Open Access

A generalized Hyers-Ulam stability of a Pexiderized logarithmic functional equation in restricted domains

Journal of Inequalities and Applications20122012:15

https://doi.org/10.1186/1029-242X-2012-15

Received: 9 May 2011

Accepted: 19 January 2012

Published: 19 January 2012

Abstract

Let + and B be the set of positive real numbers and a Banach space, respectively, f, g, h : +B and ψ : + 2 be a nonnegative function of some special forms. Generalizing the stability theorem for a Jensen-type logarithmic functional equation, we prove the Hyers-Ulam stability of the Pexiderized logarithmic functional inequality

| | f ( x y ) - g ( x ) - h ( y ) | | ψ ( x , y )

in restricted domains of the form {(x, y) : x k y s d} for fixed k, s , d > 0. We also discuss an L-version of the Hyers-Ulam stability of the inequality. 2000 MSC: 39B22.

Keywords

logarithmic functional equation Hyers-Ulam stability asymptotic behavior

1. Introduction

The Hyers-Ulam stability problems of functional equations go back to 1940 when Ulam proposed a question concerning the approximate homomorphisms from a group to a metric group (see [1]). A partial answer was given by Hyers [2, 3] under the assumption that the target space of the involved mappings is a Banach space. After the result of Hyers, Aoki [4] and Bourgin [5, 6] treated with this problem, however, there were no other results on this problem until 1978 when Rassias [7] treated again with the inequality of Aoki [4]. Following the Rassias' result a great number of articles on the subject have been published concerning numerous functional equations in various directions [819]. Among the results, the stability problem in a restricted domain was investigated by Skof, who proved the stability problem of the Cauchy functional equation in a restricted domain [20]. Developing this result, Jung, Rassias and Rassias considered the stability problems in restricted domains for the Jensen functional equation [21, 22] and Jensen-type functional equations [23]. We also refer the reader to [2429] for some interesting results on functional equations and their Hyers-Ulam stabilities in restricted conditions. In this article, generalizing the result in [8], we consider the Hyers-Ulam stability of the Pexiderized Jensen functional equation
| | f ( x y ) - g ( x ) - h ( y ) | | ψ ( x , y )
(1.1)
in the restricted domains U k,s,d = {(x, y): x > 0, y > 0, x k y s d} for fixed k, s and d > 0, where ψ(x, y) = ϕ (xy), ϕ (x) or ϕ (y). Making use of the result, we prove the asymptotic behavior of f, g and h satisfying
| | f ( x y ) - g ( x ) - h ( y ) | | 0
(1.2)
as x k y s →∞. Finally, we discuss the Hyers-Ulam stability of the inequality
| | f ( x y ) - g ( x ) - h ( y ) | | L ( U k , s , d ) ε
(1.3)

and its asymptotic behavior.

2. Stability in classical sense

We call L: +B a logarithmic function provided that
L ( x y ) - L ( x ) - L ( y ) = 0
for all x, y > 0. Let ϕ : + → [0, ∞). We assume that
Φ ( x ) : = k = 1 2 - k ϕ ( x 2 k ) + 2 ϕ ( x 2 k - 1 ) + ϕ ( 1 ) <

for all x > 0. As a direct consequence of Aoki [4] or Bourgin [5, 6], we obtain the generalized Hyers-Ulam stability for the logarithmic functional equation, viewing 〈+, ×〉 as a multiplicative group.

Theorem A. Suppose that f : +B satisfies
| | f ( x y ) - f ( x ) - f ( y ) | | ϕ ( x y ) + ϕ ( x ) + ϕ ( y ) + ϕ ( 1 )
for all x, y > 0. Then, there exists a unique logarithmic function L : +B satisfying
| | f ( x ) - L ( x ) | | Φ ( x )

for all x > 0.

In this section, we first consider the logarithmic functional inequality (1.1) in the restricted domain
U k , s , d = { ( x , y ) : x > 0 , y > 0 , x k y s d }

for fixed k, s and d > 0.

Theorem 2.1. Let d > 0, k, s , ks. Suppose that f, g, h : +B satisfy
| | f ( x y ) - g ( x ) - h ( y ) | | ϕ ( x y )
(2.1)
for all x,y U k,s,d . Then, there exists a unique logarithmic function L1 : +B such that
| | f ( x ) - L 1 ( x ) - f ( 1 ) | | Φ ( x )
(2.2)

for all x +.

Proof. For given x,y +, choosing a z > 0 such that x k y s z s-k d, x k z s-k d, y s z s-k d and z s-k d, we have
| | f ( x y ) - f ( x ) - f ( y ) + f ( 1 ) | | | | f ( x y ) - g ( x z - 1 ) - h ( y z ) | | + | | - f ( x ) + g ( x z - 1 ) + h ( z ) | | + | | - f ( y ) + g ( z - 1 ) + h ( y z ) | | + | | f ( 1 ) - g ( z - 1 ) - h ( z ) | | ϕ ( x y ) + ϕ ( x ) + ϕ ( y ) + ϕ ( 1 ) .
(2.3)

Now, by Theorem A, we get the result.

Corollary 2.2. Let ϵ,d > 0, k, s , ks. Suppose that f, g, h : +B satisfy
| | f ( x y ) - g ( x ) - h ( y ) | | ε
(2.4)
for all x,y U k,s,d . Then, there exists a unique logarithmic function L1: +B such that
| | f ( x ) - L 1 ( x ) - f ( 1 ) | | 4 ε
(2.5)

for all x +.

Remark 2.1. Note that the Corollary 2.2 fails if k = s. Indeed, let L: +B be a nonzero logarithmic function. Define g(x) = h(x) = L(x) for all x > 0 and
f ( x ) = L ( x ) , x d 1 / s , 0 , 0 < x < d 1 / s .
Then, it is easy to see that the inequality (2.4) holds for all x, y > 0, with xyd1/s. Assume that there exists a logarithmic function L1 satisfying (2.5). Then, we have
| | L 1 ( x ) | | | f ( 1 ) | + 4 ε = 4 ε
(2.6)
for all 0 < x < d1/s. The inequality (2.6) implies L1 = 0. Indeed, if L1(x0) ≠ 0 for some x0 > 0, then we have L1(1/x0) = -L1(x0) ≠ 0. Thus, we may assume that 0 < x0 < 1. Now, we encounter the contradiction
| n L 1 ( x 0 ) | = | L 1 ( x 0 n ) | 4 ε
for all large integers n. Thus, L1 = 0 and the inequality (2.5) implies
| | L ( x ) | | 4 ε
(2.7)

for all xd1/s. Similarly, using (2.7), we can show that L = 0, which contracts to the choice of L.

As a direct consequence of Corollary 2.2, we have the following.

Corollary 2.3. [8] Let p,q,P,Q be nonzero real numbers and ε , d > 0 , k , s , k p s q . Suppose that f : +B satisfies
| | f ( x p y q ) - P f ( x ) - Q f ( y ) | | ε
(2.8)
for all x,y U k,s,d . Then, there exists a unique logarithmic function L : +B such that
| | f ( x ) - L ( x ) - f ( 1 ) | | 4 ε
(2.9)

for all x +.

Proof. Replacing x by x 1 p , y by y 1 q in (2.8), we have
| | f ( x y ) - P f ( x 1 p ) - Q f ( y 1 q ) | | ε

for all x,y > 0, with x k p y s q d . Letting g ( x ) = P f ( x 1 p ) , h ( y ) = Q f ( y 1 q ) , applying Corollary 2.2 and letting L(x) = L1(x), we get the result.

Theorem 2.4. Let d > 0, s ≠ 0. Suppose that f, g, h: +B satisfy
| | f ( x y ) - g ( x ) - h ( y ) | | ϕ ( x )
(2.10)
for all x,y U k,s,d . Then, there exists a unique logarithmic function L2 : +B such that
| | g ( x ) - L 2 ( x ) - g ( 1 ) | | Φ ( x )
(2.11)

for all x +.

Proof. For given x,y +, choosing a z > 0 such that x k y k z s d, x k y s z s d, y k z s d and y s z s d, we have
| | g ( x y ) - g ( x ) - g ( y ) + g ( 1 ) | | | | - f ( x y z ) + g ( x y ) + h ( z ) | | + | | f ( x y z ) - g ( x ) - h ( y z ) | | + | | f ( y z ) - g ( y ) - h ( z ) | | + | | - f ( y z ) + g ( 1 ) + h ( y z ) | | ϕ ( x y ) + ϕ ( x ) + ϕ ( y ) + ϕ ( 1 ) .
(2.12)

Now, by Theorem A, we get the result.

Corollary 2.5. Let ϵ, d > 0, s ≠ 0. Suppose that f, g, h : +B satisfy
| | f ( x y ) - g ( x ) - h ( y ) | | ε
(2.13)
for all x, y U k,s,d . Then, there exists a unique logarithmic function L2: +B such that
| | g ( x ) - L 2 ( x ) - g ( 1 ) | | 4 ε
(2.14)

for all x +.

Remark 2.2. Similarly as in Corollary 2.2, the above result fails if s = 0. Let L : +B be a nonzero logarithmic function. Define f(x) = h(x) = L(x) for all x > 0 and
g ( x ) = L ( x ) , x d 1 / k , 0 , 0 < x < d 1 / s .

Then, the inequality (2.13) holds for all x, y > 0, with x k d but (2.14) does not hold for any logarithmic function L2.

As a direct consequence of Corollary 2.5, we have the following.

Corollary 2.6. [8] Let p, q, P, Q be nonzero real numbers and ϵ, d > 0, k, s with s ≠ 0. Suppose that f : +B satisfies
| | f ( x p y q ) - P f ( x ) - Q f ( y ) | | ε
(2.15)
for all x,y U k,s,d . Then, there exists a unique logarithmic function L : +B such that
| | f ( x ) - L ( x ) - f ( 1 ) | | 4 ε | P |
(2.16)

for all x +.

Proof. Replacing x by x 1 p , y by y 1 q in (2.15), we have
| | f ( x y ) - P f ( x 1 p ) - Q f ( y 1 q ) | | ε

for all x,y > 0, with x k p y s q d . Letting g ( x ) = P f ( x 1 p ) , h ( y ) = Q f ( y 1 q ) , applying Corollary 2.5 and dividing the result by |P|, we get the result with L ( x ) = 1 P L 2 ( x p ) .

Theorem 2.7. Let d > 0, k ≠ 0. Suppose that f, g, h : +B satisfy
| | f ( x y ) - g ( x ) - h ( y ) | | ϕ ( y )
(2.17)
for all x,y U k,s,d . Then, there exists a unique logarithmic function L3 : +B such that
| | h ( x ) - L 3 ( x ) - h ( 1 ) | | Φ ( x )
(2.18)

for all x +.

Proof. For given x,y +, choosing a z > 0 such that x s y s z k d, x k y s z k d, x s z k d and x k z k d, we have
| | h ( x y ) - h ( x ) - h ( y ) + h ( 1 ) | | | | - f ( x y z ) + g ( z ) + h ( x y ) | | + | | f ( x y z ) - g ( x z ) - h ( y ) | | + | | f ( z x ) - g ( z ) - h ( x ) | | + | | - f ( x z ) + g ( x z ) + h ( 1 ) | | ϕ ( x y ) + ϕ ( x ) + ϕ ( y ) + ϕ ( 1 ) .
(2.19)

Now, by Theorem A, we get the result.

Corollary 2.8. Let ϵ, d > 0, k ≠ 0. Suppose that f, g, h: +B satisfy
| | f ( x y ) - g ( x ) - h ( y ) | | ε
(2.20)
for all x,y U k,s,d . Then, there exists a unique logarithmic function L3 : +B such that
| | h ( x ) - L 3 ( x ) - h ( 1 ) | | 4 ε
(2.21)

for all x +.

Remark 2.3. Similarly, as in Remark 2.2, we can show that the above result fails if k = 0. Also, as a direct consequence of the result, we have the following.

Corollary 2.9. [8] Let p, q, P, Q be nonzero real numbers and ϵ, d > 0, k, s with k ≠ 0. Suppose that f : +B satisfies
| | f ( x p y q ) - P f ( x ) - Q f ( y ) | | ε
(2.22)
for all x,y U k,s,d . Then, there exists a unique logarithmic function L: +B such that
| | f ( x ) - L ( x ) - f ( 1 ) | | 4 ε | Q |
(2.23)

for all x +.

Theorem 2.10. Let ϵ, d > 0, k, s ≠ 0, ks. Suppose that f, g, h : +B satisfy
| | f ( x y ) - g ( x ) - h ( y ) | | ε
(2.24)
for all x,y U k,s,d . Then, there exists a unique logarithmic function L : +B such that
| | f ( x ) - L ( x ) - f ( 1 ) | | 4 ε , | | g ( x ) - L ( x ) - g ( 1 ) | | 4 ε , | | h ( x ) - L ( x ) - h ( 1 ) | | 4 ε

for all x +.

Proof. In view of Corollaries 2.2, 2.5 and 2.8, it suffices to prove that L1 = L2 = L3. For given x,y > 0, choose a z > 0 such that x k y s z s-k d, z s-k d. Then, in view of (2.24), we have
| | f ( x y ) - g ( x z - 1 ) - h ( y z ) | | ε ,
(2.25)
| | - f ( 1 ) + g ( z - 1 ) + h ( z ) | | ε .
(2.26)
Using the inequalities (2.10) and (2.15), we have
| | g ( x z - 1 ) - g ( x ) - g ( z - 1 ) + g ( 1 ) | | 4 ε ,
(2.27)
| | h ( y z ) - h ( z ) - h ( y ) + h ( 1 ) | | 4 ε
(2.28)
for all x,y,z > 0. From (2.25)-(2.28), using the triangle inequality, we have
| | f ( x y ) - g ( x ) - h ( y ) - f ( 1 ) + g ( 1 ) + h ( 1 ) | | 10 ε
(2.29)
for all x,y > 0. From the inequalities (2.5), (2.14), (2.21), (2.29) using the triangle inequality, we have
| | L 1 ( x y ) - L 2 ( x ) - L 3 ( y ) | | 22 ε .
(2.30)

Putting y = 1 and x = 1 in (2.30) separately, and using the fact that for all x > 0, n , L j (x n ) = nL j (x), j = 1,2,3, we can show that L1 = L2 and L1 = L3. This completes the proof.

As a direct consequence of Theorem 2.10, we have the following.

Corollary 2.11. [8] Let p, q, P ,Q be nonzero real numbers and ϵ, d > 0, k, s with k ≠ 0, s ≠ 0 and ks. Suppose that f : +B satisfies
| | f ( x p y q ) - P f ( x ) - Q f ( y ) | | ε
(2.31)
for all x,y U k,s,d . Then, there exists a unique logarithmic function L: +B such that
| | f ( x ) - L ( x ) - f ( 1 ) | | min 4 ε , 4 ε | P | , 4 ε | Q |
(2.32)

for all x +.

3. Asymptotic behaviors

In this section, we consider asymptotic behaviors of f,g, h satisfying (1.2).

Theorem 3.1. Let k, s , ks. Suppose that f, g, h : +B satisfy the asymptotic condition
| | f ( x y ) - g ( x ) - h ( y ) | | 0
(3.1)
as x k y s → ∞. Then, there exists a unique logarithmic function L : +B such that
f ( x ) = L ( x ) + f ( 1 )
(3.2)

for all x +.

Proof. By the condition (3.1), for each n , there exists d n > 0 such that
| | f ( x y ) - g ( x ) - h ( y ) | | 1 n
(3.3)
for all x, y > 0, with x k y s d n . By Corollary 2.2, there exists a unique logarithmic function L n : +B such that
| | f ( x ) - L n ( x ) - f ( 1 ) | | 4 n
(3.4)
for all x +. Replacing n by m in (3.4) and using the triangle inequality we have
| | L n ( x ) - L m ( x ) | | 4 n + 4 m 8
(3.5)
for all x +. Now, for all x > 0 and all rational numbers r > 0, we have
| | L n ( x ) - L m ( x ) | | = 1 r | | L n ( x r ) - L m ( x r ) | | 8 r .
(3.6)

Letting r → ∞ in (3.6), we have L n = L m . Letting n → ∞ in (3.4), we get the result.

Using Corollary 2.5, we obtain the following.

Theorem 3.2. Let s ≠ 0. Suppose that f, g, h : +B satisfy the asymptotic condition
| | f ( x y ) - g ( x ) - h ( y ) | | 0
(3.7)
as x k y s → ∞. Then, there exists a unique logarithmic function L: +B such that
g ( x ) = L ( x ) + g ( 1 )
(3.8)

for all x +.

Using Corollary 2.8, we obtain the following.

Theorem 3.3. Let k ≠ 0. Suppose that f, g, h : +B satisfies the asymptotic condition
| | f ( x y ) - g ( x ) - h ( y ) | | 0
(3.9)
as x k y s → ∞. Then, there exists a unique logarithmic function L : +B such that
h ( x ) = L ( x ) + h ( 1 )
(3.10)

for all x +.

Theorem 3.4. Let k, s ≠ 0 and ks. Suppose that f, g, h : +B satisfy the asymptotic condition
| | f ( x y ) - g ( x ) - h ( y ) | | 0
(3.11)
as x k y s → ∞. Then, there exists a unique logarithmic function L : +B and c1, c2 B such that
f ( x ) = L ( x ) + c 1 + c 2 , g ( x ) = L ( x ) + c 1 , h ( x ) = L ( x ) + c 2

for all x +.

Proof. By the condition (3.11), for each n , there exists d n > 0 such that
| | f ( x y ) - g ( x ) - h ( y ) | | 1 n
(3.12)
for all x, y > 0, with x k y s d n . By Theorem 2.10, there exists a unique logarithmic function L n : +B such that
| | f ( x ) - L n ( x ) - f ( 1 ) | | 4 n ,
(3.13)
| | g ( x ) - L n ( x ) - g ( 1 ) | | 4 n ,
(3.14)
| | h ( x ) - L n ( x ) - h ( 1 ) | | 4 n
(3.15)

for all x +. Similarly, as in the proof of Theorem 3.1, we have L n = L m for all n, m . Letting n → ∞ in (3.13)-(3.15), and using (3.11), we get the result.

4. Stability in L-sense and its asymptotic behavior

Let f, g, h be locally integrable functions on +. In this section, we consider the L-version of Hyers-Ulam stability of the inequality
| | f ( x y ) - g ( x ) - h ( y ) | | L ( U k , s , d ) ε ,
(4.1)
where k ≠ 0, s ≠ 0, ks, d > 0 are fixed and U k,s,d = {(x, y): x k y s d}. Let ω on be a nonnegative infinitely differentiable function satisfying the conditions
supp ω { x : | x | 1 }
and
ω ( x ) d x = 1 .

Let ω t (x): = t-1ω(x/t), t > 0 and f be a locally integrable function. Then, for each t > 0,f * ω t (x) = ∫ f(y)ω t (x - y) dy is a smooth function of x and f * ω t (x) → f(x) for almost every x as t → 0+. Now, we are in a position to prove the Hyers-Ulam stability of the inequality (3.1).

Theorem 4.1. Let f, g, h be locally integrable functions satisfying (3.1). Then, there exist c1, c2, c3, a such that
| | f ( x ) - c 1 - a ln x | | L ( + ) 4 ε , | | g ( x ) - c 2 - a ln x | | L ( + ) 4 ε , | | h ( x ) - c 3 - a ln x | | L ( + ) 4 ε .
Proof. Using the change of variables x by 2 x and y by 2 y in (4.1), we have
| | f ( 2 x + y ) - g ( 2 x ) - h ( 2 y ) | | L ( U d ) ε ,
(4.2)
where U d = { ( x , y ) : k x + s y log 2 d : = d 1 } . Now, let
F ( x ) = f ( 2 x ) , G ( x ) = g ( 2 x ) , H ( x ) = h ( 2 x ) .
(4.3)
Then, we have
| | F ( x + y ) - G ( x ) - H ( y ) | | L ( U d ) ε .
(4.4)
Convolving ω t (x)ω s (y) in (4.4) as in the proof of [8, Theorem 3.1], we have
| F * ω t * ω s ( x + y ) - G * ω t ( x ) - H * ω s ( y ) | ε
(4.5)

holds for all k x + s y d 2 : = d 1 + k 2 + s 2 and 0 < t < 1, 0 < s < 1. Using the same method as in [9, Theorem 4.3], we get the result.

Now, we discuss an asymptotic behavior of the inequality (4.1).

Theorem 4.2. Let f, g, h : +, j = 1, 2, 3, be locally integrable functions satisfying
| | f ( x y ) - g ( x ) - h ( y ) | | L ( U k , s , d ) 0
(4.6)
as d → ∞. Then, there exist a, c1, c2, c3 such that
| | f ( x ) - c 1 - a ln x | | L ( + ) = 0 , | | g ( x ) - c 2 - a ln x | | L ( + ) = 0 , | | h ( x ) - c 3 - a ln x | | L ( + ) = 0 .
Proof. By the condition (4.6), for any positive integer n, there exists d n > 0 such that
| | f ( x y ) - g ( x ) - h ( y ) | | L ( U k , s , d n ) 1 n
(4.7)
for all x , y U k , s , d n . Now, by Theorem 4.1, there exist a, c1, c2, c3 (which are independent of n) such that
| | f ( x ) - c 1 - a ln x | | L ( + ) 4 n ,
(4.8)
| | g ( x ) - c 2 - a ln x | | L ( + ) 4 n ,
(4.9)
| | h ( x ) - c 3 - a ln x | | L ( + ) 4 n .
(4.10)

Letting n → ∞ in (4.8)-(4.10), we get the result.

Declarations

Acknowledgements

This study was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (No. 2010-0016963).

Authors’ Affiliations

(1)
Department of Mathematics, Kunsan National University

References

  1. Ulam SM: A Collection of Mathematical Problems. Interscience Publishers, New York; 1960.Google Scholar
  2. Hyers DH: On the stability of the linear functional equations. Proc Natl Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
  3. Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables. Birkhauser, Basel; 1998.View ArticleGoogle Scholar
  4. Aoki T: On the stability of the linear transformation in Banach spaces. J Math Soc Japan 1950, 2: 64–66. 10.2969/jmsj/00210064MATHMathSciNetView ArticleGoogle Scholar
  5. Bourgin DG: Class of transformations and bordering transformations. Bull Amer Math Soc 1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7MATHMathSciNetView ArticleGoogle Scholar
  6. Bourgin DG: Multiplicative transformations. Proc Natl Acad Sci USA 1950, 36: 564–570. 10.1073/pnas.36.10.564MATHMathSciNetView ArticleGoogle Scholar
  7. Rassias ThM: On the stability of linear mapping in Banach spaces. Proc Amer Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1MATHMathSciNetView ArticleGoogle Scholar
  8. Chung J: Stability of a Jensen type logarithmic functional equation on restricted domains and its asymptotic behaviors. Adv Diff Equ 2010, 2010: 13. Article ID 432796View ArticleGoogle Scholar
  9. Chung J: A distributional version of functional equations and their stabilities. Nonlinear Anal 2005, 62: 1037–1051. 10.1016/j.na.2005.04.016MATHMathSciNetView ArticleGoogle Scholar
  10. Czerwik S: Stability of Functional Equations of Ulam-Hyers-Rassias Type. Hadronic Press, Inc., Palm Harbor; 2003.Google Scholar
  11. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific Publ. Co., Singapore; 2002.Google Scholar
  12. Forti GL: The stability of homomorphisms and amenability with applications to functional equations. Abh Math Sem Univ Hamburg 1987, 57: 215–226. 10.1007/BF02941612MATHMathSciNetView ArticleGoogle Scholar
  13. Jun KW, Kim HM: Stability problem for Jensen-type functional equations of cubic mappings. Acta Math Sin Engl Ser 2006, 22(6):1781–1788. 10.1007/s10114-005-0736-9MATHMathSciNetView ArticleGoogle Scholar
  14. Kim GH, Lee YH: Boundedness of approximate trigonometric functional equations. Appl Math Lett 2009, 31: 439–443.View ArticleGoogle Scholar
  15. Kannappan Pl: Functional Equations and Inequalities with Applications. Springer, New York; 2009.View ArticleGoogle Scholar
  16. Rassias JM: On the Ulam stability of mixed type mappings on restricted domains. J Math Anal Appl 2002, 276: 747–762. 10.1016/S0022-247X(02)00439-0MATHMathSciNetView ArticleGoogle Scholar
  17. Rassias JM: On approximation of approximately linear mappings by linear mappings. J Funct Anal 1982, 46: 126–130. 10.1016/0022-1236(82)90048-9MATHMathSciNetView ArticleGoogle Scholar
  18. Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Appl Math 2000, 62(1):23–130. 10.1023/A:1006499223572MATHMathSciNetView ArticleGoogle Scholar
  19. Hyers DH, Rassias ThM: Approximate homomorphisms. Aequationes Math 1992, 44: 125–153. 10.1007/BF01830975MATHMathSciNetView ArticleGoogle Scholar
  20. Skof F: Sull'approssimazione delle applicazioni localmente ω -additive. Atii Accad Sci Torino Cl Sci Fis Mat Natur 1983, 117: 377–389.MATHMathSciNetGoogle Scholar
  21. Jung SM: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York; 2011.View ArticleGoogle Scholar
  22. Jung SM: Hyers-Ulam stability of Jensen's equation and its application. Proc Amer Math Soc 1998, 126: 3137–3143. 10.1090/S0002-9939-98-04680-2MATHMathSciNetView ArticleGoogle Scholar
  23. Rassias JM, Rassias MJ: On the Ulam stability of Jensen and Jensen type mappings on restricted domains. J Math Anal Appl 2003, 281: 516–524. 10.1016/S0022-247X(03)00136-7MATHMathSciNetView ArticleGoogle Scholar
  24. Batko B: Stability of an alternative functional equation. J Math Anal Appl 2008, 339: 303–311. 10.1016/j.jmaa.2007.07.001MATHMathSciNetView ArticleGoogle Scholar
  25. Batko B: On approximation of approximate solutions of Dhombres' equation. J Math Anal Appl 2008, 340: 424–432. 10.1016/j.jmaa.2007.08.009MATHMathSciNetView ArticleGoogle Scholar
  26. Brzdȩek J: On a method of proving the Hyers-Ulam stability of functional equations on restricted domains. Austral J Math Anal Appl 2009, 6: 1–10.MathSciNetGoogle Scholar
  27. Brzdȩek J: On stability of a family of functional equations. Acta Math Hungarica 2010, 128: 139–149. 10.1007/s10474-010-9169-8MathSciNetView ArticleGoogle Scholar
  28. Sikorska J: On a Pexiderized conditional exponential functional equation. Acta Math Hun-garica 2009, 125: 287–299. 10.1007/s10474-009-9019-8MATHMathSciNetView ArticleGoogle Scholar
  29. Sikorska J: Exponential functional equation on spheres. Appl Math Lett 2010, 23: 156–160. 10.1016/j.aml.2009.09.004MATHMathSciNetView ArticleGoogle Scholar

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© Chung; licensee Springer. 2012

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