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APPROXIMATELY QUINTIC MAPPINGS IN NON-ARCHIMEDEAN 2-NORMED SPACES BY FIXED POINT THEOREM
APPROXIMATELY QUINTIC MAPPINGS IN NON-ARCHIMEDEAN 2-NORMED SPACES BY FIXED POINT THEOREM
Journal of Applied Mathematics & Informatics. 2015. May, 33(3_4): 435-445
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : December 22, 2014
  • Accepted : February 05, 2015
  • Published : May 30, 2015
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CHANG IL KIM
KAP HUN JUNG

Abstract
In this paper, using the fixed point method, we investigate the generalized Hyers-Ulam stability of the system of quintic functional equation in non-Archimedean 2-Banach spaces. AMS Mathematics Subject Classification : 39B82, 46S10.
Keywords
1. Introduction and preliminaries
In 1940, Ulam [22] posed the following problem concerning the stability of functional equations:
Let G 1 be a group and let G 2 be a metric group with the metric d (·, ·). Given ϵ > 0, does there exist a δ > 0 such that if a mapping h : G 1 G 2 satisfies the inequality d ( h ( xy ), h ( x ) h ( y )) < δ for all x, y G 1 , then there exists a homomorphism H : G 1 G 2 with d ( h ( x ), H ( x )) < ϵ for all x G 1 ?
Hyers [8] solved the Ulam's problem for the case of approximately additive functions in Banach spaces. Since then, the stability of several functional equations has been extensively investigated by several mathematicians [3 , 5 , 9 , 10 , 11 , 14 , 17] . The Hyers-Ulam stability for the quadratic functional equation
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was proved by Skof [21] for a function f : E 1 E 2 , where E 1 is a normed space and E 2 is a Banach space and later by Jung [13] on unbounded domains.
Rassias [20] investigated the stability for the following cubic functional equation
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and Jun and Kim [12] investigated the stability for the following cubic funtional equation
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A valuation is a function | · | from a field K into [0,∞) such that for any r , s ∈ K, the following conditions hold: (i) | r | = 0 if and only if r = 0, (ii) | rs | = | r s |, and (iii) | r + s | ≤ | r | + | s |. A field K is called a valued field if K carries a valuation. The usual absolute values of ℝ and ℂ are examples of valuations. If the triangle inequality is replaced by | r + s | ≤ max {| r |, | s |} for all r , s ∈ K, then the valuation | · | is called a non-Archimedean valuation and the field with a non-Archimedean valuation is called a non-Archimedean field . If | · | is a non-Archimedean valuation on K, then clearly, |1| = | − 1| and | n | ≤ 1 for all n ∈ ℕ.
Definition 1.1. Let X be a vector space over a non-Archimedean field K. A function ║ · ║ : X → ℝ is called a non-Archimedean norm if it satisfies the following conditions:
(a) ║ x ║ = 0 if and only if x = 0,
(b) ║ rx ║ = | r |║ x ║, and
(c) ║ x + y ║ ≤ max {║ x ║, ║ y ║} for all x , y X and all r ∈ K.
If ║ · ║ is a non-Archimedean norm, then ( X , ║ · ║) is called a non-Archimedean normed space .
Let ( X , ║ · ║) be a non-Archimedean normed space and { xn } a sequence in X . Then { xn } is said to be convergent in ( X , ║ · ║) if there exists an x X such that lim n→∞ xn x ║ = 0. In case, x is called the limit of the sequence { xn }, and one denotes it by lim n→∞ xn = x . A sequence { xn } is said to be Cauchy in ( X , ║ · ║) if lim n→∞ xn+p xn ║ = 0 for all p ∈ ℕ. By (c) in Definition 1.1,
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a sequence { xn } is Cauchy in ( X , ║ · ║) if and only if { x n+1 xn } converges to zero in ( X , ║ · ║). By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.
G ä hler [6 , 7] has introduced the concept of 2-normed spaces and White [23] introduced the concept of 2-Banach spaces. In 1999 to 2003, Lewandowska published a series of papers on 2-normed sets and generalized 2-normed spaces [15 , 16] .
Definition 1.2. Let X be a linear space over a non-Archimedean field K with dim X > 1 and ║·, ·║ : X × X → ℝ a function satisfying the following properties
( N A1) ║ x , y ║ = 0 if and only if x and y are linearly dependent,
( N A2) ║ x , y ║ = ║ y , x ║,
( N A3) ║ x , ay ║ = | a |║ x , y ║ , and
( N A4) ║ x , y + z ║ ≤ max {║ x , y ║, ║ x , z ║}
for all x , y , z X and all a ∈ K. Then ║· , ·║ is called a non-Archimedean 2-norm and ( X , ║· , ·║) is called a non-Archimedean 2-normed spaces .
Definition 1.3. A sequence { xn } in a non-Archimedean 2-normed space ( X , ║· , ·║) is called a Cauchy sequence if
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for all x X .
Definition 1.4. A sequence { xn } in a non-Archimedean 2-normed space ( X , ║· , ·║) is called convergent if
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for all y X and for some x X . In case, x is called the limit of the sequence { xn }, and we denoted by xn x as n → ∞ or lim n→∞ xn = x .
Let { xn } be a sequence in a non-Archimedean 2-normed space ( X , ║· , ·║). It follows from (NA4) that
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for all y X and so a sequence { xn } is a Cauchy sequence in ( X , ║· , ·║) if and only if { x m+1 xm } converges to zero in ( X , ║· , ·║).
A non-Archimedean 2-normed space ( X , ║· , ·║) is called a non-Archimedean 2-Banach space if every Cauchy sequence in ( X , ║· , ·║) is convergent. Now, we state the following results as lemma [18] .
Lemma 1.5. Let ( X , ║· , ·║) be a non-Archimedean 2-normed space. Then we have the following :
(1) |║ x , z ║ − ║ y , z ║| ≤ ║ x y , z for all x , y , z X ,
(2) ║ x , z ║ = 0 for all z X if and only if x = 0, and
(3) for any convergent sequence { xn } in ( X , ║· , ·║),
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for all z X .
In 2003, Radu [19] proposed a new method for obtaining the existence of exact solutions and error estimations, based on the fixed point alternative (see also [1 , 2] ).
We recall the following theorem by Margolis and Diaz.
Theorem 1.6 ( [4] ). Let ( X , d ) be a complete generalized metric space and let J : X X be a strictly contractive mapping with some Lipschitz constant L with 0 < L < 1. Then for each given element x X, either d ( Jnx , J n+1 x ) = ∞ for all nonnegative integers n or there exists a positive integer n 0 such that
(1) d ( Jnx , J n+1 x ) < ∞ for all 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) d ( y , y ) ≤
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for all y Y .
In this paper, we investigate the following cubic functional equation
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and using fixed point method, we inverstigate the generalized Hyers-Ulam stability for the system of the quintic functional equation
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and prove the generalized Hyers-Ulam stability for (3) in non-Archimedean 2-Banach spaces. In this paper, we will assume that ( X , ║·║) is a a non-Archimedean norm space and ( Y , ║· , ·║) is a non-Archimedean 2-Banach space.
2. Stability of quintic mappings
In this section, using the fixed point method, we investigate the generalized Hyers-Ulam stability for the system of quintic functional equation (3) in non-Archimedean 2-Banach spaces. We start the following lemma.
Lemma 2.1. Let f : X Y be a mapping with (2). Then f is a cubic mapping .
Proof . Suppose that f satisfies (2). Letting x = y = 0 in (2), we have f (0) = 0 and letting y = 0 in (2), we have
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for all x X . Letting x = 0 in (2), by (4), we have f ( y ) = − f (− y ) for all y X and so f is odd. Letting y = − y in (2), we have
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for all x , y X and by (2) and (5), we have
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for all x , y X . Interching x and y in (6), since f is odd, f satisfies (1) and hence f is cubic. □
The function f : ℝ × ℝ → ℝ given by f ( x , y ) = cx 2 y 3 is a solution of (3). In partcular, letting y = x in (3), we get a quintic function g : ℝ → ℝ in one variable given by g ( x ) = f ( x , x ) = cx 5 .
Proposition 2.2. If a mapping f : X 2 Y satisfies (3), then f ( λx , μy ) = λ 2 μ 3 f ( x , y ) for all x , y X and all rational numbers λ , μ .
Theorem 2.3. Let ϕ 1 , ϕ 2 : X 3 × Y → [0,∞) be functions such that
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for all x , y , z X , w Y and some L with 0 < L < 1. Suppose that f : X 2 Y is a mapping such that f ( x , 0) = f (0, x ) = 0 for all x X ,
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and
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for all w Y and all x , y , x 1 , x 2 , y 1 , y 2 X . Then there exists a unique quintic mapping T : X 2 Y satisfying (3) and
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for all w Y and all x , y X , where
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Proof . Putting y 2 = 0 and y 1 = y in (9), we get
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for all w Y and all x , y X . Putting x 1 = x 2 = x in (8), we get
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for all w Y and all x , y X . Thus by (11) and (12), we have
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for all w Y and all x , y X . It follows from (13) that
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for all w Y and all x , y X .
Consider the set S = { h | h : X × X Y with h ( x , 0) = h (0, x ) = 0, ∀ x X } and the generalized metric d on S defined by
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Then ( S , d ) is a complete metric space [2] . Define a mapping J : S S by Jg ( x , y ) = 2 −5 g (2 x , 2 y ) for all x , y X and all g S . Let g , h S and d ( g , h ) ≤ ε for some non-negative real number ε . Then by (7), we have
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and so d ( Jg , Jh ) ≤ εL . This mean that d ( Jg , Jh ) ≤ Ld ( g , h ) for all g , h S and so J is a strictly contractive mapping. By (14), we get d ( Jf , f ) ≤ 1 < ∞. By Theorem 1.6, there exists a mapping T : X 2 Y which is a fixed point of J such that d ( Jnf , T ) → 0 as n → ∞, which implies the equality T ( x , y ) = lim n→∞ 2 −5n f (2 nx , 2 ny ). Since d ( Jf , f ) ≤ 1 < ∞, by (4) in Theorem 1.6, we have (10). By (8) and (9), we get
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and
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for all w Y and all x , y , x 1 , x 2 , y 1 , y 2 X . Hence T satisfies (3).
To prove the uniquness of T , assume that T 1 : X 2 Y is another solution of (3) satisfying (10). Then T 1 is a fixed point of J and by (10),
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By (3) in Theorem 1.6, we have T = T 1 . □
Theorem 2.4. Let ϕ 1 , ϕ 2 : X 3 × Y → [0,∞) be functions such that
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for all x , y , z X , w Y and some L with 0 < L < 1. Suppose that f : X 2 Y is a mapping satisfying f ( x , 0) = f (0, x ) = 0 for all x X , (8) and (9). Then there exists a unique quintic mapping T : X 2 Y satisfying (3) and
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for all w Y and all x , y X , where
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Proof . Putting y 2 = 0 and
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in (9), we get
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for all w Y and all x , y X . Putting
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in (8), we get
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for all w Y and all x , y X . Thus by (17) and (18), we have
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for all x , y X and all w Y . That is, we have
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for all x , y X and all w Y .
Consider the set S = { h | h : X × X Y with h ( x , 0) = h (0, x ) = 0, ∀ x X } and the generalized metric d on S defined by
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Then ( S , d ) is a complete metric space( [2] ). Define a mapping J : S S by
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for all x , y X and all g S . Let g , h S and d ( g , h ) ≤ ε for some non-negative real number ε . Then by (15), we have
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and so d ( Jg , Jh ) ≤ εL . This mean that d ( Jg , Jh ) ≤ L d ( g , h ) for all g , h S and so J is a strictly contractive mapping. By (19), we get d ( Jf , f ) ≤ L < ∞. By Theorem 1.6, there exists a mapping T : X 2 Y which is a fixed point of J such that d ( Jnf , T ) → 0 as n → ∞, which imples the equality T ( x , y ) = lim n→∞
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. Since d ( Jf , f ) ≤ L , by (4) in Theorem 1.6, we have (16) and by (8) and (9), we get
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and
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for all w Y and all x , y , x 1 , x 2 , y 1 , y 2 X . Hence T satisfies (3).
To prove the uniquness of T , assume that T 1 : X 2 Y is another solution of (3) satisfying (16). Then T 1 is a fixed point of J and by (16),
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By (3) in Theorem 1.6, we have T = T 1 . □
As example of ϕ 1 ( x 1 , x 2 , y , w ) and ϕ 2 ( x , y 1 , y 2 , w ) in Theorem 2.3 and Theorem 2.4, we can take ϕ 1 ( x 1 , x 2 , y , w ) = θ (║ x 1 p + ║ x 2 p + ║ y p )║ w ║ and ϕ 2 ( x , y 1 , y 2 , w ) = |2| 4 θ (║ x p +║ y 1 p +║ y 2 p )║ w ║ for all x , y , x 1 , x 2 , y 1 , y 2 X , all w Y and some positive real number θ . Then we have the following corollary.
Corollary 2.5. Let θ , p be positive real numbers with p ≠ 5. Suppose that f : X 2 Y is a mapping satisfying f ( x , 0) = f (0, x ) = 0,
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and
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for all w Y and all x , y , x 1 , x 2 , y 1 , y 2 X . Then there exists a unique quintic mapping T : X 2 Y satisfying
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for all w Y and x , y X .
Proof . Let ϕ 1 ( x 1 , x 2 , y , w ) = θ (║ x 1 p +║ x 2 p +║ y p )║ w ║ and ϕ 2 ( x , y 1 , y 2 , w ) = |2| 4 θ (║ x p + ║ y 1 p + ║ y 2 p )║ w ║. Note that
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So if p > 5, by Theorem 2.3, we have (20). Note that
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So if p < 5, by Theorem 2.4, we have (20). □
As another example of ϕ 1 ( x , y , z , w ) and ϕ 2 ( x , y , z , w ) in Theorem 2.3 and Theorem 2.4, we can take ϕ 1 ( x , y , z , w ) = ϕ 2 ( x , y , z , w ) = θ x p y q z r w ║ for all x , y , z X , all w Y and some positive real number p , q , r , θ . Then we have the following corollary:
Corollary 2.6. Let p , q , r and θ be positive real numbers with p + q + r ≠ 5. Suppose that f : X 2 Y is a mapping satisfying f ( x , 0) = 0,
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and
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for all w Y and all x , y , x 1 , x 2 , y 1 , y 2 X . Then there exists a unique quintic mapping T : X 2 Y satisfying
PPT Slide
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for all w Y and all x , y X .
Proof . Let ϕ 1 ( x , y , z , w ) = ϕ 2 ( x , y , z , w ) = θ x p y q z r w ║. Then we have
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Hence if p + q + r > 5, by Theorem 2.3, we have (21). Note that
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Thus if p + q + r < 5, by Theorem 2.4, we have (21). □
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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