Open Access

Local stability of the Pexiderized Cauchy and Jensen's equations in fuzzy spaces

Journal of Inequalities and Applications20112011:78

https://doi.org/10.1186/1029-242X-2011-78

Received: 16 May 2011

Accepted: 6 October 2011

Published: 6 October 2011

Abstract

Lex X be a normed space and Y be a Banach fuzzy space. Let D = {(x, y) X × X : ||x|| + ||y|| ≥ d} where d > 0. We prove that the Pexiderized Jensen functional equation is stable in the fuzzy norm for functions defined on D and taking values in Y. We consider also the Pexiderized Cauchy functional equation.

2000 Mathematics Subject Classification: 39B22; 39B82; 46S10.

Keywords

Pexiderized Cauchy functional equationgeneralized Hyers-Ulam stabilityJensen functional equationnon-Archimedean space

1. Introduction

The functional equation (ξ) is stable if any function g satisfying the equation (ξ) approximately is near to the true solution of (ξ).

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms:

Let G1 be a group and let G2 be a metric group with the metric d(·,·). Given ε > 0, does there exist δ > 0 such that if a function h : G1G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y G1, then there exists a homomorphism H : G1G2 with d(h(x), H(x)) < ε for all x G1?

In other words, we are looking for situations when the homomorphisms are stable, i.e., if a mapping is almost a homomorphism, then there exists a true homomorphism near it. If we turn our attention to the case of functional equations, then we can ask the question: When the solutions of an equation differing slightly from a given one must be close to the true solution of the given equation.

In 1941, Hyers [2] gave a partial solution of Ulam's problem for the case of approximate additive mappings under the assumption that G1 and G2 are Banach spaces. In 1950, Aoki [3] provided a generalization of the Hyers' theorem for additive mappings, and in 1978, Th.M. Rassias [4] succeeded in extending the result of Hyers for linear mappings by allowing the Cauchy difference to be unbounded (see also [5]). The stability phenomenon that was introduced and proved by Th.M. Rassias is called the generalized Hyers-Ulam stability. Forti [6] and Gǎvruta [7] have generalized the result of Th.M. Rassias, which permitted the Cauchy difference to become arbitrary unbounded. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem. A large list of references can be found, for example, in [829].

Following [30], we give the following notion of a fuzzy norm.

Definition 1.1. [30] Let X be a real vector space. A function N : X × → [0, 1] is called a fuzzy norm on X if, for all x, y X and s, t ,

(N1) N(x, t) = 0 for all t ≤ 0;

(N2) x = 0 if and only if N(x, t) = 1 for all t > 0;

(N3) N ( c x , t ) = N ( x , t | c | ) if c ≠ 0;

(N4) N(x + y, s + t) ≥ min{N(x, s), N(y, t)};

(N5) N(x,·) is a nondecreasing function on and limt→∞N(x, t) = 1;

(N6) for x ≠ 0, N(x,·) is continuous on .

The pair (X, N) is called a fuzzy normed vector space.

Example 1.2. Let (X, ||·||) be a normed linear space and let α, β > 0. Then,
N ( x , t ) = α t α t + β x , t > 0 , x X , 0 , t 0 , x X

is a fuzzy norm on X.

Example 1.3. Let (X, ||·||) be a normed linear space and let β > α > 0. Then,
N ( x , t ) = 0 , t α x , t t + ( β - α ) x , α x < t β x ; 1 , t > β x

is a fuzzy norm on X.

Definition 1.4. Let (X, N) be a fuzzy normed space. A sequence {x n } in X is said to be convergent if there exists x X such that limn→∞N(x n - x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {x n }, and we denote it by N - lim x n = x.

The limit of the convergent sequence {x n } in (X, N) is unique. Since if N - lim x n = x and N-lim x n = y for some x, y X, it follows from (N4) that
N ( x - y , t ) min N x - x n , t 2 , N x n - y , t 2

for all t > 0 and n . So, N(x - y, t) = 1 for all t > 0. Hence, (N2) implies that x = y.

Definition 1.5. Let (X, N) be a fuzzy normed space. A sequence {x n } in X is called a Cauchy sequence if, for any ε > 0 and t > 0, there exists M such that, for all nM and p > 0,
N ( x n + p - x n , t ) > 1 - ε .

It follows from (N4) that every convergent sequence in a fuzzy normed space is a Cauchy sequence. If, in a fuzzy normed space, every Cauchy sequence is convergent, then the fuzzy norm is said to be complete, and the fuzzy normed space is called a fuzzy Banach space.

Example 1.6. [21] Let N : × → [0, 1] be a fuzzy norm on defined by
N ( x , t ) = t t + | x | , t > 0 , 0 , t 0 .

Then, (, N) is a fuzzy Banach space.

Recently, several various fuzzy stability results concerning a Cauchy sequence, Jensen and quadratic functional equations were investigated in [1720].

2. A local Hyers-Ulam stability of Jensen's equation

In 1998, Jung [16] investigated the Hyers-Ulam stability for Jensen's equation on a restricted domain. In this section, we prove a local Hyers-Ulam stability of the Pexiderized Jensen functional equation in fuzzy normed spaces.

Theorem 2.1. Let X be a normed space, (Y, N) be a fuzzy Banach space, and f, g, h : XY be mappings with f(0) = 0. Suppose that δ > 0 is a positive real number, and z0is a fixed vector of a fuzzy normed space (Z, N') such that
N 2 f x + y 2 - g ( x ) - h ( y ) , t + s min { N ( δ z 0 , t ) , N ( δ z 0 , s ) }
(2.1)
for all x, y X with ||x|| + ||y|| ≥ d and positive real numbers t, s. Then, there exists a unique additive mapping T : XY such that
N ( f ( x ) - T ( x ) , t ) N ( 4 0 δ z 0 , t ) ,
(2.2)
N ( T ( x ) - g ( x ) + g ( 0 ) , t ) N ( 3 0 δ z 0 , t ) ,
(2.3)
N ( T ( x ) - h ( x ) + h ( 0 ) , t ) N ( 3 0 δ z 0 , t )
(2.4)

for all x X and t > 0.

Proof. Suppose that ||x|| + ||y|| < d holds. If ||x|| + ||y|| = 0, let z X with ||z|| = d. Otherwise,
z : = ( d + x ) x x , i f x y , ( d + y ) y y , i f x < y .
It is easy to verify that
x - z + y + z d , 2 z + x - z d , y + 2 z d , y + z + z d , x + z d .
(2.5)
It follows from (N4), (2.1) and (2.5) that
N 2 f x + y 2 - g ( x ) - h ( y ) , t + s min N 2 f x + y 2 - g ( y + z ) - h ( x - z ) , t + s 5 , N 2 f x + z 2 - g ( 2 z ) - h ( x - z ) , t + s 5 , N 2 f y + 2 z 2 - g ( 2 z ) - h ( y ) , t + s 5 , N 2 f y + 2 z 2 - g ( y + z ) - h ( z ) , t + s 5 , N 2 f x + z 2 - g ( x ) - h ( z ) , t + s 5 min { N ( 5 δ z 0 , t ) , N ( 5 δ z 0 , s ) }
for all x, y X with ||x|| + ||y|| < d and positive real numbers t, s. Hence, we have
N 2 f x + y 2 - g ( x ) - h ( y ) , t + s min { N ( 5 δ z 0 , t ) , N ( 5 δ z 0 , s ) }
(2.6)
for all x, y X and positive real numbers t, s. Letting x = 0 (y = 0) in (2.6), we get
N 2 f y 2 - g ( 0 ) - h ( y ) , t + s min { N ( 5 δ z 0 , t ) , N ( 5 δ z 0 , s ) } , N 2 f x 2 - g ( x ) - h ( 0 ) , t + s min { N ( 5 δ z 0 , t ) , N ( 5 δ z 0 , s ) }
(2.7)
for all x, y X and positive real numbers t, s. It follows from (2.6) and (2.7) that
N 2 f x + y 2 - 2 f x 2 - 2 f y 2 , t + s min N 2 f x + y 2 - g ( x ) - h ( y ) , t + s 4 , N 2 f x 2 - g ( x ) - h ( 0 ) , t + s 4 , N 2 f y 2 - g ( 0 ) - h ( y ) , t + s 4 , N ( g ( 0 ) + h ( 0 ) , t + s 4 min { N ( 2 0 δ z 0 , t ) , N ( 2 0 δ z 0 , s ) }
for all x, y X and positive real numbers t, s. Hence,
N f ( x + y ) - f ( x ) - f ( y ) , t + s min { N ( 1 0 δ z 0 , t ) , N ( 1 0 δ z 0 , s ) }
(2.8)
for all x, y X and positive real numbers t, s. Letting y = x and t = s in (2.8), we infer that
N f ( 2 x ) 2 - f ( x ) , t N ( 1 0 δ z 0 , t )
(2.9)
for all x X and positive real number t. replacing x by 2 n x in (2.9), we get
N f ( 2 n + 1 x ) 2 n + 1 - f ( 2 n x ) 2 n , t 2 n N ( 1 0 δ z 0 , t )
(2.10)
for all x X, n ≥ 0 and positive real number t. It follows from (2.10) that
N f ( 2 n x ) 2 n - f ( 2 m x ) 2 m , k = m n - 1 t 2 k min k = m n - 1 N f ( 2 k + 1 x ) 2 k + 1 - f ( 2 k x ) 2 k , t 2 k N ( 1 0 δ z 0 , t )
(2.11)
for all x X, t > 0 and integers nm ≥ 0. For any s, ε > 0, there exist an integer l > 0 and t0 > 0 such that N'(10δz0, t0) > 1 - ε and k = m n - 1 t 0 2 k > s for all nml. Hence, it follows from (2.11) that
N f ( 2 n x ) 2 n - f ( 2 m x ) 2 m , s > 1 - ε
for all nml. So { f ( 2 n x ) 2 n } is a Cauchy sequence in Y for all x X. Since (Y, N) is complete, { f ( 2 n x ) 2 n } converges to a point T(x) Y. Thus, we can define a mapping T : XY by T ( x ) : = N - lim n f ( 2 n x ) 2 n . Moreover, if we put m = 0 in (2.11), then we observe that
N f ( 2 n x ) 2 n - f ( x ) , k = 0 n - 1 t 2 k N ( 1 0 δ z 0 , t ) .
Therefore, it follows that
N f ( 2 n x ) 2 n - f ( x ) , t N 1 0 δ z 0 , t k = 0 n - 1 2 - k )
(2.12)

for all x X and positive real number t.

Next, we show that T is additive. Let x, y X and t > 0. Then, we have
N T ( x + y ) - T ( x ) - T ( y ) , t min N T ( x + y ) - f ( 2 n ( x + y ) ) 2 n , t 4 , N f ( 2 n x ) 2 n - T ( x ) , t 4 , N f ( 2 n y ) 2 n - T ( y ) , t 4 , N f ( 2 n ( x + y ) ) 2 n - f ( 2 n x ) 2 n - f ( 2 n y ) 2 n , t 4 .
(2.13)
Since, by (2.8),
N f ( 2 n ( x + y ) ) 2 n - f ( 2 n x ) 2 n - f ( 2 n y ) 2 n , t 4 N ( 4 0 δ z 0 , 2 n t ) ,
we get
lim n N f ( 2 n ( x + y ) ) 2 n - f ( 2 n x ) 2 n - f ( 2 n y ) 2 n , t 4 = 1 .

By the definition of T, the first three terms on the right hand side of the inequality (2.13) tend to 1 as n → ∞. Therefore, by tending n → ∞ in (2.13), we observe that T is additive.

Next, we approximate the difference between f and T in a fuzzy sense. For all x X and t > 0, we have
N ( T ( x ) - f ( x ) , t ) min N T ( x ) - f ( 2 n x ) 2 n , t 2 , N f ( 2 n x ) 2 n - f ( x ) , t 2 .
Since T ( x ) : = N - lim n f ( 2 n x ) 2 n , letting n → ∞ in the above inequality and using (N) and (2.12), we get (2.2). It follows from the additivity of T and (2.7) that
N ( T ( x ) - g ( x ) + g ( 0 ) , t ) min N 2 T x 2 - 2 f x 2 , t 3 , N 2 f x 2 - g ( x ) - h ( 0 ) , t 3 , N g ( 0 ) + h ( 0 ) , t 3 N ( 3 0 δ z 0 , t )

for all x X and t > 0. So, we get (2.3). Similarly, we can obtain (2.4).

To prove the uniqueness of T, let S : XY be another additive mapping satisfying the required inequalities. Then, for any x X and t > 0, we have
N ( T ( x ) - S ( x ) , t ) min N T ( x ) - f ( x ) , t 2 , N f ( x ) - S ( x ) , t 2 N ( 8 0 δ z 0 , t ) .
Therefore, by the additivity of T and S, it follows that
N ( T ( x ) - S ( x ) , t ) = N ( T ( n x ) - S ( n x ) , n t ) N ( 8 0 δ z 0 , n t )

for all x X, t > 0 and n ≥ 1. Hence, the right hand side of the above inequality tends to 1 as n → ∞. Therefore, T(x) = S(x) for all x X. This completes the proof.    □

The following is a local Hyers-Ulam stability of the Pexiderized Cauchy functional equation in fuzzy normed spaces.

Theorem 2.2. Let X be a normed space, (Y, N) be a fuzzy Banach space, and f, g, h : XY be mappings with f(0) = 0. Suppose that δ > 0 is a positive real number, and z0is a fixed vector of a fuzzy normed space (Z, N') such that
N ( f ( x + y ) - g ( x ) - h ( y ) , t + s ) min { N ( δ z 0 , t ) , N ( δ z 0 , s ) }
(2.14)
for all x, y X with ||x|| + ||y|| ≥ d and positive real numbers t, s. Then, there exists a unique additive mapping T : XY such that
N ( f ( x ) - T ( x ) , t ) N ( 8 0 δ z 0 , t ) , N ( T ( x ) - g ( x ) + g ( 0 ) , t ) N ( 6 0 δ z 0 , t ) , N ( T ( x ) - h ( x ) + h ( 0 ) , t ) N ( 6 0 δ z 0 , t )

for all x X and t > 0.

Proof. For the case ||x|| + ||y|| < d, let z be an element of X which is defined in the proof of Theorem 2.1. It follows from (N4), (2.5) and (2.14) that
N ( f ( x + y ) - g ( x ) - h ( y ) , t + s ) min N f ( x + y ) - g ( y + z ) - h ( x - z ) , t + s 5 , N f ( x + z ) - g ( 2 z ) - h ( x - z ) , t + s 5 , N f ( y + 2 z ) - g ( 2 z ) - h ( y ) , t + s 5 , N f ( y + 2 z ) - g ( y + z ) - h ( z ) , t + s 5 , N f ( x + z ) - g ( x ) - h ( z ) , t + s 5 min { N ( 5 δ z 0 , t ) , N ( 5 δ z 0 , s ) }
for all x, y X with ||x|| + ||y|| < d and positive real numbers t, s. Hence, we have
N f ( x + y ) - g ( x ) - h ( y ) , t + s min { N ( 5 δ z 0 , t ) , N ( 5 δ z 0 , s ) }
(2.15)
for all x, y X and positive real numbers t, s. Letting x = 0 (y = 0) in (2.15), we get
N ( f ( y ) - g ( 0 ) - h ( y ) , t + s ) min { N ( 5 δ z 0 , t ) , N ( 5 δ z 0 , s ) } , N ( f ( x ) - g ( x ) - h ( 0 ) , t + s ) min { N ( 5 δ z 0 , t ) , N ( 5 δ z 0 , s ) }
(2.16)
for all x, y X and positive real numbers t, s. It follows from (2.15) and (2.16) that
N ( f ( x + y ) - f ( x ) - f ( y ) , t + s ) min N f ( x + y ) - g ( x ) - h ( y ) , t + s 4 , N f ( x ) - g ( x ) - h ( 0 ) , t + s 4 , N f ( y ) - g ( 0 ) - h ( y ) , t + s 4 , N ( g ( 0 ) + h ( 0 ) , t + s 4 ) min { N ( 2 0 δ z 0 , t ) , N ( 2 0 δ z 0 , s ) }

for all x, y X and positive real numbers t, s. The rest of the proof is similar to the proof of Theorem 2.1, and we omit the details.    □

Declarations

Acknowledgements

This work was supported by the Korea Research Foundation (KRF) grant funded by the Korea government (MEST) (no. 2009-0075850).

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili
(2)
National Institute for Mathematical Sciences, KT Daeduk 2 Research Center
(3)
Department of Mathematics Education and the RINS, Gyeongsang National University

References

  1. Ulam SM: Problems in Modern Mathematics. Volume chap. VI. Science edition. New York:Wiley; 1964.Google Scholar
  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 ArticleGoogle Scholar
  3. Aoki T: On the stability of the linear transformationin Banach spaces. J Math Soc Japan 1950, 2: 64–66. 10.2969/jmsj/00210064MathSciNetView ArticleGoogle Scholar
  4. Rassias ThM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View ArticleGoogle Scholar
  5. Bourgin DG: Classes of transformations and bordering transformations. Bull Am Math Soc 1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7MathSciNetView ArticleGoogle Scholar
  6. Forti GL: An existence and stability theorem for a class of functional equations. Stochastica 1980, 4: 23–30. 10.1080/17442508008833155MathSciNetView ArticleGoogle Scholar
  7. Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl 1994, 184: 431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleGoogle Scholar
  8. Bae J, Jun K, Lee Y: On the Hyers-Ulam-Rassias stability of an n -dimensional Pexiderized quadratic equation. Math Inequal Appl 2004, 7: 63–77.MathSciNetGoogle Scholar
  9. Faizev VA, Rassias ThM, Sahoo PK: The space of ( ψ , φ )-additive mappings on semigroups. Trans Am Math Soc 2002, 354: 4455–4472. 10.1090/S0002-9947-02-03036-2View ArticleGoogle Scholar
  10. Forti GL: Hyers-Ulam stability of functional equations in several variables. Aequ Math 1995, 50: 143–190. 10.1007/BF01831117MathSciNetView ArticleGoogle Scholar
  11. Forti GL: Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations. J Math Anal Appl 2004, 295: 127–133. 10.1016/j.jmaa.2004.03.011MathSciNetView ArticleGoogle Scholar
  12. Haruki H, Rassias ThM: A new functional equation of Pexider type related to the complex exponential function. Trans Am Math Soc 1995, 347: 3111–3119. 10.2307/2154775MathSciNetView ArticleGoogle Scholar
  13. Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.View ArticleGoogle Scholar
  14. Hyers DH, Isac G, Rassias ThM, et al.: On the asymptoticity aspect of Hyers-Ulam stability of mappings. Proc Am Math Soc 1998, 126: 425–430. 10.1090/S0002-9939-98-04060-XMathSciNetView ArticleGoogle Scholar
  15. Jun K, Lee Y: On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality. Math Inequal Appl 2001, 4: 93–118.MathSciNetGoogle Scholar
  16. Jung S-M: Hyers-Ulam-Rassias stability of Jensen's equation and its application. Proc Am Math Soc 1998, 126: 3137–3143. 10.1090/S0002-9939-98-04680-2View ArticleGoogle Scholar
  17. Miheţ D: The fixed point method for fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst 2009, 160: 1663–1667. 10.1016/j.fss.2008.06.014View ArticleGoogle Scholar
  18. Miheţ D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces. J Math Anal Appl 2008, 343: 567–572.MathSciNetView ArticleGoogle Scholar
  19. Mirmostafaee M, Mirzavaziri M, Moslehian MS: Fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst 2008, 159: 730–738. 10.1016/j.fss.2007.07.011MathSciNetView ArticleGoogle Scholar
  20. Mirmostafee AK, Moslehian MS: Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets Syst 2008, 159: 720–729. 10.1016/j.fss.2007.09.016View ArticleGoogle Scholar
  21. Najati A: Fuzzy stability of a generalized quadratic functional equation. Commun Korean Math Soc 2010, 25: 405–417. 10.4134/CKMS.2010.25.3.405MathSciNetView ArticleGoogle Scholar
  22. 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 ArticleGoogle Scholar
  23. Najati A, Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation. J Math Anal Appl 2007, 335: 763–778. 10.1016/j.jmaa.2007.02.009MathSciNetView ArticleGoogle Scholar
  24. Rassias ThM, Tabor J, (eds.): Stability of Mappings of Hyers-Ulam Type. Hadronic Press Inc. Florida; 1994.Google Scholar
  25. Rassias ThM: On the stability of the quadratic functional equation and its applications. Studia Univ Babes Bolyai Math 1998, 43: 89–124.MathSciNetGoogle Scholar
  26. Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Appl Math 2000, 62: 23–130. 10.1023/A:1006499223572MathSciNetView ArticleGoogle Scholar
  27. Rassias ThM, (ed.): Functional Equations and Inequalities. Kluwer Academic Publishers, Dordrecht; 2000.Google Scholar
  28. Rassias ThM: On the stability of functional equations in Banach spaces. J Math Anal Appl 2000, 251: 264–284. 10.1006/jmaa.2000.7046MathSciNetView ArticleGoogle Scholar
  29. Šemrl P: On quadratic functionals. Bull Aust Math Soc 1987, 37: 27–28.Google Scholar
  30. Bag T, Samanta SK: Finite dimensional fuzzy normed linear spaces. J Fuzzy Math 2003, 11: 687–705.MathSciNetGoogle Scholar

Copyright

© Najati et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.