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Nonlinear approximation of an ACQ-functional equation in nan-spaces

Fixed Point Theory and Applications20112011:60

https://doi.org/10.1186/1687-1812-2011-60

Received: 6 June 2011

Accepted: 3 October 2011

Published: 3 October 2011

Abstract

In this paper, using the fixed point and direct methods, we prove the generalized Hyers-Ulam stability of an additive-cubic-quartic functional equation in NAN-spaces.

Mathematics Subject Classification (2010)

39B52·47H10·26E30·46S10·47S10

Keywords

generalized Hyers-Ulam stabilitynon-Archimedean normed spacefixed point method

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 [1] in 1940. In the next year, Hyers [2] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Rassias [3] proved a generalization of the Hyers' theorem for additive mappings. The result of Rassias has provided a lot of influence during the last three decades in the development of a generalization of the Hyers-Ulam stability concept. This new concept is known as generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations (see [48]). Furthermore, in 1994, a generalization of the Rassias' theorem was obtained by Găvruta [9] by replacing the bound ε(||x|| p + ||y|| p ) by a general control function φ(x, y).

The functional equation
f ( x + y ) + f ( x - y ) = 2 f ( x ) + 2 f ( y )

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. In 1983, a generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [10] 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 [12] proved the generalized Hyers-Ulam stability of the quadratic functional equation.

The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [1332]).

In 1897, Hensel [33] has introduced a normed space that does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications (see [3437]).

Now, we give some definitions and lemmas for the main results in this paper.

A valuation is a function |·| from a field K into [0, ∞) such that, for all r , s K , the following conditions hold:
  1. (a)

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

     
  2. (b)

    |rs| = |r||s|;

     
  3. (c)

    |r + s| ≤ |r| + |s|.

     

A field K is called a valued field if K carries a valuation. The usual absolute values of and are examples of valuations.

Let us consider a valuation that satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by
| r + s | max { | r | , | s | }

for all r , s K , then the function |·| is called a non-Archimedean valuation and the field is called a non-Archimedean field. Clearly, |1| = | -1| = 1 and |n| ≤ 1 for all n . A trivial example of a non-Archimedean valuation is the function |·| taking everything except for 0 into 1 and |0| = 0.

Definition 1.1. Let X be a vector space over a field K with a non-Archimedean valuation |·|. A function ||·|| : X → [0, ∞) is called a non-Archimedean norm if the following conditions hold:
  1. (a)

    ||x|| = 0 if and only if x = 0 for all x X;

     
  2. (b)

    ||rx|| = |r| ||x|| for all r K and x X;

     
  3. (c)
    the strong triangle inequality holds:
    | | x + y | | max { | | x | | , | | y | | }
     

for all x, y X.

Then (X, ||·||) is called a non-Archimedean normed space (briefly NAN-space).

Definition 1.2. Let {x n } be a sequence in a non-Archimedean normed space X.
  1. (1)
    The sequence {x n } is called a Cauchy sequence if, for any ε > 0, there is a positive integer N such that
    | | x n - x m | | ε
     
for all n, mN.
  1. (2)
    The sequence {x n } is said to be convergent if, for any ε > 0, there are a positive integer N and x X such that
    | | x n - x | | ε
     
for all nN. Then, the point x X is called the limit of the sequence {x n }, which is denoted by limn→∞x n = x.
  1. (3)

    If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space.

     

Note that ||x n - x m || ≤ max{||xj+1- x j || : mjn - 1} for all m, n ≥ 1 with n > m.

Definition 1.3. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies the following conditions:
  1. (a)

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

     
  2. (b)

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

     
  3. (c)

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

     
Theorem 1.1. [38, 39]Let (X, d) be a complete generalized metric space and J : XX be a strictly contractive mapping with Lipschitz constant L < 1. Then, for all 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. (a)

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

     
  2. (b)

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

     
  3. (c)

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

     
  4. (d)

    d ( y , y * ) 1 1 - L d ( y , J y ) for all y Y.

     
In this paper, using the fixed point and direct methods, we prove the generalized Hyers-Ulam stability of the following functional equation
1 1 f ( x + 2 y ) + 1 1 f ( x - 2 y ) = 4 4 { f ( x + y ) + f ( x - y ) } + 1 2 f ( 3 y ) - 4 8 f ( 2 y ) + 6 0 f ( y ) - 6 6 f ( x )
(1.1)

in non-Archimedean normed spaces.

2. Non-Archimedean stability of the equation (1.1): a fixed point method-odd case

Using the fixed point alternative approach, we prove the generalized Hyers-Ulam stability of functional Equation (1.1) in non-Archimedean normed spaces for an odd case.

In [40], Lee et al. considered the following quartic functional equation:
f ( 2 x + y ) + f ( 2 x - y ) = 4 { f ( x + y ) + f ( x - y ) } + 2 4 f ( x ) - 6 f ( y )
(2.1)

It is easy to show that the function f(x) = x4 satisfies the functional Equation (2.1), which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping.

One can easily show that an even mapping f : XY satisfies (1.1) if and only if the even mapping f : XY is a quartic mapping, that is,
f ( 2 x + y ) + f ( 2 x - y ) = 4 { f ( x + y ) + f ( x - y ) } + 2 4 f ( x ) - 6 f ( y )
(2.2)
and an odd mapping f : XY satisfies (1.1) if and only if the odd mapping f : XY is a additive-cubic mapping, that is,
f ( 2 x + y ) + f ( 2 x - y ) = 4 { f ( x + y ) + f ( x - y ) } - 6 f ( x )
(2.3)

It was shown in [[41], Lemma 2.2] that g(x) = f(2x) - 2f(x) and h(x) = f(2x) - 8f(x) are cubic and additive, respectively, and that f ( x ) : = 1 1 6 g ( x ) - 1 1 6 h ( x ) .

For a given mapping f : XY, we define
Φ f ( x , y ) = 1 1 f ( x + 2 y ) + 1 1 f ( x - 2 y ) - 4 4 { f ( x + y ) + f ( x - y ) } - 1 2 f ( 3 y ) + 4 8 f ( 2 y ) - 6 0 f ( y ) + 6 6 f ( x )

for all x, y X.

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation Φ f (x, y) = 0 in non-Archimedean normed spaces: an odd case.

Throughout this section, let |8| ≠ 1.

Theorem 2.1. Let X be a non-Archimedean normed space and Y a non-Archimedean Banach space. Assume that γ : X2 → [0, ∞) is a function such that there exists an L < 1 with
γ x 2 , y 2 L | 8 | γ ( x , y )
(2.4)
for all x, y X. If f : XY is an odd mapping satisfying
| | Φ f ( x , y ) | | γ ( x , y )
(2.5)
for all x, y X, then the limit
C ( x ) : = lim n 8 n f x 2 n - 1 - 2 f x 2 n
exists for all x X and defines a unique cubic mapping C : XY such that
| | f ( 2 x ) - 2 f ( x ) - C ( x ) | | L | 8 | - | 8 | L m a x 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 ) .
(2.6)
Proof. Putting x = 0 in (2.5), we have
| | 1 2 f ( 3 y ) - 4 8 f ( 2 y ) + 6 0 f ( y ) | | γ ( y , 0 )
(2.7)

for all y X.

Replacing x by 2y in (2.5), we get
| | 1 1 f ( 4 y ) - 5 6 f ( 3 y ) + 1 1 4 f ( 2 y ) - 1 0 4 f ( y ) | | γ ( 2 y , y )
(2.8)
for all y X. By (2.7) and (2.8), we have
f ( 4 y ) - 1 0 f ( 2 y ) + 1 6 f ( y ) = 1 1 1 1 1 f ( 4 y ) - 5 6 f ( 3 y ) + 1 1 4 f ( 2 y ) - 1 0 4 f ( y ) + 1 4 3 3 [ 1 2 f ( 3 y ) - 4 8 f ( 2 y ) + 6 0 f ( y ) ] max 1 | 1 1 | γ ( 2 y , y ) , 1 4 3 3 γ ( y , 0 )
(2.9)
for all y X. Letting y : = x 2 and g(x) := f (2x) - 2f(x) for all x X, we get
g ( x ) - 8 g x 2 max 1 | 1 1 | γ x , x 2 , 1 4 3 3 γ x 2 , 0 .
(2.10)
Consider the set
S : = { g : X Y }
and the generalized metric d in S defined by
d ( f , g ) = inf μ ( 0 , + ) | | g ( x ) - h ( x ) | | μ max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 ) , x X ,

where inf = +∞. It is easy to show that (S, d) is complete (see [[42], Lemma 2.1]).

Now, we consider a linear mapping J : SS such that
J g ( x ) : = 8 g x 2
(2.11)
for all x X. Let g, h S be such that d(g, h) = ε. Then we have
| | g ( x ) - h ( x ) | | ε max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 )
for all x X and so
| | J g ( x ) - J h ( x ) | | = 8 g x 2 - 8 h x 2 | 8 | max 1 | 1 1 | γ x , x 2 , 1 4 3 3 γ x 2 , 0 | 8 | L | 8 | ε max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 )
for all x X. Thus d(g, h) = ε implies that d(Jg, Jh) ≤ . This means that
d ( J g , J h ) L d ( g , h )
for all g, h S. It follows from (2.10) that
d ( g , J g ) L | 8 | .
(2.12)
By Theorem 1.1, there exists a mapping C : XY satisfying the following:
  1. (1)
    C is a fixed point of J, that is,
    1 8 C ( x ) = C x 2
    (2.13)
     
for all x X. The mapping C is a unique fixed point of J in the set
Ω = { h S : d ( g , h ) < } .
This implies that C is a unique mapping satisfying (2.13) such that there exists μ (0, ∞) satisfying
| | g ( x ) - C ( x ) | | μ max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 )
for all x X.
  1. (2)
    d(J n g, C) → 0 as n → ∞. This implies the equality
    lim n 8 n g x 2 n = lim n 8 n f x 2 n - 1 - 2 f x 2 n = C ( x )
     
for all x X.
  1. (3)
    d ( g , C ) d ( g , J g ) 1 - L with g Ω, which implies the inequality
    d ( g , C ) L | 8 | - | 8 | L .
    (2.14)
     

This implies that the inequality (2.6) holds.

Since Φ g (x, y) = Φ f (2x, 2y) - 2Φ f (x, y), using (2.4) and (2.5), we have
| | Φ C ( x , y ) | | = lim n | 8 | n Φ g x 2 n , y 2 n = lim n | 8 | n Φ f x 2 n - 1 , y 2 n - 1 - 2 Φ f x 2 n , y 2 n lim n | 8 | n max Φ f x 2 n - 1 , y 2 n - 1 , | 2 | Φ f x 2 n , y 2 n lim n | 8 | n max γ x 2 n - 1 , y 2 n - 1 , | 2 | γ x 2 n , y 2 n lim n | 8 | n max L n - 1 | 8 | n - 1 γ ( x , y ) , | 2 | L n | 8 | n γ ( x , y ) = 0

for all x, y X and n ≥ 1 and so ||Φ C (x, y)|| = 0 for all x, y X. Therefore, the mapping C : XY is cubic. This completes the proof.   □

Corollary 2.1. Let θ ≥ 0 and r be a real number with r > 1. Let f : XY be an odd mapping satisfying
| | Φ f ( x , y ) | | θ ( | | x | | r + | | y | | r )
(2.15)
for all x, y X. Then the limit C ( x ) = lim n 8 n f x 2 n - 1 - 2 f x 2 n exists for all x X and C : XY is a unique cubic mapping such that
| | f ( 2 x ) - 2 f ( x ) - C ( x ) | | | 8 | r | 8 | - | 8 | r + 1 max ( | 2 | r + 1 ) θ | | x | | r | 1 1 | , 1 4 3 3 θ | | x | | r

for all x X.

Proof. The proof follows from Theorem 2.1 if we take
γ ( x , y ) = θ ( | | x | | r + | | y | | r )

for all x, y X. In fact, if we choose L = |8| r , then we get the desired result.   □

Theorem 2.2. Let X be a non-Archimedean normed space and Y a non-Archimedean Banach space. Assume that γ : X2 → [0, ∞) is a function such that there exists an L < 1 with
γ ( 2 x , 2 y ) | 8 | L γ ( x , y )
(2.16)
for all x, y X. If f : XY is an odd mapping satisfying (2.5), then the limit
C ( x ) = lim n f ( 2 n + 1 x ) - 2 f ( 2 n x ) 8 n
exists for all x X and defines a unique cubic mapping C : XY such that
| | f ( 2 x ) - 2 f ( x ) - C ( x ) | | 1 | 8 | - | 8 | L max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 ) .
(2.17)
Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.1. Consider the mapping J : (S, d) → (S, d) such that
J g ( x ) : = 1 8 g ( 2 x )
(2.18)

for all x X.

Proceeding as in the proof of Theorem 2.1, we find that d(g, h) = ε implies that d(Jg, Jh) ≤ . This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h S.

It follows from (2.10) that
g ( 2 x ) 8 - g ( x ) 1 | 8 | max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 )
for all x X. So
d ( g , J g ) 1 | 8 | .
(2.19)
By Theorem 1.1, there exists a mapping C : XY satisfying the following:
  1. (1)
    C is a fixed point of J, that is,
    8 C ( x ) =  C (2 x )
    (2.20)
     
for all x X. The mapping C is a unique fixed point of J in the set
Ω = { h S : d ( g , h ) < } .
This implies that C is a unique mapping satisfying (2.20) such that there exists μ (0, ∞) satisfying
| | g ( x ) - C ( x ) | | μ max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 )
for all x X.
  1. (2)
    d(J n g, C) → 0 as n → ∞. This implies the equality
    lim n g ( 2 n x ) 8 n = lim n f ( 2 n + 1 x ) - 2 f ( 2 n x ) 8 n = C ( x )
     
for all x X.
  1. (3)
    d ( g , C ) d ( g , J g ) 1 - L with g Ω, which implies the inequality
    d ( g , C ) 1 | 8 | - | 8 | L .
    (2.21)
     

This implies that the inequality (2.17) holds. The rest of the proof is similar to the proof of Theorem 2.1.   □

Corollary 2.2. Let θ ≥ 0 and r be a real number with 0 < r < 1. Let f : XY be an odd mapping satisfying (2.15). Then the limit C ( x ) = lim n f ( 2 n + 1 x ) - 2 f ( 2 n x ) 8 n exists for all x X and C : XY is a unique cubic mapping such that
| | f ( 2 x ) - 2 f ( x ) - C ( x ) | | | 8 | r | 8 | r + 1 - | 8 | 2 max ( | 2 | r + 1 ) θ | | x | | r | 1 1 | , 1 4 3 3 θ | | x | | r

for all x X.

Proof. The proof follows from Theorem 2.2 if we take
γ ( x , y ) = θ ( | | x | | r + | | y | | r )

for all x, y X. In fact, if we choose L = |8|1-r, then we get the desired result.   □

Theorem 2.3. Let X be a non-Archimedean normed space and Y a non-Archimedean Banach space. Assume that γ : X2 → [0, ∞) is a function such that there exists an L < 1 with
γ x 2 , y 2 L | 2 | γ ( x , y )
(2.22)
for all x, y X. If f : XY is an odd mapping satisfying (2.5), then the limit
A ( x ) : = lim n 2 n f x 2 n - 1 - 8 f x 2 n
exists for all x X and defines a unique additive mapping A : XY such that
| | f ( 2 x ) - 8 f ( x ) - A ( x ) | | L | 2 | - | 2 | L max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 ) .
(2.23)

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

Letting y : = x 2 and h(x) := f (2x) - 8f (x) for all x X in (2.9), we get
h ( x ) - 2 h x 2 max 1 1 1 γ x , x 2 , 1 4 3 3 γ x 2 , 0 .
(2.24)
Now, we consider a linear mapping J : SS such that
J h ( x ) : = 2 h x 2
(2.25)
for all x X. Let g, h S be such that d(g, h) = ε. Then we have
g ( x ) - h ( x ) ε max 1 1 1 γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 )
for all x X and so
J g ( x ) - J h ( x ) = 2 g x 2 - 2 h x 2 | 2 | max 1 | 1 1 | γ x , x 2 , 1 4 3 3 γ x 2 , 0 | 2 | L | 2 | ε max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 )
for all x X. Thus, d(g, h) = ε implies that d(Jg, Jh) ≤ . This means that
d ( J g , J h ) L d ( g , h )
for all g, h S. It follows from (2.24) that
d ( g , J g ) L | 2 | .
(2.26)
By Theorem 1.1, there exists a mapping A : XY satisfying the following:
  1. (1)
    A is a fixed point of J, that is,
    1 2 A ( x ) = A x 2
    (2.27)
     
for all x X. The mapping A is a unique fixed point of J in the set
Ω = { h S : d ( g , h ) < } .
This implies that A is a unique mapping satisfying (2.27) such that there exists μ (0, ∞) satisfying
h ( x ) - A ( x ) μ max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 )
for all x X.
  1. (2)
    d(J n h, A) → 0 as n → ∞. This implies the equality
    lim n 2 n h x 2 n = lim n 2 n f x 2 n - 1 - 8 f x 2 n = A ( x )
     
for all x X.
  1. (3)
    d ( h , A ) d ( h , J h ) 1 - L with h Ω, which implies the inequality
    d ( h , A ) L | 2 | - | 2 | L .
    (2.28)
     

This implies that the inequality (2.23) holds. The rest of the proof is similar to the proof of Theorem 2.1.   □

Corollary 2.3. Let θ ≥ 0 and r be a real number with r > 1. Let f : XY be an odd mapping satisfying (2.15). Then, the limit A ( x ) = lim n 2 n f x 2 n - 1 - 8 f x 2 n exists for all x X and A : XY is a unique additive mapping such that
f ( 2 x ) 8 f ( x ) A ( x ) | 2 | r | 2 | | 2 | r + 1 max { ( | 2 | r + 1 ) θ x r | 11 | , | 14 33 | θ x r }

for all x X.

Proof. The proof follows from Theorem 2.3 if we take
γ ( x , y ) = θ ( x r + y r )

for all x, y X. In fact, if we choose L = |2| r , then we get the desired result.   □

Theorem 2.4. Let X be a non-Archimedean normed space and Y a non-Archimedean Banach space. Assume that γ : X2 → [0, ∞) is a function such that there exists an L < 1 with
γ ( 2 x , 2 y ) | 2 | L γ ( x , y )
(2.29)
for all x, y X. If f : XY is an odd mapping satisfying (2.5), then the limit
A ( x ) = lim n f ( 2 n + 1 x ) 8 f ( 2 n x ) 2 n
exists for all x X and defines a unique additive mapping A : XY such that
f ( 2 x ) - 8 f ( x ) - A ( x ) 1 | 2 | - | 2 | L max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 ) .
(2.30)
Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.1. Consider the mapping J : (S, d) → (S, d) such that
J g ( x ) : = 1 2 g ( 2 x )
(2.31)
for all x X. By (2.24), we obtain
h ( 2 x ) 2 - g ( x ) 1 | 2 | max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 )
for all x X. So
d ( g , J g ) 1 | 2 | .
(2.32)
By Theorem 1.1, there exists a mapping A : XY satisfying the following:
  1. (1)
    A is a fixed point of J, that is,
    2 A ( x ) = A ( 2 x )
    (2.33)
     
for all x X. The mapping A is a unique fixed point of J in the set
Ω = { h S : d ( g , h ) < } .
This implies that A is a unique mapping satisfying (2.33) such that there exists μ (0, ∞) satisfying
h ( x ) - A ( x ) μ max 1 | 1 1 | γ ( 2 x , x ) , 1 4 3 3 γ ( x , 0 )
for all x X.
  1. (2)
    d(J n h, A) → 0 as n → ∞. This implies the equality
    lim n 2 n h ( x 2 n ) = lim n 2 n ( f ( x 2 n 1 ) 8 f ( x 2 n ) ) = A ( x )
     
for all x X.
  1. (3)
    d ( h , A ) d ( h , J h ) 1 - L with h Ω, which implies the inequality
    d ( h , A ) 1 | 2 | - | 2 | L .
    (2.34)
     

This implies that the inequality (2.30) holds. The rest of the proof is similar to the proof of Theorem 2.1.   □

Corollary 2.4. Let θ ≥ 0 and r be a real number with 0 < r < 1. Let f : XY be an odd mapping satisfying (2.15). Then, the limit A ( x ) = lim n f ( 2 n + 1 x ) 8 f ( 2 n x ) 2 n exists for all x X and A : XY is a unique additive mapping such that
f ( 2 x ) 8 f ( x ) A ( x ) 1 | 2 | | 2 | r + 2 max { ( | 2 | r + 1 ) θ x r | 11 | , | 14 33 | θ x r }

for all x X.

Proof. The proof follows from Theorem 2.4 if we take
γ ( x , y ) = θ ( x r + y r )

for all x, y X. In fact, if we choose L = |2|r + 1, then we get the desired result.   □

3. Non-Archimedean stability of the equation (1.1): a fixed point method-even case

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional Equation (1.1) in non-Archimedean normed spaces for an even case. Throughout this section, let |16| ≠ 1.

Theorem 3.1. Let X be a non-Archimedean normed space and Y a non-Archimedean Banach space. Assume that γ : X2 → [0, ∞) is a function such that there exists an L < 1 with
γ ( 2 x , 2 y ) | 1 6 | L γ ( x , y )
(3.1)
for all x, y X. If f : XY is an even mapping with f(0) = 0 satisfying (2.5), then the limit
Q ( x ) : = lim n f ( 2 n x ) 16 n
exists for all x X and defines a unique quartic mapping Q : XY such that
f ( x ) - Q ( x ) 1 | 1 6 | - | 1 6 | L max 1 | 2 2 | γ ( 0 , x ) , 6 1 1 γ ( x , x ) .
(3.2)
Proof. Putting x = 0 in (2.5), we have
1 2 f ( 3 y ) - 7 0 f ( 2 y ) + 1 4 8 f ( y ) γ ( 0 , y )
(3.3)

for all y X.

Substituting x = y in (2.5), we get
f ( 3 y ) - 4 f ( 2 y ) - 1 7 f ( y ) γ ( y , y )
(3.4)
for all y X. By (3.3) and (3.4), we have
f ( 2 y ) - 1 6 f ( y ) = - 1 2 2 1 2 f ( 3 y ) - 7 0 f ( 2 y ) + 1 4 8 f ( y ) + 6 1 1 [ f ( 3 y ) - 4 f ( 2 y ) - 1 7 f ( y ) ] max 1 | 2 2 | γ ( 0 , y ) , 6 1 1 γ ( y , y )
(3.5)
for all y X. Consider the set
S : = { g : X Y , g ( 0 ) = 0 }
and the generalized metric d in S defined by
d ( f , g ) = inf μ ( 0 , + ) g ( x ) - h ( x ) μ max 1 | 2 2 | γ ( 0 , x ) , 6 1 1 γ ( x , x ) , x X

where inf = +∞. It is easy to show that (S, d) is complete (see [[42], Lemma 2.1]).

Now, we consider a linear mapping J : SS such that
J g ( x ) : = 1 1 6 g ( 2 x )
(3.6)
for all x X. It follows from (3.5) that
d ( f , J f ) 1 | 1 6 | .
(3.7)
By Theorem 1.1, there exists a mapping Q : XY satisfying the following:
  1. (1)
    Q is a fixed point of J, that is,
    1 6 Q ( x ) = Q ( 2 x )
    (3.8)
     
for all x X. The mapping Q is a unique fixed point of J in the set
Ω = { h S : d ( g , h ) < } .
This implies that Q is a unique mapping satisfying (3.8) such that there exists μ (0, ∞) satisfying
f ( x ) - Q ( x ) μ max 1 | 2 2 | γ ( 0 , x ) , 6 1 1 γ ( x , x )
for all x X.
  1. (2)
    d(J n f, Q) → 0 as n → ∞. This implies the equality
    lim n f ( 2 n x ) 16 n = Q ( x )
     
for all x X.
  1. (3)
    d ( f , Q ) d ( f , J f ) 1 - L with f Ω, which implies the inequality
    d ( f , C ) 1 | 1 6 | - | 1 6 | L .
    (3.9)
     

This implies that the inequality (3.2) holds. The rest of the proof is similar to the proof of Theorem 2.1.   □

Corollary 3.1. Let θ ≥ 0 and r be a real number with r > 1. Let f : XY be an even mapping with f(0) = 0 satisfying (2.15). Then, the limit Q ( x ) = lim n f ( 2 n x ) 16 n exists for all x X and Q : XY is a unique quartic mapping such that
f ( x ) - Q ( x ) 1 | 1 6 | - | 1 6 | r + 1 max θ x r | 2 2 | , 2 6 1 1 θ x r

for all x X.

Proof. The proof follows from Theorem 3.1 if we take
γ ( x , y ) = θ ( x r + y r )

for all x, y X. In fact, if we choose L = |16| r , then we get the desired result.   □

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

Theorem 3.2. Let X be a non-Archimedean normed space and Y a non-Archimedean Banach space. Assume that γ : X2 → [0, ∞) is a function such that there exists an L < 1 with
γ x 2 , y 2 L | 1 6 | γ ( x , y )
(3.10)
for all x, y X. If f : XY is an even mapping with f(0) = 0 satisfying (2.5), then the limit
Q ( x ) : = lim n 1 6 n f x 2 n
exists for all x X and defines a unique quartic mapping Q : XY such that
f ( x ) - Q ( x ) L | 1 6 | - | 1 6 | L max 1 | 2 2 | γ ( 0 , x ) , 6 1 1 γ ( x , x ) .
(3.11)
Corollary 3.2. Let θ ≥ 0 and r be a real number with 0 < r < 1. Let f : XY be an even mapping with f(0) = 0 satisfying (2.15). Then, the limit Q ( x ) = lim n 1 6 n f x 2 n exists for all x X and Q : XY is a unique quartic mapping such that
f ( x ) - Q ( x ) | 1 6 | | 1 6 | r + 1 - | 1 6 | 2 max θ x r | 2 2 | , 2 6 1 1 θ x r

for all x X.

Proof. The proof follows from Theorem 3.2 if we take
γ ( x , y ) = θ ( x r + y r )

for all x, y X. In fact, if we choose L = |16|1-r, then we get the desired result.   □

4. Non-Archimedean stability of Equation (1.1): a direct method-odd case

Throughout this section, using direct method, we prove the generalized Hyers-Ulam stability of the functional Equation (1.1) in non-Archimedean spaces for an odd case.

Theorem 4.1. Let G be an additive semigroup and X a complete non-Archimedean space. Assume that φ : G2 → [0, +∞) is a function such that
lim n | 8 | n φ x 2 n , y 2 n = 0
(4.1)
for all x, y G. Let for all x G
Φ ( x ) = lim n max | 8 | k + 1 max 1 | 1 1 | φ x 2 k , x 2 k + 1 , 1 4 3 3 φ x 2 k + 1 , 0 ; 0 k < n
(4.2)
exist. Suppose that f : GX is an odd mapping satisfying the inequality
Φ f ( x , y ) X φ ( x , y )
(4.3)
for all x, y G. Then the limit
C ( x ) : = lim n 8 n f x 2 n - 1 - 2 f x 2 n
exists for all x G and C : GX is a cubic mapping satisfying
| | f ( 2 x ) - 2 f ( x ) - C ( x ) | | X 1 | 8 | Φ ( x )
(4.4)
for all x G. Moreover, if
lim j lim n max | 8 | k + 1 max 1 | 1 1 | φ x 2 k , x 2 k + 1 , 1 4 3 3 φ x 2 k + 1 , 0 ; j k < n + j = 0 ,

then C is the unique mapping satisfying (4.4).

Proof. Proceeding as in the proof of Theorem 2.1, we obtain
f ( 4 y ) - 1 0 f ( 2 y ) + 1 6 f ( y ) X max 1 | 1 1 | φ ( 2 y , y ) , 1 4 3 3 φ ( y , 0 )
(4.5)
for all y X. Letting y : = x 2 and g(x) := f(2x) - 2f(x) for all x X, we get
g ( x ) - 8 g x 2 X max 1 | 1 1 | φ x , x 2 , 1 4 3 3 φ x 2 , 0 .
(4.6)
Replacing x by x 2 n in (4.6), we get
8 n g x 2 n - 8 n + 1 g x 2 n + 1 X 8 n max 1 | 1 1 | φ x 2 n , x 2 n + 1 , 1 4 3 3 φ x 2 n + 1 , 0 .
(4.7)
It follows from (4.1) and (4.7) that the sequence 8 n g x 2 n n = 1 is a Cauchy sequence. Since X is complete, so 8 n g x 2 n n = 1 is convergent. Set
C ( x ) : = lim n 8 n g x 2 n = lim n 8 n f x 2 n - 1 - 2 f x 2 n .
Using induction, we see that
8 n g x 2 n - g ( x ) X 1 | 8 | max | 8 | k + 1 max 1 | 1 1 | φ x 2 k , x 2 k + 1 , 1 4 3 3 φ x 2 k + 1 , 0 ; 0 k < n .
(4.8)
By taking n to approach infinity in (4.8), one obtains (4.4). If L is another mapping satisfying (4.4), then, for x G, we get
C ( x ) - L ( x ) X = lim j 8 j L x 2 j - 8 j C x 2 j X = lim j 8 j L x 2 j ± 8 j g x 2 j - 8 j C x 2 j X lim j max 8 j L x 2 j - g x 2 j X , 8 j g x 2 j - C x 2 j X 1 | 8 | lim j lim n max | 8 | k + 1 max 1 | 1 1 | φ x 2 k , x 2 k + 1 , 1 4 3 3 φ x 2 k + 1 , 0 ; j k < n + j = 0 .

Therefore, L = C. This completes the proof.   □

Corollary 4.1. Let ξ : [0, ∞) → [0, ∞) be a function satisfying
ξ t | 2 | ξ 1 | 2 | ξ ( t ) , ξ 1 | 2 | < 1 | 8 |
for all t ≥ 0. Let δ > 0 and f : GX be an odd mapping satisfying the inequality
Φ f ( x , y ) X δ ( ξ ( | x | ) + ξ ( | y | ) )
(4.9)
for all x, y G. Then the limit C ( x ) = lim n 8 n f x 2 n - 1 - 2 f x 2 n exists for all x G and C : GX is a unique cubic mapping such that
| | f ( 2 x ) - 2 f ( x ) - C ( x ) | | X max 1 | 1 1 | δ ξ ( | x | ) 1 + 1 | 8 | , 7 1 3 2 ξ ( | x | )

for all x G.

Proof. Defining φ : G2 → [0, ∞) by φ(x, y) := δ(ξ(|x|) + ξ(|y|)). Since | 8 | ξ 1 | 2 | < 1 , we have
lim n | 8 | n φ x 2 n , y 2 n lim n | 8 | ξ 1 | 2 | n φ ( x , y ) = 0
for all x, y G. Also for all x G
Φ ( x ) = lim n max | 8 | k + 1 max 1 | 1 1 | φ x 2 k , x 2 k + 1 , 1 4 3 3 φ x 2 k + 1 , 0 ; 0 k < n = | 8 | max 1 | 1 1 | φ x , x 2 , 1 4 3 3 φ x 2 , 0 = | 8 | max 1 | 1 1 | δ ξ ( | x | ) 1 + 1 | 8 | , 7 1 3 2 ξ ( | x | )
exists for all x G. On the other hand,
lim j lim n max | 8 | k + 1 max 1 | 1 1 | φ x 2 k , x 2 k + 1 , 1 4 3 3 φ x 2 k + 1 , 0 ; j k < n + j lim j | 8 | j + 1 max 1 | 1 1 | φ x 2 j , x 2 j + 1 , 1 4 3 3 φ x 2 j + 1 , 0 = 0 .

Applying Theorem 4.1, we get the desired result.   □

Theorem 4.2. Let G be an additive semigroup and X a complete non-Archimedean space. Assume that φ : G2 → [0, +∞) is a function such that
lim n φ ( 2 n x ,2 n y ) | 8 | n = 0
(4.10)
for all x, y G. Let for each x G
Φ ( x ) = lim n max { 1 | 8 | k + 1 max { 1 | 11 | φ ( 2 k + 1 x ,2 k x ) , | 14 33 | φ ( 2 k x ,0 ) } ; 0 k < n }
(4.11)
exist. Suppose that f : GX is an odd mapping satisfying the inequality (4.3). Then the limit
C ( x ) : = lim n f ( 2 n + 1 x ) 2 f ( 2 n x ) 8 n
exists for all x G and C : GX is a cubic mapping satisfying
| | f ( 2 x ) - 2 f ( x ) - C ( x ) | | X Φ ( x )
(4.12)
for all x G. Moreover, if
lim j lim n max { 1 | 8 | k + 1 max { 1 | 11 | φ ( 2 k + 1 x ,2 k x ) , | 14 33 | φ ( 2 k x ,0 ) } ; j k < n + j } = 0,

then C is the unique mapping satisfying (4.12).

Proof. It follows from (4.5) that
g ( 2 x ) 8 - g ( x ) X 1 | 8 | max 1 | 1 1 | φ ( 2 x , x ) , 1 4 3 3 φ ( x , 0 )
(4.13)
for all x G. Replacing x by 2 n x in (4.13), we get
g ( 2 n + 1 x ) 8 n + 1 g ( 2 n x ) 8 n X 1 | 8 | n + 1 max { 1 | 11 | φ ( 2 n + 1 x ,2 n x ) , | 14 33 | φ ( 2 n x ,0 ) } .
(4.14)
It follows from (4.10) and (4.14) that the sequence { g ( 2 n x ) 8 n } n = 1 is a Cauchy sequence. Since X is complete, { g ( 2 n x ) 8 n } n = 1 is convergent. It follows from (4.14) that
g ( 2 p x ) 8 p g ( 2 q x ) 8 q X = k = p q 1 g ( 2 k + 1 x ) 8 k + 1 g ( 2 k x ) 8 k X max { g ( 2 k + 1 x ) 8 k + 1 g ( 2 k x ) 8 k X ; p k < q 1 } max { 1 | 8 | k + 1 max { 1 | 11 | φ ( 2 k + 1 x ,2 k x ) , | 14 33 | φ ( 2 k x ,0 ) } ; p k < q 1 }
(4.15)

for all x G and all non-negative integers q, p with q > p ≥ 0. Letting p = 0 and passing the limit q → ∞ in the last inequality, we obtain (4.12).

The rest of the proof is similar to the proof of Theorem 4.1.   □

Corollary 4.2. Let ξ : [0, ∞) → [0, ∞) be a function satisfying
ξ ( | 2 | t ) ξ ( | 2 | ) ξ ( t ) , ξ ( | 2 | ) < | 8 |
for all t ≥ 0. Let δ > 0 and f : GX be a mapping satisfying the inequality (4.9). Then the limit C ( x ) = lim n f ( 2 n + 1 x ) 2 f ( 2 n x ) 8 n exists for all x G and C : GX is a unique cubic mapping such that
| | f ( 2 x ) - 2 f ( x ) - C ( x ) | | X 1 | 8 | m a x 1 + | 8 | | 1 1 | δ ξ ( | x | ) , 1 4 3 3 δ ξ ( | x | )
(4.16)

for all x G.

Proof. Define φ : G2 → [0, ∞) by φ(x, y) := δ(ξ(|x|) + ξ(|y|)). Proceeding as in the proof of Corollary 4.1, we have
lim n φ ( 2 n x ,2 n y ) | 8 | n = 0
for all x, y G. Also
Φ ( x ) = l i m n max { 1 | 8 | k + 1 max { 1 | 11 | φ ( 2 k + 1 x ,2 k x ) , | 14 33 | φ ( 2 k x ,0 ) } ; 0 k < n } = 1 | 8 | max { 1 | 11 | φ ( 2 x , x ) , | 14 33 | φ ( x ,0 ) } 1 | 8 | max { 1 + | 8 | | 11 | δ ξ ( | x | ) , | 14 33 | δ ξ ( | x | ) }

exists for all x G. Applying Theorem 4.2, we get the desired result.   □

Theorem 4.3. Let G be an additive semigroup and X a complete non-Archimedean space. Assume that φ : G2 → [0, +∞) is a function such that
lim n | 2 | n φ x 2 n , y 2 n = 0
(4.17)
for all x, y G. Let for all x G
Φ ( x ) = lim n max | 2 | k max 1 | 1 1 | φ x 2 k , x 2 k + 1 , 1 4 3 3 φ x 2 k + 1 , 0 ; 0 k < n
(4.18)
exist. Suppose that f : GX is an odd mapping satisfying the inequality (4.3). Then the limit
A ( x ) : = lim n 2 n f x 2 n - 1 - 8 f x 2 n
exists for all x G and A : GX is an additive mapping satisfying
| | f ( 2 x ) - 8 f ( x ) - A ( x ) | | X Φ ( x )
(4.19)
for all x G. Moreover, if
lim j lim n max | 2 | k max 1 | 1 1 | φ x 2 k , x 2 k + 1 , 1 4 3 3 φ x 2 k + 1 , 0 ; j k < n + j = 0 ,

then A is the unique mapping satisfying (4.19).

Proof. Letting y : = x 2 and h(x) := f(2x) - 8f(x) for all x G in (4.5), we get
h ( x ) - 2 h x 2 X max 1 | 1 1 | φ x , x 2 , 1 4 3 3 φ x 2 , 0 .
(4.20)
Replacing x by x 2 n in (4.20), we obtain
2 n h x 2 n - 2 n + 1 h x 2 n + 1 X | 2 | n max 1 | 1 1 | φ x 2 n , x 2 n + 1 , 1 4 3 3 φ x 2 n + 1 , 0 .
(4.21)
Using induction, one can easily show that
2 n h x 2 n - h ( x ) X max | 2 | k max 1 | 1 1 | φ x 2 k , x 2 k + 1 , 1 4 3 3 φ x 2 k + 1 , 0 ; 0 k < n .
(4.22)

The rest of the proof is similar to the proof of Theorem 4.1.   □

Corollary 4.3. Let ξ : [0, ∞) → [0, ∞) be a function satisfying
ξ t | 2 | ξ 1 | 2 | ξ ( t ) , ξ 1 | 2 | < 1 | 2 |
for all t ≥ 0. Let δ > 0 and f : GX be an odd mapping satisfying the inequality (4.9). Then the limit A ( x ) = lim n 2 n f x 2 n - 1 - 8 f x 2 n exists for all x G and A : GX is a unique additive mapping such that
| | f ( 2 x ) - 8 f ( x ) - A ( x ) | | X max 1 + 1 | 2 | δ ξ ( | x | ) | 1 1 | , 7 3 3 ξ ( | x | )

for all x G.

Proof. Define φ : G2 → [0, ∞) by φ(x, y) := δ((ξ(|x|) + ξ(|y|)). Also
Φ ( x ) = lim n max | 2 | k max 1 | 1 1 | φ x 2 k , x 2 k + 1 , 1 4 3 3 φ x 2 k + 1 , 0 ; 0 k < n = max 1 + 1 | 2 | δ ξ ( | x | ) | 1 1 | , 7 3 3 ξ ( | x | )

exists for all x G. Applying Theorem 4.3, we get the desired result.   □

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

Theorem 4.4. Let G be an additive semigroup and X a complete non-Archimedean space. Assume that φ : G2 → [0, +∞) is a function such that
lim n φ ( 2 n x ,2 n y ) | 2 | n = 0
(4.23)
for all x, y G. Let for each x G
Φ ( x ) = lim n max { 1 | 2 | k max { 1 | 11 | φ ( 2 k + 1 x ,2 k x ) , | 14 33 | φ ( 2 k x ,0 ) } ; 0 k < n }
(4.24)
exist. Suppose that f : GX be an odd mapping satisfying the inequality (4.3). Then the limit
A ( x ) : = lim n f ( 2 n + 1 x ) 8 f ( 2 n x ) 2 n
exists for all x G and A : GX is an additive mapping satisfying
| | f ( 2 x ) - 8 f ( x ) - A ( x ) | | X 1 | 2 | Φ ( x )
(4.25)
for all x G. Moreover, if
lim j lim n max { 1 | 2 | k max { 1 | 11 | φ ( 2 k + 1 x ,2 k x ) , | 14 33 | φ ( 2 k x ,0 ) } ; j k < n + j } = 0,

then A is the unique mapping satisfying (4.25).

5. Non-Archimedean stability of Equation (1.1): a direct method-even case

Theorem 5.1. Let G be an additive semigroup and X a complete non-Archimedean space. Assume that φ : G2 → [0, +∞) is a function such that
lim n φ ( 2 n x ,2 n y ) | 16 | n = 0
(5.1)
for all x, y G. Let for all x G
Φ ( x ) = lim n max { 1 | 16 | k max { 1 | 22 | φ ( 0,2 k x ) , | 6 11 | φ ( 2 k x ,2 k x ) } ; 0 k < n }
(5.2)
exist. Suppose that f : GX is an even mapping with f(0) = 0 satisfying the inequality (4.3). Then the limit
Q ( x ) : = lim n f ( 2 n x ) 16 n
exists for all x G and Q : GX is a quartic mapping satisfying
| | f ( x ) - Q ( x ) | | X 1 | 1 6 | Φ ( x )
(5.3)
for all x G. Moreover, if
lim j lim n max { 1 | 16 | k max { 1 | 22 | φ ( 0,2 k x ) , | 6 11 | φ ( 2 k x ,2 k x ) } ; j k < n + j } = 0,

then Q is the unique mapping satisfying (5.3).

Proof. Proceeding as in the proof of Theorem 3.1, we obtain
f ( 2 x ) 1 6 - f ( x ) X 1 | 1 6 | max 1 | 2 2 | φ ( 0 , x ) , 6 1 1 φ ( x , x ) .
One can easily show that
f ( 2 n x ) 16 n f ( x ) X 1 | 16 | max { 1 | 16 | k max { 1 | 22 | φ ( 0,2 k x ) , | 6 11 | φ ( 2 k x ,2 k x ) } ; 0 k < n } .

The rest of the proof is similar to the proof of Theorem 4.1.   □

Corollary 5.1. Let ξ : [0, ∞) → [0, ∞) be a function satisfying
ξ | 2 | t ξ | 2 | ξ ( t ) , ξ | 2 | < | 1 6 |
for all t ≥ 0. Let δ > 0 and f : GX be an even mapping with f(0) = 0 satisfying the inequality (4.9). Then the limit Q ( x ) = lim n f ( 2 n x ) 16 n exists for all x G and Q : GX is a unique quartic mapping such that
| | f ( x ) - Q ( x ) | | X 1 | 1 6 | max 1 | 2 2 | δ ξ ( | x | ) , 2 6 1 1 ξ ( | x | )

for all x G.

Proof. Define φ : G2 → [0, ∞) by φ(x, y) := δ(ξ(|x|) + ξ(|y|)). Also
Φ ( x ) = max 1 | 2 2 | δ ξ ( | x | ) , 2 6 1 1 ξ ( | x | )

exists for all x G. Applying Theorem 5.1, we get the desired result.   □

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

Theorem 5.2. Let G be an additive semigroup and X a complete non-Archimedean space. Assume that φ : G2 → [0, +∞) is a function such that
lim n 1 6 | n φ x 2 n , y 2 n = 0
for all x, y G. Let for all x G
Φ ( x ) = lim n max | 1 6 | k max 1 | 2 2 | φ 0 , x 2 k + 1 , 6 1 1 φ x 2 k + 1 , x 2 k + 1 ; 0 k < n
exist. Suppose that f : GX is an even mapping satisfying the inequality (4.3). Then the limit
Q ( x ) : = lim n 1 6 n f x 2 n
exists for all x G and Q : GX is a quartic mapping satisfying
| | f ( x ) - Q ( x ) | | X Φ ( x )
(5.4)
for all x G. Moreover, if
lim j lim n max | 1 6 | k max 1 | 2 2 | φ 0 , x 2 k + 1 , 6 1 1 φ x 2 k + 1 , x 2 k + 1 ; j k < n + j = 0 ,

then Q is the unique mapping satisfying (5.4).

6. Conclusion

We linked here three different disciplines, namely, the non-Archimedean normed spaces, functional equations and fixed point theory. We established the generalized Hyers-Ulam stability of the functional Equation (1.1) in non-Archimedean normed spaces.

Declarations

Acknowledgements

The second and third authors were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0009232) and (NRF-2009-0070788), respectively.

Authors’ Affiliations

(1)
Department of Mathematics, College of Sciences, Yasouj University, Yasouj, Iran
(2)
Department of Mathematics, Daejin University, Kyeonggi, Korea
(3)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, Korea

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© Azadi Kenary et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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