Hyers-Ulam stability of a finite variable mixed type quadratic-additive functional equation in quasi-Banach spaces

1.   Introduction

The stability problem of functional equations originated from a question of Ulam ^[27] in 1940, concerning the stability of group homomorphisms. Let $(G_{1}, \cdot)$ be a group and let $(G_{2}, *)$ be a metric group with the metric $d(\cdot, \cdot)$. Given $\epsilon > 0$, does there exist a $\delta > 0$, such that if a mapping $h:G_{1}\rightarrow G_{2}$ satisfies the inequality $d(h(x, y), h(x)*h(y)) < \delta$ for all $x, y \in G_{1}$, then there exists a homomorphism $H:G_{1} \rightarrow G_{2}$ with $d(h(x), H(x)) < \epsilon$ for all $x \in G_{1}$? 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 ^[12] considered the case of approximately additive mappings $f:E \rightarrow E^{'}$, where $E$ and $E^{'}$ are Banach spaces and $f$ satisfies Hyers inequality

for all $x, y \in E$. It was shown that the limit

exists for all $x \in E$ and that $L:E \rightarrow E^{'}$ is the unique additive mapping satisfying

In 1978, Rassias ^[23] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded.

Quadratic functional equation was used to characterize inner product spaces ^[1,2,13]. A square norm on an inner product space satisfies the important parallelogram equality

The functional equation

is related to a symmetric bi-additive mapping ^[1,16]. It is natural that each equation is called a quadratic functional equation. In particular, every solution of the quadratic Eq (1.1) is said to be a quadratic mapping. It is well known that a mapping $f$ between real vector spaces is quadratic if and only if there exists a unique symmetric bi-additive mapping $B$ such that $f(x) = B(x, x)$ for all $x$ (see ^[1,16]). The bi-additive mapping $B$ is given by

A Hyers-Ulam stability problem for the quadratic functional Eq (1.1) was proved by Skof ^[25] for mappings $f:E_{1} \rightarrow E_{2}$ where $E_{1}$ is a normed space and $E_{2}$ is a Banach space (^[16]). Cholewa ^[4] noticed that the theorem of Skof is still true if the relevant domain $E_{1}$ is replaced by an Abelian group. In ^[5], Czerwik proved the Hyers-Ulam stability of the quadratic functional Eq (1.1). Grabiec ^[11] generalized these results mentioned above.

Elqorachi and M. Th. Rassias ^[6] have been extensively studied the Hyers-Ulam stability of the generalized trigonometric functional equations

where $S$ is a semigroup, $\sigma :S \rightarrow S$ is an involutive morphism, and $\mu:S \rightarrow \mathbb{C}$ is a multiplicative function such that $\mu(x\sigma(x)) = 1$ for all $x \in S$. Jung ^[19] proved the stability theorems for $n$-dimensional quartic-cubic-quadratic-additive type functional equations of the form $\sum_{i = 1}^{l}c_{i}f(a_{i1}x_{1}+a_{i2}x_{2}+\cdots+a_{in}x_{n}) = 0$ by applying the direct method. These stability theorems can save us the trouble of proving the stability of relevant solutions repeatedly appearing in the stability problems for various functional equations. Lee ^[18] introduced general quintic functional equation and general sextic functional equations such as the additive functional equation and the quadratic functional equation. He investigated the Hyers-Ulam stability results. Kayal et al. ^[24] established the Hyers-Ulam stability results belonging to two different set valued functional equations in several variables, namely, additive and cubic. The results were obtained in the contexts of Banach spaces. See ^[10,15,20] for more information on functional equations and their stability.

Jun and Kim ^[14] obtained the Hyers-Ulam stability for a mixed type of cubic and additive functional equations. In addition theHyers-Ulam for a mixed type of quadratic and additive functional equations

in quasi-Banach spaces have been investigated by Najati and Moghimi ^[21]. Najati and Eskandani ^[22] introduced the following functional equation

It is easy to see that the function $f(x) = ax^{3}+bx$ is a solution of the functional Eq (1.6). They established the general solution and the Hyers-Ulam stability for the functional Eq (1.6) in quasi-Banach spaces. In 2009, Eshaghi Gordji et al. ^[7] introduced the following mixed type cubic, quadratic and additive functional equations for a fixed integer $k$ with $k\neq 0, \pm 1$:

and proved the function $f(x) = ax^{3}+bx^{2}+cx$ is a solution of the functional Eq (1.7). They investigated the general solution of (1.7) in vector spaces, and established the Hyers-Ulam stability of the functional Eq (1.7) in quasi-Banach spaces.

In this paper, we introduce the following mixed type finite variable functional equation deriving from quadratic and additive functions

where $\phi(0) = 0$ and $l \geq 4$ is a fixed positive integer, which generalizes a quadratic-additive functional equation given in ^[17,21]. It is easy to see that the function $\phi(t) = at^2+bt$ is a solution of the functional Eq (1.8). The primary goal of this paper is to obtain the general solution of the functional Eq (1.8) and investigate the Hyers-Ulam stability for the functional Eq (1.8) in quasi-Banach spaces. Our results generalize the results given by Najati and Moghimi ^[21].

Definition 1.1. (^[3]) Let $X$ be a real linear space. A quasi-norm is a real-valued function on $X$ satisfying the following:

(ⅰ) $\left\|x\right\| \geq 0$ for all $x \in X$ and $\left\|x\right\| = 0$ if and only if $x = 0$.

(ⅱ) $\left\|\lambda x\right\| = \vert \lambda \vert \left\|x\right\|$ for all $\lambda \in \mathbb{R}$ and all $x \in X$.

(ⅲ) There is a constant $K \geq 1$ such that $\left\|x+y\right\| \leq K \left(\left\|x\right\|+\left\|y\right\|\right)$ for all $x, y \in X$.

It follows from condition (ⅲ) that

for all integers $n \geq 1$ and all $x_{1}, x_{2}, \cdots, x_{2n+1} \in X$.

The pair $(X, \left\|\cdot\right\|)$ is called a quasi-normed space if $\left\|\cdot\right\|$ is a quasi-norm on $X$. The smallest possible $K$ is called the modulus of concavity of $\left\|\cdot\right\|$. A quasi-Banach space is a complete quasi-normed space.

A quasi-norm $\left\|\cdot\right\|$ is called a p-norm $(0 < p\leq 1)$ if

for all $x, y \in X$. In this case, a quasi-Banach space is called a p-Banach space.

Given a $p$-norm, the formula $d(x, y): = \left\|x-y\right\|^{p}$ gives us a translation invariant metric on $X$. By the Aoki-Rolewicz Theorem (see ^[3]), each quasi-norm is equivalent to some $p$-norm. Since it is much easier to work with $p$-norms, we restrict our attention mainly to $p$-norms. Moreover in ^[26], Tabor investiagted a version of Hyers-Ulam theorem in quasi-Banach spaces (see ^[8,9]).

2.   Solution of the functional Eq (1.8)

Throughout this section, $P$ and $Q$ will be real vector spaces.

Lemma 2.1. If an odd mapping $\phi:P \rightarrow Q$ satisfies (1.8) for all $t_{1}, t_{2}, \cdots, t_{l} \in P$, then $\phi$ is additive.

Proof. In the view of the oddness of $\phi$, we have $\phi(-t) = -\phi(t)$ for all $t \in P$. Now, (1.8) becomes

Setting $(t_{1}, t_{2}, \cdots, t_{l}) = (0, 0, \cdots, 0)$ in (2.1), we get $\phi(0) = 0$. Now, letting $(t_{1}, t_{2}, \cdots, t_{l}) = (t, 0, \cdots, 0)$ in (2.1), we obtain

for all $t \in P$. Replacing $t$ by $2t$ in (2.2), we get

for all $t \in P$. Again replacing $t$ by $2t$ in (2.3), we have

for all $t \in P$. In general, for any positive integer $l$, we obtain

for all $t \in P$. Therefore, (2.1) now becomes

for all $t_{1}, t_{2}, \cdots, t_{l} \in P$. Replacing $(t_{1}, t_{2}, \cdots, t_{l})$ by $(x, y, x, y, 0, \cdots, 0)$ in (2.4), we get

for all $x, y \in P$. Therefore the mapping $\phi:P \rightarrow Q$ is additive.

Lemma 2.2. If an even mapping $\phi:P \rightarrow Q$ satisfies $\phi(0) = 0$ and (1.8) for all $t_{1}, t_{2}, \cdots, t_{l} \in P$, then $\phi$ is quadratic.

Proof. In view of the evenness of $\phi$, we have $\phi(-t) = \phi(t)$ for all $t \in P$. Now, (1.8) becomes

for all $t_{1}, t_{2}, \cdots, t_{l} \in P$. Replacing $(t_{1}, t_{2}, \cdots, t_{l})$ by $(t, 0, \cdots, 0)$ in (2.5), we obtain

for all $t \in P$. Replacing $t$ by $2t$ in (2.6), we have

for all $t \in P$. Replacing $t$ by $2t$ in (2.7), we obtain

for all $t \in P$. In general, for any positive integer $l$, we get

for all $t \in P$. Therefore, (2.5) becomes

for all $t_{1}, t_{2}, \cdots, t_{l} \in P$. Replacing g $(t_{1}, t_{2}, \cdots, t_{l})$ by $(x, y, -x, -y, 0, \cdots, 0)$ in (2.8), we get

for all $x, y \in P$. Therefore the mapping $\phi:P\rightarrow Q$ is quadratic.

Lemma 2.3. A mapping $\phi:P \rightarrow Q$ satisfies $\phi(0) = 0$ and (1.8) for all $t_{1}, t_{2}, \cdots, t_{l} \in P$ if and only if there exist a symmetric bi-additive mapping $B:P \times P \rightarrow Q$ and an additive mapping $A: P \rightarrow Q$ such that $\phi(t) = B(t, t)+A(t)$ for all $t \in P$.

Proof. Let $\phi$ with $\phi(0) = 0$ satisfy (1.8). We decompose $\phi$ into the even part and odd part by putting

for all $t \in P$. It is clear that $\phi(t) = \phi_{e}(t)+\phi_{o}(t)$ for all $t \in P$. It is easy to show that the mappings $\phi_{e}$ and $\phi_{o}$ satisfy (1.8). Hence by Lemmas 2.1 and 2.2, we obtain that $\phi_{e}$ and $\phi_{o}$ are quadratic and additive, respectively. Therefore, there exists a symmetric bi-additive mapping $B:P \times P \rightarrow Q$ such that $\phi_{e}(t) = B(t, t)$ for all $t \in P$. So $\phi(t) = B(t, t)+A(t)$ for all $t \in P$, where $A(t) = \phi_{o}(t)$ for all $t \in P$.

Conversely, assume that there exist a symmetric bi-additive mapping $B: P \times P \rightarrow Q$ and an additive mapping $A: P \rightarrow Q$ such that $\phi(t) = B(t, t)+A(t)$ for all $t \in P$. By a simple computation one can show that the mappings $t \mapsto B(t, t)$ and $A$ satisfy the functional Eq (1.8). So the mapping $\phi$ satisfies (1.8).

3.   Hyers-Ulam stability of (1.8)

Throughout this section, assume that $E$ is a quasi-Banach space with quasi-norm $\left\| \cdot \right\|$ and that $F$ is a $p-$Banach space with $p$-norm $\left\| \cdot \right\|$. Let $K$ be the modulus of concavity of $\left\| \cdot \right\|$.

In this section, using an idea of Gavruta we prove the Hyers-Ulam stability of the functional Eq (1.8) in the spirit of Hyers, Ulam and Rassias. For convenience, we use the following abbreviation for a given mapping $\phi:E \rightarrow F:$

for all $t_{1}, t_{2}, \cdots, t_{l} \in E$.

We will use the following lemma in this section.

Lemma 3.1. ^[21] Let $0 \leq p \leq 1$ and let $x_{1}, x_{2}, \cdots, x_{n}$ be nonnegative real numbers. Then

Theorem 3.2. Let $v \in \{-1, 1\}$ be fixed and let $\chi:E^{l} \rightarrow [0, \infty)$ be a function such that

for all $t_{1}, t_{2}, \cdots, t_{l} \in E$ and

for all $t \in E$. Suppose that an even mapping $\phi:E \rightarrow F$ with $\phi(0) = 0$ satisfies the inequality

for all $t_{1}, t_{2}, \cdots, t_{l} \in E$. Then the limit

exists for all $t \in E$ and $\Phi:E \rightarrow F$ is a unique quadratic mapping satisfying

for all $t \in E$.

Proof. Let $v = 1$. Replacing $(t_{1}, t_{2}, \cdots, t_{l})$ by $(t, 0, \cdots, 0)$ in (3.3), we obtain

for all $t \in E$. Let us take $\psi_{e}(t) = \chi(t, 0, \cdots, 0)$ for all $t \in E$. Then by (3.6), we have

for all $t \in E$. If we replace $t$ by $\frac{t}{2^{l+1}}$ in (3.7) and multiply both sides of (3.7) by $2^{2l}$, then we get

for all $t \in E$ and all nonnegative integers $l$. Since $F$ is a $p$-Banach space, by (3.8) we obtain

for all nonnegative integers $l$ and $k$ with $l \geq k$ and all $t \in E$. Since $\psi_{e}^{p}(t) = \chi^{p}(t, 0, \cdots, 0)$ for all $t \in E$, by (3.2), we have

for all $t \in E$. Therefore, it follows from (3.9) and (3.10) that the sequence $\{2^{2l}\phi\left(\frac{t}{2^{l}}\right)\}$ is a Cauchy sequence for each $t \in E$. Since $F$ is complete, the sequence $\{2^{2l}\phi\left(\frac{t}{2^{l}}\right)\}$ converges for each $t \in E$. So one can define the mapping $\Phi:E \rightarrow F$ given by (3.4) for all $t \in E$. Letting $k = 0$ and passing the limit $l \rightarrow \infty$ in (3.9), we have

for all $t \in E$. Therefore, (3.5) follows from (3.2) and (3.11). Now, we show that $\Phi$ is quadratic. It follows from (3.1), (3.3) and (3.4) that

for all $t_{1}, t_{2}, \cdots, t_{l} \in E$. Therefore, the mapping $\Phi:E \rightarrow F$ satisfies (1.8). Since $\phi$ is an even mapping, (3.4) implies that the mapping $\Phi:E \rightarrow F$ is even. Therefore, by Lemma 2.2, we get that the mapping $\Phi:E\rightarrow F$ is quadratic.

To prove the uniqueness of $\Phi$, let $\Phi^{'}:E \rightarrow F$ be another quadratic mapping satisfying (3.5). Since

for all $t \in E$,

for all $t \in E$. Therefore, it follows from (3.5) and the last equation that

for all $t \in E$. Hence $\Phi = \Phi^{'}$.

For $v = -1$, we can prove this theorem by a similar manner.

Corollary 3.3. Let $\lambda$ and $r_{1}, r_{2}, \cdots, r_{l}$ be nonnegative real numbers such that $r_{1}, r_{2}, \cdots, r_{l} > 2$ or $0 \leq r_{1}, r_{2}, \cdots, r_{l} < 2$. Suppose that an even mapping $\phi:E \rightarrow F$ with $\phi(0) = 0$ satisfies the inequality

for all $t_{1}, t_{2}, \cdots, t_{l} \in E$. Then there exists a unique quadratic mapping $\phi:E \rightarrow F$ satisfying

for all $t \in E$.

Proof. It follows from Theorem 3.2.

Theorem 3.4. Let $v \in \{-1, 1\}$ be fixed and let $\chi:E^{l} \rightarrow [0, \infty)$ be a function such that

for all $t_{1}, t_{2}, \cdots, t_{l} \in E$ and

for all $t \in E$. Suppose that an odd mapping $\phi:E \rightarrow F$ satisfies the inequality

for all $t_{1}, t_{2}, \cdots, t_{l} \in E$. Then the limit

exists for all $t \in E$ and $\Psi:E \rightarrow F$ is a unique additive mapping satisfying

for all $t \in E$.

Proof. Let $v = 1$. Replacing $(t_{1}, t_{2}, \cdots, t_{l})$ by $(t, 0, \cdots, 0)$ in (3.15), we obtain

for all $t \in E$. Let us take $\psi_{o}(t) = \chi(t, 0, \cdots, 0)$ for all $t \in E$. Then by (3.18), we have

for all $t \in E$. If we replace $t$ by $\frac{t}{2^{l+1}}$ in (3.19) and multiply both sides of (3.19) by $2^{l}$, then we get

for all $t \in E$ and all nonnegative integers $l$. Since $F$ is a $p$-Banach space, by (3.20), we obtain

for all nonnegative integers $l$ and $k$ with $l \geq k$ and all $t \in E$. Since $\psi_{o}^{p}(t) = \chi^{p}(t, 0, \cdots, 0)$ for all $t \in E$, by (3.14) we have

for all $t \in E$. Therefore, it follows from (3.21) and (3.22) that the sequence $\{2^{l}\phi\left(\frac{t}{2^{l}}\right)\}$ is a Cauchy sequence for all $t \in E$. Since $F$ is complete, the sequence $\{2^{l}\phi\left(\frac{t}{2^{l}}\right)\}$ converges for all $t \in E$. So one can define the mapping $\Psi:E \rightarrow F$ given by (3.16) for all $t \in E$. Letting $k = 0$ and passing the limit $l \rightarrow \infty$ in (3.21), we have

for all $t \in E$. Therefore, (3.17) follows from (3.14) and (3.23). Now, we show that $\Psi$ is additive. It follows from (3.20), (3.22) and (3.17) that

for all $t \in E$. So $\Psi(2t) = 2\Psi(t)$ for all $t \in E$. On the other hand, it follows from (3.13), (3.15) and (3.16) that

for all $t_{1}, t_{2}, \cdots, t_{l} \in E$. Therefore, the mapping $\Psi:E \rightarrow F$ satisfies (1.8). Since $\phi$ is an odd mapping, (3.16) implies that the mapping $\Psi:E \rightarrow F$ is odd. Therefore, by Lemma 2.1, we get that the mapping $\psi:E \rightarrow F$ is additive.

To prove the uniqueness of $\Psi$, let $\Psi^{'}:E \rightarrow F$ be another additive mapping satisfying (3.17). Since

for all $t \in E$,

for all $t \in E$. Therefore, it follows from (3.17) and the last equation that

for all $t \in E$. Hence $\Psi = \Psi^{'}$.

For $v = -1$, we can prove this theorem by a similar manner.

Corollary 3.5. Let $\lambda$ and $r_{1}, r_{2}, \cdots, r_{l}$ be nonnegative real numbers such that $r_{1}, r_{2}, \cdots, r_{l} > 1$ or $0 \leq r_{1}, r_{2}, \cdots, r_{l} < 1$. Suppose that an odd mapping $\phi:E \rightarrow F$ satisfies the inequality

for all $t_{1}, t_{2}, \cdots, t_{l} \in E$. Then there exists a unique additive function $\phi:E \rightarrow F$ satisfying

for all $t \in E$.

Proof. It follows from Theorem 3.4.

Proposition 3.6. Let $\chi:E^{l} \rightarrow [0, \infty)$ be a function which satisfies (3.1) and (3.2) for all $t_{1}, t_{2}, \cdots, t_{l} \in E$ and satisfies (3.13) and (3.14) for all $t_{1}, t_{2}, \cdots, t_{l} \in E$. Suppose that a mapping $\phi:E \rightarrow F$ with $\phi(0) = 0$ satisfies the inequality (3.3) for all $t_{1}, t_{2}, \cdots, t_{l} \in E$. Then there exist a unique quadratic mapping $\Phi:E \rightarrow F$ and a unique additive mapping $\Psi:E \rightarrow F$ satisfying (1.8) and

for all $t \in E$, where $\tilde{\psi}_{e}(t)$ and $\tilde{\psi}_{o}(t)$ were defined in (3.2) and (3.14), respectively, for all $t \in E$.

Proof. Let $\phi_{o}(t) = \frac{\phi(t)-\phi(-t)}{2}$ for all $t \in E$. Then

for all $t_{1}, t_{2}, \cdots, t_{l} \in E$. And let $\phi_{e}(t) = \frac{\phi(t)+\phi(-t)}{2}$ for all $t \in E$. Then

for all $t_{1}, t_{2}, \cdots, t_{l} \in E$. Let us define

for all $t \in E$. Now,

Using Theorems 3.2 and Theorem 3.4, we can prove the remaining proof of the theorem.

Corollary 3.7. Let $\lambda$ and $r_{1}, r_{2}, \cdots, r_{l}$ be nonnegative real numbers such that $r_{1}, r_{2}, \cdots, r_{l} \neq 2$ or $r_{1}, r_{2}, \cdots, r_{l} \neq 1$. Suppose that a mapping $\phi:E \rightarrow F$ with $\phi(0) = 0$ satisfies the inequality (3.12) for all $t_{1}, t_{2}, \cdots, t_{l} \in E$. Then there exists a unique quadratic mapping $\Phi:E \rightarrow F$ and a unique additive mapping $\Psi:E \rightarrow F$ satisfying (1.8) and

for all $t \in E$.

4.   Conclusions

We have introduced the mixed type finite variable additive-quadratic functional Eq (1.8) and have obtained the general solution of the mixed type finite variable additive-quadratic functional Eq (1.8) in quasi-Banach spaces. Furthermore, we have proved the Hyers-Ulam stability for the mixed type finite variable additive-quadratic functional Eq (1.8) in quasi-Banach spaces.

Conflict of interest

The authors declare that they have no competing interests.