Open Access

Stability of a generalized quadratic functional equation in various spaces: a fixed point alternative approach

  • Hassan Azadi Kenary1,
  • Choonkil Park2Email author,
  • Hamid Rezaei1 and
  • Sun Young Jang3
Advances in Difference Equations20112011:62

https://doi.org/10.1186/1687-1847-2011-62

Received: 12 June 2011

Accepted: 13 December 2011

Published: 13 December 2011

Abstract

Using the fixed point method, we prove the Hyers-Ulam stability of the following quadratic functional equation

c f i = 1 n x i + j = 2 n f i = 0 n x i - ( n + c - 1 ) x j = ( n + c - 1 ) f ( x 1 ) + c i = 2 n f ( x i ) + i < j , j = 3 n i = 2 n - 1 f ( x i - x j )

in various normed spaces.

2010 Mathematics Subject Classification: 39B52; 46S40; 34K36; 47S40; 26E50; 47H10; 39B82.

Keywords

Hyers-Ulam stabilityfuzzy Banach spaceorthogonalitynon-Archimedean normed spacesfixed point method

1. Introduction and preliminaries

In 1897, Hensel [1] introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications (see [25]).

A valuation is a function | · | from a field K into [0, ∞) such that 0 is the unique element having the 0 valuation, |rs| = |r| · |s| and the triangle inequality holds, i.e.,
| r + s | | r | + | s | , r , s K .

A field K is called a valued field if K carries a valuation. Throughout this paper, we assume that the base field is a valued field, hence call it simply a field. The usual absolute values of and are examples of valuations.

Let us consider a valuation which satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by
| r + s | max { | r | , | s | } , 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 said to be a non-Archimedean norm if it satisfies the following conditions:
  1. (i)

    ||x|| = 0 if and only if x = 0;

     
  2. (ii)

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

     
  3. (iii)
    the strong triangle inequality
    x + y max { x , y } , x , y X

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

     
Definition 1.2. (i) Let {x n } be a sequence in a non-Archimedean normed space X. Then the sequence {x n } is called Cauchy if for a given ε > 0 there is a positive integer N such that
x n - x m ε

for all n, mN.

(ii) Let {x n } be a sequence in a non-Archimedean normed space X. Then the sequence {x n } is called convergent if for a given ε > 0 there are a positive integer N and an x X such that
x n - x ε

for all nN. Then we call x X a limit of the sequence {x n }, and denote by lim n→∞ x n = x.

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

Assume that X is a real inner product space and f : X is a solution of the orthogonal Cauchy functional equation f(x + y) = f(x) + f(y), 〈x, y〉 = 0. By the Pythagorean theorem, f(x) = ||x||2 is a solution of the conditional equation. Of course, this function does not satisfy the additivity equation everywhere. Thus, orthogonal Cauchy equation is not equivalent to the classic Cauchy equation on the whole inner product space.

Pinsker [6] characterized orthogonally additive functionals on an inner product space when the orthogonality is the ordinary one in such spaces. Sundaresan [7] generalized this result to arbitrary Banach spaces equipped with the Birkhoff-James orthogonality. The orthogonal Cauchy functional equation
f ( x + y ) = f ( x ) + f ( y ) , x y ,

in which is an abstract orthogonality relation was first investigated by Gudder and Strawther [8]. They defined by a system consisting of five axioms and described the general semi-continuous real-valued solution of conditional Cauchy functional equation. In 1985, Rätz [9] introduced a new definition of orthogonality by using more restrictive axioms than of Gudder and Strawther. Moreover, he investigated the structure of orthogonally additive mappings. Rätz and Szabó [10] investigated the problem in a rather more general framework.

Let us recall the orthogonality in the sense of Rätz; cf. [9].

Suppose X is a real vector space with dim X ≥ 2 and is a binary relation on X with the following properties:

(O 1) totality of for zero: x 0, 0 x for all x X;

(O 2) independence: if x, y X - {0}, x y, then x, y are linearly independent;

(O 3) homogeneity: if x, y X, x y, then αx βy for all α, β ;

(O 4) the Thalesian property: if P is a 2-dimensional subspace of X, x P and λ +, which is the set of non-negative real numbers, then there exists y 0 P such that x y 0 and x + y 0 λx - y 0.

The pair (X, ) is called an orthogonality space. By an orthogonality normed space we mean an orthogonality space having a normed structure.

Some interesting examples are
  1. (i)

    The trivial orthogonality on a vector space X defined by (O 1), and for non-zero elements x, y X, x y if and only if x, y are linearly independent.

     
  2. (ii)

    The ordinary orthogonality on an inner product space (X, 〈., .〉) given by x y if and only if 〈x, y〉 = 0.

     
  3. (iii)

    The Birkhoff-James orthogonality on a normed space (X, ||.||) defined by x y if and only if ||x + λy|| ≥ ||x|| for all λ .

     

The relation is called symmetric if x y implies that y x for all x, y X. Clearly, examples (i) and (ii) are symmetric but example (iii) is not. It is remarkable to note, however, that a real normed space of dimension greater than 2 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. There are several orthogonality notions on a real normed space such as Birkhoff-James, Boussouis, Singer, Carlsson, unitary-Boussouis, Roberts, Phythagorean, isosceles and Diminnie (see [1117]).

The stability problem of functional equations originated from the following question of Ulam [18]: Under what condition does there exist an additive mapping near an approximately additive mapping? In 1941, Hyers [19] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces. In 1978, Rassias [20] extended the theorem of Hyers by considering the unbounded Cauchy difference ||f(x + y) - f(x) - f(y)|| ≤ ε(||x|| p + ||y|| p ), (ε > 0, p [0,1)). The reader is referred to [2123] and references therein for detailed information on stability of functional equations.

Ger and Sikorska [24] investigated the orthogonal stability of the Cauchy functional equation f(x + y) = f(x) + f(y), namely, they showed that if f is a mapping from an orthogonality space X into a real Banach space Y and ||f(x + y) - f(x) - f(y)|| ≤ ε for all x, y X with x y and some ε > 0, then there exists exactly one orthogonally additive mapping g : XY such that f ( x ) - g ( x ) 16 3 ε for all x X.

The first author treating the stability of the quadratic equation was Skof [25] by proving that if f is a mapping from a normed space X into a Banach space Y satisfying ||f(x + y) + f(x - y) - 2f(x) - 2f(y)|| ≤ ε for some ε > 0, then there is a unique quadratic mapping g : XY such that f ( x ) - g ( x ) ε 2 . Cholewa [26] extended the Skof's theorem by replacing X by an abelian group G. The Skof's result was later generalized by Czerwik [27] in the spirit of Hyers-Ulam-Rassias. The stability problem of functional equations has been extensively investigated by some mathematicians (see [2832]).

The orthogonally quadratic equation
f ( x + y ) + f ( x - y ) = 2 f ( x ) + 2 f ( y ) , x y

was first investigated by Vajzović [33] when X is a Hilbert space, Y is the scalar field, f is continuous and means the Hilbert space orthogonality. Later, Drljević [34], Fochi [35] and Szabó [36] generalized this result. See also [37].

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 [3851]).

Katsaras [52] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. In particular, Bag and Samanta [53], following Cheng and Mordeson [54], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Karmosil and Michalek type [55]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [56].

Definition 1.3. (Bag and Samanta [53]) Let X be a real vector space. A function N : X × → 0[1]is called a fuzzy norm on X if for all x, y X and all s, t ,

(N 1) N(x, t) = 0 for t ≤ 0;

(N 2) x = 0 if and only if N(x, t) = 1 for all t > 0;

(N 3) N ( c x , t ) = N x , t | c | if c ≠ 0;

(N 4) N(x + y, c + t) ≥ min{N(x, s), N(y, t)};

(N 5) N(x,.) is a non-decreasing function of and lim t→∞ N(x, t) = 1;

(N 6) for x ≠ 0, N(x,.) is continuous on .

The pair (X, N) is called a fuzzy normed vector space. The properties of fuzzy normed vector space and examples of fuzzy norms are given in (see [57, 58]).

Example 1.1. Let (X, ||.||) be a normed linear space and α, β > 0. Then
N ( x , t ) = α t α t + β | | x | | t > 0 , x X 0 t 0 , x X

is a fuzzy norm on X.

Definition 1.4. (Bag and Samanta [53]) Let (X, N) be a fuzzy normed vector space. A sequence {x n } in X is said to be convergent or converge if there exists an x X such that lim t→∞ N(x n - x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {x n } in X and we denote it by N - lim t→∞ x n = x.

Definition 1.5. (Bag and Samanta [53]) Let (X, N) be a fuzzy normed vector space. A sequence {x n } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n 0 such that for all nn 0 and all p > 0, we have N(x n+p - x n , t) > 1 - ε.

It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

We say that a mapping f : XY between fuzzy normed vector spaces X and Y is continuous at a point x X if for each sequence {x n } converging to x 0 X, then the sequence {f(x n )} converges to f(x 0). If f : XY is continuous at each x X, then f : XY is said to be continuous on X (see [56]).

Definition 1.6. 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. (1)

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

     
  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.

     
Theorem 1.1. ([59, 60]) Let (X, d) be a complete generalized metric space and J : XY 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. (1)

    d(J n x, J n+1 x) < ∞ for all n 0n 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 - L d ( y , J y ) for all y Y.

     
In this paper, we consider the following generalized quadratic functional equation
c f i = 1 n x i + j = 2 n f i = 1 n x i - ( n + c - 1 ) x j = ( n + c - 1 ) f ( x 1 ) + c i = 2 n f ( x i ) + i < j , j = 3 n i = 2 n - 1 f ( x i - x j )
(1)

and prove the Hyers-Ulam stability of the functional equation (1) in various normed spaces spaces.

This paper is organized as follows: In Section 2, we prove the Hyers-Ulam stability of the orthogonally quadratic functional equation (1) in non-Archimedean orthogonality spaces.

In Section 3, we prove the Hyers-Ulam stability of the quadratic functional equation (1) in fuzzy Banach spaces.

2. Stability of the orthogonally quadratic functional equation (1)

Throughout this section, assume that (X, ) is a non-Archimedean orthogonality space and that (Y, ||.|| Y ) is a real non-Archimedean Banach space. Assume that |2 - n - c| ≠ 0, 1. In this section, applying some ideas from [22, 24], we deal with the stability problem for the orthogonally quadratic functional equation (1) for all x 1, ..., x n X with x 2 x i for all i = 1, 3, ..., n in non-Archimedean Banach spaces.

Theorem 2.1. Let φ : X n → [0, ∞) be a function such that there exists an α < 1 with
φ ( x 1 , , x n ) | 2 - c - n | 2 α φ x 1 2 - c - n , , x n 2 - c - n
(2)
for all x 1, ..., x n X with x 2 x i (i ≠ 2). Let f : XY be a mapping with f(0) = 0 and satisfying
c f i = 1 n x i + j = 2 n f i = 1 n x i - ( n + c - 1 ) x j - ( n + c - 1 ) f ( x 1 ) + c i = 2 n f ( x i ) + i < j , j = 3 n i = 2 n - 1 f ( x i - x j ) Y φ ( x 1 , , x n )
(3)
for all x 1, ..., x n X with x 2 x i (i ≠ 2) and fixed positive real number c. Then there exists a unique orthogonally quadratic mapping Q : XY such that
f ( x ) - Q ( x ) Y φ ( 0 , x , 0 , , 0 ) | 2 - c - n | 2 - | 2 - c - n | 2 α
(4)

for all x X.

Proof. Putting x 2 = x and x 1 = x 3 = · · · = x n = 0 in (3), we get
f ( ( 2 - c - n ) x ) - ( 2 - c - n ) 2 f ( x ) Y φ ( 0 , x , 0 , , 0 )
(5)
for all x X, since x 0. So
f ( ( 2 - c - n ) x ) ( 2 - c - n ) 2 - f ( x ) Y φ ( 0 , x , 0 , , 0 ) | 2 - c - n | 2
(6)

for all x X.

Consider the set
S : = { h : X Y ; h ( 0 ) = 0 }
and introduce the generalized metric on S:
d ( g , h ) = inf { μ + : g ( x ) - h ( x ) Y μ φ ( 0 , x , 0 , , 0 ) , x X } ,

where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see [61]).

Now we consider the linear mapping J : SS such that
J g ( x ) : = 1 ( 2 - c - n ) 2 g ( ( 2 - c - n ) x )

for all x X.

Let g, h S be given such that d(g, h) = ε. Then,
g ( x ) - h ( x ) Y ε φ ( 0 , x , 0 , , 0 )
for all x X. Hence,
J g ( x ) - J h ( x ) Y = g ( ( 2 - c - n ) x ) ( 2 - c - n ) 2 - h ( ( 2 - c - n ) x ) ( 2 - c - n ) 2 Y α φ ( 0 , x , 0 , , 0 )
for all x X. So d(g, h) = ε implies that d(Jg, Jh) ≤ αε. This means that
d ( J g , J h ) α d ( g , h )

for all g, h S.

It follows from (6) that d ( f , J f ) 1 | 2 - c - n | 2 .

By Theorem 1.1, there exists a mapping Q : XY satisfying the following:
  1. (1)
    Q is a fixed point of J, i.e.,
    Q ( ( 2 c n ) x ) = ( 2 c n ) 2 Q ( x )
    (7)
    for all x X. The mapping Q is a unique fixed point of J in the set
    M = { g S : d ( h , g ) < } .
    This implies that Q is a unique mapping satisfying (7) such that there exists a μ (0, ∞) satisfying
    f ( x ) - Q ( x ) Y μ φ ( 0 , x , 0 , , 0 )

    for all x X;

     
  2. (2)
    d(J n f, Q) → 0 as n → ∞. This implies the equality
    lim m 1 ( 2 c n ) 2 m g ( ( 2 c n ) m x ) = Q ( x )

    for all x X;

     
  3. (3)
    d ( f , Q ) 1 1 - α d ( f , J f ) , which implies the inequality
    d ( f , Q ) 1 | 2 - c - n | 2 - | 2 - c - n | 2 α .
     

This implies that the inequality (4) holds.

It follows from (2) and (3) that
c Q i = 1 n x i + j = 2 n Q i = 1 n x i - ( n + c - 1 ) x j - ( n + c - 1 ) Q ( x 1 ) + c i = 2 n Q ( x i ) + i < j , j = 3 n i = 2 n - 1 Q ( x i - x j ) Y = lim n 1 | 2 - c - n | 2 m c f i = 1 n ( 2 - c - n ) m x i + j = 2 n f i = 1 n ( 2 - c - n ) m x i - ( n + c - 1 ) ( 2 - c - n ) m x j - ( n + c - 1 ) f ( ( 2 - c - n ) m x 1 ) + c i = 2 n f ( ( 2 - c - n ) m x i ) + i < j , j = 3 n i = 2 n - 1 f ( ( 2 - c - n ) m ( x i - x j ) ) Y lim m φ ( ( 2 - c - n ) m x 1 , , ( 2 - c - n ) m x n ) | 2 - c - n | 2 m lim m | 2 - c - n | 2 m α m | 2 - c - n | 2 m φ ( x 1 , , x m ) = 0

for all x 1, ..., x n X with x 2 x i . So Q satisfies (1) for all x 1, ..., x n X with x 2 x i . Hence, Q : XY is a unique orthogonally quadratic mapping satisfying (1), as desired.   □

From now on, in corollaries, assume that (X, ) is a non-Archimedean orthogonality normed space.

Corollary 2.1. Let θ be a positive real number and p a real number with 0 < p < 1. Let f : XY be a mapping with f(0) = 0 and satisfying
c f i = 1 n x i + j = 2 n f i = 1 n x i - ( n + c - 1 ) x j - ( n + c - 1 ) f ( x 1 ) + c i = 2 n f ( x i ) + i < j , j = 3 n i = 2 n - 1 f ( x i - x j ) Y θ i = 1 n x i p
(8)
for all x 1, ..., x n X with x 2 x i . Then there exists a unique orthogonally quadratic mapping Q : XY such that
f ( x ) - Q ( x ) = | 2 - c - n | p θ x | | p | 2 - c - n | 2 + p - | 2 - c - n | 3 i f | 2 - c - n | < 1 θ x p | 2 - c - n | 2 - | 2 - c - n | p + 1 i f | 2 - c - n | > 1 .

for all x X.

Proof. The proof follows from Theorem 2.1 by taking φ ( x 1 , , x n ) = θ ( i = 1 n x i p ) for all x 1, ..., x n X with x 2 x i . Then, we can choose
α = | 2 - c - n | 1 - p if | 2 - c - n | < 1 | 2 - c - n | p - 1 if | 2 - c - n | > 1 .

and we get the desired result.   □

Theorem 2.2. Let f : XY be a mapping with f(0) = 0 and satisfying (3) for which there exists a function φ : X n → [0, ∞) such that
φ ( x 1 , , x n ) α φ ( ( 2 - c - n ) x 1 , , ( 2 - c - n ) x n ) | 2 - c - n | 2
for all x 1, ..., x n X with x 2 x i and fixed positive real number c. Then there exists a unique orthogonally quadratic mapping Q : XY such that
f ( x ) - Q ( x ) Y α φ ( 0 , x , 0 , , 0 ) | 2 - c - n | 2 - | 2 - c - n | 2 α
(9)

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 : SS such that
J g ( x ) : = ( 2 c n ) 2 g ( x 2 c n )
for all x X. Let g, h S be given such that d(g, h) = ε. Then,
g ( x ) - h ( x ) Y ε φ ( 0 , x , 0 , , 0 )
for all x X. Hence,
J g ( x ) - J h ( x ) Y = ( 2 - c - n ) 2 g x 2 - c - n - ( 2 - c - n ) 2 h x 2 - c - n Y | 2 - c - n | 2 g x 2 - c - n - h x 2 - c - n Y | 2 - c - n | 2 φ 0 , x 2 - c - n , 0 , , 0 | 2 - c - n | 2 α ε | 2 - c - n | 2 φ ( 0 , x , 0 , , 0 ) = α ε φ ( 0 , x , 0 , , 0 )
for all x X. So d(g, h) = ε implies that d(Jg, Jh) ≤ αε. This means that
d ( J g , J h ) α d ( g , h )

for all g, h S.

It follows from (5) that
f ( x ) - ( 2 - c - n ) 2 f x 2 - c - n Y φ 0 , x 2 - c - n , 0 , , 0 α | 2 - c - n | 2 φ ( 0 , x , 0 , , 0 ) .
So
d ( f , J f ) α | 2 - c - n | 2 .
By Theorem 1.1, there exists a mapping Q : XY satisfying the following:
  1. (1)
    Q is a fixed point of J, i.e.,
    Q x 2 - c - n = 1 ( 2 - c - n ) 2 Q ( x )
    (10)
    for all x X. The mapping Q is a unique fixed point of J in the set
    M = { g S : d ( h , g ) < } .
    This implies that Q is a unique mapping satisfying (10) such that there exists a μ (0, ∞) satisfying
    f ( x ) - Q ( x ) Y μ φ ( 0 , x , 0 , , 0 )

    for all x X;

     
  2. (2)
    d(J n f, Q) → 0 as n → ∞. This implies the equality
    lim m ( 2 - c - n ) 2 m g x ( 2 - c - n ) m = Q ( x )

    for all x X;

     
  3. (3)
    d ( f , Q ) 1 1 - α d ( f , J f ) , which implies the inequality
    d ( f , Q ) α | 2 - c - n | 2 - | 2 - c - n | 2 α .
     

This implies that the inequality (9) holds.

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

Corollary 2.2. Let θ be a positive real number and p a real number with p > 1. Let f : XY be a mapping with f(0) = 0 and satisfying (8). Then there exists a unique orthogonally quadratic mapping Q : XY such that
f ( x ) - Q ( x ) = | 2 - c - n | p θ | | x | | p | 2 - c - n | 3 - | 2 - c - n | 2 + p i f | 2 - c - n | < 1 | 2 - c - n | θ | | x p | 2 - c - n | p + 2 - | 2 - c - n | 3 i f | 2 - c - n | > 1 .

for all x X.

Proof. The proof follows from Theorem 2.2 by taking φ ( x 1 , , x n ) = θ i = 1 n x i p for all x 1, ..., x n X with x 2 x i . Then, we can choose
α = | 2 - c - n | p - 1 if | 2 - c - n | < 1 | 2 - c - n | 1 - p if | 2 - c - n | > 1 .

and we get the desired result.   □

3. Fuzzy stability of the quadratic functional equation (1)

In this section, using the fixed point alternative approach, we prove the Hyers-Ulam stability of the functional equation (1) in fuzzy Banach spaces.

Throughout this section, assume that X is a vector space and that (Y, N) is a fuzzy Banach space. In the rest of the paper, let 2 - n - c > 1.

Theorem 3.1. Let φ : X n → [0, ∞) be a function such that there exists an α < 1 with
φ x 1 2 - c - n , , x n 2 - c - n α ( 2 - c - n ) 2 φ ( x 1 , , x n )
(11)
for all x 1, ..., x n X. Let f : XY be a mapping with f(0) = 0 and satisfying
N c f i = 1 n x i + j = 2 n f i = 1 n x i - ( n + c - 1 ) x j - ( n + c - 1 ) f ( x 1 ) + c i = 2 n f ( x i ) + i < j , j = 3 n i = 2 n - 1 f ( x i - x j ) , t t t + φ ( x 1 , , x n )
(12)
for all x 1, ..., x n X and all t > 0. Then the limit
Q ( x ) : = N - lim m ( 2 - c - n ) 2 m f x ( 2 - c - n ) m
exists for each x X and defines a unique quadratic mapping Q : XY such that
N ( f ( x ) Q ( x ) , t ) ( ( 2 c n ) 2 ( 2 c n ) 2 α ) t ( ( 2 c n ) 2 ( 2 c n ) 2 α ) t + α φ ( 0, x ,0, ,0 ) .
(13)
Proof. Putting x 2 = x and x 1 = x 3 = . . . = x n = 0 in (12), we have
N f ( ( 2 - c - n ) x ) - ( 2 - c - n ) 2 f ( x ) , t t t + φ ( 0 , x , 0 , , 0 )
(14)

for all x X and t > 0.

Replacing x by x 2 - c - n in (14), we obtain
N f ( x ) - ( 2 - c - n ) 2 f x 2 - c - n , t t t + φ ( 0 , x 2 - c - n , 0 , , 0 ) .
(15)

for all y X and t > 0.

By (15), we have
N f ( x ) - ( 2 - c - n ) 2 f x 2 - c - n , α t ( 2 - c - n ) 2 t t + φ ( 0 , x , 0 , , 0 ) .
(16)
Consider the set
S : = { g : X Y ; g ( 0 ) = 0 }
and the generalized metric d in S defined by
d ( f , g ) = inf μ + N ( g ( x ) - h ( x ) , μ t ) t t + φ ( 0 , x , 0 , , 0 ) , x X , t > 0 ,

where inf = +∞. It is easy to show that (S, d) is complete (see [[61], Lemma 2.1]).

Now, we consider a linear mapping J : SS such that
J g ( x ) : = ( 2 - c - n ) 2 g x 2 - c - n
for all x X. Let g, h S satisfy d(g, h) = ϵ. Then,
N ( g ( x ) - h ( x ) , ε t ) t t + φ ( 0 , x , 0 , , 0 )
for all x X and t > 0. Hence,
N ( J g ( x ) - J h ( x ) , α ε t ) = N ( 2 - c - n ) 2 g x 2 - c - n - ( 2 - c - n ) 2 h x 2 - c - n , α ε t = N g x 2 - c - n - h x 2 - c - n , α ε t ( 2 - c - n ) 2 α t ( 2 - c - n ) 2 α t ( 2 - c - n ) 2 + φ ( 0 , x 2 - c - n , 0 , , 0 ) α t ( 2 - c - n ) 2 α t ( 2 - c - n ) 2 + α ( 2 - c - n ) 2 φ ( 0 , x , 0 , , 0 ) = t t + φ ( 0 , x , 0 , , 0 )
for all x X and t > 0. Thus, d(g, h) = ϵ implies that d(Jg, Jh) ≤ αϵ. This means that
d ( J g , J h ) α d ( g , h )
for all g, h S. It follows from (16) that
d ( f , J f ) α ( 2 - c - n ) 2
By Theorem 1.1, there exists a mapping Q : XY satisfying the following:
  1. (1)
    Q is a fixed point of J, that is,
    Q x 2 - c - n = 1 ( 2 - c - n ) 2 Q ( x )
    (17)
    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 (17) such that there exists μ (0, ∞) satisfying
    N ( f ( x ) - Q ( x ) , μ t ) t t + φ ( 0 , x , 0 , , 0 )

    for all x X and t > 0.

     
  2. (2)
    d(J m f, Q) → 0 as m → ∞. This implies the equality
    N  -  lim m ( 2 - n - c ) 2 m f x ( 2 - n - c ) m = Q ( x )

    for all x X.

     
  3. (3)
    d ( f , Q ) d ( f , J f ) 1 - α with f Ω, which implies the inequality
    d ( f , Q ) α ( 2 - c - n ) 2 ( 1 - α ) .
     

This implies that the inequality (13) holds.

Using (11) and (12), we obtain
N ( 2 c n ) 2 m [ c f ( i = 1 n x i ( 2 c n ) m ) + j = 2 n f ( i = 1 n x i ( 2 c n ) m ( n + c 1 ) x j ( 2 c n ) m ) ( n + c 1 ) ( f ( x 1 ( 2 c n ) m ) + c i = 2 n f ( x i ( 2 c n ) m ) + i < j , j = 3 n ( i < j , j = 3 n 1 f ( x i x j ( 2 c n ) m ) ) ) ] , ( 2 c n ) 2 m t ) t t + φ ( x 1 ( 2 c n ) m , , x n ( 2 c n ) m )
(18)

for all x 1, ..., x n X, t > 0 and all n .

So by (11) and (18), we have
N ( 2 - c - n ) 2 m c f i = 1 n x i ( 2 - c - n ) m + j = 2 n f i = 1 n x i ( 2 - c - n ) m - ( n + c - 1 ) x j ( 2 - c - n ) m - ( n + c - 1 ) f x 1 ( 2 - c - n ) m + c i = 2 n f x i ( 2 - c - n ) m + i < j , j = 3 n i = 2 n - 1 f x i - x j ( 2 - c - n ) m , t t ( 2 - c - n ) 2 m t ( 2 - c - n ) 2 m + α m ( 2 - c - n ) 2 m φ ( x 1 , , x n )
for all x 1, ..., x n X, t > 0 and all n . Since
lim n t ( 2 - c - n ) 2 m t ( 2 - c - n ) 2 m + α m ( 2 - c - n ) 2 m φ ( x 1 , , x n ) = 1
for all x 1, ..., x n X and all t > 0, we deduce that
N c Q i = 1 n x i + j = 2 n Q i = 1 n x i - ( n + c - 1 ) x j - ( n + c - 1 ) Q ( x 1 ) + c i = 2 n Q ( x i ) + i < j , j = 3 n i = 2 n - 1 Q ( x i - x j ) , t = 1

for all x 1, ..., x n X and all t > 0. Thus the mapping Q : XY satisfying (1), as desired. This completes the proof.   □

Corollary 3.1. Let θ ≥ 0 and let r be a real number with r > 1. Let X be a normed vector space with norm || · ||. Let f : XY be a mapping with f(0) = 0 and satisfying
N c f i = 1 n x i + j = 2 n f i = 1 n x i - ( n + c - 1 ) x j - ( n + c - 1 ) f ( x 1 ) + c i = 2 n f ( x i ) + i < j , j = 3 n i = 2 n - 1 f ( x i - x j ) , t t t + θ i = 1 n x i r
(19)
for all x 1, ..., x n X and all t > 0. Then
Q ( x ) : = N - lim m ( 2 - n - c ) 2 m f x ( 2 - n - c ) m
exists for each x X and defines a unique quadratic mapping Q : XY such that
N ( f ( x ) Q ( x ) , t ) ( ( 2 c n ) 2 r ( 2 c n ) 2 ) t ( ( 2 c n ) 2 r ( 2 c n ) 2 ) t + θ x r

for all x X and all t > 0.

Proof. The proof follows from Theorem 3.1 by taking
φ ( x 1 , , x n ) : = θ i = 1 n x i r

for all x 1, ..., x n X. Then, we can choose α = (2 - c - n)2-2r and we get the desired result.   □

Theorem 3.2. Let φ : X n → [0, ∞) be a function such that there exists an α < 1 with
φ ( ( 2 - c - n ) x 1 , , ( 2 - c - n ) x n ) ( 2 - c - n ) 2 α φ ( x 1 , , x n )
(20)
for all x 1, ..., x n X. Let f : XY be a mapping with f(0) = 0 and satisfying (12). Then the limit
Q ( x ) : = N lim m f ( ( 2 c n ) m x ) ( 2 c n ) 2 m
exists for each x X and defines a unique quadratic mapping Q : XY such that
N ( f ( x ) - Q ( x ) , t ) ( 2 - c - n ) 2 ( 1 - α ) t ( 2 - c - n ) 2 ( 1 - α ) t + φ ( 0 , x , 0 , , 0 )
(21)
Proof. Let (S, d) be the generalized metric space defined as in the proof of Theorem 3.1. Consider the linear mapping J : SS such that
J g ( x ) : = 1 ( 2 - c - n ) 2 g ( ( 2 - c - n ) x )
for all x X. Let g, h S be such that d(g, h) = ϵ. Then,
N ( g ( x ) - h ( x ) , ε t ) t t + φ ( 0 , x , 0 , , 0 )
for all x X and t > 0. Hence,
N ( J g ( x ) J h ( x ) , α ε t ) = N ( g ( ( 2 c n ) x ) ( 2 c n ) 2 h ( ( 2 c n ) x ) ( 2 c n ) 2 , α ε t ) = N ( g ( ( 2 c n ) x ) h ( ( 2 c n ) x ) , ( 2 c n ) 2 α ε t ) ( 2 c n ) 2 α t ( 2 c n ) 2 α t + ( 2 c n ) 2 α φ ( 0, x ,0, ,0 ) = t t + φ ( 0, x ,0, ,0 )
for all x X and t > 0. Thus, d(g, h) = ϵ implies that d(Jg, Jh) ≤ αϵ. This means that
d J g , J h α d g , h

for all g, h S.

It follows from (14) that
N f ( ( 2 - c - n ) x ) ( 2 - c - n ) 2 - f ( x ) , t ( 2 - c - n ) 2 t t + φ ( 0 , x , 0 , , 0 )

for all x X and t > 0. So d ( f , J f ) 1 ( 2 - c - n ) 2 .

By Theorem 1.1, there exists a mapping Q : XY satisfying the following:
  1. (1)
    Q is a fixed point of J, that is,
    ( 2 - c - n ) 2 Q ( x ) = Q ( ( 2 - c - n ) x )
    (22)
    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 (22) such that there exists μ (0, ∞) satisfying
    N ( f ( x ) - Q ( x ) , μ t ) t t + φ ( 0 , x , 0 , , 0 )

    for all x X and t > 0.

     
  2. (2)
    d(J m f, Q) → 0 as m → ∞. This implies the equality
    lim m N f ( ( 2 c n ) m x ) ( 2 c n ) 2 m = Q ( x )

    for all x X.

     
  3. (3)
    d ( f , Q ) d ( f , J f ) 1 - α with f Ω, which implies the inequality
    d ( f , Q ) 1 ( 2 - c - n ) 2 ( 1 - α ) .
     

This implies that the inequality (21) holds.

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

Corollary 3.2. Let θ ≥ 0 and let r be a real number with 0 < r < 1. Let X be a normed vector space with norm || · ||. Let f : XY be a mapping with f(0) = 0 and satisfying (19). Then the limit
Q ( x ) : = N lim m f ( ( 2 c n ) m x ) ( 2 c n ) 2 m
exists for each x X and defines a unique quadratic mapping Q : XY such that
N ( f ( x ) Q ( x ) , t ) ( ( 2 c n ) 2 ( 2 c n ) 2 r ) t ( ( 2 c n ) 2 ( 2 c n ) 2 r ) t + θ x r .

for all x X and all t > 0.

Proof. The proof follows from Theorem 3.2 by taking
φ ( x 1 , , x n ) : = θ i = 1 n x i r

for all x 1, ..., x n X. Then, we can choose α = (2 - c - n)2r-2and we get the desired result.   □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, College of Sciences, Yasouj University
(2)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University
(3)
Department of Mathematics, University of Ulsan

References

  1. Hensel K: Ubereine news Begrundung der Theorie der algebraischen Zahlen. Jahresber Deutsch Math Verein 1897, 6: 83-88.Google Scholar
  2. Deses D: On the representation of non-Archimedean objects. Topol Appl 2005, 153: 774-785. 10.1016/j.topol.2005.01.010MathSciNetView ArticleGoogle Scholar
  3. 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
  4. Khrennikov A: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. In Mathematics and it's Applications. Volume 427. Kluwer Academic Publishers, Dordrecht; 1997.Google Scholar
  5. 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
  6. Pinsker AG: Sur une fonctionnelle dans l'espace de Hilbert. C R (Dokl) Acad Sci URSS, n Ser 1938, 20: 411-414.Google Scholar
  7. Sundaresan K: Orthogonality and nonlinear functionals on Banach spaces. Proc Amer Math Soc 1972, 34: 187-190. 10.1090/S0002-9939-1972-0291835-XMathSciNetView ArticleGoogle Scholar
  8. Gudder S, Strawther D: Orthogonally additive and orthogonally increasing functions on vector spaces. Pacific J Math 1975, 58: 427-436.MathSciNetView ArticleGoogle Scholar
  9. Rätz J: On orthogonally additive mappings. Aequationes Math 1985, 28: 35-49. 10.1007/BF02189390MathSciNetView ArticleGoogle Scholar
  10. Rätz J, Szabó Gy: On orthogonally additive mappings IV . Aequationes Math 1989, 38: 73-85. 10.1007/BF01839496MathSciNetView ArticleGoogle Scholar
  11. Alonso J, Benítez C: Orthogonality in normed linear spaces: a survey I . Main properties Extracta Math 1988, 3: 1-15.Google Scholar
  12. Alonso J, Benítez C: Orthogonality in normed linear spaces: a survey II . Relations between main orthogonalities Extracta Math 1989, 4: 121-131.Google Scholar
  13. Birkhoff G: Orthogonality in linear metric spaces. Duke Math J 1935, 1: 169-172. 10.1215/S0012-7094-35-00115-6MathSciNetView ArticleGoogle Scholar
  14. Carlsson SO: Orthogonality in normed linear spaces. Ark Mat 1962, 4: 297-318. 10.1007/BF02591506MathSciNetView ArticleGoogle Scholar
  15. Diminnie CR: A new orthogonality relation for normed linear spaces. Math Nachr 1983, 114: 197-203. 10.1002/mana.19831140115MathSciNetView ArticleGoogle Scholar
  16. James RC: Orthogonality in normed linear spaces. Duke Math J 1945, 12: 291-302. 10.1215/S0012-7094-45-01223-3MathSciNetView ArticleGoogle Scholar
  17. James RC: Orthogonality and linear functionals in normed linear spaces. Trans Amer Math Soc 1947, 61: 265-292. 10.1090/S0002-9947-1947-0021241-4MathSciNetView ArticleGoogle Scholar
  18. Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1960.Google Scholar
  19. Hyers DH: On the stability of the linear functional eq. Proc Natl Acad Sci USA 1941, 27: 222-224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
  20. Rassias ThM: On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc 1978, 72: 297-300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleGoogle Scholar
  21. Czerwik S: Stability of Functional Equations of Ulam-Hyers-Rassias Type. Hadronic Press, Palm Harbor, Florida; 2003.Google Scholar
  22. Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.View ArticleGoogle Scholar
  23. Jung S: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Florida; 2001.Google Scholar
  24. Ger R, Sikorska J: Stability of the orthogonal additivity. Bull Pol Acad Sci Math 1995, 43: 143-151.MathSciNetGoogle Scholar
  25. Skof F: Proprietà locali e approssimazione di operatori. Rend Sem Mat Fis Milano 1983, 53: 113-129. 10.1007/BF02924890MathSciNetView ArticleGoogle Scholar
  26. Cholewa PW: Remarks on the stability of functional equations. Aequationes Math 1984, 27: 76-86. 10.1007/BF02192660MathSciNetView ArticleGoogle Scholar
  27. Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh Math Sem Univ Hamburg 1992, 62: 59-64. 10.1007/BF02941618MathSciNetView ArticleGoogle Scholar
  28. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific Publishing Company, New Jersey, London, Singapore and Hong Kong; 2002.Google Scholar
  29. Park C, Park J: Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping. J Differ Equ Appl 2006, 12: 1277-1288. 10.1080/10236190600986925View ArticleGoogle Scholar
  30. Rassias JM: Solution of the Ulam stability problem for quartic mappings. Glas Mat Ser III 1999,34(54):243-252.MathSciNetGoogle Scholar
  31. Rassias JM: Solution of the Ulam stability problem for cubic mappings. Glas Mat Ser III 2001,36(56):63-72.MathSciNetGoogle Scholar
  32. Rassias ThM: The problem of S.M. Ulam for approximately multiplicative mappings. J Math Anal Appl 2000, 246: 352-378. 10.1006/jmaa.2000.6788MathSciNetView ArticleGoogle Scholar
  33. Vajzović F: Über das Funktional H mit der Eigenschaft: ( x , y ) = 0) H ( x + y ) + H ( x - y ) = 2 H ( x ) + 2 H ( y ). Glasnik Mat Ser III 1967,2(22):73-81.MathSciNetGoogle Scholar
  34. Drljević F: On a functional which is quadratic on A -orthogonal vectors. Publ Inst Math (Beograd) 1986, 54: 63-71.Google Scholar
  35. Fochi M: Functional equations in A -orthogonal vectors. Aequationes Math 1989, 38: 28-40. 10.1007/BF01839491MathSciNetView ArticleGoogle Scholar
  36. Szabó Gy: Sesquilinear-orthogonally quadratic mappings. Aequationes Math 1990, 40: 190-200. 10.1007/BF02112295MathSciNetView ArticleGoogle Scholar
  37. Paganoni P, Rätz J: Conditional function equations and orthogonal additivity. Aequationes Math 1995, 50: 135-142. 10.1007/BF01831116MathSciNetView ArticleGoogle Scholar
  38. Kenary HA: Non-Archimedean stability of Cauchy-Jensen type functional equation. Int J Nonlinear Anal Appl 2010, 1: 1-10.Google Scholar
  39. Kenary HA, Shafaat Kh, Shafei M, Takbiri G: Hyers-Ulam-Rassias stability of the Appollonius quadratic mapping in RN-spaces. J Nonlinear Sci Appl 2011, 4: 110-119.MathSciNetGoogle Scholar
  40. 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
  41. Cădariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl 2008, 2008: 15. Art. ID 749392Google Scholar
  42. Eshaghi Gordji M, Bavand Savadkouhi M: Stability of mixed type cubic and quartic functional equations in random normed spaces. J Inequal Appl 2009, 2009: 9. Art. ID 527462View ArticleGoogle Scholar
  43. Eshaghi Gordji M, Bavand Savadkouhi M, Park C: Quadratic-quartic functional equations in RN -spaces. J Inequal Appl 2009, 2009: 14. Art. ID 868423View ArticleGoogle Scholar
  44. Eshaghi-Gordji M, Kaboli-Gharetapeh S, Park C, Zolfaghri S: Stability of an additive-cubic-quartic functional equation. Adv Diff Equ 2009, 2009: 20. Art. ID 395693Google Scholar
  45. Isac G, Rassias ThM: Stability of ψ -additive mappings: Appications to nonlinear analysis. Int J Math Math Sci 1996, 19: 219-228. 10.1155/S0161171296000324MathSciNetView ArticleGoogle Scholar
  46. Jun K, Kim H: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J Math Anal Appl 2002, 274: 867-878. 10.1016/S0022-247X(02)00415-8MathSciNetView ArticleGoogle Scholar
  47. Jung Y, Chang I: The stability of a cubic type functional equation with the fixed point alternative. J Math Anal Appl 2005, 306: 752-760. 10.1016/j.jmaa.2004.10.017MathSciNetView ArticleGoogle Scholar
  48. Park C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory Appl 2007, 2007: 15. Art. ID 50175View ArticleGoogle Scholar
  49. Park C: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl 2008, 2008: 9. Art. ID 493751Google Scholar
  50. Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4: 91-96.MathSciNetGoogle Scholar
  51. Schin SW, Ki D, Chang JC, Kim MJ: Random stability of quadratic functional equations: A fixed point approach. J Nonlinear Sci Appl 2011, 4: 37-49.MathSciNetGoogle Scholar
  52. Katsaras AK: Fuzzy topological vector spaces. Fuzzy Sets Syst 1984, 12: 143-154. 10.1016/0165-0114(84)90034-4MathSciNetView ArticleGoogle Scholar
  53. Bag T, Samanta SK: Finite dimensional fuzzy normed linear spaces. J Fuzzy Math 2003, 11: 687-705.MathSciNetGoogle Scholar
  54. Cheng SC, Mordeson JN: Fuzzy linear operators and fuzzy normed linear spaces. Bull Calcutta Math Soc 1994, 86: 429-436.MathSciNetGoogle Scholar
  55. Karmosil I, Michalek J: Fuzzy metric and statistical metric spaces. Kybernetica 1975, 11: 326-334.Google Scholar
  56. Bag T, Samanta SK: Fuzzy bounded linear operators. Fuzzy Sets Syst 2005, 151: 523-547.MathSciNetView ArticleGoogle Scholar
  57. Mirzavaziri M, Moslehian MS: A fixed point approach to stability of a quadratic equation. Bull Braz Math Soc 2006, 37: 361-376. 10.1007/s00574-006-0016-zMathSciNetView ArticleGoogle Scholar
  58. Mirmostafaee AK, Mirzavaziri M, Moslehian MS: Fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst 2008, 159: 730-738. 10.1016/j.fss.2007.07.011MathSciNetView ArticleGoogle Scholar
  59. Cădariu L, Radu V: Fixed points and the stability of Jensen's functional equation. J Inequal Pure Appl Math 2003,4(1):15. Art. ID 4MathSciNetGoogle Scholar
  60. 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
  61. Miheţ D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces. J Math Anal Appl 2008, 343: 567-572.MathSciNetView ArticleGoogle Scholar

Copyright

© 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.