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Additive functional inequalities in 2-Banach spaces

Abstract

We prove the Hyers-Ulam stability of the Cauchy functional inequality and theCauchy-Jensen functional inequality in 2-Banach spaces.

Moreover, we prove the superstability of the Cauchy functional inequality and theCauchy-Jensen functional inequality in 2-Banach spaces under someconditions.

MSC: 39B82, 39B52, 39B62, 46B99, 46A19.

1 Introduction and preliminaries

In 1940, Ulam [1] suggested the stability problem of functional equations concerning thestability of group homomorphisms as follows: Let(G,∘)be a group and let(H,⋆,d)be a metric group with the metricd(⋅,⋅). Givenε>0, does there exist aδ=δ(ε)>0such that if a mappingf:G→Hsatisfies the inequality

d ( f ( x ∘ y ) , f ( x ) ⋆ f ( y ) ) <δ

for all x,y∈G , then a homomorphism F:G→H exists with

d ( f ( x ) , F ( x ) ) <ε

for all x∈G ?

In 1941, Hyers [2] gave a first (partial) affirmative answer to the question of Ulam forBanach spaces. Thereafter, we call that type the Hyers-Ulam stability.

Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. Ageneralization of the Rassias theorem was obtained by Gǎvruta [5] by replacing the unbounded Cauchy difference by a general controlfunction.

Gähler [6, 7] introduced the concept of linear 2-normed spaces.

Definition 1.1 Let be a real linear space with dimX>1, and let ∥⋅,⋅∥:X×X→ R ≥ 0 be a function satisfying the following properties:

  1. (a)

    ∥x,y∥=0 if and only if x and y are linearly dependent,

  2. (b)

    ∥x,y∥=∥y,x∥,

  3. (c)

    ∥αx,y∥=|α|∥x,y∥,

  4. (d)

    ∥x,y+z∥≤∥x,y∥+∥x,z∥

for all x,y,z∈X and α∈R. Then the function ∥⋅,⋅∥ is called 2-norm on and the pair (X,∥⋅,⋅∥) is called a linear 2-normed space.Sometimes condition (d) is called the triangle inequality.

See [8] for examples and properties of linear 2-normed spaces.

White [9, 10] introduced the concept of 2-Banach spaces. In order to definecompleteness, the concepts of Cauchy sequences and convergence are required.

Definition 1.2 A sequence { x n } in a linear 2-normed space is called a Cauchy sequence if

lim m , n → ∞ ∥ x n − x m ,y∥=0

for all y∈X.

Definition 1.3 A sequence { x n } in a linear 2-normed space is called a convergent sequenceif there is an x∈X such that

lim n → ∞ ∥ x n −x,y∥=0

for all y∈X. If { x n } converges to x, write x n →x as n→∞ and call x the limit of{ x n }. In this case, we also write lim n → ∞ x n =x.

The triangle inequality implies the following lemma.

Lemma 1.4[11]

For a convergent sequence{ x n }in a linear 2-normed space,

lim n → ∞ ∥ x n ,y∥= ∥ lim n → ∞ x n , y ∥

for ally∈X.

Definition 1.5 A linear 2-normed space, in which every Cauchy sequence is aconvergent sequence, is called a 2-Banach space.

Eskandani and GÇŽvruta [12] proved the Hyers-Ulam stability of a functional equation in 2-Banachspaces.

In [13], Gilányi showed that if f satisfies the functionalinequality

∥ 2 f ( x ) + 2 f ( y ) − f ( x y − 1 ) ∥ ≤ ∥ f ( x y ) ∥ ,
(1.1)

then f satisfies the Jordan-von Neumann functional equation

2f(x)+2f(y)=f(xy)+f ( x y − 1 ) .

See also [14]. Gilányi [15] and Fechner [16] proved the Hyers-Ulam stability of functional inequality (1.1).

Park et al.[17] proved the Hyers-Ulam stability of the following functional inequalities:

∥ f ( x ) + f ( y ) + f ( z ) ∥ ≤ ∥ f ( x + y + z ) ∥ ,
(1.2)
∥ f ( x ) + f ( y ) + 2 f ( z ) ∥ ≤ ∥ 2 f ( x + y 2 + z ) ∥ .
(1.3)

In this paper, we prove the Hyers-Ulam stability of Cauchy functional inequality(1.2) and Cauchy-Jensen functional inequality (1.3) in 2-Banach spaces.

Moreover, we prove the superstability of Cauchy functional inequality (1.2) andCauchy-Jensen functional inequality (1.3) in 2-Banach spaces under someconditions.

Throughout this paper, let be a normed linear space, and let be a 2-Banach space.

2 Hyers-Ulam stability of Cauchy functional inequality (1.2) in 2-Banachspaces

In this section, we prove the Hyers-Ulam stability of Cauchy functional inequality(1.2) in 2-Banach spaces.

Proposition 2.1 Let f:X→Y be a mapping satisfying

∥ f ( x ) + f ( y ) + f ( z ) , w ∥ ≤ ∥ f ( x + y + z ) , w ∥
(2.1)

for allx,y,z∈Xand allw∈Y. Then the mappingf:X→Yis additive.

Proof Letting x=y=z=0 in (2.1), we get 3∥f(0),w∥≤∥f(0),w∥ and so ∥f(0),w∥=0 for all w∈Y. Hence f(0)=0.

Letting y=−x and z=0 in (2.1), we get ∥f(x)+f(−x),w∥≤∥f(0),w∥=0 and so ∥f(x)+f(−x),w∥=0 for all x∈X and all w∈Y. Hence f(x)+f(−x)=0 for all x∈X.

Letting z=−x−y in (2.1), we get

∥ f ( x ) + f ( y ) + f ( − x − y ) , w ∥ ≤ ∥ f ( 0 ) , w ∥ =0

and so

∥ f ( x ) + f ( y ) + f ( − x − y ) , w ∥ =0

for all x,y∈X and all w∈Y. Hence

0=f(x)+f(y)+f(−x−y)=f(x)+f(y)−f(x+y)

for all x,y∈X. So, f:X→Y is additive. □

Theorem 2.2 Letθ∈[0,∞), p,q,r∈(0,∞)withp+q+r<1, and letf:X→Ybe a mapping satisfying

∥ f ( x ) + f ( y ) + f ( z ) , w ∥ ≤ ∥ f ( x + y + z ) , w ∥ +θ ∥ x ∥ p ∥ y ∥ q ∥ z ∥ r ∥w∥
(2.2)

for allx,y,z∈Xand allw∈Y. Then there is a unique additive mappingA:X→Ysuch that

∥ f ( x ) − A ( x ) , w ∥ ≤ 2 r θ 2 − 2 p + q + r ∥ x ∥ p + q + r ∥w∥
(2.3)

for allx∈Xand allw∈Y.

Proof Letting x=y=z=0 in (2.2), we get 3∥f(0),w∥≤∥f(0),w∥ and so ∥f(0),w∥=0 for all w∈Y. Hence f(0)=0.

Letting y=−x and z=0 in (2.2), we get ∥f(x)+f(−x),w∥≤∥f(0),w∥=0 and so ∥f(x)+f(−x),w∥=0 for all x∈X and all w∈Y. Hence f(x)+f(−x)=0 for all x∈X.

Putting y=x and z=−2x in (2.2), we get

∥ f ( 2 x ) − 2 f ( x ) , w ∥ ≤ ∥ f ( 0 ) , w ∥ + 2 r θ ∥ x ∥ p + q + r ∥w∥= 2 r θ ∥ x ∥ p + q + r ∥w∥
(2.4)

for all x∈X and all w∈Y. So, we get

∥ f ( x ) − 1 2 f ( 2 x ) , w ∥ ≤ 2 r θ 2 ∥ x ∥ p + q + r ∥w∥
(2.5)

for all x∈X and all w∈Y. Replacing x by 2 j x in (2.5) and dividing by 2 j , we obtain

∥ 1 2 j f ( 2 j x ) − 1 2 j + 1 f ( 2 j + 1 x ) , w ∥ ≤ 2 ( p + q + r − 1 ) j + r − 1 θ ∥ x ∥ p + q + r ∥w∥

for all x∈X, all w∈Y and all integers j≥0. For all integers l, m with0≤l<m, we get

∥ 1 2 l f ( 2 l x ) − 1 2 m f ( 2 m x ) , w ∥ ≤ ∑ j = l m − 1 2 ( p + q + r − 1 ) j + r − 1 θ ∥ x ∥ p + q + r ∥w∥
(2.6)

for all x∈X and all w∈Y. So, we get

lim l → ∞ ∥ 1 2 l f ( 2 l x ) − 1 2 m f ( 2 m x ) , w ∥ =0

for all x∈X and all w∈Y. Thus the sequence { 1 2 j f( 2 j x)} is a Cauchy sequence in for each x∈X. Since is a 2-Banach space, the sequence { 1 2 j f( 2 j x)} converges for each x∈X. So, one can define the mappingA:X→Y by

A(x):= lim j → ∞ 1 2 j f ( 2 j x )

for all x∈X. That is,

lim j → ∞ ∥ 1 2 j f ( 2 j x ) − A ( x ) , w ∥ =0

for all x∈X and all w∈Y.

By (2.2), we get

lim j → ∞ ∥ 1 2 j ( f ( 2 j x ) + f ( 2 j y ) + f ( 2 j z ) ) , w ∥ ≤ lim j → ∞ ( 1 2 j ∥ f ( 2 j x + 2 j y + 2 j z ) , w ∥ + 2 ( p + q + r ) j 2 j θ ∥ x ∥ p ∥ y ∥ q ∥ z ∥ r ∥ w ∥ ) ≤ lim j → ∞ 1 2 j ∥ f ( 2 j x + 2 j y + 2 j z ) , w ∥

for all x,y,z∈X and all w∈Y. So,

∥ A ( x ) + A ( y ) + A ( z ) , w ∥ ≤ ∥ A ( x + y + z ) , w ∥

for all x,y,z∈X and all w∈Y. By Proposition 2.1, A:X→Y is additive.

By Lemma 1.4 and (2.6), we have

∥ f ( x ) − A ( x ) , w ∥ = lim m → ∞ ∥ f ( x ) − 1 2 m f ( 2 m x ) , w ∥ ≤ 2 r θ 2 − 2 p + q + r ∥ x ∥ p + q + r ∥w∥

for all x∈X and all w∈Y.

Now, let B:X→Y be another additive mapping satisfying (2.3). Then wehave

∥ A ( x ) − B ( x ) , w ∥ = 1 2 j ∥ A ( 2 j x ) − B ( 2 j x ) , w ∥ ≤ 1 2 j [ ∥ A ( 2 j x ) − f ( 2 j x ) , w ∥ + ∥ f ( 2 j x ) − B ( 2 j x ) , w ∥ ] ≤ 2 ⋅ 2 r θ 2 − 2 p + q + r ∥ x ∥ p + q + r ∥ w ∥ ⋅ 2 ( p + q + r ) j 2 j ,

which tends to zero as j→∞ for all x∈X and all w∈Y. By Definition 1.1, we can conclude thatA(x)=B(x) for all x∈X. This proves the uniqueness ofA. □

Theorem 2.3 Letθ∈[0,∞), p,q,r∈(0,∞)withp+q+r>1, and letf:X→Ybe a mapping satisfying (2.2). Then there is a unique additivemappingA:X→Ysuch that

∥ f ( x ) − A ( x ) , w ∥ ≤ 2 r θ 2 p + q + r − 2 ∥ x ∥ p + q + r ∥w∥

for allx∈Xand allw∈Y.

Proof It follows from (2.4) that

∥ f ( x ) − 2 f ( x 2 ) , w ∥ ≤ θ 2 p + q ∥ x ∥ p + q + r ∥w∥
(2.7)

for all x∈X and all w∈Y. Replacing x by x 2 j in (2.7) and multiplying by 2 j , we obtain

∥ 2 j f ( x 2 j ) − 2 j + 1 f ( x 2 j + 1 ) , w ∥ ≤ 2 j θ 2 p + q ⋅ 2 ( p + q + r ) j ∥ x ∥ p + q + r ∥w∥

for all x∈X and all w∈Y and all integers j≥0. For all integers l, m with0≤l<m, we get

∥ 2 l f ( x 2 l ) − 2 m f ( x 2 m ) , w ∥ ≤ ∑ j = l m − 1 2 j θ 2 p + q ⋅ 2 ( p + q + r ) j ∥ x ∥ p + q + r ∥w∥

for all x∈X and all w∈Y. So, we get

lim l → ∞ ∥ 2 l f ( x 2 l ) − 2 m f ( x 2 m ) , w ∥ =0

for all x∈X and all w∈Y. Thus the sequence { 2 j f( x 2 j )} is a Cauchy sequence in . Since is a 2-Banach space, the sequence{ 2 j f( x 2 j )} converges. So, one can define the mappingA:X→Y by

A(x):= lim j → ∞ 2 j f ( x 2 j )

for all x∈X. That is,

lim j → ∞ ∥ 2 j f ( x 2 j ) − A ( x ) , w ∥ =0

for all x∈X and all w∈Y.

The further part of the proof is similar to the proof of Theorem2.2. □

Now we prove the superstability of the Cauchy functional inequality in 2-Banachspaces.

Theorem 2.4 Letθ∈[0,∞), p,q,r,t∈(0,∞)witht≠1, and letf:X→Ybe a mapping satisfying

∥ f ( x ) + f ( y ) + f ( z ) , w ∥ ≤ ∥ f ( x + y + z ) , w ∥ +θ ∥ x ∥ p ∥ y ∥ q ∥ z ∥ r ∥ w ∥ t
(2.8)

for allx,y,z∈Xand allw∈Y. Thenf:X→Yis an additive mapping.

Proof Replacing w by sw in (2.8) fors∈R∖{0}, we get

∥ f ( x ) + f ( y ) + f ( z ) , s w ∥ ≤ ∥ f ( x + y + z ) , s w ∥ +θ ∥ x ∥ p ∥ y ∥ q ∥ z ∥ r ∥ s w ∥ t

and so

∥ f ( x ) + f ( y ) + f ( z ) , w ∥ ≤ ∥ f ( x + y + z ) , w ∥ +θ ∥ x ∥ p ∥ y ∥ q ∥ z ∥ r ∥ w ∥ t | s | t | s |
(2.9)

for all x,y,z∈X, all w∈Y and all s∈R∖{0}.

If t>1, then the right-hand side of (2.9) tends to∥f(x+y+z),w∥ as s→0.

If t<1, then the right-hand side of (2.9) tends to∥f(x+y+z),w∥ as s→+∞.

Thus

∥ f ( x ) + f ( y ) + f ( z ) , w ∥ ≤ ∥ f ( x + y + z ) , w ∥

for all x,y,z∈X and all w∈Y. By Proposition 2.1, f:X→Y is additive. □

3 Hyers-Ulam stability of Cauchy-Jensen functional inequality (1.3) in 2-Banachspaces

In this section, we prove the Hyers-Ulam stability of Cauchy-Jensen functionalinequality (1.3) in 2-Banach spaces.

Proposition 3.1 Let f:X→Y be a mapping satisfying

∥ f ( x ) + f ( y ) + 2 f ( z ) , w ∥ ≤ ∥ 2 f ( x + y 2 + z ) , w ∥
(3.1)

for allx,y,z∈Xand allw∈Y. Then the mappingf:X→Yis additive.

Proof Letting x=y=z=0 in (3.1), we get 4∥f(0),w∥≤2∥f(0),w∥ and so ∥f(0),w∥=0 for all w∈Y. Hence f(0)=0.

Letting y=−x and z=0 in (3.1), we get ∥f(x)+f(−x),w∥≤2∥f(0),w∥=0 and so ∥f(x)+f(−x),w∥=0 for all x∈X and all w∈Y. Hence f(x)+f(−x)=0 for all x∈X.

Letting z=− x + y 2 in (3.1), we get

∥ f ( x ) + f ( y ) + 2 f ( − x + y 2 ) , w ∥ ≤2 ∥ f ( 0 ) , w ∥ =0

and so

∥ f ( x ) + f ( y ) + 2 f ( − x + y 2 ) , w ∥ =0

for all x,y∈X and all w∈Y. Hence

0=f(x)+f(y)+2f ( − x + y 2 ) =f(x)+f(y)−2f ( x + y 2 )

for all x,y∈X. Since f(0)=0, f:X→Y is additive. □

Theorem 3.2 Letθ∈[0,∞), p,q,r∈(0,∞)withp+q+r<1, and letf:X→Ybe a mapping satisfying

∥ f ( x ) + f ( y ) + 2 f ( z ) , w ∥ ≤ ∥ 2 f ( x + y 2 + z ) , w ∥ +θ ∥ x ∥ p ∥ y ∥ q ∥ z ∥ r ∥w∥
(3.2)

for allx,y,z∈Xand allw∈Y. Then there is a unique additive mappingA:X→Ysuch that

∥ f ( x ) − A ( x ) , w ∥ ≤ θ 2 − 2 p + q + r ∥ x ∥ p + q + r ∥w∥

for allx∈Xand allw∈Y.

Proof Letting x=y=z=0 in (3.2), we get 4∥f(0),w∥≤2∥f(0),w∥ and so ∥f(0),w∥=0 for all w∈Y. Hence f(0)=0.

Letting y=−x and z=0 in (3.2), we get ∥f(x)+f(−x),w∥≤2∥f(0),w∥=0 and so ∥f(x)+f(−x),w∥=0 for all x∈X and all w∈Y. Hence f(x)+f(−x)=0 for all x∈X.

Letting y=x and z=−x in (3.2), we get

∥ 2 f ( x ) − f ( 2 x ) , w ∥ ≤ ∥ 2 f ( 0 ) , w ∥ +θ ∥ x ∥ p + q + r ∥w∥=θ ∥ x ∥ p + q + r ∥w∥
(3.3)

for all x∈X and all w∈Y. Replacing x by 2 j x in (3.3) and dividing by 3 j , we obtain

∥ 1 2 j f ( 2 j x ) − 1 2 j + 1 f ( 2 j + 1 x ) , w ∥ ≤ 2 ( p + q + r ) j 2 ⋅ 2 j θ ∥ x ∥ p + q + r ∥w∥

for all x∈X and all w∈Y and all integers j≥0. For all integers l, m with0≤l<m, we get

∥ 1 2 l f ( 2 l x ) − 1 2 m f ( 2 m x ) , w ∥ ≤ ∑ j = l m − 1 2 ( p + q + r ) j 2 ⋅ 2 j θ ∥ x ∥ p + q + r ∥w∥

for all x∈X and all w∈Y. So, we get

lim l → ∞ ∥ 1 2 l f ( 2 l x ) − 1 2 m f ( 2 m x ) , w ∥ =0

for all x∈X and all w∈Y. Thus the sequence { 1 2 j f( 2 j x)} is a Cauchy sequence in for each x∈X. Since is a 2-Banach space, the sequence { 1 2 j f( 2 j x)} converges for each x∈X. So, one can define the mappingA:X→Y by

A(x):= lim j → ∞ 1 2 j f ( 2 j x ) = lim j → ∞ 1 2 j f ( 2 j x )

for all x∈X. That is,

lim j → ∞ ∥ 1 2 j f ( 2 j x ) − A ( x ) , w ∥ = lim j → ∞ ∥ 1 2 j f ( 2 j x ) − A ( x ) , w ∥ =0

for all x∈X and all w∈Y.

The further part of the proof is similar to the proof of Theorem2.2. □

Theorem 3.3 Letθ∈[0,∞), p,q,r∈(0,∞)withp+q+r>1, and letf:X→Ybe a mapping satisfying (3.2). Then there is a unique additivemappingA:X→Ysuch that

∥ f ( x ) − A ( x ) , w ∥ ≤ θ 2 p + q + r − 2 ∥ x ∥ p + q + r ∥w∥

for allx∈Xand allw∈Y.

Proof It follows from (3.3) that

∥ f ( x ) − 2 f ( x 2 ) , w ∥ ≤ 1 2 p + q + r θ ∥ x ∥ p + q + r ∥w∥
(3.4)

for all x∈X and all w∈Y. Replacing x by x 2 j in (3.4) and multiplying by 2 j , we obtain

∥ 2 j f ( x 2 j ) − 2 j + 1 f ( x 2 j + 1 ) , w ∥ ≤ 2 j 2 ( p + q + r ) ( j + 1 ) θ ∥ x ∥ p + q + r ∥w∥

for all x∈X and all w∈Y and all integers j≥0. For all integers l, m with0≤l<m, we get

∥ 2 l f ( x 2 l ) − 2 m f ( x 2 m ) , w ∥ ≤ ∑ j = l m − 1 2 j 2 ( p + q + r ) ( j + 1 ) θ ∥ x ∥ p + q + r ∥w∥

for all x∈X and all w∈Y. So, we get

lim l → ∞ ∥ 2 l f ( x 2 l ) − 2 m f ( x 2 m ) , w ∥ =0

for all x∈X and all w∈Y. Thus the sequence { 2 j f( x 2 j )} is a Cauchy sequence in for each x∈X. Since is a 2-Banach space, the sequence { 2 j f( x 2 j )} converges for each x∈X. So, one can define the mappingA:X→Y by

A(x):= lim j → ∞ 2 j f ( x 2 j )

for all x∈X. That is,

lim j → ∞ ∥ 2 j f ( x 2 j ) − A ( x ) , w ∥ =0

for all x∈X and all w∈Y.

The further part of the proof is similar to the proof of Theorem2.2. □

Now we prove the superstability of the Jensen functional equation in 2-Banachspaces.

Theorem 3.4 Letθ∈[0,∞), p,q,r,t∈(0,∞)witht≠1, and letf:X→Ybe a mapping satisfying

∥ f ( x ) + f ( y ) + 2 f ( z ) , w ∥ ≤ ∥ 2 f ( x + y 2 + z ) , w ∥ +θ ∥ x ∥ p ∥ y ∥ q ∥ z ∥ r ∥ w ∥ t
(3.5)

for allx,y,z∈Xand allw∈Y. Thenf:X→Yis an additive mapping.

Proof Replacing w by sw in (3.5) fors∈R∖{0}, we get

∥ f ( x ) + f ( y ) + 2 f ( z ) , s w ∥ ≤ ∥ 2 f ( x + y 2 + z ) , s w ∥ +θ ∥ x ∥ p ∥ y ∥ q ∥ z ∥ r ∥ s w ∥ t

and so

∥ f ( x ) + f ( y ) + 2 f ( z ) , w ∥ ≤ ∥ 2 f ( x + y 2 + z ) , w ∥ +θ ∥ x ∥ p ∥ y ∥ q ∥ z ∥ r ∥ w ∥ t | s | t | s |

for all x,y,z∈X, all w∈Y and all s∈R∖{0}.

The rest of the proof is similar to the proof of Theorem 2.4. □

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CP conceived of the study, participated in its design and coordination, drafted themanuscript, participated in the sequence alignment, and read and approved the finalmanuscript.

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Park, C. Additive functional inequalities in 2-Banach spaces. J Inequal Appl 2013, 447 (2013). https://doi.org/10.1186/1029-242X-2013-447

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  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-447

Keywords