Abstract
We solve the additive -random operator inequality , in which are fixed and . Finally, we get an approximation of the mentioned additive -random operator inequality by direct technique.
1. Introduction
Park [1] introduced and solved the following additive -functional inequality:where and are fixed nonzero complex numbers with . Next, he proved the Hyers–Ulam stability of the additive -functional inequality (1) in Banach spaces.
In this paper, we get a generalization from Park’s results [1] in MB-spaces. Let be a probability measure space. Assume that and are Borel measureable spaces, in which U and are MB-spaces and is a random operator. In MB-spaces, first we solve the additive -random operator inequality:in which are fixed and .
Next, we get a random approximation of the additive-random operator (2).
2. Preliminaries
Let be the set of distribution mappings, i.e., the set of all mappings , writing for , such that is left continuous and increasing on . includes all mappings for which is one and is the left limit of the mapping at the point x, i.e., .
In , we define “” as follows:for each s in (partially ordered). Note that the function defined byis an element of , and is the maximal element in this space (for some more details, see [2–4]).
Definition 1 (see [2, 5]). Let . A continuous triangular norm (shortly, a -norm) is a continuous mapping κ from to I such that(a) and for all (b) for all (c) whenever and for all Some examples of the t-norms are as follows:(1)(2)(3) (: the Lukasiewicz t-norm)
Definition 2. (see [3]). Suppose that κ is a -norm, is a linear space and ξ is a mapping from to . In this case, the ordered tuple is called a Menger normed space (in short, MN-space) if the following conditions are satisfied: (MN1) for all if and only if (MN2) for all and with (MN3) for all and Let be a linear normed space. Then,defines a Menger norm, and the ordered tuple is an MN-space.
Let be a probability measure space. Assume that and are Borel measureable spaces, in which U and are complete FB-spaces. A mapping is said to be a random operator if for all u in U and . Also, T is a random operator, if be a -valued random variable for every u in U. A random operator is called linear if , almost everywhere for each in U and scalers, and Menger random bounded (in short, MR-bounded) if there exists a nonnegative real-valued random variable such thatalmost everywhere for each in U and .
Recently, some authors have published some papers on stability of functional equations in several spaces by the direct method and the fixed point method, for example, Banach spaces [6–8], fuzzy Menger normed algebras [9], fuzzy normed spaces [10], non-Archimedean random Lie -algebras [11], non-Archimedean random normed spaces [12], random multinormed space [13], random lattice normed spaces, and random normed algebras [14, 15]. In [16, 17], the authors studied the stability problem for fractional equations. Next, Cădariu et al. [18–20] applied the fixed point method to solve the stability problem, and their work was continued by Keltouma et al. [21], Park et al. [22, 23], Jung and Lee [24], and Brzdʁk and Ciepliński [25], see also [26, 27].
3. Approximation of Additive -Random Operator Inequality: Direct Technique
Now, we modify and generalize Park’s results [1]. First, we solve and investigate the additive -random operator inequality (2) in MN-spaces.
Lemma 1. Let be an MN-space. Let be a random operator satisfying and (2), and then T is additive.
Proof. Replacing by u in (2), we getSince , for each and ,almost everywhere for each and .
(2) and (8) imply thatand soalmost everywhere for each , , and .
Putting and in (10), we getalmost everywhere for each , , and .
Now, (10) and (11) imply thatfor all . Since , , almost everywhere for each and , which implies that T is additive.
We get an approximation of the additive -random operator inequality (2) in MN-spaces, by applying the direct technique.
Theorem 1. Let be an MN-space. Assume that be a distribution function such that there exists withalmost everywhere for each , , and . Suppose that be a random operator satisfying andin which are fixed and . Therefore, there is a unique additive random operator such thatalmost everywhere for each , , and .
Proof. Putting in (15), we have thatalmost everywhere for each , , and . Replacing u by in (17) and applying (13), we getwhich implies thatReplacing u by in (19), we getwhich tends to when tend to , and so the sequence is Cauchy in the complete MN-space and converges to a point . Now, for every , we have thatWhen tends to in (21), we have thatSince is arbitrary in (22), we have thatReplacing u and by and in (15) and using (14) imply that S satisfies Lemma 1 and hence is an additive random operator. Now, let be another additive random operator satisfies (16). For an arbitrary and , we have that and for each natural element m. Using (16), we have thatwhich implies that shows the uniqueness.
Corollary 1. Let be an MN-space, and . Suppose that be a random operator satisfying andin which are fixed and . Then, there exists a unique additive random operator such thatfor each , , and .
Proof. In Theorem 1, put for each and , and .
Theorem 2. Let be an MN-space. Assume that be a distribution function such that there exists an withalmost everywhere for each , , and . Suppose that be a random operator satisfying and (15). Therefore, there is a unique additive random operator such thatalmost everywhere for each , , and .
Proof. Putting in (15), we have thatalmost everywhere for each , , and . Replacing u by in (30) and applying (27), we getwhich implies thatReplacing u by in (32), we getwhich tends to when tend to , and so the sequence is Cauchy in the complete MN-space and converges to a point . Now, for every , we have thatWhen tends to in (34), we haveSince is arbitrary in (35), we haveReplacing u and by and in (15) and using (14) imply that S satisfies Lemma 1 and hence is an additive random operator. Now, let be another additive random operator satisfies (29). For an arbitrary and , we have that , and for each positive integer m. Using (29), we havewhich implies that shows the uniqueness.
Corollary 2. Let be an MN-space, and . Suppose that be a random operator satisfying and (25). Therefore, there is a unique additive random operator such thatfor each , , and .
Proof. In Theorem 2, put for each , and .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
S. Y. Jang was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-201807042748).