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Hyers-Ulam-Rassias stability of the additive-quadratic mappings in non-Archimedean Banach spaces

Journal of Inequalities and Applications20122012:174

https://doi.org/10.1186/1029-242X-2012-174

Received: 29 September 2011

Accepted: 27 July 2012

Published: 6 August 2012

Abstract

Using the fixed point and direct methods, we prove the generalized Hyers-Ulam stability of the following additive-quadratic functional equation in non-Archimedean normed spaces

r [ f ( x + y + z s ) + f ( x y + z s ) + f ( x + y z s ) + f ( x + y + z s ) ] = γ f ( x ) + γ f ( y ) + γ f ( z ) ,

where r, s, γ are positive real numbers.

MSC:39B55, 46S10.

Keywords

Hyers-Ulam stabilityfixed point methodnon-Archimedean normed spaces

1 Introduction and preliminaries

A classical question in the theory of functional equations is the following: ‘When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?’ If the problem accepts a solution, we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [44] in 1940. In the next year, Hyers [23] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Rassias [39] proved a generalization of Hyers’ theorem for additive mappings. This new concept is known as generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. Furthermore, in 1994, a generalization of Rassias’ theorem was obtained by Gǎvruta [21] by replacing the bound ϵ ( x p + y p ) by a general control function φ ( x , y ) .

In 1983, a generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [43] for mappings f : X Y , where X is a normed space and Y is a Banach space. In 1984, Cholewa [11] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group and, in 2002, Czerwik [13] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The reader is referred to [142] and references therein for detailed information on stability of functional equations.

In 1897, Hensel [22] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications (see [16, 2527, 33]).

Definition 1.1 By a non-Archimedean field, we mean a field K equipped with a function (valuation) | | : K [ 0 , ) such that for all r , s K , the following conditions hold:
  1. (1)

    | r | = 0 if and only if r = 0 ;

     
  2. (2)

    | r s | = | r | | s | ;

     
  3. (3)

    | r + s | max { | r | , | s | } .

     
Definition 1.2 Let X be a vector space over a scalar field K with a non-Archimedean non-trivial valuation | | . A function : X R is a non-Archimedean norm (valuation) if it satisfies the following conditions:
  1. (1)

    x = 0 if and only if x = 0 ;

     
  2. (2)

    r x = | r | x ( r K , x X );

     
  3. (3)
    The strong triangle inequality (ultrametric); namely,
    x + y max { x , y } , x , y X .
     

Then ( X , ) is called a non-Archimedean space.

Due to the fact that
x n x m max { x j + 1 x j : m j n 1 } ( n > m ) .

Definition 1.3 A sequence { x n } is Cauchy if and only if { x n + 1 x n } converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.

Definition 1.4 Let X be a set. A function d : X × X [ 0 , ] is called a generalized metric on X if d satisfies
  1. (1)

    d ( x , y ) = 0 if and only if x = y ;

     
  2. (2)

    d ( x , y ) = d ( y , x ) for all x , y X ;

     
  3. (3)

    d ( x , z ) d ( x , y ) + d ( y , z ) for all x , y , z X .

     

We recall a fundamental result in fixed point theory.

Theorem 1.5 ([13, 17])

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
d ( J n x , J n + 1 x ) =
for all nonnegative integers n or there exists a positive integer n 0 such that
  1. (1)

    d ( J n x , J n + 1 x ) < , n n 0 ;

     
  2. (2)

    the sequence { J n x } converges to a fixed point y of J;

     
  3. (3)

    y is the unique fixed point of J in the set Y = { y X d ( J n 0 x , y ) < } ;

     
  4. (4)

    d ( y , y ) 1 1 α d ( y , J y ) for all y Y .

     

In 1996, G. Isac and Th. M. Rassias [24] 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 [14, 15, 35, 36, 40]).

This paper is organized as follows: In Section 2, using the fixed-point method, we prove the Hyers-Ulam stability of the following additive-quadratic functional equation:
r f ( x + y + z s ) + r f ( x y + z s ) + r f ( x + y z s ) + r f ( x + y + z s ) = γ f ( x ) + γ f ( y ) + γ f ( z ) ,
(1.1)

where x , y , z X , in non-Archimedean normed space. In Section 3, using direct methods, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.1) in non-Archimedean normed spaces.

It is easy to see that a mapping f with f ( 0 ) = 0 is a solution of equation (1.1) if and only if f is of the form f ( x ) = A ( x ) + Q ( x ) for all x X .

2 Stability of functional equation (1.1): a fixed point method

In this section, we deal with the stability problem for the additive-quadratic functional equation (1.1). In the rest of the present article, let | 2 | 1 .

Theorem 2.1 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let φ : X 3 [ 0 , ) be a function such that there exists an α < 1 with
φ ( 2 x , 2 y , 2 z ) | 2 | α φ ( x , y , z )
(2.1)
for all x , y , z X . Let f : X Y be an odd mapping satisfying
r f ( x + y + z s ) + r f ( x y + z s ) + r f ( x + y z s ) + r f ( x + y + z s ) γ f ( x ) γ f ( y ) γ f ( z ) Y φ ( x , y , z )
(2.2)
for all x , y , z X . Then there exists a unique additive mapping A : X Y such that
f ( x ) A ( x ) Y max { φ ( 2 x , 0 , 0 ) , φ ( x , x , 0 ) } | 2 γ | ( 1 α )
(2.3)

for all x X .

Proof Putting y = z = 0 in (2.2) and replacing x by 2x, we get
r f ( 2 x s ) γ 2 f ( 2 x ) Y 1 | 2 | φ ( 2 x , 0 , 0 )
(2.4)
for all x X . Putting y = x and z = 0 in (2.2), we have
r f ( 2 x s ) γ f ( x ) Y 1 | 2 | φ ( x , x , 0 )
(2.5)
for all x X . By (2.4) and (2.5), we get
f ( 2 x ) 2 f ( x ) Y = 1 | γ | γ 2 f ( 2 x ) ± r f ( 2 x s ) γ f ( x ) Y 1 | γ | max { r f ( 2 x s ) γ 2 f ( 2 x ) Y , r f ( 2 x s ) γ f ( x ) Y } 1 | 2 γ | max { φ ( 2 x , 0 , 0 ) , φ ( x , x , 0 ) } .
(2.6)
Consider the set S : = { h : X Y } and introduce the generalized metric on S:
d ( g , h ) = inf { μ ( 0 , + ) : g ( x ) h ( x ) Y μ max { φ ( 2 x , 0 , 0 ) , φ ( x , x , 0 ) } , x X } ,
where, as usual, inf ϕ = + . It is easy to show that ( S , d ) is complete (see [30]). Now we consider the linear mapping J : S S such that J g ( x ) : = 1 2 g ( 2 x ) for all x X . Let g , h S be given such that d ( g , h ) = ε . Then
g ( x ) h ( x ) Y ϵ max { φ ( 2 x , 0 , 0 ) , φ ( x , x , 0 ) }
for all x X . Hence,
J g ( x ) J h ( x ) Y = 1 2 g ( 2 x ) 1 2 h ( 2 x ) Y = g ( 2 x ) h ( 2 x ) Y | 2 | ϵ | 2 | max { φ ( 4 x , 0 , 0 ) , φ ( 2 x , 2 x , 0 ) } α ϵ max { φ ( 2 x , 0 , 0 ) , φ ( x , x , 0 ) }

for all x X . So d ( g , h ) = ε implies that d ( J g , J h ) α ε . This means that d ( J g , J h ) α d ( g , h ) for all g , h S .

It follows from (2.6) that d ( f , J f ) 1 | 2 γ | . By Theorem 1.5, there exists a mapping A : X Y satisfying the following:
  1. (1)
    A is a fixed point of J, i.e.,
    2 A ( x ) = A ( 2 x )
    (2.7)
     
for all x X . The mapping A is a unique fixed point of J in the set M = { g S : d ( h , g ) < } . This implies that A is a unique mapping satisfying (2.7) such that there exists a μ ( 0 , ) satisfying f ( x ) A ( x ) Y μ max { φ ( 2 x , 0 , 0 ) , φ ( x , x , 0 ) } for all x X ;
  1. (2)
    d ( J n f , A ) 0 as n . This implies the equality
    lim n f ( 2 n x ) 2 n = A ( x ) for all  x X ;
     
  2. (3)

    d ( f , A ) 1 1 α d ( f , J f ) , which implies the inequality d ( f , A ) 1 | 2 γ | ( 1 α ) . This implies that the inequalities (2.3) holds.

     
It follows from (2.1) and (2.2) that
r A ( x + y + z s ) + r A ( x y + z s ) + r A ( x + y z s ) + r A ( x + y + z s ) γ A ( x ) γ A ( y ) γ A ( z ) Y = lim n 1 | 2 | n r f ( 2 n ( x + y + z ) s ) + r f ( 2 n ( x y + z ) s ) + r f ( 2 n ( x + y z ) s ) + r f ( 2 n ( x + y + z ) s ) γ f ( 2 n x ) γ f ( 2 n y ) γ f ( 2 n z ) Y lim n 1 | 2 | n φ ( 2 n x , 2 n y , 2 n z ) lim n 1 | 2 | n | 2 | n α n φ ( x , y , z ) = 0
for all x , y , z X . So
r A ( x + y + z s ) + r A ( x y + z s ) + r A ( x + y z s ) + r A ( x + y + z s ) γ A ( x ) γ A ( y ) γ A ( z ) = 0

for all x , y , z X . Hence, A : X Y satisfying (1.1). This completes the proof. □

Corollary 2.2 Let θ be a positive real number and q is a real number with q > 1 . Let f : X Y be an odd mapping satisfying
r f ( x + y + z s ) + r f ( x y + z s ) + r f ( x + y z s ) + r f ( x + y + z s ) γ f ( x ) γ f ( y ) γ f ( z ) Y θ ( x q + y q + z q )
(2.8)
for all x , y , z X . Then there exists a unique additive mapping A : X Y such that
f ( x ) A ( x ) Y 2 | 2 | θ x q | 2 γ | ( | 2 | | 2 | q )

for all x X .

Proof The proof follows from Theorem 2.1 by taking φ ( x , y , z ) = θ ( x q + y q + z q ) for all x , y , z X . Then we can choose α = | 2 | q 1 and we get the desired result. □

Theorem 2.3 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let φ : X 3 [ 0 , ) be a function such that there exists an α < 1 with
φ ( x 2 , y 2 , z 2 ) α | 2 | φ ( x , y , z )
(2.9)
for all x , y , z X . Let f : X Y be an odd mapping satisfying (2.2). Then there exists a unique additive mapping A : X Y such that
f ( x ) A ( x ) Y α max { φ ( 2 x , 0 , 0 ) , φ ( x , x , 0 ) } | 2 γ | ( 1 α )

for all x X .

Proof Let ( S , d ) be the generalized metric space defined in the proof of Theorem 2.1.

Now we consider the linear mapping J : S S such that
J g ( x ) : = 2 g ( x 2 )

for all x X .

Replacing x by x 2 in (2.6) and using (2.9), we have
f ( x ) 2 f ( x 2 ) Y 1 | γ | max { φ ( x , 0 , 0 ) , φ ( x 2 , x 2 , 0 ) } α | 2 γ | max { φ ( 2 x , 0 , 0 ) , φ ( x , x , 0 ) } .
(2.10)

So d ( f , J f ) α | 2 γ | .

The rest of the proof is similar to the proof of Theorem 2.1. □

Corollary 2.4 Let θ be a positive real number and q is a real number with 0 < q < 1 . Let f : X Y be an odd mapping satisfying (2.8). Then there exists a unique additive mapping A : X Y such that
f ( x ) A ( x ) Y 2 | 2 | θ x q | 2 γ | ( | 2 | q | 2 | )

for all x X .

Proof The proof follows from Theorem 2.3 by taking φ ( x , y , z ) = θ ( x q + y q + z q ) for all x , y , z X . Then we can choose α = | 2 | 1 q and we get the desired result. □

Theorem 2.5 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let φ : X 3 [ 0 , ) be a function such that there exists an α < 1 with
φ ( 2 x , 2 y , 2 z ) | 4 | α φ ( x , y , z )
(2.11)
for all x , y , z X . Let f : X Y be an even mapping with f ( 0 ) = 0 and satisfying (2.2). Then there exists a unique quadratic mapping Q : X Y such that
f ( x ) Q ( x ) Y max { φ ( 2 x , 0 , 0 ) , | 2 | φ ( x , x , 0 ) } | 4 γ | ( 1 α )
(2.12)

for all x X .

Proof Consider the set S = { g : X Y ; g ( 0 ) = 0 } and the generalized metric d in S defined by
d ( g , h ) = inf { μ ( 0 , + ) : g ( x ) h ( x ) Y μ max { φ ( 2 x , 0 , 0 ) , | 2 | φ ( x , x , 0 ) } , x X } ,
where, as usual, inf ϕ = + . It is easy to show that ( S , d ) is complete (see [30]). Now we consider the linear mapping J : ( S , d ) ( S , d ) such that
J g ( x ) : = 1 4 g ( 2 x )

for all x X .

Putting y = x and z = 0 in (2.2), we have
2 r f ( 2 x s ) 2 γ f ( x ) Y φ ( x , x , 0 )
(2.13)

for all x X .

Substituting y = z = 0 and then replacing x by 2x in (2.2), we obtain
4 r f ( 2 x s ) γ f ( 2 x ) Y φ ( 2 x , 0 , 0 ) .
(2.14)
By (2.13) and (2.14), we get
f ( 2 x ) 4 f ( x ) Y = 1 | 4 γ | 2 ( 2 r f ( 2 x s ) 2 γ f ( x ) ) ( 4 r f ( 2 x s ) γ f ( 2 x ) ) Y 1 | 4 γ | max { | 2 | 2 r f ( 2 x s ) 2 γ f ( x ) Y , 4 r f ( 2 x s ) γ f ( 2 x ) Y } 1 | 4 γ | max { φ ( 2 x , 0 , 0 ) , | 2 | φ ( x , x , 0 ) } .
(2.15)

The rest of the proof is similar to the proof of Theorem 2.1. □

Corollary 2.6 Let θ be a positive real number and q is a real number with q > 2 . Let f : X Y be an even mapping with f ( 0 ) = 0 and satisfying (2.8). Then there exists a unique quadratic mapping Q : X Y such that
f ( x ) Q ( x ) Y | 4 | 2 | 2 | θ x q | 4 γ | ( | 4 | | 2 | q )

for all x X .

Proof The proof follows from Theorem 2.5 by taking φ ( x , y , z ) = θ ( x q + y q + z q ) for all x , y , z X . Then we can choose α = | 2 | q 2 and we get the desired result. □

Theorem 2.7 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let φ : X 3 [ 0 , ) be a function such that there exists an α < 1 with
φ ( x 2 , y 2 , z 2 ) α | 4 | φ ( x , y , z )
(2.16)
for all x , y , z X . Let f : X Y be an even mapping with f ( 0 ) = 0 and satisfying (2.2). Then there exists a unique quadratic mapping Q : X Y such that
f ( x ) Q ( x ) Y α max { φ ( 2 x , 0 , 0 ) , | 2 | φ ( x , x , 0 ) } | 4 γ | ( 1 α )
(2.17)

for all x X .

Proof It follows from (2.15) that
f ( x ) 4 f ( x 2 ) Y 1 | γ | max { φ ( x , 0 , 0 ) , | 2 | φ ( x 2 , x 2 , 0 ) } α | 4 γ | max { φ ( 2 x , 0 , 0 ) , | 2 | φ ( x , x , 0 ) } .

The rest of the proof is similar to the proof of Theorems 2.1 and 2.5. □

Corollary 2.8 Let θ be a positive real number and q is a real number with 0 < q < 2 . Let f : X Y be an even mapping with f ( 0 ) = 0 and satisfying (2.8). Then there exists a unique quadratic mapping Q : X Y such that
f ( x ) Q ( x ) Y | 4 | 2 | 2 | θ x q | 4 γ | ( | 2 | q | 4 | )

for all x X .

Proof The proof follows from Theorem 2.7 by taking φ ( x , y , z ) = θ ( x q + y q + z q ) for all x , y , z X . Then we can choose α = | 2 | 2 q and we get the desired result. □

Let f : X Y be a mapping satisfying f ( 0 ) = 0 and (1.1). Let f e ( x ) : = f ( x ) + f ( x ) 2 and f o ( x ) = f ( x ) f ( x ) 2 . Then f e is an even mapping satisfying (1.1) and f o is an odd mapping satisfying (1.1) such that f ( x ) = f e ( x ) + f o ( x ) .

On the other hand
D f o ( x , y , z ) max { D f ( x , y , z ) , D f ( x , y , z ) } | 2 | max { φ ( x , y , z ) , φ ( x , y , z ) } | 2 |
and
D f e ( x , y , z ) max { D f ( x , y , z ) , D f ( x , y , z ) } | 2 | max { φ ( x , y , z ) , φ ( x , y , z ) } | 2 |

for all x , y , z X , where D f ( x , y , z ) is the difference operator of the functional equation (1.1). So we obtain the following theorem.

Theorem 2.9 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let φ : X 3 [ 0 , ) be a function such that there exists an α < 1 with
φ ( 2 x , 2 y , 2 z ) | 4 | α φ ( x , y , z )
for all x , y , z X . Let f : X Y be a mapping with f ( 0 ) = 0 and satisfying (2.2). Then there exist a unique additive mapping A : X Y and a unique quadratic mapping Q : X Y such that
f ( x ) A ( x ) Q ( x ) Y max { f ( x ) f ( x ) 2 A ( x ) Y , f ( x ) + f ( x ) 2 Q ( x ) Y } max { max { max { φ ( 2 x , 0 , 0 ) , φ ( 2 x , 0 , 0 ) } , max { φ ( x , x , 0 ) , φ ( x , x , 0 ) } } | 4 γ | ( 1 α ) , max { max { φ ( 2 x , 0 , 0 ) , φ ( 2 x , 0 , 0 ) } , | 2 | max { φ ( x , x , 0 ) , φ ( x , x , 0 ) } } | 8 γ | ( 1 α ) }

for all x X .

Theorem 2.10 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let φ : X 3 [ 0 , ) be a function such that there exists an α < 1 with
φ ( x 2 , y 2 , z 2 ) α φ ( x , y , z ) | 2 |
for all x , y , z X . Let f : X Y be a mapping with f ( 0 ) = 0 and satisfying (2.2). Then there exist a unique additive mapping A : X Y and a unique quadratic mapping Q : X Y such that
f ( x ) A ( x ) Q ( x ) Y α max { max { max { φ ( 2 x , 0 , 0 ) , φ ( 2 x , 0 , 0 ) } , max { φ ( x , x , 0 ) , φ ( x , x , 0 ) } } | 4 γ | ( 1 α ) , max { max { φ ( 2 x , 0 , 0 ) , φ ( 2 x , 0 , 0 ) } , | 2 | max { φ ( x , x , 0 ) , φ ( x , x , 0 ) } } | 8 γ | ( 1 α ) }

for all x X .

3 Stability of functional equation (1.1): a direct method

In this section, using direct method, we prove the generalized Hyers-Ulam stability of the additive-quadratic functional equation (1.1) in non-Archimedean space.

Theorem 3.1 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that φ : G 3 [ 0 , + ) be a function such that
lim n | 2 | n φ ( x 2 n , y 2 n , z 2 n ) = 0
(3.1)
for all x , y , z G . Suppose that, for any x G , the limit
Ω ( x ) = lim n max { | 2 | k max { φ ( x 2 k , 0 , 0 ) , φ ( x 2 k + 1 , x 2 k + 1 , 0 ) } ; 0 k < n }
(3.2)
exists and f : G X be an odd mapping satisfying
r f ( x + y + z s ) + r f ( x y + z s ) + r f ( x + y z s ) + r f ( x + y + z s ) γ f ( x ) γ f ( y ) γ f ( z ) X φ ( x , y , z ) .
(3.3)
Then the limit
A ( x ) : = lim n 2 n f ( x 2 n )
exists for all x G and defines an additive mapping A : G X such that
f ( x ) A ( x ) 1 | γ | Ω ( x ) .
(3.4)
Moreover, if
lim j lim n max { | 2 | k max { φ ( x 2 k , 0 , 0 ) , φ ( x 2 k + 1 , x 2 k + 1 , 0 ) } ; j k < n + j } = 0

then A is the unique additive mapping satisfying (3.4).

Proof By (2.10), we know
f ( x ) 2 f ( x 2 ) X 1 | γ | max { φ ( x , 0 , 0 ) , φ ( x 2 , x 2 , 0 ) }
(3.5)
for all x G . Replacing x by x 2 n in (3.5), we obtain
2 n f ( x 2 n ) 2 n + 1 f ( x 2 n + 1 ) X | 2 | n | γ | max { φ ( x 2 n , 0 , 0 ) , φ ( x 2 n + 1 , x 2 n + 1 , 0 ) } .
(3.6)
Thus, it follows from (3.1) and (3.6) that the sequence { 2 n f ( x 2 n ) } n 1 is a Cauchy sequence. Since X is complete, it follows that { 2 n f ( x 2 n ) } n 1 is convergent. Set
A ( x ) : = lim n 2 n f ( x 2 n ) .
By induction on n, one can show that
2 n f ( x 2 n ) f ( x ) X 1 | γ | max { | 2 | k max { φ ( x 2 k , 0 , 0 ) , φ ( x 2 k + 1 , x 2 k + 1 , 0 ) } ; 0 k < n }
(3.7)
for all n 1 and x G . By taking n in (3.7) and using (3.2), one obtains (3.4). By (3.1) and (3.3), we get
r A ( x + y + z s ) + r A ( x y + z s ) + r A ( x + y z s ) + r A ( x + y + z s ) γ A ( x ) γ A ( y ) γ A ( z ) X = lim n | 2 | n r f ( x + y + z 2 n s ) + r f ( x y + z 2 n s ) + r f ( x + y z 2 n s ) + r f ( x + y + z 2 n s ) γ f ( x 2 n ) γ f ( y 2 n ) γ f ( z 2 n ) X lim n | 2 | n φ ( x 2 n , y 2 n , z 2 n ) = 0

for all x , y , z X . Therefore, the mapping A : G X satisfies (1.1).

To prove the uniqueness property of A, let L be another mapping satisfying (3.4). Then we have
A ( x ) L ( x ) X = lim n | 2 | n A ( x 2 n ) L ( x 2 n ) X lim k | 2 | n max { A ( x 2 n ) f ( x 2 n ) X , f ( x 2 n ) L ( x 2 n ) X } lim j lim n max { | 2 | k max { φ ( x 2 k , 0 , 0 ) , φ ( x 2 k + 1 , x 2 k + 1 , 0 ) } ; j k < n + j } = 0

for all x G . Therefore, A = L . This completes the proof. □

Corollary 3.2 Let ξ : [ 0 , ) [ 0 , ) be a function satisfying
ξ ( | 2 | 1 t ) ξ ( | 2 | 1 ) ξ ( t ) , ξ ( | 2 | 1 ) < | 2 | 1
for all t 0 . Assume that κ > 0 and f : G X be a mapping with f ( 0 ) = 0 such that
r f ( x + y + z s ) + r f ( x y + z s ) + r f ( x + y z s ) + r f ( x + y + z s ) γ f ( x ) γ f ( y ) γ f ( z ) X κ ( ξ ( x ) + ξ ( y ) + ξ ( z ) )
(3.8)
for all x , y , z G . Then there exists a unique additive mapping A : G X such that
f ( x ) A ( x ) X 1 | γ | max { κ ζ ( x ) , 2 | 2 | κ ζ ( x ) } .
Proof Defining φ : G 3 [ 0 , ) by φ ( x , y , z ) : = κ ( ξ ( x ) + ξ ( y ) + ξ ( z ) ) , then we have
lim n | 2 | n φ ( x 2 n , y 2 n , z 2 n ) lim n ( | 2 | ξ ( | 2 | 1 ) ) n φ ( x , y , z ) = 0
for all x , y , z G . The last equality comes form the fact that | 2 | ξ ( | 2 | 1 ) < 1 . On the other hand, it follows that
Ω ( x ) = lim n max { | 2 | k max { φ ( x 2 k , 0 , 0 ) , φ ( x 2 k + 1 , x 2 k + 1 , 0 ) } ; 0 k < n } max { φ ( x , 0 , 0 ) , φ ( x 2 , x 2 , 0 ) } = max { κ ζ ( x ) , 2 | 2 | κ ζ ( x ) }
exists for all x G . Also, we have
lim j lim n max { | 2 | k max { φ ( x 2 k , 0 , 0 ) , φ ( x 2 k + 1 , x 2 k + 1 , 0 ) } ; j k < n + j } = lim j | 2 | j max { φ ( x 2 j , 0 , 0 ) , φ ( x 2 j + 1 , x 2 j + 1 , 0 ) } = 0 .

Thus, applying Theorem 3.1, we have the conclusion. This completes the proof. □

Theorem 3.3 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that φ : G 3 [ 0 , + ) be a function such that
lim n φ ( 2 n x , 2 n y , 2 n z ) | 2 | n = 0
(3.9)
for all x , y , z G . Suppose that, for any x G , the limit
Ω ( x ) = lim n max { max { φ ( 2 k + 1 x , 0 , 0 ) , φ ( 2 k x , 2 k x , 0 ) } | 2 | k ; 0 k < n }
(3.10)
exists and f : G X be an odd mapping satisfying (3.3). Then the limit A ( x ) : = lim n f ( 2 n x ) 2 n exists for all x G and
f ( x ) A ( x ) X 1 | 2 γ | Ω ( x )
(3.11)
for all x G . Moreover, if
lim j lim n max { max { φ ( 2 k + 1 x , 0 , 0 ) , φ ( 2 k x , 2 k x , 0 ) } | 2 | k ; j k < n + j } = 0 ,

then A is the unique mapping satisfying (3.11).

Proof By (2.6), we get
f ( 2 x ) 2 f ( x ) X max { φ ( 2 x , 0 , 0 ) , φ ( x , x , 0 ) } | 2 γ |
(3.12)
for all x G . Replacing x by 2 n x in (3.12), we obtain
f ( 2 n + 1 x ) 2 n + 1 f ( 2 n x ) 2 n X max { φ ( 2 n + 1 x , 0 , 0 ) , φ ( 2 n x , 2 n x , 0 ) } | 2 γ | | 2 | n .
(3.13)
Thus, it follows from (3.9) and (3.13) that the sequence { f ( 2 n x ) 2 n } n 1 is convergent. Set
A ( x ) : = lim n f ( 2 n x ) 2 n .
On the other hand, it follows from (3.13) that

for all x G and p , q 0 with q > p 0 . Letting p = 0 , taking q in the last inequality and using (3.10), we obtain (3.11).

The rest of the proof is similar to the proof of Theorem 3.1. This completes the proof. □

Theorem 3.4 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that φ : G 3 [ 0 , + ) be a function such that
lim n | 4 | n φ ( x 2 n , y 2 n , z 2 n ) = 0
(3.14)
for all x , y , z G . Suppose that, for any x G , the limit
Θ ( x ) = lim n max { | 4 | k max { φ ( x 2 k , 0 , 0 ) , | 2 | φ ( x 2 k + 1 , x 2 k + 1 , 0 ) } ; 0 k < n }
(3.15)
exists and f : G X be an even mapping with f ( 0 ) = 0 and satisfying (3.3). Then the limit Q ( x ) : = lim n 4 n f ( x 2 n ) exists for all x G and defines a quadratic mapping Q : G X such that
f ( x ) Q ( x ) X 1 | γ | Θ ( x ) .
(3.16)
Moreover, if
lim j lim n max { | 4 | k max { φ ( x 2 k , 0 , 0 ) , | 2 | φ ( x 2 k + 1 , x 2 k + 1 , 0 ) } ; j k < n + j } = 0

then Q is the unique additive mapping satisfying (3.16).

Proof It follows from (2.15) that
f ( x ) 4 f ( x 2 ) X 1 | γ | max { φ ( x , 0 , 0 ) , | 2 | φ ( x 2 , x 2 , 0 ) } .
(3.17)
Replacing x by x 2 n in (3.18), we have
4 n f ( x 2 n ) 4 n + 1 f ( x 2 n + 1 ) X | 4 | n | γ | max { φ ( x 2 n , 0 , 0 ) , | 2 | φ ( x 2 n + 1 , x 2 n + 1 , 0 ) } .
(3.18)

It follows from (3.14) and (3.18) that the sequence { 4 n f ( x 2 n ) } n 1 is Cauchy sequence. The rest of the proof is similar to the proof of Theorem 3.1. □

Similarly, we can obtain the followings. We will omit the proof.

Theorem 3.5 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that φ : G 3 [ 0 , + ) be a function such that
lim n φ ( 2 n x , 2 n y , 2 n z ) | 4 | n = 0
(3.19)
for all x , y , z G . Suppose that, for any x G , the limit
Θ ( x ) = lim n max { max { φ ( 2 k + 1 x , 0 , 0 ) , φ ( 2 k x , 2 k x , 0 ) } | 4 | k ; 0 k < n }
(3.20)
exists and f : G X be an even mapping with f ( 0 ) = 0 and satisfying (3.3). Then the limit Q ( x ) : = lim n f ( 2 n x ) 4 n exists for all x G and
f ( x ) Q ( x ) X 1 | 4 γ | Θ ( x )
(3.21)
for all x G . Moreover, if
lim j lim n max { max { φ ( 2 k + 1 x , 0 , 0 ) , φ ( 2 k x , 2 k x , 0 ) } | 4 | k ; j k < n + j } = 0 ,

then Q is the unique mapping satisfying (3.21).

Let f : X Y be a mapping satisfying f ( 0 ) = 0 and (1.1). Let f e ( x ) : = f ( x ) + f ( x ) 2 and f o ( x ) = f ( x ) f ( x ) 2 . Then f e is an even mapping satisfying (1.1) and f o is an odd mapping satisfying (1.1) such that f ( x ) = f e ( x ) + f o ( x ) . On the other hand,
D f o ( x , y , z ) max { φ ( x , y , z ) , φ ( x , y , z ) } | 2 |
and
D f e ( x , y , z ) max { φ ( x , y , z ) , φ ( x , y , z ) } | 2 |

for all x , y , z X , where D f ( x , y , z ) is the difference operator of the functional equation (1.1). So we obtain the following theorem.

Theorem 3.6 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that φ : G 3 [ 0 , + ) be a function such that
lim n φ ( 2 n x , 2 n y , 2 n z ) | 4 | n = 0
for all x , y , z G . Suppose that the limits
Ω ( x ) = lim n max 0 k < n { max { max { φ ( 2 k + 1 x , 0 , 0 ) , φ ( 2 k + 1 x , 0 , 0 ) } , max { φ ( 2 k x , 2 k x , 0 ) , φ ( 2 k x , 2 k x , 0 ) } } / | 2 | k + 1 }
and
Θ ( x ) = lim n max 0 k < n { max { max { φ ( 2 k + 1 x , 0 , 0 ) , φ ( 2 k + 1 x , 0 , 0 ) } , max { φ ( 2 k x , 2 k x , 0 ) , φ ( 2 k x , 2 k x , 0 ) } } / ( | 2 | | 4 | k ) }
exist for all x G and f : G X be a mapping with f ( 0 ) = 0 and satisfying (3.3). Then there exist an additive mapping A : G X and a quadratic mapping Q : G X such that
f ( x ) A ( x ) Q ( x ) X max { f ( x ) + f ( x ) 2 Q ( x ) X , f ( x ) f ( x ) 2 A ( x ) X } max { 1 | 2 γ | Ω ( x ) , 1 | 4 γ | Θ ( x ) }
(3.22)
for all x G . Moreover, if
lim j lim n max j k < n + j { max { max { φ ( 2 k + 1 x , 0 , 0 ) , φ ( 2 k + 1 x , 0 , 0 ) } , max { φ ( 2 k x , 2 k x , 0 ) , φ ( 2 k x , 2 k x , 0 ) } } / | 2 | k + 1 } = 0
and
lim j lim n max j k < n + j { max { max { φ ( 2 k + 1 x , 0 , 0 ) , φ ( 2 k + 1 x , 0 , 0 ) } , max { φ ( 2 k x , 2 k x , 0 ) , φ ( 2 k x , 2 k x , 0 ) } } / ( | 2 | | 4 | k ) } = 0

then A, Q are the unique mappings satisfying (3.22).

Theorem 3.7 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that φ : G 3 [ 0 , + ) be a function such that
lim n | 2 | n φ ( x 2 n , y 2 n , z 2 n ) = 0
for all x , y , z G . Suppose that the limits
Ω ( x ) = 1 | 2 | lim n max 0 k < n { | 2 | k max { max { φ ( x 2 k , 0 , 0 ) , φ ( x 2 k , 0 , 0 ) } , max { φ ( x 2 k + 1 , x 2 k + 1 , 0 ) , φ ( x 2 k + 1 , x 2 k + 1 , 0 ) } } }
and
Θ ( x ) = 1 | 2 | lim n max 0 k < n { | 4 | k max { max { φ ( x 2 k , 0 , 0 ) , φ ( x 2 k , 0 , 0 ) } , | 2 | max { φ ( x 2 k + 1 , x 2 k + 1 , 0 ) , φ ( x 2 k + 1 , x 2 k + 1 , 0 ) } } }
exist for all x G and f : G X be a mapping with f ( 0 ) = 0 and satisfying (3.3). Then there exist an additive mapping A : G X and a quadratic mapping Q : G X such that
f ( x ) A ( x ) Q ( x ) X max { Ω ( x ) , Θ ( x ) } | γ |
(3.23)
for all x G . Moreover, if
lim j lim n max j k < n + j { | 2 | k max { max { φ ( x 2 k , 0 , 0 ) , φ ( x 2 k , 0 , 0 ) } , max { φ ( x 2 k + 1 , x 2 k + 1 , 0 ) , φ ( x 2 k + 1 , x 2 k + 1 , 0 ) } } } = 0
and
lim j lim n max j k < n + j { | 4 | k max { max { φ ( x 2 k , 0 , 0 ) , φ ( x 2 k , 0 , 0 ) } , | 2 | max { φ ( x 2 k + 1 , x 2 k + 1 , 0 ) , φ ( x 2 k + 1 , x 2 k + 1 , 0 ) } } } = 0

then A, Q are the unique mappings satisfying (3.23).

4 Conclusion

We linked here two different disciplines, namely, the non-Archimedean normed spaces and functional equations. We established the generalized Hyers-Ulam stability of the functional equation (1.1) in non-Archimedean normed spaces.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Hanyang University, Seoul, South Korea
(2)
Department of Mathematics, College of Sciences, Yasouj University, Yasouj, Iran
(3)
Department of Mathematics, National Technical University of Athens, Athens, Greece

References

  1. Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64–66. 10.2969/jmsj/00210064View ArticleGoogle Scholar
  2. Arriola LM, Beyer WA: Stability of the Cauchy functional equation over p -adic fields. Real Anal. Exch. 2005/06, 31: 125–132.MathSciNetGoogle Scholar
  3. Azadi Kenary H: Stability of a pexiderial functional equation in random normed spaces. Rend. Circ. Mat. Palermo 2011. doi:10.1007/s12215–011–0027–5Google Scholar
  4. Azadi Kenary H: Non-Archimedean stability of Cauchy-Jensen type functional equation. Int. J. Nonlinear Anal. Appl. 2011, 2(2):93–103.MathSciNetGoogle Scholar
  5. Azadi Kenary H: Approximate additive functional equations in closed convex cone. J. Math. Ext. 2011, 5(2):51–65.MathSciNetGoogle Scholar
  6. Azadi Kenary H: On the stability of a cubic functional equation in random normed spaces. J. Math. Ext. 2009, 4(1):105–113.MathSciNetGoogle Scholar
  7. Azadi Kenary H, Rezaei H, Talebzadeh S, Lee SJ: Stabilities of cubic mappings in various normed spaces: direct and fixed point methods. J. Appl. Math. 2012., 2012: Article ID 546819. doi:10.1155/2012/546819Google Scholar
  8. Azadi Kenary H, Rezaei H, Ebadian A, Zohdi AR: Hyers-Ulam-Rassias RNS approximation of Euler-Lagrange-type additive mappings. Math. Probl. Eng. 2012., 2012: Article ID 672531. doi:10.1155/2012/672531Google Scholar
  9. Azadi Kenary H, Shafaat K, Shafiee M, Takbiri G: Hyers-Ulam-Rassias stability of the Apollonius type quadratic mapping in RN-spaces. J. Nonlinear Sci. Appl. 2011, 4(1):82–91.MathSciNetGoogle Scholar
  10. Cădariu L, Radu V: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 2003., 4(1): Article ID 4Google Scholar
  11. Cholewa PW: Remarks on the stability of functional equations. Aequ. Math. 1984, 27: 76–86. 10.1007/BF02192660MathSciNetView ArticleGoogle Scholar
  12. Chung J, Sahoo PK: On the general solution of a quartic functional equation. Bull. Korean Math. Soc. 2003, 40: 565–576.MathSciNetView ArticleGoogle Scholar
  13. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge; 2002.View ArticleGoogle Scholar
  14. Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 2004, 346: 43–52.Google Scholar
  15. Cădariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008., 2008: Article ID 749392Google Scholar
  16. Deses D: On the representation of non-Archimedean objects. Topol. Appl. 2005, 153: 774–785. 10.1016/j.topol.2005.01.010MathSciNetView ArticleGoogle Scholar
  17. Diaz J, Margolis B: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 1968, 74: 305–309. 10.1090/S0002-9904-1968-11933-0MathSciNetView ArticleGoogle Scholar
  18. Eshaghi-Gordji M, Abbaszadeh S, Park C: On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces. J. Inequal. Appl. 2009., 2009: Article ID 153084Google Scholar
  19. Eshaghi-Gordji M, Kaboli-Gharetapeh S, Park C, Zolfaghri S: Stability of an additive-cubic-quartic functional equation. Adv. Differ. Equ. 2009., 2009: Article ID 395693Google Scholar
  20. Fechner W: Stability of a functional inequality associated with the Jordan-von Neumann functional equation. Aequ. Math. 2006, 71: 149–161. 10.1007/s00010-005-2775-9MathSciNetView ArticleGoogle Scholar
  21. Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleGoogle Scholar
  22. Hensel K: Ubereine news Begrundung der Theorie der algebraischen Zahlen. Jahresber. Dtsch. Math.-Ver. 1897, 6: 83–88.Google Scholar
  23. Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
  24. Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.View ArticleGoogle Scholar
  25. Katsaras AK, Beoyiannis A: Tensor products of non-Archimedean weighted spaces of continuous functions. Georgian Math. J. 1999, 6: 33–44. 10.1023/A:1022926309318MathSciNetView ArticleGoogle Scholar
  26. Khrennikov A Mathematics and Its Applications 427. In Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic, Dordrecht; 1997.View ArticleGoogle Scholar
  27. Kominek Z: On a local stability of the Jensen functional equation. Demonstr. Math. 1989, 22: 499–507.MathSciNetGoogle Scholar
  28. Lee S, Im S, Hwang I: Quartic functional equations. J. Math. Anal. Appl. 2005, 307: 387–394. 10.1016/j.jmaa.2004.12.062MathSciNetView ArticleGoogle Scholar
  29. Mohammadi M, Cho YJ, Park C, Vetro P, Saadati R: Random stability of an additive-quadratic-quartic functional equation. J. Inequal. Appl. 2010., 2010: Article ID 754210Google Scholar
  30. Mihet D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 2008, 343: 567–572. 10.1016/j.jmaa.2008.01.100MathSciNetView ArticleGoogle Scholar
  31. Mursaleen M, Mohiuddine SA: On stability of a cubic functional equation in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 42: 2997–3005. 10.1016/j.chaos.2009.04.041MathSciNetView ArticleGoogle Scholar
  32. Najati A, Park C:The pexiderized Apollonius-Jensen type additive mapping and isomorphisms between C -algebras. J. Differ. Equ. Appl. 2008, 14: 459–479. 10.1080/10236190701466546MathSciNetView ArticleGoogle Scholar
  33. Nyikos PJ: On some non-Archimedean spaces of Alexandrof and Urysohn. Topol. Appl. 1999, 91: 1–23. 10.1016/S0166-8641(97)00239-3MathSciNetView ArticleGoogle Scholar
  34. Park C: Generalized Hyers-Ulam-Rassias stability of n -sesquilinear-quadratic mappings on Banach modules over C -algebras. J. Comput. Appl. Math. 2005, 180: 279–291. 10.1016/j.cam.2004.11.001MathSciNetView ArticleGoogle Scholar
  35. Park C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory Appl. 2007., 2007: Article ID 50175Google Scholar
  36. Park C: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl. 2008., 2008: Article ID 493751Google Scholar
  37. Parnami JC, Vasudeva HL: On Jensen’s functional equation. Aequ. Math. 1992, 43: 211–218. 10.1007/BF01835703MathSciNetView ArticleGoogle Scholar
  38. Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4: 91–96.MathSciNetGoogle Scholar
  39. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View ArticleGoogle Scholar
  40. Rätz J: On inequalities associated with the Jordan-von Neumann functional equation. Aequ. Math. 2003, 66: 191–200. 10.1007/s00010-003-2684-8View ArticleGoogle Scholar
  41. Saadati R, Park C:Non-Archimedean L -fuzzy normed spaces and stability of functional equations. Comput. Math. Appl. 2010, 60: 2488–2496. 10.1016/j.camwa.2010.08.055MathSciNetView ArticleGoogle Scholar
  42. Saadati R, Park JH: On the intuitionistic fuzzy topological spaces. Chaos Solitons Fractals 2006, 27: 331–344. 10.1016/j.chaos.2005.03.019MathSciNetView ArticleGoogle Scholar
  43. Skof F: Local properties and approximation of operators. Rend. Semin. Mat. Fis. Milano 1983, 53: 113–129. 10.1007/BF02924890MathSciNetView ArticleGoogle Scholar
  44. Ulam SM Science Editions. In Problems in Modern Mathematics. Wiley, New York; 1964.Google Scholar

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