On the Generalized Ulam-Hyers-Rassias Stability of Quadratic Mappings in Modular Spaces without Δ 2-Conditions

The question of stability for a generic functional equation was originated in 1940 by Ulam [1]. Concerning a group homomorphism, Ulam posted the question asking how likely to an automorphism a function should behave in order to guarantee the existence of an automorphism near such functions. Under the setting of Banach spaces, Hyers [2] was the first to give an affirmative answer to Ulam’s question the following year. It was extended to the cases of additive mappings by Aoki [3] and linear mappings by Rassias [4], the latter of which has influenced many developments in the stability theory. This area is then referred to as the Ulam-HyersRassias stability or, briefly, the UHR stability. The UHR stability and its relaxations play an important role in the studies of functional equations. Many functional equations are known for their complication in finding solutions. Knowing that a particular equation is stable makes it easier to find the solution.We can obviously see that to find an approximate solution is far less difficult than finding the exact solution. The stability then says that we can actually restrict ourselves to a neighborhood of the approximate solution. In most cases, a functional equation is algebraic in nature whereas the stability is rather metrical. Hence, a normed linear space is a suitable choice to work with. However, several results in the literature have revealed that there are a great number of linear topological spaces whose appropriate topologies fail to be normable, especially in case of function spaces. Nakano [5] and Musielak and Orlicz [6] successfully considered replacing a norm with a so-called modular. A modular yields less properties than a norm does, but it makes more sense in many special situations. It is still rational to assume some additional properties like some relaxed continuities or some Δ 2 -related conditions on a modular. Thus, it is reasonable to extend the framework of stability of functional equations into a more general setting of modular spaces, as considered by Sadeghi [7] in case of Cauchy and Jensen functional equations. Note that the stability results in [7] are obtained only in the cases where the inducedmodulars are convex and lsc and satisfy a typical class ofΔ 2 -conditions. LetX and Y be two groups; a mapping f : X → Y is said to be quadratic if it satisfies the following functional equation:


Introduction
The question of stability for a generic functional equation was originated in 1940 by Ulam [1].Concerning a group homomorphism, Ulam posted the question asking how likely to an automorphism a function should behave in order to guarantee the existence of an automorphism near such functions.
Under the setting of Banach spaces, Hyers [2] was the first to give an affirmative answer to Ulam's question the following year.It was extended to the cases of additive mappings by Aoki [3] and linear mappings by Rassias [4], the latter of which has influenced many developments in the stability theory.This area is then referred to as the Ulam-Hyers-Rassias stability or, briefly, the UHR stability.
The UHR stability and its relaxations play an important role in the studies of functional equations.Many functional equations are known for their complication in finding solutions.Knowing that a particular equation is stable makes it easier to find the solution.We can obviously see that to find an approximate solution is far less difficult than finding the exact solution.The stability then says that we can actually restrict ourselves to a neighborhood of the approximate solution.
In most cases, a functional equation is algebraic in nature whereas the stability is rather metrical.Hence, a normed linear space is a suitable choice to work with.However, several results in the literature have revealed that there are a great number of linear topological spaces whose appropriate topologies fail to be normable, especially in case of function spaces.Nakano [5] and Musielak and Orlicz [6] successfully considered replacing a norm with a so-called modular.A modular yields less properties than a norm does, but it makes more sense in many special situations.It is still rational to assume some additional properties like some relaxed continuities or some Δ 2 -related conditions on a modular.Thus, it is reasonable to extend the framework of stability of functional equations into a more general setting of modular spaces, as considered by Sadeghi [7] in case of Cauchy and Jensen functional equations.Note that the stability results in [7] are obtained only in the cases where the induced modulars are convex and lsc and satisfy a typical class of Δ 2 -conditions.
Let  and  be two groups; a mapping  :  →  is said to be quadratic if it satisfies the following functional equation: The UHR and generalized UHR stabilities have always been questioned in various settings, but none takes place in modular spaces.Skof [8] has proved that quadratic mappings are generalized UHR stable provided that  and  are normed and Banach spaces, respectively.It was later realized [9] that even when  is an Abelian group, the same behavior is still guaranteed.

Journal of Function Spaces
In the present paper, we consider the case where  is a linear space and  is a -complete modular space, where the scalar fields are arbitrary.Our main results are obtained by using the fixed point method under the assumptions that the modular is lsc and convex but not necessarily satisfies any Δ 2conditions.

Preliminaries
In this section, we recollect some basic definitions and properties of a modular space.Conventionally, we write throughout the paper R, C, and N to denote, respectively, the set of all reals, complexes, and nonnegative integers.Definition 1.Let  be a vector space over a field K (R or C).A generalized functional  :  → [0, ∞] is called a modular if for arbitrary ,  ∈  (m1) () = 0 if and only if  = 0, (m2) () = () for every scalar  with || = 1, (m3) () ≤ () + () whenever  is a convex combination of  and .
The corresponding modular space, denoted by   , is then defined by Remark 2. Note that, for a fixed  ∈   , the valuation  ∈ K  → () is increasing.
Unlike a norm, a modular needs not be continuous or convex in general.However, it often occurs that some weaker forms of them are assumed.Remark 3. In case a modular  is convex, one has () ≤ ((1/)) for all  ∈   , provided that 0 <  ≤ 1. Definition 4. Let   be a modular space and let {  } be a sequence in   .Then, Another unnatural behavior one usually encounter is that the convergence of a sequence {  } to  does not imply that {  } converges to , where  is chosen from the corresponding scalar field.Thus, many mathematicians imposed some additional conditions for a modular to meet in order to make the multiples of {  } converge naturally.Such preferences are referred to mostly under the term related to the Δ 2 -conditions.
A modular  is said to satisfy the Δ 2 -condition if there exists  ≥ 2 such that (2) ≤ () for all  ∈   .Some authors varied the notion so that only  > 0 is required and called it the Δ 2 -type condition.In fact, one may see that these two notions coincide.There are still a number of equivalent notions related to the Δ 2 -conditions.
Remark 5. We have to be very careful about the convergence behaviors on multiples and sums of -convergent sequences.In general, we suppose that { 1  }, { 2  }, . . ., { 2  }, for some  ∈ N, are sequences in   in which they -converge to the points  1 ,  2 , . . .,  2 ∈   , respectively.Then, the averaged sequence In [10], Khamsi proved a series of fixed point theorems in modular spaces where the modulars do not satisfy Δ 2conditions.His results exploit one unifying hypothesis in which the boundedness of an orbit is assumed.Definition 6.Given a modular space   , a nonempty subset  ⊂   , and a mapping  :  → .The orbit of  around a point  ∈   is the set O () := {, ,  2 , . ..} . ( The quantity   () := sup{( − V) : , V ∈ O()} is then associated and is called the orbital diameter of  at .In particular, if   () < ∞, one says that  has a bounded orbit at .
Lemma 7 (see [10]).Let   be a modular space whose induced modular is lsc and let  ⊂   be a -complete subset.If  :  →  is a -contraction, that is, there is a constant  ∈ [0, 1) such that and  has a bounded orbit at a point  0 ∈   , then the sequence {   0 } is -convergent to a point  ∈ .

Generalized UHR Stability of Quadratic Mappings
This section is contributed to the stability behavior of quadratic mappings in modular spaces.Unlike in the original UHR stability where the likeliness of being a solution is guaranteed by the difference being bounded with a sufficiently small positive constant, we rather prefer using a weaker setting where the difference is being dominated by a particular real-valued function of class Φ , defined below.
Definition 8.For a constant  ≥ 0 and a linear space , one defines Φ , to be the collection of all nonnegative real-valued functions  defined on  with the following properties for all ,  ∈ : Theorem 9. Let  be linear space,   be a -complete modular space where  is lsc and convex, and  :  →   be a mapping Journal of Function Spaces 3 with (0) = 0. Suppose that, for each ,  ∈ , the following dominating condition holds: where  ∈ Φ , with  < 1/2.Then, there exists the quadratic mapping  :  →   such that for all  ∈ .Equivalently, the quadratic mapping is generalized UHR stable.
To prove this stability result, we will need the following lemma.
Lemma 10.Suppose that every assumption of Theorem 9 holds.Then, the following statements hold.(S2) It is also easy to verify that ρ satisfies the axioms (m1) and (m2) of a modular.We will next show that ρ is convex, and hence (m3) is satisfied.Let  > 0 be given.Then there exist  1 > 0 and  2 > 0 such that Consecutively, we have This concludes that ρ is a convex modular on M.
(S3) The fact that the corresponding modular space M ρ is the whole space M is trivial, so we only show that M ρ is ρ-complete.Let {  } be a ρ-Cauchy sequence in M ρ and let  > 0 be given.There exists a positive integer  0 ∈ N such that ρ(  −   ) <  for all ,  ≥  0 .By the definition, we may see that for all  ∈  and ,  ≥  0 .Thus, at each fixed  ∈ , the sequence {  ()} is a -Cauchy sequence.Since   is -complete, {  ()} is -convergent in   for each  ∈ .Hence, we can define a function  :  →   by for any  ∈ .Since  is lsc, it follows from (14) that provided that  ≥  0 .Thus, Since 2 ∈ , the above inequality yields Since  is convex, we obtain Inductively, we may deduce for all  ∈ N that Moreover, we may see that
Lemma 11.Suppose that every assumption of Theorem 9 holds and  is defined as in (21).Then,  has some fixed point.Proof.