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 PDF e-PUB PubReader PPT Export by style
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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  concerning the stability of group homomorphisms.
The functional equation PPT Slide
Lager Image
is called the Cauchy equation . In particular, every solution of the Cauchy equation is said to be an additive mapping . Hyers  gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki  for additive mappings and by Rassias  for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Gǎvruta  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 PPT Slide
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 PPT Slide
Lager Image
for all x X .
It follows from (2.1) and (2.2) that PPT Slide
Lager Image
and so f ( x + y + z ) = f ( x ) + f ( y ) + f ( z ) for all x, y, z X . Since f (0) = 0, PPT Slide
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 PPT Slide
Lager Image
is Cauchy for all x X . Since Y is a Banach space, the sequence PPT Slide
Lager Image
converges. So one can define the mapping A : X Y by PPT Slide
Lager Image
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 PPT Slide
Lager Image
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 PPT Slide
Lager Image
for all x , y , z X . So PPT Slide
Lager Image
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 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 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 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 (2.6) that 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 (2.10) that the sequence PPT Slide
Lager Image
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 PPT Slide
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 PPT Slide
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 PPT Slide
Lager Image
and so PPT Slide
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