Skip to main content

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

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.

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 2f 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 2f 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 2nx 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 10δ 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.    □

References

  1. 1.

    Ulam SM: Problems in Modern Mathematics. Volume chap. VI. Science edition. New York:Wiley; 1964.

    Google Scholar 

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 4.

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

    Article  Google Scholar 

  5. 5.

    Bourgin DG: Classes of transformations and bordering transformations. Bull Am Math Soc 1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7

    MathSciNet  Article  Google Scholar 

  6. 6.

    Forti GL: An existence and stability theorem for a class of functional equations. Stochastica 1980, 4: 23–30. 10.1080/17442508008833155

    MathSciNet  Article  Google Scholar 

  7. 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.1994.1211

    MathSciNet  Article  Google Scholar 

  8. 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.

    MathSciNet  Google Scholar 

  9. 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/S0002-9947-02-03036-2

    Article  Google Scholar 

  10. 10.

    Forti GL: Hyers-Ulam stability of functional equations in several variables. Aequ Math 1995, 50: 143–190. 10.1007/BF01831117

    MathSciNet  Article  Google Scholar 

  11. 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.2004.03.011

    MathSciNet  Article  Google Scholar 

  12. 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. 10.2307/2154775

    MathSciNet  Article  Google Scholar 

  13. 13.

    Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.

    Google Scholar 

  14. 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-X

    MathSciNet  Article  Google Scholar 

  15. 15.

    Jun K, Lee Y: On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality. Math Inequal Appl 2001, 4: 93–118.

    MathSciNet  Google Scholar 

  16. 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/S0002-9939-98-04680-2

    Article  Google Scholar 

  17. 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.2008.06.014

    Article  Google Scholar 

  18. 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.

    MathSciNet  Article  Google Scholar 

  19. 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.2007.07.011

    MathSciNet  Article  Google Scholar 

  20. 20.

    Mirmostafee AK, Moslehian MS: Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets Syst 2008, 159: 720–729. 10.1016/j.fss.2007.09.016

    Article  Google Scholar 

  21. 21.

    Najati A: Fuzzy stability of a generalized quadratic functional equation. Commun Korean Math Soc 2010, 25: 405–417. 10.4134/CKMS.2010.25.3.405

    MathSciNet  Article  Google Scholar 

  22. 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.2007.03.104

    MathSciNet  Article  Google Scholar 

  23. 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.2007.02.009

    MathSciNet  Article  Google Scholar 

  24. 24.

    Rassias ThM, Tabor J, (eds.): Stability of Mappings of Hyers-Ulam Type. Hadronic Press Inc. Florida; 1994.

  25. 25.

    Rassias ThM: On the stability of the quadratic functional equation and its applications. Studia Univ Babes Bolyai Math 1998, 43: 89–124.

    MathSciNet  Google Scholar 

  26. 26.

    Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Appl Math 2000, 62: 23–130. 10.1023/A:1006499223572

    MathSciNet  Article  Google Scholar 

  27. 27.

    Rassias ThM, (ed.): Functional Equations and Inequalities. Kluwer Academic Publishers, Dordrecht; 2000.

  28. 28.

    Rassias ThM: On the stability of functional equations in Banach spaces. J Math Anal Appl 2000, 251: 264–284. 10.1006/jmaa.2000.7046

    MathSciNet  Article  Google Scholar 

  29. 29.

    Šemrl P: On quadratic functionals. Bull Aust Math Soc 1987, 37: 27–28.

    Google Scholar 

  30. 30.

    Bag T, Samanta SK: Finite dimensional fuzzy normed linear spaces. J Fuzzy Math 2003, 11: 687–705.

    MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jung Im Kang.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Najati, A., Kang, J.I. & Cho, Y.J. Local stability of the Pexiderized Cauchy and Jensen's equations in fuzzy spaces. J Inequal Appl 2011, 78 (2011). https://doi.org/10.1186/1029-242X-2011-78

Download citation

Keywords

  • Pexiderized Cauchy functional equation
  • generalized Hyers-Ulam stability
  • Jensen functional equation
  • non-Archimedean space