Normed Spaces
1M. Jeyaraman, 2A.N. Mangayarkkarasi, 3V. Jeyanthi and 4 R. Pandiselvi
1PG and Research Department of Mathematics,
Raja Doraisingam Govt. Arts College, Sivagangai,
Affiliated to Alagappa University, Karaikudi, Tamil Nadu, India.
ORCID: orcid.org/0000-0002-0364-1845
2 Department of Mathematics, Nachiappa Swamigal Arts & Science College, Karaikudi. Affiliated to Alagappa University, Karaikudi, Tamilnadu, India.
3Government Arts College for Women, Sivagangai. Affiliated to Alagappa University, Karaikudi, Tamilnadu, India.
4 PG and Research Department of Mathematics, The Madura College, Madurai 625011, Tamilnadu, India. ORCID: orcid.org/0000-0002-8236-964X.
Email1 jeya.math@gmail.com, Email2 murugappan.mangai@gmail.com, Email3 jeykaliappa@gmail.com, Email4rpselvi@gmail.com .
In Neutrosophic Normed spaces, we investigate a unique quadratic function and a unique additive quadratic function of the Hyers-Ulam-Rassias stability for the functional equation 
which is said to be a functional equation associated with inner products
space.
Keywords: Hyers-Ulam-Rassias stability, Functional equation, Neutrosophic, Normed Space. Mathematical Subject Classification: 47H10, 39B72, 39A30.
The aim of this article is to prove an Neutrosophic version of the Hyers-Ulam-Rassias stability for the functional equation:
!
(A)
which is said to be a functional equation associated with inner product spaces. It was shown by Rassias [1] that the norm defined over a real vector space X is induced by an inner product if and only if for a fixed integer n ≥ 2 it follows
(B)
for all xi,...,xn ∈ X. Interesting new results concerning functional equations associated with inner product spaces have recently been obtained by Park et al. [2, 3] and Najati and Rassias [4] as well as for the fuzzy stability of a functional equation associated with inner product spaces [5].
Stability problem of a functional equation was first posed by Ulam [6] which was answered by Hyers [7] on approximately additive mappings and then generalized by Aoki [8] and Rassias [9] for additive mappings and linear mappings, respectively. Later there have been proved several new results on stability of various classes of functional equations in the Hyers-Ulam sense (cf. the following books and papers [10-18] and the references cited therein), as well as various fuzzy stability results concerning Cauchy, Jensen, quadratic and cubic functional equations. Furthermore some stability results concerning Jensen, cubic, mixed-type additive and cubic functional equations were investigated in the spirit of intuitionistic fuzzy normed spaces, while the idea of intuitionistic fuzzy normed space was introduced and further studied.
In future studies on this subject, it is also possible to work with the idea of “Probabilistic metric space” using neutrosophic probability. In 1940, Ulam raised the following question. Under what conditions does there exists an additive mapping near an approximately addition mapping? The case of approximately additive functions was solved by Hyers under certain assumption. In 1978, a generalized version of the theorem of Hyers for approximately linear mapping was given by Rassias. The stability concept that was introduced and investigated by Rassias is called the Hyers-Ulam-Rassias stability. During the last decades, the stability problems of several functional equations have been extensively investigated by a number of authors. Neutrosophic set(NS) is a new version of the idea of the classical set which is defined by Smarandache [26]. The first world publication related to the concept of neutrosophy was published in 1998 and included in the literature [24].
Definition 2.1. [23] A binary operation ∗ : [0,1] × [0,1] → [0,1] is a continuous t-norm [CTN] if it satisfies the following conditions :
1. ∗ is commutative and associative,
2. ∗ is continuous,
3. α ∗ 1 = α for all α ∈ [0,1],
4. α ∗ β ≤ γ ∗ δ whenever α ≤ γ and β ≤ δ, for each α,β,γ,δ ∈ [0,1].
Definition 2.2. [23] A binary operation 3 : [0,1]×[0,1] → [0,1] is a continuous t-norm [CTCN] if it satisfies the following conditions :
1. 3 is commutative and associative,
2. 3 is continuous,
3. α 3 0 = α for all α ∈ [0,1],
4. α 3 β ≤ γ 3 δ whenever α ≤ γ and β ≤ δ, for each α,β,γ,δ ∈ [0,1].
Definition 2.3. The six-tuple (X,µ,ν,ω,∗,3,⊗) is said to be an Neutrosophic Normed Spaces(NNS) if X is a vector space, Let ∗ and 3,⊗ be the CTN and CTCN, respectively. µ,ν,ω are Normed spaces on X ×(0,∞) fulfilling the conditions below: For each x,β ∈ X and for each s,t > 0, Φ ̸= 0,
1. 0 ≤ µ(x,t) ≤ 1,0 ≤ ν(x,t) ≤ 1,0 ≤ ω(x,t) ≤ 1, for all t ∈ (0,∞);
2. µ(x,t) + ν(x,t) + ω(x,t) ≤ 3;
3. µ(x,t) > 0;
4. µ(x,t) = 1 iff x = 0;
5. 
for each Φ ̸= 0;
6. µ(x,t) ∗ µ(β,s) ≤ µ(x + β,t + s);
7. µ(x,.) : (0,∞) → [0,1] is continuous increasing function;
8. 
lim µ(x,t) = 1 and; t→∞
9. ν(x,t) < 1;
10. ν(x,t) = 0 iff x = 0;
for each Φ ̸= 0;
12. ν(x,t)3 ν(β,s) ≥ ν(x + β,t + s);
13. ν(x,.) : (0,∞) → [0,1] is continuous increasing function;
14. 
lim ν(x,t) = 0 and.; t→∞
15. ω(x,t) < 1;
16. ω(x,t) = 0 iff x = 0;
for each Φ ̸= 0;
18. ω(x,t) ⊗ ω(β,s) ≥ ω(x + β,t + s);
19. ω(x,.) : (0,∞) → [0,1] is continuous increasing function;
20. 
lim ω(x,t) = 0 and.; t→∞
Then (µ,ν,ω) is called Neutrosophic Norm(NN).
Example 2.4. Let (X,∥·∥) be a NS. Define CTN and CTCN as follows x ∗β = xβ and x 3 β = x +β−xβ. For t > ∥x∥,
,
for all x,β ∈ X and t > 0. If t ≤ ∥x∥, then µ(x,t) = 0,ν(x,t) = 1,ω(x,t) = 1. Then (X,µ,ν,ω,∗,3,⊗) is NNS
Definition 2.5. Let (X,µ,ν,ω,∗,3,⊗) be a NNS. 1. A sequence (xn) in X is Neutrosophic convergent to x ∈ X if lim µ(xn − x,t) = 1, lim ν(xn − n→∞ n→∞
x,t) = 0 and lim ω(xn − x,t) = 0 as t > 0.
n→∞
2. A sequence (xn) is said to be Neutrosophic Cauchy sequence if lim µ(xn+p−xn,t) = 1, lim ν(xn+p− n→∞ n→∞
xn,t) = 0 and lim ω(xn+p − xn,t) = 0 to each t > 0 and p = 1,2,.... n→∞
3. A (X,µ,ν,ω,∗,3,⊗) is said to be Complete if every Neutrosophic Cauchy sequence in (X,µ,ν,ω,∗,3,⊗) is Neutrosophic convergent in (X,µ,ν,ω,∗,3,⊗).
Throughout this section, assume that X,(Z,µ′,ν′,ω′), and (Y,µ,ν,ω) are linear space, NNS, and Neutrosophic Banach space, respectively. For convenience, we use the following abbreviation for a given function f : X → Y :
. (C)
We begin with the Hyers-Ulam-Rassias type theorem in NNS for the functional (A) which is said to be a functional equation associated with inner product spaces.
Theorem 3.1. Let φ : X → Z be a function such that φ(2x) = αφ(x) for some real number α with 0 < |α| < 4. Suppose that an even function f : X → Y with f(0) = 0 satisfies the inequality
µ(∆f(x1,...,xn),t1 + ··· + tn) ≥ µ′(φ(x1),t1) ∗ ··· ∗ µ′(φ(xn),tn),
ν(∆f(x1,...,xn),t1 + ··· + tn) ≤ ν′(φ(x1),t1) ⋄ ··· ⋄ ν′(φ(xn),tn) and
ω(∆f(x1,...,xn),t1 + ··· + tn) ≤ ω′(φ(x1),t1) ⊗ ··· ⊗ ω′(φ(xn),tn) (3.1.1)
for all x1,...,xn ∈ X and all t1,...,tn > 0. Then there exists a unique quadratic function Q : X → Y such
that
,
and
(3.1.2)
for all x ∈ X and t > 0, where
and
(3.1.3)
Proof. Put x1 = nx1,xi = nx2(i = 2,...,n),ti = t(i = 1,...,n) in (3.1.1), and, using the evenness of f, we obtain
,
and
for all x1,x2 ∈ X and t > 0. Interchanging x1 with x2 in (3.1.4) and using the evenness of f, we get
,
and
for all x1,x2 ∈ X and t > 0. It follows from (3.1.4) and (3.1.5) that
|
nf((n − 1)x1 + x2) + nf(x1 + (n − 1)x2) µ +2f((n − 1)(x1 − x2)) + 2(n − 1)f(x1 − x2) ≥ µ′(φ(nx1),t) ∗ µ′(φ(nx2),t), −nf(nx1) − nf(nx2),2nt nf((n − 1)x1 + x2) + nf(x1 + (n − 1)x2) ν +2f((n − 1)(x1 − x2)) + 2(n − 1)f(x1 − x2) ≤ ν′(φ(nx1),t) ⋄ ν′(φ(nx2),t) and −nf(nx1) − nf(nx2),2nt |
|
|
|
(3.1.6) |
−nf(nx1) − nf(nx2),2nt
for all x1,x2 ∈ X and t > 0. Putting x1 = nx2,x2 = −nx2, xi = 0 (i = 3,...,n), ti = t (i = 1,...,n) in (3.1.1) and using the evenness of f, we obtain
!
µ nf((n − 1)x1 + x2) + f(x1 + (n − 1)x2) ≥ µ′(φ(nx1),t) ∗ µ′(φ(nx2),t) ∗ µ′(φ(0),t),
+2(n − 1)f(x1 − x2) − f(nx1) − f(nx2),nt
!
ν nf((n − 1)x1 + x2) + f(x1 + (n − 1)x2) ≤ ν′(φ(nx1),t) ⋄ ν′(φ(nx2),t) ⋄ ν′(φ(0),t) +2(n − 1)f(x1 − x2) − f(nx1) − f(nx2),nt
(3.1.7)
for all x1,x2 ∈ X and t¿0. Hence, we obtain from (3.1.6) and (3.1.7) that
and
(3.1.8)
for all x1,x2 ∈ X and t > 0. So
and
(3.1.9)
for all x ∈ X and t > 0. Putting x1 = nx,xi = 0(i = 2,...,n),ti = t(i = 1,...,n) in (3.1.1), we obtain
µ(f(nx) − f((n − 1)x) − (2n − 1)f(x),nt) ≥ µ′(φ(nx),t) ∗ µ′(φ(0),t),
ν (f(nx) − f((n − 1)x) − (2n − 1)f(x),nt) ≤ ν′(φ(nx),t) ⋄ ν′(φ(0),t) and
ω (f(nx) − f((n − 1)x) − (2n − 1)f(x),nt) ≤ ω′(φ(nx),t) ⊗ ω′(φ(0),t) (3.1.10)
for all x ∈ X and t > 0. It follows from (3.1.9) and (3.1.10) that
,
and
(3.1.11)
for all x ∈ X and t > 0. Letting x2 = −(n−1)x1 in (3.1.7) and replacing
in the obtained inequality, we get
!
f((n − 1)x) − f((n − 2)x) ′ ∗ µ′(φ((n − 1)x),t) ∗ µ′(φ(0),t),
µ ≥ µ (φ(nx),t)
−(2n − 3)f(x),nt
!
f((n − 1)x) − f((n − 2)x) ′ ⋄ ν′(φ((n − 1)x),t) ⋄ ν′(φ(0),t) and
ν ≤ ν (φ(nx),t)
−(2n − 3)f(x),nt
(3.1.12)
for all x ∈ X and t > 0. It follows from (3.1.9), (3.1.10), (3.1.11) and (3.1.12) that
,
and
(3.1.13)
for all x ∈ X and t > 0. Applying (3.1.11) and (3.1.13), we obtain
and
(3.1.14)
for all x ∈ X and t > 0. Setting x1 = x2 = nx,xi = 0(i = 3,...,n),ti = t(i = 1,...,n) in (3.1.1), we
obtain
and
(3.1.15)
for all x ∈ X and t > 0. It follows from (3.1.14) and (3.1.15) that
,
and
It follows that
,
and
. (3.1.17)
Define
and
. (3.1.18)
Then, by our assumption,
and
(3.1.19)
Replacing x by 2nx in (3.1.17) and applying (3.1.19), we get
,
and
Thus for each n > m, we have
,
and
(3.1.21)
where
and
and δ > 0 be given. Since
and 
there exists some t0 > 0 such that
. Since
, there is some
n0 ∈ N such that
for each n > m ≥ n0. It follows that
and
(3.1.22)
for all t > t0. This shows that the sequence
is Cauchy in (Y,µ,ν,ω). Since (Y,µ,ν,ω) is Neutrosophic Banach space, 
converges to some point Q(x) ∈ Y . Thus, we can define a mapping
Q(x) : X → Y such that
. Moreover, if we put m = 0 in (3.1.21), we
get
,
, and
(3.1.23)
Thus,
,
and
(3.1.24)
Now, we will show that Q is quadratic. Setting xi = 2mxi(i = 1,...,n) and
in
(3.1.1), we obtain
and
(3.1.25)
for all x1,...,xn ∈ X and t > 0. Letting n → ∞ in (3.1.25), we obtain µ(∆Q(x1,...,xn),t) = 1,ν(∆Q(x1,...,xn),t) = 0 and ω(∆Q(x1,...,xn),t) = 0 (3.1.26)
for all x1,...,xn ∈ X and all t > 0. This means that Q satisfies the functional (A) and so it is quadratic (see Lemma 2.2 of [4]).
Next, we approximate the difference between f and Q in Neutrosophic sense. By (3.1.24), we have
and
(3.1.27)
for every x ∈ X,t > 0 and large enough n. To prove the uniqueness of Q, assume that Q′ is another quadratic mapping from X to Y , which satisfies the required inequality. Then, for each x ∈ X and t > 0,
,
and
Since Q and Q′ are quadratic, we have
,
and
(3.1.29)
for all x ∈ X,t > 0 and n ∈ N. Since 0 ≤ α < 4 and 
, we get
,
and
(3.1.30)
Therefore µ(Q(x) − Q′(x),t) = 1, ν(Q(x) − Q′(x),t) = 0 and ω(Q(x) − Q′(x),t) = 0 for all x ∈ X and t > 0. Hence Q(x) = Q′(x) for all x ∈ X. This completes the proof of the theorem. 
Theorem 3.2. Let φ : X → Z be a function such that φ(2x) = αφ(x) for some real number α with 0 < |α| < 2. Suppose that an odd function f : X → Y satisfies the inequality
µ(∆f(x1,...,xn),t1 + ··· + tn) ≥ µ′(φ(x1),t1) ∗ ··· ∗ µ′(φ(xn),tn),
ν(∆f(x1,...,xn),t1 + ··· + tn) ≤ ν′(φ(x1),t1) ⋄ ··· ⋄ ν′(φ(xn),tn) and
ω(∆f(x1,...,xn),t1 + ··· + tn) ≤ ω′(φ(x1),t1) ⊗ ··· ⊗ ω′(φ(xn),tn) (3.2.1)
for all x1,...,xn ∈ X and all t1,...,tn > 0. Then there exists a unique additive quadratic function A :
X → Y such that
and
(3.2.2)
for all x ∈ X and t > 0, where
and
(3.2.3)
Proof. Put
in (3.2.1) and using the oddness of f, we
obtain
,
and
for all
. Interchanging x1 with
in (3.2.4) and using the oddness of f, we get
,
and
for all
. It follows from (3.2.4) and (3.2.5) that
|
|
|
|
|
and |
|
|
(3.2.6) |
for all
. Setting
in
(3.2.1) and using the oddness of f, we get
,
and
for all
. It follows from (3.2.6) and (3.2.7) that
(3.2.8)
for all x ∈ X and t > 0. Putting
in (3.2.1), we obtain µ(f(n(x1 − x1′ )) − f((n − 1)(x1 − x1′ )) − f(x1 − x1′ ),nt) ≥ µ′(φ(n(x1 − x′1 )),t) ∗ µ′(φ(0),t), ν(f(n(x1 − x1′ )) − f((n − 1)(x1 − x1′ )) − f(x1 − x1′ ),nt) ≤ ν′(φ(n(x1 − x′1 )),t) ⋄ ν′(φ(0),t) and
for all. It follows from (3.2.8) and (3.2.9) that
,
and
for all x ∈ X and t > 0. Replacing x1 and
and 
in (3.2.10)respectively, we have
,
and
. (3.2.11)
It follows that
and
. (3.2.12)
Define
and
. (3.2.13)
Then, by the assumption,
and
(3.2.14) Replacing x by 2nx in (3.2.12) and applying (3.2.14), we get
,
and
Thus for each n > m, we have
,
and
(3.2.16)
where
and
and δ > 0 be given. Since
and 
there exists some t0 > 0 such that
. Since
, there is some
n0 ∈ N such that
for each n > m ≥ n0. It follows that
and
(3.2.17)
for all t > t0. This shows that the sequence
is Cauchy in (Y,µ,ν,ω). Since (Y,µ,ν,ω) is Neutrosophic Banach space, 
converges to some point A(x) ∈ Y . Thus, we can define a mapping
A(x) : X → Y such that
. Moreover, if we put m = 0 in (3.2.16), we
get
,
and
. (3.2.18)
Thus,
,
and
(3.2.19)
Next, we will show that A is Additive. Putting xi = 2mxi(i = 1,...,n) and
in
(3.2.1), we obtain
and
(3.2.20)
for all x1,...,xn ∈ X and t > 0. Letting n → ∞ in (3.2.20), we obtain µ(∆A(x1,...,xn),t) = 1,ν(∆A(x1,...,xn),t) = 0 and ω(∆A(x1,...,xn),t) = 0 (3.2.21) for all x1,...,xn ∈ X and all t > 0. This means that A satisfies the functional (A) and so it is additive (see Lemma 2.1 of [4]).
Next, we approximate the difference between f and A in Neutrosophic sense. For every x ∈ X,t > 0 and sufficiently large n, by (3.2.19), we have
and
(3.2.22)
To prove the uniqueness of A, assume that A′ is another additive mapping from X to Y , which satisfies the required inequality. Then, for each x ∈ X and t > 0,
,
and
(3.2.23)
Therefore, by the additivity of A and A′, we have
,
and
(3.2.24)
for all x ∈ X,t > 0 and n ∈ N. Since 0 ≤ α < 2 and 
, we get
,
(3.2.25)
Therefore µ(A(x) − A′(x),t) = 1, ν(A(x) − A′(x),t) = 0 and ω(A(x) − A′(x),t) = 0 for all x ∈ X and t > 0. Hence A(x) = A′(x) for all x ∈ X. This completes the proof of the theorem.
Theorem 3.3. Let φ : X → Z be a function such that φ(2x) = αφ(x) for some real number α with 0 < |α| < 2. Suppose that a function f : X → Y with f(0) = 0 satisfies the inequality
µ(∆f(x1,...,xn),t1 + ··· + tn) ≥ µ′(φ(x1),t1) ∗ ··· ∗ µ′(φ(xn),tn),
ν(∆f(x1,...,xn),t1 + ··· + tn) ≤ ν′(φ(x1),t1) ⋄ ··· ⋄ ν′(φ(xn),tn) and
ω(∆f(x1,...,xn),t1 + ··· + tn) ≤ ω′(φ(x1),t1) ⊗ ··· ⊗ ω′(φ(xn),tn) (3.3.1)
for all x1,...,xn ∈ X and all t1,...,tn > 0. Then there exists a unique quadratic function Q : X → Y and a unique additive function A : X → Y such that
,
and
(3.3.2)
for all x ∈ X and t > 0, where
,
and
,
(3.3.3)
Proof. Passing to the even part fe and odd part f0 of f, we deduce from (3.3.1) that
µ(∆fe(x1,...,xn),t1 + ··· + tn)
≥ µ′(φ(x1),t1) ∗ µ′(φ(−x1),t1) ∗ ··· ∗ µ′(φ(xn),tn) ∗ µ′(φ(−xn),tn), ν(∆fe(x1,...,xn),t1 + ··· + tn)
≤ ν′(φ(x1),t1) ⋄ ν′(φ(−x1),t1) ⋄ ··· ⋄ ν′(φ(xn),tn) ⋄ ν′(φ(−xn),tn) and
|
ω(∆fe(x1,...,xn),t1 + ··· + tn) |
|
|
≤ ω′(φ(x1),t1) ⊗ ω′(φ(−x1),t1) ⊗ ··· ⊗ ω′(φ(xn),tn) ⊗ ω′(φ(−xn),tn) On the other hand, µ(∆f0(x1,...,xn),t1 + ··· + tn) ≥ µ′(φ(x1),t1) ∗ µ′(φ(−x1),t1) ∗ ··· ∗ µ′(φ(xn),tn) ∗ µ′(φ(−xn),tn), ν(∆f0(x1,...,xn),t1 + ··· + tn) ≤ ν′(φ(x1),t1) ⋄ ν′(φ(−x1),t1) ⋄ ··· ⋄ ν′(φ(xn),tn) ⋄ ν′(φ(−xn),tn) and ω(∆f0(x1,...,xn),t1 + ··· + tn) |
(3.3.4) |
|
≤ ω′(φ(x1),t1) ⊗ ω′(φ(−x1),t1) ⊗ ··· ⊗ ω′(φ(xn),tn) ⊗ ω′(φ(−xn),tn) |
(3.3.5) |
Applying the proofs of the Theorem 3.1 and 3.2, we get a unique quadratic function Q and a unique additive function A satisfying
and
. (3.3.6)
Also,
,
and
(3.3.7)
Therefore,
,
and
(3.3.8)
This completes the proof of the theorem. 
In this article, we prove existence of unique quadratic function and unique additive quadratic function between linear space and Neutrosophic Banach Space.
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