Research ArticleHyers–Ulam Stability of Additive Functional Equation UsingDirect and Fixed-Point Methods

K. Tamilvanan ,1 G. Balasubramanian,1 Nazek Alessa ,2 and K. Loganathan 3

1Department of Mathematics, Government Arts College for Men, Krishnagiri 635 001, Tamilnadu, India2Department of Mathematical Sciences, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia3Department of Mathematics, Erode Arts & Science College, Erode 638009, Tamilnadu, India

Correspondence should be addressed to K. Loganathan; loganathankaruppusamy304@gmail.com

Received 30 October 2020; Revised 15 November 2020; Accepted 23 November 2020; Published 14 December 2020

Academic Editor: Kottakkaran Sooppy Nisar

Copyright © 2020 K. Tamilvanan et al.*is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this present work, we obtain the solution of the generalized additive functional equation and also establish Hyers–Ulam stabilityresults by using alternative fixed point for a generalized additive functional equationχ(lg�1 vg) � 1≤g

χ l

g�1vg

⎛⎝ ⎞⎠ � 1≤g< h< i≤ l

χ vg + vh + vi − 1≤g< h≤ l

χ vg + vh −l2

− 5l + 22

l

g�1

χ vg − χ − vg 2

, (2)

where l≥ 5 is a nonnegative integer in Banach spaces.

2. General Solution of the FunctionalEquation (2)

In this section, we derive the general solution of the gen-eralized additive functional equation (2).

Here, we consider Φ and Ω be real vector spaces.

Theorem 1. If a mapping χ: Φ⟶Ω satisfies the functionalequation (2) for all v1, v2, . . . , vn ∈ Φ, then the mappingχ: Φ⟶Ω satisfies the functional equation (1) for allx, y ∈ Φ.

Proof. Suppose a mapping χ: Φ⟶Ω satisfies the func-tional equation (2). Replacing (v1, v2, v3, . . . , vl) by(0, 0, 0, . . . , 0) in the functional equation (2), we haveχ(0) � 0. Replacing (v1, v2, v3, . . . , vl) by (v, 0, 0, . . . , 0) inequation (2), we get χ(− v) � − χ(v) for all v ∈ Φ. *erefore,the function χ is an odd function. Replacing(v1, v2, v3, . . . , vl) by (v, v, 0, 0, . . . , 0) in equation (2) andusing the property of odd function, we have

χ(2v) � 2χ(v), (3)

for all v ∈ Φ. Replacing v by 2v in (3), we obtain

χ 22v � 22χ(v), (4)

for all v ∈ Φ. Again, replacing v by 2v in (5) and using (3), wehave

χ 23v � 23χ(v), (5)

for all v ∈ Φ. We can generalize for any nonnegative integern and we get

χ 2nv( � 2nχ(v), (6)

for all v ∈ Φ. Now, replacing (v1, v2, v3, . . . , vl) by(x, y, 0, . . . , 0) in (2), we obtain our desired result ofequation (1).

Remark 1. Let Ω be a linear space and a functionχ: Φ⟶Ω satisfies the functional equation (2). *en, thefollowing claims hold:

(1) χ(rkv) � rkχ(v) for all v ∈ R, r ∈ Q, k integers(2) χ(v) � vχ(1) for all v ∈ R if χ is continuous

In Sections 3 and 4, we take Φ be a normed space and Ωbe a Banach space. For our convincing effortlessness, wedescribe a function Θ: Φ⟶Ω as

Θ v1, v2, v3, . . . , vl( � χ l

g�1vg

⎛⎝ ⎞⎠ − 1≤g< h< i≤ l

χ vg + vh + vi

− 1≤g

‖(l − 4)χ(2v) − 2(l − 4)χ(v)‖≤ ξ(v, v, 0, . . . , 0), (12)

for all v ∈ Φ. From equality (12), we get

χ(2v)2

− χ(v)�������

�������≤ξ(v, v, 0, . . . , 0)

2(l − 4), (13)

for all v ∈ Φ. Exchanging v through 2v in (13), we obtain

χ 22v 2

− χ(2v)

���������

���������≤ξ(2v, 2v, 0, . . . , 0)

2(l − 4), (14)

for all v ∈ Φ. From (14), we achieve

χ 22v

22−χ(2v)2

���������

���������≤12ξ(2v, 2v, 0, . . . , 0)

2(l − 4), (15)

for all v ∈ Φ. Adding together (13) and (15), we get thefollowing outcome:

χ 22v

22− χ(v)

���������

���������≤

12(l − 4)

ξ(v, v, 0, . . . , 0) +ξ(2v, 2v, 0, . . . , 0)

2 , (16)

for all v ∈ Φ. It follows from (13), (15), and (16), and we cangeneralize that as follows:

χ 2lv

2l− χ(v)

���������

���������≤

12(l − 4)

l− 1

w�0

ξ 2wv, 2wv, 0, . . . , 0( 2w

, ∀v ∈ Φ,

⟹χ 2lv

2l− χ(v)

���������

���������≤

12(l − 4)

∞

w�0

ξ 2wv, 2wv, 0, . . . , 0( 2w

,

(17)

for all v ∈ Φ. In order to establish the convergence of thesequence χ(2wv)/2w , switch v through 2sv and also divideby 2s in (17). We conclude that, for some w, s> 0,

χ 2w+sv( 2(w+s)

−χ 2sv( 2s

��������

���������

12s

χ 2w+sv( 2w

− χ 2sv( ��������

��������

≤1

2(l − 4)

l− 1

w�0

ξ 2w+sv, 2w+sv, 0, . . . , 0( 2(w+s)

≤1

2(l − 4)

∞

w�0

ξ 2w+sv, 2w+sv, 0, . . . , 0( 2(w+s)

⟶ 0, as s⟶∞,

(18)

for all v ∈ Φ. *erefore, the sequence χ(2wv)/2w is aCauchy. As Ω is complete, there exists Ψ: Φ⟶Ω so thatΨ(v) � limw⟶∞(χ(2wv)/2w) for all v ∈ Φ. Taking the limitw⟶∞ in (17), we obtain that result (11) holds for allv ∈ Φ. To prove that the function Ψ satisfies equation (2),replacing (v1, v2, . . . , vl) by (2wv1, 2wv2, . . . , 2wvl) and alsodividing by 2w in (10), we get

12wΘ 2wv1, 2

wv2, . . . , 2

wvl(

��������≤

12w

ξ 2wv1, 2w

v2, . . . , 2w

vl( ,

(19)

for all v1, v2, . . . , vl ∈ Φ. Taking the limit w⟶∞ in theabove inequality and using the definition of Ψ(v), we haveΨ(v1, v2, . . . , vl) � 0 for all v1, v2, . . . , vl ∈ Φ. *us, thefunction Ψ satisfies equation (2). To prove that the function

Journal of Mathematics 3

Ψ is unique, let φ: Φ⟶Ω be another additive mappingsatisfying the functional equation (2) and (11). Hence,

‖Ψ(v) − φ(v)‖≤12sΨ 2sv( − χ 2sv(

�������� + χ 2sv( − φ 2sv(

��������

≤1

2(l − 4)

∞

w�0

ξ 2w+sv, 2w+sv, 0, . . . , 0( 2(w+s)

, ∀v ∈ Φ

⟶ 0 as s⟶∞.

(20)

Hence, Ψ is unique. Now, replacing v through (v/2) in(12), we have

(l − 4)χ(v) − 2(l − 4)χv

2

������

������≤ ξv

2,v

2, 0, . . . , 0 , (21)

for all v ∈ Φ. *e rest of the proof is similar to that whenζ � 1. So for ζ � − 1, we can prove the results by a similarmanner. Hence, the proof is completed. □

Corollary 1. Let ϕ and ϑ be positive real numbers. If thereexists a mapping Θ: Φ⟶Ω satisfying the inequality

Θ v1, v2, . . . , vl( ����

����≤

ϕ,

ϕ l

i�1vi

��������ϑ⎧⎨

⎩

⎫⎬

⎭,

ϕ l

i�1vi

��������ϑ

+ l

i�1vi

��������

lϑ⎧⎨

⎩

⎫⎬

⎭,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

for all v1, v2, . . . , vl ∈ Φ, then there exists a unique additivemapping Ψ: Φ⟶Ω such that

‖χ(v) − Ψ(v)‖≤

ϕ|l − 4|

,

2ϕ‖v‖ϑ

(l − 4) 2 − 2ϑ

; ϑ≠ 1,

2ϕ‖v‖lϑ

(l − 4) 2 − 2lϑ

; ϑ≠

1l,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

for all v ∈ Φ.

4. Hyers–Ulam Stability of the FunctionalEquation (2): Fixed-Point Method

In this section, we examined the Hyers–Ulam stability of thegeneralized additive functional equation (2) in Banach spaceby using the fixed-point method.

Theorem 3. Let Ψ: Φ⟶Ω be a mapping for which thereexists a mapping ξ: Φl⟶ [0,∞) and

limw⟶∞

ξ ηwg v1, ηwg v2, . . . , η

wg vl

ηwg� 0, (24)

where ηg �2, if g � 0,(1/2), if g � 1, and such that it satisfies theinequality

Ψ v1, v2, . . . , vl( ����

����≤ ξ v1, v2, . . . , vl( , (25)

for all v1, v2, . . . , vl ∈ Φ. If there exists a Lipschitz constantL � L(g) such that

v⟶ λ(v) �1

(l − 4)ξ

v

2,v

2, 0, . . . , 0 , (26)

has the property

λ ηgv ηg

� Lλ(v), (27)

for all v ∈ Φ. Ben, there exists a unique additive mappingΨ: Φ⟶Ω satisfying equation (2) and

‖χ(v) − Ψ(v)‖≤L1− g

1 − Lλ(v), (28)

for all v ∈ Φ.

Proof. Consider a set ψ � m/m: Φ⟶Ω, m(0) �{ 0} andinitiate the generalized metric on ψ, d(m, n) � infw ∈ (0,∞): ‖m(v)−{ n(v)‖ ≤wλ(v), v ∈ Ψ}. It is easy toview that (ψ, d) is complete. Fix ]: ψ⟶ ψ as ]m(v) �(1/ηg)m(ηgv) for all v ∈ Φ. For m, n ∈ ψ and v ∈ Φ, we have

d(m, n) � w⟹‖m(v) − n(v)‖≤wλ(v)⟹m ηgv

ηg−

n ηgv ηg

���������

���������≤1ηg

wλ ηgv

⟹‖]m(v) − ]n(v)‖≤1ηg

wλ ηgv ⟹d(]m(v), ]n(v)) ≤wL.

(29)

*at is, d(]m, ]n)≤L d(m, n). Accordingly, ] is a strictlycontractive mapping on ψ with Lipschitz constant L. Frominequality (12), we have

‖(l − 4)χ(2v) − 2(l − 4)χ(v)‖≤ ξ(v, v, 0, . . . , 0), (30)

for all v ∈ Φ. It follows from (30) that

4 Journal of Mathematics

χ(2v)2

− χ(v)�������

�������≤ξ(v, v, 0, . . . , 0)

2(l − 4), (31)

for all v ∈ Φ. Using inequality (27) when g � 0, we get

χ(v) −χ(2v)2

�������

�������≤12λ(v)⟹‖χ(v) − ]m(v)‖ ≤Lλ(v), (32)

for all v ∈ Φ. So we obtain

d(]χ(v), χ(v))≤L � L1− g

Also,

λ(v) �1

(l − 4)ξ

v

2,v

2, 0, . . . , 0 �

ϕ(l − 4)

,

2ϕ‖v‖ϑ

(l − 4)2ϑ

2ϕ‖v‖lϑ

(l − 4)2lϑ.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

And also,

1ηg

λ ηgv �

ϕ(l − 4)ηg

,

2ηg

ϕ‖v‖ϑηϑg(l − 4)2ϑ

2ηg

ϕ‖v‖lϑηlϑg(l − 4)2lϑ

,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

�

η− 1g λ(v),

ηϑ− 1g λ(v),

ηlϑ− 1g λ(v),

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(44)

for all v ∈ Φ. Inequity (2) holds for the following cases:

(i) L � 2− 1 if g � 0 and L � 21 if g � 1(ii) L � 2ϑ− 1 for ϑ< 1 if g � 0 and L � 21− ϑ for ϑ> 1 if

g � 1(iii) L � 2lϑ− 1 for ϑ < (1/l) if g � 0 and L � 21− lϑ for

ϑ > (1/l) if g � 1

Now, from (28), we verify the following cases:

Case 1: L � 2− 1 if g � 0:

‖χ(v) − Ψ(v)‖≤L1− g

1 − Lλ(v) �

2− 1ϕ(l − 4) 1 − 2− 1

�ϕ

(l − 4).

(45)

Case 2: L � 2 if g � 1:

‖χ(v) − Ψ(v)‖≤L1− g

1 − Lλ(v) �

ϕ(l − 4)(1 − 2)

� −ϕ

(l − 4).

(46)

Case 3: L � 2ϑ− 1 for ϑ< 1 if g � 0:

‖χ(v) − Ψ(v)‖≤L1− g

1 − Lλ(v) �

2ϑ− 1

1 − 2ϑ− 12ϕ‖v‖ϑ

(l − 4)2ϑ�

2ϕ‖v‖ϑ

(l − 4) 2 − 2ϑ . (47)

Case 4: L � 21− ϑ for ϑ> 1 if g � 1:

‖χ(v) − Ψ(v)‖≤L1− g

1 − Lλ(v) �

11 − 21− ϑ

2ϕ‖v‖ϑ

(l − 4)2ϑ�

2ϕ‖v‖ϑ

(l − 4) 2ϑ − 2 . (48)

Case 5: L � 2lϑ− 1 for ϑ< (1/l) if g � 0:

‖χ(v) − Ψ(v)‖≤L1− g

1 − Lλ(v) �

2lϑ− 1

1 − 2lϑ− 12ϕ‖v‖lϑ

(l − 4)2lϑ�

2ϕ‖v‖lϑ

(l − 4) 2 − 2lϑ . (49)

Case 6: L � 21− lϑ for ϑ> (1/l) if g � 1:

‖χ(v) − Ψ(v)‖≤L1− g

1 − Lλ(v) �

11 − 21− lϑ

2ϕ‖v‖lϑ

(l − 4)2lϑ�

2ϕ‖v‖lϑ

(l − 4) 2lϑ − 2 . (50)

6 Journal of Mathematics

Hence, the proof is accomplished. □

5. Counterexample

In this section, we examine the following counterexamplechanged by the well-known counterexample of Gajda [42] to(2).

Example 1. Let a mapping Θ: Φ⟶Ω be defined by

Θ(v) � ∞

l�0

ϕ 2lv

2l, (51)

where

ϕ(v) �μv, − 1< v< 1,

μ, otherwise, (52)

where μ is a constant, and then, the mapping Θ: Φ⟶Ωsatisfies the inequality

Θ v1, v2, . . . , vl(

≤l2

− 5l + 82

8μ n

i�1vi

⎛⎝ ⎞⎠, (53)

for all v1, v2, . . . , vl ∈ Φ, but there exists an additive functionΨ: Φ⟶Ω with a constant α such that

|χ(v) − Ψ(v)|≤ α|v|, (54)

for all v ∈ Φ.

Proof. It is easy to show that χ is bounded by 2μ on Φ. If

li�1 |vi|≥ (1/2) or 0, then the left side of (53) is less than

((l2 − 5l + 8)/2)2μ, and thus, (53) is true. Assume that0< li�1 |vi|≥ (1/2). *en there exists an integer m such that

12(m+2)≤

l

i�1vi

<

12(m+1)

. (55)

So that 2m|v1|< (1/2), 2m|v2|< (1/2), . . . , 2m|vl|< (1/2)and 2lv1, 2lv2, . . . , 2lvl ∈ (− 1, 1) for all l � 0, 1, 2, . . . , m − 1.So, for l � 0, 1, . . . , m − 1

ϕ l

g�12lvg⎛⎝ ⎞⎠ −

1≤g< h< i≤ lϕ 2l vg + vh + vi +

1≤g< h≤ lϕ 2l vg + vh

+l2

− 5l + 22

l

g�1

ϕ 2l vg + ϕ 2l

− vg

2� 0.

(56)

From the definition of χ, we have

Θ v1, v2, . . . , vl(

≤ ∞

i�m

12i

ϕ 2iv1, 2iv2, . . . , 2

ivl

≤ ∞

i�m

12i

l2

− 5l + 82

μ

≤l2

− 5l + 82

2(1− m)μ.

(57)

It follows from (55) that

Θ v1, v2, . . . , vl(

≤l2

− 5l + 82

8μ n

i�1vi

⎛⎝ ⎞⎠, (58)

for all v1, v2, . . . , vl ∈ Φ. *us, χ satisfies (53) for allv1, v2, . . . , vl ∈ Φ.

Assume on the contrary that there exists a mappingΨ: Φ⟶Ω additive with a constant α> 0 satisfying (54). Asχ is bounded and continuous for all v ∈ Φ, Ψ is bounded onany open interval containing the origin and continuous atthe origin.

By Remark 1, Ψ defined by Ψ(v) � av for all v ∈ Φ.Hence

|χ(v)|≤ (α +|a|)|v|, (59)

for all v ∈ Φ. However, we can select a nonnegative integer mand mμ> α + |a|. If v ∈ (0, (1/2m− 1)), then 2lv ∈ (0, 1) for alll � 0, 1, . . . , m − 1, and for this v, we get

χ(v) � ∞

l�0

ϕ 2lv

2l≥

m− 1

l�0

μ 2lv

2l� mμv>(α +|a|)v, (60)

which contradicts. □

6. Conclusion

In this work, we introduced the generalized finite-dimensionaladditive functional equation (2) and obtain its general solutionin Section 2. In Section 3, we investigated the Hyers–Ulamstability results in Banach space by using the direct method,and in Section 4, we examined the Hyers–Ulam stability resultsin Banach space by using the fixed-point method. In Section 5,we proved the counterexample changed by the well-knowncounterexample of Gajda [42] to show the nonstability of thegeneralized additive functional equation (2).

Data Availability

No data were used to support the findings of the study.

Journal of Mathematics 7

Conflicts of Interest

*e authors declare that they have no conflicts of interest.

Acknowledgments

*is research was funded by the Deanship of ScientificResearch at Princess Nourah Bint Abdulrahman Universitythrough the Fast-track Research Funding Program.

References

[1] S. M. Ulam, Problems in Modern Mathematics, Wiley, NewYork, NY, USA, Science Editions, 1964.

[2] D. H. Hyers, “On the stability of the linear functionalequation,” Proceedings of the National Academy of Sciences,vol. 27, no. 4, pp. 222–224, 1941.

[3] D. H. Hyers, G. Isac, and T. M. Rassias, Stability of FunctionalEquations in Several Variables, Birkhauser, Basel, Switzerland,1998.

[4] J. Aczél, Lectures on Functional Equations and BeirApplications, p. 509, Academic Press, Cambridge, MA, USA,1966.

[5] J. Aczél, A Short Course on Functional Equations, Beory andDecision Library. Series B: Mathematical and StatisticalMethods, D. Reidel Publishing Co., Dordrecht, Netherlands,1987.

[6] T. Aoki, “On the stability of the linear transformation inBanach spaces,” Journal of the Mathematical Society of Japan,vol. 2, no. 1-2, pp. 64–66, 1950.

[7] J. M. Rassias, “On approximation of approximately linearmappings by linear mappings,” Journal of Functional Analysis,vol. 46, no. 1, pp. 126–130, 1982.

[8] J. M. Rassias and H.-M. Kim, “Generalized Hyers-Ulamstability for general additive functional equations in quasi-β-normed spaces,” Journal of Mathematical Analysis andApplications, vol. 356, no. 1, pp. 302–309, 2009.

[9] T. M. Rassias, “On the stability of functional equations inBanach spaces,” Journal of Mathematical Analysis and Ap-plications, vol. 251, no. 1, pp. 264–284, 2000.

[10] T. M. Rassias, Functional Equations, Inequalities and Appli-cations, Kluwer Acedamic Publishers, Dordrecht, Nether-lands, 2003.

[11] S. Czerwik, “On the stability of the quadratic mapping innormed spaces,” Abhandlungen aus dem MathematischenSeminar der Universität Hamburg, vol. 62, no. 1, pp. 59–64,1992.

[12] S. Czerwik, Functional Equations and Inequalities in SeveralVariables, p. 420, World Scientific Publishing Co., Inc., RiverEdge, NJ, USA, 2002.

[13] A. M. Alanazi, G. Muhiuddin, K. Tamilvanan, E. N. Alenze,A. Ebaid, and K. Loganathan, “Fuzzy stability results of finitevariable additive functional equation: direct and fixed pointmethods,” Mathematics, vol. 8, no. 7, p. 1050, 2020.

[14] A. L. Cauchy, Analyse Algébrique: Cours d’Analyse de l’ÉcoleRoyale Polytechnique, Éditions Jacques Gabay, Paris, France, 1989.

[15] G. Darboux, “Sur la composition des forces en statique,”Bulletin des Sciences Mathématiques et Astronomiques, vol. 9,pp. 281–288, 1875.

[16] P. Gavruta, “A generalization of the Hyers-Ulam-Rassiasstability of approximately additive mappings,” Journal ofMathematical Analysis and Applications, vol. 184, no. 3,pp. 431–436, 1994.

[17] K.-W. Jun and H.-M. Kim, “On the hyers-ulam stability of ageneralized quadratic and additive functional equation,”Bulletin of the Korean Mathematical Society, vol. 42, no. 1,pp. 133–148, 2005.

[18] S. M. Jung, Hyers-Ulam-Rassias Stability of FunctionalEquations in Mathematical Analysis, Hadronic Press, PlanHarbor, FL, USA, 2001.

[19] P. Kannappan, Functional Equations and Inequalities withApplications, Springer Monographs in Mathematics, 2009.

[20] M. Kuczma, An Introduction to the Beory of FunctionalEquations and Inequalities, Uniwersytet Slaski, Katowice,Poland, 1985.

[21] D. O. Lee, “Hyers-Ulam stability of an additive functionalequation,” Journal of Applied Mathematics and Computing,vol. 13, no. 1-2, pp. 471–477, 2003.

[22] M. M. Sadr, “On the stability of a generalized additivefunctional equation,” Aequationes mathematicae, vol. 90,pp. 597–606, 2016.

[23] M. S. Moslehian and T. M. Rassias, “Orthogonal stability ofadditive type equations,” Aequationes Mathematicae, vol. 73,no. 3, p. 249, 2007.

[24] C. Park, K. Tamilvanan, G. Balasubramanian, B. Noori, andA. Najati, “On a functional equation that has the quadratic-multiplicative property,” Open Mathematics, vol. 18, no. 1,pp. 837–845, 2020.

[25] C. Park, K. Tamilvanan, B. Noori, M. B. Moghimi, andA. Najati, “Fuzzy normed spaces and stability of a generalizedquadratic functional equation,” AIMS Mathematics, vol. 5,no. 6, p. 7161, 2020.

[26] K. Tamilvanan, J. R. Lee, and C. Park, “Hyers-Ulam stability ofa finite variable mixed type quadratic-additive functionalequation in quasi-Banach spaces,” AIMS Mathematics, vol. 5,no. 6, pp. 5993–6005, 2020.

[27] K. Tamilvanan, J. R. Lee, and C. Park, “Ulam stability of afunctional equation deriving from quadratic and additivemappings in random normed spaces,” AIMS Mathematics,vol. 6, no. 1, pp. 908–924, 2020.

[28] W. Kavitha, V. Vijayakumar, R. Udhayakumar, andK. S. Nisar, “A new study on existence and uniqueness ofnonlocal fractional delay differential systems of order 1< r

[34] M. Almahalebi, “Stability of a generalization of Cauchy’s andthe quadratic functional equations,” Journal of Fixed PointBeory and Applications, vol. 20, no. 1, p. 12, 2018.

[35] V. Radu, “*e fixed point alternative and the stability offunctional equations,” Fixed Point Beory, vol. 4, no. 1,pp. 91–96, 2003.

[36] C. Park and *.M. Rassias, “Fixed points and stability of theCauchy functional equation,” Australian Journal of Mathe-matical Analysis and Applications, vol. 6, no. 1, pp. 1–9, 2009.

[37] K. Ravi and M. Arunkumar, “On a n-dimensional additivefunctional equation with fixed point alternative,” in Pro-ceedings of the ICMS, pp. 314–330, Malaysia, 2007.

[38] W. S. S. Jin, “On the Hyers-Ulam stability of additive qua-dratic functional equation,” International Journal of In-equalities in Pure and AppliedMathematics, vol. 3, no. 3, p. 33,2002.

[39] I.-i. EL-Fassi, “New stability results for the radical sexticfunctional equation related to quadratic mappings in (2,β)-Banach spaces,” Journal of Fixed Point Beory and Ap-plications, vol. 20, no. 4, p. 138, 2018.

[40] K. Tamilvanan, G. Balasubramanian, and A. CharlesSagayaraj, “Finite dimensional even-quadratic functionalequation and its Ulam-Hyers stability,” AIP ConferenceProceedings, vol. 2261, p. 030002, 2020.

[41] K. Tamilvanan, A. Nandhakumar, and P. Muthukumar,“Solution and stability of finite dimensional quadratic func-tional equation in Banach spaces,” AIP Conference Proceed-ings, vol. 2261, p. 030057, 2020.

[42] Z. Gajda, “On stability of additive mappings,” InternationalJournal of Mathematics and Mathematical Sciences, vol. 14,Article ID 817959, 4 pages, 1991.

Journal of Mathematics 9