11,12,13]. Before presenting our results, we introduce some basic facts concerning of random normed spaces as in [14,15]. (RN3) + ( + ) ≥ ( ( ) , ( ) ) for all , ∈ , and all , ≥ 0.
3. The random -normed space ( , , ) is said to be complete if ever y Cauchy sequence in is convergent to a point in . Throughout this paper, let X be a real linear space, (X, ′ , ) be a random p-normed space and (Y, μ, ) be a complete random p-normed space. For any mapping f ∶ X → Y, we define for all , ∈ and any fixed positive integer ∈ + .

Directed Method.
In this subsection, we prove the generalized Hyers-Ulam stability of the equation ∆ f ( x, y ) = 0 in a random p-normed space for 0 < p ≤ 1, by the standard Hyers' direct method.
and lim →∞ ′ ( , ) ( 2 ) = 1 for all , ∈ and all > 0. If ∶ → is a mapping with for all , ∈ and all > 0, then there exists a unique quadratic mapping ∶ → such that for all ∈ and all > 0.
for all ∈ and all > 0. Replacing x by in (2.4), we have for all ∈ and > 0. Thus, the condition (2.3) holds for all ∈ and all > 0.
Since for all ∈ and all > 0, then there exists a unique quadratic mapping ∶ → such that for all ∈ and all > 0. for all , ∈ and all > 0, then there exists a unique quadratic mapping ∶ → such that

Proof. Let
(2) the sequence { } is convergent to a fixed point * of ; (3) * is the unique fixed point of in the set = { ∈ Ω: ( 0 , ) < ∞ } ; (4) ( , * ) ≤ for all ∈ and all > 0. Let Ω be a set of all mappings from X into Y and introduce a generalized metric on Ω as follows: where, as usual, inf ∅ = +∞. It is easy to show that ( Ω, ) is a complete metric space ( [12]). Now, let us consider the mapping ∶ Ω → Ω defined by for all , ℎ ∈ Ω. Then, J is a strictly contractive self-mapping on Ω with the Lipschitz constant = 2 < 1.
To prove the uniqueness, let us assume that there exists a quadratic mapping Q ′ : → which satisfies (2.21). Then Q ′ is a fixed point of J in 1 . However, it follows from Theorem 2.5 that J has only one fixed point in for all ∈ and all > 0.

Proof.
Let Ω and d be as in the proof of Theorem 2.2. Then ( , ) becomes a complete metric space and the mapping ∶ Ω → Ω defined by