# On the Generalized Ulam-Hyers-Rassias Stability of Quadratic Mappings in Modular Spaces withoutΔ

Authors: Kittipong Wongkum; Parin Chaipunya; Poom Kumam
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We approach the generalized Ulam-Hyers-Rassias (briefly, UHR) stability of quadratic functional equations via the extensive studies of fixed point theory. Our results are obtained in the framework of modular spaces whose modulars are lower semicontinuous (briefly, lsc) but do not satisfy any relatives ofΔ
2-conditions.


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## Body

## 1. 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, theUHR 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.LetX and Y be two groups; a mapping f
:
X
→
Y is said to bequadratic if it satisfies the following functional equation:
(1)
f
(
x
+
y
)
+
f
(
x
-
y
)
=
2
f
(
x
)
+
2
f
(
y
)
.
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 X and Y are normed and Banach spaces, respectively. It was later realized [9] that even when X is an Abelian group, the same behavior is still guaranteed.In the present paper, we consider the case whereX is a linear space and Y is a ρ-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
2-conditions.


## 2. Preliminaries

In this section, we recollect some basic definitions and properties of a modular space. Conventionally, we write throughout the paperR, C, and N to denote, respectively, the set of all reals, complexes, and nonnegative integers.Definition 1.
LetX be a vector space over a field K (R or C). A generalized functional ρ
:
X
→
[
0
,
∞
] is called amodular if for arbitrary x
,
y
∈
X
(
m
1
)
ρ
(
x
)
=
0 if and only if x
=
0,
(
m
2
)
ρ
(
α
x
)
=
ρ
(
x
) for every scalar α with α
=
1,
(
m
3
)
ρ
(
z
)
≤
ρ
(
x
)
+
ρ
(
y
) whenever z is a convex combination of x and y.The correspondingmodular space, denoted by X
ρ, is then defined by
(2)
X
ρ
:
=
{
x
∈
X
:
ρ
(
λ
x
)
⟶
0
as
λ
⟶
0
}
.Remark 2.
Note that, for a fixedx
∈
X
ρ, the valuation γ
∈
K
↦
ρ
(
γ
x
) 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 modularρ is convex, one has ρ
(
x
)
≤
δ
ρ
(
(
1
/
δ
)
x
) for all x
∈
X
ρ, provided that 0
<
δ
≤
1.Definition 4.
LetX
ρ be a modular space and let {
x
n
} be a sequence in X
ρ. Then,(i)
{
x
n
} is ρ
-convergent to a point x
∈
X
ρ and write x
n
→
ρ
x if ρ
(
x
n
-
x
)
→
0 as n
→
∞.
(ii)
{
x
n
} is called ρ
-Cauchy if for any ϵ
>
0 one has ρ
(
x
n
-
x
m
)
<
ϵ for sufficiently large m
,
n
∈
N.
(iii)
A subsetK
⊂
X
ρ is called ρ
-complete if any ρ-Cauchy sequence is ρ-convergent.Another unnatural behavior one usually encounter is that the convergence of a sequence{
x
n
} to x does not imply that {
c
x
n
} converges to c
x, where c 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 {
x
n
} converge naturally. Such preferences are referred to mostly under the term related to the Δ
2-conditions.A modularρ is said to satisfy the Δ
2-condition if there exists κ
≥
2 such that ρ
(
2
x
)
≤
κ
ρ
(
x
) for all x
∈
X
ρ. Some authors varied the notion so that only κ
>
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 ofρ-convergent sequences. In general, we suppose that {
x
n
1
}
,
{
x
n
2
}
,
…
,
{
x
n
2
k
}, for some k
∈
N, are sequences in X
ρ in which they ρ-converge to the points x
1
,
x
2
,
…
,
x
2
k
∈
X
ρ, respectively. Then, the averaged sequence {
(
1
/
2
k
)
∑
i
=
1
2
k
x
n
i
}
ρ-converges to (
1
/
2
k
)
∑
i
=
1
2
k
x
i.In [10], Khamsi proved a series of fixed point theorems in modular spaces where the modulars do not satisfy Δ
2-conditions. His results exploit one unifying hypothesis in which the boundedness of an orbit is assumed.Definition 6.
Given a modular spaceX
ρ, a nonempty subset C
⊂
X
ρ, and a mapping T
:
C
→
C. Theorbit of T around a point x
∈
X
ρ is the set
(3)
O
(
x
)
:
=
{
x
,
T
x
,
T
2
x
,
…
}
.
The quantity δ
ρ
(
x
)
:
=
sup
⁡
{
ρ
(
u
-
v
)
:
u
,
v
∈
O
(
x
)
} is then associated and is called theorbital diameter of T at x. In particular, if δ
ρ
(
x
)
<
∞, one says that T has a bounded orbit at x.Lemma 7 (see [10]).
LetX
ρ be a modular space whose induced modular is lsc and let C
⊂
X
ρ be a ρ-complete subset. If T
:
C
→
C is a ρ-contraction, that is, there is a constant k
∈
[
0,1
) such that
(4)
ρ
(
T
x
-
T
y
)
≤
k
ρ
(
x
-
y
)
,
∀
x
,
y
∈
X
ρ
,
and T has a bounded orbit at a point x
0
∈
X
ρ, then the sequence {
T
n
x
0
} is ρ-convergent to a point w
∈
C.


## 3. 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 classΦ
V
,
L defined below.Definition 8.
For a constantL
≥
0 and a linear space V, one defines Φ
V
,
L to be the collection of all nonnegative real-valued functions ψ defined on V with the following properties for all x
,
y
∈
V:
(5)
lim
⁡
n
→
∞
ψ
(
2
n
x
,
2
n
y
)
4
n
=
0
,
ψ
(
2
x
,
2
x
)
≤
4
L
ψ
(
x
,
x
)
.Theorem 9.
LetV be linear space, X
ρ be a ρ-complete modular space where ρ is lsc and convex, and f
:
V
→
X
ρ be a mapping with f
(
0
)
=
0. Suppose that, for each x
,
y
∈
V, the following dominating condition holds:
(6)
ρ
(
4
f
(
x
+
y
)
+
4
f
(
x
-
y
)
-
8
f
(
x
)
-
8
f
(
y
)
)
≤
ϕ
(
x
,
y
)
,
where ϕ
∈
Φ
V
,
L with L
<
1
/
2. Then, there exists the quadratic mapping w
:
V
→
X
ρ such that
(7)
ρ
(
w
(
x
)
-
f
(
x
)
)
≤
1
16
(
1
-
L
)
ϕ
(
x
,
x
)
for all x
∈
V. 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 Theorem9 holds. Then, the following statements hold.(
S
1
)
The setM
:
=
{
g
:
V
→
X
ρ
:
g
(
0
)
=
0
} is a linear space.
(
S
2
)
A generalized functionρ
~ defined for each g
∈
M by
(8)
ρ
~
(
g
)
:
=
inf
⁡
{
c
>
0
:
ρ
(
g
(
x
)
)
≤
c
ϕ
(
x
,
x
)
,
∀
x
∈
V
}
is a convex modular onM.
(
S
3
)
The corresponding modular spaceM
ρ
~ is the whole space M and is ρ
~-complete.
(
S
4
)
ρ
~ is lscProof.
(
S
1
) is trivial.
(
S
2
) 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 ϵ
>
0 be given. Then there exist c
1
>
0 and c
2
>
0 such that
(9)
ρ
~
g
≤
c
1
≤
ρ
~
g
+
ϵ
,
ρ
~
(
h
)
≤
c
2
≤
ρ
~
(
h
)
+
ϵ
.
Consecutively, we have
(10)
ρ
g
x
≤
c
1
ϕ
x
,
x
,
ρ
(
h
(
x
)
)
≤
c
2
ϕ
(
x
,
x
)
.
Thus, if α
+
β
=
1 and α
,
β
≥
0, we get
(11)
ρ
α
g
x
+
β
h
x
≤
α
ρ
(
g
(
x
)
)
+
β
ρ
(
h
(
x
)
)
≤
(
α
c
1
+
β
c
2
)
ϕ
(
x
,
x
)
,
so that
(12)
ρ
~
(
α
g
+
β
h
)
≤
α
ρ
~
(
g
)
+
β
ρ
~
(
h
)
+
(
α
+
β
)
ϵ
.
Hence, we have
(13)
ρ
~
(
α
g
+
β
h
)
≤
α
ρ
~
(
g
)
+
β
ρ
~
(
h
)
.
This concludes that ρ
~ is a convex modular on M.
(
S
3
) 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 {
g
n
} be a ρ
~-Cauchy sequence in M
ρ
~ and let ϵ
>
0 be given. There exists a positive integer n
0
∈
N such that ρ
~
(
g
n
-
g
m
)
<
ϵ for all n
,
m
≥
n
0. By the definition, we may see that
(14)
ρ
(
g
n
(
x
)
-
g
m
(
x
)
)
≤
ϵ
ϕ
(
x
,
x
)
for all x
∈
V and n
,
m
≥
n
0. Thus, at each fixed x
∈
V, the sequence {
g
n
(
x
)
} is a ρ-Cauchy sequence. Since X
ρ is ρ-complete, {
g
n
(
x
)
} is ρ-convergent in X
ρ for each x
∈
V. Hence, we can define a function g
:
V
→
X
ρ by
(15)
g
(
x
)
=
lim
⁡
n
→
∞
⁡
g
n
(
x
)
for any x
∈
V. Since ρ is lsc, it follows from 14 that
(16)
ρ
(
g
n
(
x
)
-
g
(
x
)
)
≤
lim
⁡
inf
⁡
m
→
∞
⁡
ρ
(
g
n
(
x
)
-
g
m
(
x
)
)
≤
ϵ
ϕ
(
x
,
x
)
,
provided that n
≥
n
0. Thus, g
n
ρ
~-converges, so that M
ρ
~ is ρ
~-complete.
(
S
4
) Suppose that {
g
n
} is a sequence in M
ρ
~ which is ρ
~-convergent to an element g
∈
M
ρ
~. Let ϵ
>
0 be given. For each n
∈
N, let c
n be a constant such that
(17)
ρ
~
(
g
n
)
≤
c
n
≤
ρ
~
(
g
n
)
+
ϵ
.
Again, we have
(18)
ρ
(
g
n
(
x
)
)
≤
c
n
ϕ
(
x
,
x
)
,
∀
x
∈
V
.
Now, observe from the lower semicontinuity of ρ that
(19)
ρ
g
x
≤
lim
⁡
inf
⁡
n
→
∞
⁡
ρ
(
g
n
(
x
)
)
≤
lim
⁡
inf
⁡
n
→
∞
⁡
c
n
ϕ
(
x
,
x
)
≤
lim
⁡
inf
⁡
n
→
∞
⁡
ρ
~
(
g
n
)
+
ϵ
ϕ
(
x
,
x
)
.
Thus, we have
(20)
ρ
~
(
g
)
≤
lim
⁡
inf
⁡
n
→
∞
⁡
ρ
~
(
g
n
)
+
ϵ
.
Since ϵ
>
0 is arbitrary, we can finally conclude that ρ
~ is lscNext, we show that a self-mappingT
:
M
ρ
~
→
M
ρ
~ defined by
(21)
T
g
(
x
)
:
=
1
4
g
(
2
x
)
,
∀
g
∈
M
ρ
~
,
∀
x
∈
V
.
has some fixed point accordingly to Lemma 7.Lemma 11.
Suppose that every assumption of Theorem9 holds and T is defined as in 21. Then, T has some fixed point.Proof.
We first show thatT is a ρ
~-contraction. Let x
∈
V, g
,
h
∈
M
ρ
~, and c
∈
[
0
,
∞
] be an arbitrary constant with ρ
~
(
g
-
h
)
≤
c. Observe that we have
(22)
ρ
(
g
(
x
)
-
h
(
x
)
)
≤
c
ϕ
(
x
,
x
)
,
so that
(23)
ρ
g
(
2
x
)
4
-
h
(
2
x
)
4
≤
1
4
ρ
(
g
(
2
x
)
-
h
(
2
x
)
)
≤
1
4
c
ϕ
(
2
x
,
2
x
)
≤
L
c
ϕ
(
x
,
x
)
.
Therefore, we have ρ
~
(
T
g
-
T
h
)
≤
L
ρ
~
(
g
-
h
). Since g
,
h
∈
M
ρ
~ are arbitrary, T is a ρ
~-contraction.
Next, we show thatT has a bounded orbit at f, where f is taken from the assumption. Let x
∈
V be arbitrary and set y
=
x in 6; we get
(24)
ρ
(
4
f
(
2
x
)
-
16
f
(
x
)
)
≤
ϕ
(
x
,
x
)
,
∀
x
∈
V
.
Since 2
x
∈
V, the above inequality yields
(25)
ρ
(
4
f
(
4
x
)
-
16
f
(
2
x
)
)
≤
ϕ
(
2
x
,
2
x
)
,
∀
x
∈
V
.
Since ρ is convex, we obtain
(26)
ρ
f
2
2
x
4
-
4
f
x
≤
1
2
ρ
2
f
(
2
2
x
)
4
-
8
f
(
x
)
≤
1
2
1
2
ρ
f
2
2
x
-
4
f
2
x
m
m
m
m
+
1
2
ρ
(
4
f
(
2
x
)
-
16
f
(
x
)
)
≤
1
4
2
ϕ
(
2
x
,
2
x
)
+
1
4
ϕ
(
x
,
x
)
,
∀
x
∈
V
.
Inductively, we may deduce for all n
∈
N that
(27)
ρ
f
(
2
n
x
)
4
n
-
1
-
4
f
(
x
)
≤
∑
i
=
1
n
1
4
i
ϕ
(
2
i
-
1
x
,
2
i
-
1
x
)
,
∀
x
∈
V
.
Moreover, we may see that
(28)
ρ
f
(
2
n
x
)
4
n
-
1
-
4
f
(
x
)
≤
∑
i
=
1
n
1
4
i
ϕ
(
2
i
-
1
x
,
2
i
-
1
x
)
≤
1
4
ϕ
(
x
,
x
)
∑
i
=
1
n
L
i
-
1
≤
1
4
ϕ
(
x
,
x
)
∑
i
=
1
∞
L
i
-
1
=
1
4
(
1
-
L
)
ϕ
(
x
,
x
)
,
∀
x
∈
V
.
Now, for each m
,
n
∈
N, we have
(29)
ρ
f
(
2
n
x
)
4
n
-
f
(
2
m
x
)
4
m
≤
1
2
ρ
2
f
2
n
x
4
n
-
2
f
x
+
1
2
ρ
2
f
(
x
)
-
2
f
(
2
m
x
)
4
m
≤
1
4
ρ
2
f
2
n
x
4
n
-
1
-
4
f
x
+
1
4
ρ
4
f
(
x
)
-
f
(
2
m
x
)
4
m
-
1
≤
1
8
(
1
-
L
)
ϕ
(
x
,
x
)
,
∀
x
∈
V
.
By the definition of ρ
~, we conclude that
(30)
ρ
~
(
T
n
f
-
T
m
f
)
≤
1
8
(
1
-
L
)
<
∞
,
which implies the boundedness of an orbit of T at f. According to Lemma 7, the sequence {
T
n
f
}
ρ
~-converges to some element, say w
∈
M
ρ
~.
Now, by theρ
~-contractivity of T, one has
(31)
ρ
~
(
T
w
-
T
n
+
1
f
)
≤
L
ρ
~
(
w
-
T
n
f
)
.
Passing n towards ∞ and applying the lower semicontinuity of ρ
~, we obtain that
(32)
ρ
~
T
w
-
w
≤
lim
⁡
inf
⁡
n
→
∞
⁡
ρ
~
(
T
w
-
T
n
+
1
f
)
≤
lim
⁡
inf
⁡
n
→
∞
⁡
L
ρ
~
(
w
-
T
n
f
)
=
0
.
Therefore, w is a fixed point of T.Now, with the two lemmas above, we can finally give a simple proof to our main stability result, namely, Theorem9.Proof of Theorem9.
Since2
n
x and 2
n
y are in V provided that x
,
y
∈
V, we deduce from 6 that
(33)
ρ
4
f
(
2
n
(
x
+
y
)
)
+
4
f
(
2
n
(
x
-
y
)
)
m
m
-
8
f
(
2
n
(
x
)
)
-
8
f
(
2
n
(
y
)
)
≤
ϕ
(
2
n
x
,
2
n
y
)
,
∀
x
,
y
∈
V
.
Furthermore, we have
(34)
ρ
f
(
2
n
(
x
+
y
)
)
4
n
+
f
(
2
n
(
x
-
y
)
)
4
n
m
m
c
-
2
f
(
2
n
(
x
)
)
4
n
-
2
f
(
2
n
(
y
)
)
4
n
≤
ρ
4
f
2
n
x
+
y
4
n
+
4
f
2
n
x
-
y
4
n
m
m
m
m
c
-
8
f
(
2
n
(
x
)
)
4
n
-
8
f
(
2
n
(
y
)
)
4
n
≤
1
4
n
ρ
4
f
2
n
x
+
y
+
4
f
2
n
x
-
y
m
m
m
m
m
C
-
8
f
(
2
n
(
x
)
)
-
8
f
(
2
n
(
y
)
)
≤
ϕ
(
2
n
x
,
2
n
y
)
4
n
,
∀
x
,
y
∈
V
.
As from Lemma 11 and Remark 5, letting n
→
∞ and applying the lower semicontinuity of ρ, we deduce that
(35)
w
(
x
+
y
)
+
w
(
x
-
y
)
=
2
w
(
x
)
+
2
w
(
y
)
,
∀
x
,
y
∈
V
.
That is, w is a quadratic mapping, since every quadratic map is a fixed point of T. On the other hand, it follows from inequality 28 that
(36)
ρ
~
(
w
-
f
)
≤
1
16
(
1
-
L
)
.If domination6 is put to slightly stronger condition of a boundedness, we conclude the following classical UHR stability statement in modular spaces.Corollary 12.
LetV be a linear space, X
ρ be a ρ-complete modular space where ρ is lsc and convex, and f
:
V
→
X
ρ be a mapping with f
(
0
)
=
0. If there exist a constant δ
>
0 such that
(37)
ρ
(
4
f
(
x
+
y
)
+
4
f
(
x
-
y
)
-
8
f
(
x
)
-
8
f
(
y
)
)
≤
δ
,
∀
x
,
y
∈
V
,
then there exists the quadratic mapping w
:
V
→
X
ρ such that
(38)
ρ
(
w
(
x
)
-
f
(
x
)
)
≤
δ
12
,
∀
x
∈
V
.For the next two corollaries, we will consider the case whereρ is actually a norm. Before stating the results, we have to note that if ψ
∈
Φ
V
,
L, then we also have l
ψ
∈
Φ
V
,
L for all constant l
≥
0. Thus, there is no loss of generality asserting the upcoming corollaries in the following form(s).Corollary 13.
LetV be a linear space, (
X
,
·
) be a Banach space, and f
:
V
→
X be a mapping with f
(
0
)
=
0. Suppose that, for each x
,
y
∈
V, there holds the inequality
(39)
f
(
x
+
y
)
+
f
(
x
-
y
)
-
2
f
(
x
)
-
2
f
(
y
)
≤
ϕ
(
x
,
y
)
,
where ϕ
∈
Φ
V
,
L with L
<
1
/
2. Then, there exists a unique quadratic mapping w
:
V
→
X such that
(40)
w
(
x
)
-
f
(
x
)
≤
1
4
(
1
-
L
)
ϕ
(
x
,
x
)
,
∀
x
∈
V
.Corollary 14.
LetV be a linear space, (
X
,
·
) be a Banach space, and f
:
V
→
X be a mapping with f
(
0
)
=
0. It there exists a constant δ
>
0 such that
(41)
f
(
x
+
y
)
+
f
(
x
-
y
)
-
2
f
(
x
)
-
2
f
(
y
)
≤
δ
,
∀
x
,
y
∈
V
,
then there exists a unique quadratic mapping w
:
V
→
X such that
(42)
w
(
x
)
-
f
(
x
)
≤
δ
4
,
∀
x
∈
V
.


## 4. Concluding Remarks

Our results guarantee the stability of quadratic mappings, whose codomain is equipped with a convex and lsc modular, in both generalized and original senses. In contrast to the existing study of Sadeghi [7], our proofs contain different techniques to avoid the usage of Δ
2-conditions.Technically, comparing the results in modular and normed spaces, we may see that the coefficient in the case of modular is significantly smaller (1
/
16 to 1
/
4). However, the δ appeared in modular space cases are apparently larger.We are also curious whether the multiple of4 on the left of the inequality 6 can be dropped. This leaves a considerable interesting question for future research.



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*Source: 461719-2015-05-18.xml*