Solution of a 3-D cubic functional equation and its stability

1.   Introduction

Stability problem of a functional equation was first posed in ^[31] which was answered in ^[7] and then generalized in ^[1,28] for additive mappings and linear mappings respectively. Since then several stability problems for various functional equations have been investigated in ^[9,10,12,23]. Fuzzy version was discussed in ^[13,14]. Recently, the stability problem for Jensen functional equation and cubic functional equation were considered in ^[15,20] respectively in intuitionistic fuzzy normed spaces; while the idea of intuitionistic fuzzy normed space was introduced in ^[30] and further studied in ^{[16,17,18,19,21,22,24,25,26,27,29]} to deal with some summability problems. Several results for the Hyers-Ulam stability of many functional equations have been proved by several researchers ^{[4,5,6,8,11,23]}

In modeling applied problems only partial information may be known (or) there may be a degree of uncertainty in the parameters used in the model or some measurments may be imprecise. Due to such features, many authors have considered the study of functional equations in the fuzzy setting. Jun et al. introduced the following functional equations

and

and investigated its general solution and the Hyers-Ulam stability respectively. The functional equations (1.1) and (1.2) are called cubic functional equations because the function $f(x) = cx^{3}$ is a solution of the above functional equations (1.1) and (1.2).

Rassias introduced the following new cubic equation $f : X\rightarrow Y$ satisfying the cubic functional equation

for all $x, y \in X$, with $X$ a linear space, $Y$ a real complete linear space, and then solved the Hyers-Ulam stability problem for the above functional equation.

In this paper, we define and find the general solution of the 3-D cubic functional equation

We also prove the Hyers-Ulam stability of this functional equation in fuzzy normed spaces by using the direct method and the fixed point method.

2.   Preliminaries

Definition 2.1. Let X be a real linear space. A function $N: X\times \mathbb{R} \rightarrow$ $[0, 1]$ is said to be a fuzzy norm on $X$ if for all $x, y\in X$ and $a, b\in \mathbb{R}$,

$(N_1)\quad N(x, c) = 0$ for$c\leq 0$;

$(N_2)\quad x = 0$ if and only if $N(x, c) = 1$ for all $c > 0$;

$(N_3)\quad N(cx, b) = N\left(x, \frac{b}{|c|}\right) \quad if \quad c\neq 0$;

$(N_4)\quad N(x+y, a+b)\geq \min\{N(x, a), N(y, b)\}$;

$(N_5)\quad N(x, \cdot)$ is a non-decreasing function on $\mathbb{R}$ and $\lim_{b\rightarrow \infty} N(x, b) = 1;$

$(N_6)\quad$for $x\neq0, N(x, \cdot)$ is (upper semi) continuous on $\mathbb{R}$.

The pair $(X, N)$ is called a fuzzy normed linear space. One may regard $N(x, b)$ as the truth value of the statement the norm of $x$ is less than or equal to the real number $b$.

Definition 2.2. Let $\left(X, N\right)$ be a fuzzy normed linear space. Let $\{x_{n}\}$ be a sequence in $X$. Then $x_{n}$ is said to be convergent if there exists $x\in X$ such that $\lim_{n \rightarrow \infty} N\left(x_{n}-x, b\right) = 1$ for all $b > 0$. In that case, $x$ is called the limit of the sequence $\{x_{n}\}$ and we denote it by $N-\lim_{n \rightarrow \infty}x_{n} = x$.

Definition 2.3. A sequence $\{x_{n}\}$ in $X$ is called Cauchy if for each $\epsilon > 0$ and each $b > 0$ there exists $n_{0}$ such that for all $n \geq n_{0}$ and all $p > 0$, we have $N\left(x_{n+p}-x_{n}, b\right) > 1-\epsilon$.

Every convergent sequence in a fuzzy normed space is Cauchy.

Definition 2.4. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space.

Definition 2.5. A mapping $f: X \rightarrow Y$ between fuzzy normed spaces X and Y is continuous at a point $x_{0}$ if for each sequence $\{ x_{n}\}$ converging to $x_{0}$ in X, the sequence $\{f(x_{n})\}$ converges to $f(x_{0})$. If $f$ is continuous at each point of $x_{0} \in X$, then $f$ is said to be continuous on $X$.

We bring the following theorems which some results in fixed point theory. These results play a fundamental role to arrive our purpose of this paper.

Theorem 2.6. (Banach Contraction Principle) Let $\left(X, d\right)$ be a complete metric space and consider a mapping $T:X\to X$ which is strictly contractive mapping, that is,

(A1) if $d\left(T_{x}, T_{y} \right)\le Ld\left(x, y\right)$ for some (Lipschitz constant) $L < 1$, then

1) the mapping T has one and only fixed point $x^{*} = T\left(x^{*} \right);$

2) the fixed point for each given element $x^{*}$ is globally attractive, that is,

(A2) ${\mathop{\lim }\limits_{n\to \infty }} T^{n} x = x^{*}$ for any starting point $x\in X$;

1) One has the following estimation inequalities:

(A3) $d\left(T^{n} x, x^{*} \right)\le \frac{1}{1-L} d\left(T^{n} x, T^{n+1} x\right)$ for all $n\ge 0, \, \, \, x\in X,$

(A4) $d\left(x, x^{*} \right)\le \frac{1}{1-L} d\left(x, x^{*} \right), \, \, \, \, \, \forall \, \, x\in X$.

Theorem 2.7. (The Alternative of fixed point) For a complete generalized metric space $(X, d)$ and a strictly contractive mapping $T:X\rightarrow X$ with Lipschitz constant $L$, and for each given element $x\in X$, either

(B1) $d(T^n x, T^{n+1} x) = +\infty,$ for all $n\geq 0$, or

(B2) There exists a natural number $n_0$ such that

i) $d(T^n x, T^{n+1} x) < \infty$ for all $n\geq n_0$;

ii) the sequence $(T^n x)$ is convergent to a fixed point $y^*$ of T;

iii) $y^*$ is the unique fixed point of T in the set $Y = \left\{y\in X; d(T^{n_{0}} x, y) < \infty \right\}$;

iv) $d(y^*, y)\leq \frac{1}{1-L}d(y, Ty)$ for all $y\in Y$.

3.   General solution of the functional equation (1.4)

In this section, we discuss the general solution of the functional equation (1.4).

Theorem 3.1. If an odd mapping $f:X \rightarrow Y$ satisfies the functional equation

for all $x, y \in X$ if and only if $f:X \rightarrow Y$ satisfies the functional equation

for all $x_{1}, x_{2}, x_{3} \in X$.

Proof. Let $f:X \rightarrow Y$ satisfy the functional equation (3.1). Setting $(x, y) = (0, 0)$ in (3.1), we get $f(0) = 0$. Replacing $(x, y)$ by $(x, 0)$, $(x, x)$ and $(x, 2x)$ respectively in (3.1), we obtain

for all $x \in X$. In general for any positive integer $a$, we have

for all $x \in X$. It follows from (3.4) that

for all $x \in X$. Replacing $(x, y)$ by $(x_{1}, x_{2}+x_{3})$ in (3.1), we get

for all $x_{1}, x_{2}, x_{3} \in X$. Again replacing $(x, y)$ by $(x_{2}+x_{3}, -2x_{1})$ in (3.1), we get

for all $x_{1}, x_{2}, x_{3} \in X$. Substituting (3.7) in (3.6), we get

for all $x_{1}, x_{2}, x_{3} \in X$. Setting $(x, y)$ by $(x_{2}, 2x_{1})$ in (3.1). we obtain

for all $x_{1}, x_{2} \in X$. Switching $(x, y)$ by $(x_{3}, 2x_{1})$ in (3.1), we obtain

for all $x_{1}, x_{3} \in X$. Adding (3.9) and (3.10), we get

for all $x_{1}, x_{2}, x_{3} \in X$. Adding (3.8) and (3.11), we get

for all $x_{1}, x_{2}, x_{3} \in X$. Replacing $(x, y)$ by $(-x_{1}, x_{2})$ in (3.1), we obtain

for all $x_{1}, x_{2} \in X$. Switching $(x, y)$ by $(-x_{1}, x_{3})$ in (3.1), we have

for all $x_{1}, x_{3} \in X$. Adding (3.14) and (3.15), we obtain

for all $x_{1}, x_{2}, x_{3} \in X$. Substituting (3.16) in (3.12), we have

for all $x_{1}, x_{2}, x_{3} \in X$.

Conversely, let $f:X \rightarrow Y$ satisfy the functional equation (3.2). Replacing $(x_{1}, x_{2}, x_{3})$ by $(x, 0, 0)$, $(0, x, 0)$ and $(0, 0, x)$ respectively in (3.17), we get

One can easy to verify from (3.18) that, replacing $(x_{1}, x_{2}, x_{3})$ by $(x, y, 0)$ in (3.2), we have

for all $x, y \in X$. Again replacing $(x_{1}, x_{2}, x_{3})$ by $(x, 0, -y)$ in (3.2), we obtain

for all $x, y \in X$. Adding the equations (3.19) and (3.20), we get our result.

Throughout the upcoming sections, assume that $X$, $(Z, N^{'})$ and $(Y, N)$ are linear space, fuzzy normed space and fuzzy Banach space, respectively. Let us denote

4.   Stability results for the functional equation (1.4): Direct method

In this section, we investigate the Hyers-Ulam stability of the functional equation (1.4) in fuzzy normed space via direct method.

Theorem 4.1. Let $\omega \in \{-1, 1\}$ be fixed and $\Gamma:X^{3} \rightarrow Z$ be a mapping such that for some $\rho > 0$ with $\left(\frac{\rho}{2^{3}}\right)^{\omega} < 1$

for all $x \in X$ and all $\varepsilon > 0$ and

for all $x_{1}, x_{2}, x_{3} \in X$ and all $\varepsilon > 0$. Suppose an odd mapping $f:X \rightarrow Y$ satisfies the inequality

for all $\varepsilon > 0$ and all $x_{1}, x_{2}, x_{3} \in X$. Then the limit

exists for all $x \in X$ and the mapping $C:X \rightarrow Y$ is a unique cubic mapping such that

for all $x \in X$ and all $\varepsilon > 0$.

Proof. First assume $\omega = 1$. Replacing $(x_{1}, x_{2}, x_{3})$ by $(x, 0, 0)$ in (4.2), we get

for all $x \in X$ and all $\varepsilon > 0$. From (4.4), we have

for all $x \in X$ and all $\varepsilon > 0$. Replacing $x$ by $2^{n}x$ in (4.5), we obtain

for all $x \in X$ and $\varepsilon > 0$. Using (4.1), $(N_{3})$ in (4.6) we get

for all $x \in X$ and all $\varepsilon > 0$. It is easy to verify from (4.7), that

holds for all $x \in X$ and all $\varepsilon > 0$. Replacing $\varepsilon$ by $\rho^{n} \varepsilon$ in (4.8), we get

for all $x \in X$ and all $\varepsilon > 0$. It is easy to see that

for all $x \in X$. From (4.9) and (4.10), we have

for all $x \in X$ and all $\varepsilon > 0$. Replacing $x$ by $2^{m}x$ in (4.11) and using (4.1), $(N_{3})$, we get

and so

for all $x \in X$, $\varepsilon > 0$ and all $m, n \geq 0$. Replacing $\varepsilon$ by $\frac{\varepsilon}{\sum_{i = m}^{n+m-1}\frac{\rho^{i}}{3(2^{3(i+1)})}}$ in (4.12), we get

for all $x \in X$, $\varepsilon > 0$ and all $m, n \geq 0$. Since $0 < \rho < 2^{3}$ and $\sum_{i = 0}^{n}\left(\frac{\rho}{2^{3}}\right)^{i} < \infty$, the Cauchy criterion for convergence and $(N_{5})$ imply that $\{\frac{f(2^{n}x)}{2^{3n}}\}$ is a Cauchy sequence in $(Y, N)$. Since $(Y, N)$ is complete, this sequence converges to some point $C(x) \in Y$. So one can define the mapping $C:X \rightarrow Y$ by

for all $x \in X$. Since $f$ is odd, $C$ is odd. Letting $m = 0$ in (4.13), we obtain

for all $x \in X$ and all $\varepsilon > 0$. Taking the limit as $n \rightarrow \infty$ in (4.14) and using $(N_{6})$, we get

for all $x \in X$ and all $\varepsilon > 0$. Now we claim that $C$ is cubic. Replacing $(x_{1}, x_{2}, x_{3})$ by $(2^{n}x_{1}, 2^{n}x_{2}, 2^{n}x_{3})$ in (4.2) respectively, we have

for all $x \in X$ and all $\varepsilon > 0$. Since

$A$ satisfies the functional equation (1.4). Hence $C:X \rightarrow Y$ is cubic. To prove the uniqueness of $C$, let $D:X \rightarrow Y$ be another cubic mapping satisfying (4.3). Fix $x \in X$. Clearly, $C(2^{n}x) = 2^{3n}C(x)$ and $D(2^{n}x) = 2^{3n}D(x)$ for all $x \in X$ and all $n \in \mathbb{N}$. It follows from (4.3) that

for all $x \in X$ and all $\varepsilon > 0$. Since $\lim_{n \rightarrow \infty}\frac{3(2^{n})\varepsilon (2^{3}-\rho)}{2\rho^{n}} = \infty$, we have

Thus $N(C(x)-D(x), \varepsilon) = 1$ for all $x \in X$ and all $\varepsilon > 0$, and so $C(x) = D(x)$.

For $\omega = -1$, we can prove the result by a similar method. This completes the proof of the theorem.

The following corollary is an immediate consequence of Theorem 4.1, concerning the stability for the functional equation (1.4).

Corollary 4.2. Suppose that the mapping $f:X\rightarrow Y$ satisfies the inequality

for all $x_1, x_2, x_3\in X$ and all $\varepsilon > 0,$ where $\theta,$ s are constants with $\theta > 0$. Then there exists a unique cubic mapping $C:X\rightarrow Y$ such that

for all $x \in X$ and all $r > 0$.

5.   Stability results for the functional equation (1.4): Fixed point method

In this section, we establish the Hyers-Ulam stability of the functional equation (1.4) in fuzzy normed space via fixed point method.

To prove the stability result, we define the following: $\eta_i$ is a constant such that

and $\Omega$ is the set such that $\Omega = \left\{t: X\rightarrow Y, t(0) = 0\right\}.$

Theorem 5.1. Let $f:X\rightarrow Y$ be a mapping for which there exists a mapping $\Gamma:X^{3}\rightarrow Z$ with condition

for all $x_1, x_2, x_3\in X$ and all $\varepsilon > 0$ and satisfying the inequality

for all $x_{1}, x_{2}, x_{3}\in X$ and $\varepsilon > 0.$ If there exists $L = L[i]$ such that the function $x\rightarrow \beta(x) = \frac{1}{3} \Gamma \left(\frac{x}{2}, 0, 0\right)$ has the property

for all $x\in X$ and $\varepsilon > 0,$ then there exists a unique cubic function $C:X\rightarrow Y$ satisfying the functional equation (1.4) and

for all $x\in X$ and $\varepsilon > 0.$

Proof. Set

for all $x_1, x_2, x_3\in X$. Then

Thus (5.1) holds. But we have

has the property

for all $x\in X$ and $\varepsilon > 0$. Hence

Thus

Now we can decide the Lipschitz constant $0 < L < 1$ by $\eta_i$, given in the previous statement of Theorem 5.1. So we divide into the following 6 cases for the conditions of $\eta_{i}$ as follows:

Case (ⅰ): $L = 2^{-3} \quad for \quad s = 0 \quad if \quad i = 0$:

$N(f(x)-C(x), \varepsilon)\geq N'\left(\frac{L^{1-i}}{1-L}\beta(x), \varepsilon\right)\geq N'\left(\frac{\theta(2^{-3})}{1-2^{-3}}, 3 \varepsilon\right)\geq N'\left(\theta, 21 \varepsilon\right)$.

Case (ⅱ): $L = 2^{3} \quad for \quad s = 0 \quad if \quad i = 1$:

$N(f(x)-C(x), \varepsilon)\geq N'\left(\frac{L^{1-i}}{1-L}\beta(x), \varepsilon\right)\geq N'\left(\frac{ \theta}{1-2}, 3 \varepsilon\right)\geq N'\left(\theta, -21\varepsilon\right)$.

Case (ⅲ): $L = 2^{s-3} \quad for \quad s < 3 \quad if \quad i = 0$:

Case (ⅳ): $L = 2^{3-s} \quad for \quad s > 3 \quad if \quad i = 1$:

Case (ⅴ): $L = 2^{ns-3} \quad for \quad s < \frac{3}{n} \quad if \quad i = 0$:

Case (ⅵ): $L = 2^{3-ns} \quad for \quad s < \frac{3}{n} \quad if \quad i = 1$:

Hence the proof is completed.

The following corollary is an immediate consequence of Theorem 5.1, concerning the stability of the functional equation (1.4).

Corollary 5.2. Suppose a function $f:X\rightarrow Y$ satisfies the inequality

for all $x_1, x_2, x_3\in X$ and $\varepsilon > 0,$ where $\theta, s$ are constants with $\theta > 0.$ Then there exists a unique cubic mapping $C:X\rightarrow Y$ such that

for all $x\in X$ and $\varepsilon > 0.$

Remark 5.3. To prove the stability of functional equations and functional inequalities, we have two methods: direct method and fixed point method. For the direct method, we use the Hyers-Ulam method, which is a traditional method, and for the fixed point method, we use the Isac-Rassias method, which is a more recent method. The proofs for the stability of functional equations and functional inequalities are similar to the orginal direct method and/or the fixed point method.

In general, to prove the stability, we divide two cases, for an example, $p > 3$ and $0 < p < 3$ in $\|x\|^p$, appeared in a control function of cubic functional equations. In this paper, we just use one control function to prove the stability of a new 3-D cubic functional equation by using the direct method and by using the fixed point method.

6.   Conclusion

We have introduced the following 3-D cubic functional equation

We have solved the 3-D cubic functional equation and we have proved the Hyers-Ulam stability of the 3-D functional equation in fuzzy normed spaces by using the direct method and the fixed point method.

Acknowledgments

C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF -2017R1D1A1B04032937).

Conflict of interest

The authors declare that they have no competing interests.