Abstract
Using the fixed point method and direct method, we prove the Hyers-Ulam stability of an additive-quadratic-cubic-quartic functional equation in paranormed spaces.
MSC:35A17, 39B52, 47H10, 39B72.
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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 Fréchet spaces.
Definition 1.1 [9]
Let X be a vector space. A paranorm is a function on X such that
-
(1)
;
-
(2)
;
-
(3)
(triangle inequality);
-
(4)
If is a sequence of scalars with and with , then (continuity of multiplication).
The pair is called a paranormed space if P is a paranorm on X.
The paranorm is called total if, in addition, we have
-
(5)
implies .
A Fréchet space is a total and complete paranormed space.
The stability problem of functional equations originated from the question of Ulam [10] concerning the stability of group homomorphisms. Hyers [11] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [12] for additive mappings and by Th.M. Rassias [13] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by Găvruta [14] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach.
In 1990 during the 27th International Symposium on Functional Equations, Th.M. Rassias [15] asked the question whether such a theorem can also be proved for . In 1991, following the same approach as in Th.M. Rassias [13], Gajda [16] gave an affirmative solution to this question for . It was shown by Gajda [16], as well as by Th.M. Rassias and Šemrl [17], that one cannot prove a Th.M. Rassias’ type theorem when (cf. the books of P. Czerwik [18], D.H. Hyers, G. Isac and Th.M. Rassias [19]).
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 [20] for mappings , where X is a normed space and Y is a Banach space. Cholewa [21] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [22] proved the 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 [23–29]).
In [30], Jun and Kim considered the following cubic functional equation:
It is easy to show that the function 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.
In [31], Lee et al. considered the following quartic functional equation:
It is easy to show that the function 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 paper, assume that is a Fréchet space and that is a Banach space.
In this paper, we prove the Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation
in paranormed spaces by using the fixed point method and direct method.
One can easily show that an odd mapping satisfies (1.3) if and only if the odd mapping is an additive-cubic mapping, i.e.,
It was shown in [32], Lemma 2.2] that and are cubic and additive, respectively, and that .
One can easily show that an even mapping satisfies (1.3) if and only if the even mapping is a quadratic-quartic mapping, i.e.,
It was shown in [33], Lemma 2.1] that and 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
Using the fixed point method and direct method, we prove the Hyers-Ulam stability of the functional equation in paranormed spaces: an odd mapping case.
Let S be a set. A function is called a generalized metric on S if m satisfies
-
(1)
if and only if ;
-
(2)
for all ;
-
(3)
for all .
We recall a fundamental result in the fixed point theory.
Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either
for all nonnegative integers n or there exists a positive integer such that
-
(1)
, ;
-
(2)
the sequence converges to a fixed point of J;
-
(3)
is the unique fixed point of J in the set ;
-
(4)
for all .
In 1996, Isac and Th.M. Rassias [36] 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 [37–45]).
Note that for all .
Theorem 2.2 Let be a function such that there exists an with
for all . Let be an odd mapping such that
for all . Then there exists a unique additive mapping such that
for all .
Proof Letting in (2.2), we get
for all .
Replacing x by 2y in (2.2), we get
for all .
By (2.4) and (2.5),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-1847-2012-148/MediaObjects/13662_2012_Article_286_Equ9_HTML.gif)
for all . Replacing y by x and letting in (2.6), we get
for all .
Consider the set
and introduce the generalized metric on S:
where, as usual, . It is easy to show that is complete (see [44], Lemma 2.1]).
Now we consider the linear mapping such that
for all .
Let be given such that . Since
for all , implies that . This means that
for all .
It follows from (2.7) that .
By Theorem 2.1, there exists a mapping satisfying the following:
-
(1)
A is a fixed point of J, i.e.,
(2.8)for all . The mapping A is a unique fixed point of J in the set
This implies that A is a unique mapping satisfying (2.8) such that there exists a satisfying
for all ;
-
(2)
as . This implies the equality
for all ;
-
(3)
, which implies the inequality
This implies that the inequality (2.3) holds true.
It follows from (2.1) and (2.2) that
for all . So for all . By [32], Lemma 2.2], is an additive mapping, as desired. □
Corollary 2.3 Let r be a positive real number with , and let be an odd mapping such that
for all . Then there exists a unique additive mapping such that
for all .
Proof Taking for all and choosing in Theorem 2.2, we get the desired result. □
Theorem 2.4 Let be a function such that
for all . Let be an odd mapping satisfying (2.2). Then there exists a unique additive mapping such that
for all .
Proof The proof is similar to the proof of [45], Theorem 2.2]. □
Remark 2.5 Let . Letting for all in Theorem 2.4, we obtain the inequality (2.10). The proof is given in [45], Theorem 2.2].
Theorem 2.6 Let be a function such that there exists an with
for all . Let be an odd mapping such that
for all . Then there exists a unique additive mapping such that
for all .
Proof Letting in (2.12), we get
for all .
Replacing x by 2y in (2.12), we get
for all .
By (2.14) and (2.15),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-1847-2012-148/MediaObjects/13662_2012_Article_286_Equ19_HTML.gif)
for all . Replacing y by and letting in (2.16), we get
for all .
Consider the set
and introduce the generalized metric on S:
where, as usual, . It is easy to show that is complete (see [44], Lemma 2.1]).
Now we consider the linear mapping such that
for all .
Let be given such that . Since
for all , implies that . This means that
for all .
It follows from (2.17) that .
By Theorem 2.1, there exists a mapping satisfying the following:
-
(1)
A is a fixed point of J, i.e.,
(2.18)for all . The mapping A is a unique fixed point of J in the set
This implies that A is a unique mapping satisfying (2.18) such that there exists a satisfying
for all ;
-
(2)
as . This implies the equality
for all ;
-
(3)
, which implies the inequality
This implies that the inequality (2.13) holds true.
It follows from (2.11) and (2.12) that
for all . So for all . By [32], Lemma 2.2], is an additive mapping, as desired. □
Corollary 2.7 Let r, θ be positive real numbers with , and let be an odd mapping such that
for all . Then there exists a unique additive mapping such that
for all .
Proof Taking for all and choosing in Theorem 2.6, we get the desired result. □
Theorem 2.8 Let be a function such that
for all . Let be an odd mapping satisfying (2.12). Then there exists a unique additive mapping such that
for all .
Proof The proof is similar to the proof of [45], Theorem 2.1]. □
Remark 2.9 Let . Letting for all in Theorem 2.8, we obtain the inequality (2.20). The proof is given in [45], Theorem 2.1].
Theorem 2.10 Let be a function such that there exists an with
for all . Let be an odd mapping satisfying (2.2). Then there exists a unique cubic mapping such that
for all .
Proof Replacing y by x and letting in (2.6), we get
for all .
Consider the set
and introduce the generalized metric on S:
where, as usual, . It is easy to show that is complete (see [44], Lemma 2.1]).
Now we consider the linear mapping such that
for all .
Let be given such that . Since
for all , implies that . This means that
for all .
It follows from (2.23) that .
By Theorem 2.1, there exists a mapping satisfying the following:
-
(1)
C is a fixed point of J, i.e.,
(2.24)for all . The mapping C is a unique fixed point of J in the set
This implies that C is a unique mapping satisfying (2.24) such that there exists a satisfying
for all ;
-
(2)
as . This implies the equality
for all ;
-
(3)
, which implies the inequality
This implies that the inequality (2.22) holds true.
It follows from (2.2) and (2.21) that
for all . So for all . By [32], Lemma 2.2], is a cubic mapping, as desired. □
Corollary 2.11 Let r be a positive real number with , and let be an odd mapping satisfying (2.9). Then there exists a unique cubic mapping such that
for all .
Proof Taking for all and choosing in Theorem 2.10, we get the desired result. □
Theorem 2.12 Let be a function such that
for all . Let be an odd mapping satisfying (2.2). Then there exists a unique cubic mapping such that
for all .
Proof The proof is similar to the proof of [45], Theorem 2.4]. □
Remark 2.13 Let . Letting for all in Theorem 2.12, we obtain the inequality (2.25). The proof is given in [45], Theorem 2.4].
Theorem 2.14 Let be a function such that there exists an with
for all . Let be an odd mapping satisfying (2.12). Then there exists a unique cubic mapping such that
for all .
Proof Replacing y by and letting in (2.16), we get
for all .
Consider the set
and introduce the generalized metric on S:
where, as usual, . It is easy to show that is complete (see [44], Lemma 2.1]).
Now we consider the linear mapping such that
for all .
Let be given such that . Since
for all , implies that . This means that
for all .
It follows from (2.28) that .
By Theorem 2.1, there exists a mapping satisfying the following:
-
(1)
C is a fixed point of J, i.e.,
(2.29)for all . The mapping C is a unique fixed point of J in the set
This implies that C is a unique mapping satisfying (2.29) such that there exists a satisfying
for all ;
-
(2)
as . This implies the equality
for all ;
-
(3)
, which implies the inequality
This implies that the inequality (2.27) holds true.
It follows from (2.12) and (2.26) that
for all . So for all . By [32], Lemma 2.2], is a cubic mapping, as desired. □
Corollary 2.15 Let r, θ be positive real numbers with , and let be an odd mapping satisfying (2.19). Then there exists a unique cubic mapping such that
for all .
Proof Taking for all and choosing in Theorem 2.14, we get the desired result. □
Theorem 2.16 Let be a function such that
for all . Let be an odd mapping satisfying (2.12). Then there exists a unique additive mapping such that
for all .
Proof The proof is similar to the proof of [45], Theorem 2.3]. □
Remark 2.17 Let . Letting for all in Theorem 2.16, we obtain the inequality (2.30). The proof is given in [45], Theorem 2.3].
3 Hyers-Ulam stability of the functional equation (1.3): an even mapping case
Using the fixed point method and direct method, we prove the Hyers-Ulam stability of the functional equation in paranormed spaces: an even mapping case.
Note that for all .
Theorem 3.1 Let be a function such that there exists an with
for all . Let be an even mapping satisfying and (2.2). Then there exists a unique quadratic mapping such that
for all .
Proof Letting in (2.2), we get
for all .
Replacing x by 2y in (2.2), we get
for all .
By (3.1) and (3.2),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-1847-2012-148/MediaObjects/13662_2012_Article_286_Equ36_HTML.gif)
for all . Replacing y by x and in (3.3), we get
for all .
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 3.2 Let r be a positive real number with , and let be an even mapping satisfying and (2.9). Then there exists a unique quadratic mapping such that
for all .
Proof Taking for all and choosing in Theorem 3.1, we get the desired result. □
Theorem 3.3 Let be a function such that
for all . Let be an even mapping satisfying and (2.2). Then there exists a unique quadratic mapping such that
for all .
Proof The proof is similar to the proof of [45], Theorem 3.2]. □
Remark 3.4 Let . Letting for all in Theorem 3.3, we obtain the inequality (3.4). The proof is given in [45], Theorem 3.2].
Theorem 3.5 Let be a function such that there exists an with
for all . Let be an even mapping satisfying and (2.12). Then there exists a unique quadratic mapping such that
for all .
Proof Letting in (2.12), we get
for all .
Replacing x by 2y in (2.12), we get
for all .
By (3.5) and (3.6),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-1847-2012-148/MediaObjects/13662_2012_Article_286_Equ40_HTML.gif)
for all . Replacing y by and in (3.7), we get
for all .
The rest of the proof is similar to the proof of Theorem 2.6. □
Corollary 3.6 Let r, θ be positive real numbers with , and let be an even mapping satisfying and (2.19). Then there exists a unique quadratic mapping such that
for all .
Proof Taking for all and choosing in Theorem 3.5, we get the desired result. □
Theorem 3.7 Let be a function such that
for all . Let be an even mapping satisfying and (2.12). Then there exists a unique quadratic mapping such that
for all .
Proof The proof is similar to the proof of [45], Theorem 3.1]. □
Remark 3.8 Let . Letting for all in Theorem 3.7, we obtain the inequality (3.8). The proof is given in [45], Theorem 3.1].
Theorem 3.9 Let be a function such that there exists an with
for all . Let be an even mapping satisfying and (2.2). Then there exists a unique quartic mapping such that
for all .
Proof Replacing y by x and letting in (3.3), we get
for all .
The rest of the proof is similar to the proof of Theorem 2.10. □
Corollary 3.10 Let r be a positive real number with , and let be an even mapping satisfying and (2.9). Then there exists a unique quartic mapping such that
for all .
Proof Taking for all and choosing in Theorem 3.9, we get the desired result. □
Theorem 3.11 Let be a function such that
for all . Let be an even mapping satisfying and (2.2). Then there exists a unique quartic mapping such that
for all .
Proof The proof is similar to the proof of [45], Theorem 3.4]. □
Remark 3.12 Let . Letting for all in Theorem 3.11, we obtain the inequality (3.9). The proof is given in [45], Theorem 3.4].
Theorem 3.13 Let be a function such that there exists an with
for all . Let be an even mapping satisfying and (2.12). Then there exists a unique quartic mapping such that
for all .
Proof Replacing y by and letting in (3.7), we get
for all .
The rest of the proof is similar to the proof of Theorem 2.14. □
Corollary 3.14 Let r, θ be positive real numbers with , and let be an even mapping satisfying and (2.19). Then there exists a unique quartic mapping such that
for all .
Proof Taking for all and choosing in Theorem 3.13, we get the desired result. □
Theorem 3.15 Let be a function such that
for all . Let be an even mapping satisfying and (2.12). Then there exists a unique quartic mapping such that
for all .
Proof The proof is similar to the proof of [45], Theorem 3.3]. □
Remark 3.16 Let . Letting for all in Theorem 3.15, we obtain the inequality (3.10). The proof is given in [45], Theorem 3.3].
We can summarize the corollaries as follows.
Let and . Then is odd and is even. , satisfy the functional equation (1.3). Let and . Then . Let and . Then . Thus
Theorem 3.17 Let r be a positive real number with . Let be a mapping satisfying and (2.9). Then there exist an additive mapping , a quadratic mapping , a cubic mapping and a quartic mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-1847-2012-148/MediaObjects/13662_2012_Article_286_Equbw_HTML.gif)
for all .
Theorem 3.18 Let r, θ be positive real numbers with . Let be a mapping satisfying and (2.19). Then there exist an additive mapping , a quadratic mapping , a cubic mapping and a quartic mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-1847-2012-148/MediaObjects/13662_2012_Article_286_Equbx_HTML.gif)
for all .
4 Conclusions
Using the fixed point method and direct method, we have proved the Hyers-Ulam stability of an additive-quadratic-cubic-quartic functional equation in paranormed spaces.
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Park, C. Stability of an AQCQ-functional equation in paranormed spaces. Adv Differ Equ 2012, 148 (2012). https://doi.org/10.1186/1687-1847-2012-148
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DOI: https://doi.org/10.1186/1687-1847-2012-148