On the Stability of a Generalized Quadratic and Quartic Type Functional Equation in Quasi-Banach Spaces

Abstract

We establish the general solution of the functional equation for fixed integers with and investigate the generalized Hyers-Ulam stability of this equation in quasi-Banach spaces.

1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. Let be a group and let be a metric group with the metric Given , does there exist a , such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that

(1.1)

for all and for some Then there exists a unique additive mapping such that

(1.2)

for all Moreover, if is continuous in for each fixed then is -linear. In 1978, Th. M. Rassias [3] provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded. The functional equation

(1.3)

is related to a symmetric biadditive mapping [4–7]. It is natural that this functional equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.3) is said to be a quadratic mapping. It is well known that a mapping between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive mapping such that for all (see [4, 7]). The biadditive mapping is given by

(1.4)

The generalized Hyers-Ulam stability problem for the quadratic functional equation (1.3) was proved by Skof for mappings , where is a normed space and is a Banach space (see [8]). Cholewa [9] noticed that the theorem of Skof is still true if relevant domain is replaced by an abelian group. In [10] , Czerwik proved the generalized Hyers-Ulam stability of the functional equation (1.3). Grabiec [11] has generalized these results mentioned above.

In [12], Park and Bae considered the following quartic functional equation:

(1.5)

In fact, they proved that a mapping between two real vector spaces and is a solution of (1.5) if and only if there exists a unique symmetric multiadditive mapping such that for all . It is easy to show that satisfies the functional equation (1.5), which is called a quartic functional equation (see also [13]).

In addition, Kim [14] has obtained the generalized Hyers-Ulam stability for a mixed type of quartic and quadratic functional equation between two real linear Banach spaces. Najati and Zamani Eskandani [15] have established the general solution and the generalized Hyers-Ulam stability for a mixed type of cubic and additive functional equation, whenever is a mapping between two quasi-Banach spaces (see also [16, 17]).

Now we introduce the following functional equation for fixed integers with :

(1.6)

in quasi-Banach spaces. It is easy to see that the function is a solution of the functional equation (1.6). In the present paper we investigate the general solution of the functional equation (1.6) when is a mapping between vector spaces, and we establish the generalized Hyers-Ulam stability of this functional equation whenever is a mapping between two quasi-Banach spaces.

We recall some basic facts concerning quasi-Banach space and some preliminary results.

Definition 1.1 (See [18, 19]).

Let be a real linear space. A quasinorm is a real-valued function on satisfying the following.

(1) for all and if and only if

(2) for all and all

(3)There is a constant such that for all

It follows from the condition (3) that

(1.7)

for all and all

The pair is called a quasinormed space if is a quasinorm on . The smallest possible is called the modulus of concavity of . A quasi-Banach space is a complete quasi-normed space.

A quasi-norm is called a -norm if

(1.8)

for all . In this case, a quasi-Banach space is called a -Banach space.

Given a -norm, the formula gives us a translation invariant metric on X. By the Aoki-Rolewicz theorem [19] (see also [18]), each quasi-norm is equivalent to some -norm. Since it is much easier to work with -norms, henceforth we restrict our attention mainly to -norms. In [20], Tabor has investigated a version of Hyers-Rassias-Gajda theorem (see [3, 21]) in quasi-Banach spaces.

2. General Solution

Throughout this section, and will be real vector spaces. We here present the general solution of (1.6).

Lemma 2.1.

If a mapping satisfies the functional equation (1.6), then is a quadratic and quartic mapping.

Proof.

Letting in (1.6), we get . Setting in (1.6), we get for all . So the mapping is even. Replacing by in (1.6) and then by in (1.6), we get

(2.1)

(2.2)

for all . Interchanging and in (1.6) and using the evenness of , we get the relation

(2.3)

for all . Replacing by in (1.6) and then using (2.3), we have

(2.4)

for all . If we add (2.1) to (2.2) and use (2.4), then we have

(2.5)

for all . Replacing by in (1.6) and then by in (1.6) and using the evenness of , we obtain that

(2.6)

(2.7)

for all . Interchanging with in (2.6) and (2.7) and using the evenness of , we get the relations

(2.8)

(2.9)

for all . Replacing by in (1.6) and then by in (1.6), we have

(2.10)

(2.11)

for all . Replacing by in (1.6), we obtain

(2.12)

for all . Adding (2.10) to (2.11) and using (2.8), (2.9), and (2.12), we get

(2.13)

for all . By (2.5) and (2.13), we obtain

(2.14)

for all . Interchanging and in (2.14) and using the evenness of , we get the relation

(2.15)

for all .

Now we show that (2.15) is a quadratic and quartic functional equation. To get this, we show that the mappings , defined by , and , defined by , are quadratic and quartic, respectively.

Replacing by in (2.15) and using the evenness of , we have

(2.16)

for all . Interchanging with in (2.16) and then using (2.15), we obtain by the evenness of

(2.17)

for all . By (2.17), we have

(2.18)

for all . This means that

(2.19)

for all . Thus the mapping is quadratic.

To prove that is quartic, we have to show that

(2.20)

for all . Replacing and by and in (2.15), respectively, we get

(2.21)

for all . Since for all and is a quadratic mapping, we have

(2.22)

for all . So it follows from (2.15), (2.21), and (2.22) that

(2.23)

for all . Thus is a quartic mapping.

Theorem 2.2.

A mapping satisfies (1.6) if and only if there exist a unique symmetric multiadditive mapping and a unique symmetric bi-additive mapping such that

(2.24)

for all

Proof.

We first assume that the mapping satisfies (1.6). Let be mappings defined by

(2.25)

for all By Lemma 2.1, we achieve that the mappings and are quadratic and quartic, respectively, and

(2.26)

for all Thus there exist a unique symmetric multiadditive mapping and a unique symmetric bi-additive mapping such that and for all (see citead, ki). So

(2.27)

for all

Conversely assume that

(2.28)

for all where the mapping is symmetric multi-additive and is bi-additive. By a simple computation, one can show that the mappings and satisfy the functional equation (1.6), so the mapping f satisfies (1.6).

3. Generalized Hyers-Ulam Stability of (1.6)

From now on, let and be a quasi-Banach space with quasi-norm and a -Banach space with -norm , respectively. Let be the modulus of concavity of . In this section, using an idea of Gvruta [22], we prove the stability of (1.6) in the spirit of Hyers, Ulam, and Rassias. For convenience we use the following abbreviation for a given mapping :

(3.1)

for all . Let . We will use the following lemma in this section.

Lemma 3.1 (see [15]).

Let and let be nonnegative real numbers. Then

(3.2)

Theorem 3.2.

Let be a function such that

(3.3)

for all and

(3.4)

for all and all Suppose that a mapping with satisfies the inequality

(3.5)

for all Then the limit

(3.6)

exists for all and is a unique quadratic mapping satisfying

(3.7)

for all where

(3.8)

Proof.

Setting in (3.5) and then interchanging and , we get

(3.9)

for all . Replacing by , , , and in (3.5), respectively, we get

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

for all . Combining (3.9) and (3.11)–(3.17), respectively, yields the following inequalities:

(3.18)

(3.19)

(3.20)

(3.21)

(3.22)

(3.23)

(3.24)

for all .

Replacing and by and in (3.5), respectively, we obtain

(3.25)

for all . Putting and instead of and in (3.5), respectively, we have

(3.26)

for all . It follows from (3.10), (3.18), (3.19), (3.20), (3.21), and (3.25) that

(3.27)

for all . Also, from (3.10), (3.18), (3.19), (3.22), (3.23), (3.24), and (3.26), we conclude

(3.28)

for all . Finally, combining (3.27) and (3.28) yields

(3.29)

for all . Let

(3.30)

Then the inequality (3.29) implies that

(3.31)

for all

Let be a mapping defined by for all From (3.31), we conclude that

(3.32)

for all If we replace in (3.32) by and multiply both sides of (3.32) by then we get

(3.33)

for all and all nonnegative integers . Since is a p-Banach space, the inequality (3.33) gives

(3.34)

for all nonnegative integers and with and all Since , by Lemma 3.1 and (3.30), we conclude that

(3.35)

for all Therefore, it follows from (3.4) and (3.35) that

(3.36)

for all It follows from (3.34) and (3.36) that the sequence is Cauchy for all Since is complete, the sequence converges for all So one can define the mapping by

(3.37)

for all Letting and passing the limit in (3.34), we get

(3.38)

for all Thus (3.7) follows from (3.4) and (3.38).

Now we show that is quadratic. It follows from (3.3), (3.33) and (3.37) that

(3.39)

for all So

(3.40)

for all On the other hand, it follows from (3.3), (3.5), (3.6) and (3.37) that

(3.41)

for all Hence the mapping satisfies (1.6). By Lemma 2.1, the mapping is quadratic. Hence (3.40) implies that the mapping is quadratic.

It remains to show that is unique. Suppose that there exists another quadratic mapping which satisfies (1.6) and (3.7). Since and for all , we conclude from (3.7) that

(3.42)

for all On the other hand, since

(3.43)

for all and all then

(3.44)

for all . Using (3.44) and (3.42), we get as desired.

Theorem 3.3.

Let be a function such that

(3.45)

for all and

(3.46)

for all and all Suppose that a mapping with satisfies the inequality

(3.47)

for all Then the limit

(3.48)

exists for all and is a unique quadratic mapping satisfying

(3.49)

for all x∈ X, where

(3.50)

Proof.

The proof is similar to the proof of Theorem 3.2.

Corollary 3.4.

Let be nonnegative real numbers such that or . Suppose that a mapping with satisfies the inequality

(3.51)

for all Then there exists a unique quadratic mapping satisfying

(3.52)

for all where

(3.53)

Proof.

In Theorem 3.2, putting for all , we get the desired result.

Corollary 3.5.

Let and be nonnegative real numbers such that . Suppose that a maping with satisfies the inequality

(3.54)

for all Then there exists a unique quadratic mapping satisfying

(3.55)

for all

Proof.

In Theorem 3.2, putting for all , we get the desired result.

Theorem 3.6.

Let be a function such that

(3.56)

for all and

(3.57)

for all and all Suppose that a mapping with satisfies the inequality

(3.58)

for all Then the limit

(3.59)

exists for all and is a unique quartic mapping satisfying

(3.60)

for all where

(3.61)

Proof.

Similar to the proof Theorem 3.2, we have

(3.62)

for all where

(3.63)

Let be a mapping defined by . Then we conclude that

(3.64)

for all If we replace in (3.65) by and multiply both sides of (3.65) by then we get

(3.65)

for all and all nonnegative integers . Since is a -Banach space, the inequality (3.66) gives

(3.66)

for all nonnegative integers and with and all Since , by Lemma 3.1, we conclude from (3.64) that

(3.67)

for all It follows from (3.57) and (3.67) that

(3.68)

for all Thus we conclude from (3.67) and (3.69) that the sequence is Cauchy for all Since is complete, the sequence converges for all So one can define the mapping by

(3.69)

for all Letting and passing the limit in (3.67), we get

(3.70)

for all Thus (3.60) follows from (3.58) and (3.70).

Now we show that is quartic. From (3.57), (3.66), and (3.70), it follows that

(3.71)

for all So

(3.72)

for all On the other hand, by (3.59), (3.69), and (3.70), we have

(3.73)

for all Hence the mapping satisfies (1.6). By Lemma 2.1, the mapping is quartic. Therefore, (3.75) implies that the mapping is quartic.

To prove the uniqueness property of let be another quartic mapping satisfying (3.61). Since

(3.74)

for all and all then

(3.75)

for all . It follows from (3.61) and (3.86) that

(3.76)

for all So as desired.

Theorem 3.7.

Let be a function such that

(3.77)

for all and

(3.78)

for all and all Suppose that a mapping with satisfies the inequality

(3.79)

for all Then the limit

(3.80)

exists for all and is a unique quartic mapping satisfying

(3.81)

for all where

(3.82)

Proof.

The proof is similar to the proof of Theorem 3.6.

Corollary 3.8.

Let be nonnegative real numbers such that or . Suppose that a mapping with satisfies the inequality(3.51)for all Then there exists a unique quartic mapping satisfying

(3.83)

for all where

(3.84)

Proof.

In Theorem 3.6, putting for all , we get the desired result.

Corollary 3.9.

Let and be nonnegative real numbers such that . Suppose that a mapping with satisfies the inequality (3.56) for all Then there exists a unique quartic mapping satisfying

(3.85)

for all

Proof.

In Theorem 3.6, putting for all , we get the desired result.

Theorem 3.10.

Let be a function such that

(3.86)

for all and

(3.87)

for all and all . Suppose that a mapping with satisfies the inequality

(3.88)

for all Then there exist a unique quadratic mapping and a unique quartic mapping such that

(3.89)

for all where and are defined in Theorems 3.2 and 3.7, respectively.

Proof.

By Theorems 3.2 and 3.7, there exist a quadratic mapping and a quartic mapping such that

(3.90)

for all It follows from the last inequalities that

(3.91)

for all So we obtain (3.92) by letting and for all

To prove the uniqueness property of and we first show the uniqueness property for and and then we conclude the uniqueness property of and Let be another quadratic and quartic mappings satisfying (3.92) and let , , and So

(3.92)

for all Since

(3.93)

for all (3.62) implies that for all Thus But is only a quartic function and is only a quadratic function.

Therefore, we have and this completes the uniqueness property of and We can prove the other results similarly.

Corollary 3.11.

Let be nonnegative real numbers such that or or . Suppose that a mapping satisfies the inequality(3.51)for all Then there exist a unique quadratic mapping and a unique quartic mapping such that

(3.94)

for all where , and are defined as in Corollaries 3.4 and 3.8.

Corollary 3.12.

Let and be nonnegative real numbers such that . Suppose that a mapping satisfies the inequality (3.56) for all Then there exist a unique quadratic mapping and a unique quartic function such that

(3.95)

for all