ADDITIVE ρ-FUNCTIONAL EQUATIONS IN BANACH SPACES
ADDITIVE ρ-FUNCTIONAL EQUATIONS IN BANACH SPACES
The Pure and Applied Mathematics. 2015. Nov, 22(4): 375-382
• Received : October 01, 2015
• Accepted : October 12, 2015
• Published : November 30, 2015
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IN WHAN, JUN
DEPARTMENT OF MATHEMATICS, HANYANG UNIVERSITY, SEOUL 04763, KOREAEmail address:zhanggua@@naver.com
JEONG PIL, SEO
OHSANG HIGH SCHOOL, GUMI 730-842, KYEONGSANGBUK-DO, KOREAEmail address:sjp4829@@hanmail.net
SUNGJIN, LEE
DEPARTMENT OF MATHEMATICS, DAEJIN UNIVERSITY, KYEONGGI 11159, KOREAEmail address:hyper@@daejin.ac.kr

Abstract
In this paper, we solve the additive ρ -functional equations where ρ is a fixed number with ρ ≠ 1, 2, and where ρ is a fixed number with ρ ≠ 1. Using the direct method, we prove the Hyers-Ulam stability of the additive ρ -functional equations (0.1) and (0.2) in Banach spaces.
Keywords
1. INTRODUCTION
The stability problem of functional equations originated from a question of Ulam [5] concerning the stability of group homomorphisms.
The functional equation
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is called the Cauchy equation . In particular, every solution of the Cauchy equation is said to be an additive mapping . Hyers [3] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Gǎvruta [2] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.
In Section 2, we solve the additive functional equation (0.1) and prove the Hyers-Ulam stability of the additive functional equation (0.1) in Banach spaces.
In Section 3, we solve the additive functional equation (0.2) and prove the Hyers-Ulam stability of the additive functional equation (0.2) in Banach spaces.
Throughout this paper, assume that X is a normed space and that Y is a Banach space.
Let ρ be a number with ρ ≠ 1, 2.
We solve and investigate the additive ρ -functional equation (0.1) in normed spaces.
Lemma 2.1. If a mapping f : X Y satisfies
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Lager Image
for all x , y , z X , then f : X Y is additive .
Proof . Assume that f : X Y satisfies (2.1).
Letting x = y = z = 0 in (2.1), we get −2 f (0) = − ρf (0). So f (0) = 0.
Letting y = x and z = 0 in (2.1), we get f (2 x )−2 f ( x ) = 0 and so f (2 x ) = 2 f ( x ) for all x X . Thus
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for all x X .
It follows from (2.1) and (2.2) that
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and so f ( x + y + z ) = f ( x ) + f ( y ) + f ( z ) for all x, y, z X . Since f (0) = 0,
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Lager Image
for all x , y X .   ☐
We prove the Hyers-Ulam stability of the additive ρ -functional equation (2.1) in Banach spaces.
Theorem 2.2. Let φ : X 3 → [0, ∞) be a function and let f : X Y be a mapping satisfying f (0) = 0 and
PPT Slide
Lager Image
PPT Slide
Lager Image
for all x , y , z X . Then there exists a unique additive mapping A : X Y such that
PPT Slide
Lager Image
for all x X .
Proof . Letting y = x and z = 0 in (2.4), we get
PPT Slide
Lager Image
for all x X . So
PPT Slide
Lager Image
for all x X . Hence
PPT Slide
Lager Image
for all nonnegative integers m and l with m > l and all x X . It follows from (2.7) that the sequence
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Lager Image
is Cauchy for all x X . Since Y is a Banach space, the sequence
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converges. So one can define the mapping A : X Y by
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for all x X . Moreover, letting l = 0 and passing the limit m → ∞ in (2.7), we get (2.5).
Now, let T : X Y be another additive mapping satisfying (2.5). Then we have
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which tends to zero as q → ∞ for all x X . So we can conclude that A ( x ) = T ( x ) for all x X . This proves the uniqueness of A .
It follows from (2.3) and (2.4) that
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for all x , y , z X . So
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for all x , y , z X . By Lemma 2.1, the mapping A : X Y is additive.   ☐
Corollary 2.3. Let r > 1 and θ be nonnegative real numbers, and let f : X Y be a mapping satisfying f (0) = 0 and
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Lager Image
for all x , y , z X . Then there exists a unique additive mapping A : X Y such that
PPT Slide
Lager Image
for all x X .
Proof . Letting φ ( x , y , z ) :=
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Lager Image
in Theorem 2.2, we get the desired result.   ☐
Theorem 2.4. Let φ : X 3 → [0, ∞) be a function and let f : X Y be a mapping satisfying f (0) = 0, (2.4) and
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Lager Image
for all x , y , z X . Then there exists a unique additive mapping A : X Y such that
PPT Slide
Lager Image
for all x X .
Proof . It follows from (2.6) that
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Lager Image
for all x X . Hence
PPT Slide
Lager Image
for all nonnegative integers m and l with m > l and all x X . It follows from (2.10) that the sequence
PPT Slide
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is a Cauchy sequence for all x X . Since Y is complete, the sequence
PPT Slide
Lager Image
converges. So one can define the mapping A : X Y by
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Lager Image
for all x X . Moreover, letting l = 0 and passing the limit m → ∞ in (2.10), we get (2.9).
The rest of the proof is similar to the proof of Theorem 2.2.   ☐
Corollary 2.5. Let r < 1 and θ be nonnegative real numbers, and let f : X Y be a mapping satisfying f (0) = 0 and (2.8). Then there exists a unique additive mapping A : X Y such that
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Lager Image
for all x X .
Proof . Letting φ ( x , y , z ) :=
PPT Slide
Lager Image
in Theorem 2.4, we get the desired result.   ☐
Let ρ be a number with ρ ≠ 1.
We solve and investigate the additive ρ -functional equation (0.2) in normed spaces.
Lemma 3.1. If a mapping f : X Y satisfies
PPT Slide
Lager Image
for all x , y , z X , then f : X Y is additive .
Proof . Assume that f : X Y satisfies (3.1).
Letting x = y = z = 0 in (2.1), we get -2 f (0) = -2 ρf (0). So f (0) = 0.
Letting y = x and z = 0 in (2.1), we get f (2 x )-2 f ( x ) = 0 and so f (2 x ) = 2 f ( x ) for all x X . Thus
PPT Slide
Lager Image
for all x X .
It follows from (3.1) and (3.2) that
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Lager Image
and so
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Lager Image
for all x , y X .   ☐
We prove the Hyers-Ulam stability of the additive ρ -functional equation (3.1) in Banach spaces.
Theorem 3.2. Let φ : X 3 → [0, ∞) be a function and let f : X Y be a mapping satisfying f (0) = 0 and
PPT Slide
Lager Image
for all x , y , z X . Then there exists a unique additive mapping A : X Y such that
PPT Slide
Lager Image
for all x X .
Proof . Letting y = x and z = 0 in (3.3), we get
PPT Slide
Lager Image
for all x X .
The rest of the proof is similar to the proof of Theorem 2.2.   ☐
Corollary 3.3. Let r > 1 and θ be nonnegative real numbers, and let f : X Y be a mapping satisfying f (0) = 0 and
PPT Slide
Lager Image
for all x , y , z X . Then there exists a unique additive mapping A : X Y such that
PPT Slide
Lager Image
for all x X .
Proof . Letting φ ( x , y , z ) :=
PPT Slide
Lager Image
in Theorem 3.2, we get the desired result.   ☐
Theorem 3.4. Let φ : X 3 → [0, ∞) be a function and let f : X Y be a mapping satisfying f (0) = 0, (3.3) and
PPT Slide
Lager Image
for all x , y , z X . Then there exists a unique additive mapping A : X Y such that
PPT Slide
Lager Image
for all x X .
Proof . It follows from (3.4) that
PPT Slide
Lager Image
for all x X .
The rest of the proof is similar to the proofs of Theorems 2.2 and 2.4.   ☐
Corollary 3.5. Let r < 1 and θ be nonnegative real numbers, and let f : X Y be a mapping satisfying f (0) = 0 and (3.5). Then there exists a unique additive mapping A : X Y such that
PPT Slide
Lager Image
for all x X .
Proof . Letting φ ( x , y , z ) :=
PPT Slide
Lager Image
in Theorem 3.4, we get the desired result.   ☐
References
Aoki T. (1950) On the stability of the linear transformation in Banach spaces J. Math. Soc. Japan 2 64 - 66    DOI : 10.2969/jmsj/00210064
Gǎvruta P. (1994) A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings J. Math. Anal. Appl. 184 431 - 443    DOI : 10.1006/jmaa.1994.1211
Hyers D.H. (1941) On the stability of the linear functional equation Proc. Natl. Acad. Sci. U.S.A. 27 222 - 224    DOI : 10.1073/pnas.27.4.222
Rassias Th.M. (1978) On the stability of the linear mapping in Banach spaces Proc. Amer. Math. Soc. 72 297 - 300    DOI : 10.1090/S0002-9939-1978-0507327-1
Ulam S.M. 1960 A Collection of the Mathematical Problems Interscience Publ. New York