Approximate multi-variable bi-Jensen-type mappings

Jae-Hyeong Bae; Won-Gil Park

doi:10.1515/dema-2023-0105

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Approximate multi-variable bi-Jensen-type mappings

Jae-Hyeong Bae and Won-Gil Park

Published/Copyright: December 20, 2023

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From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

In this study, we obtained the stability of the multi-variable bi-Jensen-type functional equation:

n 2 f x 1 + ⋯ + x n n , y 1 + ⋯ + y n n = ∑ i = 1 n ∑ j = 1 n f ( x i , y j ) .

Keywords: linear 2-normed space; quasi-normed space; multi-variable bi-Jensen-type mapping

MSC 2010: 39B52; 39B82

1 Introduction

In 1940, Ulam [1] suggested the stability problem of functional equations concerning the stability of group homomorphisms:

Let G be a group and let ℋ be a metric group with the metric d ( ⋅ , ⋅ ) . Given ε > 0 , does there exist a δ > 0 such that if a mapping h : G → ℋ satisfies the inequality d ( h ( x y ) , h ( x ) h ( y ) ) < δ for all x , y ∈ G then there is a homomorphism H : G → ℋ with d ( h ( x ) , H ( x ) ) < ε for all x ∈ G ?

Hyers-Ulam stability is a mathematical result that deals with the variation of approximations of a function under small perturbations. Research on the stability of functional equations has been continuously conducted, and rich results are coming out [2–8].

Throughout this article, let X and Y be the vector spaces.

Definition 1

A mapping f : X × X → Y is called a bi-Jensen mapping if f satisfies the system of equations:

(1) 2 f x + y 2 , z = f ( x , z ) + f ( y , z ) , and 2 f x , y + z 2 = f ( x , y ) + f ( x , z ) .

Let f : X × X → Y be a mapping. In 2006, Bae and Park [9] obtained the general solution of the bi-Jensen functional equation

(2) 4 f x + y 2 , z + w 2 = f ( x , z ) + f ( x , w ) + f ( y , z ) + f ( y , w )

and proved its stability. Subsequent articles have been published since 2008 by several authors [10–13].

For an integer n greater than 1, consider the multi-variable bi-Jensen-type functional equation:

(3) n 2 f x 1 + ⋯ + x n n , y 1 + ⋯ + y n n = ∑ i = 1 n ∑ j = 1 n f ( x i , y j ) .

Equation (2) is a special case of equation (3).

In 2011, Park [14] investigated the approximate additive, Jensen, and quadratic mappings in 2-Banach spaces. In 2018, EL-Fassi [15] investigated the generalized hyperstability of bi-Jensen functional equation in (2, β )-Banach spaces.

In this study, we solved the solution and investigated the stability of the multi-variable bi-Jensen-type functional equation (3) in 2-Banach spaces and quasi-Banach spaces.

2 Main results

We introduce some definitions on 2-Banach spaces [16,17].

Definition 2

Let X be a real vector space with dim X ≥ 2 and ‖ ⋅ , ⋅ ‖ : X 2 → R be a function. Then, ( X , ‖ ⋅ , ⋅ ‖ ) is called a linear 2-normed space if the following conditions hold:

‖ x , y ‖ = 0 if and only if x and y are linearly dependent,
‖ x , y ‖ = ‖ y , x ‖ ,
‖ α x , y ‖ = ∣ α ∣ ‖ x , y ‖ ,
‖ x , y + z ‖ ≤ ‖ x , y ‖ + ‖ x , z ‖ ,

for all α ∈ R and x , y , z ∈ X . In this case, the function ‖ ⋅ , ⋅ ‖ is called a 2-norm on X .

Definition 3

Let { x n } be a sequence in a linear 2-normed space X . The sequence { x n } is said to be convergent in X if there exists an element x ∈ X such that

lim n → ∞ ‖ x n − x , y ‖ = 0

for all y ∈ X . In this case, we say that the sequence { x n } converges to the limit x , simply, denoted by lim n → ∞ x n = x .

Definition 4

A sequence { x n } in a linear 2-normed space X is called a Cauchy sequence if there are two linearly independent points y , z ∈ X such that for any ε > 0 , there exists N ∈ N such that for all m , n ≥ N , ‖ x m − x n , y ‖ < ε and ‖ x m − x n , z ‖ < ε . A 2-Banach space is defined to be a linear 2-normed space in which every Cauchy sequence is convergent.

In the following lemma, we obtain some basic properties in a linear 2-normed space, which will be used to prove the stability results.

Lemma 1

[14] Let ( X , ‖ ⋅ , ⋅ ‖ ) be a linear 2-normed space and x ∈ X .

If ‖ x , y ‖ = 0 for all y ∈ X , then x = 0 .
∣ ‖ x , z ‖ − ‖ y , z ‖ ∣ ≤ ‖ x − y , z ‖ for all x , y , z ∈ X .
If a sequence { x n } is convergent in X , then lim n → ∞ ‖ x n , y ‖ = ‖ lim n → ∞ x n , y ‖ for all y ∈ X .

Let X be a normed space and Y be a 2-Banach space.

Lemma 2

Let f : X × X → Y satisfy (3). And let g x , g y ′ : X → Y be given by g x ( y ) ≔ f ( x , y ) − f ( x , 0 ) and g y ′ ( x ) ≔ f ( x , y ) − f ( 0 , y ) for all x , y ∈ X . Then, g x is additive for all x ∈ X and g y ′ is additive for all y ∈ X .

Proof

Letting x 1 = x 2 = ⋯ = x n = x , y 1 = y , and y 2 = ⋯ = y n = 0 in (3), we have

n f x , y n = f ( x , y ) + ( n − 1 ) f ( x , 0 )

for all x , y ∈ X . So, we have

(4) n g x y n = n f x , y n − n f ( x , 0 ) = f ( x , y ) − f ( x , 0 ) = g x ( y )

for all x , y ∈ X . Putting x 1 = x 2 = ⋯ = x n = x and y 3 = ⋯ = y n = 0 in (3), we have

n f x , y 1 + y 2 n = f ( x , y 1 ) + f ( x , y 2 ) + ( n − 2 ) f ( x , 0 )

for all x , y 1 , y 2 ∈ X . So, we have

n g x y 1 + y 2 n = n f x , y 1 + y 2 n − n f ( x , 0 ) = f ( x , y 1 ) + f ( x , y 2 ) − 2 f ( x , 0 ) = g x ( y 1 ) + g x ( y 2 )

for all x , y 1 , y 2 ∈ X . By equation (4) and the aforementioned equation, we know that g x is additive for all x ∈ X . Similarly, we also know that g y ′ is additive for all y ∈ X .□

Theorem 1

A mapping f : X × X → Y satisfies (1) if and only if it satisfies (3).

Proof

First, we assume that f satisfies (1). By [9], there exist a bi-additive mapping B : X × X → Y and two additive mappings A , A ′ : X → Y such that f ( x , y ) = B ( x , y ) + A ( x ) + A ′ ( y ) + f ( 0 , 0 ) for all x , y ∈ X . Thus, we have

n 2 f x 1 + ⋯ + x n n , y 1 + ⋯ + y n n = n 2 B 1 n ∑ i = 1 n x i , 1 n ∑ j = 1 n y j + n 2 A 1 n ∑ i = 1 n x i + n 2 A ′ 1 n ∑ j = 1 n y j + n 2 f ( 0 , 0 ) = ∑ i = 1 n ∑ j = 1 n B ( x i , y j ) + n ∑ i = 1 n A ( x i ) + n ∑ j = 1 n A ′ ( y j ) + n 2 f ( 0 , 0 ) = ∑ i = 1 n ∑ j = 1 n B ( x i , y j ) + ∑ i = 1 n ∑ j = 1 n A ( x i ) + ∑ i = 1 n ∑ j = 1 n A ′ ( y j ) + ∑ i = 1 n ∑ j = 1 n f ( 0 , 0 ) = ∑ i = 1 n ∑ j = 1 n f ( x i , y j )

for all x 1 , … , x n , y 1 , … , y n ∈ X , i.e., f satisfies (3).

Conversely, assume that f satisfies (3). Define g x , g y ′ : X → Y by g x ( y ) ≔ f ( x , y ) − f ( x , 0 ) and g y ′ ( x ) ≔ f ( x , y ) − f ( 0 , y ) for all x , y ∈ X . By Lemma 2, g x is additive for all x ∈ X and g y ′ is additive for all y ∈ X . Thus, we obtain

2 g x y + z 2 = g x ( y ) + g x ( z )

and

2 f x , y + z 2 = 2 g x y + z 2 + 2 f ( x , 0 ) = f ( x , y ) + f ( x , z )

for all x , y , z ∈ X . Similarly, we obtain

2 f x + y 2 , z = f ( x , z ) + f ( y , z )

for all x , y , z ∈ X , i.e., f satisfies (1).□

The following theorem proves the stability of equation (3) in 2-Banach spaces.

Theorem 2

Let r ∈ ( 0 , 2 ) , ε > 0 , δ , η ≥ 0 , and let f : X × X → Y be a surjection satisfying f ( x , 0 ) = 0 such that

(5) n 2 f x 1 + ⋯ + x n n , y 1 + ⋯ + y n n − ∑ i = 1 n ∑ j = 1 n f ( x i , y j ) , f ( s , t ) ≤ ε + δ ∑ i = 1 n ‖ x i ‖ r + ∑ j = 1 n ‖ y j ‖ r + η ( ‖ s ‖ + ‖ t ‖ )

for all x , x 1 , … , x n , y 1 , … , y n , s , t ∈ X . Then, there exists a unique bi-additive mapping F : X × X → Y such that

(6) ‖ f ( x , y ) − f ( 0 , y ) − F ( x , y ) , f ( s , t ) ‖ ≤ 2 n 2 − 1 [ ε + η ( ‖ s ‖ + ‖ t ‖ ) ] + n r δ n 2 − n r ( ‖ x ‖ r + 2 ‖ y ‖ r )

for all x , y , s , t ∈ X .

Proof

Let g ( x , y ) ≔ f ( x , y ) − f ( 0 , y ) for all x , y ∈ X . Then, g ( 0 , y ) = 0 for all y ∈ X . By (5), g satisfies

(7) n 2 g x 1 + ⋯ + x n n , y 1 + ⋯ + y n n − ∑ i = 1 n ∑ j = 1 n g ( x i , y j ) , f ( s , t ) ≤ 2 ε + δ ∑ i = 1 n ‖ x i ‖ r + 2 ∑ j = 1 n ‖ y j ‖ r + 2 η ( ‖ s ‖ + ‖ t ‖ )

for all x , x 1 , … , x n , y 1 , … , y n , s , t ∈ X . Putting x 1 = n k + 1 x , x 2 = ⋯ = x n = 0 , y 1 = n k + 1 y , y 2 = ⋯ = y n = 0 in (7), we gain

(8) 1 n 2 k g ( n k x , n k y ) − 1 n 2 ( k + 1 ) g ( n k + 1 x , n k + 1 y ) , f ( s , t ) ≤ 1 n 2 ( k + 1 ) [ 2 ε + δ n r ( k + 1 ) ( ‖ x ‖ r + 2 ‖ y ‖ r ) + 2 η ( ‖ s ‖ + ‖ t ‖ ) ]

for all x , y , s , t ∈ X and all k . Thus, we have

(9) 1 n 2 l g ( n l x , n l y ) − 1 n 2 m g ( n m x , n m y ) , f ( s , t ) ≤ ∑ k = l m − 1 1 n 2 ( k + 1 ) [ 2 ε + δ n r ( k + 1 ) ( ‖ x ‖ r + 2 ‖ y ‖ r ) + 2 η ( ‖ s ‖ + ‖ t ‖ ) ]

for all integers l , m ( 0 ≤ l < m ) and all x , y , s , t ∈ X . By (9), the sequence { 1 n 2 k g ( n k x , n k y ) } is a Cauchy sequence for each x , y ∈ X . Since Y is complete, the sequence { 1 n 2 k g ( n k x , n k y ) } converges for each x , y ∈ X .

Define F : X × X → Y by:

(10) F ( x , y ) ≔ lim k → ∞ 1 n 2 k g ( n k x , n k y )

for all x , y ∈ X . By (7), we have

1 n 2 k n 2 g n k − 1 ∑ i = 1 n x i , n k − 1 ∑ j = 1 n y j − ∑ i = 1 n ∑ j = 1 n g ( n k x i , n k y j ) , f ( s , t ) ≤ 1 n 2 k 2 ε + n k r δ ∑ i = 1 n ‖ x i ‖ r + 2 ∑ j = 1 n ‖ y j ‖ r + 2 η ( ‖ s ‖ + ‖ t ‖ )

for all x 1 , … , x n , y 1 , … , y n , s , t ∈ X and all k . Letting k → ∞ in the aforementioned inequality, we obtain that F satisfies (3). By Theorem 1, F is a bi-Jensen mapping. Setting l = 0 and taking m → ∞ in (9), one can obtain inequality (6).

Define G x , G y ′ : X → Y by G x ( y ) ≔ F ( x , y ) − F ( x , 0 ) and G y ′ ( x ) ≔ F ( x , y ) − F ( 0 , y ) for all x , y ∈ X . By Lemma 2, G x is additive for all x ∈ X and G y ′ is additive for all y ∈ X . Since F ( x , 0 ) = F ( 0 , y ) = 0 for all x , y ∈ X , we have G x ( y ) = G y ′ ( x ) = F ( x , y ) for all x , y ∈ X . Hence, F is bi-additive.

Let G : X × X → Y be another bi-additive mapping satisfying (6). Then, we have

‖ F ( x , y ) − G ( x , y ) , f ( s , t ) ‖ = lim k → ∞ 1 n 2 k ∥ f ( n k x , n k y ) − f ( 0 , n k y ) − G ( n k x , n k y ) , f ( s , t ) ∥ ≤ lim k → ∞ 1 n 2 k 2 n 2 − 1 [ ε + η ( ‖ s ‖ + ‖ t ‖ ) ] + n r ( k + 1 ) δ n 2 − n r ( ‖ x ‖ r + 2 ‖ y ‖ r ) = 0

for all x , y , s , t ∈ X . So F = G .□

In [18–20], one can find the concept of quasi-Banach spaces.

Definition 5

Let X be a real vector space. A quasi-norm is a real-valued function on X satisfying the following:

‖ x ‖ ≥ 0 for all x ∈ X and ‖ x ‖ = 0 if and only if x = 0 .
‖ λ x ‖ = ∣ λ ∣ ‖ x ‖ for all λ ∈ R and all x ∈ X .
There is a constant K ≥ 1 such that ‖ x + y ‖ ≤ K ( ‖ x ‖ + ‖ y ‖ ) for all x , y ∈ X .

The pair ( X , ‖ ⋅ ‖ ) is called a quasi-normed space if ‖ ⋅ ‖ is a quasi-norm on X . The smallest possible K is called the modulus of concavity of ‖ ⋅ ‖ . A quasi-Banach space is a complete quasi-normed space. A quasi-norm ‖ ⋅ ‖ is called a p-norm ( 0 < p ≤ 1 ) if

‖ x + y ‖ p ≤ ‖ x ‖ p + ‖ y ‖ p

for all x , y ∈ X . In this case, a quasi-Banach space is called a p-Banach space.

From now on, assume that X is a quasi-normed space with quasi-norm ‖ ⋅ ‖ and that Y is a p -Banach space with p -norm ‖ ⋅ ‖ Y . Let K be the modulus of concavity of ‖ ⋅ ‖ Y .

We will use the following lemma in the proof of the next theorem.

Lemma 3

[21] Let 0 ≤ p ≤ 1 and let x 1 , x 2 , … , x n be non-negative real numbers. Then,

( x 1 + x 2 + ⋯ + x n ) p ≤ x 1 p + x 2 p + ⋯ + x n p .

The following theorem proves the stability of equation (3) in quasi-Banach spaces.

Theorem 3

Let r ∈ ( 0 , 2 ) , ε > 0 , δ ≥ 0 and let f : X × X → Y be a mapping satisfying f ( x , 0 ) = 0 such that

(11) n 2 f x 1 + ⋯ + x n n , y 1 + ⋯ + y n n − ∑ i = 1 n ∑ j = 1 n f ( x i , y j ) Y ≤ ε + δ ∑ i = 1 n ‖ x i ‖ r + ∑ j = 1 n ‖ y j ‖ r

for all x , x 1 , … , x n , y 1 , … , y n ∈ X . Then, there exists a unique bi-additive mapping F : X × X → Y such that

(12) ‖ f ( x , y ) − f ( 0 , y ) − F ( x , y ) ‖ Y ≤ 2 ε p n 2 p − 1 + n p r δ p n 2 p − n p r ( ‖ x ‖ p r + 2 ‖ y ‖ p r ) 1 p

for all x , y ∈ X .

Proof

Letting x 1 = n k + 1 x , x 2 = ⋯ = x n = 0 , y 1 = n k + 1 y , y 2 = ⋯ = y n = 0 in (11), we gain

1 n 2 k f ( n k x , n k y ) − 1 n 2 ( k + 1 ) f ( n k + 1 x , n k + 1 y ) − n − 1 n 2 ( k + 1 ) f ( 0 , n k + 1 y ) Y ≤ 1 n 2 ( k + 1 ) [ ε + δ n r ( k + 1 ) ( ‖ x ‖ r + ‖ y ‖ r ) ]

for all x , y ∈ X and all k . Putting x = 0 in the aforementioned inequality, we obtain

1 n 2 k f ( 0 , n k y ) − n n 2 ( k + 1 ) f ( 0 , n k + 1 y ) Y ≤ 1 n 2 ( k + 1 ) [ ε + δ n r ( k + 1 ) ‖ y ‖ r ]

for all y ∈ X and all k . By the aforementioned two inequalities, we have

(13) 1 n 2 k [ f ( n k x , n k y ) − f ( 0 , n k y ) ] − 1 n 2 ( k + 1 ) [ f ( n k + 1 x , n k + 1 y ) − f ( 0 , n k + 1 y ) ] Y p ≤ 1 n 2 p ( k + 1 ) [ 2 ε p + δ p n p r ( k + 1 ) ( ‖ x ‖ p r + 2 ‖ y ‖ p r ) ]

for all x , y ∈ X and all k . Thus, we have

(14) 1 n 2 l [ f ( n l x , n l y ) − f ( 0 , n l y ) ] − 1 n 2 m [ f ( n m x , n m y ) − f ( 0 , n m y ) ] Y p ≤ ∑ k = l m − 1 1 n 2 p ( k + 1 ) [ 2 ε p + δ p n p r ( k + 1 ) ( ‖ x ‖ p r + 2 ‖ y ‖ p r ) ]

for all integers l , m ( 0 ≤ l < m ) and all x , y ∈ X . By (14), the sequence { 1 n 2 k [ f ( n k x , n k y ) − f ( 0 , n k y ) ] } is a Cauchy sequence for all x , y ∈ X . Since Y is complete, the sequence { 1 n 2 k [ f ( n k x , n k y ) − f ( 0 , n k y ) ] } converges for all x , y ∈ X .

Define F : X × X → Y by:

F ( x , y ) ≔ lim k → ∞ 1 n 2 k [ f ( n k x , n k y ) − f ( 0 , n k y ) ]

for all x , y ∈ X . Setting x 1 = ⋯ = x n = 0 in (11), we gain

n 2 f 0 , 1 n ∑ j = 1 n y j − n ∑ j = 1 n f ( 0 , y j ) Y ≤ ε + δ ∑ j = 1 n ‖ y j ‖ r

for all y 1 , … , y n ∈ X . By (11), the aforementioned inequality and Lemma 3, we have

1 n 2 p k n 2 f n k − 1 ∑ i = 1 n x i , n k − 1 ∑ j = 1 n y j − n 2 f 0 , n k − 1 ∑ j = 1 n y j − ∑ i = 1 n ∑ j = 1 n [ f ( n k x i , n k y j ) − f ( 0 , n k y j ) ] Y p ≤ 1 n 2 p k 2 ε p + n k r p δ p ∑ i = 1 n ‖ x i ‖ p r + 2 ∑ j = 1 n ‖ y j ‖ p r = 2 ε p 1 n 2 p k + n ( r − 2 ) p k δ p ∑ i = 1 n ‖ x i ‖ p r + 2 ∑ j = 1 n ‖ y j ‖ p r

for all x 1 , … , x n , y 1 , … , y n ∈ X and all k . Letting k → ∞ in the aforementioned inequality, we obtain that F satisfies (3).

To prove the uniqueness of F , let G : X × X → Y be another bi-additive mapping satisfying (12). Then, we have

‖ F ( x , y ) − G ( x , y ) ‖ Y = lim k → ∞ 1 n 2 k ∥ f ( n k x , n k y ) − f ( 0 , n k y ) − G ( n k x , n k y ) ∥ Y ≤ lim k → ∞ 1 n 2 k 2 ε p n 2 p − 1 + n p r ( k + 1 ) δ p n 2 p − n p r ( ‖ x ‖ p r + 2 ‖ y ‖ p r ) 1 p = 0

for all x , y ∈ X . So F = G .□

Taking n = 2 and δ = 0 in Theorem 3, we obtain the following corollary. The result coincides with the one of Corollary 4 in [22].

Corollary 1

Let ε > 0 be fixed. Suppose that f : X × X → Y be a mapping satisfying f ( x , 0 ) = 0 such that

4 f x + y 2 , z + w 2 − f ( x , z ) − f ( x , w ) − f ( y , z ) − f ( y , w ) Y ≤ ε

for all x , y , z , w ∈ X . Then, there exists a unique bi-additive mapping F : X × X → Y satisfying

‖ f ( x , y ) − f ( 0 , y ) − F ( x , y ) ‖ Y ≤ ε 2 4 p − 1 1 p

for all x , y ∈ X .

3 Conclusion

We demonstrated the stability of the multi-variable bi-Jensen functional equation (3) as the duplicative fusion equation of the multi-variable Jensen functional equation:

n f x 1 + ⋯ + x n n = f ( x 1 ) + ⋯ + f ( x n ) .

Acknowledgement

The authors would like to thank the handling editor and the referees for their helpful comments and suggestions.

Funding information: This research received no external funding.
Author contributions: All authors contributed equally to the writing of this article. All authors have accepted responsibility for entire content of the manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to the article as no datasets were generated or analyzed during this study.
Ethical approval: The conducted research is not related to either human or animal use.

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Received: 2022-12-08

Revised: 2023-04-28

Accepted: 2023-07-19

Published Online: 2023-12-20

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2023-0105

Keywords for this article

linear 2-normed space; quasi-normed space; multi-variable bi-Jensen-type mapping

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