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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:

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 stability
  • fixed point method
  • non-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, 75914-353, Iran
(3)
Department of Mathematics, National Technical University of Athens, Zografou, Campus, Athens, 15780, Greece

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