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

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.

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:XY, 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,sK, the following conditions hold:

  1. (1)

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

  2. (2)

    |rs|=|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 :XR is a non-Archimedean norm (valuation) if it satisfies the following conditions:

  1. (1)

    x=0 if and only if x=0;

  2. (2)

    rx=|r|x (rK, xX);

  3. (3)

    The strong triangle inequality (ultrametric); namely,

    x+ymax { x , y } ,x,yX.

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,yX;

  3. (3)

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

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:XX be a strictly contractive mapping with Lipschitz constant α<1. Then for each given element xX, 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={yXd( J n 0 x,y)<};

  4. (4)

    d(y, y ) 1 1 α d(y,Jy) for all yY.

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,zX, 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 xX.

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(2x,2y,2z)|2|αφ(x,y,z)
(2.1)

for all x,y,zX. Let f:XY 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,zX. Then there exists a unique additive mapping A:XY 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 xX.

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(2x,0,0)
(2.4)

for all xX. 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 xX. 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:XY} 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, infϕ=+. It is easy to show that (S,d) is complete (see [30]). Now we consider the linear mapping J:SS such that Jg(x):= 1 2 g(2x) for all xX. Let g,hS be given such that d(g,h)=ε. Then

g ( x ) h ( x ) Y ϵmax { φ ( 2 x , 0 , 0 ) , φ ( x , x , 0 ) }

for all xX. 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 xX. So d(g,h)=ε implies that d(Jg,Jh)αε. This means that d(Jg,Jh)αd(g,h) for all g,hS.

It follows from (2.6) that d(f,Jf) 1 | 2 γ | . By Theorem 1.5, there exists a mapping A:XY satisfying the following:

  1. (1)

    A is a fixed point of J, i.e.,

    2A(x)=A(2x)
    (2.7)

for all xX. The mapping A is a unique fixed point of J in the set M={gS: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 μmax{φ(2x,0,0),φ(x,x,0)} for all xX;

  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 xX;
  2. (3)

    d(f,A) 1 1 α d(f,Jf), 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,zX. 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,zX. Hence, A:XY 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:XY 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,zX. Then there exists a unique additive mapping A:XY such that

f ( x ) A ( x ) Y 2 | 2 | θ x q | 2 γ | ( | 2 | | 2 | q )

for all xX.

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,zX. 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,zX. Let f:XY be an odd mapping satisfying (2.2). Then there exists a unique additive mapping A:XY such that

f ( x ) A ( x ) Y α max { φ ( 2 x , 0 , 0 ) , φ ( x , x , 0 ) } | 2 γ | ( 1 α )

for all xX.

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

Now we consider the linear mapping J:SS such that

Jg(x):=2g ( x 2 )

for all xX.

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,Jf) 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:XY be an odd mapping satisfying (2.8). Then there exists a unique additive mapping A:XY such that

f ( x ) A ( x ) Y 2 | 2 | θ x q | 2 γ | ( | 2 | q | 2 | )

for all xX.

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,zX. 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(2x,2y,2z)|4|αφ(x,y,z)
(2.11)

for all x,y,zX. Let f:XY be an even mapping with f(0)=0 and satisfying (2.2). Then there exists a unique quadratic mapping Q:XY 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 xX.

Proof Consider the set S ={g:XY;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, infϕ=+. 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

Jg(x):= 1 4 g(2x)

for all xX.

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 xX.

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(2x,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:XY be an even mapping with f(0)=0 and satisfying (2.8). Then there exists a unique quadratic mapping Q:XY such that

f ( x ) Q ( x ) Y | 4 | 2 | 2 | θ x q | 4 γ | ( | 4 | | 2 | q )

for all xX.

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,zX. 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,zX. Let f:XY be an even mapping with f(0)=0 and satisfying (2.2). Then there exists a unique quadratic mapping Q:XY 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 xX.

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:XY be an even mapping with f(0)=0 and satisfying (2.8). Then there exists a unique quadratic mapping Q:XY such that

f ( x ) Q ( x ) Y | 4 | 2 | 2 | θ x q | 4 γ | ( | 2 | q | 4 | )

for all xX.

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,zX. Then we can choose α= | 2 | 2 q and we get the desired result. □

Let f:XY 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,zX, 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(2x,2y,2z)|4|αφ(x,y,z)

for all x,y,zX. Let f:XY be a mapping with f(0)=0 and satisfying (2.2). Then there exist a unique additive mapping A:XY and a unique quadratic mapping Q:XY 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 xX.

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,zX. Let f:XY be a mapping with f(0)=0 and satisfying (2.2). Then there exist a unique additive mapping A:XY and a unique quadratic mapping Q:XY 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 xX.

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,zG. Suppose that, for any xG, 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:GX 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 xG and defines an additive mapping A:GX 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 xG. 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 n1 and xG. 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,zX. Therefore, the mapping A:GX 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 xG. 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 t0. Assume that κ>0 and f:GX 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,zG. Then there exists a unique additive mapping A:GX 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,zG. 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 xG. 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,zG. Suppose that, for any xG, 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:GX be an odd mapping satisfying (3.3). Then the limit A(x):= lim n f ( 2 n x ) 2 n exists for all xG and

f ( x ) A ( x ) X 1 | 2 γ | Ω(x)
(3.11)

for all xG. 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 xG. 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 xG and p,q0 with q>p0. 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,zG. Suppose that, for any xG, 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:GX 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 xG and defines a quadratic mapping Q:GX 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,zG. Suppose that, for any xG, 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:GX 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 xG and

f ( x ) Q ( x ) X 1 | 4 γ | Θ(x)
(3.21)

for all xG. 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:XY 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,zX, 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,zG. 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 xG and f:GX be a mapping with f(0)=0 and satisfying (3.3). Then there exist an additive mapping A:GX and a quadratic mapping Q:GX 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 xG. 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,zG. 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 xG and f:GX be a mapping with f(0)=0 and satisfying (3.3). Then there exist an additive mapping A:GX and a quadratic mapping Q:GX such that

f ( x ) A ( x ) Q ( x ) X max { Ω ( x ) , Θ ( x ) } | γ |
(3.23)

for all xG. 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.

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Correspondence to H Azadi Kenary.

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Keywords

  • Hyers-Ulam stability
  • fixed point method
  • non-Archimedean normed spaces