Hyperstability of the Drygas Functional Equation

Abstract

We use the fixed point theorem for functional spaces to obtain the hyperstability result for the Drygas functional equation on a restricted domain. Namely, we show that a function satisfying the Drygas equation approximately must be exactly the solution of it.

1. Introduction

We say that a function 𝑓∶ℝ→ℝsatisfies the Drygas equation if 𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+𝑓𝑦+𝑓−𝑦,𝑥,𝑦∈ℝ. (1) The above equation was introduced in [1] in order to obtain a characterization of the quasi-inner-product spaces. The general solution of (1), obtained by Ebanks et al. in [2] (see also [3]), is of the following form: 𝑓(𝑥)=𝑎(𝑥)+𝑞(𝑥),𝑥∈ℝ, (2) where 𝑎∶ℝ→ℝis an additive function and 𝑞∶ℝ→ℝis a quadratic function; that is, 𝑞 satisfies the quadratic functional equation 𝑞𝑥+𝑦+𝑞𝑥−𝑦=2𝑞(𝑥)+2𝑞𝑦,𝑥,𝑦∈ℝ. (3) A set-valued version of (1) was considered by Smajdor in [4].

The stability in the Hyers-Ulam sense of the Drygas equation has been investigated, for example, in [5–8]. For more information and numerous references on the stability of functional equations in general, we refer to [9, 10].

The term hyperstability was used for the first time probably in [11]; however, it seems that the first hyperstability result was published in [12] and concerned the ring homomorphisms. The hyperstability results for the Cauchy equation were investigated by Brzdȩk in [13, 14]. Gselmann in [15] studied the hyperstability of the parametric fundamental equation of information.

In the paper, we prove the hyperstability of the Drygas equation on a restricted domain. Let 𝑋 be a nonempty subset of an Abelian group and 𝑌 a semigroup. We say that a function 𝑓∶𝑋→𝑌 satisfies the Drygas functional equation on 𝑋 if 𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+𝑓𝑦+𝑓−𝑦 (4) for all 𝑥,𝑦∈𝑋 such that −𝑦,𝑥+𝑦,𝑥−𝑦∈𝑋. We will show that (4) is hyperstable; that is, each function 𝑓∶𝑋→𝑌 (under some additional assumptions on 𝑋) satisfying the inequality 𝑓(𝑥)𝑦‖𝑥‖𝑥+𝑦+𝑓𝑥−𝑦−2𝑓−𝑓−𝑓−𝑦≤𝑐^𝑝+𝑦^𝑝 (5) for all 𝑥,𝑦∈𝑋 such that −𝑦,𝑥+𝑦,𝑥−𝑦∈𝑋 with 𝑝<0 must satisfy the Drygas equation (4).

The method of the proof of the main theorem is motivated by an idea used by Brzdȩk in [13] and further by Piszczek in [16]. It is based on a fixed point theorem for functional spaces obtained by Brzdęk et al. (see [17, Theorem  1]). Some generalizations of their result were proved by Cădariu et al. in [18]. The case of fixed point theorem for non-Archimedean metric spaces was also studied by Brzdęk and Ciepliński in [19]. It is worth mentioning that using fixed point theorems is now one of the most popular methods of investigating the stability of functional equations in single as well as in several variables. Let us recall a few recent approaches of Jung in [20], Lee and Jung in [21], or Cădariu and Radu in [22, 23]. More information on the application of the fixed point method was collected by Ciepliński in [24].

First, we take the following three hypotheses (all notations come from [17]).(H1)𝑋 is a nonempty set, 𝑌 is a Banach space, and 𝑓₁,…,𝑓_𝑘∶𝑋→𝑋 and 𝐿₁,…,𝐿_𝑘∶𝑋→ℝ₊ are given.(H2)𝒯∶𝑌^𝑋→𝑌^𝑋 is an operator satisfying the inequality 𝒯𝜉(𝑥)−𝒯𝜇(𝑥)≤^𝑘_𝑖=1𝐿_𝑖(𝑥)𝜉𝑓_𝑖(𝑥)−𝜇𝑓_𝑖(𝑥) (6) for all 𝜉,𝜇∈𝑌^𝑋, 𝑥∈𝑋. (H3)Λ∶ℝ₊^𝑋→ℝ₊^𝑋 is defined by Λ𝛿(𝑥)∶=^𝑘_𝑖=1𝐿_𝑖(𝑥)𝛿𝑓_𝑖(𝑥),𝛿∈ℝ₊^𝑋,𝑥∈𝑋. (7)

The mentioned fixed point theorem is stated as follows.

Theorem 1. Let hypotheses (H1)–(H3) be valid and functions 𝜀∶𝑋→ℝ₊ and let 𝜑∶𝑋→𝑌 fulfil the following two conditions: (𝑥)(𝑥(𝑥)𝜀𝒯𝜑−𝜑)≤𝜀,𝑥∈𝑋,_∗(𝑥)∶=^∞_𝑛=0Λ^𝑛𝜀(𝑥)<∞,𝑥∈𝑋. (8) Then, there exists a unique fixed point 𝜓 of 𝒯 with 𝜑(𝑥)−𝜓(𝑥)≤𝜀^∗(𝑥),𝑥∈𝑋. (9) Moreover, 𝜓(𝑥)∶=lim_𝑛→∞𝒯^𝑛𝜑(𝑥),𝑥∈𝑋. (10)

Throughout the paper, ℕ, ℕ₀, and ℕ_𝑚0 denote the set of all positive integers, the set of all nonnegative integers, and the set of all integers greater than or equal to 𝑚₀, respectively.

2. The Main Result

Theorem 2. Assume that 𝑋 is a nonempty, symmetric with respect to 0 subset of a normed space such that 0∉𝑋 and there exists 𝑛₀∈ℕwith 𝑛𝑥∈𝑋 for 𝑥∈𝑋, 𝑛∈ℕ_𝑛0. Let 𝑌 be a Banach space, 𝑐≥0, and 𝑝<0. If 𝑓∶𝑋→𝑌 satisfies 𝑓(𝑥)𝑦‖𝑥‖𝑥+𝑦+𝑓𝑥−𝑦−2𝑓−𝑓−𝑓−𝑦≤𝑐^𝑝+𝑦^𝑝 (11) for all 𝑥,𝑦∈𝑋 such that 𝑥+𝑦,𝑥−𝑦∈𝑋, then 𝑓 satisfies the Drygas equation on 𝑋.

Proof. First observe that there exists 𝑚₀∈ℕ_𝑛0 such that 2(𝑚+1)^𝑝+2𝑚^𝑝+(2𝑚+1)^𝑝<1for𝑚≥𝑚₀. (12) Let us fix 𝑚∈ℕ_𝑚0. Replacing 𝑥 with (𝑚+1)𝑥 and 𝑦 with 𝑚𝑥 in (11), we get )𝑥)()())𝑥)(𝑥()2𝑓((𝑚+1+𝑓𝑚𝑥+𝑓−𝑚𝑥−𝑓((2𝑚+1−𝑓)≤𝑐𝑚+1^𝑝+𝑚^𝑝‖𝑥‖^𝑝,𝑥∈𝑋. (13) Further put (𝑥))𝑥)()())𝑥)𝒯𝜉∶=2𝜉((𝑚+1+𝜉𝑚𝑥+𝜉−𝑚𝑥−𝜉((2𝑚+1,𝑥∈𝑋,𝜉∈𝑌^𝑋,𝜀(𝑥)∶=𝑐(𝑚+1)^𝑝+𝑚^𝑝‖𝑥‖^𝑝,𝑥∈𝑋. (14) Inequality (13) now takes the form 𝒯𝑓(𝑥)−𝑓(𝑥)≤𝜀(𝑥),𝑥∈𝑋. (15) The operator (𝑥))𝑥)()())𝑥)Λ𝜂∶=2𝜂((𝑚+1+𝜂𝑚𝑥+𝜂−𝑚𝑥+𝜂((2𝑚+1,𝜂∈ℝ₊^𝑋,𝑥∈𝑋 (16) has the form described in (H3) with 𝑘=4 and 𝑓₁(𝑥)=(𝑚+1)𝑥, 𝑓₂(𝑥)=𝑚𝑥, 𝑓₃(𝑥)=−𝑚𝑥, 𝑓₄(𝑥)=(2𝑚+1)𝑥, 𝐿₁(𝑥)=2, and 𝐿₂(𝑥)=𝐿₃(𝑥)=𝐿₄(𝑥)=1 for 𝑥∈𝑋. Moreover, for every 𝜉,𝜇∈𝑌^𝑋,  𝑥∈𝑋, (𝑥)(𝑥=)𝑥)()())𝑥))𝑥)()())𝑥)𝑥+(+(+)𝑥=𝒯𝜉−𝒯𝜇)2𝜉((𝑚+1+𝜉𝑚𝑥+𝜉−𝑚𝑥−𝜉((2𝑚+1−2𝜇((𝑚+1−𝜇𝑚𝑥−𝜇−𝑚𝑥+𝜇((2𝑚+1)≤2𝜉−𝜇((𝑚+1)𝜉−𝜇𝑚𝑥)𝜉−𝜇−𝑚𝑥)𝜉−𝜇((2𝑚+1)_4𝑖=1𝐿_𝑖(𝑥)𝜉−𝜇𝑓_𝑖(𝑥). (17) As 2(𝑚+1)^𝑝+2𝑚^𝑝+(2𝑚+1)^𝑝<1, we have 𝜀^∗(𝑥)∶=^∞_𝑛=0Λ^𝑛𝜀(𝑥)=^∞_𝑛=0𝑐(𝑚+1)^𝑝+𝑚^𝑝2(𝑚+1)^𝑝+2𝑚^𝑝+(2𝑚+1)^𝑝𝑛‖𝑥‖^𝑝=𝑐(𝑚+1)^𝑝+𝑚^𝑝‖𝑥‖^𝑝1−2(𝑚+1)^𝑝−2𝑚^𝑝−(2𝑚+1)^𝑝. (18) Thus, according to Theorem 1 there exists a unique solution 𝐹∶𝑋→𝑌 of the equation 𝐹(𝑥)=2𝐹((𝑚+1)𝑥)+𝐹(𝑚𝑥)+𝐹(−𝑚𝑥)−𝐹((2𝑚+1)𝑥) (19) such that 𝑓(𝑥)(𝑥≤𝑐()−𝐹)𝑚+1^𝑝+𝑚^𝑝‖𝑥‖^𝑝1−2(𝑚+1)^𝑝−2𝑚^𝑝−(2𝑚+1)^𝑝,𝑥∈𝑋. (20) Moreover, 𝐹(𝑥)=lim_𝑛→∞𝒯^𝑛𝑓(𝑥). (21)
To prove that 𝐹 satisfies the Drygas equation on 𝑋, observe that 𝒯^𝑛𝑓𝑥+𝑦+𝒯^𝑛𝑓𝑥−𝑦−2𝒯^𝑛𝑓(𝑥)−𝒯^𝑛𝑓𝑦−𝒯^𝑛𝑓2()−𝑦≤𝑐𝑚+1^𝑝+2𝑚^𝑝+(2𝑚+1)^𝑝𝑛‖𝑥‖^𝑝+𝑦^𝑝 (22) for every 𝑥,𝑦∈𝑋,  𝑥+𝑦,𝑥−𝑦∈𝑋, and 𝑛∈ℕ₀.
Indeed, if 𝑛=0, then (22) is simply (11). So, fix 𝑛∈ℕ₀ and suppose that (22) holds for 𝑛 and 𝑥,𝑦∈𝑋 such that 𝑥+𝑦,𝑥−𝑦∈𝑋. Then 𝒯^𝑛+1𝑓𝑥+𝑦+𝒯^𝑛+1𝑓𝑥−𝑦−2𝒯^𝑛+1𝑓(𝑥)−𝒯^𝑛+1𝑓𝑦−𝒯^𝑛+1𝑓=−𝑦2𝒯^𝑛𝑓(𝑚+1)𝑥+𝑦+𝒯^𝑛𝑓𝑚𝑥+𝑦+𝒯^𝑛𝑓−𝑚𝑥+𝑦−𝒯^𝑛𝑓(2𝑚+1)𝑥+𝑦+2𝒯^𝑛𝑓(𝑚+1)𝑥−𝑦+𝒯^𝑛𝑓𝑚𝑥−𝑦+𝒯^𝑛𝑓−𝑚𝑥−𝑦−𝒯^𝑛𝑓(2𝑚+1)𝑥−𝑦−22𝒯^𝑛𝑓((𝑚+1)𝑥)+𝒯^𝑛𝑓(𝑚𝑥)+𝒯^𝑛𝑓(−𝑚𝑥)−𝒯^𝑛𝑓((2𝑚+1)𝑥)−2𝒯^𝑛𝑓(𝑚+1)𝑦−𝒯^𝑛𝑓𝑚𝑦−𝒯^𝑛𝑓−𝑚𝑦+𝒯^𝑛𝑓(2𝑚+1)𝑦−2𝒯^𝑛𝑓(𝑚+1)−𝑦−𝒯^𝑛𝑓𝑚−𝑦−𝒯^𝑛𝑓−𝑚−𝑦+𝒯^𝑛𝑓(2()2𝑚+1)−𝑦≤𝑐𝑚+1^𝑝+2𝑚^𝑝+(2𝑚+1)^𝑝𝑛×2‖(𝑚+1)𝑥‖^𝑝+(𝑚+1)𝑦^𝑝+‖𝑚𝑥‖^𝑝+𝑚𝑦^𝑝+‖−𝑚𝑥‖^𝑝+−𝑚𝑦^𝑝+‖(2𝑚+1)𝑥‖^𝑝+(2𝑚+1)𝑦^𝑝=𝑐2(𝑚+1)^𝑝+2𝑚^𝑝+(2𝑚+1)^𝑝𝑛+1×‖𝑥‖^𝑝+𝑦^𝑝. (23) By induction, we have shown that (22) holds for all 𝑥,𝑦∈𝑋,  𝑥+𝑦,𝑥−𝑦∈𝑋, and 𝑛∈ℕ₀. Letting 𝑛→∞ in (22), we obtain 𝐹𝑥+𝑦+𝐹𝑥−𝑦=2𝐹(𝑥)+𝐹𝑦+𝐹−𝑦,𝑥,𝑦∈𝑋,𝑥+𝑦,𝑥−𝑦∈𝑋. (24) Thus, we have proved that for every 𝑚∈ℕ_𝑚0 there exists a function 𝐹_𝑚∶𝑋→𝑌 such that 𝐹_𝑚 is a solution of the Drygas equation on 𝑋 and 𝑓(𝑥)−𝐹_𝑚(𝑥≤𝑐())𝑚+1^𝑝+𝑚^𝑝‖𝑥‖^𝑝1−2(𝑚+1)^𝑝−2𝑚^𝑝−(2𝑚+1)^𝑝,𝑥∈𝑋. (25) Since 𝑝<0, the sequence ((𝑐((𝑚+1)^𝑝+𝑚^𝑝)‖𝑥‖^𝑝)/(1−2(𝑚+1)^𝑝−2𝑚^𝑝−(2𝑚+1)^𝑝))_{𝑚∈ℕ𝑚0} tends to zero. Consequently, 𝑓 satisfies the Drygas equation on 𝑋 as the pointwise limit of (𝐹_𝑚)_{𝑚∈ℕ𝑚0}.

Remark 3. If 𝑋 is a normed space and 𝑓∶𝑋→𝑌 satisfies (11) for 𝑥,𝑦∈𝑋⧵{0}, with 𝑝<0, then by Theorem 2 we know that 𝑓 satisfies the Drygas equation on 𝑋⧵{0}. It is easy to see that if 𝑓(0)=0, then 𝑓 satisfies the Drygas equation on the whole 𝑋.

We end with two examples. First notice that the assumption of unboundedness of the set 𝑋 is necessary.

Example 4. Let 𝑋=[−1,1]⧵{0} and let 𝑓∶𝑋→ℝbe defined by 𝑓(𝑥)=|𝑥|, 𝑥∈𝑋. Then for all 𝑥,𝑦∈𝑋 such that 𝑥+𝑦,𝑥−𝑦∈𝑋, 𝑓(𝑥)𝑦≤|𝑥|𝑥+𝑦+𝑓𝑥−𝑦−2𝑓−𝑓−𝑓−𝑦_𝑝+𝑦^𝑝, (26) with 𝑝<0, but 𝑓 does not satisfy the Drygas equation on 𝑋.

In the case 𝑝≥0, the considered Drygas equation is not hyperstable.

Example 5. Let 𝑋=ℝ⧵[−1,1] and let 𝑓∶𝑋→ℝbe a constant function 𝑓(𝑥)=𝑐, 𝑥∈𝑋 for some 𝑐>0. Then 𝑓 satisfies the inequality 𝑓(𝑥)𝑦|𝑥|𝑥+𝑦+𝑓𝑥−𝑦−2𝑓−𝑓−𝑓−𝑦≤𝑐^𝑝+𝑦^𝑝, (27) for all 𝑥,𝑦∈𝑋 such that 𝑥+𝑦,𝑥−𝑦∈𝑋, with 𝑝≥0, but is not a solution of the Drygas equation on 𝑋.