Abstract
In this paper, by using fixed point theory, we investigate the generalized Hyers–Ulam stability of an \(\alpha \)-cubic functional equation in modular spaces.
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1 Introduction and preliminaries
In 1940, Ulam [23] asked the first question on the stability problem. In 1941, Hyers [9] solved the problem of Ulam. This result was generalized by Aoki [1] for additive mappings and by Rassias [20] for linear mappings by considering an unbounded Cauchy difference. In 1994, a further generalization was obtained by Găvruta [8]. Rassias [16,17,18,19] generalized Hyers result. During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [2, 5,6,7, 11, 15, 16, 21]). We also refer the readers to the books: Czerwik [3] and Hyers, Isac and Rassias [10].
The theory of modulars on linear spaces and the corresponding theory of modular linear spaces were founded by Nakano [12] and were intensively developed by Amemiya, Koshi, Shimogaki, Yamamuro [14, 25] and others.
Definition 1.1
Let X be a vector space over a field K (\(\mathbb {R}\) or \(\mathbb {C}\)). A generalized functional \(\rho : X\longrightarrow [0,\infty ]\) is called a modular if for arbitrary \(x, y \in X,\) \(\rho \) satisfies:
-
(a)
\(\rho (x) = 0\) if and only if \( x = 0\),
-
(b)
\( \rho (a x) = \rho (x)\) for every scalar a with \(|a| = 1,\)
-
(c)
\(\rho (a x+b y)\le \rho (x) + \rho (y)\), whenever \(a,b \ge 0\) and \(a+b =1\).
If we replace (c) by
(\(c'\)) \(\rho (a x+b y)\le a\rho (x) +b\rho (y)\), whenever \(a,b \ge 0\) and \(a+b =1,\) then the modular \(\rho \) is called convex. A modular \(\rho \) defines a corresponding modular space, i.e., the vector space \( X_{\rho }\) given by:
Definition 1.2
Let \(\{x_{n}\}\) and x be in \( X_{\rho }\). Then
-
(i)
The sequence \(\{x_{n}\}\), with \(x_{n} \in X_{\rho }\), is \(\rho \)-convergent to x if \(\rho (x_{n}-x)\rightarrow 0\) as \(n\rightarrow \infty \).
-
(ii)
The sequence \(\{x_{n}\}\), with \(x_{n} \in X_{\rho }\), is called \(\rho \)-Cauchy if \(\rho (x_{n}-x_{m})\rightarrow 0\) as \(n,m\rightarrow \infty \).
-
(iii)
A subset S of \( X_{\rho }\) is called \(\rho \)-complete if and only if any \(\rho \)-Cauchy sequence is \(\rho \)-convergent to an element of S.
Fatou property The modular \(\rho \) has the Fatou property if and only if \(\rho (x)\le \displaystyle \liminf _{n\rightarrow \infty }\rho (x_{n})\) whenever the sequence \(\{x_{n}\}\) is \(\rho \)-convergent to x. A function modular is said to satisfy the \(\Delta _{\alpha }\)-condition \((\alpha \in \mathbb {N}\),\(\alpha \ge 2)\) if there exists \(\kappa >0\) such that \(\rho (\alpha x)\le \kappa \rho (x),\) for all \(x \in X_{\rho }\).
Remark
\(\Delta _{\alpha }\)-condition implies \(\Delta _{2}\)-condition.
Definition 1.3
Let \( X_{\rho }\) be a modular space and C be a nonempty subset of \( X_{\rho }\). The self-map \(T:C\rightarrow C\) is said to be quasicontraction if there exists \(k < 1\) such that
for any \(x, y\in C.\)
Definition 1.4
Given a modular space \( X_{\rho }\), a nonempty subset \(C\subseteq X_{\rho }\), and a mapping \(T:C\rightarrow C\), the orbit of T around a point x is the set
the quantity
is then associated to T and is called the orbital diameter of T at x. In particular, if \(\delta _{\rho }(T)<\infty ,\) one says that T has a bounded orbit at x.
Theorem 1.5
([13]) Let \( X_{\rho }\) be a modular space such that \(\rho \) satisfies the Fatou property and let \(C\subseteq X_{\rho }\) be a \(\rho \)-complete subset. If \(T:C\rightarrow C\) is a quasicontraction and T has a bounded orbit at \(x_{0}\), then the sequence \(\{T^{n}x_{0}\}\) is \(\rho \)-convergent to a point \(\omega \in C.\)
Stability of quadratic and generalized Jensen functional equation in modular spaces have been investigated in [22] and [24].
In this paper, we investigate the generalized Hyers–Ulam stability of the \(\alpha \)- cubic functional equation
with \(\alpha \in {\mathbb {N}}, \alpha \ne 1\) via the extensive studies of fixed point theory in modular spaces.
2 Stability of \(\alpha \)-cubic functional equation (1.1)
Throughout this section, we assume that \(\rho \) is a convex modular on \(\rho \)-complete modular space \(X_{\rho }\) with the Fatou property such that satisfies the \(\Delta _{\alpha }\)-condition with \(0<\kappa \le \alpha \). In addition, let V be a linear space. For convenience, we use the following abbreviation for a given function \(f:V\longrightarrow X_{\rho }\):
with \(\alpha \in {\mathbb {N}}, \alpha \ne 1\) and for all \(x,y\in V.\) We shall need the following lemmas:
Lemma 2.1
If a mapping \(f:X\rightarrow Y\) satisfies the functional equation
with \(\alpha \in {\mathbb {N}}, \alpha \ne 1\) and for all \(x, y\in X,\) then f is cubic.
Proof
Replacing (x, y) with (0, 0) in (2.1), we get \(2\alpha (\alpha ^{2}-1)f(0)=0\) with \(\alpha \in {\mathbb {N}}, \alpha \ne 1\). Therefore \(f(0)=0\). Replacing (x, y) with (0, x) and \((0,-x)\) in (2.1), we get, respectively, equations:
By adding these two equations, one can obtain \(f(-x)=-f(x)\). By using (2.2) and \(f(-x)=-f(x)\), we get \(f(\alpha x)=\alpha ^{3}f(x)\) with \(\alpha \in {\mathbb {N}}, \alpha \ne 1\) and for all \(x\in X\) (See [4]). \(\square \)
Lemma 2.2
If a mapping \(f:X\rightarrow Y\) satisfies (1.1) for all \(x, y\in X,\) then f is cubic.
Proof
Replacing (x, y) with (0, 0) in (1.1), we get \(f(0)=0\). Replacing (x, y) with (x, 0) in (1.1), we get,
for all \(x\in X.\) By setting \(x=0\) and using (2.3), we get \(f(-y)=-f(y)\) for all \(y\in X,\) that is f is odd. Replacing (x, y) with \((x,-y)\) in (1.1) and using oddness of f, we get,
for all \(x,y\in X.\) It follows from (1.1) and (2.4) that
for all \(x,y\in X.\) It follows from Lemma 2.1 that f is cubic. \(\square \)
Theorem 2.3
Let \(\varphi : V^{2}\longrightarrow [0, +\infty )\) be a function such that
and
for all \(x,y\in V\) with \(L<1.\) Suppose that \(f:V\longrightarrow X_{\rho }\) satisfies the condition
for all \(x,y\in V\) and \(f(0)=0\). Then there exists a unique cubic mapping \(C_{\alpha }: V\longrightarrow X_{\rho }\) such that
for all \(x\in V\).
Proof
We consider the set
and define the function \(\overline{\rho }\) on M as follows,
We show that \(\overline{\rho }\) is a convex modular on M. It is also easy to verify that \(\overline{\rho }\) satisfies the axioms (a) and (b) of a modular. We will next show that \(\overline{\rho }\) is convex, and hence \((c')\) is satisfied. Let \(\epsilon >0\) be given. Then there exist real constants \(c_{1}>0\) and \(c_{2}>0\) such that
Also
for all \(x\in V.\) If \(a+b=1\) and \(a,b\ge 0\), then we get
so we get
This concludes that \(\overline{\rho }\) is a convex modular on M. Now we show that \(M_{\overline{\rho }}\) is \(\overline{\rho }\)-complete. Let \(\{g_{n}\}\) be a \(\overline{\rho }\)-Cauchy sequence in \(M_{\overline{\rho }}\) and let \(\epsilon > 0\) be given. There exists a positive integer \(n_{0}\in \mathbb {N}\) such that
for all \(n,m\ge n_{0}\). We have
for all \(x\in V\) and \(n,m\ge n_{0}\). Therefore if x is any given point of V, \(\{g_{n}(x)\}\) is a \(\rho \)-Cauchy sequence in \(X_{\rho }\). Since \(X_{\rho }\) is \(\rho \)-complete, so \(\{g_{n}(x)\}\) is convergent in \(X_{\rho }\), for each \(x\in V \). Hence, we can define a function \(g:V\rightarrow X_{\rho }\) by:
for all \(x\in V.\) Since \(\rho \) satisfies the Fatou property, it follows from (2.11) that
so
for all \(n\ge n_{0}\). Thus, \(\{g_{n}\}\) is \(\overline{\rho }\)-converges, so that \(M_{\overline{\rho }}\) is \(\overline{\rho }\)-complete.
Now we show that \(\overline{\rho }\) satisfies the Fatou property. Suppose that \(\{g_{n}\}\) is a sequence in \(M_{\overline{\rho }}\) which is \(\overline{\rho }\)- convergent to an element \(g\in M_{\overline{\rho }}\). Let \( \epsilon > 0\) be given. For each \(n\in \mathbb {N}\), let \(c_{n}\) be a constant such that
so
for all \(x\in V\). Since \(\rho \) satisfies the Fatou property, we have
Thus, we have
So \(\overline{\rho }\) satisfies the Fatou property. We consider the function \(\tau :M_{\overline{\rho }}\rightarrow M_{\overline{\rho }}\) defined by:
for all \(x\in V\) and \(g\in M_{\overline{\rho }}\). Let \(g,h\in M_{\overline{\rho }}\) and let \(c \in [0,1]\) be an arbitrary constant with \(\overline{\rho }(g-h)<c\). From the definition of \(\overline{\rho }\), we have \(\rho (g(x)-h(x))\le c\varphi (x,0)\) for all \(x\in V\). By (2.7) and the last inequality, we get
for all \(x\in V\). Hence, \(\overline{\rho }(\tau g-\tau h)\le L\overline{\rho }(g-h),\) for all \(g,h\in M_{\overline{\rho }}\), that is, \(\tau \) is a \(\overline{\rho }\)-contraction. Next, we show that \(\tau \) has a bounded orbit at f. Letting \(y =0\) in (2.8), we get
for all \(x\in V\). Replacing x with \(\alpha x\) in (2.17), we get
By using (2.17) and (2.18), we get
for all \(x\in V\). By induction, we can easily see that
for all \(x\in V\). It follows from inequality (2.20) that
for every \(x\in V\) and \(n,k \in \mathbb {N}\), By the definition of \(\overline{\rho }\), we conclude that
which implies the boundedness of an orbit of \(\tau \) at f. It follows from Theorem 1.5 that, the sequence \(\{\tau ^{n}f\}\) \(\overline{\rho }\)-converges to \(C_{\alpha }\in M_{\overline{\rho }}\). Now, by the \(\overline{\rho }\)-contractivity of \(\tau \), we have
Passing to the limit \(n\rightarrow \infty \) and applying the Fatou property of \(\overline{\rho }\), we obtain that
Therefore, \(C_{\alpha }\) is a fixed point of \(\tau \). Letting \(x = \alpha ^{n}x\) and \(y = \alpha ^{n}y\) in (2.8), we get
for all \(x,y\in V\). Therefore
Employing the limit \(n\rightarrow \infty \), we get
for all \(x,y\in V\). It follows from Lemma 2.2, that \(C_{\alpha }\) is cubic. By using (2.20), we get (2.9).
To prove the uniqueness of \(C_{\alpha },\) let \(C:V\rightarrow X_{\rho }\) be another cubic mapping satisfying (2.9). Then, C is a fixed point of \(\tau \).
which implies that \(\overline{\rho }\big (C_{\alpha }-C\big )=0\) or \(C_{\alpha }=C\). This completes the proof.
Corollary 2.4
Let X be a Banach space, \(\varphi : V^{2}\longrightarrow [0, +\infty )\) be a function such that
and
for all \(x,y\in V\) with \(L<1.\) Suppose that \(f:V\longrightarrow X\) satisfies the following condition
\(x,y\in V\) and \(f(x)=0.\) Then there exists a unique cubic mapping \(C_{\alpha }: V\longrightarrow X\) such that
for all \(x\in V\).
Proof
It is known that every normed space is modular space with the modular \(\rho (x) = \Vert x\Vert \) and satisfies the \(\Delta _{\alpha }\)-condition with \( \kappa = \alpha \). \(\square \)
Theorem 2.5
Let \(\varphi : V^{2}\longrightarrow [0, +\infty )\) be a function such that
and
for all \(x,y\in V\) with \(L<1.\) Suppose that \(f:V\longrightarrow X_{\rho }\) satisfies the condition
for all \(x,y\in V\) and \(f(0)=0.\) Then there exists a unique mapping \(C_{\alpha }: V\longrightarrow X_{\rho }\) such that
for all \(x\in V\).
Proof
We consider the set
and define the function \(\overline{\rho }\) on M as follows,
Similar to the proof of Theorem 2.3, we have:
-
1.
\(\overline{\rho }\) is a convex modular on M,
-
2.
\(M_{\overline{\rho }}\) is \(\overline{\rho }\)-complete.
-
3.
\(\overline{\rho }\) satisfies the Fatou property.
Now, we consider the function \(\tau :M_{\overline{\rho }}\rightarrow M_{\overline{\rho }}\) defined by:
for all \(x\in V\) and \(g\in M_{\overline{\rho }}.\) Let \(g,h\in M_{\overline{\rho }}\) and let \(c \in [0,1]\) be an arbitrary constant with \(\overline{\rho }(g-h)<c\). From the definition of \(\overline{\rho }\), we have \(\rho (g(x)-h(x))\le c\varphi (x,0)\) for all \(x\in V\). By the assumption and the last inequality, we get
for all \(x\in V\). Hence, \(\overline{\rho }(\tau g-\tau h)\le L\overline{\rho }(g-h),\) for all \(g,h\in \mathfrak {M_{\overline{\rho }}}\) that is, \(\tau \) is a \(\overline{\rho }\)-contraction. Next, we show that \(\tau \) has a bounded orbit at f. Letting \(y=0\) in (2.24), we get
for all \(x\in V\). Replacing x with \(\frac{x}{\alpha }\) in (2.26), we get
for all \(x\in V\). Replacing x with \(\frac{x}{\alpha }\) in (2.27), we get
for all \(x\in V\). By using (2.26), (2.27) and (2.28), we get
for all \(x\in V\). By induction, we can easily see that
for all \(x\in V\). It follows from inequality (2.30) that
for every \(x\in V\) and \(n,k \in \mathbb {N}\), By the definition of \(\overline{\rho }\), we conclude that
which implies the boundedness of an orbit of \(\tau \) at f. It follows from Theorem 1.5 that, the sequence \(\{\tau ^{n}f\}\) \(\overline{\rho }\)-converges to \(C_{\alpha }\in M_{\overline{\rho }}\). Now, by the \(\overline{\rho }\)-contractivity of \(\tau \), we have
Employing the limit \(n\rightarrow \infty \) and applying the Fatou property of \(\overline{\rho }\), we obtain that
Therefore, \(C_{\alpha }\) is a fixed point of \(\tau \). Letting \(x = \displaystyle \frac{x}{\alpha ^{n}}\) and \(y=\displaystyle \frac{y}{\alpha ^{n}}\) in (2.24), we get
for all \(x,y\in V\). Therefore
Passing to the limit \(n\rightarrow \infty \), we get
for all \(x,y\in V\). It follows from Lemma 2.2 that \(C_{\alpha }\) is cubic. By using (2.30), we get (2.25). \(\square \)
Corollary 2.6
Let X be a Banach space, \(\varphi : V^{2}\longrightarrow [0, +\infty )\) be a function such that
and
for all \(x,y\in V\) with \(L<1.\) Suppose that \(f:V\longrightarrow X\) satisfies the condition
for all \(x,y\in V\) and \(f(0)=0.\) Then there exists a unique cubic mapping \(C_{\alpha }: V\longrightarrow X\) such that
for all \(x\in V\).
Proof
It is known that every normed space is modular space with the modular \(\rho (x) = \Vert x\Vert \) and satisfies the \(\Delta _{\alpha }\)-condition with \( \kappa = \alpha \). \(\square \)
Remark 2.7
In Corollaries 2.4 and 2.6, by replacing \(\varphi \) with:
under suitable conditions, it is possible to obtain some corollaries.
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The first author was supported by University of Tabriz.
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Eskandani, G.Z., Rassias, J.M. Stability of general A-cubic functional equations in modular spaces. RACSAM 112, 425–435 (2018). https://doi.org/10.1007/s13398-017-0388-5
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DOI: https://doi.org/10.1007/s13398-017-0388-5