generalized popoviciu functional equations in … · the functional equation (^4) in banach modules...

NEW ZEALAND JOURNAL OF MATHEMATICS Volume 32 (2003), 183-193

GENERALIZED POPOVICIU FUNCTIONAL EQUATIONS IN BAN ACH MODULES OVER A C *-ALG EB R A AND

APPROXIM ATE ALGEBRA HOMOMORPHISMS

C h u n - G il P a r k

(Received March 2002)

Abstract. We prove the Hyers-Ulam—Rassias stability of generalized Popovi- ciu functional equations in Banach modules over a unital C * -algebra. It is applied to show the stability of Banach algebra homomorphisms between Banach algebras associated with generalized Popoviciu functional equations in Banach algebras.

1. Introduction

Let Ei and E 2 be Banach spaces with norms || • || and || • ||, respectively. Consider / : E\ —> E 2 to be a mapping such that f { t x ) is continuous in t G M for each fixed x G Ei. Assume that there exist constants e > 0 and p G [0,1) such that

\\f(x + y) - f ( x ) ~ f(y)\\ < £(|kir + |M|P)

for all x, y G E\. Th.M. Rassias [11] showed that there exists a unique M-linear mapping T : Ei —> E2 such that

||/(*)-T(x)|| < 2 ^ l l * r

for all x E Ei.Throughout this paper, let A be a unital C *-algebra with norm | • |, U{A) the

unitary group of A , A in the set of invertible elements in A, A sa the set of self-adjoint elements in A, Ai = {a G A | |a| = 1}, and A the set of positive elements in A\. Let a B and \C be left Banach A-modules with norms || • || and || • ||, respectively. Let n and k be integers with 2 < k < n — 1.

Recently, T. Trif [19] generalized the Popoviciu functional equation

2000 AMS Mathematics Subject Classification: Primary 47B48, 39B72, 46L05.Key words and phrases: Banach module over C * -algebra, stability, real rank 0, unitary group, generalized Popoviciu functional equation, approximate algebra homomorphism.This work was supported by grant No. 1 99 9 -2 -102 -001 -3 from the interdisciplinary Research program year of the KOSEF .

184 CHUN-GIL PARK

Lemma A. [19, Theorem 2.1] Let V and W be vector spaces. A mapping / : V —> W with /(0 ) = 0 satisfies the functional equation

n- n-2Ck-2f + n~2Ck-l f (Xj)' i=l

= *• E / (— i ' - <A>1 <Z1 < n ' '

for all x \, • • •, xn G V if and only if the mapping / : V —► W satisfies the additive Cauchy equation / ( x + y) = f (x) + f (y) for all x, y G V.

The following is useful to prove the stability of the functional equation (A).

Lemma B. [10, Theorem 1] Let a G A and |a| < 1 — ^ for some integer m greater than 2. Then there are m elements • • •, um G U(A) such that ma = uH------ bwm.

The main purpose of this paper is to prove the Hyers-Ulam-Rassias stability of the functional equation ( 4) in Banach modules over a unital (7*-algebra, and to prove the Hyers-Ulam-Rassias stability of algebra homomorphisms between Banach algebras associated with the functional equation ( 4).

2. Stability of Generalized Popoviciu Functional Equations in Banach Modules over a C*-Algebra Associated with its Unitary Group

We are going to prove the Hyers-Ulam-Rassias stability of the functional equation (A ) in Banach modules over a unital (7*-algebra associated with its unitary group.

For a given mapping / : aB —► aC and a given a G A, we set

D af { x i, ■ • • ,x „ ) := n • n- 2Ck- 2aj

+ n. 2Ck- 1 ^ a f ( x i) - k - £ f f a * u ± _ _ + a x ± \ i=l

for all x i , ■ ■ ■ , x n G a B.

Theorem 2.1. Let q = an^ r = ~n^k' ^et f : aB aC be a mappingwith / ( 0) = 0 for which there exists a function : a B71 [0, oo) such that

OO

<p(xi ,-- - ,x „) I , " ’ ,QJx n)3=0

< oo\\Duf ( x 1: --- ,x n)|| < ,x „) (2.i)

for all u G U(A) and all x \ , - - - , x n G a&- Then there exists a unique A-linear mapping T : aB —► aC such that

\\f{x)-T(x)\\ < ------- l— ---- tp(qx,rx,--- ,rx) (2.ii)K•n-lOfc-i

for all x G a B.

GENERALIZED POPOVICIU FUNCTIONAL EQUATIONS 185

P roof. Put u = 1 G U(A). By [19, Theorem 3.1], there exists a unique additive mapping T : a& —► aC satisfying (2.ii). The additive mapping T : a& aC was defined by

T(x) = lim q~3 f(q3x)j —>oc

for all x G a&-By the assumption, for each u G U(A),

q~3\\Duf(qJx , - - - ,qJx)\\ < q~3p(q3x , - ■ ■ ,q3x)

for all x G a&, and

q~3\\Duf(q3x, ,qJx)\\ -*■ 0

as n —> oo for all x G a&- So

D uT(x, • • • , x) = lim q~j D uf(q3x, ■ ■ ■ , q3x) = 0j —>oo

for all u G U(A) and all x G a&- Hence

D uT(x, • • • , x) = n • n- 2C k -2uT(x ) + n ■ n- 2Ck-\uT(x) - fc • nCkT(ux) = 0

for all u G U{A) and all x G a&- So

uT(x) = T(ux)

for all u G U{A) and all x G a&-Now let a G A (a ^ 0) and M an integer greater than 4|a|. Then

a 1 |a| 1 2 1

lM l = M |a|<^ = 4 < _ 3 = 3

By Lemma B, there exist three elements ui ,u2,u3 G U(A) such that 3-^ = ui + W2 + W3. And T(x) = T (3 • = 3T (|x) for all x G a#- So T (|a:) = |T(x) for all x E aB- Thus

T(ax) = t ( ^ - - 3 ^ - x ] = M - t ( \ ■ 3 - % ) - ~ T ( 3 - ^ x )\ 3 M J \3 M J 3 \ M J

= ^ - T ( u i x + u2x 4- u3x) = (T(uix) + T(u2x ) + T(u3x ))o oM . Srri. . M „ a rri/ .

= y O i + u2 + u3)T(x ) = — • 3 — T (x)

= aT(x)

for all x G a # - Obviously, T(0x) — 0T(x) for all x G a&- Hence

T(ax + by) = T(ax) + T(by) = aT{x) + bT(y)

for all a,b G A and all x, y G a >- So the unique additive mapping T : a & —> a C is an A-linear mapping, as desired. □

Applying the unital C*-algebra C to Theorem 2.1, one can obtain the following.

186 CHUN-GIL PARK

Corollary 2 .2 . Let E\ and E2 be complex Banach spaces with norms || • || and || • ||, respectively. Let q = and r = — Let f : E\ —> E 2 be a mappingwith / ( 0) = 0 for which there exists a function (p : E ” —> [0, oo) such that

OO

(p(x !,••■ , X n ) : = l , - * * , 9 J^ n )J=0

< ooII-Da /O t i , - - - , x „)|| < ,x n)

for all A G T 1 := {A G C | |A| = 1} and all x\, ■ • ■ , x n € E\. Then there exists a unique C-linear mapping T : E\ —> E 2 such that

||/(x)-T(x)|| < ------- l— ---- (p(qx,rx, • • • , rx)

for all x £ Ei.

Theorem 2.3. Let q = and r = — f '■ a B —> fee a continuousmapping with / ( 0) = 0 /or which there exists a function </? : —> [0, oo) satisfying(2.i) such that

\\Duf { x i , - - - ,x n)|| < <^(xi,--- ,x n)/or all u G Z7(A) and all xi, • • • ,x n G 7/ £/ie sequence {q~3 f(q3 x)} converges uniformly on a &, then there exists a unique continuous A-linear mapping T : a B —> a C satisfying (2Ai).

Proof. Put u — 1 G U(A). By Theorem 2.1, there exists a unique A-linear mapping T : a& —> satisfying (2.m). By the continuity of / , the uniform convergence and the definition of T, the A-linear mapping T : a& —> aC is continuous, as desired. □

Theorem 2.4. Let q = and r = — / : a # a C be a mappingwith / ( 0) = 0 for which there exists a function ip : —■> [0, oo) such that

OO

^(xi, • •• ,x „ ) : = y ^ g V (g ~ JXi, • ■ • ,q~3x n) < ooi3=0

\\Duf(xi , • • • ,x n)|| < </?(xi, • • • ,x n) (2.iii)

for all u G U(A) and all x i,-- - ,x n G Then there exists a unique A-linear mapping T : a& —► swc/i that

||/(x)-T(x)|| < ----- i ---- <p(x,rx, ••• ,rx) (2.iv)n —2^fc—1

/o r all x G

P roof. Put « = 1 G U(A). By [19, Theorem 3.2], there exists a unique additive mapping T : aB —► satisfying (2iv). The additive mapping T : wasdefined by

T(x) = lim q3 f {q~3x)j —>oo

for all x E a B.By the assumption, for each w G £V(A),

9J||7>u/(9_J^ , ' • • ,<7-J x)|| < qJip(q~Jx, ■ ■ ■ ,q~3x)


for all x G and

qJ\\Duf{q~Jx , - " ,q~3x )|| 0

as n —► oo for all x € a&- So

D uT (x , • • • , x) = lim qj D uf(q~j x, • • • , q~j x) = 0j—>OG

for all u G U(A) and all x € aB. Hence

D uT(x, , x) = n • n- 2C k - 2uT(x) + n • n_ 2Cfc_inT(x) - fc • nCkT(ux) = 0

for all u G W(-A) and all x G ,4# . So

uT(x) = T(ux)

for all w G and all x G a #-The rest of the proof is the same as the proof of Theorem 2.1. □

3. Stability of Generalized Popoviciu Functional Equations in Banach Modules over a C *—Algebra

Given a locally compact abelian group G and a multiplier co on G, one can associate to them the twisted group C*-algebra C * ( G , lj). (7*(Zm,u;) is said to be a noncommutative torus of rank m and denoted by Aw. The multiplier w determines a subgroup Sw of G, called its symmetry group, and the multiplier is called totally skew if the symmetry group Sw is trivial. And A w is called completely irrational if u) is totally skew (see [2]). It was shown in [2] that if G is a locally compact abelian group and a; is a totally skew multiplier on G, then C*(G, u>) is a simple C*-algebra. It was shown in [3, Theorem 1.5] that if A w is a simple noncommutative torus then Aw has real rank 0, where “real rank 0” means that the set of invertible self-adjoint elements is dense in the set of self-adjoint elements (see [3], [6]).

From now on, assume that ip : a Bn —► [0, oo) is a function satisfying (2.i), and that q = and r = - ^ .

We are going to prove the Hyers-Ulam-Rassias stability of the functional equation (A) in Banach modules over a unital C'*-algebra.

Theorem 3.1. Let q be not an integer. Let f : a& —► aC be a mapping with / ( 0) = 0 such that

\ \ D i f ( x ! , - - - , £ n )|| < <fi(Xi, ■ ■ ■ , X n )

for all x i, • • • ,x n G a&- Then there exists a unique additive mapping T : a& —> aC satisfying (2 .ii). Further, if f(Xx) is continuous in A g I for each fixed x G a&, then the additive mapping T : a& —» aC is K.-linear.

Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique additive mapping T : a& —> aC satisfying (2.ii).

Assume that /(Ax) is continuous in A G M for each fixed x G a&- By the assumption, q is a rational number which is not an integer. The additive mapping T given above is similar to the additive mapping T given in the proof of [11, Theorem]. By the same reasoning as the proof of [11, Theorem], the additive mapping T : a& —> aC is M-linear. □

188 CHUN-GIL PARK

Theorem 3.2. Let q be not an integer. Let f : a & —> a C be a continuous mapping with / ( 0) = 0 such that

\\Daf { x i , - - ’ , x n)\\ < , x n)

for all a G A f U { i } and all , x n G a &- If the sequence {q~if(q3x)} converges uniformly on a &, then there exists a unique continuous A-linear mapping T : a & —*■ a C satisfying (2.ii).

Proof. Put a = 1 G A f . By Theorem 3.1, there exists a unique M-linear mapping T : a& —+ aC satisfying (2.ii). By the continuity of / and the uniform convergence, the IR-linear mapping T : a& aC is continuous.

By the same reasoning as the proof of Theorem 2.1,

T(ax) = aT(x)

for all a G A± U {*}.For any element a G A, a = g- ° - + i q~ f , and and <l-^f ■ are self-adjoint

elements, furthermore, a — ( a\a ) + — ( a+2a ) + f ) + — i { a~2i ) > where( a~Q ) + , ( a+2 ) , ( a^ ) + , and ( a~“ ) are positive elements (see [4, Lemma 38.8]). So

/ 'a + a * \ + f a + a*\ . f a — a*\ + . ( a — a*T(ax) = T [ ( — - — I x — ( — - — I x + i [ ——— x — i

2 i J V 2 i

T ( x ) + T ( - x ) + — — T(ix)

a + a*\ + r , . f a + a*\ rri/ . . f a — a*x ++ ) T(x)

T(x)2 i

+ i {— ) - ' [ T * - ) ) T(X)— aT(x)

for all a G A and all x G a &- Hence

T(ax + by) = T(ax) + T(by) = aT(x) + bT(y)

for all a,b G A and all x, y G a&, as desired. □

Theorem 3.3. Let A be a unital C* -algebra of real rank 0, and q not an integer. Let f : a & —» a C be a continuous mapping with f ( 0) = 0 such that

\\Daf ( x i , - - - , x n)\\ < (p(xi, ■ ■ ■ , x n)


for all a G (A+ fl Ain) U { i } and all x\, • • • ,x „ G a B. If the sequence {q~3 f{q3x)} converges uniformly on a B, then there exists a unique continuous A-linear mapping T : a & ~> aC satisfying (2.ii).

P roof. By the same reasoning as the proof of Theorem 3.2, there exists a unique continuous M-linear mapping T : a B —* a C satisfying (2.ii), and

T(ax) = aT(x) (3.1)

for all a G ( A f fl Ain) U {z} and all x G aB.Let b G A~l \ Ain. Since Ain fl A sa is dense in A sa, there exists a sequence { 6m}

in Ain n A sa such that bm —> b as m —> oo. Put cm = j ^ b m. Then cm —> ^ b = b

as m —■> oo. Put am = y/c^cm. Then am —> b as m —> oo and am G A^ fl Ain. Thus there exists a sequence {am} in A~{ fl Ain such that am —> b as m —> oo, and by the continuity of T

lim T(amx) = T{ lim amx) = T(bx) (3.2)m —►oo m —►oo

for all x G aB. By (3.1),

||T(amx) - bT(x) || = ||amT(x) - bT{x) || - ||MT(x) - bT{x) || = 0 (3.3)

as m —► oo. By (3.2) and (3.3),

||T(te) - bT(x)II < ||r(te) - T(amx )II + IIT(amx) - bT(x)\\

—> 0 as m —* oo (3.4)

for all x G aB. By (3.1) and (3.4), T(ax) = aT(x) for all a G A~l U { i } and allx G a B.

The rest of the proof is similar to the proof of Theorem 3.2. □

Theorem 3.4. Let q be not an integer. Let f : a B a C be a mapping with / ( 0) = 0 such that

\ \ D a f { x i , ’ - ’ ,®n)|| < i p { X ! , - ■ ■ , X n )

for all a G A f U { i } and all x \,••• ,x n G a B. If f{Xx) is continuous in X G M for each fixed x G a B, then there exists a unique A-linear mapping T : a B —> a C satisfying (2.ii).

P roof. Put a = 1 G A f . By Theorem 3.1, there exists a unique R-linear mapping T : aB —» aC satisfying (2.ii).


Theorem 3.5. Let A be a unital C* -algebra of real rank 0, and q not an integer. Let f : a & —> a C be a mapping with /(0 ) = 0 such that

\\Daf ( x i , - " , x n)\\ < <p(xi , - - - , x n)

for all a G (A+ n A*n) U { i } and all X\,--- ,x n G a B. Assume that f{ax) is continuous m a G i i U l for each fixed x G a B, and that the sequence {q~i f (q3x)} converges uniformly on A\ for each fixed x G a B. Then there exists a unique A-linear mapping T : a B —> a C satisfying (2.ii).

190 CHUN-GIL PARK

Proof. By the same reasoning as the proof of Theorem 3.2, there exists a unique M-linear mapping T : aB —* aC- satisfying (2.ii), and

T(ax) = aT(x)

for all a G ( A f fl Ain) U {«} and all x G aB . By the continuity of / and the uniform convergence, one can show that T(ax) is continuous in a G A\ for each x G aB.


Theorem 3.6. Let f : aB aC be a mapping with /(0 ) = 0 such that

\\Daf(xi , • • • ,x„)|| < <p(xi, • • • ,x„)

for all a G A^ U { i } U l and all - , x n G aB. Then there exists a unique A-linear mapping T : aB —> aC satisfying (2.ii).

Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique additive mapping T : aB —► aC satisfying (2.ii), and

T(ax) = aT(x)

for all a G A f U {i}U lR and all x G a$- So the mapping T : a $ — aC is M—linear, and satisfies

T(ax) = aT(x) for all a G A * U { i } and all x G aB-

The rest of the proof is the same as the proof of Theorem 3.2. □

Similarly, for the case that <p : a B71 [0, oo) is a function such thatOO

Y , q 3(p(q~3xi , ' - - ,q~3x n) < oo3=0

for all xi, ■ ■ • , x n G aB, one can obtain similar results to the theorems given above.

4. Stability of Generalized Popoviciu Functional Equations in Banach Algebras and Approximate Algebra Homomorphisms Associated with (A)

In this section, let A and B be Banach algebras with norms || • || and || • ||, respectively.

D.G. Bourgin [5] proved the stability of ring homomorphisms between Banach algebras. In [1], R. Badora generalized the Bourgin’s result.

We prove the Hyers-Ulam -Rassias stability of algebra homomorphisms between Banach algebras associated with the functional equation (A).

Theorem 4.1. Let A and B be real Banach algebras, and q not an integer. Let f : A —> B be a mapping with / ( 0) = 0 for which there exists a function ip : A x A —► [0, oo) such that

OO

$(x ,y) ;= Y L q~3^ qJx y )3=0

\\D\f(x\, ■ • • , ar„)||

\\f(x-y) - f(x)f(y)\\

< oo, (4.i)

< (p{x\, • • * , x n), (4-ii)

< ^{x ,y ) (4.iii)


for all x, y, x\, ■ • • , xn G A. If / ( Ax) is continuous in A G M for each fixed x G A , then there exists a unique algebra homomorphism T : A —> B satisfying (2.ii). Further, if A and B are unital, then f itself is an algebra homomorphism.

P roof. Under the assumption (2.i) and (4.ii), in Theorem 3.1, we showed that there exists a unique E-linear mapping T : A —> B satisfying (2.ii). The IR-linear mapping T : A —* B was given by

T(x) = lim q~3f(q3x)j —►oo

for all x G A. LetR(x,y) = f ( x -y) - f {x ) f (y )

for all x , y G A. By (4 .i), we get

lim q~~3 R(q3x,y) = 0j —1-oo

for all x , y G A. So

T(x • y) — lim q~J (4-1)j —*oo

f (q J{x -y ) ) = lim q~Jf({qJx)y)j —>oo

= lim q~3(f{qJx)f{y) + R(qJx,y)) = T ( x ) f { y ) (4.2)J—>oo

for all x , y G A. Thus

T{x)f(qJy) = T(x{q3y)) = T((q3x)y) = T(q3x)f(y) = q3T(x)f(y)

for all x, y G A. Hence

T(x)q~3f(q3y) = T{x)f(y) (4.3)

for all x, y G A. Taking the limit in (4.2) as j —> oo, we obtain

T(x)T(y) = T(x)f(y)

for all x , y G A. Therefore,

T(x ■ y ) = T(x)T(y)

for all x, y G A. So T : A —> B is an algebra homomorphism.Now assume that A and B are unital. By (4.2),

T(y) = T( 1 • y) = T(l) f ( y) = f(y)

for all y G A. So / : A —> B is an algebra homomorphism, as desired. □

Theorem 4.2. Let A and B be complex Banach algebras. Let f : A —> B be a mapping with / ( 0) = 0 for which there exists a function ^ : A x A —> [0, oo) satisfying (4.i) and (4.iii) such that

\\D\f(xi,--- , x n)\\ < <p(xi , - - - ,xn) (4.iv)

for all A G T 1 and all X\, • • • , x n G A. Then there exists a unique algebra homomorphism T : A —> B satisfying (2.ii). Further, if A and B are unital, then f itself is an algebra homomorphism.

192 CHUN-GIL PARK

Proof. Under the assumption (2.i) and (4.iv), in Corollary 2.2, we showed that there exists a unique C-linear mapping T : A —> B satisfying (2.ii).

The rest of the proof is the same as the proof of Theorem 4.2. □

Theorem 4.3. Let A and B be complex Banach * -algebras. Let f : A —> B be a mapping with / ( 0 ) = 0 for which there exists a function : A x A —> [0, oo) satisfying (4.i) and (4.in) such that

\\D\f(xi, ■ ■ ■ ,x n)|| < , x n) (4.iv)

\\f(x*)~f{x)*\\ < <p(x ,- - - ,x) (4.v)

for all A G T1 and all x ,x \ , - - - , x n G A. Then there exists a unique *-algebra homomorphism T : A —> B satisfying (2.ii). Further, if A and B are unital, then f itself is a * -algebra homomorphism.

Proof. By the same reasoning as the proof of Theorem 4.2, there exists a unique C-linear mapping T : A —► B satisfying (2.ii).

Nowq~3\\f{q3x*) - f(q3x)*\\ < q~3<p{qJx, • • • , q3 x)

for all x G A. Thusq~3\\f{q3x*) - f{q3x)*\\ 0

as n —» oo for all x G A. HenceT(x*) = lim q~3f{q3x*) = lim q~3f(q3x)* = T ( x ) *

j —>oc j —> oo

for all x G A.The rest of the proof is the same as the proof of Theorem 4.2. □

Similarly, for the case that ip : A n —> [0, oo) and if) : A x A —► [0, oo) are functions such that

OO

^ 2 q 3v{q~3x i , ■ - ,q~3x n) < oo,3=0

OO

^ q 3^{ q~3x , y ) < oo3=0

for all x , y , x i ,- - - , x n G A, one can obtain similar results to the theorems given above.

References

1 . R. Badora, On approximate rinq homomorphisms, J. Math. Anal. Appl. 276 (2002), 589-597.

2. L. Baggett and A. Kleppner, Multiplier representations of abelian groups, J. Funct. Anal. 14 (1973), 299-324.

3. B. Blackadar, A. Kumjian and M. R0rdam, Approximately central matrix units and the structure of noncommutative tori, if-Theory 6 (1992), 267-284.

4. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, New York, Heidelberg and Berlin, 1973.

5. D.G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16 (1949), 385-397.


6. L. Brown and G. Pedersen, C*-algebras of real rank zero, J. Funct. Anal. 99 (1991), 131-149.

7. D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Berlin, Basel and Boston, 1998.

8. G. Isac and Th.M. Rassias, On the Hyers-Ulam stability of ip-additive mappings, J. Approx. Theory 72 (1993), 131-137.

9. B.E. Johnson, An approximately multiplicative map between Banach algebras, J. London Math. Soc. 37 (1988), 294-316.

10. R. Kadison and G. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249-266.

11. Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

12. Th.M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), 106-113.

13. Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264-284.

14. Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 123-130.

15. Th.M. Rassias and P. Semrl, On the behavior of mappings which does not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989-993.

16. Th.M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325-338.

17. H. Schroder, K -Theory for Real C*-Algebras and Applications, Pitman Research Notes in Mathematics, Longman, Essex, vol. 290, 1993.

18. T. Trif, Hyers-Ulam-Rassias stability of a Jensen type functional equation, J. Math. Anal. Appl. 250 (2000), 579-588.

19. T. Trif, On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions, J. Math. Anal. Appl. 272 (2002), 604-616.

Chun-Gil ParkDepartment of MathematicsChungnam National UniversityDaejeon 305-764K O R EAcgpar kmat h. cnu. ac. kr

generalized popoviciu functional equations in … · the functional equation (^4) in banach modules...

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