Stability results of the functional equation deriving from quadratic function in random normed spaces

1.   Introduction and preliminaries

A standard problem in the theory of functional equations is the subsequent subject:

When is it true that a function which approximately satisfies a functional equation must be close to an approximate solution of the functional equation?

We conclude that the functional equation is stable, if the problem satisfies the solution of the functional equation. In 1940, the stability problems of functional equations about group homomorphisms was introduced by Ulam ^[26]. In 1941, Hyers ^[10] gave is affirmative answer to Ulam's quaestion for additive groups (under the assumption that groups are Banach spaces). Rassias in ^[18] proved the generalized Hyers theorem for additive mappings.

The Rassias stability results provided manipulate during the last three decades in the growth of a generalization of the Hyers-Ulam stability conception. This novel method is called as generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. Also, in 1994, Rassias's generalization theorem was delivered by Gavruta ^[9] by replacing a general function $\epsilon\left(\|x\|^{p}+\|y\|^{p}\right)$ by $\varphi(x, y)$.

In 1982, J.M. Rassias ^[17] followed the modern approach of the Th.M.Rassias theorem in which he replaced the factor product of norms instead of sum of norms. Ravi et al., ^[21] investigated Gavruta's theorem for the unbounded Cauchy difference in the spirit of Rassias approach.

We mention here some quadratic functional equations papers concerning the stability (Ref. ^{[7,11,16,19,20]}). Recently, the stability of many functional equations in various spaces such as Banach spaces, fuzzy normed spaces and Random normed spaces have been broadly inspected by a numeral Mathematicians (Ref.^{[1,2,3,6,14,15,24]}). In the consequence, we take on the usual terminology, notions and conventions of the theory of random normed spaces as in ^{[4,5,12,13,22,23]}.

All through this work, $\Delta^{+}$ is the distribution functions spaces, i.e., the space of all mappings $V:R\cup\{-\infty, \infty\}\rightarrow [0, 1]$, such that $F$ is left continuous and increasing on $R, V(0) = 0$ and $V(+\infty) = 1$. $D^{+} \subset \Delta^{+}$ consisting of all functions $V\in\Delta^{+}$ for which $l^{-}V(+\infty) = 1$, where $l^{-}\phi(s)$ denotes $l^{-}\phi(s) = \lim_{t\rightarrow s^{-}}\phi(t)$. The space $\Delta^{+}$ is partially ordered by the usual point wise ordering of functions, i.e., $V\leq W \Longleftrightarrow V(t)\leq W(t)\; \forall t \in \mathbb{R}$. The maximal element for $\Delta^{+}$ in this order is the distribution function $\epsilon_{0}$ given by

Definition 1.1. A Random Normed space (briefly, RN-space) is a triple $(E, \Psi, \Upsilon)$, where $E$ is a vector space, $\Upsilon$ is a continuous $t-$norm and $\Psi:E \rightarrow D^{+}$ satisfying the following conditions:

$1. \; \Psi_{s}(t) = \varsigma_{0}(t)$ for all $t > 0$ if and only if $s = 0$;

$2. \; \Psi_{\alpha s}(t) = \Psi_{s}\left(\frac{t}{\vert \alpha\vert}\right)$ for all $s \in E, \; t\geq 0$ and $\alpha\in \mathbb{R}$ with $\alpha\neq 0$;

$3. \; \Psi_{s_{1}+s_{2}}(t+u) \geq \Upsilon\left(\Psi_{s_{1}}(t), \Psi_{s_{2}}(u)\right)$ for all $s_{1}, s_{2} \in E$ and $t, u\geq 0$.

Definition 1.2. Let $(E, \Psi, \Upsilon)$ be a RN-space.

(RN1) A sequence $\{s_{m}\}$ in $E$ is said to be convergent to a point $s \in E$ if $\lim_{m\rightarrow \infty}\Psi_{s_{m}-s}(t) = 1, \; t > 0$.

(RN2) A sequence $\{s_{m}\}$ in $E$ is called a Cauchy sequence if $\lim_{m\rightarrow \infty}\Psi_{s_{m}-s_{l}}(t) = 1, \; t > 0$.

(RN3) A RN-space $(E, \Psi, \Upsilon)$ is said to be complete if every Cauchy sequence in $E$ is convergent.

Theorem 1.3. ^[17] If $(E, \Psi, \Upsilon)$ is a RN-space and $\{s_{m}\}$ is a sequence in $E$ such that $s_{m}\rightarrow s$, then $\lim_{m\rightarrow \infty}\Psi_{s_{m}}(t) = \Psi_{s}(t)$ almost everywhere.

The authors introduce the finite variable quadratic functional equation is of the form

where $m\geq 5$, and derive its solution. Also, we investigate its Hyers-Ulam stability in Random Normed space(RN-space) by using direct and fixed point methods.

2.   General solution

Theorem 2.1. If a mapping $\phi:E\rightarrow F$ satisfies the functional equation (1.1), then the mapping $\phi:E\rightarrow F$ is quadratic.

Proof. Assume that the mapping $\phi:E\rightarrow F$ satisfies the functional equation (1.1). Replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(0, 0, \cdots, 0)$ in (1.1), we obtain $\phi(0) = 0$. Replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(s, 0, \cdots, 0)$ in (1.1), we have $\phi(-s) = \phi(s)$ for all $s \in E$. Therefore, the function $\phi$ is even. Next, replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(s, s, 0, \cdots, 0)$ in (1.1), we obtain

for all $s \in E$. Replacing $s$ by $2s$ in (2.1), we have

for all $s \in E$. Replacing $s$ by $2s$ in (2.2), we get

for all $s \in E$. In general, for any positive integer $n$, we conclude that

for all $s \in E$. Now, replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(u, v, 0, \cdots, 0)$ in (1.1), we reach our desired results of $\phi$.

Remark 2.2. Let $F$ be a linear space and a mapping $\phi : E \rightarrow F$ satisfies the functional equation (1.1), then the following results are true:

$(1) \; \phi(r^{t}s) = r^{t}\phi(s)$ for all $s \in E, r\in \mathbb{Q}$, $t$ integers.

$(2) \; \phi(s) = s\phi(1)$ for all $s \in E$ if $\phi$ is continuous.

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

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

Throughout the paper, we consider $E$ be a linear space and $(E, \Psi, \Upsilon)$ is a complete RN-space.

3.   Result and discussion: Direct method

Theorem 3.1. Let a mapping $\phi:E \rightarrow F$ for which there exists a mapping $\Phi:E^{m} \rightarrow D^{+}$ with

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and all $\epsilon > 0$ such that the functional inequality with $\phi(0) = 0$ such that

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and all $\epsilon > 0$. Then there exists a unique quadratic mapping $Q_{2}:E \rightarrow F$ satisfying the functional equation (1.1) with

for all $s \in E$ and all $\epsilon > 0$. The mapping $Q_{2}:E\rightarrow F$ is defined by

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

Proof. Replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(s, s, 0, \cdots, 0)$ in (3.1), we obtain

for all $s \in E$ and all $\epsilon > 0$. It follows from (3.5) and (RN2), we get

for all $s \in E$ and $\epsilon > 0$. Replacing $s$ by $2^{t}s$ in (3.6), we have

for all $s \in E$ and $\epsilon > 0$. Since

for all $s \in E$ and $\epsilon > 0$. From (3.7) and (3.8), we get

for all $s \in E$ and $\epsilon > 0$. Replacing $s$ by $2^{l}s$ in (3.9), we obtain

for all $s \in E$ and $\epsilon > 0$. As $\Phi_{s, s, 0, \cdots, 0} \left(\frac{\epsilon}{ \sum_{t = l}^{m+l-1}\frac{\alpha^{t}}{2^{2(t+1)}2}}\right)\rightarrow 1$ as $l, m\rightarrow \infty$, then $\Big\{\frac{\phi(2^{m}s)}{2^{2m}}\Big\}$ is a Cauchy sequence in $(F, \Psi, \Upsilon)$. Since $(F, \Psi, \Upsilon)$ is a complete RN-space, this sequence converges to some point $Q_{2}(s)\in F$. Fix $s \in E$ and put $l = 0$ in (3.10), we obtain

and so, for every $\zeta > 0$, we get

for all $s \in E$ and all $\epsilon, \zeta > 0$. Taking the limit $m\rightarrow \infty$ and using inequality (3.12), we have

for all $s \in E$ and all $\epsilon, \zeta > 0$. Since $\zeta$ was arbitrary, by taking $\zeta \rightarrow 0$ in (3.13), we get

for all $s \in E$ and all $\epsilon > 0$. Replacing $(s_{1}, s_{2}, \cdots, s_{m})$ by $(2^{m}s_{1}, 2^{m}s_{2}, \cdots, 2^{m}s_{m})$ in (3.2), we obtain

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and all $\epsilon > 0$. Since

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and all $\epsilon > 0$. We conclude that $Q_{2}$ satisfies the functional equation (1.1). To prove the uniqueness of the quadratic mapping $Q_{2}$. Assume that there exists a quadratic mapping $R_{2}:E \rightarrow F$, which satisfies the inequality(3.13). Fix $s \in E$. Clearly, $Q_{2}(2^{m}s) = 2^{2m}Q_{2}(s)$ and $R_{2}(2^{m}s) = 2^{2m}R_{2}(s)$ for all $s \in E$. It follows from (3.13) that

for all $s \in E$ and all $\epsilon > 0$. Since, $\lim_{m\rightarrow\infty}\left(\frac{2^{2m}2(2^{2}-\alpha)\epsilon}{\alpha^{m}}\right) = \infty$, we get $\lim_{m\rightarrow\infty}\Phi_{s, s, 0, \cdots, 0}\left(\frac{2^{2m}2(2^{2}-\alpha)\epsilon}{\alpha^{m}}\right) = 1$. Therefore, it follows that $\Psi_{Q_{2}(s)-R_{2}(s)}(\epsilon) = 1$ for all $s\in E$ and all $\epsilon > 0$. And so $Q_{2}(s) = R_{2}(s)$. This completes the proof.

Theorem 3.2. Let a ampping $\phi:E \rightarrow F$ for which there exists a mapping $\Phi:E^{m} \rightarrow D^{+}$ with the condition

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and all $\epsilon > 0$ such that the functional inequality with $\phi(0) = 0$ such that

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and all $\epsilon > 0$. Then there exists a unique quadratic mapping $Q_{2}:E \rightarrow F$ satisfying the functional equation (1.1) and

The mapping $Q_{2}(s)$ is defined by

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

Corollary 3.3. Let $\varsigma$ be positive real numbers. If $\phi:E\rightarrow F$ be a quadratic function which satisfies

for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and all $\epsilon > 0$. Then there exists a unique quadratic mapping $Q_{2}:E \rightarrow F$ such that

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

Proof. If $s_{1}, s_{2}, \cdots, s_{m} = \varsigma$, then the proof is true from Theorem 3.1 and 3.2 by taking $\alpha = 2^{0}$.

Corollary 3.4. Let $\varsigma$ and $\theta$ be nonnegative real numbers with $\theta \in (0, 2)\cup (2, +\infty)$. If a quadratic mapping $\phi:E\rightarrow F$ satisfies

for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and all $\epsilon > 0$. Then there exists a unique quadratic mapping $Q_{2}:E \rightarrow F$ such that

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

Proof. If $s_{1}, s_{2}, \cdots, s_{m} = \varsigma \sum_{i = 1}^{m}\|s_{i}\|^{\theta}$, then the proof is true from Theorem 3.1 and 3.2 by taking $\alpha = 2^{\theta}$.

Corollary 3.5. Let $\varsigma$ and $\theta$ be nonnegative real numbers with $m\theta \in (0, 2)\cup (2, +\infty)$. If a quadratic mapping $\phi:E\rightarrow F$ satisfies

for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and all $\epsilon > 0$. Then there exists a unique quadratic mapping $Q_{2}:E \rightarrow F$ such that

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

Proof. If $s_{1}, s_{2}, \cdots, s_{m} = \varsigma\left(\sum_{i = 1}^{m}\|s_{i}\|^{m\theta} + \prod_{i = 1}^{m}\|s_{i}\|^{\theta}\right)$, then the proof is true from Theorem 3.1 and 3.2 by taking $\alpha = 2^{m\theta}$.

4.   Result and discussion: Fixed point method

Theorem 4.1. If a mapping $\phi:E\rightarrow F$ for which there exists a function $\Phi:E^{m}\rightarrow D^{+}$ with

for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and all $\epsilon > 0$ and where $\zeta_{a} = \begin{cases} 2 \; \mathit{\mbox{if}}\; a = 0;\\ \frac{1}{2}\; \mathit{\mbox{if}}\; a = 1; \end{cases}$ satisfying the inequality

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and all $\epsilon > 0$. If there exists $L = L(a)$ such that the function $s\rightarrow \tau(s, \epsilon) = \Phi_{\frac{s}{2}, \frac{s}{2}, 0, \cdots, 0}\left(2\epsilon\right)$ has the property, that

Then there exists a unqiue quadratic mapping $Q_{2}:E\rightarrow F$ satisfies the functional equation (1.1) and satisfies

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

Proof. Consider a general metric $\rho$ on $\Delta$ such that $\rho(n_{1}, n_{2}) = \inf \Big\{v\in (0, \infty)/\Psi_{n_{1}(s)-n_{2}(s)}(v\epsilon)\geq \tau(s, \epsilon), s\in E, \epsilon > 0\Big\}$. It is easy to view that $(\Delta, \rho)$ is complete. Let us define a mapping $\Upsilon:\Delta\rightarrow\Delta$ by $\Upsilon n_{1}(s) = \frac{1}{\zeta_{a}^{2}}n_{1}(\zeta_{a}s)$, for all $s \in E$. Now for $n_{1}, n_{2}\in \Delta$, we have $\rho(n_{1}, n_{2})\leq v$.

for all $n_{1}, n_{2} \in \Delta$. Therefore, $\upsilon$ is strictly contractive mapping on $\Delta$ with Lipschitz constant $L$. If follows from (3.5) that

for all $s \in E$ and all $\epsilon > 0$. It follows from (4.6) that

for all $s \in E$ and all $\epsilon > 0$. Using (4.3) for $a = 0$, it reduces to

for all $s \in E$ and all $\epsilon > 0$. Hence, we obtain

for all $s \in E$. Replacing $s$ by $\frac{s}{2}$ in (4.7), we have

for all $s \in E$ and all $\epsilon > 0$. Using (4.3) for $a = 1$, it reduces to

for all $s \in E$ and all $\epsilon > 0$. Hence, we arrive

for all $s \in E$. From (4.8) and (4.10), we can conclude

for all $s \in E$. In order to prove $Q_{2}:E \rightarrow F$ satisfies the functional equation (1.1), the remaining proof is similar as in Theorem 3.1. As the function $Q_{2}$ is unique fixed point of $\Upsilon$ in $\Omega = \{\phi\in \Delta/\rho(\phi, Q_{2}) < \infty\}$. Finally, $Q_{2}$ is an unique function such that

for all $s \in E$ and all $\epsilon > 0$. This completes the proof.

Corollary 4.2. Let $\varsigma$ and $\theta$ be positive real numbers. If a quadratic mapping $\phi:E\rightarrow F$ satisfies

for all $s_{1}, s_{2}, \cdots, s_{m}\in E$ and all $\epsilon > 0$. Then there exists a unique quadratic mapping $Q_{2}:E \rightarrow F$ such that

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

Proof. Suppose

for all $s_{1}, s_{2}, \cdots, s_{m} \in E$ and all $\epsilon > 0$. Then

But, we have $\tau(s, \epsilon) = \Phi_{\frac{s}{2}, \frac{s}{2}, 0, \cdots, 0}\left(2\epsilon\right)$ has the property $L\frac{1}{\zeta_{a}^{2}}\tau(\zeta_{a}s, \epsilon)$ for all $s \in E$ and all $\epsilon > 0$. Now,

By using Theorem 4.1, we prove the following cases:

Case-1: $L = 2^{-2}$ if $a = 0$

Case-2: $L = 2^{2}$ if $a = 1$

Case-3: $L = 2^{\theta-2}$ for $\theta < 2$ if $a = 0$

Case-4: $L = 2^{2-\theta}$ for $\theta > 2$ if $a = 1$

Case-5: $L = 2^{m\theta-2}$ for $\theta < \frac{2}{m}$ if $a = 0$

Case-6: $L = 2^{2-m\theta}$ for $\theta > \frac{2}{m}$ if $a = 1$

Hence the proof is complete.

Counter example

Next, we show the following counter example replaced by the well-known counter example of Gajda ^[8] to the functional equation(1.1):

Example 4.3. Let a mapping $\phi:E\rightarrow F$ defined by

where

where $\psi$ is a constant, then the mapping $\phi:E\rightarrow F$ satisfies the inequality

for all $s_{1}, s_{2}, \cdots, s_{l} \in E$, but there does not exist a quadratic mapping $Q_{2}:E\rightarrow F$ with a constant $\varepsilon$ such that

for all $s \in E$.

Proof. It is easy to notice that $\phi$ is bounded by $\frac{4}{3}\psi$ on $E$. If $\sum_{j = 1}^{l}\vert s_{j}\vert^{2}\geq \frac{1}{2^{2}}$ or 0, then the left side of (4.13) is less than $\left(-m^{2}+7m-7\right) \frac{4}{3} \psi$, and thus (4.13) is true. Assume that $0 < \sum_{j = 1}^{m}\vert s_{j}\vert^{2} < \frac{1}{2^{2}}$. Then there exists an integer $l$ such that

So that $2^{2l}\vert s_{1}\vert < \frac{1}{2^{2}}, 2^{2l}\vert s_{2}\vert < \frac{1}{2^{2}}, \cdots, 2^{2l}\vert s_{m}\vert < \frac{1}{2^{2}}$ and $2^{m}s_{1}, 2^{m}s_{2}, \cdots, 2^{m}s_{m} \in (-1, 1)$ for all $m = 0, 1, 2, \cdots, l-1$. So, for $m = 0, 1, \cdots, l-1$

By the definition of $\phi$, we obtain

It follows from (4.15) that

for all $s_{1}, s_{2}, \cdots, s_{m}\in E$. Thus the function $\phi$ satisfies the inequality (4.13) for all $s_{1}, s_{2}, \cdots, s_{l}\in E$. We propose that there exists an quadratic mapping $Q_{2}:E\rightarrow F$ with a constant $\varepsilon > 0$ satisfying the inequality (4.14). Since the function $\phi$ is bounded and continuous for all $s$ in $E$, $Q_{2}$ is bounded on evry open interval containing the origin and continuous at the origin. By Remark 2.2, $Q_{2}$ must have the form $Q_{2}(v) = \gamma s^{2}$ for all $s\in E$. Thus we have

for all $s \in E$. However, we can select a non-negative integer $l$ and $l\psi > \varepsilon+\vert \gamma \vert$. If $s \in \left(0, 2^{-l}\right)$, then $2^{m}s\in (0, 1)$ for all $m = 0, 1, \cdots, l-1$ and for this $s$, we obtain

which is contradictory.

5.   Conclusions

We have introduced the generalized quadratic functional equation (1.1) and have obtained its general solution. Mainly, we have investigated Hyers-Ulam stability of the generalized quadratic functional equation (1.1) in Random Normed spaces by using direct and fixed point methods. Furthermore, we proved the counter example for the non-stability to the functional equation (1.1).

Conflict of interest

The authors declare that they have no conflict of interest.

Acknowledgement

This work was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.