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Comment on "Functional inequalities associated with Jordan-von Neumann type additive functional equations"

Abstract

Park et al. proved the Hyers-Ulam stability of some additive functional inequalities. There is a fatal error in the proof of Theorem 3.1. We revise the statements of the main theorems and prove the revised theorems.

2010 Mathematics Subject Classification: Primary 39B62; 39B72; 39B52.

1 Introduction and preliminaries

Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.

We are given a group G and a metric group G' with metric ρ(·,·). Given ϵ > 0, does there exist a δ > 0 such that if f : GG' satisfies ρ(f (xy), f(x) f(y)) < δ for all x,y G, then a homomorphism h : GG' exists with ρ(f(x), h(x)) < ϵ for all x G?

Hyers [2] considered the case of approximately additive mappings f: EE', where E and E' are Banach spaces and f satisfies Hyers' inequality

f ( x + y ) - f ( x ) - f ( y ) ε

for all x,y E. It was shown that the limit

L ( x ) = lim n f ( 2 n x ) 2 n

exists for all x E and that L : EE' is the unique additive mapping satisfying

f ( x ) - L ( x ) ε .

Rassias [3] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded.

Theorem 1.1. (Rassias). Let f : EE' be a mapping from a normed vector space E into a Banach space E' subject to the inequality

f ( x + y ) - f ( x ) - f ( y ) ε ( x p + y p )
(1.1)

for all x,y E, where ϵ and p are constants with ϵ > 0 and p < 1. Then the limit

L ( x ) = lim n f ( 2 n x ) 2 n

exists for all x E and L : EE' is the unique additive mapping which satisfies

f ( x ) - L ( x ) 2 ε 2 - 2 p x p
(1.2)

for all x E. If p < 0 then inequality (1.1) holds for x,y ≠ 0 and (1.2) for x ≠ 0.

Rassias [4] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. Gajda [5] following the same approach as in Rassias [3] gave an affirmative solution to this question for p > 1. It was shown by Gajda [5], as well as by Rassias and Šemrl [6] that one cannot prove a Rassias' type theorem when p = 1 (cf. the books of Czerwik [7] and Hyers et al. [8]).

Rassias [9] followed the innovative approach of Rassias' theorem [3] in which he replaced the factor xp+ ypby xp· yqfor p,q with p + q ≠ 1. Găvruta [10] provided a further generalization of Rassias' theorem. During the last two decades a number of papers and research monographs have been published on various generalizations and applications of the Hyers-Ulam stability to a number of functional equations and mappings (see [1113]).

Throughout this article, let G be a 2-divisible abelian group. Assume that X is a normed space with norm || · || X and that Y is a Banach space with norm || · || Y .

Gilányi [14] showed that if f satisfies the functional inequality

2 f ( x ) + 2 f ( y ) - f ( x y - 1 ) f ( x y )
(1.3)

then f satisfies the Jordan-von Neumann functional equation

2 f ( x ) + 2 f ( y ) = f ( x y ) + f ( x y - 1 ) .

See also [15]. Gilányi [16] and Fechner [17] proved the Hyers-Ulam stability of the functional inequality (1.3).

Park et al. [18] proved the Hyers-Ulam stability of the following functional inequalities

f ( x ) + f ( y ) + f ( z ) 2 f x + y + z 2 ,
(1.4)
f ( x ) + f ( y ) + f ( z ) f ( x + y + z ) ,
(1.5)
f ( x ) + f ( y ) + 2 f ( z ) 2 f x + y 2 + z .
(1.6)

But there is an error in the 8th line on the 6th page in the proof of [18, Theorem 3.1]. We revise the statements of the main theorems and prove the revised theorems.

In Section 2, we prove the Hyers-Ulam stability of the functional inequality (1.4).

In Section 3, we prove the Hyers-Ulam stability of the functional inequality (1.5).

In Section 4, we prove the Hyers-Ulam stability of the functional inequality (1.6).

2 Stability of a functional inequality associated with a 3-variable Jensen additive functional equation

Proposition 2.1. [18, Proposition 2.1] Let f : GY be a mapping such that

f ( x ) + f ( y ) + f ( z ) Y 2 f x + y + z 2 Y

for all x, y, z G. Then f is Cauchy additive.

We prove the Hyers-Ulam stability of a functional inequality associated with a Jordan-von Neumann type 3-variable Jensen additive functional equation.

Theorem 2.2. Let r > 1 and θ be nonnegative real numbers, and let f : XY be an odd mapping such that

f ( x ) + f ( y ) + f ( z ) Y 2 f x + y + z 2 Y + θ ( x X r + y X r + z X r )
(2.1)

for all x,y, z X. Then there exists a unique Cauchy additive mapping h : XY such that

f ( x ) - h ( x ) Y 2 r + 2 2 r - 2 θ x X r
(2.2)

for all x X.

Proof. Letting y = x and z = -2x in (2.1), we get

2 f ( x ) - f ( 2 x ) Y = 2 f ( x ) + f ( - 2 x ) Y ( 2 + 2 r ) θ x X r
(2.3)

for all x X. So

f ( x ) - 2 f x 2 Y 2 + 2 r 2 r θ x X r

for all x X. Hence

2 l f x 2 l - 2 m f x 2 m Y j = l m - 1 2 j f x 2 j - 2 j + 1 f x 2 j + 1 Y 2 + 2 r 2 r j = l m - 1 2 j 2 r j θ x X r
(2.4)

for all nonnegative integers m and l with m > l and all x X.

It follows from (2.4) that the sequence { 2 n f ( x 2 n ) } is a Cauchy sequence for all x X. Since Y is complete, the sequence { 2 n f ( x 2 n ) } converges. So one can define the mapping h : XY by

h ( x ) : = lim n 2 n f ( x 2 n )

for all x X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.4), we get (2.2).

It follows from (2.1) that

h ( x ) + h ( y ) + h ( z ) Y = lim n 2 n f x 2 n + f y 2 n + f z 2 n Y lim n 2 n 2 f x + y + z 2 n + 1 Y + lim n 2 n θ 2 n r ( x X r + y X r + z X r ) = 2 h x + y + z 2 Y

for all x, y, z X. So

h ( x ) + h ( y ) + h ( z ) Y 2 h x + y + z 2 Y

for all x, y, z X. By Proposition 2.1, the mapping h : XY is Cauchy additive.

Now, let T : XY be another Cauchy additive mapping satisfying (2.2). Then we have

h ( x ) - T ( x ) Y = 2 n h x 2 n - T x 2 n Y 2 n h x 2 n - f x 2 n Y + T x 2 n - f x 2 n Y 2 ( 2 r + 2 ) 2 n ( 2 r - 2 ) 2 n r θ x X r ,

which tends to zero as n → ∞ for all x X. So we can conclude that h(x) = T(x) for all x X. This proves the uniqueness of h. Thus the mapping h : XY is a unique Cauchy additive mapping satisfying (2.2).

Theorem 2.3. Let r < 1 and θ be positive real numbers, and let f : XY be an odd mapping satisfying (2.1). Then there exists a unique Cauchy additive mapping h : XY such that

f ( x ) - h ( x ) Y 2 + 2 r 2 - 2 r θ x X r
(2.5)

for all x X.

Proof. It follows from (2.3) that

f ( x ) - 1 2 f ( 2 x ) Y 2 + 2 r 2 θ x X r

for all x X. Hence

1 2 l f ( 2 l x ) - 1 2 m f ( 2 m x ) Y j = l m - 1 1 2 j f ( 2 j x ) - 1 2 j + 1 f ( 2 j + 1 x ) Y 2 + 2 r 2 j = l m - 1 2 r j 2 j θ x X r
(2.6)

for all nonnegative integers m and l with m > l and all x X.

It follows from (2.6) that the sequence { 1 2 n f ( 2 n x ) } is a Cauchy sequence for all x X. Since Y is complete, the sequence { 1 2 n f ( 2 n x ) } converges. So one can define the mapping h : XY by

h ( x ) : = lim n 1 2 n f ( 2 n x )

for all x X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), we get (2.5).

The rest of the proof is similar to the proof of Theorem 2.2.

Theorem 2.4. Let r> 1 3 and θ be nonnegative real numbers, and let f : XY be an odd mapping such that

f ( x ) + f ( y ) + f ( z ) Y 2 f x + y + z 2 Y + θ x X r y X r z X r
(2.7)

for all x, y, z X. Then there exists a unique Cauchy additive mapping h : XY such that

f ( x ) - h ( x ) Y 2 r θ 8 r - 2 x X 3 r
(2.8)

for all x X.

Proof. Letting y = x and z = - 2x in (2.7), we get

2 f ( x ) - f ( 2 x ) Y = 2 f ( x ) + f ( - 2 x ) Y 2 r θ x X 3 r
(2.9)

for all x X. So

f ( x ) - 2 f x 2 Y 2 r 8 r θ x X 3 r

for all x X. Hence

2 l f x 2 l - 2 m f x 2 m Y j = l m - 1 2 j f x 2 j - 2 j + 1 f x 2 j + 1 Y 2 r 8 r j = l m - 1 2 j 8 r j θ x X 3 r
(2.10)

for all nonnegative integers m and l with m > l and all x X.

It follows from (2.10) that the sequence { 2 n f ( x 2 n ) } is a Cauchy sequence for all x X. Since Y is complete, the sequence { 2 n f ( x 2 n ) } converges. So one can define the mapping h : XY by

h ( x ) : = lim n 2 n f ( x 2 n )

for all x X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.10), we get (2.8).

The rest of the proof is similar to the proof of Theorem 2.2.

Theorem 2.5. Let r< 1 3 and θ be positive real numbers, and let f : XY be an odd mapping satisfying (2.7). Then there exists a unique Cauchy additive mapping h:XY such that

f ( x ) - h ( x ) Y 2 r θ 2 - 8 r x X 3 r
(2.11)

for all x X.

Proof. It follows from (2.9) that

f ( x ) - 1 2 f ( 2 x ) Y 2 r 2 θ x X 3 r

for all x X. Hence

1 2 l f ( 2 l x ) - 1 2 m f ( 2 m x ) Y j = l m - 1 1 2 j f ( 2 j x ) - 1 2 j + 1 f ( 2 j + 1 x ) Y 2 r 2 j = l m - 1 8 r j 2 j θ x X r
(2.12)

for all nonnegative integers m and l with m > l and all x X.

It follows from (2.12) that the sequence { 1 2 n f ( 2 n x ) } is a Cauchy sequence for all x X. Since Y is complete, the sequence { 1 2 n f ( 2 n x ) } converges. So one can define the mapping h : XY by

h ( x ) : = lim n 1 2 n f ( 2 n x )

for all x X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.12), we get (2.11).

The rest of the proof is similar to the proof of Theorem 2.2.

3 Stability of a functional inequality associated with a 3-variable Cauchy additive functional equation

Proposition 3.1. [18, Proposition 2.2] Let f : GY be a mapping such that

f ( x ) + f ( y ) + f ( z ) Y f ( x + y + z ) Y

for all x, y, z G. Then f is Cauchy additive.

We prove the Hyers-Ulam stability of a functional inequality associated with a Jordan-von Neumann type 3-variable Cauchy additive functional equation.

Theorem 3.2. Let r > 1 and θ be nonnegative real numbers, and let f : XY be an odd mapping such that

f ( x ) + f ( y ) + f ( z ) Y f ( x + y + z ) Y + θ ( x X r + y X r + z X r )
(3.1)

for all x, y, z X. Then there exists a unique Cauchy additive mapping h : XY such that

f ( x ) - h ( x ) Y 2 r + 2 2 r - 2 θ x X r

for all x X.

Proof. Letting y = x and z = - 2x in (3.1), we get

2 f ( x ) - f ( 2 x ) Y = 2 f ( x ) + f ( - 2 x ) Y ( 2 + 2 r ) θ x X r
(3.2)

for all x X.

The rest of the proof is the same as in the proof of Theorem 2.2.

Theorem 3.3. Let r < 1 and θ be positive real numbers, and let f : XY be an odd mapping satisfying (3.1). Then there exists a unique Cauchy additive mapping h:XY such that

f ( x ) - h ( x ) Y 2 + 2 r 2 - 2 r θ x X r

for all x X.

Proof. It follows from (3.2) that

f ( x ) - 1 2 f ( 2 x ) Y 2 + 2 r 2 θ x X r

for all x X.

The rest of the proof is the same as in the proofs of Theorems 2.2 and 2.3.

Theorem 3.4. Let r> 1 3 and θ be nonnegative real numbers, and let f : XY be an odd mapping such that

f ( x ) + f ( y ) + f ( z ) Y f ( x + y + z ) Y + θ x X r y X r z X r
(3.3)

for all x, y, z X. Then there exists a unique Cauchy additive mapping h : XY such that

f ( x ) - h ( x ) Y 2 r θ 8 r - 2 x X 3 r

for all x X.

Proof Letting y = x and z = - 2x in (3.3), we get

2 f ( x ) - f ( 2 x ) Y = 2 f ( x ) + f ( - 2 x ) Y 2 r θ x X 3 r
(3.4)

for all x X.

The rest of the proof is the same as in the proofs of Theorems 2.2 and 2.4.

Theorem 3.5. Let r< 1 3 and θ be positive real numbers, and let f : XY be an odd mapping satisfying (3.3). Then there exists a unique Cauchy additive mapping h : XY such that

f ( x ) - h ( x ) Y 2 r θ 2 - 8 r x X 3 r

for all x X.

Proof. It follows from (3.4) that

f ( x ) - 1 2 f ( 2 x ) Y 2 r 2 θ x X 3 r

for all x X.

The rest of the proof is the same as in the proofs of Theorems 2.2 and 2.5.

4 Stability of a functional inequality associated with the Cauchy-Jensen functional equation

Proposition 4.1. [18, Proposition 2.3] Let f :GY be a mapping such that

f ( x ) + f ( y ) + 2 f ( z ) Y 2 f x + y 2 + z Y

for all x, y, z G. Then f is Cauchy additive.

We prove the Hyers-Ulam stability of a functional inequality associated with a Jordan-von Neumann type Cauchy-Jensen functional equation.

Theorem 4.2. Let r > 1 and θ be nonnegative real numbers, and let f : XY be an odd mapping such that

f ( x ) + f ( y ) + 2 f ( z ) Y 2 f x + y 2 + z Y + θ ( x X r + y X r + z X r )
(4.1)

for all x, y, z X. Then there exists a unique Cauchy additive mapping h : XY such that

f ( x ) - h ( x ) Y 2 r + 1 2 r - 2 θ x X r

for all x X.

Proof. Replacing x by 2x and letting y = 0 and z = -x in (4.1), we get

f ( 2 x ) - 2 f ( x ) Y = f ( 2 x ) + 2 f ( - x ) Y ( 1 + 2 r ) θ x X r
(4.2)

for all x X. So

f ( x ) - 2 f x 2 Y 1 + 2 r 2 r θ x X r

for all x X.

The rest of the proof is similar to the proof of Theorem 2.2.

Theorem 4.3. Let r < 1 and θ be positive real numbers, and let f : XY be an odd mapping satisfying (4.1). Then there exists a unique Cauchy additive mapping h : XY such that

f ( x ) - h ( x ) Y 1 + 2 r 2 - 2 r θ x X r

for all x X.

Proof. It follows from (4.2) that

f ( x ) - 1 2 f ( 2 x ) Y 1 + 2 r 2 θ x X r

for all x X.

The rest of the proof is similar to the proofs of Theorems 2.2 and 2.3.

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Correspondence to Jung Rye Lee.

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Park, C., Lee, J.R. Comment on "Functional inequalities associated with Jordan-von Neumann type additive functional equations". J Inequal Appl 2012, 47 (2012). https://doi.org/10.1186/1029-242X-2012-47

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Keywords

  • Jordan-von Neumann functional equation
  • Hyers-Ulam stability
  • functional inequality