- Email: [email protected]
On the Hyers–Ulam–Rassias Stability of Approximately Additive Mappings
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
204, 221]226 Ž1996.
0433
On the Hyers]Ulam]Rassias Stability of Approximately Additive Mappings Soon-Mo Jung Mathematical Part, College of Science and Technology, Hong-Ik Uni¨ ersity, 339-800 Chochiwon, South Korea Submitted by Joseph D. Ward Received July 25, 1995
In this paper, using an idea from the paper of P. Gavruta ˘ ˘ Ž J. Math. Anal. Appl. 184, 1994, 431]436., the results of Th. M. Rassias Ž J. Math. Anal. Appl. 158, 1991, 106]113. shall be generalized. Q 1996 Academic Press, Inc.
1. INTRODUCTION Ulam Žcf. w8x. had raised the following question: Under what conditions does there exist an additive mapping near an approximately additive mapping? In 1941 Hyers w3x proved that if f : E1 ª E2 is a mapping satisfying 5 f Ž x q y. y f Ž x. y f Ž y. 5 F d for all x, y g E1 , where E1 and E2 are Banach spaces and d is a given positive number, then there exists a unique additive mapping T : E1 ª E2 such that 5 f Ž x. y T Ž x. 5 F d for any x g E1. In addition, he also proved that if f Ž tx . is continuous in the real variable t for any fixed x g E1 , then T is linear. Rassias w4, 5x and Gajda w1x showed some generalizations of the result of Hyers in the following ways: Let f : E1 ª E2 be a mapping satisfying 5 f Ž x q y. y f Ž x. y f Ž y. 5 F u Ž 5 x5 p q 5 y5 p. 221 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
222
SOON-MO JUNG
for all x, y g E1 , where u G 0 and p / 1. Suppose f Ž tx . is continuous in t for e¨ ery fixed x g E1. Then there exists a unique linear mapping T : E1 ª E2 such that 5 f Ž x. y T Ž x. 5 F
ku
5 x5 p
for all x g E1 and any gi¨ en integer k ) 1. However, it was verified that the similar result for the case p s 1 does not hold Žsee w7x.. Furthermore, in 1993 Rassias and ˇ Semrl w6x proved a more generalized form of the above result: Let E1 be a real normed space and E2 a real Banach space. Assume that H: w0, `. = w0, `. ª w0, `. is a monotonically increasing symmetric homogeneous mapping of degree p / 1, i.e., H Ž l x, l y . s l p H Ž x, y . for l, x, y G 0. Suppose f : E1 ª E2 to satisfy 5 f Ž x q y . y f Ž x . y f Ž y . 5 F H Ž 5 x 5 , 5 y 5. for all x, y g E1. Then there exists a unique additi¨ e mapping T : E1 ª E2 satisfying 5 f Ž x. y T Ž x. 5 F
H Ž 1, 1 . <2 p y 2 <
5 x5 p
for all x g E1. Recently, Gavruta ˘ ˘ w2x verified the following theorem: Let G be an abelian group and E a Banach space. Denote by w : G = G ª w0, `. a mapping such that F Ž x, y . s
`
Ý ns1
1 2n
w Ž 2 ny 1 x, 2 ny1 y . - `
for all x, y g G. Suppose f : G ª E is a mapping satisfying 5 f Ž x q y . y f Ž x . y f Ž y . 5 F w Ž x, y . for e¨ ery x, y g G. Then there exists a unique additi¨ e mapping T : G ª E such that 5 f Ž x . y T Ž x . 5 F F Ž x, x . for all x g G. In this paper, let Ž G, q. be an abelian group and Ž E, 5 ? 5. a Banach space. Following the convention we occasionally write 2 x, 3 x, ??? instead of
HYERS ] ULAM ] RASSIAS STABILITY
223
x q x, x q x q x, . . . . Consider a mapping w : G = G ª w0, `. satisfying w Ž x, y . s w Ž y, x . for all x, y g G. For all n g N and all x, y g G define
wn1 Ž x, y . s w Ž nx, y .
wn2 Ž x, y . s w Ž x, ny . .
and
By a s Ž a1 , a2 , . . . . we denote a sequence with a n g 1, 24 for all n g N, and we define for a fixed natural number k ) 1
c ka Ž x, y . s w 1a1 Ž x, y . q ??? qw ka k Ž x, y . . Suppose that there exists a sequence a s Ž a1 , a2 , . . . . with a n g 1, 24 for all n g N such that Ck Ž x, y . s
`
Ý ns1
1 kn
a c ky1 Ž k ny1 x, k ny1 y . - `
Ž 1.
for all x, y g G. We shall prove the following theorem: THEOREM.
Suppose f : G ª E is a mapping satisfying 5 f Ž x q y . y f Ž x . y f Ž y . 5 F w Ž x, y .
Ž 2.
for all x, y g G. Then there exists a unique additi¨ e mapping T : G ª E such that 5 f Ž x . y T Ž x . 5 F Ck Ž x, x . . Moreo¨ er, if G is a Banach space, and if f Ž tx . is continuous in t for e¨ ery fixed x g G, then T is linear.
2. PROOF OF THE THEOREM Ža. We claim 1 n
f Ž nx . y f Ž x . F
1 n
a cny 1 Ž x, x .
Ž 3.
for each n ) 1 and all x g G. We verify it by induction on n. By putting y s x in Ž2., we obtain 5 f Ž 2 x . y 2 f Ž x . 5 F w Ž x, x . s c 1a Ž x, x . . This implies the validity of Ž3. for the case n s 2. Now, assume that the inequality Ž3. is valid for n s m Ž m G 2., i.e., a 5 f Ž mx . y mf Ž x . 5 F cmy 1 Ž x, x . .
Ž 4.
224
SOON-MO JUNG
For the case n s m q 1, replacing y by mx in Ž2., we get 5 f Ž x q mx . y f Ž x . y f Ž mx . 5 F w Ž x, mx . s w Ž mx, x . . It follows from Ž4. and Ž5. that
Ž 5.
5 f Ž Ž m q 1 . x . y Ž m q 1 . f Ž x . 5 F 5 f Ž Ž m q 1 . x . y f Ž x . y f Ž mx . 5 q 5 f Ž mx . y mf Ž x . 5 F cma Ž x, x . . Accordingly, the assertion Ž3. holds for all n ) 1 and all x g G. Žb. We claim n 1 1 a n f k x y f x F c k iy1 x, k iy1 x . Ž . Ž . Ý n i ky 1 Ž k k is1
Ž 6.
for each n g N. We also prove it by induction on n. The validity of Ž6. for n s 1 follows from Ž3.. By assuming the induction argument for n s m and by putting n s 1 and then substituting k m x for x in Ž6., we obtain 1 f k mq 1 x . y f Ž x . mq 1 Ž k F F F
1 k
mq1
1
1
k
m
k
mq1
s
k
1 1 m
Ý is1
f Ž k mq 1 x . y
k
1 k
f Ž kmx. q
m
m
f Ž k ? kmx. y f Ž kmx. q
is1 m
a m m c ky 1 Ž k x, k x . q
1
Ý
ki
is1
1 ki
Ý
1 km 1 ki
f Ž kmx. y f Ž x.
a iy1 c ky x, k iy1 x . 1Ž k
a c ky1 Ž k iy1 x, k iy1 x .
a iy1 c ky x, k iy1 x . . 1Ž k
Hence, the inequality Ž6. is valid for all n g N. Žc. We claim that the sequence kyn f Ž k n x .4 is a Cauchy sequence. Indeed, by Ž6. we have, for n ) m, 1 1 1 1 f knx. y m f Ž kmx. s m f k nym ? k m x . y f Ž k m x . n Ž nym Ž k k k k F F
nym
1 k
m
1
Ý
ki
is1 `
Ý ismq1
1 ki
a iy1 c ky ? k m x, k iy1 ? k m x . 1Ž k
a iy1 c ky x, k iy1 x . . 1Ž k
HYERS ] ULAM ] RASSIAS STABILITY
225
In view of Ž1. we can make the last term as small as we wish by choosing m sufficiently large. Therefore, the given sequence is a Cauchy sequence. Since E is a Banach space, the sequence kyn f Ž k n x .4 converges for every x g G. Thus we may define T Ž x . s lim
1
nª`
kn
f Ž knx. .
Žd. We claim that the mapping T is additive. By substituting k n x and k n y for x and y in Ž2. respectively, we get 5 f Ž k n x q k n y . y f Ž k n x . y f Ž k n y . 5 F w Ž k n x, k n y . .
Ž 7.
But it follows from Ž1. that lim
nª`
1 kn
a n n c ky 1 Ž k x, k y . s 0.
Therefore, dividing both sides of Ž7. by k n and letting n ª `, we obtain T Ž x q y. s T Ž x. q T Ž y. . Že. According to Ž6. it is obvious that 5 f Ž x . y T Ž x . 5 F Ck Ž x, x .
Ž 8.
for all x g G. Žf. We assert that T is unique. Let T 9: G ª E be another additive mapping with the property Ž8.. Since T and T 9 are additive mappings satisfying Ž8., we have 5T Ž x . y T 9Ž x . 5 s F F
1 k
n
1 k
n
1 k
s2
n
T Ž knx. y
1 kn
T 9Ž k n x .
5T Ž k n x . y f Ž k n x . 5 q Ck Ž k n x, k n x . q `
Ý msnq1
1 km
1 kn
1 kn
5 f Ž k n x . y T 9Ž k n x . 5
Ck Ž k n x, k n x .
a my1 c ky x, k my1 x . . 1Ž k
In view of Ž1. we can make the last term arbitrarily small by selecting sufficiently large n. Hence, if follows T s T 9. Žg. Finally, it can be also proved that T is linear if G is a Banach space and f Ž tx . is continuous in t for every fixed x g G Žcf. w4x..
226
SOON-MO JUNG
3. DISCUSSION The result of this paper was obtained independently by P. Gavruta, ˘ ˘ M. Hossu, D. Popescu, and C. Caprau ˘ ˘ in the papers: w*x P. Gavruta, ˘ ˘ M. Hossu, D. Popescu, and C. Caprau, ˘ ˘ On the stability of mappings, Bull. Appl. Math. Tech. Uni¨ . Budapest 73 Ž1994., 169]176. w**x P. Gavruta, ˘ ˘ M. Hossu, D. Popescu, and C. Caprau, ˘ ˘ On the stability of mappings and an answer to a problem of Th. M. Rassias, submitted for publication. The result of this paper is non-trivial as was proved by P. Gavruta ˘ ˘ in w***x P. Gavruta, ˘ ˘ On the Hyers-Ulam-Rassias stability of mappings, J. Math. Anal. Appl. in press.
REFERENCES 1. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 Ž1991., 431]434. 2. P. Gavruta, ˘ ˘ A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 Ž1994., 431]436. 3. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S. A. 27 Ž1941., 222]224. 4. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 Ž1978., 297]300. 5. Th. M. Rassias, On a modified Hyers-Ulam Sequence, J. Math. Anal. Appl. 158 Ž1991., 106]113. 6. Th. M. Rassias and P. ˇ Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 Ž1993., 325]338. 7. Th. M. Rassias and P. ˇ Semrl, On the behavior of mappings which do not satisfy HyersUlam stability, Proc. Amer. Math. Soc. 114 Ž1992., 989]993. 8. S. M. Ulam, ‘‘Problems in Modern Mathematics,’’ Chap. VI, Science ed., Wiley, New York, 1960.
204, 221]226 Ž1996.
0433
On the Hyers]Ulam]Rassias Stability of Approximately Additive Mappings Soon-Mo Jung Mathematical Part, College of Science and Technology, Hong-Ik Uni¨ ersity, 339-800 Chochiwon, South Korea Submitted by Joseph D. Ward Received July 25, 1995
In this paper, using an idea from the paper of P. Gavruta ˘ ˘ Ž J. Math. Anal. Appl. 184, 1994, 431]436., the results of Th. M. Rassias Ž J. Math. Anal. Appl. 158, 1991, 106]113. shall be generalized. Q 1996 Academic Press, Inc.
1. INTRODUCTION Ulam Žcf. w8x. had raised the following question: Under what conditions does there exist an additive mapping near an approximately additive mapping? In 1941 Hyers w3x proved that if f : E1 ª E2 is a mapping satisfying 5 f Ž x q y. y f Ž x. y f Ž y. 5 F d for all x, y g E1 , where E1 and E2 are Banach spaces and d is a given positive number, then there exists a unique additive mapping T : E1 ª E2 such that 5 f Ž x. y T Ž x. 5 F d for any x g E1. In addition, he also proved that if f Ž tx . is continuous in the real variable t for any fixed x g E1 , then T is linear. Rassias w4, 5x and Gajda w1x showed some generalizations of the result of Hyers in the following ways: Let f : E1 ª E2 be a mapping satisfying 5 f Ž x q y. y f Ž x. y f Ž y. 5 F u Ž 5 x5 p q 5 y5 p. 221 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
222
SOON-MO JUNG
for all x, y g E1 , where u G 0 and p / 1. Suppose f Ž tx . is continuous in t for e¨ ery fixed x g E1. Then there exists a unique linear mapping T : E1 ª E2 such that 5 f Ž x. y T Ž x. 5 F
ku
5 x5 p
for all x g E1 and any gi¨ en integer k ) 1. However, it was verified that the similar result for the case p s 1 does not hold Žsee w7x.. Furthermore, in 1993 Rassias and ˇ Semrl w6x proved a more generalized form of the above result: Let E1 be a real normed space and E2 a real Banach space. Assume that H: w0, `. = w0, `. ª w0, `. is a monotonically increasing symmetric homogeneous mapping of degree p / 1, i.e., H Ž l x, l y . s l p H Ž x, y . for l, x, y G 0. Suppose f : E1 ª E2 to satisfy 5 f Ž x q y . y f Ž x . y f Ž y . 5 F H Ž 5 x 5 , 5 y 5. for all x, y g E1. Then there exists a unique additi¨ e mapping T : E1 ª E2 satisfying 5 f Ž x. y T Ž x. 5 F
H Ž 1, 1 . <2 p y 2 <
5 x5 p
for all x g E1. Recently, Gavruta ˘ ˘ w2x verified the following theorem: Let G be an abelian group and E a Banach space. Denote by w : G = G ª w0, `. a mapping such that F Ž x, y . s
`
Ý ns1
1 2n
w Ž 2 ny 1 x, 2 ny1 y . - `
for all x, y g G. Suppose f : G ª E is a mapping satisfying 5 f Ž x q y . y f Ž x . y f Ž y . 5 F w Ž x, y . for e¨ ery x, y g G. Then there exists a unique additi¨ e mapping T : G ª E such that 5 f Ž x . y T Ž x . 5 F F Ž x, x . for all x g G. In this paper, let Ž G, q. be an abelian group and Ž E, 5 ? 5. a Banach space. Following the convention we occasionally write 2 x, 3 x, ??? instead of
HYERS ] ULAM ] RASSIAS STABILITY
223
x q x, x q x q x, . . . . Consider a mapping w : G = G ª w0, `. satisfying w Ž x, y . s w Ž y, x . for all x, y g G. For all n g N and all x, y g G define
wn1 Ž x, y . s w Ž nx, y .
wn2 Ž x, y . s w Ž x, ny . .
and
By a s Ž a1 , a2 , . . . . we denote a sequence with a n g 1, 24 for all n g N, and we define for a fixed natural number k ) 1
c ka Ž x, y . s w 1a1 Ž x, y . q ??? qw ka k Ž x, y . . Suppose that there exists a sequence a s Ž a1 , a2 , . . . . with a n g 1, 24 for all n g N such that Ck Ž x, y . s
`
Ý ns1
1 kn
a c ky1 Ž k ny1 x, k ny1 y . - `
Ž 1.
for all x, y g G. We shall prove the following theorem: THEOREM.
Suppose f : G ª E is a mapping satisfying 5 f Ž x q y . y f Ž x . y f Ž y . 5 F w Ž x, y .
Ž 2.
for all x, y g G. Then there exists a unique additi¨ e mapping T : G ª E such that 5 f Ž x . y T Ž x . 5 F Ck Ž x, x . . Moreo¨ er, if G is a Banach space, and if f Ž tx . is continuous in t for e¨ ery fixed x g G, then T is linear.
2. PROOF OF THE THEOREM Ža. We claim 1 n
f Ž nx . y f Ž x . F
1 n
a cny 1 Ž x, x .
Ž 3.
for each n ) 1 and all x g G. We verify it by induction on n. By putting y s x in Ž2., we obtain 5 f Ž 2 x . y 2 f Ž x . 5 F w Ž x, x . s c 1a Ž x, x . . This implies the validity of Ž3. for the case n s 2. Now, assume that the inequality Ž3. is valid for n s m Ž m G 2., i.e., a 5 f Ž mx . y mf Ž x . 5 F cmy 1 Ž x, x . .
Ž 4.
224
SOON-MO JUNG
For the case n s m q 1, replacing y by mx in Ž2., we get 5 f Ž x q mx . y f Ž x . y f Ž mx . 5 F w Ž x, mx . s w Ž mx, x . . It follows from Ž4. and Ž5. that
Ž 5.
5 f Ž Ž m q 1 . x . y Ž m q 1 . f Ž x . 5 F 5 f Ž Ž m q 1 . x . y f Ž x . y f Ž mx . 5 q 5 f Ž mx . y mf Ž x . 5 F cma Ž x, x . . Accordingly, the assertion Ž3. holds for all n ) 1 and all x g G. Žb. We claim n 1 1 a n f k x y f x F c k iy1 x, k iy1 x . Ž . Ž . Ý n i ky 1 Ž k k is1
Ž 6.
for each n g N. We also prove it by induction on n. The validity of Ž6. for n s 1 follows from Ž3.. By assuming the induction argument for n s m and by putting n s 1 and then substituting k m x for x in Ž6., we obtain 1 f k mq 1 x . y f Ž x . mq 1 Ž k F F F
1 k
mq1
1
1
k
m
k
mq1
s
k
1 1 m
Ý is1
f Ž k mq 1 x . y
k
1 k
f Ž kmx. q
m
m
f Ž k ? kmx. y f Ž kmx. q
is1 m
a m m c ky 1 Ž k x, k x . q
1
Ý
ki
is1
1 ki
Ý
1 km 1 ki
f Ž kmx. y f Ž x.
a iy1 c ky x, k iy1 x . 1Ž k
a c ky1 Ž k iy1 x, k iy1 x .
a iy1 c ky x, k iy1 x . . 1Ž k
Hence, the inequality Ž6. is valid for all n g N. Žc. We claim that the sequence kyn f Ž k n x .4 is a Cauchy sequence. Indeed, by Ž6. we have, for n ) m, 1 1 1 1 f knx. y m f Ž kmx. s m f k nym ? k m x . y f Ž k m x . n Ž nym Ž k k k k F F
nym
1 k
m
1
Ý
ki
is1 `
Ý ismq1
1 ki
a iy1 c ky ? k m x, k iy1 ? k m x . 1Ž k
a iy1 c ky x, k iy1 x . . 1Ž k
HYERS ] ULAM ] RASSIAS STABILITY
225
In view of Ž1. we can make the last term as small as we wish by choosing m sufficiently large. Therefore, the given sequence is a Cauchy sequence. Since E is a Banach space, the sequence kyn f Ž k n x .4 converges for every x g G. Thus we may define T Ž x . s lim
1
nª`
kn
f Ž knx. .
Žd. We claim that the mapping T is additive. By substituting k n x and k n y for x and y in Ž2. respectively, we get 5 f Ž k n x q k n y . y f Ž k n x . y f Ž k n y . 5 F w Ž k n x, k n y . .
Ž 7.
But it follows from Ž1. that lim
nª`
1 kn
a n n c ky 1 Ž k x, k y . s 0.
Therefore, dividing both sides of Ž7. by k n and letting n ª `, we obtain T Ž x q y. s T Ž x. q T Ž y. . Že. According to Ž6. it is obvious that 5 f Ž x . y T Ž x . 5 F Ck Ž x, x .
Ž 8.
for all x g G. Žf. We assert that T is unique. Let T 9: G ª E be another additive mapping with the property Ž8.. Since T and T 9 are additive mappings satisfying Ž8., we have 5T Ž x . y T 9Ž x . 5 s F F
1 k
n
1 k
n
1 k
s2
n
T Ž knx. y
1 kn
T 9Ž k n x .
5T Ž k n x . y f Ž k n x . 5 q Ck Ž k n x, k n x . q `
Ý msnq1
1 km
1 kn
1 kn
5 f Ž k n x . y T 9Ž k n x . 5
Ck Ž k n x, k n x .
a my1 c ky x, k my1 x . . 1Ž k
In view of Ž1. we can make the last term arbitrarily small by selecting sufficiently large n. Hence, if follows T s T 9. Žg. Finally, it can be also proved that T is linear if G is a Banach space and f Ž tx . is continuous in t for every fixed x g G Žcf. w4x..
226
SOON-MO JUNG
3. DISCUSSION The result of this paper was obtained independently by P. Gavruta, ˘ ˘ M. Hossu, D. Popescu, and C. Caprau ˘ ˘ in the papers: w*x P. Gavruta, ˘ ˘ M. Hossu, D. Popescu, and C. Caprau, ˘ ˘ On the stability of mappings, Bull. Appl. Math. Tech. Uni¨ . Budapest 73 Ž1994., 169]176. w**x P. Gavruta, ˘ ˘ M. Hossu, D. Popescu, and C. Caprau, ˘ ˘ On the stability of mappings and an answer to a problem of Th. M. Rassias, submitted for publication. The result of this paper is non-trivial as was proved by P. Gavruta ˘ ˘ in w***x P. Gavruta, ˘ ˘ On the Hyers-Ulam-Rassias stability of mappings, J. Math. Anal. Appl. in press.
REFERENCES 1. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 Ž1991., 431]434. 2. P. Gavruta, ˘ ˘ A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 Ž1994., 431]436. 3. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S. A. 27 Ž1941., 222]224. 4. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 Ž1978., 297]300. 5. Th. M. Rassias, On a modified Hyers-Ulam Sequence, J. Math. Anal. Appl. 158 Ž1991., 106]113. 6. Th. M. Rassias and P. ˇ Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 Ž1993., 325]338. 7. Th. M. Rassias and P. ˇ Semrl, On the behavior of mappings which do not satisfy HyersUlam stability, Proc. Amer. Math. Soc. 114 Ž1992., 989]993. 8. S. M. Ulam, ‘‘Problems in Modern Mathematics,’’ Chap. VI, Science ed., Wiley, New York, 1960.