FIXED POINTS AND ADDITIVE ρ-FUNCTIONAL EQUATIONS IN BANACH SPACES
FIXED POINTS AND ADDITIVE ρ-FUNCTIONAL EQUATIONS IN BANACH SPACES
The Pure and Applied Mathematics. 2015. Nov, 22(4): 365-374
• Received : September 26, 2015
• Accepted : September 30, 2015
• Published : November 30, 2015
PDF
e-PUB
PPT
Export by style
Article
Author
Metrics
Cited by
TagCloud
YONG HOON, CHOI
DEPARTMENT OF MATHEMATICS, HANYANG UNIVERSITY, SEOUL 04763, KOREAEmail address:etoile5131@naver.com
SUNGSIK, YUN
DEPARTMENT OF FINANCIAL MATHEMATICS, HANSHIN UNIVERSITY, GYEONGGI-DO 447-791, KOREAEmail address:ssyun@hs.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 fixed point method, we prove the Hyers-Ulam stability of the additive ρ -functional equations (0.1) and (0.2) in Banach spaces.
Keywords
1. INTRODUCTION AND PRELIMINARIES
The stability problem of functional equations originated from a question of Ulam [14] 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 [7] 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 [13] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [6] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.
We recall a fundamental result in fixed point theory.
Theorem 1.1 ( [2 , 5] ) . Let ( X, d ) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x X, either
PPT Slide
Lager Image
for all nonnegative integers n or there exists a positive integer n0 such that
(1) d ( Jnx , J n+1 x ) < ∞, ∀ n n 0 ;
(2) the sequence { Jnx } converges to a fixed point y * of J ;
(3) y is the unique fixed point of J in the set Y = { y X | d ( J n0 x, y ) < ∞};
(4)
PPT Slide
Lager Image
for all y Y .
In 1996, G. Isac and Th.M. Rassias [8] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [3 , 4 , 10 , 11 , 12] ).
In Section 2, we solve the additive functional equation (0.1) and prove the HyersUlam 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 HyersUlam 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 such that there exists an L < 1 with
PPT Slide
Lager Image
for all x , y , z X. 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 (2.4), we get
PPT Slide
Lager Image
for all x X . So
PPT Slide
Lager Image
for all x X .
Consider the set
PPT Slide
Lager Image
and introduce the generalized metric on S :
PPT Slide
Lager Image
where, as usual, inf ϕ = +∞. It is easy to show that ( S , d ) is complete (see [9] ).
Now we consider the linear mapping J : S → S such that
PPT Slide
Lager Image
for all x X .
Let g , h S be given such that d ( g , h ) = ε . Then
PPT Slide
Lager Image
for all x X . Hence
PPT Slide
Lager Image
for all x X . So d ( g , h ) = ε implies that d ( Jg , Jh ) ≤ . This means that
PPT Slide
Lager Image
for all g , h S .
It follows from (2.6) that
PPT Slide
Lager Image
for all
PPT Slide
Lager Image
By Theorem 1.1, there exists a mapping A : X → Y satisfying the following:
(1) A is a fixed point of J , i.e.,
PPT Slide
Lager Image
for all x X . The mapping A is a unique fixed point of J in the set
PPT Slide
Lager Image
This implies that A is a unique mapping satisfying (2.7) such that there exists a µ ∈ (0, ∞) satisfying
PPT Slide
Lager Image
for all x X ;
(2) d ( Jlf , A ) → 0 as l → ∞. This implies the equality
PPT Slide
Lager Image
for all x X ;
(3)
PPT Slide
Lager Image
, which implies
PPT Slide
Lager Image
for all x X .
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 . The proof follows from Theorem 2.2 by takig φ ( x , y , z ) = θ (∥ x r +∥ y r +∥ z r ) for all x , y , z X . Then we can choose L = 2 1−r and we get the desired result.   ☐
Theorem 2.4. Let φ : X 3 → [0,∞) be a function such that there exists an L < 1 with
PPT Slide
Lager Image
for all x , y , z X Let f : X → Y be a mapping satisfying f (0) = 0 and (2.4). 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 .
Let ( S , d ) be the generalized metric space defined in the proof of Theorem 2.2.
Now we consider the linear mapping J : S → S such that
PPT Slide
Lager Image
for all x X .
It follows from (2.9) that
PPT Slide
Lager Image
. So
PPT Slide
Lager Image
for all x X .
The rest of the proof is similar to the proof of Theorem 2.2.   ☐
Corollary 2.5. Let r < 1 and θ be positive 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 . The proof follows from Theorem 2.4 by takig φ ( x , y , z ) = θ (∥ x r +∥ y r +∥ z r ) for all x , y , z X . Then we can choose L = 2 r−1 and 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 (3.1), we get −2 f (0) = −2 ρf (0). So f (0) = 0.
Letting y = x and z = 0 in (3.1), 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 f ( x + y ) = f ( x ) + f ( y ) 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 such that there exists an L < 1 with
PPT Slide
Lager Image
for all x , y , z X . 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 . The proof follows from Theorem 3.2 by takig φ ( x , y , z ) = θ (∥ x r +∥ y r +∥ z r ) for all x , y , z X . Then we can choose L = 2 1−r and we get the desired result.   ☐
Theorem 3.4. Let φ : X 3 → [0,∞) be a function such that there exists an L < 1 with
PPT Slide
Lager Image
for all x , y , z X Let f : X → Y be a mapping satisfying f (0) = 0 and (3.3). 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 proof of Theorem 2.2.   ☐
Corollary 3.5. Let r < 1 and θ be positive 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 . The proof follows from Theorem 3.4 by takig φ ( x , y , z ) = θ (∥ x r +∥ y r +∥ z r ) for all x , y , z X . Then we can choose L = 2 r−1 and we get the desired result.   ☐
Acknowledgements
This research was supported by Hanshin University Research Grant.
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
Cădariu L. , Radu V. (2003) Fixed points and the stability of Jensen’s functional equation J. Inequal. Pure Appl. Math. Art. ID 4 4 (1)
Cădariu L. , Radu V. (2004) On the stability of the Cauchy functional equation: a fixed point approach Grazer Math. Ber. 346 (1) 43 - 52
Cădariu L. , Radu V. (2008) Fixed point methods for the generalized stability of functional equations in a single variable Fixed Point Theory and Applications Art. ID 749392 2008
Diaz J. , Margolis B. (1968) A fixed point theorem of the alternative for contractions on a generalized complete metric space Bull. Amer. Math. Soc. 74 305 - 309    DOI : 10.1090/S0002-9904-1968-11933-0
Găvruta P. (1994) A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings J. Math. Anal. Appl. 184 431 - 43    DOI : 10.1006/jmaa.1994.1211
D.H. Hyers (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
Isac G. , Rassias Th. M. (1996) Stability of ψ-additive mappings: Appications to nonlinear analysis Internat. J. Math. Math. Sci. 19 219 - 228    DOI : 10.1155/S0161171296000324
Miheƫ D. , Radu V. (2008) On the stability of the additive Cauchy functional equation in random normed spaces J. Math. Anal. Appl. 343 567 - 572    DOI : 10.1016/j.jmaa.2008.01.100
Park C. (2007) Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras Fixed Point Theory and Applications Art. ID 50175 2007
Park C. (2008) Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach Fixed Point Theory and Applications Art. ID 493751 2008
Radu V. (2003) The fixed point alternative and the stability of functional equations Fixed Point Theory 4 91 - 96
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