Abstract
In this paper, using Brzdȩk and Ciepliński’s fixed point theorems in a 2-Banach space, we investigate approximate solution for the generalized inhomogeneous radical quadratic functional equation of the form
where f is a mapping on the set of real numbers, \(a, b\in\mathbf {R}_{+}\) and \(D(x,y)\) is a given function. Some stability and hyperstability properties are presented.
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1 Introduction
In this paper, N and R denote the sets of all positive integers, and real numbers, respectively. We put \(\mathbf {N}_{0}:=\mathbf{N}\cup{0}\), \(\mathbf{R}_{0}:=\mathbf{R}\setminus{0}\) and \(\mathbf{R}_{+}:=[0,\infty)\). Also, \(Y^{X}\) denotes the set of all functions from a nonempty set X to a nonempty set Y.
The study of stability problems for functional equations originates from a question of Ulam [28] concerning the stability of group homomorphisms. Popularly speaking, the question was “Under what conditions a mathematical object satisfying a certain property approximately must be close to an object satisfying the property exactly?” In the following year, Hyers [20] first partially answered Ulam’s question, and proved the Ulam stability of Cauchy function in Banach spaces. Aoki [5] and Rassias [27] generalized the Hyers’ results by allowing the Cauchy difference to become unbounded. During the last decades, the Ulam–Hyers–Rassias stability of functional equations has been extensively investigated and generalized by many mathematicians (see [2, 3, 6,7,8,9, 11,12,13,14, 16, 24, 26, 29] and the references therein).
Recently, a lot of papers (see, for instance, [1, 4, 15, 17,18,19, 21,22,23]) on the stability of radical function equations have been published. The functional equation
is called a radical quadratic functional equation. Kim et al. [23] investigated the generalized Hyers–Ulam–Rassias stability problem of Eq. (1) in quasi-β-Banach spaces using the direct method. Khodaei et al. [22] introduced and solved the generalized radical quadratic functional equation
They established some stability results in 2-normed spaces by using the direct method, and proved new theorems about the generalized Ulam stability by using subadditive and subquadratic functions in p-2-normed spaces. Cho et al. [15] proved the generalized Hyers–Ulam stability results for Eq. (2) in quasi-β-Banach spaces by using subadditive and subquadratic functions. Using Brzdȩk’s fixed point theorem, Aiemsomboom et al. [1] and Kang [21] investigated the stability of Eqs. (1) and (2), respectively, where f is a self-mapping on R.
Let \((Y, \Vert \cdot,\cdot \Vert )\) be a 2-Banach space, \(D: \mathbf {R}^{2}\rightarrow Y\) a given function, and let \(a, b\in\mathbf{R}_{+}\) be fixed. The purpose of this paper is to prove stability and hyperstability results for the generalized inhomogeneous quadratic radical functional equation
in a 2-Banach space using Brzdȩk and Ciepliński’s fixed point results in [11].
2 Preliminaries
Let us recall some basic definitions and facts concerning 2-Banach spaces (see, for instance, [11, 17, 18, 25]).
Definition 1
Let X be a linear space over R with \(\dim X\geq2\) and let \(\Vert \cdot,\cdot \Vert :X\times X\rightarrow\mathbf {R}_{+}\) be a function satisfying the following properties:
-
(1)
\(\Vert x,y \Vert = 0\) if and only if x and y are linearly dependent;
-
(2)
\(\Vert x,y \Vert = \Vert y,x \Vert \) for \(x, y\in X\);
-
(3)
\(\Vert r x,y \Vert = |r| \Vert x,y \Vert \) for \(r\in\mathbf{R}\) and \(x, y\in X\);
-
(4)
\(\Vert x + y,z \Vert \leq \Vert x,z \Vert + \Vert y,z \Vert \) for \(x, y,z \in X\).
Then the pair \((X, \Vert \cdot,\cdot \Vert )\) is called a 2-normed space.
If \(x\in X\) and \(\Vert x,y \Vert =0\) for all \(y\in X\), then \(x=0\). Moreover, the functions \(x\rightarrow \Vert x,y \Vert \) are continuous functions of X into \(\mathbf{R}_{+}\) for each fixed \(y\in X\).
Definition 2
Let \(\{x_{n}\}\) be a sequence in a 2-normed space X.
-
(1)
A sequence \(\{x_{n}\}\) in a 2-normed space is called a Cauchy sequence if there are linear independent \(y,z\in X\) such that
$$\lim_{n,m\rightarrow\infty} \Vert x_{n}-x_{m},y \Vert =0= \lim_{n,m\rightarrow\infty} \Vert x_{n}-x_{m},z \Vert ; $$ -
(2)
A sequence \(\{x_{n}\}\) is said to be convergent if there exists an \(x\in X\) such that \(\lim_{n\rightarrow\infty} \Vert x_{n}-x,y \Vert =0\) for all \(y\in X\). Then, the point x is called the limit of the sequence \(\{x_{n}\}\), which is denoted by \(\lim_{n\rightarrow\infty}x_{n}=x\);
-
(3)
If every Cauchy sequence in X converges, then the 2-normed space X is called a 2-Banach space.
It is easily seen that \((\mathbf{R}^{2}, \Vert \cdot,\cdot \Vert )\) is a 2-Banach space, where the Euclidean 2-norm \(\Vert \cdot,\cdot \Vert \) is defined by
The next example following from [11, Proposition 2.3].
Example 1
If \((X, \langle\cdot, \cdot\rangle)\) is a real Hilbert space, then \((X, \Vert \cdot,\cdot \Vert )\) is a 2-Banach space, where \(\Vert \cdot,\cdot \Vert \) is given by
3 Fixed point theorems
Recently, Brzdȩk and Ciepliński [11] proved a new fixed point theorem in 2-Banach spaces and showed its applications to the Ulam stability of some single-variable equations and the most important functional equation in several variables, namely, the Cauchy equation. And they extended the fixed point result to the n-normed spaces in [10].
Let us introduce the following hypotheses:
-
(H1)
X is a nonempty set, \((Y, \Vert \cdot, \cdot \Vert )\) is a 2-Banach space, \(Y_{0}\) is a subset of Y containing two linearly independent vectors;
-
(H2)
\(j\in\mathbf{N}, f_{1},\dots,f_{j}: X\rightarrow X\), \(g_{1},\dots,g_{j}: Y_{0}\rightarrow Y_{0}\), and \(L_{1},\dots,L_{j}:X\times Y_{0}\rightarrow\mathbf{R}_{+}\) are given maps;
-
(H3)
\(\mathcal{T}: Y^{X}\rightarrow Y^{X}\) is an operator satisfying the inequality
$$ \bigl\Vert (\mathcal{T} \xi) (x)-(\mathcal{T} \eta) (x),z \bigr\Vert \leq\sum_{i=1}^{j} L_{i}(x, y) \bigl\Vert \xi \bigl(f_{i}(x) \bigr)-\eta \bigl(f_{i}(x) \bigr),g_{i}(z) \bigr\Vert , $$(4)where \(\xi, \eta\in Y^{X}, x\in X, z\in Y_{0}\);
-
(H4)
\(\varLambda:\mathbf{R}_{+}^{X\times Y_{0}}\rightarrow\mathbf {R}_{+}^{X\times Y_{0}}\) is an operator defined by
$$ (\varLambda\delta) (x, z):=\sum_{i=1}^{j} L_{i}(x,z)\delta \bigl(f_{i}(x), g_{i}(z) \bigr), \delta\in\mathbf{R}_{+}^{X\times Y_{0}},\quad x\in X, z\in Y_{0}. $$(5)
Now, we are in a position to present the above mentioned fixed point result. We use it to assert the existence of a unique fixed point of operator \(\mathcal{T}: Y^{X}\rightarrow Y^{X}\).
Theorem 1
Let hypotheses (H1)–(H4) hold and functions \(\epsilon: X\times Y_{0}\rightarrow\mathbf{R}_{+}\) and \(\varphi: X\rightarrow Y\) fulfill the following two conditions:
and
Then, there exists a unique fixed point ψ of \(\mathcal{T}\) with
Moreover,
4 The main results
In this section, we investigate the stability and hyperstability of the generalized inhomogeneous radical quadratic functional equation (3) in 2-Banach spaces by using Theorem 1. In what follows, we assume that \(a, b\in\mathbf{N}\) are fixed, \((Y, \Vert \cdot ,\cdot \Vert )\) is a 2-Banach space, and \(Y_{0}\) is a subset of Y containing two linearly independent vectors.
Theorem 2
Let \(h_{1},h_{2}:\mathbf{R}_{0}\times Y_{0}\rightarrow\mathbf{R}_{+}\) be two functions such that
where
where \(x\in\mathbf{R}_{0}, z\in Y_{0}, i=1,2, n\in\mathbf{N}\). Suppose that \(f:\mathbf{R}\rightarrow Y\) satisfies the inequality
where \(x,y\in\mathbf{R}_{0},z\in Y_{0}\). Then there exists a unique solution \(Q: \mathbf{R}\rightarrow Y\) of (2) such that
where
Proof
Putting \(y=mx\) in (10), we obtain that
where \(m\in\mathbf{N}, x\in\mathbf{R}_{0}, z\in Y_{0}\), and so
where \(m\in\mathbf{N}, x\in\mathbf{R}_{0}, z\in Y_{0}\).
For each \(m\in\mathbf{N}\), we define the operators \(\mathcal{T}_{m}: Y^{\mathbf{R}_{0}} \rightarrow Y^{\mathbf{R}_{0}}\) and \(\varLambda_{m}: \mathbf {R}_{+}^{\mathbf{R}_{0}\times Y_{0} } \rightarrow\mathbf{R}_{+}^{\mathbf {R}_{0}\times Y_{0} }\) by
where \(x\in\mathbf{R}_{0}, \xi\in Y^{\mathbf{R}_{0}}, \delta\in\mathbf {R}_{+}^{\mathbf{R}_{0}\times Y_{0} }, z\in Y_{0}\). Then the operator \(\varLambda _{m}\) has the form (5) with \(X:=\mathbf{R}_{0}\), \(j=2\), \(f_{1}(x):=\sqrt{(a+bm^{2})x^{2}},f_{2}(x):=mx\), \(g_{1}(z)=g_{2}(z):=z, L_{1}(x,z):=\frac{1}{a} \) and \(L_{2}(x,z):= \frac{b}{a} \) for all \(x\in \mathbf{R}_{0}\) and \(z\in Y_{0}\). Next, put
and observe that
where \(m\in\mathbf{N}, x\in\mathbf{R}_{0},z\in Y_{0}\). Then, inequality (13) can be rewritten as
and we have
for any \(x\in\mathbf{R}_{0}, \xi, \eta\in Y^{\mathbf{R}_{0}}, z\in Y_{0}\). Therefore,
so (4) holds for \(\mathcal{T}:= \mathcal{T}_{m}\) with \(m\in \mathbf{N}\). By the definition of \(\lambda_{i}(n)\), we have
whence, using induction, we get
Indeed, for \(n=0\), (17) is obviously true. Next, we will assume that (17) holds for \(n=j\), where \(j\in \mathbf{N}\). Then, we have
This shows that (17) holds for \(n=j+1\). Now we can conclude that inequality (17) holds for all \(n\in \mathbf{N}_{0}\). Therefore, by (17), we obtain that
for all \(x\in\mathbf{R}_{0},z\in Y_{0}\) and \(m\in M_{0}\). Thus, according to Theorem 1, for any \(m\in M_{0}\), there exists a unique fixed point \(Q_{m}': \mathbf{R}_{0}\rightarrow Y\) of \(\mathcal{T}_{m}\), which satisfies the estimate
where \(x\in\mathbf{R}_{0}, z\in Y_{0}, m\in M_{0}\). Moreover,
and for any \(m\in M_{0}\), the function \(Q_{m}: \mathbf{R}\rightarrow Y\), given by the formula
is a solution of the equation
Now, we show that
for any \(x,y\in\mathbf{R}_{0}, z\in Y_{0}, m\in M_{0}\) and \(n\in\mathbf{N}_{0}\).
Since the case \(n=0\) follows immediately from (10), take \(j\in\mathbf{N}_{0}\) and assume that (20) holds for \(n=j\), \(x,y\in\mathbf{R}_{0}\), \(m\in M_{0}\) and \(z\in Y\). Then, by (16), we get
Thus, by induction, we have shown that (20) holds for any \(n\in\mathbf{N}_{0}\), \(x,y\in\mathbf{R}_{0}, z\in Y_{0}\) and \(m\in M_{0}\). Letting \(n\rightarrow\infty\) in (20) and using Lemmas 2.1 and 2.2 in [11], we obtain that
This way, for each \(m\in M_{0}\), we obtain a function \(Q_{m}\) such that (21) holds for \(x,y\in\mathbf{R}\) and
Let \(L>0\) be a constant. Next, we will see that every generalized radical quadratic mapping \(Q:\mathbf{R}\rightarrow Y\) satisfying the inequality
is equal to \(Q_{m}\) for any \(m\in M_{0}\). To do this, fix \(s \in M_{0}\) and let \(Q:\mathbf{R}\rightarrow Y\) be a generalized radical quadratic mapping satisfying (23). By (18), we have
where \(L_{0}=aL(1-k_{s})+ \lambda_{2}(s^{2})\). Observe also that Q and \(Q_{s}\) are solutions of equation (19) for any \(m\in M_{0}\).
Now, we will see that, for any \(j\in\mathbf{N}_{0}\),
The case \(j=0\) follows from the previous inequality. Fix a \(j\in\mathbf {N}_{0}\) and assume that (25) holds. Then, by (16), we get
Thus (25) is valid for any \(j\in\mathbf{N}_{0}\). Letting \(j\rightarrow\infty\) in (25) and using Lemma 2.1 in [11], we get
which, together with \(Q(0)=Q_{s}(0)=0\), gives \(Q=Q_{s}\). This means that \(Q_{m}=Q_{s}\) for any \(m\in M_{0}\). Therefore, by (18), we have
Hence, we get inequality (11) with \(Q:= Q_{s}\). □
In a similar way, one can prove the following.
Theorem 3
Let \(H:\mathbf{R}_{0}\times Y_{0}\rightarrow\mathbf{R}_{+}\) be a function such that
where
Suppose that \(f:\mathbf{R}\rightarrow Y\) satisfies the inequality
where \(x,y\in\mathbf{R}_{0},z\in Y_{0}\). Then there exists a unique solution \(Q: \mathbf{R}\rightarrow Y\) of (2) such that
where
From the above theorems we can obtain results analogous to Theorems 2 and 3 for the inhomogeneous radical quadratic functional equation.
Corollary 1
Let \(h_{1},h_{2}:\mathbf{R}_{0}\times Y_{0}\rightarrow\mathbf{R}_{+}\), \(f:\mathbf{R}\rightarrow Y\) and \(D: \mathbf{R}^{2} \rightarrow Y\) such that (9) holds, and
where \(x,y\in\mathbf{R}_{0},z\in Y_{0}\). Assume that (3) has a solution \(f_{0}:\mathbf{R}\rightarrow Y\). Then there exists a unique solution \(F: \mathbf{R}\rightarrow Y\) of (3) such that
where \(\lambda_{0}(x,z)\) is defined as in Theorem 2.
Proof
Write \(f_{1}:=f-f_{0}\). Then, we have
and, according to Theorem 2, there is a unique solution \(Q: \mathbf{R}\rightarrow Y\) of (2) such that
where \(\lambda_{0}(x,z)\) is defined as in Theorem 2. Let \(F=f_{0}+Q\). Then F is a solution to (3) and (33) holds. The uniqueness of F follows from the uniqueness of Q (see [6, Corollary 4]). □
Corollary 2
Let \(H:\mathbf{R}_{0}\times Y_{0}\rightarrow\mathbf{R}_{+}\), \(f:\mathbf {R}\rightarrow Y\) and \(D: \mathbf{R}^{2} \rightarrow Y\) such that (29) holds, and
where \(x,y\in\mathbf{R}_{0},z\in Y_{0}\). Assume that (3) admits a solution \(f_{0}:\mathbf{R}\rightarrow Y\). Then there exists a unique solution \(F: \mathbf{R}\rightarrow Y\) of (3) such that
where \(\rho_{0}(x,z)\) is defined as in Theorem 3.
Corollaries 1 and 2 yield at once the following hyperstability results.
Corollary 3
Let \(h_{1},h_{2}:\mathbf{R}_{0}\times Y_{0}\rightarrow\mathbf{R}_{+}\) be functions such that
where \(\lambda_{i}(\cdot)\ (i=1,2)\) are defined as in Theorem 2. Assume that Eq. (3) has a solution \(f_{0}\). Then any function \(f:\mathbf{R}\rightarrow Y\), which satisfies \(f(0)=f_{0}(0)\) and inequality (32), is a solution of (3).
Corollary 4
Let \(H:\mathbf{R}_{0}\times Y_{0}\rightarrow\mathbf{R}_{+}\) be function such that
where \(\rho(\cdot)\) is defined as in Theorem 3. Assume that (3) has a solution \(f_{0}\). Then any function \(f:\mathbf{R}\rightarrow Y\), which satisfies \(f(0)=f_{0}(0)\) and inequality (34), is a solution of (3).
According to Corollaries 3 and 4, we derive the following particular cases.
Corollary 5
Let \(h_{1},h_{2}:\mathbf{R}_{0}\times Y_{0}\rightarrow\mathbf{R}_{+}\) be functions such that
where \(\lambda_{i}(\cdot)\ (i=1,2)\) are defined as in Theorem 2. Assume that (3) has a solution \(f_{0}\). Then any function \(f:\mathbf{R}\rightarrow Y\), which satisfies \(f(0)=f_{0}(0)\) and inequality (32), is a solution of (3).
Corollary 6
Let \(H:\mathbf{R}_{0}\times Y_{0}\rightarrow\mathbf{R}_{+}\) be function such that
where \(\rho(\cdot)\) is defined as in Theorem 3. Assume that (3) has a solution \(f_{0}\). Then any function \(f:\mathbf{R}\rightarrow Y\), which satisfies \(f(0)=f_{0}(0)\) and inequality (34), is a solution of (3).
Next, we derive some hyperstability results for particular forms of \(h_{1},h_{2},H\) and (3).
Corollary 7
Let \(\theta\in\mathbf{R}_{+}\) and let \(p,q\in\mathbf{R}\) be such that \(p+q<0\). Assume that Eq. (3) has a solution \(f_{0}\). If \(f:\mathbf{R}\rightarrow Y\) satisfies \(f(0)=f_{0}(0)\) and the inequality
then f is a solution of (3).
Proof
Define \(h_{1},h_{2}:\mathbf{R}_{0}\times Y_{0}\rightarrow\mathbf{R}_{+}\) by \(h_{1}(x^{2},z)=\theta_{1}|x|^{p}\) and \(h_{2}(y^{2},z)=\theta_{2} |y|^{p}\), where \(\theta_{1},\theta_{2}\in\mathbf{R}_{+}\) with \(\theta=\theta_{1} \theta_{2}\). Then, we have
Similarly, we get \(\lambda_{2}(n)=n^{q/2}\) for any \(n\in\mathbf{N}\). Thus,
As \(p,q\in\mathbf{R}\) with \(p+q<0\), either \(p<0\) or \(q<0\). Hence, by inequality (39), one can see that it is sufficient to consider only the case \(q<0\), and thus
So, we can apply Corollary 5. □
Corollary 8
Let \(\theta\in\mathbf{R}_{+}\) and let \(p,q\in\mathbf{R}\) be such that \(p+q<0\). If \(f:\mathbf{R}\rightarrow Y\) satisfies \(f(0)=0\) and the inequality
then f is a solution of (2).
Similarly, we can prove the following.
Corollary 9
Let \(\theta\in\mathbf{R}_{+}\) and consider \(p\in\mathbf{R}\) with \(p<0\). Assume that (3) has a solution \(f_{0}\). If \(f:\mathbf{R}\rightarrow Y\) satisfies \(f(0)=f_{0}(0)\) and the inequality
where \(x,y\in\mathbf{R}_{0}, z\in Y\), then f is a solution of (3).
Corollary 10
Let \(\theta\in\mathbf{R}_{+}\) and consider \(p\in\mathbf{R}\) with \(p<0\). If \(f:\mathbf{R}\rightarrow Y\) satisfies \(f(0)=0\) and the inequality
then f is a solution of (2).
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The authors would like to thank the editor and referees for careful reading the original manuscript and giving comments which were useful for improving the manuscript.
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This research is supported by the Natural Science Pre-research Foundation of Shaanxi University of Science and Technology under grant 2018BJ-35.
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Ding, Y., Xu, TZ. Approximate solution of generalized inhomogeneous radical quadratic functional equations in 2-Banach spaces. J Inequal Appl 2019, 31 (2019). https://doi.org/10.1186/s13660-019-1973-2
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DOI: https://doi.org/10.1186/s13660-019-1973-2