Open Access

A pexider difference for a pexider functional equation

  • Abbas Najati1Email author,
  • Saeid Ostadbashi2,
  • Gwang Hui Kim3 and
  • Sooran Mahmoudfakhe2
Advances in Difference Equations20122012:26

DOI: 10.1186/1687-1847-2012-26

Received: 27 June 2011

Accepted: 6 March 2012

Published: 6 March 2012

Abstract

We deal with a Pexider difference

f ( 2 x + y ) + f ( 2 x - y ) - g ( x + y ) - g ( x - y ) - 2 g ( 2 x ) + 2 g ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equa_HTML.gif

where f and g map be a given abelian group (G, +) into a sequentially complete Hausdorff topological vector space. We also investigate the Hyers-Ulam stability of the following Pexiderized functional equation

f ( 2 x + y ) + f ( 2 x - y ) = g ( x + y ) + g ( x - y ) + 2 g ( 2 x ) - 2 g ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equb_HTML.gif

in topological vector spaces.

Mathematics subject classification (2000): Primary 39B82; Secondary 34K20, 54A20.

Keywords

Hyers-Ulam stability additive mapping quadratic mapping topological vector space

1. Introduction and preliminaries

In 1940, Ulam [1] proposed the general stability problem: Let G1 be a group, G2 be a metric group with the metric d. Given ε > 0, does there exists δ > 0 such that if a function h: G1 G2 satisfies the inequality
d h ( x y ) - h ( x ) h ( y ) < δ , ( x , y G 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equc_HTML.gif
then there is a homomorphism H: G 1 G 2 with
d ( h ( x ) , H ( x ) ) < ε , ( x G 1 ) ? https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equd_HTML.gif
Hyers [2] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces. In 1950, Aoki [3] extended the theorem of Hyers by considering the unbounded cauchy difference inequality
f ( x + y ) - f ( x ) - f ( y ) ε ( x p + y p ) ( ε > 0 , p ε [ 0 , 1 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Eque_HTML.gif

In 1978, Rassias [4] also generalized the Hyers' theorem for linear mappings under the assumption t f (tx) is continuous in t for each fixed x.

Recently, Adam and Czerwik [5] investigated the problem of the Hyers-Ulam stability of a generalized quadratic functional equation in linear topological spaces. Najati and Moghimi [6] investigated the Hyers-Ulam stability of the functional equation
f ( 2 x + y ) + f ( 2 x - y ) = f ( x + y ) + f ( x - y ) + 2 f ( 2 x ) - 2 f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equf_HTML.gif
in quasi-Banach spaces. In this article, we prove that the Pexiderized functional equation
f ( 2 x + y ) + f ( 2 x - y ) = g ( x + y ) + g ( x - y ) + 2 g ( 2 x ) - 2 g ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equg_HTML.gif

is stable for functions f, g defined on an abelian group and taking values in a topological vector space.

Throughout this article, let G be an abelian group and X be a sequentially complete Hausdorff topological vector space over the field of rational numbers.

A mapping f: GX is said to be quadratic if and only if it satisfies the following functional equation
f ( x + y ) + f ( x - y ) = 2 f ( x ) + 2 f ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equh_HTML.gif
for all x y G. A mapping f GX is said to be additive if and only if it satisfies f (x + y) = f (x) + f (y) for all x y ε G. For a given f: GX, we will use the following notation
D f ( x , y ) : = f ( 2 x + y ) + f ( 2 x - y ) - f ( x + y ) - f ( x - y ) - 2 f ( 2 x ) + 2 f ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equi_HTML.gif
For given sets A B X and a number k , we define the well known operations
A + B : = { a + b : a A , b B } , k A : = { k a : a A } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equj_HTML.gif
We denote the convex hull of a set U X by conv(U) and by U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq1_HTML.gif the sequential closure of U. Moreover it is well know that:
  1. (1)

    If A X are bounded sets, then conv(A) and A ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq2_HTML.gif are bounded subsets of X.

     
  2. (2)

    If A, B X and α β , then α conv(A) + β conv(B) = conv(αA + βB).

     
  3. (3)

    Let X1 and X2 be linear spaces over . If f: X1 X2 is a additive (quadratic) function, then f (rx) = rf (x) (f (rx) = r2f (x)), for all x X1 and all r .

     

2. Main results

We start with the following lemma.

Lemma 2.1. Let G be a 2-divisible abelian group and B X be a nonempty set. If the functions f, g: G X satisfy
f ( 2 x + y ) + f ( 2 x - y ) - g ( x + y ) - g ( x - y ) - 2 g ( 2 x ) + 2 g ( x ) B https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ1_HTML.gif
(2.1)
for all x, y G, then
D f ( x , y ) 2  conv ( B - B ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ2_HTML.gif
(2.2)
D g ( x , y ) conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ3_HTML.gif
(2.3)

for all x, y G.

Proof. Putting y = 0 in (2.1), we get
2 f ( 2 x ) - 2 g ( 2 x ) B https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ4_HTML.gif
(2.4)
for all x G. If we replace x by 1 2 x https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq3_HTML.gif in (2.4), then we have
f ( x ) - g ( x ) 1 2 B https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ5_HTML.gif
(2.5)
for all x G. It follows from (2.5) and (2.1) that
D f ( x , y ) = f ( 2 x + y ) + f ( 2 x - y ) - g ( x + y ) - g ( x - y ) - 2 g ( 2 x ) + 2 g ( x ) - [ f ( x + y ) - g ( x + y ) ] - [ f ( x - y ) - g ( x - y ) ] - [ 2 f ( 2 x ) - 2 g ( 2 x ) ] + [ 2 f ( x ) - 2 g ( x ) ] 2  conv ( B - B ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equk_HTML.gif
Moreover, we have
D g ( x , y ) = f ( 2 x + y ) + f ( 2 x - y ) - g ( x + y ) - g ( x - y ) - 2 g ( 2 x ) + 2 g ( x ) - [ f ( 2 x + y ) - g ( 2 x + y ) ] - [ f ( 2 x - y ) - g ( 2 x - y ) ] conv ( B - B ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equl_HTML.gif
Theorem 2.2. Let G be a 2-divisible abelian group and B X be a bounded set. Suppose that the odd functions f, g: G X satisfy (2 1) for all x, y G. Then there exists exactly one additive function A : G X https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq4_HTML.gif such that
A ( x ) - f ( x ) 4 conv ( B - B ) ¯ , A ( x ) - g ( x ) 2 conv ( B - B ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ6_HTML.gif
(2.6)
for all x G. Moreover the function A https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq5_HTML.gif is given by
A ( x ) = lim n 1 2 n f ( 2 n x ) = lim n 1 2 n g ( 2 n x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equm_HTML.gif

for all x G. Moreover, the convergence of the sequences are uniform on G.

Proof. By Lemma 2.1, we get (2.2). Setting y = x, y = 3x and y = 4x in (2.2), we get
f ( 3 x ) - 3 f ( 2 x ) + 3 f ( x ) 2  conv ( B - B ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ7_HTML.gif
(2.7)
f ( 5 x ) - f ( 4 x ) - f ( 2 x ) + f ( x ) 2 conv ( B - B ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ8_HTML.gif
(2.8)
f ( 6 x ) - f ( 5 x ) + f ( 3 x ) - 3 f ( 2 x ) + 2 f ( x ) 2 conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ9_HTML.gif
(2.9)
for all x G. It follows from (2.7), (2.8), and (2.9) that
f ( 6 x ) - f ( 4 x ) - f ( 2 x ) 6 conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equn_HTML.gif
for all x G. So
f ( 3 x ) - f ( 2 x ) - f ( x ) 6 conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ10_HTML.gif
(2.10)
for all x G. Using (2.7) and (2.10), we obtain
1 2 f ( 2 x ) - f ( x ) 2 conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equo_HTML.gif
for all x G. Therefore
1 2 n f ( 2 n x ) - 1 2 m f ( 2 m x ) = k = m n - 1 1 2 k + 1 f ( 2 k + 1 x ) - 1 2 k f ( 2 k x ) k = m n - 1 2 2 k conv ( B - B ) 4 2 m conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ11_HTML.gif
(2.11)
for all x G and all integers n > m ≥ 0. Since B is bounded, we conclude that conv(B - B) is bounded. It follows from (2.11) and boundedness of the set conv(B - B) that the sequence { 1 2 n f ( 2 n x ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq6_HTML.gif is (uniformly) Cauchy in X for all x G. Since X is a sequential complete topological vector space, the sequence { 1 2 n f ( 2 n x ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq6_HTML.gif is convergent for all x G, and the convergence is uniform on G. Define
A 1 : G X , A 1 ( x ) : = lim n 1 2 n f ( 2 n x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equp_HTML.gif
Since conv(B - B) is bounded, it follows from (2.2) that
D A 1 ( x , y ) = lim n 1 2 n D f ( 2 n x , 2 n y ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equq_HTML.gif
for all x y G. So A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq7_HTML.gif is additive (see [6]). Letting m = 0 and n ∞ in (2.11), we get
A 1 ( x ) - f ( x ) 4 conv ( B - B ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ12_HTML.gif
(2.12)
for all x G. Similarly as before applying (2.3) we have an additive mapping A 2 : G X https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq8_HTML.gif defined by A 2 ( x ) : = lim n 1 2 n g ( 2 n x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq9_HTML.gif which is satisfying
A 2 ( x ) - g ( x ) 2 conv ( B - B ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ13_HTML.gif
(2.13)

for all x G. Since B is bounded, it follows from (2.5) that A 1 = A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq10_HTML.gif. Letting A : = A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq11_HTML.gif, we obtain (2.6) from (2.12) and (2.13).

To prove the uniqueness of A https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq5_HTML.gif, suppose that there exists another additive function A https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq12_HTML.gif: G X satisfying (2.6). So
A ( x ) - A ( x ) = [ A ( x ) - f ( x ) ] + [ f ( x ) - A ( x ) ] 8 conv ( B - B ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equr_HTML.gif
for all x G. Since A https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq12_HTML.gif and A https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq5_HTML.gif are additive, replacing x by 2 n x implies that
A ( x ) - A ( x ) 8 2 n conv ( B - B ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equs_HTML.gif

for all x G and all integers n. Since conv ( B - B ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq13_HTML.gif is bounded, we infer A = A https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq14_HTML.gif. This completes the proof of theorem.

Theorem 2.3 Let G be a 2, 3-divisible abelian group and B X be a bounded set. Suppose that the even functions f, g: G X satisfy (2 1) for all x, y G. Then there exists exactly one quadratic function Q : G X https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq15_HTML.gifsuch that
Q ( x ) - f ( x ) + f ( 0 ) 4 conv ( B - B ) ¯ , Q ( x ) - g ( x ) + g ( 0 ) 2 conv ( B - B ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equt_HTML.gif
for all x G. Moreover, the function Q https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq16_HTML.gif is given by
Q ( x ) = lim n 1 4 n f ( 2 n x ) = lim n 1 4 n g ( 2 n x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equu_HTML.gif

for all x G. Moreover, the convergence of the sequences are uniform on G.

Proof. By replacing y by x + y in (2.2), we get
f ( 3 x + y ) + f ( x - y ) - f ( 2 x + y ) - f ( y ) - 2 f ( 2 x ) + 2 f ( x ) 2 c o n v ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ14_HTML.gif
(2.14)
for all x, y G. Replacing y by - y in (2.14), we get
f ( 3 x - y ) + f ( x + y ) - f ( 2 x - y ) - f ( y ) - 2 f ( 2 x ) + 2 f ( x ) 2 conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ15_HTML.gif
(2.15)
for all x, y G. It follows from (2.2), (2.14), and (2.15) that
f ( 3 x + y ) + f ( 3 x - y ) - 2 f ( y ) - 6 f ( 2 x ) + 6 f ( x ) 6 conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ16_HTML.gif
(2.16)
for all x, y G. By letting y = 0 and y = 3x in (2.16), we get
2 f ( 3 x ) - 6 f ( 2 x ) + 6 f ( x ) - 2 f ( 0 ) 6 conv ( B - B ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ17_HTML.gif
(2.17)
f ( 6 x ) - 2 f ( 3 x ) - 6 f ( 2 x ) + 6 f ( x ) + f ( 0 ) 6 conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ18_HTML.gif
(2.18)
for all x G. Using (2.17) and (2.18), we obtain
f ( 6 x ) - 4 f ( 3 x ) + 3 f ( 0 ) 12 conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ19_HTML.gif
(2.19)
for all x G. If we replace x by 1 3 x https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq17_HTML.gif in (2.19), then
f ( 2 x ) - 4 f ( x ) + 3 f ( 0 ) 12 conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equv_HTML.gif
for all x G. Therefore
1 4 n + 1 f ( 2 n + 1 x ) - 1 4 n f ( 2 n x ) + 3 4 n + 1 f ( 0 ) 3 4 n conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ20_HTML.gif
(2.20)
for all x G and all integers n. So
1 4 n f ( 2 n x ) - 1 4 m f ( 2 m x ) = k = m n - 1 1 4 k + 1 f ( 2 k + 1 x ) - 1 4 k f ( 2 k x ) - k = m n - 1 3 4 k + 1 f ( 0 ) + k = m n - 1 3 4 k conv ( B - B ) - k = m n - 1 3 4 k + 1 f ( 0 ) + 1 4 m - 1 conv ( B - B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_Equ21_HTML.gif
(2.21)

for all x G and all integers n >m ≥ 0. It follows from (2.21) and boundedness of the set conv(B - B) that the sequence { 1 4 n f ( 2 n x ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-26/MediaObjects/13662_2011_Article_137_IEq18_HTML.gif is (uniformly) Cauchy in X for all x G. The rest of the proof is similar to proof of of Theorem 2.2.

Remark 2.4. If the functions f, g: G X satisfy (2.1), where f is even (odd) and g is odd (even), then it is easy to show that f and g are bounded.

Declarations

Authors’ Affiliations

(1)
Department of Mathematical Sciences, University of Mohaghegh Ardabili
(2)
Department of Mathematics, Urmia University
(3)
Department of Applied Mathematics, Kangnam University

References

  1. Ulam SM: Problem in Modern Mathematics, Science edn. Wiley, New York; 1960.
  2. Hyers DH: On the stability of the linear functional equation. Proc Nat Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView Article
  3. Aoki T: On the stability of linear trasformation in Banach spaces. J Math Soc Japan 1950, 2: 64–66. 10.2969/jmsj/00210064MathSciNetView Article
  4. Rassias ThM: On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView Article
  5. Adam M, Czerwik S: On the stability of the quadratic functional equation in topological spaces. Banach J Math Anal 2007, 1: 245–251.MathSciNetView Article
  6. Najati A, Moghimi MB: Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. J Math Anal Appl 2008, 337: 399–415. 10.1016/j.jmaa.2007.03.104MathSciNetView Article

Copyright

© Najati et al; licensee Springer. 2012

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