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An AQCQ-functional equation in paranormed spaces
Advances in Difference Equations volume 2012, Article number: 63 (2012)
Abstract
In this article, we prove the Hyers-Ulam stability of an additive-quadratic-cubic-quartic functional equation in paranormed spaces.
Mathematics Subject Classification (2010): Primary 39B82; 39B52; 39B72; 46A99.
1. Introduction and preliminaries
The concept of statistical convergence for sequences of real numbers was introduced by Fast [1] and Steinhaus [2] independently and since then several generalizations and applications of this notion have been investigated by various authors (see [3–7]). This notion was defined in normed spaces by Kolk [8].
We recall some basic facts concerning Fré chet spaces.
Definition 1.1. [9] Let X be a vector space. A paranorm P : X → [0, ∞) is a function on X such that
-
(1)
P(0) = 0;
-
(2)
P(-x) = P(x);
-
(3)
P(x + y) ≤ P(x) + P(y) (triangle inequality)
-
(4)
If {t n } is a sequence of scalars with t n → t and {x n } ⊂ X with P(x n - x) → 0, then P(t n x n - tx) → 0 (continuity of multiplication).
The pair (X, P) is called a paranormed space if P is a paranorm on X.
The paranorm is called total if, in addition, we have
-
(5)
P(x) = 0 implies x = 0.
A Fréchet space is a total and complete paranormed space.
The stability problem of functional equations originated from a question of Ulam [10] concerning the stability of group homomorphisms. Hyers [11] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [12] for additive mappings and by Rassias [13] for linear mappings by considering an unbounded Cauchy difference. A generalization of Rassias' theorem was obtained by Găvruta [14] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach.
In 1990, Rassias [15] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. In 1991, Gajda [16] following the same approach as in Rassias [13], gave an affirmative solution to this question for p > 1. It was shown by Gajda [16], as well as by Rassias and Šemrl [17] that one cannot prove a Rassias-type theorem when p = 1 (cf. the books of Czerwik [18], Hyers et al. [19]).
In 1982, Rassias [20] followed the innovative approach of the Rassias' theorem [13] in which he replaced the factor ∥x∥p+ ∥y∥pby ∥x∥p· ∥y∥qfor p, q ∈ ℝ with p + q ≠ 1.
The functional equation
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [21] for mappings f : X → Y, where X is a normed space and Y is a Banach space. Cholewa [22] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [23] proved the Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have extensively been investigated by a number of authors and there are many interesting results concerning this problem (see [24–30]).
Jun and Kim [31] considered the following cubic functional equation
It is easy to show that the function f(x) = x3 satisfies the functional equation (1.1), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping.
Lee et al. [32] considered the following quartic functional equation
It is easy to show that the function f(x) = x4 satisfies the functional equation (1.2), which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping.
Throughout this article, assume that (X, P) is a Fré chet space and that (Y, ∥ · ∥) is a Banach space.
In this article, we prove the Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation
in paranormed spaces.
One can easily show that an odd mapping f : X → Y satisfies (1.3) if and only if the odd mapping f : X → Y is an additive-cubic mapping, i.e.,
It was shown in [[33], Lemma 2.2] that g(x) := f(2x) - 2f(x) and h(x) := f(2x) - 8f(x) are cubic and additive, respectively, and that .
One can easily show that an even mapping f : X → Y satisfies (1.3) if and only if the even mapping f : X → Y is a quadratic-quartic mapping, i.e.,
It was shown in [[34], Lemma 2.1] that g(x) := f(2x) - 4f(x) and h(x) := f(2x) - 16f(x) are quartic and quadratic, respectively, and that .
2. Hyers-Ulam stability of the functional equation (1.3): an odd mapping case
For a given mapping f, we define
In this section, we prove the Hyers-Ulam stability of the functional equation Df(x, y) = 0 in paranormed spaces: an odd mapping case.
Note that P(2x) ≤ 2P(x) for all x ∈ Y.
Theorem 2.1. Let r, θ be positive real numbers with r > 1, and let f : Y → X be an odd mapping such that
for all x, y ∈ Y. Then there exists a unique additive mapping A : Y → X such that
for all x ∈ Y.
Proof. Letting x = y in (2.1), we get
for all y ∈ Y.
Replacing x by 2y in (2.1), we get
for all y ∈ Y.
By (2.3) and (2.4),
for all y ∈ Y. Replacing y by and letting g(x) := f(2x) - 8f(x) in (2.5), we get
for all x ∈ Y. Hence
for all nonnegative integers m and l with m > l and all x ∈ Y. It follows from (2.6) that the sequence is Cauchy for all x ∈ Y. Since X is complete, the sequence converges. So one can define the mapping A : Y → X by
for all x ∈ Y.
By (2.1),
for all x, y ∈ Y. So DA(x, y) = 0. Since g : Y → X is odd, A : Y → X is odd. So the mapping A : Y → X is additive. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), we get (2.2). So there exists an additive mapping A : Y → X satisfying (2.2).
Now, let T : Y → X be another additive mapping satisfying (2.2). Then we have
which tends to zero as q → ∞ for all x ∈ Y. So we can conclude that A(x) = T(x) for all x ∈ Y. This proves the uniqueness of A. Thus the mapping A : Y → X is a unique additive mapping satisfying (2.2).
Theorem 2.2. Let r be a positive real number with r < 1, and let f : X → Y be an odd mapping such that
for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that
for all x ∈ X.
Proof. Letting x = y in (2.7), we get
for all y ∈ X.
Replacing x by 2y in (2.7), we get
for all y ∈ X.
By (2.9) and (2.10),
for all y ∈ X. Replacing y by x and letting g(x) := f(2x) - 8f(x) in (2.11), we get
for all x ∈ X. Hence
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.12) that the sequence is Cauchy for all x ∈ X. Since Y is complete, the sequence converges. So one can define the mapping A : X → Y by
for all x ∈ X.
By (2.7),
for all x, y ∈ X. So DA(x, y) = 0. Since g : X → Y is odd, A : X → Y is odd. So the mapping A : X → Y is additive. Moreover, letting l = 0 and passing the limit m → ∞ in (2.12), we get (2.8). So there exists an additive mapping A : X → Y satisfying (2.8).
Now, let T : X → Y be another additive mapping satisfying (2.8). Then we have
which tends to zero as q → ∞ for all x ∈ X. So we can conclude that A(x) = T(x) for all x ∈ X. This proves the uniqueness of A. Thus the mapping A : X → Y is a unique additive mapping satisfying (2.8).
Theorem 2.3. Let r, θ be positive real numbers with r > 3, and let f : Y → X be an odd mapping satisfying (2.1). Then there exists a unique cubic mapping C : Y → X such that
for all x ∈ Y.
Proof. Replacing y by and letting g(x) := f(2x) - 2f(x) in (2.5), we get
for all x ∈ Y.
The rest of the proof is similar to the proof of Theorem 2.1.
Theorem 2.4. Let r be a positive real number with r < 3, and let f : X → Y be an odd mapping satisfying (2.7). Then there exists a unique cubic mapping C : X → Y such that
for all x ∈ X.
Proof. Replacing y by x and letting g(x) := f(2x) - 2f(x) in (2.11), we get
for all x ∈ X.
The rest of the proof is similar to the proof of Theorem 2.2.
3. Hyers-Ulam stability of the functional equation (1.3): an even mapping case
In this section, we prove the Hyers-Ulam stability of the functional equation Df(x, y) = 0 in paranormed spaces: an even mapping case.
Note that P(2x) ≤ 2P(x) for all x ∈ Y.
Theorem 3.1. Let r, θ be positive real numbers with r > 2, and let f : Y → X be an even mapping satisfying f(0) = 0 and (2.1). Then there exists a unique quadratic mapping Q2 : Y → X such that
for all x ∈ Y.
Proof. Letting x = y in (2.1), we get
for all y ∈ Y.
Replacing x by 2y in (2.1), we get
for all y ∈ Y.
By (3.1) and (3.2),
for all y ∈ Y. Replacing y by and letting g(x) := f(2x) - 16f(x) in (3.3), we get
for all x ∈ Y.
The rest of the proof is similar to the proof of Theorem 2.1.
Theorem 3.2. Let r be a positive real number with r < 2, and let f : X → Y be an even mapping satisfying f(0) = 0 and (2.7). Then there exists a unique quadratic mapping Q2 : X → Y such that
for all x ∈ X.
Proof. Letting x = y in (2.7), we get
for all y ∈ X.
Replacing x by 2y in (2.7), we get
for all y ∈ X.
By (3.5) and (3.6),
for all y ∈ X. Replacing y by x and letting g(x) := f(2x) - 16f(x) in (3.7), we get
for all x ∈ X.
The rest of the proof is similar to the proof of Theorem 2.2.
Theorem 3.3. Let r, θ be positive real numbers with r > 4, and let f : Y → X be an even mapping satisfying f(0) = 0 and (2.1). Then there exists a unique quartic mapping Q4 : Y → X such that
for all x ∈ Y.
Proof. Replacing y by and letting g(x) := f(2x) - 4f(x) in (3.3), we get
for all x ∈ Y.
The rest of the proof is similar to the proof of Theorem 2.1.
Theorem 3.4. Let r be a positive real number with r < 4, and let f : X → Y be an even mapping satisfying f(0) = 0 and (2.7). Then there exists a unique quartic mapping Q4 : X → Y such that
for all x ∈ X.
Proof. Replacing y by x and letting g(x) := f(2x) - 4f(x) in (3.7), we get
for all x ∈ X.
The rest of the proof is similar to the proof of Theorem 2.2.
Let and . Then f o is odd and f e is even. f o , f e satisfy the functional equation (1.3). Let g o (x) := f o (2x) - 2f o (x) and h o (x) := f o (2x) - 8f o (x). Then . Let g e (x) := f e (2x) - 4f e (x) and h e (x) := f e (2x) - 16f e (x). Then . Thus
Theorem 3.5. Let r, θ be positive real numbers with r > 4. Let f : Y → X be a mapping satisfying f(0) = 0 and (2.1). Then there exist an additive mapping A : Y → X, a quadratic mapping Q2 : Y → X, a cubic mapping C : Y → X and a quartic mapping Q4 : Y → X such that
for all x ∈ Y.
Theorem 3.6. Let r be a positive real number with r < 1. Let f : X → Y be a mapping satisfying f(0) = 0 and (2.7). Then there exist an additive mapping A : X → Y, a quadratic mapping Q2 : X → Y, a cubic mapping C : X → Y and a quartic mapping Q4 : X → Y such that
for all x ∈ X.
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Acknowledgements
This study was supported by the Daejin University Research Grants in 2012.
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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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Park, C., Lee, J.R. An AQCQ-functional equation in paranormed spaces. Adv Differ Equ 2012, 63 (2012). https://doi.org/10.1186/1687-1847-2012-63
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DOI: https://doi.org/10.1186/1687-1847-2012-63
Keywords
- Hyers-Ulam stability
- paranormed space
- additive-quadratic-cubic-quartic functional equation.