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We approach the generalized Ulam-Hyers-Rassias (briefly, UHR) stability of quadratic functional equations via the extensive studies of fixed point theory. Our results are obtained in the framework of modular spaces whose modulars are lower semicontinuous (briefly, lsc) but do not satisfy any relatives of

The question of stability for a generic functional equation was originated in 1940 by Ulam [

Under the setting of Banach spaces, Hyers [

The UHR stability and its relaxations play an important role in the studies of functional equations. Many functional equations are known for their complication in finding solutions. Knowing that a particular equation is stable makes it easier to find the solution. We can obviously see that to find an approximate solution is far less difficult than finding the exact solution. The stability then says that we can actually restrict ourselves to a neighborhood of the approximate solution.

In most cases, a functional equation is algebraic in nature whereas the stability is rather metrical. Hence, a normed linear space is a suitable choice to work with. However, several results in the literature have revealed that there are a great number of linear topological spaces whose appropriate topologies fail to be normable, especially in case of function spaces. Nakano [

Let

In the present paper, we consider the case where

In this section, we recollect some basic definitions and properties of a modular space. Conventionally, we write throughout the paper

Let

Note that, for a fixed

Unlike a norm, a modular needs not be continuous or convex in general. However, it often occurs that some weaker forms of them are assumed.

In case a modular

Let

A subset

Another unnatural behavior one usually encounter is that the convergence of a sequence

A modular

We have to be very careful about the convergence behaviors on multiples and sums of

In [

Given a modular space

Let

This section is contributed to the stability behavior of quadratic mappings in modular spaces. Unlike in the original UHR stability where the likeliness of being a solution is guaranteed by the difference being bounded with a sufficiently small positive constant, we rather prefer using a weaker setting where the difference is being dominated by a particular real-valued function of class

For a constant

Let

To prove this stability result, we will need the following lemma.

Suppose that every assumption of Theorem

The set

A generalized function

is a convex modular on

The corresponding modular space

Next, we show that a self-mapping

Suppose that every assumption of Theorem

We first show that

Next, we show that

Now, by the

Now, with the two lemmas above, we can finally give a simple proof to our main stability result, namely, Theorem

Since

If domination

Let

For the next two corollaries, we will consider the case where

Let

Let

Our results guarantee the stability of quadratic mappings, whose codomain is equipped with a convex and lsc modular, in both generalized and original senses. In contrast to the existing study of Sadeghi [

Technically, comparing the results in modular and normed spaces, we may see that the coefficient in the case of modular is significantly smaller (

We are also curious whether the multiple of

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are gratefully thankful for the referee’s valuable comments, which significantly improve materials in this paper. Mr. Parin Chaipunya was supported by the Thailand Research Fund and the King Mongkut’s University of Technology Thonburi through the Royal Golden Jubilee Ph.D. program (Grant no. PHD/0045/2555). In addition, this study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under Grant no. NRU57000621).