Approximately Quintic and Sextic Mappings Form 𝑟-Divisible Groups into Ŝerstnev Probabilistic Banach Spaces: Fixed Point Method

Abstract

Using the fixed point method, we investigate the stability of the systems of quadratic-cubic and additive-quadratic-cubic functional equations with constant coefficients form r-divisible groups into Ŝerstnev probabilistic Banach spaces.

1. Introduction and Preliminaries

The stability problem of functional equations started with the following question concerning stability of group homomorphisms proposed by Ulam [1] during a talk before a Mathematical Colloquium at the University of Wisconsin, Madison, in 1940.

Let (𝐺₁,⋅) be a group and (𝐺₂,∗) a metric group with the metric 𝑑(⋅,⋅). Given 𝜖>0, does there exist a 𝛿>0 such that if a mapping ℎ∶𝐺₁→𝐺₂ satisfies the inequality 𝑑(ℎ(𝑥⋅𝑦),ℎ(𝑥)∗ℎ(𝑦))<𝛿 for all 𝑥,𝑦∈𝐺₁, then there exists a homomorphism 𝐻∶𝐺₁→𝐺₂ with 𝑑(ℎ(𝑥),𝐻(𝑥))<𝜖 for all 𝑥∈𝐺₁?

In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces as follows.

If 𝐸 and 𝐸^ are Banach spaces and 𝑓∶𝐸→𝐸^ is a mapping for which there is 𝜀>0 such that ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜀 for all 𝑥,𝑦∈𝐸, then there is a unique additive mapping 𝐿∶𝐸→𝐸^ such that ‖𝑓(𝑥)−𝐿(𝑥)‖≤𝜀 for all 𝑥∈𝐸.

Hyers’ Theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference, respectively.

The paper of Rassias [5] has provided a lot of influence in the development of what we now call the generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations. In 1994, a generalization of the Rassias theorem was obtained by Găvruţa [6] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. For more details about the results concerning such problems, the reader is referred to [4, 5, 7–21, 21–30].

The functional equation𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦)=2𝑓(𝑥)+2𝑓(𝑦)(1.1) is related to a symmetric biadditive function [31, 32]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is called a quadratic function. The Hyers-Ulam stability problem for the quadratic functional equation was solved by Skof [33]. In [8], Czerwik proved the Hyers-Ulam-Rassias stability of (1.1). Eshaghi Gordji and Khodaei [34] obtained the general solution and the generalized Hyers-Ulam-Rassias stability of the following quadratic functional equation: for all 𝑎,𝑏∈ℤ⧵{0} with 𝑎≠±1,±𝑏,𝑓(𝑎𝑥+𝑏𝑦)+𝑓(𝑎𝑥−𝑏𝑦)=2𝑎²𝑓(𝑥)+2𝑏²𝑓(𝑦).(1.2)

Jun and Kim [35] introduced the following cubic functional equation:𝑓(2𝑥+𝑦)+𝑓(2𝑥−𝑦)=2𝑓(𝑥+𝑦)+2𝑓(𝑥−𝑦)+12𝑓(𝑥),(1.3) and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.3). Jun et al. [36] investigated the solution and the Hyers-Ulam stability for the cubic functional equation𝑓(𝑎𝑥+𝑏𝑦)+𝑓(𝑎𝑥−𝑏𝑦)=𝑎𝑏²(𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦))+2𝑎𝑎²−𝑏²𝑓(𝑥),(1.4) where 𝑎,𝑏∈ℤ⧵{0} with 𝑎≠±1,±𝑏. For other cubic functional equations, see [37].

Lee et al. [38] considered the following functional equation:𝑓(2𝑥+𝑦)+𝑓(2𝑥−𝑦)=4𝑓(𝑥+𝑦)+4𝑓(𝑥−𝑦)+24𝑓(𝑥)−6𝑓(𝑦).(1.5)

In fact, they proved that a function 𝑓 between two real vector spaces 𝑋 and 𝑌 is a solution of (1.5) if and only if there exists a unique symmetric biquadratic function 𝐵₂∶𝑋×𝑋→𝑌 such that 𝑓(𝑥)=𝐵₂(𝑥,𝑥) for all 𝑥∈𝑋. The bi-quadratic function 𝐵₂ is given by 𝐵₂(𝑥,𝑦)=112(𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦)−2𝑓(𝑥)−2𝑓(𝑦)).(1.6)

Obviously, the function 𝑓(𝑥)=𝑐𝑥⁴ satisfies the functional equation (1.5), which is called the quartic functional equation. For other quartic functional equations, see [39].

Ebadian et al. [40] considered the generalized Hyers-Ulam stability of the following systems of the additive-quartic functional equations:𝑓𝑓𝑥𝑥,₁2+𝑦₁𝑥₂,𝑦=𝑓𝑥₁,𝑦+𝑓𝑥₂,𝑦,+24𝑓+𝑥,𝑦𝑦₂₁+𝑓𝑥,2𝑦₁−𝑦₂=4𝑓𝑥,𝑦₁+𝑦₂+4𝑓𝑥,𝑦₁−𝑦₂−6𝑓𝑥,𝑦₂,(1.7) and the quadratic-cubic functional equations:𝑓𝑥,=2𝑦2₁𝑓+𝑥𝑦,₂𝑦₁+𝑓𝑥,2𝑦₁−𝑦₂𝑓𝑥,𝑦₁+𝑦₂+2𝑓𝑥,𝑦₁−𝑦₂+12𝑓𝑥,𝑦₁,+𝑦₂+𝑓𝑥,𝑦₁−𝑦₂=2𝑓𝑥,𝑦₁+2𝑓𝑥,𝑦₂.(1.8)

For more details about the results concerning mixed type functional equations, the readers are referred to [41–44].

Recently, Ghaemi et al. [45] investigated the stability of the following systems of quadratic-cubic functional equations:𝑓𝑓𝑎𝑥𝑥,𝑎₁𝑦+₁𝑏𝑥₂,𝑦+𝑓𝑎𝑥₁−𝑏𝑥₂,𝑦=2𝑎²𝑓𝑥₁,𝑦+2𝑏²𝑓𝑥₂,𝑦,=𝑎𝑏₂+𝑏𝑦₂+𝑓𝑥,𝑎𝑦₁−𝑏𝑦₂𝑓𝑥,𝑦₁+𝑦₂+𝑓𝑥,𝑦₁−𝑦₂+2𝑎𝑎²−𝑏²𝑓𝑥,𝑦₁,(1.9) and additive-quadratic-cubic functional equations:𝑓𝑓𝑎𝑥𝑥,₁𝑎𝑦+₁𝑏𝑥₂,𝑦,𝑧+𝑓𝑎𝑥₁−𝑏𝑥₂,𝑦,𝑧=2𝑎𝑓𝑥₁,𝑦,𝑧,𝑓𝑥,𝑦,𝑎𝑧+₁𝑏𝑦₂,𝑧+𝑓𝑥,𝑎𝑦₁−𝑏𝑦₂,𝑧=2𝑎²𝑓𝑥,𝑦₁,𝑧+2𝑏²𝑓𝑥,𝑦₂,𝑧,+𝑏𝑧₂+𝑓𝑥,𝑦,𝑎𝑧₁−𝑏𝑧₂=+𝑎𝑏2²𝑎𝑓𝑎₂𝑥,𝑦,𝑧₁+𝑧₂+𝑓𝑥,𝑦,𝑧₁−𝑧₂−𝑏²𝑓𝑥,𝑦,𝑧₁(1.10) in PN-spaces (see Definition 1.6), where 𝑎,𝑏∈ℤ⧵{0} with 𝑎≠±1,±𝑏. The function 𝑓∶ℝ×ℝ→ℝ given by 𝑓(𝑥,𝑦)=𝑐𝑥²𝑦³ is a solution of the system (1.9). In particular, letting 𝑦=𝑥, we get a quintic function 𝑔∶ℝ→ℝ in one variable given by 𝑔(𝑥)∶=𝑓(𝑥,𝑥)=𝑐𝑥⁵. Also, it is easy to see that the function 𝑓∶ℝ×ℝ×ℝ→ℝ defined by 𝑓(𝑥,𝑦,𝑧)=𝑐𝑥𝑦²𝑧³ is a solution of the system (1.10). In particular, letting 𝑦=𝑧=𝑥, we get a sextic function ℎ∶ℝ→ℝ in one variable given by ℎ(𝑥)∶=𝑓(𝑥,𝑥,𝑥)=𝑐𝑥⁶.

The proof of the following propositions are evident.

Proposition 1.1. Let X and Y be real linear spaces. If a function 𝑓∶𝑋×𝑋→𝑌 satisfies the system (1.9), then 𝑓(𝜆𝑥,𝜇𝑦)=𝜆²𝜇³𝑓(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and rational numbers 𝜆,𝜇.

Proposition 1.2. Let X and Y be real linear spaces. If a function 𝑓∶𝑋×𝑋×𝑋→𝑌 satisfies the system (1.10), then 𝑓(𝜆𝑥,𝜇𝑦,𝜂𝑧)=𝜆𝜇²𝜂³𝑓(𝑥,𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋 and rational numbers 𝜆,𝜇,𝜂.

For our main results, we introduce Banach’s fixed point theorem and related results. For the proof of Theorem 1.3, refer to [46] and also Chapter 5 in [29] and, for more fixed point theory and other nonlinear methods, refer to [28, 47]. Especially, in 2003, Radu [27] proposed a new method for obtaining the existence of exact solutions and error estimations, based on the fixed point alternative (see also [48–53]).

Let (𝑋,𝑑) be a generalized metric space. We say that an operator 𝑇∶𝑋→𝑋 satisfies a Lipschitz condition with Lipschitz constant 𝐿 if there exists a constant 𝐿≥0 such that 𝑑(𝑇𝑥,𝑇𝑦)≤𝐿𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋. If the Lipschitz constant 𝐿 is less than 1, then the operator 𝑇 is called a strictly contractive operator.

Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We recall the following theorem by Margolis and Diaz.

Theorem 1.3 (see [27, 46]). Suppose that (Ω,𝑑) is a complete generalized metric space and 𝑇∶Ω→Ω is a strictly contractive mapping with Lipschitz constant 𝐿. Then, for any 𝑥∈Ω, either 𝑑𝑇^𝑚𝑥,𝑇^𝑚+1𝑥=∞(1.11) for all 𝑚≥0 or there exists a natural number 𝑚₀ such that(1)𝑑(𝑇^𝑚𝑥,𝑇^𝑚+1𝑥)<∞ for all 𝑚≥𝑚₀;(2)the sequence {𝑇^𝑚𝑥} is convergent to a fixed point 𝑦^∗ of 𝑇;(3)𝑦^∗ is the unique fixed point of 𝑇 in Λ={𝑦∈Ω∶𝑑(𝑇^𝑚0𝑥,𝑦)<∞};(4)𝑑(𝑦,𝑦^∗)≤(1/1−𝐿)𝑑(𝑦,𝑇𝑦) for all 𝑦∈Λ.

The PN-spaces were first defined by Šerstnev in 1963 (see [54]). Their definition was generalized by Alsina et al. in [55]. In this paper, we follow the definition of probabilistic space briefly as given in [56] (also, see [57]).

Definition 1.4. A distance distribution function (d.d.f.) is a nondecreasing function 𝐹 from ℝ⁺ into [0,1] that satisfies 𝐹(0)=0, 𝐹(+∞)=1 and 𝐹 is left-continuous on (0,+∞), where ℝ⁺∶=[0,+∞].

Forward, the space of distance distribution functions is denoted by Δ⁺ and the set of all 𝐹 in Δ⁺ with lim_{𝑡→+∞−}𝐹(𝑡)=1 by 𝐷⁺. The space Δ⁺ is partially ordered by the usual pointwise ordering of functions, that is, 𝐹≤𝐺 if and only if 𝐹(𝑥)≤𝐺(𝑥) for all 𝑥 in ℝ⁺. For any 𝑎≥0, 𝜀^+𝑎 is the d.d.f. given by 𝜀^+𝑎(𝑡)=0,1,if𝑡≤𝑎,if𝑡>𝑎.(1.12)

Definition 1.5. A triangle function is a binary operation on Δ⁺, that is, a function 𝜏∶Δ⁺×Δ⁺→Δ⁺ that is associative, commutative, non-decreasing in each place, and has 𝜀₀ as the identity, that is, for all 𝐹,𝐺 and 𝐻 in Δ⁺,(TF1)𝜏(𝜏(𝐹,𝐺),𝐻)=𝜏(𝐹,𝜏(𝐺,𝐻));(TF2)𝜏(𝐹,𝐺)=𝜏(𝐺,𝐹);(TF3)𝐹≤𝐺⇒𝜏(𝐹,𝐻)≤𝜏(𝐺,𝐻);(TF4)𝜏(𝐹,𝜀₀)=𝜏(𝜀₀,𝐹)=𝐹.

Typical continuous triangle function is Π_𝑇(𝐹,𝐺)(𝑥)=𝑇(𝐹(𝑥),𝐺(𝑥)),(1.13) where 𝑇 is a continuous 𝑡-norm, that is, a continuous binary operation on [0,1] that is commutative, associative, non-decreasing in each variable, and has 1 as the identity. For example, we introduce the following: 𝑀(𝑥,𝑦)=min(𝑥,𝑦)(1.14) for all 𝑥,𝑦∈[0,1] is a continuous and maximal 𝑡-norm, namely, for any 𝑡-norm 𝑇, 𝑀≥𝑇. Also, note that Π_𝑀 is a maximal triangle function, that is, for all triangle function 𝜏, Π_𝑀≥𝜏.

Definition 1.6. A Šerstnev probabilistic normed space (Šerstnev PN-space) is a triple (𝑋,𝜈,𝜏), where 𝑋 is a real vector space, 𝜏 is continuous triangle function, and 𝜈 is a mapping (the probabilistic norm) from 𝑋 into Δ⁺ such that, for all choice of 𝑝,𝑞∈𝑋 and 𝑎∈ℝ⁺, the following conditions hold:(N1)𝜈(𝑝)=𝜀₀, if and only if 𝑝=𝜃(𝜃 is the null vector in 𝑋);(N2)𝜈(𝑎𝑝)(𝑡)=𝜈(𝑝)(𝑡/|𝑎|);(N3)𝜈(𝑝+𝑞)≥𝜏(𝜈(𝑝),𝜈(𝑞)).

Let (𝑋,𝜈,𝜏) be a PN-space and {𝑥_𝑛} a sequence in 𝑋. Then {𝑥_𝑛} is said to be convergent if there exists 𝑥∈𝑋 such that lim_𝑛→∞𝜈𝑥_𝑛−𝑥(𝑡)=1(1.15) for all 𝑡>0. In this case, the point 𝑥 is called the limit of {𝑥_𝑛}. The sequence {𝑥_𝑛} in (𝑋,𝜈,𝜏) is called a Cauchy sequence if, for any 𝜀>0 and 𝛿>0, there exists a positive integer 𝑛₀ such that 𝜈(𝑥_𝑛−𝑥_𝑚)(𝛿)>1−𝜀 for all 𝑚,𝑛≥𝑛₀. Clearly, every convergent sequence in a PN-space is a Cauchy sequence. If each Cauchy sequence is convergent in a PN-space (𝑋,𝜈,𝜏), then (𝑋,𝜈,𝜏) is called a probabilistic Banach space (PB-space).

For more details about the results concerning stability of the functional equations on PN-spaces, the readers are referred to [58–61].

In this paper, by using the fixed point method, we establish the stability of the systems (1.9) and (1.10) form 𝑟-divisible groups into Šerstnev PB-space.

2. Mail Results

We start our work by the following theorem which investigates the stability problem for the system of the functional equations (1.9) form 𝑟-divisible groups into Šerstnev PB-space by using fixed point methods.

Theorem 2.1. Let 𝑠∈{−1,1} be fixed. Let G be an 𝑟-divisible group and (𝑌,𝜈,Π_𝑇) a Šerstnev PB-space. Let 𝜙,𝜓∶𝐺×𝐺×𝐺→𝐷⁺ be two functions such that Φ(𝑥,𝑦)(𝑡)∶=Π_𝑇𝜙𝑎^(𝑠−1)/2𝑥,0,𝑎^(𝑠−1)/2𝑦2𝑎^{(5𝑠−1)/2}𝑡,𝜓𝑎^(𝑠+1)/2𝑥,𝑎^(𝑠−1)/2𝑦,02𝑎^(5𝑠+5)/2𝑡(2.1) for all 𝑥,𝑦∈𝐺 and, for some 0<𝑘<𝑎^10𝑠, lim_𝑛→∞Φ(𝑎^𝑠𝑥,𝑎^𝑠𝑦)𝑘𝑎^−2𝑠𝑡≥Φ(𝑥,𝑦)(𝑡),𝜙𝑎^𝑠𝑛𝑥₁,𝑎^𝑠𝑛𝑥₂,𝑎^𝑠𝑛𝑦𝑎^−5𝑠𝑛𝑡=lim_𝑛→∞𝜓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦₁,𝑎^𝑠𝑛𝑦₂𝑎^−5𝑠𝑛𝑡=1(2.2) for all 𝑥,𝑦,𝑥₁,𝑥₂,𝑦₁,𝑦₂∈𝐺 and 𝑡>0. If 𝑓∶𝐺×𝐺→𝑌 is a function such that 𝑓(0,𝑦)=0 for all 𝑦∈𝐺 and 𝜈𝑓𝑎𝑥₁+b𝑥₂,𝑦+𝑓𝑎𝑥₁−𝑏𝑥₂,𝑦−2𝑎²𝑓𝑥₁,𝑦−2𝑏²𝑓𝑥₂,𝑦(𝑡)≥𝜙𝑥₁,𝑥₂,𝑦,(2.3)𝜈𝑓−𝑥𝑎,𝑏𝑎₂𝑦₁+𝑏𝑦₂+𝑓𝑥,𝑎𝑦₁−𝑏𝑦₂−𝑎𝑏²𝑓𝑥,𝑦₁+𝑦₂𝑓𝑥,𝑦₁−𝑦₂−2𝑎𝑎²−𝑏²𝑓𝑥,𝑦₁(𝑡)≥𝜓𝑥,𝑦₁,𝑦₂(2.4) for all 𝑥,𝑦,𝑥₁,𝑥₂,𝑦₁,𝑦₂∈𝐺, then there exists a unique quintic function 𝑇∶𝐺×𝐺→𝑌 satisfying the system (1.9) and 𝜈(𝑓(𝑥,𝑦)−𝑇(𝑥,𝑦))(𝑡)≥Φ(𝑥,𝑦)1−𝑘𝑎^−10𝑠𝑡(2.5) for all 𝑥,𝑦∈𝐺.

Proof. Putting 𝑥₁=2𝑥 and 𝑥₂=0 and replacing 𝑦 by 2𝑦 in (2.3), we get 𝜈𝑓(2𝑎𝑥,2𝑦)−𝑎²𝑓(2𝑥,2𝑦)(𝑡)≥𝜙(2𝑥,0,2𝑦)(2𝑡)(2.6) for all 𝑥,𝑦∈𝐺. Putting 𝑦₁=2𝑦 and 𝑦₂=0 and replacing 𝑥 by 2𝑎𝑥 in (2.4), we get 𝜈𝑓(2𝑎𝑥,2𝑎𝑦)−𝑎³𝑓(2𝑎𝑥,2𝑦)(𝑡)≥𝜓(2𝑎𝑥,2𝑦,0)(2𝑡)(2.7) for all 𝑥,𝑦∈𝐺. Thus, we have 𝜈𝑓(2𝑎𝑥,2𝑎𝑦)−𝑎⁵𝑓(2𝑥,2𝑦)(𝑡)≥Π_𝑇𝜙(2𝑥,0,2𝑦)2𝑎⁻³𝑡,𝜓(2𝑎𝑥,2𝑦,0)(2𝑡)(2.8) for all 𝑥,𝑦∈𝐺. Replacing 𝑥, 𝑦 by 𝑥/2, 𝑦/2 in (2.8), we have 𝜈𝑓(𝑎𝑥,𝑎𝑦)−𝑎⁵𝑓(𝑥,𝑦)(𝑡)≥Π_𝑇𝜙(𝑥,0,𝑦)2𝑎⁻³𝑡,𝜓(𝑎𝑥,𝑦,0)(2𝑡)(2.9) for all 𝑥,𝑦∈𝐺. It follows from (2.9) that 𝜈𝑎𝜈𝑎⁻⁵𝑓(𝑎𝑥,𝑎𝑦)−𝑓(𝑥,𝑦)(𝑡)≥Π_𝑇𝜙(𝑥,0,𝑦)2𝑎²𝑡,𝜓(𝑎𝑥,𝑦,0)2𝑎⁵𝑡₅,𝑓𝑎⁻¹𝑥,𝑎⁻¹𝑦−𝑓(𝑥,𝑦)(𝑡)≥Π_𝑇𝜙𝑎⁻¹𝑥,0,𝑎⁻¹𝑦2𝑎⁻³𝑡,𝜓𝑎𝑥,𝑎⁻¹𝑦,0(2𝑡)(2.10) for all 𝑥,𝑦∈𝐺. So we have 𝜈𝑎^−5𝑠𝑓(𝑎^𝑠𝑥,𝑎^𝑠𝑦)−𝑓(𝑥,𝑦)(𝑡)≥Φ(𝑥,𝑦)(𝑡)(2.11) for all 𝑥,𝑦∈𝐺. Let 𝑆 be the set of all mappings ℎ∶𝐺×𝐺→𝑌 with ℎ(0,𝑥)=0 for all 𝑥∈𝐺, and define a generalized metric on 𝑆 as follows: 𝑑(ℎ,𝑘)=inf𝑢∈ℝ⁺∶𝜈(ℎ(𝑥,𝑦)−𝑘(𝑥,𝑦))(𝑢𝑡)≥Φ(𝑥,𝑦)(𝑡),∀𝑥,𝑦∈𝐺,∀𝑡>0,(2.12) where, as usual, inf∅=+∞. The proof of the fact that (𝑆,𝑑) is a complete generalized metric space, can be shown in [48, 62].
Now, we consider the mapping 𝐽∶𝑆→𝑆 defined by 𝐽ℎ(𝑥,𝑦)∶=𝑎^−5𝑠ℎ(𝑎^𝑠𝑥,𝑎^𝑠𝑦)(2.13) for all ℎ∈𝑆 and 𝑥,𝑦∈𝐺. Let 𝑓,𝑔∈𝑆 such that 𝑑(𝑓,𝑔)<𝜀. Then it follows that 𝜈(𝐽𝑔=(𝑥𝜈,𝑦𝑎)−_−5𝑠𝐽𝑓(𝑥,𝑦))𝑘𝑢𝑎^−10𝑠𝑡=𝜈(𝑔(𝑎_𝑠𝑔(𝑎^𝑠𝑥,𝑎^𝑠𝑦)−𝑎^−5𝑠𝑓(𝑎^𝑠𝑥,𝑎^𝑠𝑦)𝑘𝑢𝑎^−10𝑠𝑡≥Φ(𝑎_𝑠𝑥,𝑎^𝑠𝑦)−𝑓(𝑎^𝑠𝑥,𝑎^𝑠𝑦))𝑘𝑢𝑎^−2𝑠𝑡≥Φ(𝑥,𝑥𝑦,)𝑎(𝑡^𝑠)𝑦,)𝑘𝑎^−2𝑠𝑡(2.14) that is, if 𝑑(𝑓,𝑔)<𝜀, then we have 𝑑(𝐽𝑓,𝐽𝑔)<𝑘𝑎^−10𝑠𝜀. This means that 𝑑(𝐽𝑓,𝐽𝑔)≤𝑘𝑎^−10𝑠𝑑(𝑓,𝑔)(2.15) for all 𝑓,𝑔∈𝑆; that is, 𝐽 is a strictly contractive self-mapping on 𝑆 with the Lipschitz constant 𝑘𝑎^−10𝑠. It follows from (2.11) that 𝜈(𝐽𝑓(𝑥,𝑦)−𝑓(𝑥,𝑦))(𝑡)≥Φ(𝑥,𝑦)(𝑡)(2.16) for all 𝑥,𝑦∈𝐺 and 𝑡>0, which implies that 𝑑(𝐽𝑓,𝑓)≤1. From Theorem 1.3, it follows that there exists a unique mapping 𝑇∶𝐺×𝐺→𝑌 such that 𝑇 is a fixed point of 𝐽, that is, 𝑇(𝑎^𝑠𝑥,𝑎^𝑠𝑦)=𝑎^5𝑠𝑇(𝑥,𝑦) for all 𝑥,𝑦∈𝐺. Also, we have 𝑑(𝐽^𝑚𝑔,𝑇)→0 as 𝑚→∞, which implies the equality lim_𝑚→∞𝑎^−5𝑠𝑚𝑓(𝑎^𝑠𝑚𝑥,𝑎^𝑠𝑚𝑦)=𝑇(𝑥)(2.17) for all 𝑥,𝑦∈𝐺. It follows from (2.3) that 𝜈𝑇=𝑎𝑥li₁m_𝑛→∞+𝑏𝑥₂,𝑦+𝑇𝑎𝑥₁−𝑏𝑥₂,𝑦−2𝑎²𝑇𝑥₁,𝑦−2𝑏²T𝑥₂,𝑦(𝑡)−2𝑎𝜈_−5𝑠𝑛𝑎^−5𝑠𝑛𝑓𝑎^𝑠𝑛𝑎𝑥₁+𝑏𝑥₂,𝑎^𝑠𝑛𝑦+𝑎^−5𝑠𝑛𝑓𝑎^𝑠𝑛𝑎𝑥₁−𝑏𝑥₂,𝑎^𝑠𝑛𝑦≥lim_𝑛→∞𝑎²𝑓𝑎^𝑠𝑛𝑥₁,𝑎^𝑠𝑛𝑦−2𝑎^−5𝑠𝑛𝑏²𝑓𝑎^𝑠𝑛𝑥₂,𝑎^𝑠𝑛𝑦(𝑡)=1𝜙𝑎^𝑠𝑛𝑥₁,𝑎^𝑠𝑛𝑥₂,𝑎^𝑠𝑛𝑦𝑎^−5𝑠𝑛𝑡(2.18) for all 𝑥₁,𝑥₂,𝑦∈𝐺. Also, it follows from (2.4) that 𝜈𝑇𝑥−,𝑎𝑎𝑏𝑦₁₂+𝑏𝑦₂+𝑇𝑥,𝑎𝑦₁−𝑏𝑦₂=lim_𝑛→∞𝑇𝑥,𝑦₁+𝑦₂−𝑇𝑥,𝑦₁−𝑦₂−2𝑎𝑎²−𝑏²𝑇𝑥,𝑦₁(𝑡)𝜈−𝑎𝑎^−5𝑠𝑛𝑓_−5𝑠𝑛𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑎𝑦₁+𝑏𝑦₂+𝑎^−5𝑠𝑛𝑓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑎𝑦₁−𝑏𝑦₂−2𝑎_−5𝑠𝑛𝑎𝑏²𝑓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦₁+𝑦₂−𝑎^−5𝑠𝑛𝑎𝑏²𝑓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦₁−𝑦₂≥lim_𝑛→∞𝑎𝑎²−𝑏²𝑓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦₁(𝑡)=1𝑎^−5𝑠𝑛𝜓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦₁,𝑎^𝑠𝑛𝑦₂(𝑡)(2.19) for all 𝑥,𝑦₁,𝑦₂∈𝐺. This means that 𝑇 satisfies (1.9); that is, 𝑇 is quintic.
According to the fixed point alternative, since 𝑇 is the unique fixed point of 𝐽 in the set Ω={𝑔∈𝑆∶𝑑(𝑓,𝑔)<∞}, 𝑇 is the unique mapping such that 𝜈(𝑓(𝑥,𝑦)−𝑇(𝑥,𝑦))(𝑢𝑡)≥Φ(𝑥,𝑦)(𝑡)(2.20) for all 𝑥,𝑦∈𝐺 and 𝑡>0. Using the fixed point alternative, we obtain 𝑑(𝑓,𝑇)≤11−𝐿𝑑(𝑓,𝐽𝑓)≤1−𝑘1𝑎^−10𝑠,(2.21) which implies the inequality 𝜈(𝑓(𝑥,𝑦)−𝑇(𝑥,𝑦))𝑡1−𝑘𝑎^−10𝑠≥Φ(𝑥,𝑦)(𝑡)(2.22) for all 𝑥,𝑦∈𝐺 and 𝑡>0. Therefore, we have 𝜈(𝑓(𝑥,𝑦)−𝑇(𝑥,𝑦))(𝑡)≥Φ(𝑥,𝑦)1−𝑘𝑎^−10𝑠𝑡(2.23) for all 𝑥,𝑦∈𝑋𝐺 and 𝑡>0. This completes the proof.

Now, we investigate the stability problem for the system of the functional equations (1.10) form 𝑟-divisible groups into Šerstnev PB-space by using the fixed point theorem.

Theorem 2.2. Let 𝑠∈{−1,1} be fixed. Let G be an 𝑟-divisible group and (𝑌,𝜈,Π_𝑇) a Šerstnev PB-space. Let Φ,Ψ,Υ∶𝐺×G×𝐺×𝐺→𝐷⁺ be functions such that Θ(𝑥,𝑦,𝑧)(𝑡)∶=Π_𝑇Π_𝑇Υ𝑎^(𝑠+1)/2𝑥,𝑎^(𝑠+1)/2𝑦,𝑎^(𝑠+1)/2𝑧,02𝑎^3𝑠+3𝑡,ΦΨ𝑎𝑎^(𝑠+1)/2_(𝑠−1)/2𝑥,𝑎^(𝑠+1)/2𝑦,0,𝑎^(𝑠−1)/2𝑧2𝑎^3𝑠+6𝑡,𝑥,0,𝑎^(𝑠−1)/2𝑦,𝑎^(𝑠−1)/2𝑧2𝑎^3𝑠+8𝑡(2.24) for all 𝑥,𝑦,𝑧∈𝐺 and, for some 0<𝑘<𝑎^6𝑠, Φlim(_𝑛→∞𝑎^𝑠𝑥,𝑎^𝑠𝑦,𝑎^𝑠𝑧)(𝑘𝑡)≥Φ(𝑥,𝑦,𝑧)(𝑡);=Φl𝑎im^𝑠𝑛_𝑛→∞𝑥₁,𝑎^𝑠𝑛𝑥₂,𝑎^𝑠𝑛𝑦,𝑎^𝑠𝑛𝑧𝑎^−6𝑠𝑛𝑡=lim_𝑛→∞Ψ𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦₁,𝑎^𝑠𝑛𝑦₂,𝑎^𝑠𝑛𝑧𝑎^−6𝑠𝑛𝑡=1Υ𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦,𝑎^𝑠𝑛𝑧₁,𝑎^𝑠𝑛𝑧₂𝑎^−6𝑠𝑛𝑡(2.25) for all 𝑥,𝑦,𝑥₁,𝑥₂,𝑦₁,𝑦₂,𝑧₁,𝑧₂∈𝐺. If 𝑓∶𝐺×𝐺×𝐺→𝑌 is a function such that 𝑓(𝑥,0,𝑧)=0 for all 𝑥,𝑧∈𝐺 and 𝜈𝑓𝑎𝑥₁+𝑏𝑥₂,𝑦,𝑧+𝑓𝑎𝑥₁−𝑏𝑥₂,𝑦,𝑧−2𝑎𝑓𝑥₁,𝑦,𝑧(𝑡)≥Φ𝑥₁,𝑥₂,𝑦,𝑧(𝑡),(2.26)𝜈𝑓≥𝑥Ψ,𝑎𝑦𝑥₁,+𝑦₁𝑏𝑦₂,𝑧+𝑓𝑥,𝑎𝑦₁−𝑏𝑦₂,𝑧−2𝑎²𝑓𝑥,𝑦₁,𝑧−2𝑏²𝑓𝑥,𝑦₂,𝑧(𝑡),𝑦₂,𝑧(𝑡),(2.27)𝜈𝑓𝑥−,𝑎𝑦𝑏,₂𝑎𝑧₁+𝑏𝑧₂+𝑓𝑥,𝑦,𝑎𝑧₁−𝑏𝑧₂≥Υ𝑥,𝑦𝑓,𝑧₁𝑥,𝑦,𝑧₁+𝑧₂+𝑓𝑥,𝑦,𝑧₁−𝑧₂−2𝑎𝑎²−𝑏²𝑓𝑥,𝑦,𝑧₁(𝑡),𝑧₂(𝑡)(2.28) for all 𝑥,𝑦,𝑥₁,𝑥₂,𝑦₁,𝑦₂,𝑧₁,𝑧₂∈𝐺, then there exists a unique quintic function 𝑇∶𝐺×𝐺×𝐺→𝑌 satisfying (1.10) and 𝜈(𝑓(𝑥,𝑦,𝑧)−𝑇(𝑥,𝑦,𝑧))(𝑡)≥Θ(𝑥,𝑦,𝑧)1−𝑘𝑎^−6𝑠𝑡(2.29) for all 𝑥,𝑦,𝑧∈𝐺.

Proof. Putting 𝑥₁=2𝑥 and 𝑥₂=0 and replacing 𝑦,𝑧 by 2𝑦,2𝑧 in (2.26), we get 𝜈(𝑓(2𝑎𝑥,2𝑦,2𝑧)−𝑎𝑓(2𝑥,2𝑦,2𝑧))12𝑡≥Φ(2𝑥,0,2𝑦,2𝑧)(𝑡)(2.30) for all 𝑥,𝑦,𝑧∈𝐺. Putting 𝑦₁=2𝑦 and 𝑦₂=0 and replacing 𝑥,𝑧 by 2𝑎𝑥,2𝑧 in (2.27), we get 𝜈𝑓(2𝑎𝑥,2𝑎𝑦,2𝑧)−𝑎²𝑓(2𝑎𝑥,2𝑦,2𝑧)12𝑡≥Ψ(2𝑎𝑥,2𝑦,0,2𝑧)(𝑡)(2.31) for all 𝑥,𝑦,𝑧∈𝐺. Putting 𝑧₁=2𝑧 and 𝑧₂=0 and replacing 𝑥,𝑦 by 2𝑎𝑥,2𝑎𝑦 in (2.28), we get 𝜈𝑓(2𝑎𝑥,2𝑎𝑦,2𝑎𝑧)−𝑎³𝑓(2𝑎𝑥,2𝑎𝑦,2𝑧)12𝑡≥Υ(2𝑎𝑥,2𝑎𝑦,2𝑧,0)(𝑡),(2.32) for all 𝑥,𝑦,𝑧∈𝐺. Thus, 𝜈𝑓≥(2𝑎Π𝑥_𝑇,2𝑎𝑦,2𝑎𝑧)−𝑎⁶𝑓(2𝑥,2𝑦,2𝑧)(𝑡)Υ(2𝑎𝑥,2𝑎𝑦,2𝑧,0)(2𝑡),Π_𝑇Ψ(2𝑎𝑥,2𝑦,0,2𝑧)2𝑎³𝑡,Φ(2𝑥,0,2𝑦,2𝑧)2𝑎⁵𝑡(2.33) for all 𝑥,𝑦,𝑧∈𝐺. Replacing 𝑥, 𝑦, and 𝑧 by 𝑥/2, 𝑦/2, and 𝑧/2 in (2.33), we have 𝜈𝑓≥(𝑎𝑥Π,_𝑇𝑎𝑦,𝑎𝑧)−𝑎⁶𝑓(𝑥,𝑦,𝑧)(𝑡)Υ(𝑎𝑥,𝑎𝑦,𝑧,0)(2𝑡),Π_𝑇Ψ(𝑎𝑥,𝑦,0,𝑧)2𝑎³𝑡,Φ(𝑥,0,𝑦,𝑧)2𝑎⁵𝑡(2.34) for all 𝑥,𝑦,𝑧∈𝐺. It follows from (2.34) that 𝜈𝑎≥⁻⁶𝑓Π(_𝑇𝑎𝑥,𝑎𝑦,𝑎𝑧)−𝑓(𝑥,𝑦,𝑧)(𝑡)𝜈𝑎Υ(𝑎𝑥,𝑎𝑦,𝑧,0)2𝑎⁶𝑡,Π_𝑇Ψ(𝑎𝑥,𝑦,0,𝑧)2𝑎⁹𝑡,Φ(𝑥,0,𝑦,𝑧)2𝑎¹¹𝑡₆,≥𝑓Π𝑎⁻¹_𝑇𝑥,𝑎⁻¹𝑦,𝑎⁻¹𝑧−𝑓(𝑥,𝑦,𝑧)(𝑡)Υ𝑥,𝑦,𝑎⁻¹𝑧,0(2𝑎𝑡)2,Π𝑎_𝑇5Ψ𝑥,𝑎⁻¹𝑦,0,𝑎⁻¹𝑧2𝑎³𝑡Φ𝑎⁻¹𝑥,0,𝑎⁻¹𝑦,𝑎⁻¹𝑧,𝑡(2.35) for all 𝑥,𝑦,𝑧∈𝐺. Thus, we have 𝜈𝑎^−6𝑠𝑓(𝑎^𝑠𝑥,𝑎^𝑠𝑦,𝑎^𝑠𝑧)−𝑓(𝑥,𝑦,𝑧)(𝑡)≥Θ(𝑥,𝑦,𝑧)(𝑡)(2.36) for all 𝑥,𝑦,𝑧∈𝐺. Let 𝑆 be the set of all mappings ℎ∶𝑋×𝑋×𝑋→𝑌 with ℎ(𝑥,0,𝑧)=0 for all 𝑥,𝑧∈𝐺, and define a generalized metric on 𝑆 as follows: 𝑑(ℎ,𝑘)=inf𝑢∈ℝ⁺∶𝜈(ℎ(𝑥,𝑦,𝑧)−𝑘(𝑥,𝑦,𝑧))(𝑢𝑡)≥Θ(𝑥,𝑦,𝑧)(𝑡),∀𝑥,𝑦,𝑧∈𝐺,𝑡>0,(2.37) where, as usual, inf∅=+∞. The proof of the fact that (𝑆,𝑑) is a complete generalized metric space can be shown in [48, 62].
Now, we consider the mapping 𝐽∶𝑆→𝑆 defined by 𝐽ℎ(𝑥,𝑦,𝑧)∶=𝑎^−6𝑠ℎ(𝑎^𝑠𝑥,𝑎^𝑠𝑦,𝑎^𝑠𝑧)(2.38) for all ℎ∈𝑆 and 𝑥,𝑦,𝑧∈𝐺. Let 𝑓,𝑔∈𝑆 be such that 𝑑(𝑓,𝑔)<𝜀. Then we have 𝜈(𝐽𝑔=(𝑥𝜈,𝑦𝑎,𝑧)_−6𝑠−𝐽𝑓(𝑥,𝑦,𝑧))𝑘𝑢𝑎^−6𝑠𝑡=𝜈(𝑔(𝑎_𝑠𝑔(𝑎^𝑠𝑥,𝑎^𝑠𝑦,𝑎^𝑠𝑧)−𝑎^−6𝑠𝑓(𝑎^𝑠𝑥,𝑎^𝑠𝑦,𝑎^𝑠𝑧)𝑘𝑢𝑎^−6𝑠𝑡≥Θ(𝑎_𝑠𝑥,𝑎^𝑠𝑦,𝑎^𝑠𝑧)−𝑓(𝑎^𝑠𝑥,𝑎^𝑠𝑦,𝑎^𝑠𝑧))(𝑘𝑢𝑡)𝑥,𝑎^𝑠𝑦,𝑎^𝑠𝑧)(𝑘𝑡)≥Θ(𝑥,𝑦,𝑧)(𝑡),(2.39) that is, if 𝑑(𝑓,𝑔)<𝜀, then we have 𝑑(𝐽𝑓,𝐽𝑔)<𝑘𝑎^−6𝑠𝜀. This means that 𝑑(𝐽𝑓,𝐽𝑔)≤𝑘𝑎^−6𝑠𝑑(𝑓,𝑔)(2.40) for all 𝑓,𝑔∈𝑆; that is, 𝐽 is a strictly contractive self-mapping on 𝑆 with the Lipschitz constant 𝑘𝑎^−6𝑠. It follows from (2.36) that 𝜈(𝐽𝑓(𝑥,𝑦,𝑧)−𝑓(𝑥,𝑦,𝑧))(𝑡)≥Θ(𝑥,𝑦,𝑧)(𝑡)(2.41) for all 𝑥,𝑦,𝑧∈𝐺 and all 𝑡>0, which implies that 𝑑(𝐽𝑓,𝑓)≤1. From Theorem 1.3, it follows that there exists a unique mapping 𝑇∶𝐺×𝐺×𝐺→𝑌 such that 𝑇 is a fixed point of 𝐽, that is, 𝑇(𝑎^𝑠𝑥,𝑎^𝑠𝑦,𝑎^𝑠𝑧)=𝑎^6𝑠𝑇(𝑥,𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝐺. Also, 𝑑(𝐽^𝑚𝑔,𝑇)→0 as 𝑚→∞, which implies the equality lim_𝑚→∞𝑎^−6𝑠𝑚𝑓(𝑎^𝑠𝑚𝑥,𝑎^𝑠𝑚𝑦,𝑎^𝑠𝑚𝑧)=𝑇(𝑥)(2.42) for all 𝑥∈𝑋. It follows from (2.26), (2.27), and (2.28) that 𝜈𝑇=𝑎𝑥li₁m_𝑛→∞+𝑏𝑥₂,𝑦,𝑧+𝑇𝑎𝑥₁−𝑏𝑥₂,𝑦,𝑧−2𝑎𝑇𝑥₁,𝑦,𝑧(𝑡)𝜈+𝑎𝑎^−6𝑠𝑛_−6𝑠𝑛𝑓𝑎^𝑠𝑛𝑎𝑥₁+𝑏𝑥₂,𝑎^𝑠𝑛𝑦,𝑎^𝑠n𝑧≥lim_𝑛→∞𝑓𝑎^𝑠𝑛𝑎𝑥₁−𝑏𝑥₂,𝑎^𝑠𝑛𝑦,𝑎^𝑠𝑛𝑧−2𝑎𝑎^−6𝑠𝑛𝑓𝑎^𝑠𝑛𝑥₁,𝑎^𝑠𝑛𝑦,𝑎^𝑠𝑛𝑧(𝑡)𝜈𝑇=𝑥1,,𝑎𝑦₁Φ𝑎^𝑠𝑛𝑥₁,𝑎^𝑠𝑛𝑥₂,𝑎^𝑠𝑛𝑦,𝑎^𝑠𝑛𝑧𝑎^−6𝑠𝑛𝑡=lim_𝑛→∞+𝑏𝑦₂,𝑧+𝑇𝑥,𝑎𝑦₁−𝑏𝑦₂,𝑧−2𝑎²𝑇𝑥,𝑦₁,𝑧−2𝑏²𝑇𝑥,𝑦₂,𝑧(𝑡)𝜈−𝑎2𝑎^−6𝑠𝑛₂𝑓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑎𝑦₁+𝑏𝑦₂,𝑎^𝑠𝑛𝑧+𝑓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑎𝑦₁−𝑎^𝑠𝑛𝑏𝑦₂,𝑎^𝑠𝑛𝑧≥lim_𝑛→∞𝑎^−6𝑠𝑛𝑓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦₁,𝑎^𝑠𝑛𝑧+2𝑏²𝑎^−6𝑠𝑛𝑓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦₂,𝑎^𝑠𝑛𝑧(𝑡)𝜈𝑇=𝑥1,,𝑦,𝑎Ψ𝑧₁𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦₁,𝑎^𝑠𝑛𝑦₂,𝑎^𝑠𝑛𝑧𝑎^−6𝑠𝑛𝑡−𝑇𝑥,𝑦+,𝑧𝑏₁𝑧₂+𝑇𝑥,𝑦,𝑎𝑧₁−𝑏𝑧₂−𝑎𝑏²𝑇𝑥,𝑦,𝑧₁+𝑧₂=lim_𝑛→∞−𝑧₂−2𝑎𝑎²−𝑏²𝑇𝑥,𝑦,𝑧₁(𝑡)+𝑎𝜈_−6𝑠𝑛𝑎^−6𝑠𝑛𝑓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦,𝑎^𝑠𝑛𝑎𝑧₁+𝑏𝑧₂+𝑓𝑎𝑓_𝑠𝑛𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦,𝑎^𝑠𝑛𝑎𝑧₁−𝑏𝑧₂−𝑎𝑏²𝑎^−6𝑠𝑛𝑓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦,𝑎^𝑠𝑛𝑧₁+𝑧₂≥lim_𝑛→∞𝑥,𝑎^𝑠𝑛𝑦,𝑎^𝑠𝑛𝑧₁−𝑧₂−2𝑎𝑎^−6𝑠𝑛𝑎²−𝑏²𝑓𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦,𝑎^𝑠𝑛𝑧₁(𝑡)=1Υ𝑎^𝑠𝑛𝑥,𝑎^𝑠𝑛𝑦,𝑎^𝑠𝑛𝑧₁,𝑎^𝑠𝑛𝑧₂𝑎^−6𝑠𝑛𝑡(2.43) for all 𝑥,𝑦,𝑥₁,𝑥₂,𝑦₁,𝑦₂,𝑧₁,𝑧₂∈𝐺. This means that 𝑇 satisfies (1.10); that is, 𝑇 is sextic.
According to the fixed point alternative, since 𝑇 is the unique fixed point of 𝐽 in the set Ω={𝑔∈𝑆∶𝑑(𝑓,𝑔)<∞}, 𝑇 is the unique mapping such that 𝜈(𝑓(𝑥,𝑦,𝑧)−𝑇(𝑥,𝑦,𝑧))(𝑢𝑡)≥Θ(𝑥,𝑦,𝑧)(𝑡)(2.44) for all 𝑥,𝑦,𝑧∈𝐺 and 𝑡>0. Using the fixed point alternative, we obtain 𝑑(𝑓,𝑇)≤11−𝐿𝑑(𝑓,𝐽f)≤11−𝑘𝑎^−6𝑠,(2.45) which implies the inequality 𝜈(𝑓(𝑥,𝑦,𝑧)−𝑇(𝑥,𝑦,𝑧))𝑡1−𝑘𝑎^−6𝑠≥Θ(𝑥,𝑦,𝑧)(𝑡)(2.46) for all 𝑥,𝑦,𝑧∈𝐺 and 𝑡>0. So 𝜈(𝑓(𝑥,𝑦,𝑧)−𝑇(𝑥,𝑦,𝑧))(𝑡)≥Θ(𝑥,𝑦,𝑧)1−𝑘𝑎^−6𝑠𝑡(2.47) for all 𝑥,𝑦,𝑧∈𝐺 and 𝑡>0. This completes the proof.

Acknowledgment

The second author was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).