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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:GHsatisfies the inequality

d ( f ( x y ) , f ( x ) f ( y ) ) <δ

for all x,yG , then a homomorphism F:GH exists with

d ( f ( x ) , F ( x ) ) <ε

for all xG ?

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+zx,y+x,z

for all x,y,zX and αR. 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 yX.

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

lim n x n x,y=0

for all yX. 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 allyX.

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

Eskandani and Gǎvruta [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:XY be a mapping satisfying

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

for allx,y,zXand allwY. Then the mappingf:XYis additive.

Proof Letting x=y=z=0 in (2.1), we get 3f(0),wf(0),w and so f(0),w=0 for all wY. Hence f(0)=0.

Letting y=x and z=0 in (2.1), we get f(x)+f(x),wf(0),w=0 and so f(x)+f(x),w=0 for all xX and all wY. Hence f(x)+f(x)=0 for all xX.

Letting z=xy 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,yX and all wY. Hence

0=f(x)+f(y)+f(xy)=f(x)+f(y)f(x+y)

for all x,yX. So, f:XY is additive. □

Theorem 2.2 Letθ[0,), p,q,r(0,)withp+q+r<1, and letf:XYbe 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,zXand allwY. Then there is a unique additive mappingA:XYsuch that

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

for allxXand allwY.

Proof Letting x=y=z=0 in (2.2), we get 3f(0),wf(0),w and so f(0),w=0 for all wY. Hence f(0)=0.

Letting y=x and z=0 in (2.2), we get f(x)+f(x),wf(0),w=0 and so f(x)+f(x),w=0 for all xX and all wY. Hence f(x)+f(x)=0 for all xX.

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 xX and all wY. So, we get

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

for all xX and all wY. 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 xX, all wY and all integers j0. For all integers l, m with0l<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 xX and all wY. So, we get

lim l 1 2 l f ( 2 l x ) 1 2 m f ( 2 m x ) , w =0

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

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

for all xX. That is,

lim j 1 2 j f ( 2 j x ) A ( x ) , w =0

for all xX and all wY.

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,zX and all wY. So,

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

for all x,y,zX and all wY. By Proposition 2.1, A:XY 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 xX and all wY.

Now, let B:XY 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 xX and all wY. By Definition 1.1, we can conclude thatA(x)=B(x) for all xX. This proves the uniqueness ofA. □

Theorem 2.3 Letθ[0,), p,q,r(0,)withp+q+r>1, and letf:XYbe a mapping satisfying (2.2). Then there is a unique additivemappingA:XYsuch that

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

for allxXand allwY.

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 xX and all wY. 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 xX and all wY and all integers j0. For all integers l, m with0l<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 xX and all wY. So, we get

lim l 2 l f ( x 2 l ) 2 m f ( x 2 m ) , w =0

for all xX and all wY. 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:XY by

A(x):= lim j 2 j f ( x 2 j )

for all xX. That is,

lim j 2 j f ( x 2 j ) A ( x ) , w =0

for all xX and all wY.

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,)witht1, and letf:XYbe 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,zXand allwY. Thenf:XYis an additive mapping.

Proof Replacing w by sw in (2.8) forsR{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,zX, all wY and all sR{0}.

If t>1, then the right-hand side of (2.9) tends tof(x+y+z),w as s0.

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

Thus

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

for all x,y,zX and all wY. By Proposition 2.1, f:XY 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:XY be a mapping satisfying

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

for allx,y,zXand allwY. Then the mappingf:XYis additive.

Proof Letting x=y=z=0 in (3.1), we get 4f(0),w2f(0),w and so f(0),w=0 for all wY. Hence f(0)=0.

Letting y=x and z=0 in (3.1), we get f(x)+f(x),w2f(0),w=0 and so f(x)+f(x),w=0 for all xX and all wY. Hence f(x)+f(x)=0 for all xX.

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,yX and all wY. Hence

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

for all x,yX. Since f(0)=0, f:XY is additive. □

Theorem 3.2 Letθ[0,), p,q,r(0,)withp+q+r<1, and letf:XYbe 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,zXand allwY. Then there is a unique additive mappingA:XYsuch that

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

for allxXand allwY.

Proof Letting x=y=z=0 in (3.2), we get 4f(0),w2f(0),w and so f(0),w=0 for all wY. Hence f(0)=0.

Letting y=x and z=0 in (3.2), we get f(x)+f(x),w2f(0),w=0 and so f(x)+f(x),w=0 for all xX and all wY. Hence f(x)+f(x)=0 for all xX.

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 xX and all wY. 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 xX and all wY and all integers j0. For all integers l, m with0l<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 xX and all wY. So, we get

lim l 1 2 l f ( 2 l x ) 1 2 m f ( 2 m x ) , w =0

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

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

for all xX. 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 xX and all wY.

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:XYbe a mapping satisfying (3.2). Then there is a unique additivemappingA:XYsuch that

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

for allxXand allwY.

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 xX and all wY. 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 xX and all wY and all integers j0. For all integers l, m with0l<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 xX and all wY. So, we get

lim l 2 l f ( x 2 l ) 2 m f ( x 2 m ) , w =0

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

A(x):= lim j 2 j f ( x 2 j )

for all xX. That is,

lim j 2 j f ( x 2 j ) A ( x ) , w =0

for all xX and all wY.

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,)witht1, and letf:XYbe 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,zXand allwY. Thenf:XYis an additive mapping.

Proof Replacing w by sw in (3.5) forsR{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,zX, all wY and all sR{0}.

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

References

  1. Ulam SM: A Collection of the Mathematical Problems. Interscience, New York; 1960.

    Google Scholar 

  2. Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222

    MathSciNet  Article  Google Scholar 

  3. Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64–66. 10.2969/jmsj/00210064

    MATH  Article  Google Scholar 

  4. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1

    MATH  Article  Google Scholar 

  5. Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximatelyadditive mappings. J. Math. Anal. Appl. 1994, 184: 431–436. 10.1006/jmaa.1994.1211

    MathSciNet  Article  Google Scholar 

  6. Gähler S: 2-metrische Räume und ihre topologische Struktur. Math. Nachr. 1963, 26: 115–148. 10.1002/mana.19630260109

    MATH  MathSciNet  Article  Google Scholar 

  7. Gähler S: Lineare 2-normierte Räumen. Math. Nachr. 1964, 28: 1–43. 10.1002/mana.19640280102

    MATH  MathSciNet  Article  Google Scholar 

  8. Cho Y, Lin PSC, Kim S, Misiak A: Theory of 2-Inner Product Spaces. Nova Science Publishers, New York; 2001.

    MATH  Google Scholar 

  9. White, A: 2-Banach spaces. Dissertation, St. Louis University (1968)

  10. White A: 2-Banach spaces. Math. Nachr. 1969, 42: 43–60. 10.1002/mana.19690420104

    MATH  MathSciNet  Article  Google Scholar 

  11. Park W: Approximate additive mappings in 2-Banach spaces and related topics. J. Math. Anal. Appl. 2011, 376: 193–202. 10.1016/j.jmaa.2010.10.004

    MATH  MathSciNet  Article  Google Scholar 

  12. Eskandani GZ, Gǎvruta P: Hyers-Ulam-Rassias stability of Pexiderized Cauchy functional equation in2-Banach spaces. J. Nonlinear Sci. Appl. 2012, 5: 459–465.

    MATH  MathSciNet  Google Scholar 

  13. Gilányi A: Eine zur Parallelogrammgleichung äquivalente Ungleichung. Aequ. Math. 2001, 62: 303–309. 10.1007/PL00000156

    MATH  Article  Google Scholar 

  14. Rätz J: On inequalities associated with the Jordan-von Neumann functionalequation. Aequ. Math. 2003, 66: 191–200. 10.1007/s00010-003-2684-8

    MATH  Article  Google Scholar 

  15. Gilányi A: On a problem by K. Nikodem. Math. Inequal. Appl. 2002, 5: 707–710.

    MATH  MathSciNet  Google Scholar 

  16. Fechner W: Stability of a functional inequalities associated with the Jordan-von Neumannfunctional equation. Aequ. Math. 2006, 71: 149–161. 10.1007/s00010-005-2775-9

    MATH  MathSciNet  Article  Google Scholar 

  17. Park C, Cho Y, Han M: Functional inequalities associated with Jordan-von Neumann-type additivefunctional equations. J. Inequal. Appl. 2007., 2007: Article ID 41820

    Google Scholar 

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Correspondence to Choonkil Park.

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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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Keywords

  • Hyers-Ulam stability
  • linear 2-normed space
  • additive mapping
  • additive functional inequality
  • superstability