Operator Methods in Approximation TheoryView this Special Issue
On a New Stability Problem of Radical th-Degree Functional Equation by Brzdęk’s Fixed-Point Method
In this paper, we introduce the radical th-degree functional equation of the form with a positive integer , discuss its general solutions, and prove new Hyers-Ulam-type stability results for the equation by using Brzdęk’s fixed-point method.
In mathematical analysis, we often deal with the following question: Under what conditions should a mathematical object satisfying certain properties approximately be close to the one satisfying the properties exactly? If we consider a functional equation, then we can ask the same question: When could the approximates to a functional equation be close to the solution of the equation? Then, there would be an issue of error estimation between the approximates and the solution of a functional equation that we will investigate in this paper, not only the process of finding the solution of the equation. Such a fundamental question for functional equations led to the theory of Hyers-Ulam stability.
The Hyers-Ulam stability problem of functional equations was first raised in a talk at the University of Wisconsin. In , a Polish-American mathematician called Ulam  proposed the stability problem of a group homomorphism: When does a linear mapping near an approximately linear mapping exist?
In , Hyers  gave the first, affirmative, and partial solution to Ulam’s question with an additive function, , in Banach spaces as in the following theorem.
Theorem 1. Assume that and are Banach spaces. If a function satisfies the inequality for some and for all , then the limit exists for each and is the unique additive function (or the solution to Cauchy function) such that for any . Moreover, if is continuous in for each fixed , then is linear.
In , Aoki  provided a generalization of Hyers’ theorem with a positive monotone nondecreasing symmetric function of and involving a power , where Theorem 1 is a special case when . Also see [4, 5] for the generalization in terms of bordering transformations and an approximately linear mapping of addition and scalar multiplication, respectively.
For the last decades, stability problems of various functional equations, not only linear case, have been extensively investigated and generalized by many mathematicians (see [6–10]). One of the functional equations of the form
is called a radical quadratic functional equation and every solution to this functional equation is referred to as a radical quadratic function or mapping. The question of existence and uniqueness of the general solution of the functional equation (3) was answered by Brzdęk (see , p.196); i.e., a real-valued function ( stands for the set of real numbers) is a solution to (3) if and only if , , where the function is additive; i.e., it satisfies for all . Gordji and Parviz  and Cho et al.  investigated the Hyers-Ulam stability problems of the functional equations of the radical type as (3). In particular, Baker  applied for the first time a variant of Banach’s fixed-point theorem to obtain the stability of a functional equation in a single variable. For the applications and surveys of this approach in detail, see [15–17] where the paper  is an updated version of survey . Moreover, Brzdęk and Ciepliński  introduced the following existence theorem of the fixed point for nonlinear operator in metric spaces.
Theorem 2 (see ). Let be a nonempty set, a complete metric space, and a nondecreasing operator satisfying the hypothesis Suppose that is an operator satisfying the inequality where is a mapping which is defined by If there exist functions and such that and for all , then the limit exists for each Moreover, the function defined by is a fixed point of with for all
Then they used this result to prove the stability problem of functional equations in non-Archimedean metric spaces and obtained the fixed-point results in arbitrary metric spaces. Another version of the Brzdęk fixed-point method was also obtained from Theorem 2 (see  for details) as follows.
Theorem 3 (see ). Let be a nonempty set, a complete metric space, and given mappings. Suppose that and are two operators satisfying the following conditions: and for all , and If there exist and such that for all , then the limit exists for each Moreover, the function is a fixed point of with for all
Recently, Aiemsomboon and Sintunavarat  used Theorem 3 above to investigate a new type of stability for the radical quadratic functional equation of the form (3). We refer to [21, 22] for more results from the Brzdęk fixed-point method in the stability problems of various functional equations such as Drygas functional equations and the general linear equations. Also Kang  studied the stability problem for generalized quadratic radical functional equations by using Brzdęk’s fixed-point approach.
In this paper, we consider the radical th-degree functional equation of the form
for all positive integers and give an application of Brzdęk’s fixed-point method for the stability problem of the radical th-degree functional equation (16) (see [24–26] for the cases of of (16) with various approaches). As the radical quadratic functional equation (3), a function satisfies (16) if and only if it is of the form
with an additive function . For more extensions and generalizations of the solutions to functional equations of this radical type (16) and a different type of the radical th-degree functional equation of the form , we refer to [27, 28].
The stability results in this article will be an improvement and generalization of the stability problem of the radical quadratic, cubic, and quartic functional equations like (3). Throughout this paper, , and denote the set of nonnegative integers, the set of positive integers, and the set of nonnegative real numbers, respectively.
2. Stability of the Radical th-Degree Functional Equation
In this section, we will investigate the stability problems of the radical th-degree functional equation (16) as introduced earlier; i.e.,
for a positive integer by using Brzdęk’s fixed-point method; see Theorem 3 in the introduction.
Theorem 4. Let be a complete metric in which is invariant (i.e., for ). Assume that for each positive integer , is a function such that where for Also suppose that a function satisfies the inequality for all Then, there exists a unique radical th-degree mapping such that for all , where
Proof. Let be a positive integer. On taking in the inequality (21), we will see easily that where for all Now, let us define two operators and by and for all , (or a function ), and , respectively. Then we note that for each , as in Theorem 3 with for all Hence, (13) in Theorem 3 is satisfied by (26). With the properties of the metric , two inequalities (24) and (25) imply that and also we have for all and Thus, the metric satisfies (12) in Theorem 3. Let We; then, note that for all Applying the mathematical induction, it is not hard to show that for all and each Hence, for each and , we conclude that where , which means it satisfies the inequalities (14) in Theorem 3. Therefore, Brzdęk’s fixed-point method implies that exists for each and , and we have for all and (refer to Theorem 3 in the introduction). By using the mathematical induction on , we will show that for all and , where The case of follows from the inequality (24). Assume that it holds when By using the properties of , we will see that for all Letting in the inequality (35), we may obtain the following equality: for all and For each , the mapping is a solution of the radical th-degree functional equation; that is, for all Let be constant. Then, the mapping satisfying for all should be equal to for each Let be fixed and satisfy the inequality (39). We, then, note that where (the case is trivial, so we may exclude it here). Observe that and are solutions for (38) for each Now, we will show that for each , for all To show this, we will use the mathematical induction, again. The case follows from the previous inequality. Assume that it holds when the case Now, ; (37) and (38) imply that Hence, the inequality (41) holds whenever Letting in the inequality (41), we have where This means that for each Hence, we get that for all and Thus, we may conclude that the inequality (22) holds with and also the uniqueness follows from the equality (43).
Corollary 5. Let be a positive integer and a mapping such that Suppose that satisfies for all Then, there exists a unique radical th-degree functional equation such that for all .
Proof. By the definition as in Theorem 4, we will see that and These inequalities imply that for all Now, for each , let for each By our assumption, it is a sequence with subsequence such that ; that is, The inequalities (50) and (52) imply that that is, Thus, we have Letting as in Theorem 4, the inequality (47) follows from the inequality (22).
There were no data used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
The present research was conducted by the research fund of Dankook University in 2018.
S. M. Ulam, Problems in Modern Mathematics, Science Editions, Chapter 6, Wiley, New York, NY, USA, 1960.View at: MathSciNet
Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431–434, 1991.View at: Google Scholar
J. Brzdek, “Remarks on solutions to the functional equations of the radical type,” Advances in the Theory of Nonlinear Analysis and its Applications, vol. 1, pp. 125–135, 2017.View at: Google Scholar