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Approximate lie brackets: a fixed point approach

Abstract

The aim of this article is to investigate the stability and superstability of Lie brackets on Banach spaces by using fixed point methods.

2010 Mathematics Subject Classification: 46L06; 39B82; 39B52.

1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Rassias [3] considered the stability problem with unbounded Cauchy differences. The stability problems of several functional equations have extensively been investigated by a number of authors and there are many interesting results concerning this problem (see [418]).

In 2003, Cǎdariu and Radu applied the fixed point method and they could present a short and simple proof (different from the "direct method", initiated by Hyers in 1941) for the generalized Hyers-Ulam stability of Jensen functional equation.

In this article, by using the fixed point method, we prove that, if there exists an approximately Lie bracket f : A × A → A on Banach spaces A, then there exists a Lie bracket T : A × AA which is near to f. Moreover, under some conditions on f, the Banach space A has a Lie algebra structure with Lie bracket T.

We recall a Lie algebra consists of a (finite dimensional) vector space A over a field F and a multiplication in A (usually, the product of x, y A is denoted by [x, y] and called a Lie bracket or commutator) with the following two properties:

  1. (1)

    Anti-commutativity: [x, x] = 0 for any x A;

  2. (2)

    Jacobi identity: [z, [x, y]] = [[z, x], y] + [x, [z, y]] for any x, y, z A.

For more details about Lie algebras, the readers are referred to [1922]. Throughout this article, we assume that n0 is a positive integer,

T 1 : = { z : ǀ z ǀ = 1 } , T 1 n o 1 : = { e i θ : 0 θ 2 π n 0 } .

It is easy to see that T 1 = T 1 1 1 . Moreover, we suppose that A is a complex Banach space. For any mapping f : A × AA, we define

D μ f ( x , y , z , t ) : = 4 μ f x + y 2 , z + t 2 + 4 μ f x - y 2 , z + t 2 + 4 μ f x + y 2 + z - t 2 + 4 μ f x - y 2 , z - t 2 - 4 f ( μ x , z )

for all μ T 1 n o 1 and x, y, z, t A.

2. Main results

We need the following theorem to prove the main result of this article.

Theorem 2.1. (The alternative of fixed point theorem [23, 24]) Suppose that (Ω, d) is a complete generalized metric space and T : Ω → Ω is a strictly contractive mapping with Lipschitz constant L. Then, for any x Ω, either d(Tmx, Tm+1x) = ∞ for all m ≥ 0 or there exists a natural number m0 such that

  1. (1)

    d(T mx, T m+1x) < 1 for all mm0;

  2. (2)

    the sequence {Tmx} is convergent to a fixed point y* of T;

  3. (3)

    y* is the unique fixed point of T in the set Λ= { y Ω : d ( T m 0 x , y ) < ;

  4. (4)

    d ( y , y * ) 1 1 - L d ( y , T y ) for all y Λ.

Now, we give our main results by using. Theorem 2.1.

Theorem 2.2. Let f : A × AA be a continuous mapping and let ϕ : A4 = A × A × A × A → [0, ∞) be a mapping such that

ǁ D μ f ( x , y , z , t ) ǁ ϕ ( x , y , z , t ) ,
(2.1)
lim n 4 - n f ( 2 n z , 2 n f ( x , y ) ) - f ( 4 - n f ( 2 n z , 2 n x ) , y ) - f ( x , 4 - n f ( 2 n z , 2 n y ) ) ϕ ( x , y , 0 , 0 ) ,
(2.2)
lim n 4 - n f ( 2 n x , 2 n x ) = 0
(2.3)

for all μ T 1 n o 1 and x, y, z, t A. If there exists L < 1 such that ϕ ( x , y , z , t ) 4Lϕ ( x 2 , y 2 , z 2 , t 2 ) for all x, y, z, t A, then there exists a unique bilinear mapping T : A × AA such that

f ( x , z ) - T ( x , z ) L 1 - L ϕ ( x , 0 , z , 0 )
(2.4)

for all x, z M. Moreover, for any sequence {a m } in A, if

lim m lim n 4 - n f ( 2 n x , 2 n a m ) = lim n lim m 4 - n f ( 2 n x , 2 n a m )
(2.5)

for all x A, then A is a Lie algebra with Lie bracket [x, y] = T (x, y) for all x, y A.

Proof. Putting μ = 1 and y = t = 0 in (2.1), we get

4 f ( x 2 , z 2 ) - f ( x , z ) ϕ ( x , 0 , z , 0 )

for all x, z A and so

1 4 f ( 2 x , 2 z ) - f ( x , z ) 1 4 ϕ ( 2 x , 0 , 2 z , 0 ) L ϕ ( x , 0 , z , 0 )
(2.6)

for all x, z A. Consider the set X : = {g : g : A × AA} and introduce the generalized metric on X by:

d ( h , g ) : = inf { C + : ǁ g ( x , z ) - h ( x , z ) ǁ C ϕ ( x , 0 , z , 0 ) for all  x , z A } .

It is easy to show that (X, d) is a complete generalized metric space. Now, we define the mapping J : XX by

J ( h ) ( x , z ) = 1 4 h ( 2 x , 2 z )

for all x, z A. For any g, h X, we have

d ( g , h ) < C g ( x , z ) - h ( x , z ) C ϕ ( x , 0 , z , 0 ) 1 4 g ( 2 x , 2 z ) - 1 4 h ( 2 x , 2 z ) 1 4 C ϕ ( 2 x , 0 , 2 z , 0 ) 1 4 g ( 2 x , 2 z ) - 1 4 h ( 2 x , 2 z ) L C ϕ ( x , 0 , z , 0 ) d ( J ( g ) , J ( h ) ) L C

for all x, z A, which means that

d ( J ( g ) , J ( h ) ) Ld ( g , h )

for all g, h X. It follows from (2.6) that

d ( f , J ( f ) ) L.

From Theorem 2.1, it follows that J has a unique fixed point in the set X1:= {I X: d(f, T) < ∞}. Let T be the fixed point of J. Then we have limn→∞d(Jn (f), T) = 0 and

lim n 1 4 n f ( 2 n x , 2 n z ) = T ( x , z )
(2.7)

for all x, z A. By the inequality d(f, J(f)) ≤ L and J(T) = T, we have

d ( f , T ) d ( f , J ( f ) ) +d ( J ( f ) , J ( T ) ) L+Ld ( f , T )

and so

d ( f , T ) L 1 - L .

This implies the inequality (2.4). From ϕ ( x , y , z , t ) 4 L ϕ x 2 , y 2 , z 2 , t 2 , we have

lim j 4 - j ϕ ( 2 j x , 2 j y , 2 j z , 2 j t ) = 0
(2.8)

for all x, y, z A. Thus it follows from (2.1), (2.7) and (2.8) that

4 T x + y 2 , z + t 2 + 4 T x - y 2 , z + t 2 - 4 T x + y 2 , z - t 2 + 4 T x - y 2 , z - t 2 - 4 T ( x , z ) = lim n 1 4 n 4 f 2 n x + 2 n y 2 , 2 n z + 2 n t 2 + 4 f 2 n x - 2 n y 2 - 2 n z + 2 n t 2 - 4 f 2 n x + 2 n y 2 , 2 n z - 2 n t 2 - 4 f 2 n x - 2 n y 2 , 2 n z - 2 n t 2 - 4 f ( 2 n x , 2 n z ) lim n 1 4 n ϕ ( 2 n x , 2 n y , 2 n z , 2 n t ) = 0

for all x, y, z M and so

T x + y 2 , z + t 2 + T x - y 2 , z + t 2 - T x + y 2 , z - t 2 + T x - y 2 , z - t 2 = T ( x , z )

for all x, y, z, t A. This shows that

T ( x + y , z + t ) =T ( x , z ) +T ( y , z ) +T ( x , t ) +T ( y , t )

for all x, y, z, t A. Hence, T is Cauchy additive with respect to the first and second variables. By putting y : = x and t : = z in (2.1), we have

4 μ f ( x , z ) - 4 f ( μ x , z ) ϕ ( x , x , z , z )
(2.9)

for all x, z A. and μ T 1 n o 1 and so

4 μ T ( x , z ) - 4 T ( μ x , z ) = lim n 1 4 n 4 μ f ( 2 n x , 2 n z ) - 4 f ( 2 n μ x , 2 n z ) lim n 1 4 n ϕ ( 2 n x , 2 n x , 2 n z , 2 n z ) = 0

for all x, z A and μ T 1 n o 1 , that is,

T ( μ x , z ) =μT ( x , z )
(2.10)

for all x, z A.

If λ belongs to T 1 , then there exists θ [0, 2π] such that λ = e. If we set λ 1 = e i θ n o , then λ1 belongs to T 1 n o 1 . By using (2.10), we have

T ( λ x , z ) =T ( λ 1 n 0 x , z ) = λ 1 n 0 T ( x , z ) =λT ( x , z )

for all x, z M.

If λ belongs to n T 1 = { n z : z T 1 } for some n , then, by (2.9), we have

T ( λ x , z ) = T ( n λ 1 x , z ) = T ( λ 1 ( n x ) , z ) = λ 1 T ( n x , z ) = λ 1 n T ( x , z ) = λ T ( x , z )

for all x, z A. Let s (0, ∞). Then, by Archimedean property of , there exists a positive real number n such that the point (s, 0) 2 lies in the interior of circle with center at origin and radius n in 2. Putting s 1 :=s+ n 2 - s 2 i and s 2 :=t- n 2 - s 2 i, we have s = s 1 + s 2 2 and s 1 , s 2 n T 1 . Thus, by (2.9), we have

T ( s x , z ) = T s 1 + s 2 2 x , z = T s 1 x 2 , z + T s 2 x 2 , z = s 1 T x 2 , z + s 2 T ( x 2 , z ) = 4 s 1 + s 2 2 T x 2 , z 2 = s T ( x , z )

for all x, z s. Moreover, there exists θ [0, 2π] such that λ = ǀλ ǀ e.

Therefore, we have

T ( λ x , z ) =T ( λ e i θ x , z ) = λ T ( e i θ x , z ) = λ e i θ T ( x , z ) =λT ( x , z )
(2.11)

for all x, z A and so T : A × AA is homogeneous with respect to the first variable. It follows from (2.9) and (2.11) that T is -Linear with respect to the first variable.

Moreover, by (2.3), T (x, x) = 0 for all x A, whence

0=T ( x + y , x + y ) =T ( x , x ) +T ( x , y ) +T ( y , x ) +T ( y , y ) =T ( x , y ) +T ( y , x )

for all x, y A and so

T ( x , y ) =-T ( y , x )

for all x, y A, that is, T is skew symmetric. Let z A and define a mapping ad(z): AA by

ad ( z ) ( x ) =T ( z , x )

for all x A. It is clear that ad(z) is a linear and continuous mapping at zero. In fact, if {a m } is a sequence in A such that limn→∞a m = 0, then, by (2.5), we have

lim m a d ( z ) ( a m ) = lim m lim n 4 - n f ( 2 n z , 2 n a m ) = lim n lim m 4 - n f ( 2 n z , 2 n a m ) = lim n 4 - n f ( 2 n z , 0 ) = a d ( z ) ( 0 ) = 0 .

Thus, for all z A, ad(z) is continuous at zero and so ad(z) is a continuous and linear mapping. Substituting x with 2mx and y with 2my in (2.2) and multiplying by 4-mboth sides of the inequality, we have

lim n 4 - m 4 - n f ( 2 n z , 2 n f ( 2 m x , 2 m y ) ) - f ( 4 - n f ( 2 n z , 2 n + m x ) , 2 m y ) - f ( 2 m x , 4 - n f ( 2 n z , 2 n + m y ) ) 4 - m ϕ ( 2 m x , 2 m y , 0 , 0 )

for all x, y, z A and m . Since f is continuous, we have

4 - m a d ( z ) ( f ( 2 m x , 2 m y ) ) - f ( a d ( z ) ( 2 m x ) , 2 m y ) - f ( 2 m x , a d ( z ) 2 m y ) 4 - m ϕ ( 2 m x , 2 m y , 0 , 0 )

for all x, y, z A. Since, for all z A, ad(z) is a linear and continuous mapping, we get

ad ( z ) T ( x , y ) -T ( a d ( z ) ( x ) , y ) -T ( x , a d ( z ) ( y ) ) =0

for all x, y, z A. Since T is skew symmetric, it is easy to show that T is satisfies in the Jacobi identity condition. Thus T is a Lie bracket satisfies in (2.4) and (A, T) is a Lie algebra.

To prove the uniqueness property of T, let Q : A × AA be another bilinear mapping satisfying (2.7). Then we have

T ( x , z ) - Q ( x , z ) = lim n f ( 2 n x , 2 n z ) 4 n - Q ( 2 n x , 2 n z ) 4 n lim n 1 4 n L 1 - L ϕ ( 2 n x , 0 , 2 n z , 0 ) = 0

for all x, z A. This means that T = Q. This completes the proof. □

Corollary 2.3. Let p (0, 1) and θ [0, ∞) be real numbers. Suppose that f : A × AA is a mapping such that

D μ f ( x , y , z , t ) θ x p + y p + z p + t p ,
lim n 4 - n f ( 2 n z , 2 n f ( x , y ) ) - f ( 4 - n f ( 2 n z , 2 n x ) , y ) - f ( x , 4 - n f ( 2 n z , 2 n y ) ) θ ( ǁ x ǀ ǀ p + ǁ y ǀ ǀ p ) ,
lim n 4 - n f ( 2 n x , 2 n x ) =0

for all μ T 1 n o 1 and x, y, z, t A. Then there exists a unique bilinear mapping T : A × AA such that

ǀf ( x , z ) -T ( x , z ) ǁ 4 p θ 4 - 4 p ( ǁ x ǀ ǀ p + ǁ z ǀ ǀ p )

for all x, z A. Moreover, for any sequence {a m } in A, if

lim m lim n 4 - n f ( 2 n x , 2 n a m ) = lim n lim m 4 - n f ( 2 n x , 2 n a m )

for all x A, then A is a Lie algebra with Lie bracket [x, y] = T(x, y) for all x, y A.

Proof. It follows from Theorem 2.2 by putting ϕ(x, y, z): = θx ǁ py ǁ pz ǁ pt ǁ p ) for all x, y, z , M and L = 4p -1. □

Finally, we prove the superstability of Lie brackets as follows:

Corollary 2.4. Let p 0 , 1 4 and θ [0, ∞) be real numbers. Suppose that f: A × AA is a mapping such that

ǁ D μ f ( x , y , z , t ) ǁθ ( ǁ x ǀ ǀ p ǁ y ǀ ǀ p ǁ z ǀ ǀ p ǁ t ǀ ǀ p ) ,
lim n ǁ 4 - n f ( 2 n z , 2 n f ( x , y ) ) - f ( 4 - n f ( 2 n z , 2 n x ) , y ) - f ( x , 4 - n f ( 2 n z , 2 n y ) ) ǁ θ ( ǁ x ǀ ǀ p ǁ y ǀ ǀ p ) ,
lim n 4 - n f ( 2 n x , 2 n x ) =0

for all μ T 1 n o 1 and x, y, z, t A. Moreover, for any sequence {a m } in A, if

lim m lim n 4 - n f ( 2 n x , 2 n a m ) = lim n lim m 4 - n f ( 2 n x , 2 n a m )

for all x A, then A is a Lie algebra with Lie bracket [x, y] = f(x, y) for all x, y A.

Proof. Putting ϕ(x, y, z, t): = θx ǁp ǁy ǁp ǁz ǁp ǁt ǁp) for all x, y, z M and L= 1 2 in Theorem 2.2, the conclusion follows. □

References

  1. Ulam SM: Problems in Modern Mathematics, Chapter VI. Wiley, New York; 1940. Science Ed

    Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  4. Agarwal RP, Cho YJ, Saadati R, Wang S: Nonlinear L -fuzzy stability of cubic functional equations. J Inequal Appl 2012, 2012: 77. 10.1186/1029-242X-2012-77

    MathSciNet  Article  Google Scholar 

  5. Baktash E, Cho YJ, Jalili M, Saadati R, Vaezpour SM: On the stability of cubic mappings and quadratic mappings in random normed spaces. J Inequal Appl 2008: 11. Article ID 902187

    Google Scholar 

  6. Brzdek J, Popa D, Xu B: Hyers-Ulam stability for linear equations of higher orders. Acta Math Hungar 2008, 120: 1–8. 10.1007/s10474-007-7069-3

    MathSciNet  Article  Google Scholar 

  7. Brzdek J: On stability of a family of functional equations. Acta Math Hungar 2010, 128: 139–149. 10.1007/s10474-010-9169-8

    MathSciNet  Article  Google Scholar 

  8. Brzdek J: On approximately microperiodic mappings. Acta Math Hungar 2007, 117: 179–186. 10.1007/s10474-007-6087-5

    MathSciNet  Article  Google Scholar 

  9. Cǎdariu L, Radu V: The fixed points method for the stability of some functional equations. Carpathian J Math 2007, 23: 63–72.

    MathSciNet  Google Scholar 

  10. Cho YJ, Eshaghi Gordji M, Zolfaghari S: Solutions and stability of generalized mixed type QC functional equations in random normed spaces. J Inequal Appl 2010, 2010: 16. Article ID 403101

    MathSciNet  Google Scholar 

  11. Cho YJ, Park C, Rassias ThM, Saadati R: Inner product spaces and functional equations. J Comput Anal Appl 2011, 13: 296–304.

    MathSciNet  Google Scholar 

  12. Cho YJ, Kang JI, Saadati R: Fixed points and stability of additive functional equations on the Banach algebras. J Comput Anal Appl 2012, 14: 1103–1111.

    MathSciNet  Google Scholar 

  13. Cho YJ, Kang SM, Sadaati R: Nonlinear random stability via fixed-point method. J Appl Math 2012: 44. Article ID 902931

    Google Scholar 

  14. Cho YJ, Park C, Saadati R: Functional inequalities in non-Archimedean in Banach spaces. Appl Math Lett 2010, 60: 1994–2002.

    MathSciNet  Google Scholar 

  15. Cho YJ, Saadati R: Lattice non-Archimedean random stability of ACQ functional equation. Advan in Diff Equat 2011, 2011: 31. 10.1186/1687-1847-2011-31

    MathSciNet  Article  Google Scholar 

  16. Cho YJ, Saadati R, Vahidi J: Approximation of homomorphisms and derivations on non-Archimedean Lie C* -algebras via fixed point method. Discrete Dynamics in Nature and Society 2012, 2012: 9. Article ID 373904

    Google Scholar 

  17. Eshaghi Gordji M, Khodaei H: Stability of Functional Equations. LAP Lambert Academic Publishing, Saarbrucken; 2010.

    Google Scholar 

  18. Khodaei H, Rassias ThM: Approximately generalized additive functions in several variables. Internat J Nonlinear Anal Appl 2010, 1: 22–41.

    Google Scholar 

  19. Bourbaki N: Lie Groups and Lie Algebras--Chapters 1–3. Springer, New York; 1989. ISBN 3–540–64242–0

    Google Scholar 

  20. Erdmann K, Wildon M: Introduction to Lie Algebras. 1st edition. Springer, New York; 2006. ISBN 1–84628–040–0

    Book  Google Scholar 

  21. Humphreys JE: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. Volume 9. 2nd edition. Springer-Verlag, New York; 1978. ISBN 0–387–90053–5

    Google Scholar 

  22. Varadarajan VS: Lie Groups, Lie Algebras, and Their Representations. 1st edition. Springer, New York; 2004. ISBN 0–387–90969–9

    Google Scholar 

  23. Margolis B, Diaz JB: A fixed point theorem of the alternative for contractions on the generalized complete metric space. Bull Am Math Soc 1968, 126: 305–309.

    MathSciNet  Google Scholar 

  24. Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4: 91–96.

    MathSciNet  Google Scholar 

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Acknowledgements

This study was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2011-0021821).

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Correspondence to Madjid Eshaghi Gordji or Yeol JE Cho.

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Gordji, M.E., Ramezani, M., Cho, Y.J. et al. Approximate lie brackets: a fixed point approach. J Inequal Appl 2012, 125 (2012). https://doi.org/10.1186/1029-242X-2012-125

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Keywords

  • generalized Hyers-Ulam stability
  • fixed point
  • superstability
  • Lie algebra
  • skew-symmetry
  • Jacobi identity.