# On a New Stability Problem of Radicalnth-Degree Functional Equation by Brzdęk’s Fixed-Point Method

Authors: Dongseung Kang; Hoewoon B. Kim
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this paper, we introduce the radicalnth-degree functional equation of the form f(xn+ynn)=f(x)+f(y) with a positive integer n, discuss its general solutions, and prove new Hyers-Ulam-type stability results for the equation by using Brzdęk’s fixed-point method.


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## Body

## 1. Introduction

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. In1940, a Polish-American mathematician called Ulam [1] proposed the stability problem of a group homomorphism: When does a linear mapping near an approximately linear mapping exist?In1941, Hyers [2] gave the first, affirmative, and partial solution to Ulam’s question with an additive function, f(x+y)=f(x)+f(y), in Banach spaces as in the following theorem.Theorem 1.
Assume thatE1 and E2 are Banach spaces. If a function f:E1→E2 satisfies the inequality (1)fx+y-fx-fy≤ϵ for some ϵ≥0 and for all x,y∈E1, then the limit a(x)=limn→∞2-nf(2nx) exists for each x∈E1 and a:E1→E2 is the unique additive function (or the solution to Cauchy function) such that (2)fx-ax≤ϵ for any x∈E1. Moreover, if f(tx) is continuous in t for each fixed x∈E1, then a is linear.In1950, Aoki [3] provided a generalization of Hyers’ theorem with a positive monotone nondecreasing symmetric function of x and y involving a power 0≤p<0, where Theorem 1 is a special case when p=0. 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 (3)fx2+y2=fx+fyis 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 [11], p.196); i.e., a real-valued function f:R→R (R stands for the set of real numbers) is a solution to (3) if and only if f(x)=L(x2), x∈R, where the function L:R→R is additive; i.e., it satisfies L(x+y)=L(x)+L(y) for all x,y∈R. Gordji and Parviz [12] and Cho et al. [13] investigated the Hyers-Ulam stability problems of the functional equations of the radical type as (3). In particular, Baker [14] 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 [16] is an updated version of survey [15]. Moreover, Brzdęk and Ciepliński [18] introduced the following existence theorem of the fixed point for nonlinear operator in metric spaces.Theorem 2 (see [18]).
LetX be a nonempty set, (Y,d) a complete metric space, and Λ:YX→YX a nondecreasing operator satisfying the hypothesis (4)limn→∞Λδn=0foreverysequenceδnn∈Nwithlimn→∞δn=0inYX.
Suppose thatT:YX→YX is an operator satisfying the inequality (5)dTεx,Tμx≤Λ△ε,μx,ε,μ∈YX,x∈X
where△:(YX)2→R+X is a mapping which is defined by (6)△ε,μx≔dεx,μxforε,μ∈YX,x∈X.
If there exist functionsε:X→R+ and ϕ:X→Y such that (7)dTϕx,ϕx≤εx
and(8)ε∗x≔∑n∈N0Λnεx<∞
for allx∈X, then the limit (9)limn→∞Tnϕx
exists for eachx∈X. Moreover, the function ϕ∈YX defined by (10)ψx≔limn→∞Tnϕx
is a fixed point ofT with (11)dϕx,ψx≤ε∗x
for allx∈X.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 Theorem2 (see [19] for details) as follows.Theorem 3 (see [19]).
LetX be a nonempty set, (Y,d) a complete metric space, and f1,f2:X→X given mappings. Suppose that T:YX→YX and Λ:R+X→R+X are two operators satisfying the following conditions: (12)dTξx,Tμx≤dξf1x,μf1x+dξf2x,μf2x
and(13)Λδx≔δf1x+δf2x
for allξ,μ∈YX,δ∈R+X, and x∈X. If there exist ε:X→R+ and ϕ:X→Y such that (14)dTϕx,ϕx≤εxε∗x≔∑n=0∞Λnεx<∞
for allx∈X, then the limit limn→∞(Tnϕ)(x) exists for each x∈X. Moreover, the function ψ(x)≔limn→∞(Tnϕ)(x) is a fixed point of T with (15)dϕx,ψx≤ε∗x
for allx∈X.Recently, Aiemsomboon and Sintunavarat [20] 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 [23] studied the stability problem for generalized quadratic radical functional equations by using Brzdęk’s fixed-point approach.In this paper, we consider the radicalnth-degree functional equation of the form (16)fxn+ynn=fx+fyfor all positive integersn>0 and give an application of Brzdęk’s fixed-point method for the stability problem of the radical nth-degree functional equation (16) (see [24–26] for the cases of n=2,3,4 of (16) with various approaches). As the radical quadratic functional equation (3), a function f:R→R satisfies (16) if and only if it is of the form (17)fx=axn,x∈Rwith an additive functiona:R→R. For more extensions and generalizations of the solutions to functional equations of this radical type (16) and a different type of the radical nth-degree functional equation of the form f(xn+ynn)+f(|xn-ynn|)=2f(x)+2f(y), 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, N0,N, and R+ denote the set of nonnegative integers, the set of positive integers, and the set of nonnegative real numbers, respectively.


## 2. Stability of the Radicalnth-Degree Functional Equation

In this section, we will investigate the stability problems of the radicalnth-degree functional equation (16) as introduced earlier; i.e., (18)fxn+ynn=fx+fyfor a positive integern by using Brzdęk’s fixed-point method; see Theorem 3 in the introduction.Theorem 4.
Letd be a complete metric in R which is invariant (i.e., d(x+z,y+z)=d(x,y) for x,y,z∈R). Assume that for each positive integer n∈N, h:R+→R+ is a function such that (19)M0≔m∈N:smn+s1+mn<1≠∅
where(20)sl≔infr∈R+:hlxn≤rhxnforallx∈R,
forl∈N. Also suppose that a function f:R+→R satisfies the inequality (21)dfxn+ynn,fx+fy≤hxn+hyn
for allx,y∈R+. Then, there exists a unique radical nth-degree mapping R:R+→R such that (22)dfx,Rx≤s0hxn
for allx∈R+, where (23)s0≔inf1+smn1-smn-s1+mn:m∈M0.Proof.
Letm∈N be a positive integer. On taking y=mx in the inequality (21), we will see easily that (24)df1+mnxnn,fx+fmx≤cmx
wherecm(x)=(1+s(mn))h(xn)for all x∈R+. Now, let us define two operators Tm:RR+→RR+ and Λm:R+R+→R+R+ by (25)Tmεx≔ε1+mnxnn-εmx
and(26)Λmμx≔μ1+mnxnn+μmx
for allx∈R+, ε∈RR+(or a function ε:R+→R), and μ∈R+R+, respectively. Then we note that for each m∈N, Λ=Λm as in Theorem 3 with (27)f1x=1+mnxnn,f2x=mx
for allx∈R+. Hence, (13) in Theorem 3 is satisfied by (26). With the properties of the metric d, two inequalities (24) and (25) imply that (28)dTmfx,fx=df1+mnxnn-fmx,fx≤cmx
and also we have(29)dTmεx,Tmμx=dε1+mnxnn-εmx,μ1+mnxnn-μmx≤dε1+mnxnn,μ1+mnxnn+dεmx,μmx=dεf1x,μf1x+dεf2x,μf2x
for allx∈R+and ε,μ∈RR+. Thus, the metric d satisfies (12) in Theorem 3. Let m∈M0. We; then, note that (30)Λmcmx=cm1+mnxnn+cmmx=1+smnh1+mnxn+hmnxn≤1+smns1+mn+smnhxn
for allx∈R+. Applying the mathematical induction, it is not hard to show that (31)Λmkcmx≤1+smns1+mn+smnkhxn
for allx∈R+ and each k∈N. Hence, for each m∈M0 and x∈R+, we conclude that (32)cm∗x≔∑j=0∞Λmjcmx≤1+smn1-s1+mn-smnhxn
whereΛm0cm(x)=cm(x), which means it satisfies the inequalities (14) in Theorem 3. Therefore, Brzdęk’s fixed-point method implies that (33)Tmx≔limk→∞Tmkfx
exists for eachm∈M0 and x∈R+, and we have(34)dfx,Tmx≤cm∗x
for allm∈M0 and x∈R+(refer to Theorem 3 in the introduction). By using the mathematical induction on k∈N0, we will show that (35)dTmkfxn+ynn,Tmkfx+Tmkfy≤s1+mn+smnkhxn+hyn
for allx,y∈R+ and m∈M0, where Tm0f(x)=f(x). The case of k=0 follows from the inequality (24). Assume that it holds when k=t. By using the properties of d, we will see that (36)dTmt+1fxn+ynn,Tmt+1fx+Tmt+1fy=dTmtf1+mnxn+ynn-Tmtfmnxn+ynn,Tmtf1+mnxnn-Tmtfmx+Tmtf1+mnynn-Tmtfmy≤dTmtf1+mnxn+ynn,Tmtf1+mnxnn+Tmtf1+mnynn+dTmtfmnxn+ynn,Tmtfmx+Tmtfmy≤s1+mn+smnth1+mnxn+h1+mnyn+hmnxn+hmnyn≤s1+mn+smnt+1hxn+hyn
for allx,y∈R+. Letting k→∞ in the inequality (35), we may obtain the following equality: (37)Tmxn+ynn=Tmx+Tmy
for allx,y∈R+and m∈M0. For each m∈M0, the mapping Tm is a solution of the radical nth-degree functional equation; that is, (38)Rx=R1+mnxnn-Rmx
for allx∈R+. Let L>0 be constant. Then, the mapping R:R+→R satisfying (39)dfx,Rx≤Lhxn
for allx∈R+ should be equal to Tm for each m∈M0. Let m0∈M0 be fixed and R:R+→R satisfy the inequality (39). We, then, note that (40)dRx,Tm0x≤dRx,fx+dfx,Tm0x≤L+1+sm0n1-s1+m0n-sm0nhxn≤hxnL0∑j=0∞s1+m0n+sm0nj
whereL0=(1-s(1+m0n)-s(m0n))L+(1+s(m0n))(the case h(x)=0 is trivial, so we may exclude it here). Observe that R and Tm0 are solutions for (38) for each m0∈M0. Now, we will show that for each l∈N0, (41)dRx,Tm0x≤hxnL0∑j=l∞sm0n+s1+m0nj
for allx∈R+. To show this, we will use the mathematical induction, again. The case l=0 follows from the previous inequality. Assume that it holds when the case l∈N0. Now, m,m0∈M0; (37) and (38) imply that (42)dRx,Tm0x=dR1+m0nxnn-Rm0x,Tm01+m02xnn-Tm0m0x≤dR1+m0nxnn,Tm01+m0nxnn+dRm0x,Tm0m0x≤h1+m0nxnL0∑j=l∞sm0n+s1+m0nj+hm0nxnL0∑j=l∞sm0n+s1+m0nj≤sm0n+s1+m0nhxnL0∑j=l∞sm0n+s1+m0nj=hxnL0∑j=l+1∞sm0n+s1+m0nj.
Hence, the inequality (41) holds whenever l∈N0. Letting l→∞ in the inequality (41), we have (43)R=Tm0
wherem0∈M0. This means that Tm=Tm0 for each m∈M0. Hence, we get that (44)dfx,Tm0x≤1+smn1-s1+mn-smnhxn
for allm∈M0 and x∈R+. Thus, we may conclude that the inequality (22) holds with R≔Tm0 and also the uniqueness follows from the equality (43).Let us give a generalized classical Cauchy-difference-type stability of the radicaln-th degree functional equation (16) from Theorem 4.Corollary 5.
Letn be a positive integer and h:R+→(0,∞) a mapping such that (45)limk→∞infsupx∈R+hknxn+h1+knxnhxn=0.
Suppose thatf:R+→R satisfies (46)dxn+ynn,fx+fy≤hxn+hyn
for allx,y∈R+. Then, there exists a unique radical nth-degree functional equation R:R+→R such that (47)dfx,Rx≤hxn for all x∈R+.Proof.
By the definitions(m) as in Theorem 4, we will see that (48)smn≤supx∈R+hmnxnhxn≤supx∈R+hmnxn+h1+mnxnhxn
and(49)s1+mn≤supx∈R+h1+mnxnhxn≤supx∈R+hmnxn+h1+mnxnhxn
These inequalities imply that(50)smn+s1+mn≤2supx∈R+hmnxn+h1+mnxnhxn
for allx∈R+. Now, for each m∈N, let (51)am≔supx∈R+hmnxn+h1+mnxnhxn
for eachx∈R+. By our assumption, it is a sequence {am} with subsequence {amk} such that limk→∞amk=0; that is, (52)limk→∞supx∈R+⁡hmknxn+h1+mknxnhxn=0
The inequalities (50) and (52) imply that (53)limk→∞smkn+s1+mkn=0;
that is,limk→∞s(mkn)=0. Thus, we have (54)limk→∞1+smkn1-smkn-s1+mkn=1.
Lettings0=1 as in Theorem 4, the inequality (47) follows from the inequality (22).



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