Additive $\rho$-functional inequalities in non-Archimedean 2-normed spaces

1.   Introduction and preliminaries

The study of stability problems for functional equations is related to a question of Ulam ^[23] in 1940 concerning the stability of group homomorphisms.

The functional equation

is called Cauchy functional equation. Every solution of the Cauchy functional equation is said to be an additive mapping. In $1941$, Hyers ^[10] gave the first affirmative answer to the question of Ulam for Banach spaces. Hyers' result was generalized by Aoki ^[1] for additive mappings and by Rassias ^[20] for linear mappings by considering an unbounded Cauchy difference. In $1994$, Găvruţă ^[3] provided a further generalization of the Rassias' theorem in which he replaced the unbounded Cauchy difference by a general control function.

The following functional equation

is called Jensen functional equation. See ^{[15,16,19,22]} for more information on functional equations.

Gilányi ^[7] and Rätz ^[21] showed that if $f$ satisfies the functional inequality

then $f$ satisfies the Jordan-Von Neumann functional equation

Fechner ^[2] and Gilányi ^[8] proved the generalized Hyers-Ulam stability of the functional inequality (1.1). Park et al. ^[17] investigated the generalized Hyers-Ulam stability of functional inequalities associated with Jordon-Von Neumann type additive functional equations. Kim et al. ^[11] solved the additive $\rho$-functional inequalities in complex normed spaces and proved the Hyers-Ulam stability of the additive $\rho$-functional inequalities in complex Banach spaces. In $2014$, Park ^[14] considered the following two additive $\rho$-functional inequalities

in non-Archimedean Banach spaces and in complex Banach spaces, where $\rho$ is a fixed non-Archimedean number with $|\rho| < 1$ or $\rho$ is a fixed complex number with $|\rho| < 1$.

In this paper, we establish the solution of the additive $\rho$-functional inequalities (1.4) and (1.5), and prove the Hyers-Ulam stability of the additive $\rho$-functional inequalities (1.4) and (1.5) in non-Archimedean $2$-Banach spaces. Moreover, we prove the Hyers-Ulam stability of additve $\rho$-functional equations associated with the additive $\rho$-functional inequalities (1.4) and (1.5) in non-Archimedean $2$-Banach spaces.

Gähler ^[4,5] has introduced the concept of linear $2$-normed spaces in the middle of the 1960s. Then Gähler ^[6] and White ^[24,25] introduced the concept of $2$-Banach spaces. Following ^[9,12,13,18], we recall some basic facts concerning non-Archimedean normed space and non-Archimedean $2$-normed space and some preliminary results.

By a non-Archimedean field we mean a field ${\mathbb{K}}$ equipped with a function (valuation) $|\cdot|$ from ${\mathbb{K}}$ into $[0, \infty)$ such that $|r| = 0$ if and only if $r = 0$, $|rs| = |r||s|$, and $|r+s|\leq \max\{|r|, |s|\}$ for $r, s \in {\mathbb{K}}$. Clearly $|1| = |-1| = 1$ and $|n|\leq 1$ for all $n\in {\mathbb{N}}$. By the trivial valuation we mean the function $|\cdot|$ taking everything but $0$ into $1$ and $|0| = 0$.

Definition 1.1. (cf. ^[9,13]) Let $X$ be a linear space over a scalar field ${\mathbb{K}}$ with a non-Archimedean non-trivial valuation $|\cdot|$. A function $\|\cdot\|:X\to {\mathbb{R}}$ is called a non-Archimedean norm (valuation) if it satisfies the following conditions:

(i) $\|x\| = 0$ if and only if $x = 0$;

(ii) $\|rx\| = |r|\|x\|$ for all $r\in {\mathbb{K}}$, $x\in X$;

(iii) the strong triangle inequality; namely,

for all $x, y\in X$. Then $(X, \|\cdot\|)$ is called a non-Archimedean normed space.

Definition 1.2. (cf. ^[12,18]) Let $X$ be a linear space over a scalar field ${\mathbb{K}}$ with a non-Archimedean non-trivial valuation $|\cdot|$ with $\dim X > 1$. A function $\|\cdot, \cdot\|:X\to {\mathbb{R}}$ is called a non-Archimedean $2$-norm (valuation) if it satisfies the following conditions:

(NA1) $\|x, y\| = 0$ if and only if $x, y$ are linearly dependent;

(NA2) $\|x, y\| = \|y, x\|$;

(NA3) $\|rx, y\| = |r|\|x, y\|$;

(NA4) $\|x, y+z\|\leq\max\{\|x, y\|, \|x, z\|\}$;

for all $r\in {\mathbb{K}}$ and all $x, y, z\in X$. Then $(X, \|\cdot, \cdot\|)$ is called a non-Archimedean $2$-normed space.

According to the conditions in Definition 1.2, we have the following lemma.

Lemma 1.3. Let $(X, \|\cdot, \cdot\|)$ be a non-Archimedean $2$-normed space. If $x\in X$ and $\|x, y\| = 0$ for all $y\in X$, then $x = 0$.

Definition 1.4. A sequence $\{x_{n}\}$ in a non-Archimedean $2$-normed space $(X, \|\cdot, \cdot\|)$ is called a Cauchy sequence if there are two linearly independent points $y, z\in X$ such that

Definition 1.5. A sequence $\{x_{n}\}$ in a non-Archimedean $2$-normed space $(X, \|\cdot, \cdot\|)$ is called a convergent sequence if there exists an $x\in X$ such that

for all $y\in X$. In this case, we call that $\{x_{n}\}$ converges to $x$ or that $x$ is the limit of $\{x_{n}\}$, write $\{x_{n}\}\to x$ as $n\to\infty$ or $\lim\limits_{n\to\infty}x_{n} = x$.

By ${\rm{(NA4)}}$, we have

for all $y\in X$. Hence, a sequence $\{x_{n}\}$ is Cauchy in $(X, \|\cdot, \cdot\|)$ if and only if $\{x_{n+1}-x_{n}\}$ converges to zero in a non-Archimedean $2$-normed space $(X, \|\cdot, \cdot\|)$.

Remark 1.6. Let $(X, \|\cdot, \cdot\|)$ be a non-Archimedean $2$-normed space. One can show that conditions (NA2) and (NA4) in Definition 1.2 imply that

for all $x, y, z\in X$.

We can easily get the following lemma by Remark 1.6.

Lemma 1.7. For a convergent sequence $\{x_{n}\}$ in a non-Archimedean $2$-normed space $(X, \|\cdot, \cdot\|)$,

for all $y\in X$.

Definition 1.8. A non-Archimedean $2$-normed space, in which every Cauchy sequence is a convergent sequence, is called a non-Archimedean $2$-Banach space.

Throughout this paper, let $X$ be a non-Archimedean $2$-normed space with $\dim X > 1$ and $Y$ be a non-Archimedean $2$-Banach space with $\dim Y > 1$. Let ${\mathbb{N}} = \{0, 1, 2, \ldots, \}$, and $\rho$ be a fixed non-Archimedean number with $|\rho| < 1$.

2.   Solutions and stability of the inequality (1.4)

In this section, we solve and investigate the additive $\rho$-functional inequality (1.4) in non-Archimedean $2$-normed spaces.

Lemma 2.1. A mapping $f:X\to Y$ satisfies

for all $x, y\in X$ and all $\omega\in Y$ if and only if $f:X\to Y$ is additive.

Proof. Suppose that $f$ satisfies (2.1). Setting $x = y = 0$ in (2.1), we have $\|f(0), \omega\|\leq \|0, \omega\|$ = 0 for all $\omega\in Y$ and so $\|f(0), \omega\| = 0$ for all $\omega\in Y$. Hence we get

Putting $y = x$ in (2.1), we get

for all $x\in X$ and all $\omega\in Y$. Thus we have

for all $x\in X$. It follows from (2.1) and (2.3) that

for all $x, y\in X$ and all $\omega\in Y$. Hence, we obtain

for all $x, y\in X$.

The converse is obviously true. This completes the proof of the lemma.

The following corollary can be found in [14,Corollary 2.2].

Corollary 2.2. A mapping $f:X\to Y$ satisfies

for all $x, y\in X$ if and only if $f:X\to Y$ is additive.

Theorem 2.3. Let $\varphi:X^{2}\to [0, \infty)$ be a function such that

for all $x, y\in X$. Suppose that $f:X\to Y$ be a mapping satisfying

for all $x, y\in X$ and all $\omega\in Y$. Then there exists a unique additive mapping $A:X\to Y$ such that

for all $x\in X$ and all $\omega\in Y$.

Proof. Letting $y = x$ in (2.6), we get

for all $x\in X$ and all $\omega\in Y$. So

for all $x\in X$ and all $\omega\in Y$. Hence

for all nonnegative integers $m, l$ with $m > l$ and for all $x\in X$ and all $\omega\in Y$. It follows from (2.11) that

for all $x\in X$ and all $\omega\in Y$. Thus the sequence $\{\frac{f(2^{n}x)}{2^{n}}\}$ is a Cauchy sequence in $Y$. Since $Y$ is a non-Archimedean $2$-Banach space, the sequence $\{\frac{f(2^{n}x)}{2^{n}}\}$ converges for all $x\in X$. So one can define the mapping $A:X\to Y$ by

for all $x\in X$. That is,

for all $x\in X$ and all $\omega\in Y$.

By Lemma 1.7, (2.6) and (2.7), we get

for all $x, y\in X$ and all $\omega\in Y$. Thus, the mapping $A:X\to Y$ is additive by Lemma 2.1.

By Lemma 1.7 and (2.11), we have

for all $x\in X$ and all $\omega\in Y$. Hence, we obtain (2.8), as desired.

To prove the uniqueness property of $A$, Let $A^{\prime}: X\to Y$ be an another additive mapping satisfying (2.8). Then we have

which tends to zero as $n\to \infty$ for all $x\in X$ and all $\omega\in Y$. By Lemma 1.3, we can conclude that $A(x) = A^{\prime}(x)$ for all $x\in X$. This proves the uniqueness of $A$.

Corollary 2.4. Let $r, \theta$ be positive real numbers with $r > 1$, and let $f:X\to Y$ be a mapping such that

for all $x, y\in X$ and all $\omega\in Y$. Then there exists a unique additive mapping $A:X\to Y$ such that

for all $x\in X$ and all $\omega\in Y$.

Proof. The proof follows from Theorem 2.3 by taking $\varphi(x, y) = \theta(\|x\|^{r}+\|y\|^{r})\|\omega\|$ for all $x, y\in X$ and all $\omega\in Y$, as desired.

Theorem 2.5. Let $\varphi:X^{2}\to [0, \infty)$ be a function such that

for all $x, y\in X$. Suppose that $f:X\to Y$ be a mapping satisfying

for all $x, y\in X$ and all $\omega\in Y$. Then there exists a unique additive mapping $A:X\to Y$ such that

for all $x\in X$ and all $\omega\in Y$.

Proof. It follows from (2.9) that

for all $x\in X$ and all $\omega\in Y$. Hence

for all nonnegative integers $m, l$ with $m > l$ and for all $x\in X$ and all $\omega\in Y$. It follows from (2.19) that

for all $x\in X$ and all $\omega\in Y$. Thus the sequence $\{2^{n}f(\frac{x}{2^{n}})\}$ is a Cauchy sequence in $Y$. Since $Y$ is a non-Archimedean $2$-Banach space, the sequence $\{2^{n}f(\frac{x}{2^{n}})\}$ converges for all $x\in X$. So one can define the mapping $A:X\to Y$ by

for all $x\in X$. That is,

for all $x\in X$ and all $\omega\in Y$. By Lemma 1.7 and (2.19), we have

for all $x\in X$ and all $\omega\in Y$. Hence, we obtain (2.17), as desired. The rest of the proof is similar to that of Theorem 2.3 and thus it is omitted.

Corollary 2.6. Let $r, \theta$ be positive real numbers with $r < 1$, and let $f:X\to Y$ be a mapping satisfying (2.13) for all $x, y\in X$ and all $\omega\in Y$. Then there exists a unique additive mapping $A:X\to Y$ such that

for all $x\in X$ and all $\omega\in Y$.

Let $A(x, y): = f(x+y)-f(x)-f(y)$ and $B(x, y): = \rho(2f(\frac{x+y}{2})-f(x)-f(y))$ for all $x, y\in X$. For $x, y\in X$ and $\omega\in Y$ with $\|A(x, y), \omega\|\leq \|B(x, y), \omega\|$, we have

For $x, y\in X$ and $\omega\in Y$ with $\|A(x, y), \omega\| > \|B(x, y), \omega\|$, we have

So we can obtain

As corollaries of Theorems 2.3 and 2.5, we obtain the Hyers-Ulam stability results for the additive $\rho$-functional equation associated with the additive $\rho$-functional inequality (1.4) in non-Archimedean $2$-Banach spaces.

Corollary 2.7. Let $\varphi:X^{2}\to [0, \infty)$ be a function and let $f:X\to Y$ be a mapping satisfying (2.6) and

for all $x, y\in X$ and all $\omega\in Y$. Then there exists a unique additive mapping $A:X\to Y$ satisfying (2.8) for all $x\in X$ and all $\omega\in Y$.

Corollary 2.8. Let $r, \theta$ be positive real numbers with $r > 1$, and let $f:X\to Y$ be a mapping such that

for all $x, y\in X$ and all $\omega\in Y$. Then there exists a unique additive mapping $A:X\to Y$ satisfying (2.14) for all $x\in X$ and all $\omega\in Y$.

Corollary 2.9. Let $\varphi:X^{2}\to [0, \infty)$ be a function and let $f:X\to Y$ be a mapping satisfying (2.15) and (2.21) for all $x, y\in X$ and all $\omega\in Y$. Then there exists a unique additive mapping $A:X\to Y$ satisfying (2.17) for all $x\in X$ and all $\omega\in Y$.

Corollary 2.10. Let $r, \theta$ be positive real numbers with $r < 1$, and let $f:X\to Y$ be a mapping satisfying (2.22) for all $x, y\in X$ and all $\omega\in Y$. Then there exists a unique additive mapping $A:X\to Y$ satisfying (2.20) for all $x\in X$ and all $\omega\in Y$.

3.   Solutions and stability of the inequality (1.5)

In this section, we solve and investigate the additive $\rho$-functional inequality (1.5) in non-Archimedean $2$-normed spaces.

Lemma 3.1. A mapping $f:X\to Y$ satisfies $f(0) = 0$ and

for all $x, y\in X$ and all $\upsilon \in Y$ if and only if $f:X\to Y$ is additive.

Proof. Suppose that $f$ satisfies (3.1). Letting $y = 0$ in (3.1), we have

for all $x\in X$ and all $\upsilon\in Y$. Thus we have

for all $x\in X$. It follows from (3.1) and (3.3) that

for all $x, y\in X$ and all $\upsilon\in Y$. Hence, we obtain

for all $x, y\in X$.

The converse is obviously true. This completes the proof of the lemma.

The following corollary can be found in [14,Corollary 3.2].

Corollary 3.2. A mapping $f:X\to Y$ satisfies $f(0) = 0$ and

for all $x, y\in X$ if and only if $f:X\to Y$ is additive.

Theorem 3.3. Let $\phi:X^{2}\to [0, \infty)$ be a function such that

for all $x, y\in X$. Suppose that $f:X\to Y$ be a mapping satisfying $f(0) = 0$ and

for all $x, y\in X$ and all $\upsilon\in Y$. Then there exists a unique additive mapping $A:X\to Y$ such that

for all $x\in X$ and all $\upsilon\in Y$.

Proof. Letting $y = 0$ in (3.6), we get

for all $x\in X$ and all $\upsilon\in Y$. So

for all $x\in X$ and all $\upsilon\in Y$. Hence

for all nonnegative integers $m, l$ with $m > l$ and for all $x\in X$ and all $\upsilon\in Y$. It follows from (3.11) that

for all $x\in X$ and all $\upsilon\in Y$. Thus the sequence $\{\frac{f(2^{n}x)}{2^{n}}\}$ is a Cauchy sequence in $Y$. Since $Y$ is a non-Archimedean $2$-Banach space, the sequence $\{\frac{f(2^{n}x)}{2^{n}}\}$ converges for all $x\in X$. So one can define the mapping $A:X\to Y$ by

for all $x\in X$. That is,

for all $x\in X$ and all $\upsilon\in Y$.

By Lemma 1.7, (3.6) and (3.7), we get

for all $x, y\in X$ and all $\upsilon\in Y$. Thus, the mapping $A:X\to Y$ is additive by Lemma 3.1.

By Lemma 1.7 and (3.11), we have

for all $x\in X$ and all $\upsilon\in Y$. Hence, we obtain (3.8), as desired.

To prove the uniqueness property of $A$, Let $A^{\prime}: X\to Y$ be an another additive mapping satisfying (3.8). Then we have

which tends to zero as $n\to \infty$ for all $x\in X$ and all $\upsilon\in Y$. By Lemma 1.3, we can conclude that $A(x) = A^{\prime}(x)$ for all $x\in X$. This proves the uniqueness of $A$.

Corollary 3.4. Let $s, \delta$ be positive real numbers with $s > 1$, and let $f:X\to Y$ be a mapping satisfying $f(0) = 0$ and

for all $x, y\in X$ and all $\upsilon\in Y$. Then there exists a unique additive mapping $A:X\to Y$ such that

for all $x\in X$ and all $\upsilon\in Y$.

Proof. The proof follows from Theorem 3.3 by taking $\phi(x, y) = \delta(\|x\|^{s}+\|y\|^{s})\|\upsilon\|$ for all $x, y\in X$ and all $\upsilon\in Y$, as desired.

Theorem 3.5. Let $\phi:X^{2}\to [0, \infty)$ be a function such that

for all $x, y\in X$. Suppose that $f:X\to Y$ be a mapping satisfying $f(0) = 0$ and

for all $x, y\in X$ and all $\upsilon\in Y$. Then there exists a unique additive mapping $A:X\to Y$ such that

for all $x\in X$ and all $\upsilon\in Y$.

Proof. It follows from (3.9) that

for all $x\in X$ and all $\upsilon\in Y$. Hence

for all nonnegative integers $m, l$ with $m > l$ and for all $x\in X$ and all $\upsilon\in Y$. It follows from (3.19) that

for all $x\in X$ and all $\upsilon\in Y$. Thus the sequence $\{2^{n}f(\frac{x}{2^{n}})\}$ is a Cauchy sequence in $Y$. Since $Y$ is a non-Archimedean $2$-Banach space, the sequence $\{2^{n}f(\frac{x}{2^{n}})\}$ converges for all $x\in X$. So one can define the mapping $A:X\to Y$ by

for all $x\in X$. That is,

for all $x\in X$ and all $\upsilon\in Y$. By Lemma 1.7 and (3.19), we have

for all $x\in X$ and all $\upsilon\in Y$. Hence, we obtain (3.17), as desired. The rest of the proof is similar to that of Theorem 3.3 and thus it is omitted.

Corollary 3.6. Let $s, \delta$ be positive real numbers with $s < 1$, and let $f:X\to Y$ be a mapping satisfying $f(0) = 0$ and (3.13) for all $x, y\in X$ and all $\upsilon\in Y$. Then there exists a unique additive mapping $A:X\to Y$ such that

for all $x\in X$ and all $\upsilon\in Y$.

Let $\tilde{A}(x, y): = 2f(\frac{x+y}{2})-f(x)-f(y)$ and $\tilde{B}(x, y): = \rho(f(x+y)-f(x)-f(y))$ for all $x, y\in X$. For $x, y\in X$ and $\upsilon\in Y$ with $\|\tilde{A}(x, y), \upsilon\|\leq \|\tilde{B}(x, y), \upsilon\|$, we have

For $x, y\in X$ and $\upsilon\in Y$ with $\|\tilde{A}(x, y), \upsilon\| > \|\tilde{B}(x, y), \upsilon\|$, we have

So we can obtain

As corollaries of Theorems 3.3 and 3.5, we obtain the Hyers-Ulam stability results for the additive $\rho$-functional equation associated with the additive $\rho$-functional inequality (1.5) in non-Archimedean $2$-Banach spaces.

Corollary 3.7. Let $\phi:X^{2}\to [0, \infty)$ be a function and let $f:X\to Y$ be a mapping satisfying $f(0) = 0$, (3.6) and

for all $x, y\in X$ and all $\upsilon\in Y$. Then there exists a unique additive mapping $A:X\to Y$ satisfying (3.8) for all $x\in X$ and all $\upsilon\in Y$.

Corollary 3.8. Let $s, \delta$ be positive real numbers with $s > 1$, and let $f:X\to Y$ be a mapping satisfying $f(0) = 0$ and

for all $x, y\in X$ and all $\upsilon\in Y$. Then there exists a unique additive mapping $A:X\to Y$ satisfying (3.14) for all $x\in X$ and all $\upsilon\in Y$.

Corollary 3.9. Let $\phi:X^{2}\to [0, \infty)$ be a function and let $f:X\to Y$ be a mapping satisfying $f(0) = 0$, (3.15) and (3.21) for all $x, y\in X$ and all $\upsilon\in Y$. Then there exists a unique additive mapping $A:X\to Y$ satisfying (3.17) for all $x\in X$ and all $\upsilon\in Y$.

Corollary 3.10. Let $s, \delta$ be positive real numbers with $s < 1$, and let $f:X\to Y$ be a mapping satisfying $f(0) = 0$ and (3.22) for all $x, y\in X$ and all $\upsilon\in Y$. Then there exists a unique additive mapping $A:X\to Y$ satisfying (3.20) for all $x\in X$ and all $\upsilon\in Y$.

4.   Conclusion

In this paper, we have solved the additive $\rho$-functional inequalities:

where $\rho$ is a fixed non-Archimedean number with $|\rho| < 1$. More precisely, we have investigated the solutions of these inequalities in non-Archimedean $2$-normed spaces, and have proved the Hyers-Ulam stability of these inequalities in non-Archimedean $2$-normed spaces. Furthermore, we have also proved the Hyers-Ulam stability of additive $\rho$-functional equations associated with these inequalities in non-Archimedean $2$-normed spaces.

Acknowledgments

We would like to express our sincere gratitude to the anonymous referees for their helpful comments that will help to improve the quality of the manuscript.

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.

Conflict of interest

The authors declare that they have no competing interests.