Refined stability of additive and quadratic functional equations in modular spaces

The purpose of this paper is to obtain refined stability results and alternative stability results for additive and quadratic functional equations using direct method in modular spaces.


Introduction
The theory of modulars on linear spaces and the related theory of modular linear spaces have been established by Nakano in  []. Since then, these have been thoroughly developed by several mathematicians, for example, Amemiya  First of all, we introduce to adopt the usual terminologies, notations, definitions and properties of the theory of modular spaces.
A modular ρ defines the following vector space: and we say that X ρ is a modular space.
Definition  Let X ρ be a modular space and let {x n } be a sequence in X ρ . Then: It is said that the modular ρ has the Fatou property if and only if ρ(x) ≤ lim inf n→∞ ρ(x n ) whenever the sequence {x n } is ρ-convergent to x in modular space X ρ .

Proposition  In modular spaces,
It is noticed that the convergence of a sequence {x n } to x does not imply that {cx n } converges to cx if c is chosen from the corresponding scalar field with |c| >  in modular spaces. Thus, additional conditions on modular spaces were imposed by many mathematicians so that the multiples of convergent sequence {x n } in the modular spaces converge naturally. A modular ρ is said to satisfy the  -condition if there exists k >  such that ρ(x) ≤ kρ(x) for all x ∈ X ρ . Throughout this paper, we say that this constant k is a  -constant related to  -condition.

Remark  Suppose that ρ is convex and satisfies
, which implies ρ = . Therefore, we must have the  -constant k ≥  if ρ is convex modular. On the other hand, many authors have investigated the stability using fixed point theorem of quasicontraction mappings in modular spaces without  -condition, which has been introduced by Khamsi

Stability of additive functional equations in modular spaces
Throughout this paper, we assume that V is a linear space and X ρ is a ρ-complete convex modular space. We present a main theorem, which concerns Hyers-Ulam stability of an additive functional equation in modular spaces without using the Fatou property.
for all x, y ∈ V , then there exists a unique additive mapping A : for all x ∈ V .
Proof By letting x, y by x  in (), respectively, we get for all x ∈ V , and then it follows from the  -condition and the convexity of the modular ρ that for all x ∈ V . Since the right-hand side of the above inequality tends to zero as m goes to infinity, the sequence { n f ( x  n )} is a ρ-Cauchy sequence in X ρ and so the sequence is a ρ-convergent sequence on X ρ . Thus, we may define a mapping A : V → X ρ as According to the  -condition without using the Fatou property, we obtain the following inequality: for all x ∈ V . Taking n → ∞, we conclude that the estimation () of f by A holds for all x ∈ V . Now, we claim that the mapping A is additive. Setting (x, y) := ( -n x,  -n y) in () and using the  -condition, we see that for all x, y ∈ V . Thus, it follows from the  -condition and ρ(αx for all x, y ∈ V and all positive integers n. Taking the limit as n → ∞, one sees that A is additive.
To show the uniqueness of A, we assume that there exists an additive mapping A : V → X ρ which satisfies the inequality for all x ∈ V . Then, since A and A are additive mappings, we see from the equality for all x ∈ V and all positive integers n. Hence A is a unique additive mapping near f satisfying the approximation () in the modular space X ρ . This completes the proof.

Corollary  Suppose V is a normed space with norm · and X ρ satisfies  -condition. For given real numbers θ >  and p
for all x, y ∈ V , then there exists a unique additive mapping A : V → X ρ such that Next, we are going to prove an alternative stability theorem of additive functional equations in modular spaces without using the  -condition.
Theorem  Let X ρ satisfy the Fatou property. Suppose that a mapping f : V → X ρ satisfies for all x, y ∈ V . Then there exists a unique additive mapping A : V → X ρ such that for all x ∈ V .
Proof We let y = x in () and have so we observe without using the  -condition that for all x ∈ V and all positive integers n > . This yields for all x ∈ V and all n, m ∈ N with n > m. Thus, we see that the sequence for all x ∈ V . Then, it follows from the Fatou property that the inequality holds for all x ∈ V . Now, we claim that A satisfies the additive functional equation. Note that for all x, y ∈ V and all n ∈ N. Thus, we observe by convexity of ρ that holds for all x, y ∈ V , and then taking n → ∞, one obtains ρ(   (A(x + y) -A(x) -A(y))) = . This implies that A is additive.
To show the uniqueness of A, we assume that there exists another additive mapping A : V → X ρ near f satisfying the approximation (). Since A and A are additive mappings, we see from the equality A( n x) =  n A(x) and A ( n x) =  n A (x) that for all x ∈ V . Taking n → ∞, we find that A = A . Hence A is a unique additive mapping near f satisfying the approximation ().
Remark  In particular, if X ρ is a Banach space with norm ρ, then ρ(x) = ρ(x), k = , and so Theorem  is equivalent to the result of Gǎvruta [] in this case.
The following corollary, which does not use  -condition of ρ, is a refined version of Sadeghi's stability result (Theorem . in []) in modular space X ρ .
Corollary  Let X ρ satisfy the Fatou property. Suppose that a mapping f : V → X ρ satisfies for all x, y ∈ V . Then there exists a unique additive mapping A : V → X ρ such that for all x ∈ V .
Corollary  Let V be a normed space with norm · and X ρ satisfy the Fatou property. For given real numbers θ , ε >  and p ∈ (-∞, ), if f : V → X ρ is a mapping such that for all x, y ∈ V , then there exists a unique additive mapping A : V → X ρ such that

Stability of quadratic functional equations in modular spaces
In this section, we investigate refined stability results of the original quadratic functional equation in modular space X ρ . We present the Hyers-Ulam stability of a quadratic functional equation in modular spaces without using the Fatou property.
Theorem  Suppose X ρ satisfies the  -condition. If there exists a function ϕ : for all x, y ∈ V , then there exists a unique quadratic mapping B : V → X ρ , defined as B(x) = lim n→∞  n f ( x  n ) and for all x ∈ V .
Proof First, we observe that f () =  because of φ(, ) =  by the convergence of for all x ∈ V . By the  -condition of ρ and n i=   i ≤ , one can prove the following functional inequality: for all x ∈ V . Now, replacing x by  -m x in (), we obtain To show the uniqueness of B, we assume that there exists a quadratic mapping B : V → X ρ satisfying the approximation Then we see from the equality B( -n x) =  -n B(x) and B ( -n x) =  -n B (x) that for all x ∈ V and all sufficiently large positive integers n. Taking n → ∞, we arrive at the uniqueness of B. This completes the proof.
Corollary  Suppose V is a normed space with norm · and X ρ satisfies  -condition. For given real numbers θ >  and p > log  k   , if f : V → X ρ is a mapping such that ρ f (x + y) + f (xy) -f (x) -f (y) ≤ θ x p + y p for all x, y ∈ V , then there exists a unique quadratic mapping B : V → X ρ such that Next, we provide an alternative stability theorem of Theorem  without using both the  -condition and the Fatou property in modular spaces.