International Journal of Analysis and Applications

Volume 16, Number 2 (2018), 232-238

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-232

STABILITY OF EULER-LAGRANGE-JENSEN’S (a,b)- SEXTIC FUNCTIONAL

EQUATION IN MULTI-BANACH SPACES

JOHN MICHAEL RASSIAS1, R. MURALI2,∗ AND A. ANTONY RAJ2

1Pedagogical Department E.E., National and Kapodistrian University of Athens, Section of Mathematics

and Informatics, Athens, 15342, Greece

2PG and Research Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur - 635 601,

Tamil Nadu, India

∗Corresponding author: shcrmurali@yahoo.co.in

Abstract. In this paper, we prove the Hyers-Ulam Stability of Euler-Lagrange-Jensen’s (a,b)-Sextic Func-

tional Equation in Multi-Banach Spaces.

1. Introduction and Preliminaries

The theory of stability is an important branch of the qualitative theory of functional equations. The

concept of stability for a functional equation arises when one replaces a functional equation by an inequality

which acts as a perturbation of the equation. The first stability problem of functional equation was raised by

S.M. Ulam [17] about seventy seven years ago. Since then, this question has attracted the attention of many

researchers. Note that the affirmtive solution to this question was given in the next year by D.H. Hyers [5] in

1941. In the year 1950, T. Aoki [1] generalized Hyers theorem for additive mappings. The result of Hyers was

generalized independently by Th.M.Rassias [14] for linear mappings by considering an unbounded Cauchy

difference. In 1994, a further generalization of Th.M. Rassias theorem was obtained by P.Gavruta [4].

Then, the stability problem of several functional equations has been extensively investigated by a number

Received 2017-10-25; accepted 2018-01-04; published 2018-03-07.

2010 Mathematics Subject Classification. 39B52, 39B72,39B82.

Key words and phrases. Hyers-Ulam stability; multi-Banach Spaces; sextic functional equation; direct method.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

232

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-232


Int. J. Anal. Appl. 16 (2) (2018) 233

of authors, and there are many interesting results concerning this problem ( [3, 6, 7, 9, 11–13, 15, 16, 18, 19]).

The Hyers-Ulam stability of functional equation is investigated and the investigation is following. Here, we

establish the Hyers-Ulam Stability of Euler-Lagrange-Jensen’s (a,b)- Sextic Functional Equation is of the

form

f(ax + by) + f(bx + ay) + (a− b)6
[
f

(
ax− by
a− b

)
+ f

(
bx−ay
b−a

)]
= 64(ab)2(a2 + b2)

[
f(
x + y

2
) + f(

x−y
2

)

]
+ 2(a2 − b2)(a4 − b4) [f(x) + f(y)] (1.1)

where a 6= b such that k ∈ R, h = a + b 6= 0, ±1 in Multi-Banach Spaces by using direct and fixed point

method.

Definition 1.1. [2] A Multi- norm on
{
Ak : k ∈ N

}
is a sequence (‖.‖) = (‖.‖k : k ∈ N) such that ‖.‖k is

a norm on Ak for each k ∈ N,‖x‖1 = ‖x‖ for each x ∈ A, and the following axioms are satisfied for each

k ∈ N with k ≥ 2 :

(1)
∥∥(xσ(1), ...,xσ(k))∥∥k = ‖(x1...xk)‖k , for σ ∈ Ψk,x1, ...,xk ∈A;

(2) ‖(α1x1, ...,αkxk)‖k ≤ (maxi∈Nk |αi|)‖(x1...xk)‖k
for α1...αk ∈ C,x1, ...,xk ∈A;

(3) ‖(x1, ...,xk−1, 0)‖k = ‖(x1, ...,xk−1)‖k−1 , for x1, ...,xk−1 ∈A;

(4) ‖(x1, ...,xk−1,xk−1)‖k = ‖(x1, ...,xk−1)‖k−1 for x1, ...,xk−1 ∈A.

In this case, we say that
(
(Ak,‖.‖k) : k ∈ N

)
is a multi - normed space.

Suppose that
(
(Ak,‖.‖k) : k ∈ N

)
is a multi - normed space, and take k ∈ N. We need the following two

properties of multi - norms. They can be found in [2].

(a)‖(x,...,x)‖k = ‖x‖ , for x ∈A,

(b) max
i∈Nk
‖xi‖≤‖(x1, ...,xk)‖k ≤

k∑
i=1

‖xi‖≤ k max
i∈Nk
‖xi‖ ,∀x1, ...,xk ∈A.

It follows from (b) that if (A,‖.‖) is a Banach space, then (Ak,‖.‖k) is a Banach space for each k ∈ N.

In this case,
(
(Ak,‖.‖k) : k ∈ N

)
is a multi - Banach space.

2. Stability of Functional Equation (1.1) in Multi-Banach Spaces: Direct Method

Theorem 2.1. Let X be a linear space and ((Y n,‖.‖n) : n ∈ N) be a multi-Banach Spaces. Let f : X → Y

be a mapping satisfying f(0) = 0 such that

sup
k∈N
‖(Df(x1,y1), ...,Df(xk,yk))‖k ≤ � (2.1)



Int. J. Anal. Appl. 16 (2) (2018) 234

∀x1, ...,xk,y1, ...,yk ∈ Y. Then there exists a unique sextic mapping S : X → Y such that

sup
k∈N
‖(f(x1) −S(x1), ...,f(xk) −S(xk))‖≤

�

h6
(2.2)

Proof. Letting yi = xi where i = 1, 2, ...k in (2.1), we arrive at

sup
k∈N

∥∥∥∥
(

1

h6
f(hx1) −f(x1), ...,

1

h6
f(hxk) −f(xk)

)∥∥∥∥ ≤ �2h6 (2.3)
Now, Replacing xi by 2xi where i = 1, 2, ..,k and dividing by 2 in above equation, we get

sup
k∈N

∥∥∥∥
(
f(2hx1)

h6
−f(x1), ...,

f(2hxk)

h6
−f(xk)

)∥∥∥∥ ≤ �22h6 + �2h6 (2.4)
By using induction for a positive integer n, we obtain

sup
k∈N

∥∥∥∥
(
f(2nhx1)

2nh6
−f(x1), ...,

f(2nhxk)

2nh6
−f(xk)

)∥∥∥∥ ≤ 1h6
n−1∑
i=0

�

2i+1
≤

1

h6

∞∑
i=0

�

2i+1
(2.5)

Now, we have to show that the sequence

{
f(2nhx)

2nh6

}
is a Cauchy sequence, by fixing x ∈ X and replacing

x1, ...xk by x, 2x,..., 2
k−1x such that

sup
k∈N

∥∥∥∥
(
f(2nhx)

2nh6
−
f(2mx)

2m
, ...,

f(2n+k−1hx)

2n+k−1h6
−
f(2m+k−1x)

2m+k−1

)∥∥∥∥
≤ sup

k∈N

∥∥∥∥
(
f(2nhx)

2nh6
−
f(2mx)

2m
, ...,

1

2k−1

[
f(2n(2k−1hx))

2nh6
−
f(2m(2k−1x))

2m

])∥∥∥∥
Using the definition of Multi-norm, we arrive at

sup
k∈N

∥∥∥∥
(
f(2nhx)

2nh6
−
f(2mx)

2m
, ...,

f(2n(2k−1hx))

2nh6
−
f(2m(2k−1x))

2m

)∥∥∥∥
≤

1

h6

n−1∑
i=m

�

2i+1
. (2.6)

Hence the above inequality (2.6), shows that

{
f(2nhx)

2nh6

}
is a Cauchy sequence as n → ∞. Since Y is

complete, then the sequence

{
f(2nhx)

2nh6

}
converges to a fixed point S(x) ∈ Y such that

S(x) = lim
n→∞

f(2nhx)

2nh6
.

Therefore, as n →∞, the inequality (2.5) implies the inequality (2.2). Obviously, one can find the uniqueness

of the mapping S : X → Y, using the definition of multi-norm. That is, we can prove S = S′. �

Corollary 2.1. Let X be a linear space and ((Y n,‖.‖n) : n ∈ N) be a multi-Banach space. Let f : X → Y

be a mapping satisfying f(0) = 0 such that

sup
k∈N
‖Df(x1,y1, ...,xk,yk)‖k ≤ φ(x1,y1, ...,xk,yk) (2.7)



Int. J. Anal. Appl. 16 (2) (2018) 235

for all x1, ..,xk,y1, ..,yk ∈ X. Then there exists a unique sextic mapping S : X → Y such that

sup
k∈N
‖f(x1) −S(x1), ...,f(xk) −S(xk)‖k ≤

1

h6

∞∑
i=1

1

2i+1
φ
(
2ix1,x1, ..., 2

ixk,xk
)

(2.8)

for all x1, ..,xk ∈ X.

Proof. Proof is similar to that of Theorem 2.1 by replacing the condition φ(x1,y1, ...,xk,yk) in place of �. �

Corollary 2.2. Let X be a linear space and ((Y n,‖.‖n) : n ∈ N) be a multi-Banach space. Let 0 < p <

6 ,θ ≥ 0 and f : X → Y be a mapping satisfying f(0) = 0 such that

sup
k∈N
‖Df(x1,y1, ...,xk,yk)‖k ≤ θ (‖x1‖

p
+ ‖y1‖

p
, ...,‖xk‖

p
+ ‖yk‖

p
) (2.9)

for all x1, ..,xk,y1, ..,yk ∈ X. Then there exists a unique sextic mapping S : X → Y such that

sup
k∈N
‖f(x1) −S(x1), ...,f(xk) −S(xk)‖k ≤

θ

h6(2p − 1)
(‖x1‖

p
, ...,‖xk‖

p
) (2.10)

for all x1, ..,xk ∈ X.

Proof. Proof is similar to that of Theorem 2.1 by replacing the condition

θ (‖x1‖
p

+ ‖y1‖
p
, ...,‖xk‖

p
+ ‖yk‖

p
) in place of �. �

3. Stability of Functional Equation (1.1) in Multi-Banach Spaces: Fixed Point Method

Theorem 3.1. Let X be a linear space and ((Y n,‖.‖n) : n ∈ N) be a multi-Banach Spaces. Let f : X → Y

be a mapping satisfying f(0) = 0 such that

sup
k∈N
‖(Df(x1,y1), ...,Df(xk,yk))‖k ≤ � (3.1)

∀x1, ...,xk,y1, ...,yk ∈ Y. Then there exists a unique sextic mapping S : X → Y such that

sup
k∈N
‖(f(x1) −S(x1), ...,f(xk) −S(xk))‖≤

�

2(h6 − 1)
(3.2)

Proof. Letting yi = xi where i = 1, 2, ...k in (2.1), we arrive at

sup
k∈N

∥∥∥∥
(

1

h6
f(hx1) −f(x1), ...,

1

h6
f(hxk) −f(xk)

)∥∥∥∥ ≤ �2h6 (3.3)
Let Ψ = {l : X → Y |l(0) = 0} and introduce the generalized metric d defined on Ψ by

d(l,m) = inf

{
Ψ ∈ [0,∞]|sup

k∈N
‖l(x1) −m(x1), ..., l(xk) −m(xk)‖k ≤ Ψ ∀ x1, ...,xk ∈ X

}
Then it is easy to show that Ψ,d is a generalized complete metric space, See [8].

We define an operator J : Ψ → Ψ by

J l(x) =
1

h6
l(hx) x ∈ X



Int. J. Anal. Appl. 16 (2) (2018) 236

We assert that J is a strictly contractive operator. Given l,m ∈ Ψ, let Ψ ∈ [0,∞] be an arbitary constant

with d(l,m) ≤ Ψ. From the definition if follows that

sup
k∈N
‖l(x1) −m(x1), ..., l(xk) −m(xk)‖k ≤ Ψ x1, ...,xk ∈ X.

Therefore, supk∈N ‖(J l(x1) −Jm(x1), ...,J l(xk) −Jm(xk))‖k ≤
1

h6
Ψ

x1, ...,xk ∈ X. Hence,it holds that

d(J l,Jm) ≤
1

h6
Ψd(J l,Jm) ≤

1

h6
d(l,m)

∀l,m ∈ Ψ.

This Means that J is strictly contractive operator on Ψ with the Lipschitz constant L =
1

h6
.

By (3.3), we have d(Jf,f) ≤
�

2h6
. Applying the Theorem 2.2 in [10], we deduce the existence of a fixed

point of J that is the existence of mapping S : X → Y such that

S(hx) = h6S(x) ∀x ∈ X.

Moreover, we have d (Jnf,S) → 0, which implies

S(x) = lim
n→∞

Jnf(x) = lim
n→∞

f(hnx)

h6n

for all x ∈ X.

Also, d(f,S) ≤
1

1 −L
d(Jf,f) implies the inequality

≤
�

2(h6 − 1)
.

Doing x1 =, ..., = xk = h
nx, and y1 =, ..., = yk = h

ny in (1.1) and dividing by h6n. Now, applying the

property (a) of multi-norms, we have

‖DS(x,y)‖ = lim
n→∞

1

h6n
‖Df (hnx,hny)‖

≤ lim
n→∞

1

h6n
= 0

for all x,y ∈ X. The uniqueness of S follows from the fact that S is the unique fixed point of J with the

property that there exists ` ∈ (0,∞) such that

sup
k∈N
‖(f(x1) −S(x1), ...,f(xk) −S(xk))‖k ≤ `

for all x1, ...,xk ∈ X.

Hence the proof. �



Int. J. Anal. Appl. 16 (2) (2018) 237

Corollary 3.1. Let X be a linear space and ((Y n,‖.‖n) : n ∈ N) be a multi-Banach space. Let 0 < p <

6 ,θ ≥ 0 and f : X → Y be a mapping satisfying f(0) = 0 such that

sup
k∈N
‖Df(x1,y1, ...,xk,yk)‖k ≤ θ (‖x1‖

p
+ ‖y1‖

p
, ...,‖xk‖

p
+ ‖yk‖

p
) (3.4)

for all x1, ..,xk,y1, ..,yk ∈ X. Then there exists a unique sextic mapping S : X → Y such that

sup
k∈N
‖f(x1) −S(x1), ...,f(xk) −S(xk)‖k ≤

2θ

h6 − 2hp
(‖x1‖

p
, ...,‖xk‖

p
) (3.5)

for all x1, ..,xk ∈ X.

Proof. Proof is similar to that of Theorem 3.1 by replacing the condition

θ (‖x1‖
p

+ ‖y1‖
p
, ...,‖xk‖

p
+ ‖yk‖

p
) in place of �. �

Corollary 3.2. Let X be a linear space and ((Y n,‖.‖n) : n ∈ N) be a multi-Banach space. Let f : X → Y

be a mapping satisfying f(0) = 0 such that

sup
k∈N
‖Df(x1,y1, ...,xk,yk)‖k ≤ φ(x1,y1, ...,xk,yk) (3.6)

for all x1, ..,xk,y1, ..,yk ∈ X. Then there exists a unique sextic mapping S : X → Y such that

sup
k∈N
‖f(x1) −S(x1), ...,f(xk) −S(xk)‖k ≤

1

2(h6 − 1)
φ(x1,x1, ...,xk,xk) (3.7)

for all x1, ..,xk ∈ X.

Proof. Proof is similar to that of Theorem 3.1 by replacing the condition φ(x1,y1, ...,xk,yk) in place of �. �

References

[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J.Math. Soc. Japan 2 (1950), 64-66.

[2] Dales, H.G and Moslehian, Stability of mappings on multi-normed spaces, Glasgow Math. J. 49 (2007), 321-332.

[3] Fridoun Moradlou, Approximate Euler-Lagrange-Jensen type Additive mapping in Multi-Banach Spaces: A Fixed point

Approach, Commun. Korean Math. Soc. 28 (2013), 319-333.

[4] P. Gavruta, On a problem of G.Isac and Th.M.Rassias concerning the stability of mappings, J. Math. Anal. Appl. 261

(2001), 543-553.

[5] D.H.Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222-224.

[6] John Michael Rassias, R. Murali, Matina John Rassias, A. Antony Raj, General Solution, stability and Non-sstability of

Quattuorvigintic functional equation in Multi-Banach spaces, Int. J. Math. Appl. 5 (2017), 181-194.

[7] Manoj Kumar and Ashish Kumar, Stability of Jenson Type Quadratic Functional Equations in Multi-Banach Spaces, Int.

J. Math. Arch. 3(4) (2012), 1372-1378.

[8] D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math.

Anal. Appl. 343 (2008), 567-572.

[9] R. Murali, Matina J. Rassias and V. Vithya, The General Solution and stability of Nonadecic Functional Equation in

Matrix Normed Spaces, Malaya J. Mat. 5(2) (2017), 416-427.

[10] V.Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91-96.



Int. J. Anal. Appl. 16 (2) (2018) 238

[11] K. Ravi, J.M. Rassias and B.V. Senthil Kumar, Ulam-Hyers stability of undecic functional equation in quasi-beta normed

spaces fixed point method, Tbilisi Math. Sci. 9 (2) (2016), 83-103.

[12] K. Ravi, J.M. Rassias, S. Pinelas and S.Suresh, General solution and stability of Quattuordecic functional equation in

quasi beta normed spaces, Adv. Pure Math. 6 (2016), 921-941.

[13] J. M. Rassias and M. Eslamian, Fixed points and stability of nonic functional equation in quasi-β-normed spaces, Cont.

Anal. Appl. Math. 3 (2) (2015), 293-309.

[14] Th.M. Rassias, On the stability of the linear mappings in Banach Spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.

[15] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta. Appl. Math. 62 (2000) 23 - 130.

[16] Tian Zhou Xu, John Michael Rassias and Wan Xin Xu, Generalized Ulam - Hyers Stability of a General Mixed AQCQ

functional equation in Multi-Banach Spaces: A Fixed point Approach, Eur. J. Pure Appl. Math. 3 (2010), 1032-1047.

[17] S.M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, (1960).

[18] T. Z. Xu, J. M. Rassias, M. J. Rassias and W. X. Xu, A fixed point approach to the stability of quintic and sextic

functional equations in quasi-β-normed spaces, J. Inequal. Appl. 2010 (2010), Article ID 423231.

[19] Xiuzhong Wang, Lidan Chang, Guofen Liu, Orthogonal Stability of Mixed Additive-Quadratic Jenson Type Functional

Equation in Multi-Banach Spaces, Adv. Pure Math. 5 (2015), 325-332.


	1. Introduction and Preliminaries
	2. Stability of Functional Equation (1.1) in Multi-Banach Spaces: Direct Method
	3. Stability of Functional Equation (1.1) in Multi-Banach Spaces: Fixed Point Method
	References