Ulam stability of a functional equation deriving from quadratic and additive mappings in random normed spaces

1.   Introduction

In 1940, Ulam ^[32] raised a question concerning the stability of homomorphisms: Given a group $G_{1}$, a metric group $G_{2}$ with the metric $d(\cdot, \cdot)$, and a nonnegative real number $\epsilon$, does there exist a $\delta > 0$ such that if a mapping $f:G_{1}\rightarrow G_{2}$ satisfies the inequality

for all $x, y \in G_{1}$ then there exists a homomorphism $F:G_{1}\rightarrow G_{2}$ with

for all $x \in G_{1}$? As mentioned above, when this problem has a solution, we say that the homomorphisms from $G_{1}$ to $G_{2}$ are stable.

In 1941, Hyers ^[12] gave a partial solution of Ulam's problem for the case of approximately additive mappings $f:X \rightarrow Y$, where $X$ and $Y$ are Banach spaces and $f$ satisfies Hyers inequality

for all $x, y \in X$. The limit

exists for all $x \in X$ and the mapping $A:X \rightarrow Y$ is a unique additive mapping which satisfies

for all $x \in X$.

The outcome declared that the Cauchy functional equation is stable for any pair of Banach space. The technique which was providing through Hyers, forming the additive function $A(x)$, is called direct method. This is called as Ulam stability for the Cauchy additive functional equation. We refer the interested readers for more information on such problems to the articles ^{[6,9,11,13,15,16,18,26,33]}.

Hyers' result was generalized by Aoki ^[1] for additive mappings and Rassias ^[25] for linear mappings by considering the stability problem with unbounded Cauchy differences. Furthermore, in 1994, a generalization of Rassias' theorem was obtained by Gavruta ^[10] by replacing the bound $\epsilon\left(\|x\|^{p}+\|y\|^{p}\right)$ by a general control function $\varphi(x, y)$.

In 2008, Mihet and Radu ^[19] applied fixed point alternative method to prove the stability theorems of the Cauchy functional equation:

in random normed spaces. In 2008, Najati and Moghimi ^[22] obtained a stability of the functional equation deriving from quadratic and additive function:

by using the direct method. After that, Jin and Lee ^[14] proved the stability of the above mentioned functional equation in random normed spaces.

In 2011, Saadati et al. ^[24] proved the nonlinear stability of the quartic functional equation of the form

in the setting of random normed spaces. Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, the fixed point theory, the theory of intuitionistic spaces and the theory of functional equations were also presented. Azadi Kenary ^[4] investigated the Ulam stability of the following nonlinear function equation

in random normed spaces. Recently, the stability problems of several functional equations in various spaces such as random normed spaces, intuitionistic random normed spaces, quasi-Banach spaces, fuzzy normed spaces have been extensively investigated by a number of mathematicians such as Azadi Kenary ^[2,3], Chang et al. ^[5], Eshaghi Gordji et al. ^[7,8], Mihet et al. ^[20], Saadati et al. ^[21,27,28] and Tamilvanan et al. ^[23,31].

In this paper, we introduce a new mixed type quadratic-additive functional equation of the form

where $\phi(0) = 0$ and $m$ is an integer greater than 4. The main aim of this work is to obtain its general solution and to investigate the Ulam stability by using the Hyers method in random normed spaces. It is easy to see that the mapping $\phi(s) = as^{2}+bs$ is a solution of the functional equation (1.2). Every solution of the functional equation deriving from quadratic and additive function (1.2) is said to be a general quadratic mapping.

2.   Preliminaries

In this section, we state the usual terminology, notions and conventions of the theory of random normed spaces as in ^[29].

Let $\Gamma^{+}$ denote the set of all probability distribution functions $F:\mathbb{R}\cup[-\infty, +\infty]\rightarrow [0, 1]$ such that $F$ is left-continuous and nondecreasing on $\mathbb{R}$ and $F(0) = 0$, $F(+\infty) = 1$. It is clear that the set $D^{+} = \{F \in \Gamma^{+}:l^{-}F(-\infty) = 1\}$, where $l^{-}f(x) = \lim_{t\rightarrow x^{-}}f(t)$, is a subset of $\Gamma^{+}$. The set $\Gamma^{+}$ is partially ordered by the usual pointwise ordering of functions, that is, $F\leq G$ if and only if $F(t)\leq G(t)$ for all $t \in \mathbb{R}$. For any $a\geq 0$, the element $H_{a}(t)$ of $D^{+}$ is defined by

We can easily show that the maximal element in $\Gamma^{+}$ is the distribution function $H_{0}(t)$.

Definition 2.1. ^[29] A function $T:[0, 1]^{2}\rightarrow [0, 1]$ is a continuous triangular norm (briefly, a t-norm) if $T$ satisfies the following conditions:

(a) $T$ is commutative and associative;

(b) $T$ is continuous;

(c) $T(x, 1) = x$ for all $x\in [0, 1]$;

(d) $T(x, y)\leq T(z, w)$ whenever $x \leq z$ and $y \leq w$ for all $x, y, z, w \in [0, 1]$.

Three typical examples of continuous t-norms are $T_{P}(x, y) = xy$, $T_{\max}(x, y) = \max \{a+b-1, 0\}$, $T_{M}(x, y) = \min \{a, b\}$.

Recall that, if $T$ is a t-norm and $\{x_{n}\}$ is a sequence in $[0, 1]$, then $T_{i = 1}^{n}x_{i}$ is defined recursively by $T_{i = 1}^{1}x_{i} = x_{1}$ and $T_{i = 1}^{n}x_{i} = T\left(T_{i = 1}^{n-1}x_{i}, x_{n}\right)$ for all $n \geq 2$. $T_{i = n}^{\infty}x_{i}$ is defined by $T_{i = 1}^{\infty}x_{n+i}$.

Definition 2.2. ^[30] A random normed space (briefly, RNS) is a triple $(X, \mu, T)$, where $X$ is a vector space, $T$ is a continuous t-norm and $\mu:X \rightarrow D^{+}$ is a mapping such that the following conditions hold:

(RN1) $\mu_{x}(t) = H_{0}(t)$ for all $x \in X$ and $t > 0$ if and only if $x = 0$;

(RN2) $\mu_{\alpha x}(t) = \mu_{x}\left(\frac{t}{\vert \alpha \vert}\right)$ for all $\alpha\in \mathbb{R}$ with $\alpha \neq 0$, $x \in X$ and $t \geq 0$;

(RN3) $\mu_{x+y}(t+s)\geq T\left(\mu_{x}(t), \mu_{y}(t)\right)$ for all $x, y \in X$ and $t, s \geq 0$.

Every normed space $\left(X, \|\cdot \|\right)$ defines a random normed space $\left(X, \mu, T_{M}\right)$, where $\mu_{u}(t) = \frac{t}{t+\|u\|}$ for all $t > 0$ and $T_{M}$ is the minimum t-norm. This space $X$ is called the induced random normed space. If the t-norm $T$ is such that $\sup_{0 < a < 1}T(a, a) = 1$, then every random normed space $(X, \mu, T)$ is a metrizable linear topological space with the topology $\tau$ (called the $\mu$-topology or the ($\epsilon, \delta$)-topology, where $\epsilon > 0$ and $\lambda\in (0, 1)$) induced by the base $\{U(\epsilon, \lambda)\}$ of neighbourhoods of $\theta$, where $U(\epsilon, \lambda) = \{x \in X:\Psi_{x}(\epsilon) > 1-\lambda\}$.

Definition 2.3. Let $(X, \mu, T)$ be a random normed space.

(ⅰ) A sequence $\{x_{n}\}$ in $X$ is said to be convergent to a point $x \in X$ (write $x_{n}\rightarrow x$ as $n \rightarrow \infty$) if $\lim_{n\rightarrow \infty}\mu_{x_{n}-x}(t) = 1$ for all $t > 0$.

(ⅱ) A sequence $\{x_{n}\}$ in $X$ is called a Cauchy sequence in $X$ if $\lim_{n\rightarrow \infty}\mu_{x_{n}-x_{m}}(t) = 1$ for all $t > 0$.

(ⅲ) The random normed space $\left(X, \mu, T\right)$ is said to be complete if every Cauchy sequence in $X$ is convergent.

Theorem 2.4. ^[29] If $(X, \mu, T)$ is a random normed space and $\{x_{n}\}$ is a sequence such that $x_{n}\rightarrow x$, then $\lim_{n\rightarrow \infty}\mu_{x_{n}}(t) = \mu_{x}(t)$.

3.   Solution of the functional equation (1.2)

Throughout this section, assume that $E$ and $F$ are real vector spaces.

Theorem 3.1. If $\phi:E\rightarrow F$ is an odd mapping which satisfies the functional equation (1.2) for all $s_{1}, s_{2}, \cdots, s_{m}\in E$, then $\phi$ is additive.

Proof. In the sense of oddness of $\phi$, $\phi(-s) = -\phi(s)$ for all $s \in E$. Then (1.2) turns into

for all $s_{1}, s_{2}, \cdots, s_{m}\in E$. Now, setting $s_{1} = s_{2} = \cdots = s_{m} = 0$ in (3.1), we obtain that $\phi(0) = 0$. Replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(0, s, 0, \cdots)$ in (3.1), we obtain that

for all $s \in E$. Again replacing $s$ by $2s$ in (3.2), we get

for all $s \in E$. Also, changing $s$ by $2s$ in (3.3), we have

for all $s \in E$. From (3.2)–(3.4), we conclude, for a positive integer $m$,

for all $s \in E$. Now, replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(u, \frac{v}{2}, 0, \cdots, 0)$ in (3.1), we get

for all $u, v \in E$. Therefore, the mapping $\phi$ is additive.

Theorem 3.2. If $\phi:E\rightarrow F$ is an even mapping which satisfies the functional equation (1.2) for all $s_{1}, s_{2}, \cdots, s_{m}\in E$, then $\phi$ is quadratic.

Proof. In the sense of evenness of $\phi$, $\phi(-s) = \phi(s)$ for all $s \in E$. Then (1.2) becomes

for all $s_{1}, s_{2}, \cdots, s_{m}\in E$. Now, setting $s_{1} = s_{2} = \cdots = s_{m} = 0$ in (3.5), we get $\phi(0) = 0$. Letting $(s_{1}, s_{2}, \cdots, s_{m}) = (0, s, 0, \cdots)$ in (3.5), we have

for all $s \in E$. Replacing $s$ by $2s$ in (3.6), we obtain

for all $s \in E$. Replacing $s$ by $2s$ in (3.7), we get

for all $s \in E$. From (3.6)–(3.8), we conclude, for a positive integer $m$,

for all $s \in E$. Now, replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(u, \frac{v}{2}, 0, \cdots, 0)$ in (3.5), we get

for all $u, v \in E$. Therefore, the mapping $\phi$ is quadratic.

Theorem 3.3. A mapping $\phi:E\rightarrow F$ satisfies $\phi(0) = 0$ and (1.2) for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ if and only if there exist a symmetric bi-additive mapping $Q:E \times E \rightarrow F$ and an additive mapping $A:E \rightarrow F$ such that $\phi(s) = Q(s, s)+A(s)$ for all $s \in E$.

Proof. Let $\phi$ satisfy (1.2) and $\phi(0) = 0$. We split $\phi$ into the odd part and even part as follows

for all $s \in E$, respectively. It is clear that $\phi(s) = \phi_{e}(s)+\phi_{o}(s)$ for all $s \in E$. It is easy to show that the mappings $\phi_{o}$ and $\phi_{e}$ satisfy (1.2). Hence by Theorems 3.1 and 3.2, we have that $\phi_{o}$ and $\phi_{e}$ are additive and quadratic, respectively. So there exist a symmetric bi-additive mapping $Q:E \times E \rightarrow F$ such that $\phi_{e}(s) = Q(s, s)$ and an additive mapping $A:E \rightarrow F$ such that $\phi_{o}(s) = A(s)$ for all $s \in E$. Hence $\phi(s) = Q(s, s)+A(s)$ for all $s \in E$.

Conversely, assume that there exist a symmetric bi-additive mapping $Q:E \times E \rightarrow F$ and an additive mapping $A:E \rightarrow F$ such that $\phi(s) = Q(s, s)+A(s)$ for all $s \in E$. One can easily show that the mappings $s\mapsto Q(s, s)$ and the mapping $A:E \rightarrow F$ satisfy the functional equation (1.2). Therefore, the mapping $\phi:E \rightarrow F$ satisfies the functional equation (1.2).

For our notational handiness, for a mapping $\phi:E \rightarrow F$, we define

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$.

4.   Main results for odd case

In this section, we investigate the Ulam stability of the finite variable functional equation (1.2) for odd case in random normed spaces by using the Hyers method.

Theorem 4.1. Let $E$ be a real linear space, $\left(Z, \mu^{'}, \min\right)$ be a random normed space and $\varphi:E^{m}\rightarrow Z$ be a function such that there exists $0 < \rho < \frac{1}{2}$ such that

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and $t > 0$ and $\lim_{m\rightarrow \infty}\mu^{'}_{\varphi\left(\frac{s_{1}}{2^{m}}, \frac{s_{2}}{2^{m}}, \cdots, \frac{s_{m}}{2^{m}} \right)}\left(\frac{t}{2^{m}}\right) = 1$ for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and $t > 0$. Let $\left(F, \mu, \min\right)$ be a complete random normed space. If $\phi:E \rightarrow F$ is a mapping such that

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and $t > 0$, then the limit $A_{1}(s) = \lim_{m\rightarrow \infty}2^{m}\phi\left(\frac{s}{2^{m}}\right)$ exists for all $s \in E$ and defines a unique additive mapping $A_{1}:E \rightarrow F$ such that

for all $s \in E$ and $t > 0$.

Proof. Replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(0, s, 0, \cdots, 0)$ in (4.2), we get

for all $s \in E$. From (4.4), we get

for all $s \in E$. Again, replacing $s$ by $\frac{s}{2}$ in (4.5), we have

for all $s \in E$. Replacing $s$ by $\frac{s}{2^{n}}$ in (4.6) and using (4.1), we obtain

for all $s \in E$. We know that

and so

for all $s \in E$. Replacing $s$ by $\frac{s}{2^{q}}$ in (4.8), we have

for all $s \in E$. Since $\lim_{q, n\rightarrow \infty}\mu^{'}_{\varphi(0, s, 0, \cdots, 0)}\left(\frac{(m^{2}-5m+4)}{\sum_{l = q}^{n+q-1\;\;}2^{l}\rho^{l+1}} t\right) = 1$, it follows that $\{2^{n}\phi\left(\frac{s}{2^{n}}\right)\}_{n = 1}^{\infty}$ is a Cauchy sequence in a complete random normed space $(F, \mu, \min)$ and so there exists a point $A_{1}(s)\in F$ such that $\lim_{n\rightarrow \infty}2^{n}\phi\left(\frac{s}{2^{n}}\right) = A_{1}(s)$. Fix $s \in E$ and put $q = 0$ in (4.9). Then we obtain

and so, for any $\delta > 0$,

for all $s \in E$ and $t > 0$. Taking $n\rightarrow \infty$ in (4.10), we have

for all $s\in E$. Since $\delta$ is arbitrary, by taking $\delta\rightarrow 0$ in (4.11), we obtain

for all $s \in E$. Now, replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $\left(\frac{s_{1}}{2^{n}}, \frac{s_{2}}{2^{n}}, \cdots, \frac{s_{m}}{2^{n}}\right)$ in (4.2), we have

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and $t > 0$. Since $\lim_{n\rightarrow \infty}\mu^{'}_{\varphi\left(\frac{s_{1}}{2^{n}}, \frac{s_{2}}{2^{n}}, \cdots, \frac{s_{m}}{2^{n}}\right)} \left(\frac{t}{2^{n}}\right) = 1$, we conclude that $A_{1}$ satisfies the functional equation (1.2). On the other hand,

for all $s \in E$. This implies that $A_{1}:E \rightarrow F$ is an additive mapping. To prove the uniqueness of the additive mapping $A_{1}$, assume that there exists another additive mapping $A_{2}:E\rightarrow F$ which satisfies the inequality (4.3). Then we get

for all $s \in E$ and $t > 0$. Since $\lim_{n\rightarrow\infty}(m^{2}-5m+4)\left(\frac{1-2\rho}{2^{n+1\;\;}\rho^{n}}\right)t = \infty$, we have

It follows that $\mu_{A_{1}(s)-A_{2}(s)}(t) = 1$ for all $t > 0$ and so $A_{1}(s) = A_{2}(s)$. This completes the proof.

Corollary 4.2. Let $E$ be a real normed linear space, $(Z, \mu^{'}, \min)$ be a random normed space and $(F, \mu, \min)$ be a complete random normed space. Let $p$ be a positive real number with $p > 1$, $z_{0}\in Z$ and $\phi:E \rightarrow F$ be a mapping satisfying

for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and $t > 0$. Then the limit $A_{1}(s) = \lim_{n\rightarrow\infty}2^{n}\phi\left(\frac{s}{2^{n}}\right)$ exists for all $s\in E$ and defines a unique additive mapping $A_{1}:E\rightarrow F$ such that

for all $s\in E$ and $t > 0$.

Proof. Let $\rho = 2^{-p}$ and $\varphi:E^{m}\rightarrow Z$ be a mapping defined by $\varphi(s_{1}, s_{2}, \cdots, s_{m}) = \left(\sum_{j = 1}^{m}\|s_{j}\|^{p}\right)z_{0}$. Then, from Theorem 4.1, the conclusion follows.

Theorem 4.3. Let $E$ be a real linear space, $\left(Z, \mu^{'}, \min\right)$ be a random normed space and $\varphi:E^{m}\rightarrow Z$ be a function for which t there exists $0 < \rho < 2$ such that

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and $t > 0$ and $\lim_{n\rightarrow \infty}\mu^{'}_{\varphi\left(2^{n}s_{1}, 2^{n}s_{2}, \cdots, 2^{n}s_{m} \right)}\left(2^{n}t\right) = 1$ for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and $t > 0$. Let $\left(F, \mu, \min\right)$ be a complete random normed space. If $\phi:E \rightarrow F$ is a mapping satisfying (4.2), then the limit $A_{1}(s) = \lim_{n\rightarrow \infty}\frac{\phi\left(2^{n}s\right)}{2^{n}}$ exists for all $s \in E$ and defines a unique additive mapping $A_{1}:E \rightarrow F$ such that

for all $s \in E$ and $t > 0$.

Proof. Replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(0, s, 0, \cdots, 0)$ in (4.2), we get

for all $s \in E$. From (4.14), we obtain

for all $s \in E$. Replacing $s$ by $2^{n}s$ in (4) and using (4.13), we obtain

for all $s \in E$. The rest of the proof is similar to the proof of Theorem 4.1.

Corollary 4.4. Let $E$ be a real normed linear space, $(Z, \mu^{'}, \min)$ be a random normed space and $(F, \mu, \min)$ be a complete random normed space. Let $p$ be a positive real number with $0 < p < 1$, $z_{0}\in Z$ and $\phi:E \rightarrow F$ be a mapping satisfying (4.12). Then the limit $A_{1}(s) = \lim_{n\rightarrow\infty}\frac{\phi\left(2^{n}s\right)}{2^{n}}$ exists for all $s\in E$ and defines a unique additive mapping $A_{1}:E\rightarrow F$ such that

for all $s\in E$ and $t > 0$.

Proof. Let $\rho = 2^{p}$ and $\varphi:E^{m}\rightarrow Z$ be a mapping defined by $\varphi(s_{1}, s_{2}, \cdots, s_{m}) = \left(\sum_{j = 1}^{m}\|s_{j}\|^{p}\right)z_{0}$. Then, from Theorem 4.3, the conclusion follows.

5.   Main results for even case

In this section, we investigate the Ulam stability of the finite variable functional equation (1.2) for even case in random normed spaces by using the Hyers method.

Theorem 5.1. Let $E$ be a real linear space, $\left(Z, \mu^{'}, \min\right)$ be a random normed space and $\varphi:E^{m}\rightarrow Z$ be a function for which there exists $0 < \rho < \frac{1}{2^{2}}$ such that

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and $t > 0$ and $\lim_{n\rightarrow \infty}\mu^{'}_{\varphi\left(\frac{s_{1}}{2^{n}}, \frac{s_{2}}{2^{n}}, \cdots, \frac{s_{m}}{2^{n}} \right)}\left(\frac{t}{2^{2n}}\right) = 1$ for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and $t > 0$. Let $\left(F, \mu, \min\right)$ be a complete random normed space. If $\phi:E \rightarrow F$ is a mapping with $\phi(0) = 0$ such that

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and $t > 0$, then the limit $Q_{2}(s) = \lim_{n\rightarrow \infty}2^{2n}\phi\left(\frac{s}{2^{n}}\right)$ exists for all $s \in E$ and defines a unique quadratic mapping $Q_{2}:E \rightarrow F$ such that

for all $s \in E$ and $t > 0$.

Proof. Replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(0, s, 0, \cdots, 0)$ in (5.2), we obtain

for all $s \in E$. From (5.4), we have

for all $s \in E$. Replacing $s$ by $\frac{s}{2}$ in (5.5), we get

for all $s \in E$. Again, replacing $s$ by $\frac{s}{2^{n}}$ in (5.6) and using (5.1), we have

for all $s \in E$. We know that

and so

for all $s \in E$. Replacing $s$ by $\frac{s}{2^{q}}$ in (5.7), we get

for all $s \in E$. Since $\lim_{q, n\rightarrow \infty}\mu^{'}_{\varphi(0, s, 0, \cdots, 0)}\left(\frac{(m^{2}-5m+2)}{\sum_{l = q}^{n+q-1}2^{2l}\rho^{l+1}} t\right) = 1$, it follows that $\{2^{2n}\phi\left(\frac{s}{2^{n}}\right)\}_{n = 1}^{\infty}$ is a Cauchy sequence in a complete random normed space $(F, \mu, \min)$ and so there exists a point $Q_{2}(s)\in F$ such that $\lim_{n\rightarrow \infty}2^{2n}\phi\left(\frac{s}{2^{n}}\right) = Q_{2}(s)$. Fix $s \in E$ and put $q = 0$ in (5.8). Then we have

and so, for any $\delta > 0$,

for all $s \in E$ and $t > 0$. Passing the limit $n\rightarrow \infty$ in (5.9), we get

for all $s\in E$. Since $\delta$ is arbitrary, by taking $\delta\rightarrow 0$ in (5.10), we obtain

for all $s \in E$. Now, replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $\left(\frac{s_{1}}{2^{n}}, \frac{s_{2}}{2^{n}}, \cdots, \frac{s_{m}}{2^{n}}\right)$ in (5.2), we have

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and $t > 0$. Since $\lim_{n\rightarrow \infty}\mu^{'}_{\varphi\left(\frac{s_{1}}{2^{n}}, \frac{s_{2}}{2^{n}}, \cdots, \frac{s_{m}}{2^{n}}\right)} \left(\frac{t}{2^{2n}}\right) = 1$, we conclude that $Q_{2}$ satisfies the functional equation (1.2). On the other hand,

for all $s \in E$. This implies that $Q_{2}$ is a quadratic mapping. To prove the uniqueness of the quadratic mapping $Q_{2}$, assume that there exists another quadratic mapping $Q_{2}^{'}:E\rightarrow F$ which satisfies (5.3). Then we have

for all $s \in E$ and $t > 0$. Since $\lim_{n\rightarrow\infty}(m^{2}-5m+2)\frac{(1-2^{2}\rho)}{2^{2(n+1)}\rho^{n}}t = \infty$, we have

It follows that $\mu_{Q_{2}(s)-Q_{2}^{'}(s)}(t) = 1$ for all $t > 0$ and so $Q_{2}(s) = Q_{2}^{'}(s)$. This completes the proof.

Corollary 5.2. Let $E$ be a real normed linear space, $(Z, \mu^{'}, \min)$ be a random normed space and $(F, \mu, \min)$ be a complete random normed space. Let $p$ be a positive real number with $p > 2$, $z_{0}\in Z$ and $\phi:E \rightarrow F$ be a mapping satisfying (4.12) for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and $t > 0$. Then the limit $Q_{2}(s) = \lim_{n\rightarrow\infty}2^{2n}\phi\left(\frac{s}{2^{n}}\right)$ exists for all $s\in E$ and defines a unique quadratic mapping $Q_{2}:E\rightarrow F$ such that

for all $s\in E$ and $t > 0$.

Proof. Let $\rho = 2^{-p}$ and $\varphi:E^{m}\rightarrow Z$ be a mapping defined by $\varphi(s_{1}, s_{2}, \cdots, s_{m}) = \left(\sum_{j = 1}^{m}\|s_{j}\|^{p}\right)z_{0}$. Then, from Theorem 5.1, the conclusion follows.

Theorem 5.3. Let $E$ be a real linear space, $\left(Z, \mu^{'}, \min\right)$ be a random normed space and $\varphi:E^{m}\rightarrow Z$ be a function such that there exists $0 < \rho < 2^{2}$ such that

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and $t > 0$ and $\lim_{n\rightarrow \infty}\mu^{'}_{\varphi\left(2^{n}s_{1}, 2^{n}s_{2}, \cdots, 2^{n}s_{m} \right)}\left(2^{2n}t\right) = 1$ for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and $t > 0$. Let $\left(F, \mu, \min\right)$ be a complete random normed space. If $\phi:E \rightarrow F$ is a mapping with $\phi(0) = 0$ sstisfyinf (4.2), then the limit $Q_{2}(s) = \lim_{n\rightarrow \infty}\frac{\phi\left(2^{n}s\right)}{2^{2n}}$ exists for all $s \in E$ and defines a unique quadratic mapping $Q_{2}:E \rightarrow F$ such that

for all $s \in E$ and $t > 0$.

Proof. Replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(0, s, 0, \cdots, 0)$ in (5.2), we obtain

for all $s \in E$. From (5.13), we have

for all $s \in E$. Replacing $s$ by $2^{n}s$ in (5.14) and using (5.11), we get

for all $s \in E$. The rest of the proof is similar to the proof of Theorem 5.1.

Corollary 5.4. Let $E$ be a real normed linear space, $(Z, \mu^{'}, \min)$ be a random normed space and $(F, \mu, \min)$ be a complete random normed space. Let $p$ be a positive real number with $0 < p < 2$, $z_{0}\in Z$ and $\phi:E \rightarrow F$ be a mapping satisfying (4.12). Then the limit $Q_{2}(s) = \lim_{n\rightarrow\infty}\frac{\phi\left(2^{n}s\right)}{2^{2n}}$ exists for all $s\in E$ and defines a unique quadratic mapping $Q_{2}:E\rightarrow F$ such that

for all $s\in E$ and $t > 0$.

Proof. Let $\rho = 2^{p}$ and $\varphi:E^{m}\rightarrow Z$ be a mapping defined by $\varphi(s_{1}, s_{2}, \cdots, s_{m}) = \left(\sum_{j = 1}^{m}\|s_{j}\|^{p}\right)z_{0}$. Then, from Theorem 5.3, the conclusion follows.

6.   Main results for mixed case

In this section, we investigate the Ulam stability of the finite variable functional equation (1.2) for mixed case in random normed spaces by using the Hyers method.

Theorem 6.1. Let $E$ be a real linear space, $\left(Z, \mu^{'}, \min\right)$ be a random normed space and $\varphi:E^{m}\rightarrow Z$ be a function for which there exists $0 < \rho < \frac{1}{2^{2}}$ such that (4.1) and $\lim_{n\rightarrow \infty}\mu^{'}_{\varphi\left(\frac{s_{1}}{2^{n}}, \frac{s_{2}}{2^{n}}, \cdots, \frac{s_{m}}{2^{n}} \right)}\left(\frac{t}{2^{n}}\right) = 1$ and $\lim_{n\rightarrow \infty}\mu^{'}_{\varphi\left(\frac{s_{1}}{2^{n}}, \frac{s_{2}}{2^{n}}, \cdots, \frac{s_{m}}{2^{n}} \right)}\left(\frac{t}{2^{2n}}\right) = 1$ for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and $t > 0$. Let $\left(F, \mu, \min\right)$ be a complete random normed space. If $\phi:E \rightarrow F$ is a mapping with $\phi(0) = 0$ satisfying (4.2), then the limits $Q_{2}(s) = \lim_{n\rightarrow \infty}2^{2n}\phi\left(\frac{s}{2^{n}}\right)$ and $A_{1}(s) = \lim_{n\rightarrow \infty}2^{n}\phi\left(\frac{s}{2^{n}}\right)$ exist for all $s \in E$ and define a unique quadratic mapping $Q_{2}: E \rightarrow F$ and a unique additive mapping $A_{1}:E \rightarrow F$ such that

for all $s \in E$ and $t > 0$.

Proof. By Theorem 3.3, $\phi(s) = \phi_{e}(s)+\phi_{o}(s)$, where

for all $s\in E$, respectively. So

and

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$. Now,

for all $s\in E$ and $t > 0$. Using Theorems 4.1 and 5.1, we can complete the remaining proof of the theorem.

Corollary 6.2. Let $E$ be a real normed linear space, $(Z, \mu^{'}, \min)$ be a random normed space and $(F, \mu, \min)$ be a complete random normed space. Let $p$ be a positive real number with $p > 2$, $z_{0}\in Z$ and $\phi:E \rightarrow F$ be a mapping with $\phi(0) = 0$ satisfying (4.12) for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and $t > 0$. Then the limits $Q_{2}(s) = \lim_{n\rightarrow\infty}2^{2n}\phi\left(\frac{s}{2^{n}}\right)$ and $A_{1}(s) = \lim_{n\rightarrow\infty}2^{n}\phi\left(\frac{s}{2^{n}}\right)$ exist for all $s\in E$ and define a unique quadratic mapping $Q_{2}:E\rightarrow F$ and a unique additive mapping $A_{1}:E\rightarrow F$ such that

for all $s\in E$ and $t > 0$.

Proof. Let $\rho = 2^{-p}$ and $\varphi:E^{m}\rightarrow Z$ be a mapping defined by $\varphi(s_{1}, s_{2}, \cdots, s_{m}) = \left(\sum_{j = 1}^{m}\|s_{j}\|^{p}\right)z_{0}$. Then, from Theorem 6.1, the conclusion follows.

Theorem 6.3. Let $E$ be a real linear space, $\left(Z, \mu^{'}, \min\right)$ be a random normed space and $\varphi:E^{m}\rightarrow Z$ be a function for which there exists $0 < \rho < 2$ such that $\mu^{'}_{\varphi\left(2s_{1}, 2s_{2}, \cdots, 2s_{m}\right)}\left(t\right) \geq \mu^{'}_{\rho \varphi\left(s_{1}, s_{2}, \cdots, s_{m}\right)}\left(t\right)$ for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and $t > 0$ and $\lim_{n\rightarrow \infty}\mu^{'}_{\varphi\left(2^{n}s_{1}, 2^{n}s_{2}, \cdots, 2^{n}s_{m} \right)}\left(2^{2n}t\right) = 1$ and $\lim_{n\rightarrow \infty}\mu^{'}_{\varphi\left(2^{n}s_{1}, 2^{n}s_{2}, \cdots, 2^{n}s_{m} \right)}\left(2^{n}t\right) = 1$ for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and $t > 0$. Let $\left(F, \mu, \min\right)$ be a complete random normed space. If $\phi:E \rightarrow F$ is a mapping with $\phi(0) = 0$ satisfying (4.2), then the limits $Q_{2}(s) = \lim_{n\rightarrow \infty}\frac{\phi\left(2^{n}s\right)}{2^{2n}}$ and $A_{1}(s) = \lim_{n\rightarrow \infty}\frac{\phi\left(2^{n}s\right)}{2^{n}}$ exist for all $s \in E$ and define a unique quadratic mapping $Q_{2}:E \rightarrow F$ and a unique additive mapping $A_{1}:E \rightarrow F$ such that

for all $s \in E$ and $t > 0$.

Proof. Using Theorems 4.3 and 5.3, in a similar manner of Theorem 6.1, we can complete the proof of the theorem.

Corollary 6.4. Let $E$ be a real normed linear space, $(Z, \mu^{'}, \min)$ be a random normed space and $(F, \mu, \min)$ be a complete random normed space. Let $p$ be a positive real number with $0 < p < 1$, $z_{0}\in Z$ and $\phi:E \rightarrow F$ be a mapping with $\phi(0) = 0$ satisfying (4.12). Then the limits $Q_{2}(s) = \lim_{n\rightarrow\infty}\frac{\phi\left(2^{n}s\right)}{2^{2n}}$ and $A_{1}(s) = \lim_{n\rightarrow\infty}\frac{\phi\left(2^{n}s\right)}{2^{n}}$ exist for all $s\in E$ and define a unique quadratic mapping $Q_{2}:E \rightarrow F$ and a unique additive mapping $A_{1}:E\rightarrow F$ such that

for all $s\in E$ and $t > 0$.

Proof. Let $\rho = 2^{p}$ and $\varphi:E^{m}\rightarrow Z$ be a mapping defined by $\varphi(s_{1}, s_{2}, \cdots, s_{m}) = \left(\sum_{j = 1}^{m}\|s_{j}\|^{p}\right)z_{0}$. Then, from Theorem 6.3, the conclusion follows.

7.   Conclusions

We have dealt with a new finite variable mixed type quadratic-additive functional equation (1.2) to obtain its solution. We employed the algorithm of the powerful tool (direct method) devised by Hyers to achieve our main results of Ulam stability of a finite variable mixed type functional equation (1.2).

Authors' contributions

The authors equally 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.

Acknowledgements

We would like to express our sincere gratitude to the anonymous referee for his/her helpful comments that will help to improve the quality of the manuscript.

Conflict of interest

The authors declare that they have no competing interests.