STABILITY OF A CUBIC FUNCTIONAL EQUATION IN 2-NORMED SPACES
STABILITY OF A CUBIC FUNCTIONAL EQUATION IN 2-NORMED SPACES
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(5_6): 817-825
• Received : November 10, 2013
• Accepted : February 27, 2014
• Published : September 30, 2014
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CHANG IL KIM
KAP HUN JUNG

Abstract
In this paper, we prove the generalized Hyers-Ulam stability of the following cubic funtional equation by the direct method in 2-normed spaces. AMS Mathematics Subject Classification : 39B52, 39B72, 39B82.
Keywords
1. Introduction and preliminaries
G ä hler [4 , 5] has introduced the concept of 2-normed spaces and G ä hler and White [16] introduced the concept of 2-Banach spaces. Lewandowska published a series of papers on 2-normed sets and generalized 2-normed spaces [10 , 11] . Recently, Park [12] investigated approximate additive mappings, approximate Jensen mappings and approximate quadratic mappings in 2-Banach spaces.
We list some definitions related to 2-normed spaces.
Definition 1.1. Let X be a linear space over ℝ with dim X > 1 and let ∥· , ·∥ : X × X → ℝ be a function satisfying the following properties :
(1) ∥ x, y ∥ = 0 if and only if x and y are linearly dependent,
(2) ∥ x, y ∥ = ∥ y, x ∥,
(3) ∥ ax, y ∥ = | a |∥ x, y ∥, and
(4) ∥ x, y + z ∥ ≤ ∥ x, y ∥ + ∥ x, z
for all x, y, z X and a ∈ ℝ. Then the function ∥· , ·∥ is called a 2 - norm on X and ( X , ∥· , ·∥) is called a 2 - normed space .
Let ( X , ∥· , ·∥) be a 2-normed space. Suppose that x X and ∥ x, y ∥ = 0 for all y X . Suppose that x ≠ 0. Since dim X > 1, choose y X such that { x, y } is linearly independent and so by (1) in Definition1.1, we have ∥ x, y ∥ ≠ 0, which is a contradiction. Hence we have the following lemma.
Lemma 1.2. Let ( X , ∥· , ·∥) be a 2 - normed space. If x X and x, y ∥ = 0 for all y X, then x = 0.
Definition 1.3. A sequence { xn } in a 2-normed space ( X , ∥· ; ·∥) is called a 2 - Cauchy sequence if
PPT Slide
Lager Image
for all x X .
Definition 1.4. A sequence { xn } in a 2-normed space ( X , ∥· , ·∥) is called 2 - convergent if
PPT Slide
Lager Image
for all y X and for some x X . In case, { xn } said to be converge to x and denoted by xn x as n → ∞ or lim n→∞ xn = x .
A 2-normed space ( X , ∥· , ·∥) is called a 2 - Banach space if every 2-Cauchy sequence in X is 2-convergent. Now, we state the following results as lemma [12] .
Lemma 1.5. Let ( X , ∥· , ·∥) be a 2 - normed space. Then we have the following :
(1) |∥ x, z ∥ − ∥ y, z ∥| ≤ ∥ x y, z for all x, y, z X ,
(2) if x, z ∥ = 0 for all z X, then x = 0, and
(3) for any 2 - convergent sequence { xn } in X ,
PPT Slide
Lager Image
for all z X .
In 1940, S.M.Ulam [15] proposed the following stability problem :
“Let G 1 be a group and G 2 a metric group with the metric d . Given a constant δ > 0, does there exists a constant c > 0 such that if a mapping f : G 1 G 2 satisfies d ( f ( xy ); f ( x ) f ( y )) < 0 for all x, y G 1 , then there exists a unique homomorphism h : G 1 G 2 with d ( f ( x ); h ( x )) < δ for all x G 1 ?”
In the next year, D. H. Hyers [7] gave a partial solution of Ulam’s problem for the case of approximate additive mappings. Subsequently, his result was generalized by T. Aoki [1] for additive mappings and by TH. M. Rassias [14] for linear mappings, to consider the stability problem with unbounded Cauchy differences. During the last decades, the stability problems of funtional equations have been extensively investigated by a number of mathematicians.
Rassias [13] introduced the cubic functional equation
PPT Slide
Lager Image
and Jun and Kim [8] introduced the following cubic funtional equation
PPT Slide
Lager Image
In this paper, we inverstigate the following cubic funtional equation
PPT Slide
Lager Image
which is a linear combination of (1) and (2) and proved the generalized Hyers-Ulam stability of (3) in 2-normed spaces.
2. Stability of (3) from a normed space to a 2-Banach space
Thoughout this section, ( X , ∥·∥) or simply X is a real normed space and ( Y , ∥·,·∥) or simply Y is a 2-Banach space. We start the following theorem.
Theorem 2.1. A mapping f : X Y satisfies (3) if and only if f is cubic .
Proof . Suppose that f satisfies (3). Letting x = y = 0 in (3), we have f (0) = 0 and letting y = 0 in (3), we have
PPT Slide
Lager Image
for all x X . Letting x = 0 in (3), by (4), we have f ( y ) = − f (− y ) for all y X and so f is odd. Letting y = − y in (3), we have
PPT Slide
Lager Image
for all x, y X and by (3) and (5), we have
PPT Slide
Lager Image
for all x, y X . Hence by (3) and (6), we have
PPT Slide
Lager Image
for all x, y X . Interching x and y in (7), since f is odd, f satisfies (2) and hence f is cubic.
For any function f : X Y , we define the difference operator Df : X × X Y by
PPT Slide
Lager Image
Now we prove the generalized Hyers-Ulam stability of (3).
Theorem 2.2. Let ε ≥ 0, p and q be positive real numbers with p + q < 3 and r > 0. Suppose that f : X Y is a function such that
PPT Slide
Lager Image
for all x, y X and z Y. Then there exists a unique cubic function C : X Y satisfying (3) and
PPT Slide
Lager Image
for all x X and z Y .
Proof . Letting x = y = 0 in (8), we have ∥2 f (0); z ∥ = 0 for all z Y and by the definition of 2-norm, we have f (0) = 0. Putting y = 0 in (8), we have
PPT Slide
Lager Image
for all x X and z Y and so
PPT Slide
Lager Image
for all x X and z Y . Replacing x by 2 x in (11), we get
PPT Slide
Lager Image
for all x X and z Y . By (11) and (12), we get
PPT Slide
Lager Image
for all x X and z Y . By induction on n , we can show that
PPT Slide
Lager Image
for all x X and z Y . For m, n ∈ ℕ with n < m and x X , by (13), we have
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
is a 2- Cauchy sequence in Y for all x X . Since Y is a 2-Banach space, the sequence
PPT Slide
Lager Image
is a 2-convergent in Y for all x X and so we can define a mapping C : X Y as
PPT Slide
Lager Image
for all x X . By (14), we have
PPT Slide
Lager Image
for all x X and z Y and by Lemma 1.5 , we have
PPT Slide
Lager Image
for all x X and z Y . Next we will show that C satisfies (3). By (8), we have
PPT Slide
Lager Image
for all z Y , because p < 3, q < 3, p + q < 3 and so DC ( x, y ) = 0 for all x, y X . By Theoem 2.1, C is cubic.
To show that C is unique, suppose there exists another cubic function C′ : X Y which satisfies (3) and (9). Since C and C′ are cubic,
PPT Slide
Lager Image
for all x X . It follows that
PPT Slide
Lager Image
So ∥ C′ ( x )− C ( x ), z ∥= 0 for all z Y and hence C′ ( x ) = C ( x ) for all x X .
Related with Theorem 2.2, we can also the following theorem.
Theorem 2.3. Let ε ≥ 0, p and q be positive real numbers with p, q > 3 and r > 0. Suppose that f : X Y is a function satisfying (8). Then there exists a unique cubic function C : X Y satisfying (3) and
PPT Slide
Lager Image
for all x X and z Y .
Proof . Letting x = y = 0 in (8), we have ∥2 f (0), z ∥ = 0 for all z Y and so we have f (0) = 0. Putting y = 0 and replacing x by
PPT Slide
Lager Image
in (8), we get
PPT Slide
Lager Image
for all x X and z Y and so
PPT Slide
Lager Image
for all x X and z Y . Replacing x by
PPT Slide
Lager Image
in (16), we get
PPT Slide
Lager Image
for all x X and z Y . By (16) and (17), we get
PPT Slide
Lager Image
for all x X and z Y . By induction on n , we can show that
PPT Slide
Lager Image
for all x X and z Y . For m, n ∈ ℕ with n < m and x X , by (18), we have
PPT Slide
Lager Image
and since p > 3,
PPT Slide
Lager Image
is a 2- Cauchy sequence in Y for all x X . Since Y is a 2-Banach space, the sequence
PPT Slide
Lager Image
is a 2-convergent in Y for all x X . Define C : X Y as
PPT Slide
Lager Image
for all x X . By (18), we have
PPT Slide
Lager Image
for all x X and z Y and by Lemma 1.5, we have
PPT Slide
Lager Image
for all x X and z Y . Next we will show that C satisfies (3).
PPT Slide
Lager Image
for all z Y , because p, q > 3 and so DC ( x, y ) = 0 for all x, y X . By Theoem 2.1, C is cubic.
To show that C is unique, suppose there exists another cubic function C′ : X Y which satisfies (3) and (15). Since C and C′ are cubic,
PPT Slide
Lager Image
for all x X . It follows that
PPT Slide
Lager Image
So ∥ C′ ( x )− C ( x ), z ∥= 0 for all z Y and hence C′ ( x ) = C ( x ) for all x X .
3. Stability of (3) from a 2- normed space to a 2-Banach space
In this section, we study similar problems of (3). Let ( X , ∥· , ·∥) be a 2-normed space and ( Y , ∥· , ·∥) a 2- Banach space.
Theorem 3.1. Let ε ≥ 0 and p and q be positive real numbers with p + q < 3. Suppose that f : X Y is a function such that
PPT Slide
Lager Image
for all x, y X and z Y. Then there exists a unique cubic function C : X X satisfying (3) and
PPT Slide
Lager Image
for all x X and z Y .
Proof . Letting x = y = 0 in (19). We have ∥2 f (0), z ∥ = 0 for all z Y , so we have f (0) = 0. Putting y = 0 in (19), we have
PPT Slide
Lager Image
for all x X and z Y . Therefore
PPT Slide
Lager Image
for all x X and z Y . Replacing x by 2 x in (21), we get
PPT Slide
Lager Image
for all x X and z Y . By induction on n , we can show that
PPT Slide
Lager Image
for all x X and z Y . For m, n ∈ ℕ with n < m and x X , by (22), we get
PPT Slide
Lager Image
for all x X and z Y . Since p < 3,
PPT Slide
Lager Image
is a 2- Cauchy sequence in Y for all x X . Since Y is a 2-Banach space, the sequence
PPT Slide
Lager Image
is a 2-convergent in Y for all x X . Define C : X Y as
PPT Slide
Lager Image
for all x X and by Lemma 1.5 and (22), we have (20). Next we show that C satisfies (3). By (19), we have
PPT Slide
Lager Image
for all z Y and so DC ( x, y ) = 0 for all x, y X . By Theoem 2.1, C is cubic.
To show that C is unique, suppose that there exists another cubic function C′ : X Y which satisfies (3) and (20). Since C and C′ are cubic,
PPT Slide
Lager Image
for all x X . Since p < 3,
PPT Slide
Lager Image
So ∥ C′ ( x )− C ( x ), z ∥= 0 for all z Y and hence C′ ( x ) = C ( x ) for all x X .
Similar to Theorem 3.1, we have the following theorem.
Theorem 3.2. Let ( X , ∥· , ·∥) be a 2 - Banach space. Let ε ≥ 0, p and q be positive real numbers with p, q > 3. Suppose that f : X X is a function satisfying (19). Then there exists a unique cubic function C : X X satisfying (3) and
PPT Slide
Lager Image
for all x, z X .
BIO
Chang Il Kim received M.Sc. from Sogang Uninversity and Ph.D at Sogang Uninersity. Since 1993 he has been at Dankook University. His research interests include general topology and functional analysis.
Department of Mathematics Education, Dankook University, 152, Jukjeon, Suji, Yongin, Gyeonggi, 448-701, Korea.
e-mail:kci206@hanmail.net
Kap Hun Jung received M.Sc. and Ph.D. from Dankook University. He is now teaching at Seoul National University of Science and Technology as a lecturer. His research interests include functional analysis.
School of Liberal Arts, Seoul National University of Science and Technology, Seoul 139-743, Korea.
e-mail:jkh58@hanmail.net
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