Abstract
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 -conditions.
1. Introduction
The question of stability for a generic functional equation was originated in 1940 by Ulam [1]. Concerning a group homomorphism, Ulam posted the question asking how likely to an automorphism a function should behave in order to guarantee the existence of an automorphism near such functions.
Under the setting of Banach spaces, Hyers [2] was the first to give an affirmative answer to Ulam’s question the following year. It was extended to the cases of additive mappings by Aoki [3] and linear mappings by Rassias [4], the latter of which has influenced many developments in the stability theory. This area is then referred to as the Ulam-Hyers-Rassias stability or, briefly, the UHR stability.
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 [5] and Musielak and Orlicz [6] successfully considered replacing a norm with a so-called modular. A modular yields less properties than a norm does, but it makes more sense in many special situations. It is still rational to assume some additional properties like some relaxed continuities or some -related conditions on a modular. Thus, it is reasonable to extend the framework of stability of functional equations into a more general setting of modular spaces, as considered by Sadeghi [7] in case of Cauchy and Jensen functional equations. Note that the stability results in [7] are obtained only in the cases where the induced modulars are convex and lsc and satisfy a typical class of -conditions.
Let and be two groups; a mapping is said to be quadratic if it satisfies the following functional equation: The UHR and generalized UHR stabilities have always been questioned in various settings, but none takes place in modular spaces. Skof [8] has proved that quadratic mappings are generalized UHR stable provided that and are normed and Banach spaces, respectively. It was later realized [9] that even when is an Abelian group, the same behavior is still guaranteed.
In the present paper, we consider the case where is a linear space and is a -complete modular space, where the scalar fields are arbitrary. Our main results are obtained by using the fixed point method under the assumptions that the modular is lsc and convex but not necessarily satisfies any -conditions.
2. Preliminaries
In this section, we recollect some basic definitions and properties of a modular space. Conventionally, we write throughout the paper , , and to denote, respectively, the set of all reals, complexes, and nonnegative integers.
Definition 1. Let be a vector space over a field ( or ). A generalized functional is called a modular if for arbitrary if and only if , for every scalar with , whenever is a convex combination of and .The corresponding modular space, denoted by , is then defined by
Remark 2. Note that, for a fixed , the valuation is increasing.
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.
Remark 3. In case a modular is convex, one has for all , provided that .
Definition 4. Let be a modular space and let be a sequence in . Then,(i) is -convergent to a point and write if as .(ii) is called -Cauchy if for any one has for sufficiently large .(iii)A subset is called -complete if any -Cauchy sequence is -convergent.
Another unnatural behavior one usually encounter is that the convergence of a sequence to does not imply that converges to , where is chosen from the corresponding scalar field. Thus, many mathematicians imposed some additional conditions for a modular to meet in order to make the multiples of converge naturally. Such preferences are referred to mostly under the term related to the -conditions.
A modular is said to satisfy the -condition if there exists such that for all . Some authors varied the notion so that only is required and called it the -type condition. In fact, one may see that these two notions coincide. There are still a number of equivalent notions related to the -conditions.
Remark 5. We have to be very careful about the convergence behaviors on multiples and sums of -convergent sequences. In general, we suppose that , for some , are sequences in in which they -converge to the points , respectively. Then, the averaged sequence -converges to .
In [10], Khamsi proved a series of fixed point theorems in modular spaces where the modulars do not satisfy -conditions. His results exploit one unifying hypothesis in which the boundedness of an orbit is assumed.
Definition 6. Given a modular space , a nonempty subset , and a mapping . The orbit of around a point is the set The quantity is then associated and is called the orbital diameter of at . In particular, if , one says that has a bounded orbit at .
Lemma 7 (see [10]). Let be a modular space whose induced modular is lsc and let be a -complete subset. If is a -contraction, that is, there is a constant such that and has a bounded orbit at a point , then the sequence is -convergent to a point .
3. Generalized UHR Stability of Quadratic Mappings
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 defined below.
Definition 8. For a constant and a linear space , one defines to be the collection of all nonnegative real-valued functions defined on with the following properties for all :
Theorem 9. Let be linear space, be a -complete modular space where is lsc and convex, and be a mapping with . Suppose that, for each , the following dominating condition holds: where with . Then, there exists the quadratic mapping such that for all . Equivalently, the quadratic mapping is generalized UHR stable.
To prove this stability result, we will need the following lemma.
Lemma 10. Suppose that every assumption of Theorem 9 holds. Then, the following statements hold.The set is a linear space.A generalized function defined for each by is a convex modular on .The corresponding modular space is the whole space and is -complete. is lsc
Proof. is trivial.
It is also easy to verify that satisfies the axioms (m1) and (m2) of a modular. We will next show that is convex, and hence (m3) is satisfied. Let be given. Then there exist and such that
Consecutively, we have
Thus, if and , we get
so that
Hence, we have
This concludes that is a convex modular on .
The fact that the corresponding modular space is the whole space is trivial, so we only show that is -complete. Let be a -Cauchy sequence in and let be given. There exists a positive integer such that for all . By the definition, we may see that
for all and . Thus, at each fixed , the sequence is a -Cauchy sequence. Since is -complete, is -convergent in for each . Hence, we can define a function by
for any . Since is lsc, it follows from 14 that
provided that . Thus, -converges, so that is -complete.
Suppose that is a sequence in which is -convergent to an element . Let be given. For each , let be a constant such that
Again, we have
Now, observe from the lower semicontinuity of that
Thus, we have
Since is arbitrary, we can finally conclude that is lsc
Next, we show that a self-mapping defined by has some fixed point accordingly to Lemma 7.
Lemma 11. Suppose that every assumption of Theorem 9 holds and is defined as in 21. Then, has some fixed point.
Proof. We first show that is a -contraction. Let , , and be an arbitrary constant with . Observe that we have
so that
Therefore, we have . Since are arbitrary, is a -contraction.
Next, we show that has a bounded orbit at , where is taken from the assumption. Let be arbitrary and set in 6; we get
Since , the above inequality yields
Since is convex, we obtain
Inductively, we may deduce for all that
Moreover, we may see that
Now, for each , we have
By the definition of , we conclude that
which implies the boundedness of an orbit of at . According to Lemma 7, the sequence -converges to some element, say .
Now, by the -contractivity of , one has
Passing towards and applying the lower semicontinuity of , we obtain that
Therefore, is a fixed point of .
Now, with the two lemmas above, we can finally give a simple proof to our main stability result, namely, Theorem 9.
Proof of Theorem 9. Since and are in provided that , we deduce from 6 that Furthermore, we have As from Lemma 11 and Remark 5, letting and applying the lower semicontinuity of , we deduce that That is, is a quadratic mapping, since every quadratic map is a fixed point of . On the other hand, it follows from inequality 28 that
If domination 6 is put to slightly stronger condition of a boundedness, we conclude the following classical UHR stability statement in modular spaces.
Corollary 12. Let be a linear space, be a -complete modular space where is lsc and convex, and be a mapping with . If there exist a constant such that then there exists the quadratic mapping such that
For the next two corollaries, we will consider the case where is actually a norm. Before stating the results, we have to note that if , then we also have for all constant . Thus, there is no loss of generality asserting the upcoming corollaries in the following form(s).
Corollary 13. Let be a linear space, be a Banach space, and be a mapping with . Suppose that, for each , there holds the inequality where with . Then, there exists a unique quadratic mapping such that
Corollary 14. Let be a linear space, be a Banach space, and be a mapping with . It there exists a constant such that then there exists a unique quadratic mapping such that
4. Concluding Remarks
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 [7], our proofs contain different techniques to avoid the usage of -conditions.
Technically, comparing the results in modular and normed spaces, we may see that the coefficient in the case of modular is significantly smaller ( to ). However, the appeared in modular space cases are apparently larger.
We are also curious whether the multiple of on the left of the inequality 6 can be dropped. This leaves a considerable interesting question for future research.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
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).