Skip to main content

Stability of a functional equation connected with Reynolds operator

Abstract

Let G be a commutative semigroup, K=R or and F:G K n . Generalizing the stability of the functional equation F(xg(y))F(x)F(y)=0 with bounded difference (Najdecki in J. Inequal. Appl. 2007:79816, 2007), we prove the stability of the above functional equation with unbounded differences. We also give a more precise description for bounded components of F=( f 1 , f 2 ,, f n ).

MSC:39B82.

1 Main results

Throughout this paper, G, is a commutative semigroup with an identity e, the set of real numbers, the set of complex numbers, K=R or , ϵ0, and g:GG and ϕ:G[0,) are given functions. For ( a 1 , a 2 ,, a n ),( b 1 , b 2 ,, b n ) K n , we define ( a 1 , a 2 ,, a n )( b 1 , b 2 ,, b n )=( a 1 b 1 , a 2 b 2 ,, a n b n ). A function σ:GG is said to be an involution if σ(xy)=σ(x)σ(y) and σ(σ(x))=x for all x,yG. A function m:G K n is called an exponential function provided that m(xy)=m(x)m(y) for all x,yG.

Generalizing the result of Ger and Šemrl [1], Najdecki [2] proved the stability of the functional equation

F ( x g ( y ) ) F(x)F(y)=0
(1.1)

in the class of functions F:G K n . The particular cases of (1.1) are the exponential equation f(xy)=f(x)f(y) (see Aczél and Dhombres [3] and Baker [4]) and the equation

f ( x f ( y ) ) =f(x)f(y)
(1.2)

for all x,yK{0}, where f:K{0}K{0} (see Brzdȩk [5], Brzdȩk, Najdecki and Xu [6] and Chudziak and Tabor [7] for related equations). As mentioned in [2, 5], (1.2) arises in averaging theory applied to the turbulent fluid motion and is connected with the Reynolds operator (see Marias [8]), the averaging operator and the multiplicatively symmetric operator (see [3]). Moreover, the equation (1.2) is connected with a description of some associative operations, i.e., the binary operation :(K{0})×(K{0})K{0} defined by xf(y)=xf(y) is associative if and only if f satisfies (1.2) (see [5] for more details). We also refer the reader to Belluot, Brzdȩk and Ciepliński [9] and Brzdȩk and Ciepliński [10] for some recent developments on the issues of stability and superstability for functional equations.

The main result of Najdecki [2] is the following.

Theorem 1.1 Let F:G K n , F=( f 1 , f 2 ,, f n ) satisfy

F ( x g ( y ) ) F ( x ) F ( y ) ϵ
(1.3)

for all x,yG with any norm in K n . Then there exist ideals I,J K n such that K n =IJ, PF is bounded and QF satisfies (1.1), where P: K n I, Q: K n J are the natural projections.

In this paper, generalizing the above result we consider the functional inequalities

F ( x g ( y ) ) F ( x ) F ( y ) ϕ(y),
(1.4)
F ( x g ( y ) ) F ( x ) F ( y ) ϕ(x)
(1.5)

for all x,yG with any norm in K n (see [6] for related results).

Throughout this paper we denote

L = { j : f j  is bounded , j = 1 , 2 , , n } , K = { j : f j  is unbounded , j = 1 , 2 , , n } ,

where F=( f 1 , f 2 ,, f n ).

Theorem 1.2 Let F:G K n , F=( f 1 , f 2 ,, f n ) satisfy (1.4) for all x,yG with any norm in K n . Assume that one of the following two conditions is fulfilled.

  1. (i)

    g is an involution,

  2. (ii)

    for each jK, there exists a sequence x n , n=1,2,3, (possibly depending on j) such that

    | f j ( x n ) | 1 + ϕ ( x n ) as n.
    (1.6)

Then there exist ideals I,J K n such that K n =IJ, PF is bounded and QF satisfies (1.1), where P: K n I, Q: K n J are the natural projections. Moreover, Q(F g 1 ) is exponential provided g is bijective.

Remark The case (ii) of Theorem 1.2 includes Theorem 1.1.

Theorem 1.3 Let F:G K n , F=( f 1 , f 2 ,, f n ) satisfy (1.5) for all x,yG with any norm in K n . Assume that g is an involution. Then there exist ideals I,J K n such that K n =IJ, PF is bounded, QF satisfies (1.1), where P: K n I, Q: K n J are the natural projections.

If we replace by the usual norm u on K n defined by

( a 1 , a 2 , , a n ) u = | a 1 | 2 + | a 2 | 2 + + | a n | 2 ,

we can estimate PF (in Theorem 1.2 and Theorem 1.3) as follows.

Theorem 1.4 The following two statements are valid.

  1. (a)

    If F:G K n , F=( f 1 , f 2 ,, f n ) satisfies (1.4), then PF satisfies

    P F ( y ) u | L | 2 ( 1 + 1 + 4 ϕ ( y ) )
    (1.7)

for all yG, where |L| denotes the number of the elements of L. In particular, if |L|=1 and G is a group, then PF satisfies either

1 2 ( 1 + 1 4 ϕ ( y ) ) P F ( y ) u 1 2 ( 1 + 1 + 4 ϕ ( y ) )
(1.8)

for all yB:={yG:ϕ(y)< 1 4 }, or

P F ( y ) u 1 2 ( 1 1 4 ϕ ( y ) )
(1.9)

for all yB.

  1. (b)

    If F:G K n , F=( f 1 , f 2 ,, f n ) satisfies (1.5), then PF satisfies (1.7). In particular if G is a group, g is surjective and |L|=1, then PF satisfies (1.8) or (1.9).

2 Proofs

Let g:GG and ϕ:G[0,) be given. We first consider the stability of the functional equation

f ( x g ( y ) ) f(x)f(y)=0
(2.1)

in the class of functions f:GK, i.e., we investigate both bounded and unbounded functions f:GK satisfying the functional inequalities

|f ( x g ( y ) ) f(x)f(y)|ϕ(y),
(2.2)
| f ( x g ( y ) ) f ( x ) f ( y ) | ϕ(x)
(2.3)

for all x,yG.

Lemma 2.1 Assume that g=σ is an involution and f:GK is an unbounded function satisfying the inequality (2.2). Then f is exponential and satisfies (2.1). In particular, if G is 2-divisible, then f has the form

f(x)=m ( x σ ( x ) 2 )
(2.4)

for all xG, where m:GK is an exponential function.

Proof Choose a sequence x n G, n=1,2,3, , such that |f( x n )| as n. Putting x= x n , n=1,2,3, , in (2.2), dividing the result by |f( x n )| and letting n we have

f(y)= lim n f ( x n σ ( y ) ) f ( x n )
(2.5)

for all yG. Multiplying both sides of (2.5) by f(x) and using (2.2) and (2.5) we have

f ( y ) f ( x ) = lim n f ( x n σ ( y ) ) f ( x ) f ( x n ) = lim n f ( x n σ ( y ) σ ( x ) ) f ( x n ) = lim n f ( x n σ ( y x ) ) f ( x n ) = f ( y x )
(2.6)

for all x,yG. Thus, f is an exponential function, say f=m. From (2.2) and (2.6) we have

| f ( x ) | | f ( σ ( y ) ) f ( y ) | ϕ(y)
(2.7)

for all x,yG. Since f is unbounded, from (2.7) we have

f ( σ ( y ) ) =f(y)
(2.8)

for all yG. Replacing y by σ(y) in (2.6) and using (2.8) we get the equation (2.1). In particular, if G is 2-divisible, then we can write

f ( x ) = f ( x 2 x 2 ) = f ( x 2 σ ( x 2 ) ) = f ( x 2 σ ( x ) 2 ) = m ( x σ ( x ) 2 )
(2.9)

for all xG. This completes the proof. □

Lemma 2.2 Let f:GK be an unbounded function satisfying (2.2). Assume that there exists a sequence x n , n=1,2,3, , satisfying

lim n | f ( x n ) | 1 + ϕ ( x n ) =.
(2.10)

Then f satisfies (2.1).

Proof Note that (2.10) implies

lim n 1 | f ( x n ) | =0and lim n ϕ ( x n ) | f ( x n ) | =0.

Putting y= x n , n=1,2,3, , in (2.2) and dividing the result by |f( x n )| we have

| f ( x ) f ( x g ( x n ) ) f ( x n ) | ϕ ( x n ) | f ( x n ) |
(2.11)

for all xG, n=1,2,3, . Letting n in (2.11) we have

f(x)= lim n f ( x g ( x n ) ) f ( x n )
(2.12)

for all xG. Multiplying both sides of (2.12) by f(y) and using (2.2) and (2.12) we have

f ( x ) f ( y ) = lim n f ( x g ( x n ) ) f ( y ) f ( x n ) = lim n f ( x g ( x n ) g ( y ) ) f ( x n ) = lim n f ( x g ( y ) g ( x n ) ) f ( x n ) = f ( x g ( y ) )
(2.13)

for all x,yG. This completes the proof. □

Lemma 2.3 Assume that g is bijective and f:GK is an unbounded function satisfying the inequality (2.2). Then f g 1 is an exponential function.

Proof Choose a sequence x n G, n=1,2,3, , such that |f( x n )| as n. Putting x= x n , n=1,2,3, , in (2.2), dividing the result by |f( x n )|, replacing y by g 1 (y) and letting n we have

f ( g 1 ( y ) ) = lim n f ( x n y ) f ( x n )
(2.14)

for all yG. Multiplying both sides of (2.14) by f( g 1 (x)) and using (2.2) and (2.14) we have

f ( g 1 ( y ) ) f ( g 1 ( x ) ) = lim n f ( x n y ) f ( g 1 ( x ) ) f ( x n ) = lim n f ( x n y x ) f ( x n ) = f ( g 1 ( y x ) )
(2.15)

for all x,yG. Thus, f g 1 is an exponential function. This completes the proof. □

Proof of Theorem 1.2 Since every two norms in K n are equivalent, from (1.4) there exists α>0 such that

| f j ( x g ( y ) ) f j ( x ) f j ( y ) | F ( x g ( y ) ) F ( x ) F ( y ) u α F ( x g ( y ) ) F ( x ) F ( y ) α ϕ ( y )
(2.16)

for all x,yG and all j{1,2,,n}. For the case (i), by Lemma 2.1, f j satisfies (2.1) for all jK. For the case (ii), by Lemma 2.2, f j satisfies (2.1) for all jK. Let I={( a 1 , a 2 ,, a n ): a i =0 for iK}, J={( a 1 , a 2 ,, a n ): a i =0 for iL}. Then it follows that K n =IJ, PF is bounded and QF satisfies (1.1). If g is bijective, then by Lemma 2.3, f j g 1 are exponential function for all jK, which implies Q(F g 1 ) is an exponential function. This completes the proof. □

Lemma 2.4 Assume that g=σ is an involution and f:GK is an unbounded function satisfying the inequality (2.3). Then f satisfies (2.1). In particular, if G is 2-divisible, then f has the form

f(x)=m ( x σ ( x ) 2 )
(2.17)

for all xG, where m:GK is an exponential function.

Proof Choose a sequence y n G, n=1,2,3, , such that |f( y n )| as n. Putting y= y n , n=1,2,3, , in (2.3), dividing the result by |f( y n )| and letting n we have

f(x)= lim n f ( x σ ( y n ) ) f ( y n ) .
(2.18)

Putting x=e in (2.3) and replacing y by σ(y) in the result we have

| f ( y ) f ( e ) f ( σ ( y ) ) | ϕ(e)
(2.19)

for all x,yG. Multiplying both sides of (2.18) by f(y) and using (2.3), (2.18), and (2.19) we have

f ( y ) f ( x ) = lim n f ( y ) f ( x σ ( y n ) ) f ( y n ) = lim n f ( y σ ( x σ ( y n ) ) ) f ( y n ) = lim n f ( e ) f ( σ ( y ) x σ ( y n ) ) f ( y n ) = f ( e ) f ( σ ( y ) x )
(2.20)

for all x,yG. Putting x=e in (2.20) we have

f(y)=f ( σ ( y ) )
(2.21)

for all yG. From (2.19) and (2.21) we have

| f ( y ) | | 1 f ( e ) | ϕ(e)
(2.22)

for all yG. Since f is unbounded, from (2.22) we have f(e)=1. Thus, f satisfies (2.1). This completes the proof. □

Proof of Theorem 1.3 From (1.5), as in (2.16) there exists α>0 such that

| f j ( x g ( y ) ) f j ( x ) f j ( y ) | αϕ(x)
(2.23)

for all x,yG, j{1,2,,n}. Applying Lemma 2.4 to (2.23) for each jK we find that f j satisfies (2.1) for all jK, which implies that QF satisfies (1.1). This completes the proof. □

Now, we investigate bounded functions satisfying each of (2.2) and (2.3) (see [4, 1113] for bounded solutions of an exponential functional equation).

Lemma 2.5 Let f:GK be a bounded function satisfying (2.2). Then f satisfies

| f ( y ) | 1 2 ( 1 + 1 + 4 ϕ ( y ) )
(2.24)

for all yG. In particular, G is a group and let B={yG:ϕ(y)< 1 4 }, then f satisfies either

1 2 ( 1 + 1 4 ϕ ( y ) ) | f ( y ) | 1 2 ( 1 + 1 + 4 ϕ ( y ) )
(2.25)

for all yB, or

| f ( y ) | 1 2 ( 1 1 4 ϕ ( y ) )
(2.26)

for all yB.

Proof Let M f = sup x G |f(x)|. Using the triangle inequality with (2.2) we have

| f ( x ) f ( y ) | | f ( x g ( y ) ) | +ϕ(y) M f +ϕ(y)
(2.27)

for all x,yG. Taking the supremum of the left hand side of (2.27) with respect to xG we get M f |f(y)| M f +ϕ(y) for all yG. Thus, we have

M f ( | f ( y ) | 1 ) ϕ(y)
(2.28)

for all yG. From (2.28) we have

| f ( y ) | ( | f ( y ) | 1 ) ϕ(y)
(2.29)

for all yG. Solving the inequality (2.29) we get (2.24). Now, we assume that G is a group. Replacing x by xg ( y ) 1 in (2.2) and using the triangle inequality we have

| f ( x ) | | f ( x g ( y ) 1 ) f ( y ) | +ϕ(y) M f | f ( y ) | +ϕ(y)
(2.30)

for all x,yG. Taking the supremum of the left hand side of (2.30) with respect to xG we get M f M f |f(y)|+ϕ(y) for all yG. Thus, we have

M f ( 1 | f ( y ) | ) ϕ(y)
(2.31)

for all yG. From (2.28) and (2.31) we have

| f ( y ) | | 1 | f ( y ) | | M f | 1 | f ( y ) | | ϕ(y)
(2.32)

for all yG. For each fixed yB, solving the inequality (2.32) we get

1 2 ( 1 + 1 4 ϕ ( y ) ) | f ( y ) | 1 2 ( 1 + 1 + 4 ϕ ( y ) ) ,
(2.33)

or

| f ( y ) | 1 2 ( 1 1 4 ϕ ( y ) ) .
(2.34)

Now, assume that there exist a bounded function f and y 1 , y 2 B such that

| f ( y 1 ) | 1 2 ( 1 1 4 ϕ ( y 1 ) ) , | f ( y 2 ) | 1 2 ( 1 + 1 4 ϕ ( y 2 ) ) .
(2.35)

Then from (2.31) we have

| f ( y 2 ) | ( 1 | f ( y 1 ) | ) M f ( 1 | f ( y 1 ) | ) ϕ( y 1 ).
(2.36)

On the other hand, from (2.35) we have

| f ( y 2 ) | ( 1 | f ( y 1 ) | ) 1 2 ( 1 + 1 4 ϕ ( y 2 ) ) ( 1 1 2 ( 1 1 4 ϕ ( y 1 ) ) ) > 1 2 ( 1 1 4 ϕ ( y 1 ) ) ( 1 1 2 ( 1 1 4 ϕ ( y 1 ) ) ) = ϕ ( y 1 ) ,

which contradicts (2.36). Thus, f satisfies (2.25) for all yB, or it satisfies (2.26) for all yB. This completes the proof. □

Lemma 2.6 Let f:GK be a bounded function satisfying (2.3). Then f satisfies (2.24) for all yG. In particular, if G is a group and g is surjective, then f satisfies (2.25) for all yB:={yG:ϕ(y)< 1 4 }, or satisfies (2.26) for all yB.

Proof Using the triangle inequality with (2.3) we have

| f ( x ) f ( y ) | | f ( x g ( y ) ) | +ϕ(x) M f +ϕ(x)
(2.37)

for all x,yG. Taking the supremum of the left hand side of (2.37) with respect to yG we get M f |f(x)| M f +ϕ(x) for all xG. Thus, we have

M f ( | f ( x ) | 1 ) ϕ(x)
(2.38)

for all xG. From (2.38) we get (2.24) as in the proof of Lemma 2.5. We assume that G is a group. For given x,zG, choosing wG such that g(w)= x 1 z, putting y=w in (2.3) and using the triangle inequality we have

| f ( z ) | | f ( x ) f ( w ) | +ϕ(x) | f ( x ) | M f +ϕ(x)
(2.39)

for all x,zG. Taking the supremum of the left hand side of (2.39) we get M f M f |f(x)|+ϕ(x) for all xG. Thus, we have

M f ( 1 | f ( x ) | ) ϕ(x)
(2.40)

for all xG. Now, the remaining parts of the proof are the same as those of Lemma 2.5. □

Proof of Theorem 1.4 From Lemma 2.5 and Lemma 2.6, for each jL we have

| f j ( y ) | 1 2 ( 1 + 1 + 4 ϕ ( y ) )
(2.41)

for all yG. Thus, from (2.41) we have

P F ( y ) u = j L | f j ( y ) | 2 | L | 2 ( 1 + 1 + 4 ϕ ( y ) )

for all yG, which gives (1.7). Now, if |L|=1, say L={ j 0 } we have

P F ( y ) u = | f j 0 ( y ) |

for all yG. Thus, the inequalities (1.8) and (1.9) follow immediately from (2.25) and (2.26). This completes the proof. □

Author’s contributions

The author is the only person who is responsible to this work.

References

  1. 1.

    Ger R, Šemrl P: The stability of exponential equation. Proc. Am. Math. Soc. 1996, 124: 779–787. 10.1090/S0002-9939-96-03031-6

    Article  MathSciNet  MATH  Google Scholar 

  2. 2.

    Najdecki A: On stability of functional equation connected with the Reynolds operator. J. Inequal. Appl. 2007., 2007: Article ID 79816

    Google Scholar 

  3. 3.

    Aczél J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, New York; 1989.

    Google Scholar 

  4. 4.

    Baker JA: The stability of cosine functional equation. Proc. Am. Math. Soc. 1980, 80: 411–416. 10.1090/S0002-9939-1980-0580995-3

    Article  MATH  Google Scholar 

  5. 5.

    Brzdȩk J: On solutions of a generalization of the Reynolds functional equation. Demonstr. Math. 2008, 41: 859–868.

    MathSciNet  Google Scholar 

  6. 6.

    10.1007/s0010-014-0266-6

  7. 7.

    Chudziak J, Tabor J: On the stability of the Gołąb-Schinzel functional equation. J. Math. Anal. Appl. 2005, 302: 196–200. 10.1016/j.jmaa.2004.07.053

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Marras Y:Sur l’équation fonctionnelle f[xf(y)]=f(x)f(y). Bull. Cl. Sci., Acad. R. Belg. 1969, 55: 779–787. 5e Série

    Google Scholar 

  9. 9.

    Brillouët-Belluot N, Brzdȩk J, Ciepliński K: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012., 2012: Article ID 716936

    Google Scholar 

  10. 10.

    Brzdȩk J, Ciepliński K: Hyperstability and superstability. Abstr. Appl. Anal. 2013., 2013: Article ID 401756

    Google Scholar 

  11. 11.

    Albert M, Baker JA: Bounded solutions of a functional inequality. Can. Math. Bull. 1982, 25: 491–495. 10.4153/CMB-1982-071-9

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Chung J: On an exponential functional inequality and its distributional version. Can. Math. Bull. 2012. 10.4153/CMB-2014-012-x

    Google Scholar 

  13. 13.

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

    Google Scholar 

Download references

Acknowledgements

The author is very thankful to the referees for valuable suggestions that improved the presentation of the paper. This work was supported by Basic Science Research Program through the National Foundation of Korea (NRF) funded by the Korea Government (no. 2012R1A1A008507).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jaeyoung Chung.

Additional information

Competing interests

The author declares that he has no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chung, J. Stability of a functional equation connected with Reynolds operator. Adv Differ Equ 2014, 158 (2014). https://doi.org/10.1186/1687-1847-2014-158

Download citation

Keywords

  • bounded solution
  • exponential function
  • involution
  • Reynolds operator
  • stability