Abstract
First we investigate the Hyers–Ulam stability of the Cauchy functional equation for mappings from bounded (unbounded) intervals into Banach spaces. Then we study the Hyers–Ulam stability of the functional equation \(f(xy)=xg(y)+h(x)y\) for mappings from bounded (unbounded) intervals into multi-normed spaces.
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1 Introduction
The concept of stability for a functional equation \((*)\) arises when the functional equation \((*)\) is replaced by an inequality that acts as the equation perturbation. In 1940, Ulam [16] posed the first question concerning the stability of homomorphisms between groups. Hyers [4] answered the question of Ulam in the context of Banach spaces. Hyers’s stability theorem was generalized by some authors, and they considered cases where the Cauchy difference was unbounded (see [1, 3, 8, 10, 12, 13]). The stability problem for the Cauchy functional equation on a bounded domain was first proved by Skof [14]. The stability problem for the functional equation
on the interval \((0, 1]\) was posed by Maksa [7]. Tabor [15] and Páles [11] proved the Hyers–Ulam stability of functional equation (1.1) for real-valued functions on the intervals \((0,1]\) and \([1,+\infty )\), respectively. In this paper, we use some ideas from the works [6, 9, 14] to investigate the Hyers–Ulam stability of the Cauchy functional equation and (1.1) for mappings from bounded (unbounded) intervals into multi-normed spaces.
Let \((E, \|\cdot\|)\) be a complex linear space. For given \(k\in \mathbb{N}\), we denote by \(E^{k}\) the linear space consisting of k-tuples \((x_{1}, \ldots , x_{k})\), where \(x_{1}, \ldots , x_{k}\in E\). The linear operations on \(E^{k}\) are defined coordinatewise. We write \((0, \ldots , 0, x_{i}, 0, \ldots , 0)\) for an element in \(E^{k}\), when \(x_{i}\) appears in the ith coordinate. We denote the zero element of either E or \(E^{k}\) by 0.
Definition 1.1
([2])
A multi-norm on \(\{E^{n}: n \in \mathbb{N}\}\) is a sequence \(\{\|\cdot\|_{n}\}_{n}\) such that \(\|\cdot\|_{n}\) is a norm on \(E^{n}\) for each \(n\in \mathbb{N}\) and the following axioms are satisfied for each \(n\in \mathbb{N}\):
- \((A_{1})\):
-
\(\|x\|_{1}=\|x\|\) for each \(x\in E\), and \(\|\cdot\|_{n}\) is a norm on \(E^{n}\);
- \((A_{2})\):
-
\(\|(\alpha _{1}x_{1}, \ldots , \alpha _{n}x_{n})\|_{n}\leqslant ( \max_{1\leq i\leqslant n}|\alpha _{i}| )\|(x_{1}, \ldots , x_{n}) \|_{n}\) for each \(\alpha _{1}, \ldots , \alpha _{n}\in \mathbb{C}\) and each \(x_{1}, \ldots , x_{n}\in E\);
- \((A_{3})\):
-
\(\|(x_{\sigma (1)},\ldots ,x_{\sigma (n)})\|_{n}=\|(x_{1}, \ldots , x_{n}) \|_{n}\) for each permutation σ on \(\{1,\ldots ,n\}\);
- \((A_{4})\):
-
\(\|(x_{1}, \ldots , x_{n},0)\|_{n+1}=\|(x_{1}, \ldots , x_{n})\|_{n}\) for each \(x_{1}, \ldots , x_{n}\in E\);
- \((A_{5})\):
-
\(\|(x_{1}, \ldots , x_{n-1}, x_{n}, x_{n})\|_{n+1}=\|(x_{1}, \ldots , x_{n-1}, x_{n})\|_{n}\) for each \(x_{1}, \ldots , x_{n}\in E\);
In this case \(\{(E^{n}, \|\cdot\|_{n}): n\in \mathbb{N}\}\) is called a multi-normed space.
Example 1.2
([2])
Let \((E, \|\cdot\|)\) be a normed space. For given \(n\in \mathbb{N}\), define \(\|\cdot\|_{n}\) on \(E^{n}\) by \(\|x_{1},\ldots ,x_{n}\|_{n}=\max_{1\leq i\leqslant n}\|x_{i}\|\). This gives a multi-norm on \(\{E^{n}: n \in \mathbb{N}\}\).
For details and many other examples, we refer the readers to [2]. We now have the following consequences of the axioms.
Proposition 1.3
([2])
Let\(\{(E^{n}, \|\cdot\|_{n}): n\in \mathbb{N}\}\)be a multi-normed space. Then
- \((i)\):
-
\(\|(x,\ldots ,x)\|_{n}=\|x\|\)for each\(x\in E\);
- \((\mathit{ii})\):
-
\(\|(\alpha _{1}x_{1}, \ldots , \alpha _{n}x_{n})\|_{n}=\|(x_{1}, \ldots , x_{n})\|_{n}\)for each\(x_{1}, \ldots , x_{n}\in E\)and\(\alpha _{1}, \ldots , \alpha _{n}\in \mathbb{C}\)with\(|\alpha _{1}|=\cdots =|\alpha _{n}|=1\);
- \((\mathit{iii})\):
-
\(\max_{1\leq i\leqslant n}\|x_{i}\|\leqslant \|(x_{1},\ldots ,x_{n}) \|_{n}\leqslant \sum_{i=1}^{n}\|x_{i}\|\leqslant n\max_{1\leqslant i \leqslant n}\|x_{i}\|\)for each\(x_{1}, \ldots , x_{n}\in E\).
Proof
\((i)\) follows from \((A_{1})\) and \((A_{5})\). To prove \((\mathit{ii})\), it follows from \((A_{2})\) that
This proves \((\mathit{ii})\). To prove \((\mathit{iii})\), since \(\|\cdot\|_{n}\) is a norm on \(E^{n}\), we have by \((A_{4})\)
On the other hand, for each \(1\leqslant i\leqslant n\), we have by \((\mathit{ii})\)
Hence we get \(\max_{1\leq i\leqslant n}\|x_{i}\|\leqslant \|(x_{1},\ldots ,x_{n}) \|_{n}\). □
Item \((\mathit{iii})\) of Proposition 1.3 implies that if \((E,\|\cdot\|)\) is a Banach space, then \((E^{n}, \|\cdot\|_{n})\) is a Banach space for each \(n\in \mathbb{N}\). We use the term multi-Banach space for \(\{(E^{n}, \|\cdot\|_{n}): n\in \mathbb{N}\}\) when \((E, \|\cdot\|)\) is Banach.
Definition 1.4
Let \(\{(E^{n}, \|\cdot\|_{n}): n\in \mathbb{N}\}\) be a multi-normed space. A sequence \(\{x_{n}\}_{n}\) in E is said to be a multi-Cauchy sequence in E if, for each \(\varepsilon >0\), there exists \(n_{0}\in \mathbb{N}\) such that
A sequence \(\{x_{n}\}_{n}\) in E is called multi-convergent to a in E if, for each \(\varepsilon >0\), there exists \(m\in \mathbb{N}\) such that
In this case we write
Applying the triangle inequality for the norm \(\|\cdot\|_{k}\) and property \((\mathit{iii})\) of Proposition 1.3, we deduce the following result.
Lemma 1.5
([9])
Suppose that\(k \in \mathbb{N}\)and\((x_{1}, \ldots , x_{k}), (y_{1}, \ldots , y_{k}) \in E^{k}\). For each\(j \in \{1, \ldots , k\}\), let\(\{x_{n,j} \}_{n}\)be a sequence inEsuch that\(\lim_{n\to \infty }x_{n,j}=x_{j}\). Then
It is clear that each multi-convergent sequence is a multi-Cauchy sequence and convergent. In multi-Banach spaces a multi-Cauchy sequence is multi-convergent.
In this paper, using some ideas from [6, 9, 14], we investigate the Hyers–Ulam stability of functional equation (1.1) for mappings from subsets of \(\mathbb{R}\) into multi-normed spaces.
2 Stability of functional equation (1.1)
Theorem 2.1
Let\(f:[0,c) \rightarrow E\)be a function satisfying
for some\(\delta >0\)and all\(x_{1}, \ldots , x_{n}, y_{1}, \ldots , y_{n}\in (0,c]\)with\(x_{i}+y_{i}\in (0,c]\)for all\(i\in \{1,\ldots ,n\}\). Then there exists an additive function\(A:\mathbb{R} \rightarrow E\)such that
Proof
We extend the function f to \([0,+\infty )\). For this we represent arbitrary \(x\geqslant 0\) by \(x=n(c/2)+\alpha \), where n is an integer and \(0\leqslant \alpha < c/2\). Then we define a function \(\varphi :[0,+\infty )\to E\) by \(\varphi (x)=nf(c/2)+f(\alpha )\). It is clear that \(\varphi (x)=f(x)\) for all \(x\in [0,c/2)\). If \(x\in [c/2,c)\), then \(\varphi (x)=f(c/2)+f(x-c/2)\). We claim that
Let \(n\in \mathbb{N}\) and \(x_{1},\ldots , x_{n}\in [0,c)\). We set \(\varOmega =\{i: 1\leqslant i\leqslant n, x_{i}\in [c/2,c)\}\) and \(|\varOmega |=m\). If Ω is empty, then \(\varphi (x_{i})=f(x_{i})\) for all \(1\leqslant i\leqslant n\), and consequently the claim is true. For the case \(m\geqslant 1\), we have \(\varphi (x_{i})-f(x_{i})=f(c/2)+f(x_{i}-c/2)-f(x_{i})\) for all \(i\in \varOmega \). Let \(j_{1},\ldots , j_{m}\in \varOmega \). Then \((A_{4})\) and (2.1) imply
which proves (2.3). We now prove that
for all \(x_{1},\ldots , x_{n}, y_{1},\ldots , y_{n}\in [0,+\infty )\). For given \(n\in \mathbb{N}\) and \(x_{i},y_{i}\geqslant 0\), let \(x_{i}=n_{i}(c/2)+\alpha _{i}\) and \(y_{i}=m_{i}(c/2)+\beta _{i}\), where \(m_{i}\) and \(n_{i}\) are integers and \(0\leqslant \alpha _{i}, \beta _{i}< c/2\). We set \(\Delta =\{i: 1\leqslant i\leqslant n, \alpha _{i}+\beta _{i}\in [c/2,c) \}\). Then it is easy to show that
To prove (2.4), we need to consider three cases as follows.
Case 1. Suppose that Δ is empty. Then (2.4) follows from (2.1) and (2.5).
Case 2. Suppose \(|\Delta |=n\). Then, by (2.1), (2.3), and (2.6), we have
Case 3. Suppose that Δ is not empty and \(|\Delta |=m< n\). Then
Hence we have proved (2.4). Letting \(y_{i}=x_{i}\) for \(1\leqslant i\leqslant n\) in (2.4), we get
for all \(x_{1},\ldots , x_{n}, y_{1},\ldots , y_{n}\in [0,+\infty )\). Replacing \(x_{1},\ldots , x_{n}\) by \(2^{k}x_{1},\ldots , 2^{k}x_{n}\) in the above inequality and dividing both sides of the resulting inequality by \(2^{k+1}\), we obtain
for all \(x_{1},\ldots , x_{n}\in [0,+\infty )\). Then
for all \(x_{1},\ldots , x_{n}\in [0,+\infty )\). For fixed \(x\in [0,+\infty )\), replacing \(x_{j}\) by \(2^{j-1}x\) for all \(1\leqslant j\leqslant n\) in the above inequality, we obtain
Then \((A_{2})\) and (2.8) yield
Hence \(\{\frac{\varphi (2^{m}x)}{2^{m}}\}_{m}\) is a multi-Cauchy sequence in E for all \(x\in [0,+\infty )\). So it is multi-convergent in the multi-Banach space \(\{(E^{n}, \|\cdot\|_{n}): n\in \mathbb{N}\}\). Consider the function \(A_{0}:[0,+\infty )\to E\) defined by
Letting \(m=0\) in (2.7), we get
for all \(x_{1},\ldots , x_{n}\in [0,+\infty )\). Letting \(k\to \infty \) and utilizing Lemma 1.5, we infer that
Using (2.3), we have
We now extend \(A_{0}\) to a function \(A:\mathbb{R}\to E\) given by
We show that A is additive. For given \(x,y\in \mathbb{R}\), since \(A(x+y)-A(x)-A(y)\) is symmetric in x and y, we may assume the following cases:
- \((i)\):
-
If \(x,y\geqslant 0\) or \(x,y<0\), then we get \(A(x+y)=A(x)+A(y)\).
- \((\mathit{ii})\):
-
If \(x\geqslant 0, y<0\) and \(x+y\geqslant 0\), then
$$\begin{aligned} A(x+y)-A(x)-A(y) &= A_{0}(x+y)-A_{0}(x)+A_{0}(-y) \\ & =\bigl[A_{0}(x+y)+A_{0}(-y)\bigr]-A_{0}(x) \\ &=A_{0}(x)-A_{0}(x)=0. \end{aligned}$$ - \((\mathit{iii})\):
-
If \(x\geqslant 0, y<0\) and \(x+y< 0\), then
$$\begin{aligned} A(x+y)-A(x)-A(y) &= -A_{0}(-x-y)-A_{0}(x)+A_{0}(-y) \\ & =A_{0}(-y)-\bigl[A_{0}(-x-y)+A_{0}(x)\bigr] \\ &=A_{0}(-y)-A_{0}(-y)=0. \end{aligned}$$
Hence \(A:\mathbb{R}\to E\) is an additive function satisfying (2.2), which completes the proof. □
Theorem 2.2
Let\(c\geqslant 0\)and\(f:[c,+\infty ) \rightarrow E\)be a function satisfying
for some\(\delta >0\)and all\(x_{1}, \ldots , x_{n}, y_{1}, \ldots , y_{n}\in [c,+\infty )\). Then there exists an additive function\(A:\mathbb{R} \rightarrow E\)such that
Proof
Using the same argument as in the proof of Theorem 2.1, there exists an additive function \(T:[c,+\infty ) \rightarrow E\) such that
We extend T from \([c,+\infty )\) to \(\mathbb{R}\). First, we extend T from \([c,+\infty )\) to \([0,+\infty )\) by defining
It is easy to see that \(\widetilde{T}:[0,+\infty )\to E\) is additive. Now, we extend T̃ to the additive function \(A:\mathbb{R}\to E\) by defining
□
For convenience, we use the following abbreviation for a given mapping \(f:(0,1]\to E\):
Theorem 2.3
Let\(\{(E^{n}, \|\cdot\|_{n}): n\in \mathbb{N}\}\)be a multi-Banach space. Suppose thatεis a nonnegative real number and\(f:(0,1]\to E\)is a mapping satisfying
for all\(n\in \mathbb{N}\)and all\(x_{1}, \ldots , x_{n}, y_{1}, \ldots , y_{n}\in (0,1]\). Then there exists a function\(\Delta : (0,1]\to E\)satisfying functional equation (1.1) and the inequality
Proof
For each \(n\in \mathbb{N}\), it follows from \((A_{2})\) and (2.11) that
for all \(x_{1}, \ldots , x_{n}, y_{1}, \ldots , y_{n}\in (0,1]\). Then
If we define the mapping \(g:(0,1]\to E\) by
then (2.12) means
for all \(x_{1}, \ldots , x_{n}, y_{1}, \ldots , y_{n}\in (0,1]\). Let us define the mapping \(G:[0,+\infty )\to E\) by \(G(t)=g(e^{-t})\). Replacing \((x_{1},\ldots , x_{n})\) and \((y_{1},\ldots , y_{n})\) by \((e^{-t_{1}},\ldots , e^{-t_{n}})\) and \((e^{-s_{1}},\ldots , e^{-s_{n}})\), respectively, in (2.13), we get
for all \(t_{1}, \ldots , t_{n}, s_{1}, \ldots , s_{n}\in [0,+\infty )\). As in the proof of Theorem 2.1, there exists an additive mapping \(A :\mathbb{R}\to E\) such that
Letting \(x_{i}=-\ln t_{i}\) in the above inequality and using the definitions of G and g, we obtain
Applying \((A_{2})\) and (2.14), we get
If we define the function \(\Delta :(0,1]\to E\) by \(\Delta (x)=xA(-\ln x)\), then \(\Delta (xy)=\Delta (x)y+x\Delta (y)\) for all \(x,y\in (0,1]\) and (2.15) implies
□
Theorem 2.4
Let\(\mathbb{X}\)be a normed space. Suppose that\(f, g, h:(0,+\infty )\to \mathbb{X}\)are mappings satisfying
where\(\psi :(0,+\infty )\to [0,+\infty )\)is a mapping satisfying and
Then
Moreover, if
then
Proof
We define the functions \(F, G, H:(0,+\infty )\to \mathbb{X}\) by
By (2.16), we have
Then from (2.19) we get
Therefore, for all \(a,b\in (0,+\infty )\), we have
Moreover, if \(\lim_{x\to \infty }\psi (a/x,x)=0\), then by replacing x with \(x/y\) in (2.19) and letting \(y\to \infty \) in the resulting inequality, we get
Consequently,
Hence we get \(f(ab)=af(b)+f(a)b-abf(1)\) for all \(a,b\in (0,+\infty )\). By a similar argument, we get the result if \(\lim_{x\to \infty }\psi (ax,1/x)=0\) for all \(a\in (0,+\infty )\). □
Corollary 2.5
Let\(\mathbb{X}\)be a normed space. Suppose that\(f, h:(0,+\infty )\to \mathbb{X}\)are mappings satisfying
where\(\psi :(0,+\infty )\to [0,+\infty )\)is a mapping satisfying and
Then
for all\(a,b\in (0,+\infty )\).
Corollary 2.6
Let\(\mathbb{X}\)be a normed space and\(p,q,r,s\in (-\infty ,1)\). Suppose thatε, δ, θare nonnegative real numbers and\(f, g:(0,+\infty )\to \mathbb{X}\)are mappings satisfying
Thenfandgsatisfy (2.21).
Corollary 2.7
Let\(\mathbb{X}\)be a normed space and\(p,q\in (-\infty ,1)\)with\(pq<0\). Suppose thatεis a nonnegative real number and\(f, g, h:(0,+\infty )\to \mathbb{X}\)are mappings satisfying
Thenf, g, andhsatisfy (2.17) and (2.18).
Remark 2.8
By similar reasoning as in the proof of Theorem 2.4, it can be shown that Theorem 2.4 is also valid if the domains of functions f, g, ψ are \((-\infty , 0)\) or \(\mathbb{R}\setminus \{0\}\).
The following theorem is an improved version of the main result in [6].
Theorem 2.9
Let\(\mathbb{X}\)be a normed space. Suppose thatεis a nonnegative real number and\(f, g, h:(0,+\infty )\to \mathbb{X}\)are mappings satisfying
Then
- \((i)\):
-
f, gsatisfy (2.17).
- \((\mathit{ii})\):
-
there exists a function\(\varphi : (0,+\infty )\to \mathbb{X}\)satisfying (1.1) and
$$ \bigl\Vert f(x)-\varphi (x)-xf(1) \bigr\Vert \leqslant 4x \varepsilon ,\quad x\in [1,+ \infty ). $$(2.23)
Moreover, φis unique on the domain\([1,+\infty )\).
Proof
\((i)\) follows from Theorem 2.4. It suffices to prove \((\mathit{ii})\). Letting \(x=1\), \(y=1\), and \(x=y=1\) in (2.22), respectively, we obtain
for all \(x,y\in (0,+\infty )\). Therefore
Define the functions \(F:(0,+\infty )\to \mathbb{X}\) and \(T:\mathbb{R}\to \mathbb{X}\) by
Then by (2.24) we have
Replacing x and y by \(e^{t}\) and \(e^{s}\) in (2.25), respectively, we get
Hence
Then by [5, Lemma 2.27] there exists a unique additive function \(A: \mathbb{R}\to \mathbb{X}\) such that
Therefore
We put \(\varphi (x):=xA(\ln x)\), \(x\in (0,+\infty )\). Then we get \(\varphi (xy)=x\varphi (y)+\varphi (x)y\) and (2.23). To prove the uniqueness of φ, let ψ be another function satisfying (2.23) and \(\psi (xy)=x\psi (y)+\psi (x)y\) for all \(x\in [1,+\infty )\). It is easy to see that \(\varphi (x^{n})=nx^{n-1}\varphi (x)\) and \(\psi (x^{n})=nx^{n-1}\psi (x)\) for all \(n\in \mathbb{N}\) and \(x\in [1,+\infty )\). Then
Hence
Letting \(n\to \infty \), we conclude that \(\varphi (x)=\psi (x)\) for all \(x\in [1,+\infty )\). □
For convenience, we use the following abbreviation for given mappings \(f, g, h:[c,+\infty )\to E\), where \(c\geqslant 0\):
Theorem 2.10
Let\(\{(E^{n}, \|\cdot\|_{n}): n\in \mathbb{N}\}\)be a multi-Banach space. Suppose thatεis a nonnegative real number and\(f, g, h:[1,+\infty )\to E\)are mappings satisfying
for all\(x_{1}, \ldots , x_{n}, y_{1}, \ldots , y_{n}\in [1,+\infty )\). Then there exists a function\(\Delta : (0,+\infty )\to E\)satisfying functional equation (1.1) and the following inequalities:
for all\(x_{1}, \ldots , x_{n}\in [1,+\infty )\)and
for all\(x_{1}, \ldots , x_{n}\in [1,c]\).
Proof
It follows from (2.26) that \(\|D_{g,h}f(x,y)\|\leqslant \varepsilon \) for all \(x, y\in [1,+\infty )\). Then g and h satisfy (2.17) by Theorem 2.9. Applying axiom \((A_{2})\) in (2.26), we infer
for all \(x_{1}, \ldots , x_{n}, y_{1}, \ldots , y_{n}\in [1,+\infty )\). Letting \(x_{1}= \cdots = x_{n}=1\), \(y_{1}=\cdots = y_{n}=1\), and \(x_{1}= \cdots = x_{n}=y_{1}=\cdots = y_{n}=1\) in (2.27), respectively, we obtain
for all \(x_{1}, \ldots , x_{n}, y_{1}, \ldots , y_{n}\in [1,+\infty )\). Let \(F, G, H:[1,+\infty )\to E\) be mappings defined by
Therefore we conclude from (2.27) and the above inequalities that
Letting \(T(t)=F(e^{t})\) and replacing \(x_{i}\) and \(y_{i}\) by \(e^{t_{i}}\) and \(e^{s_{i}}\) in the above inequality, respectively, we get
for all \(s_{1}, \ldots , s_{n}, t_{1}, \ldots , t_{n}\in [0,+\infty )\). By Theorem 2.2 there exists an additive function \(A:\mathbb{R} \rightarrow E\) such that
for all \(t_{1}, \ldots , t_{n}\in [0,+\infty )\). Hence
for all \(x_{1}, \ldots , x_{n}\in [1,+\infty )\). Define \(\Delta :(0,+\infty )\to E\) by \(\Delta (x)=x^{-1}A(\ln x)\). Then \(\Delta (xy)=x\Delta (y)+\Delta (x)y\) for all \(x, y\in (0,+\infty )\) and
for all \(x_{1}, \ldots , x_{n}\in [1,+\infty )\). Using axiom \((A_{2})\), we get
for all \(x_{1}, \ldots , x_{n}\in [1,c]\). □
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Park, C., Noori, B., Moghimi, M.B. et al. Local stability of mappings on multi-normed spaces. Adv Differ Equ 2020, 395 (2020). https://doi.org/10.1186/s13662-020-02858-9
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DOI: https://doi.org/10.1186/s13662-020-02858-9