R E S E A R C H
Open Access
Additive functional inequalities in -Banach
spaces
Choonkil Park
**Correspondence: baak@hanyang.ac.kr Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea
Abstract
We prove the Hyers-Ulam stability of the Cauchy functional inequality and the Cauchy-Jensen functional inequality in 2-Banach spaces.
Moreover, we prove the superstability of the Cauchy functional inequality and the Cauchy-Jensen functional inequality in 2-Banach spaces under some conditions. MSC: 39B82; 39B52; 39B62; 46B99; 46A19
Keywords: Hyers-Ulam stability; linear 2-normed space; additive mapping; additive functional inequality; superstability
1 Introduction and preliminaries
In , Ulam [] suggested the stability problem of functional equations concerning the stability of group homomorphisms as follows: Let (G, ◦) be a group and let (H, , d) be a
metric group with the metric d(·, ·). Given ε > , does there exist a δ = δ(ε) > such that if a mapping f:G → H satisfies the inequality
df(x◦ y), f (x) f (y)< δ
for all x, y∈G, then a homomorphism F : G → H exists with
df(x), F(x)< ε
for all x∈G?
In , Hyers [] gave a first (partial) affirmative answer to the question of Ulam for Banach spaces. Thereafter, we call that type the Hyers-Ulam stability.
Hyers’ theorem was generalized by Aoki [] for additive mappings and by Rassias [] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Gˇavruta [] by replacing the unbounded Cauchy difference by a general control function.
Gähler [, ] introduced the concept of linear -normed spaces.
Definition . LetX be a real linear space with dim X > , and let ·, · : X × X → R≥ be a function satisfying the following properties:
(a) x, y = if and only if x and y are linearly dependent, (b) x, y = y, x,
©2013 Park; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribuAttribu-tion, and reproducAttribu-tion in any medium, provided the original work is properly cited.
(c) αx, y = |α|x, y, (d) x, y + z ≤ x, y + x, z
for all x, y, z∈X and α ∈ R. Then the function ·, · is called -norm on X and the pair (X , ·, ·) is called a linear -normed space. Sometimes condition (d) is called the triangle
inequality.
See [] for examples and properties of linear -normed spaces.
White [, ] introduced the concept of -Banach spaces. In order to define complete-ness, the concepts of Cauchy sequences and convergence are required.
Definition . A sequence{xn} in a linear -normed spaceX is called a Cauchy sequence
if
lim
m,n→∞xn– xm, y =
for all y∈X .
Definition . A sequence{xn} in a linear -normed spaceX is called a convergent
se-quenceif there is an x∈X such that lim
n→∞xn– x, y =
for all y∈X . If {xn} converges to x, write xn→ x as n → ∞ and call x the limit of {xn}. In
this case, we also write limn→∞xn= x.
The triangle inequality implies the following lemma.
Lemma .[] For a convergent sequence{xn} in a linear -normed spaceX ,
lim n→∞xn, y = lim n→∞xn, y for all y∈X .
Definition . A linear -normed space, in which every Cauchy sequence is a convergent sequence, is called a -Banach space.
Eskandani and Gˇavruta [] proved the Hyers-Ulam stability of a functional equation in -Banach spaces.
In [], Gilányi showed that if f satisfies the functional inequality
f (x) + f (y) – fxy– ≤f(xy), (.)
then f satisfies the Jordan-von Neumann functional equation f (x) + f (y) = f (xy) + fxy–.
See also []. Gilányi [] and Fechner [] proved the Hyers-Ulam stability of functional inequality (.).
Park et al. [] proved the Hyers-Ulam stability of the following functional inequalities: f(x) + f (y) + f (z) ≤f(x + y + z), (.) f(x) + f (y) + f (z) ≤f x+ y + z . (.)
In this paper, we prove the Hyers-Ulam stability of Cauchy functional inequality (.) and Cauchy-Jensen functional inequality (.) in -Banach spaces.
Moreover, we prove the superstability of Cauchy functional inequality (.) and Cauchy-Jensen functional inequality (.) in -Banach spaces under some conditions.
Throughout this paper, let X be a normed linear space, and let Y be a -Banach space.
2 Hyers-Ulam stability of Cauchy functional inequality (1.2) in 2-Banach spaces
In this section, we prove the Hyers-Ulam stability of Cauchy functional inequality (.) in -Banach spaces.
Proposition . Let f :X → Y be a mapping satisfying
f(x) + f (y) + f (z), w ≤f(x + y + z), w (.)
for all x, y, z∈X and all w ∈ Y. Then the mapping f : X → Y is additive.
Proof Letting x = y = z = in (.), we get f (), w ≤ f (), w and so f (), w = for all w∈Y. Hence f () = .
Letting y = –x and z = in (.), we getf (x) + f (–x), w ≤ f (), w = and so f (x) +
f(–x), w = for all x ∈X and all w ∈ Y. Hence f (x) + f (–x) = for all x ∈ X .
Letting z = –x – y in (.), we get
f(x) + f (y) + f (–x – y), w ≤f(), w= and so
f(x) + f (y) + f (–x – y), w= for all x, y∈X and all w ∈ Y. Hence
= 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 . Let θ ∈ [, ∞), p, q, r ∈ (, ∞) with p + q + r < , and let f :X → Y be a mapping satisfying
for all x, y, z∈X and all w ∈ Y. Then there is a unique additive mapping A : X → Y such that f(x) – A(x), w ≤ rθ – p+q+rx p+q+rw (.)
for all x∈X and all w ∈ Y.
Proof Letting x = y = z = in (.), we get f (), w ≤ f (), w and so f (), w = for all w∈Y. Hence f () = .
Letting y = –x and z = in (.), we getf (x) + f (–x), w ≤ f (), w = and so f (x) +
f(–x), w = for all x ∈X and all w ∈ Y. Hence f (x) + f (–x) = for all x ∈ X .
Putting y = x and z = –x in (.), we get
f(x) – f (x), w ≤f(), w+ rθxp+q+rw = rθxp+q+rw (.) for all x∈X and all w ∈ Y. So, we get
f(x) –f(x), w ≤
rθ
x
p+q+rw (.)
for all x∈X and all w ∈ Y. Replacing x by jxin (.) and dividing by j, we obtain
jf jx– j+f j+x, w ≤ (p+q+r–)j+r–θxp+q+rw
for all x∈X , all w ∈ Y and all integers j ≥ . For all integers l, m with ≤ l < m, we get lf lx– mf mx, w ≤ m– j=l (p+q+r–)j+r–θxp+q+rw (.)
for all x∈X and all w ∈ Y. So, we get
lim l→∞ lf lx– mf mx, w =
for all x∈X and all w ∈ Y. Thus the sequence {
jf(jx)} is a Cauchy sequence inY for
each x∈X . Since Y is a -Banach space, the sequence {jf(jx)} converges for each x ∈X .
So, one can define the mapping A :X → Y by
A(x) := lim j→∞ jf jx
for all x∈X . That is,
lim j→∞ jf jx– A(x), w =
By (.), we get lim j→∞ j fjx+ fjy+ fjz, w ≤ lim j→∞ jf jx+ jy+ jz, w+(p+q+r)j j θx pyqzrw ≤ lim j→∞ jf jx+ jy+ jz, 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 ., A : X → Y is additive. By Lemma . and (.), we have
f(x) – A(x), w= lim m→∞ f(x) –mf mx, w ≤ rθ – p+q+rx p+q+rw
for all x∈X and all w ∈ Y.
Now, let B :X → Y be another additive mapping satisfying (.). Then we have A(x) – B(x), w= jA jx– Bjx, w ≤ jA jx– fjx, w+fjx– Bjx, w ≤ · rθ – p+q+rx p+q+rw ·(p+q+r)j j ,
which tends to zero as j→ ∞ for all x ∈X and all w ∈ Y. By Definition ., we can conclude
that A(x) = B(x) for all x∈X . This proves the uniqueness of A.
Theorem . Let θ ∈ [, ∞), p, q, r ∈ (, ∞) with p + q + r > , and let f :X → Y be a mapping satisfying(.). Then there is a unique additive mapping A :X → Y such that
f(x) – A(x), w ≤
rθ
p+q+r– x
p+q+rw
for all x∈X and all w ∈ Y. Proof It follows from (.) that
f(x) – fx, w ≤ θ p+qx
p+q+rw (.)
for all x∈X and all w ∈ Y. Replacing x byxj in (.) and multiplying by j, we obtain
jf x j – j+f x j+ , w ≤ jθ p+q· (p+q+r)jx p+q+rw
for all x∈X and all w ∈ Y and all integers j ≥ . For all integers l, m with ≤ l < m, we get lf x l – mf x m , w ≤ m– j=l jθ p+q· (p+q+r)jx p+q+rw
for all x∈X and all w ∈ Y. So, we get lim l→∞ lf x l – mf x m , w =
for all x∈X and all w ∈ Y. Thus the sequence {jf(xj)} is a Cauchy sequence inY. Since
Y is a -Banach space, the sequence {jf(x
j)} converges. So, one can define the mapping
A:X → Y by A(x) := lim j→∞ jf x j
for all x∈X . That is, lim j→∞ jf x j – A(x), w = for all x∈X and all w ∈ Y.
The further part of the proof is similar to the proof of Theorem .. Now we prove the superstability of the Cauchy functional inequality in -Banach spaces. Theorem . Let θ∈ [, ∞), p, q, r, t ∈ (, ∞) with t = , and let f :X → Y be a mapping satisfying
f(x) + f (y) + f (z), w ≤f(x + y + z), w+ θxpyqzrwt (.)
for all x, y, z∈X and all w ∈ Y. Then f : X → Y is an additive mapping.
Proof Replacing w by sw in (.) for s∈ R \ {}, we get
f(x) + f (y) + f (z), sw ≤f(x + y + z), sw+ θxpyqzrswt and so
f(x) + f (y) + f (z), w ≤f(x + y + z), w+ θxpyqzrwt|s|
t
|s| (.)
for all x, y, z∈X , all w ∈ Y and all s ∈ R \ {}.
If t > , then the right-hand side of (.) tends tof (x + y + z), w as s → . If t < , then the right-hand side of (.) tends tof (x + y + z), w as s → +∞. Thus
f(x) + f (y) + f (z), w ≤f(x + y + z), w
3 Hyers-Ulam stability of Cauchy-Jensen functional inequality (1.3) in 2-Banach spaces
In this section, we prove the Hyers-Ulam stability of Cauchy-Jensen functional inequality (.) in -Banach spaces.
Proposition . Let f :X → Y be a mapping satisfying f(x) + f (y) + f (z), w ≤f x+ y + z , w (.)
for all x, y, z∈X and all w ∈ Y. Then the mapping f : X → Y is additive.
Proof Letting x = y = z = in (.), we get f (), w ≤ f (), w and so f (), w = for
all w∈Y. Hence f () = .
Letting y = –x and z = in (.), we getf (x) + f (–x), w ≤ f (), w = and so f (x) +
f(–x), w = for all x ∈X and all w ∈ Y. Hence f (x) + f (–x) = for all x ∈ X .
Letting z = –x+y in (.), we get f(x) + f(y) + f–x+ y , w ≤ f(), w= and so f(x) + f(y) + f–x+ y , w =
for all x, y∈X and all w ∈ Y. Hence = f (x) + f (y) + f –x+ y = f (x) + f (y) – f x+ y
for all x, y∈X . Since f () = , f : X → Y is additive.
Theorem . Let θ ∈ [, ∞), p, q, r ∈ (, ∞) with p + q + r < , and let f :X → Y be a mapping satisfying f(x) + f (y) + f (z), w ≤f x+ y + z , w + θxpyqzrw (.)
for all x, y, z∈X and all w ∈ Y. Then there is a unique additive mapping A : X → Y such
that
f(x) – A(x), w ≤ θ
– p+q+rx
p+q+rw
for all x∈X and all w ∈ Y.
Proof Letting x = y = z = in (.), we get f (), w ≤ f (), w and so f (), w = for all w∈Y. Hence f () = .
Letting y = –x and z = in (.), we getf (x) + f (–x), w ≤ f (), w = and so f (x) +
Letting y = x and z = –x in (.), we get
f (x) – f (x), w ≤f (), w+ θxp+q+rw = θxp+q+rw (.) for all x∈X and all w ∈ Y. Replacing x by jxin (.) and dividing by j, we obtain
jf jx– j+f j+x, w ≤ (p+q+r)j · j θx p+q+rw
for all x∈X and all w ∈ Y and all integers j ≥ . For all integers l, m with ≤ l < m, we get lf lx– mf mx, w ≤ m– j=l (p+q+r)j · j θx p+q+rw
for all x∈X and all w ∈ Y. So, we get lim l→∞ lf lx– mf mx, w =
for all x∈X and all w ∈ Y. Thus the sequence {
jf(jx)} is a Cauchy sequence inY for
each x∈X . Since Y is a -Banach space, the sequence {
jf(jx)} converges for each x ∈X .
So, one can define the mapping A :X → Y by
A(x) := lim j→∞ jf jx= lim j→∞ jf jx
for all x∈X . That is, lim j→∞ jf jx– A(x), w = lim j→∞ jf jx– A(x), w = for all x∈X and all w ∈ Y.
The further part of the proof is similar to the proof of Theorem .. Theorem . Let θ∈ [, ∞), p, q, r ∈ (, ∞) with p + q + r > , and let f :X → Y be a mapping satisfying(.). Then there is a unique additive mapping A :X → Y such that
f(x) – A(x), w ≤ θ
p+q+r– x
p+q+rw
for all x∈X and all w ∈ Y. Proof It follows from (.) that
f(x) – fx, w ≤ p+q+rθx
p+q+rw (.)
for all x∈X and all w ∈ Y. Replacing x byxj in (.) and multiplying by j, we obtain
jf x j – j+f x j+ , w ≤ j (p+q+r)(j+)θx p+q+rw
for all x∈X and all w ∈ Y and all integers j ≥ . For all integers l, m with ≤ l < m, we get lf x l – mf x m , w ≤ m– j=l j (p+q+r)(j+)θx p+q+rw
for all x∈X and all w ∈ Y. So, we get lim l→∞ lf x l – mf x m , w =
for all x∈X and all w ∈ Y. Thus the sequence {jf(x
j)} is a Cauchy sequence inY for each
x∈X . Since Y is a -Banach space, the sequence {jf(x
j)} converges for each x ∈X . So,
one can define the mapping A :X → Y by
A(x) := lim j→∞ jf x j
for all x∈X . That is, lim j→∞ jf x j – A(x), w = for all x∈X and all w ∈ Y.
The further part of the proof is similar to the proof of Theorem .. Now we prove the superstability of the Jensen functional equation in -Banach spaces. Theorem . Let θ∈ [, ∞), p, q, r, t ∈ (, ∞) with t = , and let f :X → Y be a mapping satisfying f(x) + f (y) + f (z), w ≤f x+ y + z , w + θxpyqzrwt (.)
for all x, y, z∈X and all w ∈ Y. Then f : X → Y is an additive mapping.
Proof Replacing w by sw in (.) for s∈ R \ {}, we get f(x) + f (y) + f (z), sw ≤f x+ y + z , sw + θxpyqzrswt and so f(x) + f (y) + f (z), w ≤f x+ y + z , w + θxpyqzrwt|s| t |s| for all x, y, z∈X , all w ∈ Y and all s ∈ R \ {}.
The rest of the proof is similar to the proof of Theorem ..
Competing interests
Authors’ contributions
CP conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Received: 26 February 2013 Accepted: 30 August 2013 Published:04 Nov 2013
References
1. Ulam, SM: A Collection of the Mathematical Problems. Interscience, New York (1960)
2. Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222-224 (1941) 3. Aoki, T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64-66 (1950) 4. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297-300 (1978) 5. Gˇavruta, P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal.
Appl. 184, 431-436 (1994)
6. Gähler, S: 2-metrische Räume und ihre topologische Struktur. Math. Nachr. 26, 115-148 (1963) 7. Gähler, S: Lineare 2-normierte Räumen. Math. Nachr. 28, 1-43 (1964)
8. Cho, Y, Lin, PSC, Kim, S, Misiak, A: Theory of 2-Inner Product Spaces. Nova Science Publishers, New York (2001) 9. White, A: 2-Banach spaces. Dissertation, St. Louis University (1968)
10. White, A: 2-Banach spaces. Math. Nachr. 42, 43-60 (1969)
11. Park, W: Approximate additive mappings in 2-Banach spaces and related topics. J. Math. Anal. Appl. 376, 193-202 (2011)
12. Eskandani, GZ, Gˇavruta, P: Hyers-Ulam-Rassias stability of Pexiderized Cauchy functional equation in 2-Banach spaces. J. Nonlinear Sci. Appl. 5, 459-465 (2012)
13. Gilányi, A: Eine zur Parallelogrammgleichung äquivalente Ungleichung. Aequ. Math. 62, 303-309 (2001) 14. Rätz, J: On inequalities associated with the Jordan-von Neumann functional equation. Aequ. Math. 66, 191-200
(2003)
15. Gilányi, A: On a problem by K. Nikodem. Math. Inequal. Appl. 5, 707-710 (2002)
16. Fechner, W: Stability of a functional inequalities associated with the Jordan-von Neumann functional equation. Aequ. Math. 71, 149-161 (2006)
17. Park, C, Cho, Y, Han, M: Functional inequalities associated with Jordan-von Neumann-type additive functional equations. J. Inequal. Appl. 2007, Article ID 41820 (2007)
10.1186/1029-242X-2013-447
Cite this article as: Park: Additive functional inequalities in 2-Banach spaces. Journal of Inequalities and Applications 2013, 2013:447
