International Conference on Mathematical Sciences
Brahmagupta Quadratic Functional Equations Connected with Homomorphisms and Derivations on Non-Archimedean Algebras: Direct and Fixed Point Methods M. Arunkumar a and S. Karthikeyan b * a b
Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, TamilNadu, India.
Department of Mathematics, R.M.K. Engineering College, Kavaraipettai - 601 206, Tamil Nadu, India.
Abstract In this paper, we obtain the general solution and the generalized Ulam-Hyers stability of Brahmagupta quadratic functional equations of the form
f ( x1 ) nf ( x2 ) f ( x3 ) nf ( x4 ) = f x1 x3 nx2 x4 nf x1 x4 x2 x3
on non-Archimedean Banach algebras using direct and fixed point methods. An application of this functional equation is also studied. Keywords: quadratic functional equations, generalized Ulam-Hyers stability, non-Archimedean Banach algebras, fixed point.
1.
Hyers-Ulam stability of homomorphisms and derivations on non-Archimedean Banach algebras. In a survey, the Brahmagupta identity is actually first found in Diophantus’ Arithmetica (III, 19), of the third century BC. It was rediscovered by Brahmagupta (598668), an Indian Mathematician and Astronomer, who generalized it (to the Brahmagupta identity) and used it in his study of what is now called Pell’s equation [19]. His Brahmasphutasiddanta was translated from Sanskrit into Arabic by Mohammad al-Fazari, and was subsequently translated into Latin in 1126 [16]. The identity later appeared in Fibonacci’s Book of Squares in 1225. Brahmagupta proved and used a more general identity (the Brahmagupta identity), equivalent to
Introduction
The stability problem of a functional equation was first posed by Ulam [30] concerning the stability of group homomorphism which was answered by Hyers [18] for Banach spaces. Hyers theorem was generalized by Aoki [2] for additive mappings and by Th. M. Rassias [26] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th. M. Rassias theorem was obtained by Gavruta [15] by replacing the unbounded Cauchy difference by a general control function. On the other hand, J. M. Rassias [24] generalized the Hyers stability result by presenting a weaker condition controlled by a product of different powers of norms. The control function x
p
y
q
x
pq
y
pq
( BI1) a 2 nb2 c2 nd 2 = ac nbd n ad bc 2
was introduced by
2
( BI 2) a2 nb2 c2 nd 2 = ac nbd n ad bc . 2
Ravi et al. [27]. The functional equation
f x y f x y = 2 f x 2 f y
(1.1) is said to be quadratic functional equation because the quadratic function f ( x) ax 2 is a solution of the functional equation (1.1). The stability problems of several quadratic functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem S. Czerwik [9], S.M. Jung [20], PL. Kannappan [21], Y.H. Bae, K. W. Jun [6], M. Arunkumar et al., [3] and I.S. Chang, H.M. Kim [10]. Recently, Eshaghi and Khodaei [13] considered the following quadratic functional equation: (1.2) f (ax by) f (ax by) = 2a 2 f ( x) 2b 2 f ( y) and proved the Hyers-Ulam stability of the above functional equation in classical Banach spaces. Recently, Eshaghi [12] and Eshaghi et al. [14] proved the _____________________________
2
Both (BI1) and (BI2) can be verified by expanding each side of the equation. Also (BI2) can be obtained from (BI1), or (BI1) from (BI2),by changing b by b . The above two identities can be transformed into Brahmagupta quadratic functional equations (BQFES) of the form f ( x1 ) nf ( x2 ) f ( x3 ) nf ( x4 ) (1.3) = f x1 x3 nx2 x4 nf x1 x4 x2 x3
f ( x1 ) nf ( x2 ) f ( x3 ) nf ( x4 ) (1.4) = f x1 x3 nx2 x4 nf x1 x4 x2 x3 , where n is set of all real numbers. The above two BQFES can be framed into a single functional equation as f ( x1 ) nf ( x2 ) f ( x3 ) nf ( x4 ) = f x1 x3 nx2 x4 nf x1 x4
x2 x3 .
(1.5)
In this paper, the authors proved the general solution and established generalized Ulam-Hyers stability of (1.5) over non-Archimedean Banach algebras using
*Corresponding author Tel: +91-9976137200 E-mail:
[email protected] (S.Karthikeyan)
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ISBN – 978-93-5107-261-4
International Conference on Mathematical Sciences
for all x X . Letting ( x1 , x2 , x3 , x4 ) by ( x,0,0, x) in (1.4), we obtain
direct and fixed point methods. An application of this functional equation is also studied.
f (x2 )
f ( x) =
even function. Setting ( x1 , x2 , x3 , x4 ) by ( x ,0, y ,0) in (1.4), we get
In this section, the authors investigate the general solution of (1.3) and (1.4). Through out this section, let us consider X and Y be real vector spaces.
f
2
for
( x1 , x2 , x3 , x4 ) by ( x,0, x,0) in (1.3), we obtain (2.4)
f (x2 )
for all x X . Replacing x by x in (2.4), we get f ( x) = f ( x) is an even function. Setting ( x1 , x2 , x3 , x4 ) n
n
f ( x) f ( y ) 2 f
x f y = f ( x y)
for all x, y X . Letting
by
( x1 , x2 , x3 , x4 )
n
n
all
x f y = f ( x y)
x, y X .
(2.11)
Letting
3. Preliminaries on Non-Archimedean Banach Algebras
by x , y , x , y in (1.3) and using (2.1), (2.3), we get
Replacing
x, y X . Replacing ( x1 , x2 , x3 , x4 ) by ( x , x , x , x ) in (1.3) and using (2.1), we arrive (2.3) f (n 1) x = (n 1) 2 f ( x) for all and where n = 1, 2,... . Letting x X all
f ( x) =
(2.9)
by ( x1 , x2 , x3 , x4 ) ( x , y , y , x ) in (1.4) and using (2.7), (2.9), (2.10) , we obtain (2.12) f ( x) f ( y ) 2 f x f y = f ( x y ) for all x, y X . Adding (2.11) and (2.12), we derive (1.1) for all x, y X . Corollary 2.3 If the mapping f : X Y satisfies the functional equation (1.5) for all x1 , x2 , x3 , x4 X if and only if f : X Y satisfying the functional equation (1.1) for all x, y X . Proof. The proof follows from Theorems 2.1 and 2.2.
for all x X . Setting ( x1 , x2 , x3 , x4 ) by ( x ,0, y ,0) in (1.3), we get (2.2) f x f y = f x y for
f ( x) f ( y ) 2 f
(2.1)
x, y X .
all
by x , y , x , y in (1.4) and using (2.7), (2.10), we get
( x1 , x2 , x3 , x4 ) by ( x ,0, x ,0) in (1.3),
= f ( x)
x y
x , x , x , x in (1.4) and using (2.7), we arrive (2.10) f (n 1) x = (n 1) 2 f ( x) for all x X and where n = 1, 2,... . Setting ( x1 , x2 , x3 , x4 )
we obtain
f x
x f y = f
for
f : X Y satisfies the functional equation (1.3) for all x1 , x2 , x3 , x4 X then f : X Y satisfying the functional equation (1.1) for all x, y X . Theorem 2.1 If the mapping
Proof. Replacing
(2.8)
for all x X . Replacing x by x in (2.8), we get f ( x) = f ( x) is an
2. General Solution of the Functional Equations (1.3) and (1.4)
Let K denote a field and function (valuation absolute) . from K into [0, ) . A non-Archimedean
(2.5)
( x1 , x2 , x3 , x4 ) by
valuation is a function . that satisfies the strong triangle
( x , y , y , x ) in (1.3) and using (2.1), (2.2), (2.3) , we obtain (2.6) f ( x) f ( y ) 2 f x f y = f ( x y )
inequality; namely,
| x y | max | x |,| y | | x | | y |
for all x, y K . The associated field K is referred to as a non-Archimedean field. Clearly, | 1 |=| 1 |= 1 and | n | 1 for all n 1. A trivial example of a non-Archimedean
for all x, y X . Adding (2.5) and (2.6), we derive (1.1) for all x, y X . Theorem 2.2 If the mapping f : X Y satisfies the functional equation (1.4) for all x1 , x2 , x3 , x4 X then
valuation is the function
.
taking everything except 0
into 1 and | 0 |= 0 . We always assume in addition that . is
f : X Y satisfying the functional equation (1.1) for all x, y X .
nontrivial, i.e., there is a z K such that | z | 0,1 . Let X be a linear space over a field K with a nonArchimedean nontrivial valuation . . A function
Proof. Replacing ( x1 , x2 , x3 , x4 ) by ( x ,0,0, x ) in (1.4), we get 2 (2.7) f x = f ( x)
. : X [0, ) is said to be a non-Archimedean norm if it is a norm over K with the strong triangle inequality
(ultrametric); namely,
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ISBN – 978-93-5107-261-4
International Conference on Mathematical Sciences
x y max x , y
Theorem 4.1 Let j {1,1} . Let : A4 [0, ) be a function such that
for all x, y X . Then X , . is called a non-Archimedean space. In any such a space a sequence xn is Cauchy
nN
if and only if
xn1 xn n N
k =0
converges to zero. By a
lim
kj
kj
x1 ,
x2 ,
kj
x3 ,
x4
2 kj
(4.1)
0 , we arrive
In this section, the authors present the generalized Ulam - Hyers stability of the functional equations (1.5).
f(
k
m
x)
2k 2m
33
f(
m 2m
x)
=
1 2m
f(
k 2k
m
x)
f(
m
x)
ISBN – 978-93-5107-261-4
International Conference on Mathematical Sciences
k 1
1 2
1
2( i m )
i =0
1
2
im
im
x,
im
x,
im
im
x,
im
x,
i m
x,
im
x,
x
x
, 4 || xi ||s , i =1 F ( x1 , x2 , x3 , x4 ) 4 || xi ||s , i =1 4 4 s 4s || xi || || xi || , i =1 i =1
(4.9)
2( i m )
i =0
for all x A. Since the right hand side of the inequality (4.9) tends to 0 as m , the sequence f ( k x) is a
2k
f ( k x) 2k
k
quadratic
, x A.
in (4.2), we arrive
1 2k
F
k
x1 , k x2 , k x3 , k x4
s 1;
H : AB
homomorphism
such
2 1 , 4 || x ||s , s | 2 2 | f ( x) H ( x) || x ||2 s , 2 2s | | 5 || x ||2 s 2 2s | |
Letting k in (4.8), we see that (4.4) holds for all x A . Now, we need to prove H satisfies (1.5), replacing ( x1 , x2 , x3 , x4 ) by ( k x1 , k x2 , k x3 , k x4 ) and divided by 2k
s 1;
(4.10) for all x1 , x2 , x3 , x4 A . Then there exists a unique
Cauchy sequence. Since B is complete, there exists a mapping H : A B such that
H ( x) = lim
s 4;
k x1 , k x2 , k x3 , k x4 2k
that
(4.11)
for all x1 , x2 , x3 , x4 A . Letting k in the above inequalities, we arrive
for all x A.
Hence H satisfies (1.5) for all x1 , x2 , x3 , x4 A . This
Theorem 4.3 Let j {1,1} . Let : A4 [0, ) be a function such that
H x1 , x2 , x3 , x4 = 0.
shows that H is quadratic. Also 1 H ( x1 x3 ) H ( x1 ) H ( x3 ) = lim 4 k f k
1
lim
4k
k
for
2k
x1 x3 f
x1 f
k
x3
lim
1
2
2k
2
1
H(
i =0
k
x) f ( x) f ( x) H ( x)
1 2( i k )
k
ik
k
x,
ik
k
x,
ik
x,
kj
x2 ,
x3 ,
kj
x4
2 kj
kj
kj
x1 ,
x2 ,
kj
kj
x3 ,
x4
2 kj
converges to R and (4.12)
0. Finally by (iv) , we obtain 1 d( f ,H) d ( f , Tf ) 1 L this implies L1i d( f , H) 1 L which yields L1i f ( x) H ( x) ( x) 1 L this completes the proof of the theorem. The following Corollary is an immediate consequence of Theorem 5.2 concerning the stability of (1.9).
for all x A . Again
is
f ( x) H ( x) K ( x )
f ( x)
H
function such that
for all x A . Using (5.4) and (5.5) for the case reduces to
Therefore,
T in the set H : d f , H , H is the unique
for all x A . Hence from the above inequality, we have f ( x) 1 (5.8) f ( x) 2 2 x , x , x , x
x1 , x3 A .
homomorphism. By
i.e., T is a strictly contractive mapping on with Lipschitz constant L . Replacing ( x1 , x2 , x3 , x4 ) by ( x , x , x , x ) in (1.5), we get, 2 (5.7) f ( x) f ( x ) x , x , x , x
all
ik x1 , 0, ik x3 , 0 = 0
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ISBN – 978-93-5107-261-4
International Conference on Mathematical Sciences
for all x1 , x3 A . If there exists the function
for all x1 , x2 , x3 , x4 A . Now, ( ik x1 , ik x2 , ik x3 , ik x4 ) i2 k
i2 k , 4 0 as i( s 2)k || x i || s , 0 as i =1 = (4s 2)k 4 = s i || x i || , 0 as i =1 0 as 4 4 i(4s 2)k || x i || s || x i || 4 s , i =1 i =1 Thus, (5.1) is holds.
x x x x x ( x) = , , , ,
k , k ,
( x) = L
1
i2
( x) =
x
k .
,
,
,
, s 4 || x || 2 , s x 2 = 1 2s 2 s || x || , 1 4 2s 2 s 2 s || x || .
lim
k
lim
i=0
k
f ( x1 x3 ) x12 f ( x3 ) f ( x1 ) x32 ( x1 ,0, x3 ,0)
1
1
4k i
f i2 k x1 x3 i x1 f ik x3 f ik x1 ik x3
i4 k
2
2
ik x1 , 0, ik x3 , 0 = 0
Corollary 5.5 Let f : A B be a mapping and there exits real numbers
and
s such that
, 4 || xi ||s , i =1 F ( x1 , x2 , x3 , x4 ) 4 || xi ||s , i =1 4 4 s 4s || xi || || xi || , i =1 i =1
(5.13)
s 4;
(5.19) s 1; s 1;
for all x1 , x2 , x3 , x4 A , then there exists a unique quadratic Derivation D : A A such that
i = 1 such that the
functional inequality with F ( x1 , x2 , x3 , x4 ) ( x1 , x2 , x3 , x4 ) for all x1 , x2 , x3 , x4 A and
for all x A . It follows
for all x1 , x3 A . Therefore, D : A A is a quadratic derivation satisfying. The rest of the proof is similar to that of theorem 5.2. The following Corollary is an immediate consequence of Theorem 5.4 concerning the stability of (1.5).
Theorem 5.4 Let f : A A be a mapping for which there
if
2k i
D ( x1 x3 ) x12 D ( x3 ) D ( x1 ) x32
from (5.6), we arrive (5.12). Hence the proof is complete
where i =
f ik x
from (5.14) that
i 0 and L (4 s )/ 2 for s 4 if i 1, L 2 s 2 for and L 2 2s for s 1 if i 1 . Thus, s 1 if i 0
and i = 1 if
(5.17)
k
if
exist a function : A4 [0, ) with the condition 1 k k k k lim 2 k ( i x1 , i x2 , i x3 , i x4 ) = 0 k i
i x .
given by D( x) = lim
2 , i s i2 ( x), 4 2 2 s || i x || , s 4 2 1 i 2 ( i x ) = = i ( x), 2 i 1 i2 s 2 ( x), 2s 2 2 s || i x || , 2 s 2 i ( x). i 5 2s 2 2 s || i x || i for all x A . Hence the inequality (5.5) holds either, L 2 if i 0 and L 2 if i 1, L ( s 4) / 2 for s 4
i2
for all x A Then there exists a unique quadratic derivation D : A A satisfying the functional equation (1.5) and L1i (5.18) f ( x) D( x) ( x) 1 L for all x A . Proof. By the same reasoning as that in the proof of Theorem 5.2, there exists a unique quadratic mapping D : A A satisfying (5.18). The mapping D : A A is
x
1
( x) = L
k ,
i x for all x A . Hence
x
(5.16)
has the property
But we have ( x) = x , x , x , x has the property
L = L(i) < 1 such that
(5.14) (5.15)
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ISBN – 978-93-5107-261-4
International Conference on Mathematical Sciences
| 2 1| , s 4 || x || 2 , s 2 f ( x) D( x) | 2 | || x ||2 s 2 , 2s | | 2s 5 || x || . 2 2s | |
for all
(5.20)
[7]
D.G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16, 385397 (1949).
[8]
P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76-86, 1984.
[9]
S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002.
[10] I.S. Chang, H.M. Kim, On the Hyers-Ulam-Rassias stability of a quadratic functional equations, J. Ineq. Appl. Math, 33 (2002), 1-12.
x A.
[11] JB. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on the generalized complete metric space, Bull. Am. Math. Soc. 126, 305-309 (1968).
6. Application of the Functional Equation (1.5) Consider the functional equation (1.5), that is
f ( x1 ) nf ( x2 ) f ( x3 ) nf ( x4 ) = f x1x3 nx2 x4 nf x1x4
x2 x3 .
[12] M. Eshaghi Gordji, Nearly ring homomorphisms and nearly ring derivations on non-Archimedean Banach algebras, Abstr. Appl. Anal. 2010, Article ID 393247 (2010).
The above equation (1.5) can be written in the form
[13] M. Eshaghi Gordji, H. Khodaei, On the generalized Hyers-UlamRassias stability of quadratic functional Equations, Abstr. Appl. Anal. 2009, Article ID 923476 (2009).
Since f ( x) = x 2 is the solution of the above two functional equations, then the above equations can be written as follows 2 2 ( BI1) x12 nx22 x32 nx42 = x1 x3 nx2 x4 nx1 x4 x2 x3
[14] M. Eshaghi Gordji, H. Khodaei, R. Khodabakhsh, C. Park, Fixed points and quadratic equations connected with homomorphisms and derivations on non-Archimedean algebras. Advances in Difference Equations, 2012, 2012:128.
f ( x1 ) nf ( x2 ) f ( x3 ) nf ( x4 ) = f x1x3 nx2 x4 nf x1x4 x2 x3 f ( x1 ) nf ( x2 ) f ( x3 ) nf ( x4 ) = f x1x3 nx2 x4 nf x1x4 x2 x3 .
( BI 2)x
nx x
nx = x x
[15] P. G a vruta, A generalization of the Hyers-Ulam- Rassias stability of approximately additive mappings , J. Math. Anal. Appl., 184 (1994), 431-436.
nx2 x4 nx1 x4 x2 x3 where n is set of all real numbers. The above identities shows that "set of all numbers of the form x 2 ny2 is closed under multiplication". (BI1) and (BI2) identities holds in both the ring of integers and the ring of rational numbers, and more generally in any commutative ring. In the integer case the above two identities finds applications in number theory. For example when used in conjunction with one of Fermat’s theorems it proves that the product of a square and any number of primes of the form 4n 1 is also a sum of two squares. 2 1
2 2
2 3
2 4
2
2
1 3
[16] George G. Joseph, The Crest of the Peacock, Princeton University Press, p.306, ISBN 0-691-00659- 8, 2000. [17] A. Grabiec, The generalized Hyers-Ulam stability of a class of functional equations, Publicationes Mathematicae Debrecen, vol. 48, no. 3-4, pp. 217-235, 1996. [18]
[19] John Stillwell, Mathematics and its history, Springer, pp.72-76, ISBN 978-0-387-95336-6, 2002. [20] S.M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126-137.
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J. Aczel and J. Dhombres, Functional Equations in Variables, Cambridge Univ, Press, 1989.
[21] Pl. Kannappan, Quadratic functional equation inner product spaces, Results Math. 27, No.3-4, (1995), 368-372.
Several
[22] GJ. Murphy, Non-Archimedean Banach algebras, Ph.D thesis, University of Cambridge (1977).
[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66. [3]
[4]
[23]
M. Arunkumar, S. Jayanthi, S. Hemalatha, Solution Quadratic Derivations of Arun-quadratic Functional Equation, International Journal of Mathematical Sciences and Engineering Applications, Vol. 5, No.4, September 2011, 433-443.
L. Narici, Non-Archimedean Banach spaces and algebras, Arch. Math. 19, 428-435 (1968).
[24] J.M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46, (1982) 126-130.
R. Badora, On approximate ring homomorphisms, J. Math. Anal.Appl. 276, 589-597 (2002).
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[5] R. Badora, On approximate derivations, Math. Inequal. Appl. 9, 167173 (2006). [6]
D.H. Hyers, On the stability of the linear functional equation, Proc.Nat. Acad.Sci.,U.S.A.,27 (1941) 222- 224.
J.M. Rassias, On approximately of approximately linear mappings by linear mappings, Bull. Sc. Math, 108, (1984) 445-446.
[26] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc.Amer.Math. Soc., 72 (1978), 297-300.
Bae. Y.H, Jun. K. W, On the Hyer-Ulam-Rassias stability of a quadratic functional equation, Bull.Korean.Math.Soc., 38(2) (2001), 325-336.
[27] K. Ravi, M. Arunkumar and J.M. Rassias, On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation,
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ISBN – 978-93-5107-261-4
International Conference on Mathematical Sciences
International Journal of Mathematical Sciences, Autumn 2008, Vol.3, no. 08, 36-47. [28]
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ISBN – 978-93-5107-261-4
