Abstract
We investigate the Hyers–Ulam–Rassias stability property of a quadratic functional equation. The analysis is done in the context of modular spaces. The type of stability considered here is very general in character which has been considered in various domains of mathematics. The speciality of the functional equation considered here is that it has a geometrical background behind its introduction. We approach the problem by applying a fixed point method for which a version of the contraction mapping principle in modular spaces is utilized. Also the results in this paper are established without using some familiar conditions on modular spaces.
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1. Introduction
We consider a quadratic functional equation in order to investigate the Hyers–Ulam–Rassias stability in modular metric spaces. The reason for considering the present problem is twofold. First, it is mathematically interesting to see how stability studies of functional equations on traditional Banach and Hilbert spaces can be extended to the relatively new structure of modular spaces. Second, the functional equation we consider here, that is, Pappus’ functional equation arises from some interesting geometrical considerations [1] and is supposed to be of further interest in future works due to its geometrical background.
Stability studies in various forms appear in almost all branches of mathematics. Here our purpose is to establish a stability result for a certain type of quadratic functional equation in modular metric spaces. The type of stability we investigate here is known as Hyers–Ulam–Rassias stability or briefly H-U-R stability. It is of general character and has been studied in several branches of mathematics, such as differential equations [2], fixed point theory [3], studies on isometries [4], etc. Such stability studies for functional equations are quite extensive; see [5]–[10].
As to the development of H-U-R stability, it was initiated as a mathematical question raised by Ulam [11] along with its extensions and partial answer given by Hyers [12] and Rassias [13]. Gruber [14] described the problem in its full generality, which is that H-U-R stability is said to exist for a mathematical object having certain property approximately when there is an approximation of the object by an object which satisfies the property exactly. It has a history of development over the years that followed. Several works in this line of development are noted in the references [15]–[18].
The present work is in the framework of modular spaces [19], [20]. The concept stems from the concept of “modular function,” which a function on a linear space with specific properties and in turn defines the corresponding space known as modular space. It has a general structure and several studies in many areas of functional analysis have been very justifiably extended to this space. For the motivations behind the development and technical details of this important concept of mathematics, we refer to [21], [22].
The study of functional equations in modular spaces was initiated in the work of W. M. Kozlowski [22] in 1988 following which several other works on this line of research were published in modular spaces [23]– [30].
In this paper, our approach to the problem of H-U-R stability is through a fixed point methodology.
2. Preliminaries
In this section, we recall some definitions and results concerning modular spaces.
Definition 1 ([19], [20]).
Let \(X\) be a vector space over a field \(\mathbb{K}\) (\(\mathbb{R}\) or \(\mathbb{C}\)). A generalized functional \(\rho: X\rightarrow[0,\infty]\) is called a modular if, for arbitrary \(x,y\in X\), it satisfies
-
(i)
\(\rho(x)=0\) if and only if \(x=0\);
-
(ii)
\(\rho(\alpha x)=\rho(x)\) for every scalar \(\alpha\) with \(|\alpha|=1\);
-
(iii)
\(\rho(z)\leq \rho(x)+\rho(y)\) whenever \(z\) is a convex combination of \(x\) and \(y\).
If (iii) is replaced by
-
(iii)*
\(\rho(\alpha x+\beta y)\leq \alpha\rho(x)+\beta\rho(y)\) if and only if \(\alpha+\beta=1\) and \(\alpha,\beta \geq 0\),
then, we say that \(\rho\) is a convex modular.
The corresponding modular space, denoted by \(X_\rho\), is defined by
The modular space \(X_\rho\) can be equipped with a norm called the Luxemburg norm defined by
Remark 1.
We have the following properties:
-
(i)
For a fixed \(x\in X_\rho\), the valuation [31] \(\gamma\in K \to \rho (\gamma x)\) is increasing.
-
(ii)
\(\rho(x)\leq \delta\rho((1/\delta)x)\) for all \(x\in X_\rho\), provided that \(\rho\) is a convex modular and \(0<\delta\leq 1\).
-
(iii)
Every norm defined on \(X\) is a modular on \(X\). The converse is not always true. There are modulars which are not subadditive and, therefore, not norms.
-
(iv)
Also, if \(\rho\) is a convex modular on \(X\) and \(|\alpha|\leq1\), then \(\rho(\alpha\,x)\leq\, \alpha\,\rho(x)\) and also \(\rho(x)\leq\,1/2 \rho(2x)\) for all \(x\in\,X.\)
Definition 2.
Let \(X_\rho\) be a modular space, and let \(\{x_n\}\) be a sequence in \(X_\rho\). Then
-
(i)
\(\{x_n\}\) is said to be \(\rho\)-convergent to a point \(x\in X_\rho\), and we write \(x_n\xrightarrow{\rho} x\) if \(\rho(x_n-x)\rightarrow 0\) as \(n\rightarrow\infty\) [32].
-
(ii)
\(\{x_n\}\) is said to be \(\rho\)-Cauchy if, for any \(\epsilon>0\), \(\rho(x_n-x_m)<\epsilon\) for sufficiently large \(m,n\in\mathbb{N}\) [32].
-
(iii)
A subset \(K (\subset X_\rho\) ) is said to be \(\rho\)-complete if any \(\rho\)-Cauchy sequence is \(\rho\)-convergent [32].
Definition 3 ([32]).
Fatou Property. A modular \(\rho\) has the Fatou property if \(\rho(x)\leq \lim_{n\rightarrow\infty}\text{inf}\rho(x_{n})\) whenever the sequence \(\{x_n\}\) is \(\rho\)-convergent to \(x\).
Definition 4 ([33]).
\(\Delta_{\alpha}\)-Condition. A modular \(\rho\) is said to satisfy the \(\Delta_\alpha\)-condition if there exists a \(\kappa \geq 0\) such that \(\rho(\alpha\,x)\leq \kappa \rho(x)\) for all \(x\in X_\rho\) and \(\alpha\,\in\,N,\,\alpha\geq\,2\).
Remark 2.
In particular, a modular function \(\rho\) is said to satisfy the \(\Delta_2\)-condition if there exists a \(k>0\) such that \(\rho(2x)\leq k\rho(x)\) for all \(x\in X_\rho\) [8]. The \(\Delta_2\)-condition has been used in several works. Note that the theorem we prove below is obtained without the \(\Delta_2\)-condition.
Definition 5 ([29]).
Given a modular space \(X_\rho\), a nonempty subset \(C\subset X_\rho\), and a mapping \(T:C\rightarrow C\), the orbit of \(T\) around a point \(x\in X_\rho\) is the set
The quantity \(\delta_\rho(x) := \sup \{\rho(u-v): u,v\in \mathbb{O}(x)\}\) is then associated with the orbit and is called the orbit diameter of \(T\) at \(x\), In particular, if \(\delta_\rho(x)<\infty\), one says that \(T\) has a bounded orbit at \(x\).
Definition 6 ([23]).
Let \(\rho\) be a modular defined on a vector space \(X\). Let \(C\subset X_\rho\) be nonempty. A mapping \(T:C\rightarrow C\) is a called \(\rho\)-Lipschitzian if there exists a constant \(L\geq 0\) such that
If \(L<1\), then \(T\) is called a \(\rho\)-contraction.
The following result is the modular space version of the Banach Contraction Principle.
Theorem 1 ([23]).
Assume \(X_\rho\) is \(\rho\) -complete. Let \(C\) be a nonempty \(\rho\) -closed subset of \(X_\rho\) . Let \(T:C\rightarrow C\) be a \(\rho\) -contraction mapping. Then \(T\) has a fixed point \(z\) if and only if \(T\) has a \(\rho\) -bounded orbit. Moreover if \(\rho(x-z)<\infty\) , then \(\{T^n(x)\}\) \(\rho\) -converges to \(z\) , for any \(x\in C\) .
If \(z_1\) and \(z_2\) are two fixed points of \(T\) such that \(\rho(z_1-z_2)<\infty\), then we have \(z_1=z_2\). In particular, if \(C\) is \(\rho\)-bounded, then \(T\) has a unique fixed point in \(C\).
Definition 7 ([1]).
A mapping \(Q : X\to Y\) is called a Pappus-type quadratic functional equation if
for all \(x,\,y \in X\) with \(X, Y\) are linear spaces.
In particular, if \(m=n=1\), then the above equation reduces to the quadratic functional equation
Remark 3.
There is a geometric motivation behind the definition of this functional equation. In fact, it was defined by Jun et al. in their work [1] where they described the motivation behind this definition. For details, we refer to the above-mentioned work [1].
3. Main Results: Stability of Generalized Pappus-Type Quadratic Equation in Modular Spaces
Throughout this paper, we assume that \(X\) is a linear space, \(\rho\) is a modular on \(X,\) \(X_\rho\) is a complete convex modular space, and also that the convex modular \(\rho\) has the Fatou property.
Theorem 2.
Let \(f: X \to X_\rho\) is a mapping which satisfies \(f(0)=0\) and the functional inequality
for all \(x,y\in X\) with \(\phi: X^2 \rightarrow[0,\infty)\) is a mapping satisfying
for all \(x, y\in X\) and some \(L\) with \(0<L<1\) and \(\lambda=(m+n)/n\) .
Then there exists a unique Pappus-type quadratic mapping \(Q: X\rightarrow X_\rho\) such that
Proof.
Putting \(y=0\) in (1), we obtain
Further, putting \(y=\frac{x}{n}\) in (1), we obtain
that is,
Therefore,
Thus,
Consider the set \(M=\{g: X\rightarrow X_\rho : g(0)=0 \}\) and define a mapping on \(M\) by
Now it follows from [34] that is a complete convex modular on \(M\).
Next, we will consider the corresponding modular space (see Definition 1 above) and the mapping defined by
We will now prove that \(J\) is a -contractive.
Let , and let \(c\in [0, \infty)\) be a constant with . Then
Now
Therefore
Hence \(J\) is a -contractive.
Now let us show that \(J\) has a bounded orbit at \({f}/{2}\), that is,
It follows from (4) that
Now
We used (4) and (6) to obtain the third line in the expression above/
Repeating the above steps \(n\) times, we obtain
Then, by the Fatou property and since \(m,\,n\in \mathbb{N}\), \( 0 < L < 1,\) for all \(x\in X\), we have
Thus, it follows from (8) that, for any \(m,\,n\in N\), we have
Here we have used (8).
Therefore, by the definition of we conclude that
According to Definition 6, we see that \(J\) has a bounded orbit at \(f/2\).
Moreover,
that is,
for all \(x\in X\) and for all \(m,n\in \mathbb{N}\).
Therefore, by Theorem 1, we have the following.
-
(i)
\(J\) has a fixed point \(Q\in M,\) that is, \(JQ=Q\), and hence \(Q(x)={Q(\lambda x)}/{\lambda^2}\) for all \(x\in X\).
-
(ii)
The sequence \(\{J^n({f}/{2})\}\) -converges to \(Q\).
Therefore,
So we can define
Again replacing \(x\) and \(y\) by \(\lambda^n x\) and \(\lambda^n y\) respectively in (1), we obtain
for all \(x\in X\) and \(m,\,n\in \mathbb{N}\).
Hence, taking limit as \(n\rightarrow \infty\) and using the Fatou property with \(0<L<1\), we see from the definition of \(Q\) that \(Q\) satisfies the Pappus equation
Therefore, from (9), taking the limit as \(n\to \infty\) and applying the fact that \(\rho\) has the Fatou property, we obtain
Uniqueness. Let \(Q^\prime : X\to X_\rho\) be another Pappus-type quadratic mapping satisfying (3). Then we have
for all \(x\in X\) and for all \(k\in \mathbb{N}\). Thus,
Therefore, by Theorem 1, we conclude that \(Q(x)=Q^\prime (x)\) for all \(x\in X\).
This completes the proof of the theorem.
Corollary 1.
Let \(\theta \geq\,0\) and \(X\) be a normed linear space and a mapping \(f:X\rightarrow X_\rho\) with \(f(0)=0\) satisfying inequality
for all \(x,y\in X\) and \(m,\,n\in \mathbb{N}, 0\leq p <1\) . Then there exists a unique Pappus-type quadratic mapping \(Q:X\rightarrow X_\rho\) such that
for all \(x\in X\) .
Proof.
Define \(\phi(x,y)=\theta(\|x\|^p+\|y\|^p)\) for all \(x,y\in X\). Then the result follows from the above theorem by taking \(L=2^{p-1}\).
Corollary 2.
Let \(X\) be a Banach space and a mapping \(f:X\rightarrow X_\rho\) with \(f(0)=0\) satisfying inequality
for all \(x,y\in X \) and \(m,\,n\in \mathbb{N}\) with \(L<1\) and \(\phi: X^2 \rightarrow[0,\infty)\) is a mapping satisfying (2). Then there exists a unique Pappus-type quadratic mapping \(Q:X\rightarrow X_\rho\) such that
for all \(x\in X\).
Proof.
It is known that every normed space is modular space with the modular \(\rho(x)=||x||\) then the corollary is proving exactly as Theorem 2.
Theorem 3.
Let \(f: X \to X_\rho\) is a mapping which satisfies (1) and \(\phi: X^2 \rightarrow[0,\infty)\) is a mapping satisfying
for all \(x, y\in X\) and some \(L\) with \(0<L<1\) and \(\lambda=(m+n)/n\).
Then there exists a unique Pappus-type quadratic mapping \(Q: X\rightarrow X_\rho\) such that
Proof.
The proof of the theorem is proved exactly as the proof of Theorem 2 only by considering the mapping by
Conclusions
In this paper, we do not use the \(\Delta_\alpha\)-condition (Definition 5) on the modular metric space. The use of this condition is often convenient for the proofs of similar results in modular spaces. For example, Sadeghi [34] established the Hyers–Ulam stability of a generalized Jensen functional equation in convex modular spaces fulfilling the \(\Delta_2\)-condition with \(0<k\leq 2\) and S. S. Kim et al. [35] proved the same fact for \(0<k<2\). It is a special trait of our theorem that it falls outside the category of the above-mentioned works insofar as the method of proof is concerned.
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Saha, P., Mondal, P. & Choudhury, B.S. Stability Property of Functional Equations in Modular Spaces: A Fixed-Point Approach. Math Notes 109, 262–269 (2021). https://doi.org/10.1134/S0001434621010302
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DOI: https://doi.org/10.1134/S0001434621010302