Inclusions in a Single Variable in Ultrametric Spaces and Hyers-Ulam Stability

Abstract

We present some properties of set-valued inclusions in a single variable in ultrametric spaces. As a consequence, we obtain stability results for the corresponding functional equations.

1. Introduction

A metric space (𝑋,𝑑) is called an ultrametric space (or non-Archimedean metric space), if 𝑑, called an ultrametric, satisfies the strong triangle inequality 𝑑(𝑥,𝑧)≤max𝑑𝑥,𝑦,𝑑𝑦,𝑧for𝑥,𝑦,𝑧∈𝑋. (1)

One of the typical ultrametrics is a 𝑝-adic metric. Let 𝑝 be a fixed prime. For 𝑛,𝑚∈ℕ, we define 𝑑()=𝑝𝑚,𝑛0if𝑚=𝑛,_−𝑟if𝑚≠𝑛, (2) where 𝑟 is the largest nonnegative integer such that 𝑝^𝑟 divides 𝑚−𝑛. This example is the introduction to the 𝑝-adic numbers which play the essential role because of their connections with some problem coming from quantum physics, 𝑝-adic string or superstring (see [1]).

The inequality 𝑑𝑥_𝑚,𝑥_𝑛≤max𝑑𝑥_𝑚,𝑥_𝑚−1,…,𝑑𝑥_𝑛+1,𝑥_𝑛for𝑚>𝑛 (3) yields and implies the following lemma.

Lemma 1. A sequence (𝑥_𝑛)_𝑛∈ℕ in an ultrametric space (𝑋,𝑑) is a Cauchy sequence if and only if lim_𝑛→∞𝑑(𝑥_𝑛+1,𝑥_𝑛)=0.

Let (𝑋,𝑑) be an ultrametric space. The number 𝛿(𝐴)=sup{𝑑(𝑥,𝑦)∶𝑥,𝑦∈𝐴} is called the diameter of 𝐴⊂𝑋. We will denote by 𝑛(𝑋) the family of all nonempty subsets of 𝑋. Moreover, let 𝑏𝑑(𝑋) stand for the family of all bounded sets of 𝑛(𝑋) let and 𝑏cl(𝑋) denote the family of all closed sets of 𝑏𝑑(𝑋). We understand the convergence of sets with respect to the Hausdorff metric ℎ derived from the metric 𝑑. It is easy to see that (𝑏cl(𝑋),ℎ) is also an ultrametric space, that is, ℎ satisfies the strong triangle inequality ℎ(𝐴,𝐶)≤max{ℎ(𝐴,𝐵),ℎ(𝐵,𝐶)}for𝐴,𝐵,𝐶∈𝑏cl(𝑋). (4)

We say that (𝑋,+,𝑑) is a complete ultrametric commutative groupoid with 0, if (𝑋,+) is a commutative groupoid with a neutral element 0, (𝑋,𝑑) is a complete ultrametric space and the operation + is continuous with respect to the metric 𝑑.

From now on, we assume that 𝐾 is a nonempty set and (𝑋,+,𝑑) is a complete ultrametric commutative groupoid with 0. For 𝐴,𝐵∈𝑛(𝑋), we define 𝐴+𝐵={𝑎+𝑏∶𝑎∈𝐴,𝑏∈𝐵}. (5)

The aim of the paper is to obtain some results concerning the following inclusion: Ψ(𝐹(𝑎(𝑥)))⊂𝐹(𝑥)+𝐺(𝑥), (6) where Ψ∶𝑋→𝑋, 𝐹,𝐺∶𝐾→𝑛(𝑋), and 𝑎∶𝐾→𝐾 and its generalization in an ultrametric space. In ultrametric spaces, it is possible to get better estimation with weaker assumptions, than in metric spaces. The ideas of proofs are based on the ideas in [2]. As a consequence we obtain stability results for the corresponding functional equation Ψ∘𝑓∘𝑎=𝑓 and its generalization. Some results of the stability of functional equations in non-Archimedean spaces can be found in [3–8].

2. Main Results

Theorem 2.  Let 𝐹,𝐺∶𝐾→𝑛(𝑋), 0∈𝐺(𝑥) for all 𝑥∈𝐾, Ψ∶𝑋→𝑋, 𝑎∶𝐾→𝐾, 𝜆∈(0,1), and 𝑑Ψ(𝑥)𝑦{𝛿(𝐹(𝑥)(𝑥}Ψ(𝐹(𝑎(𝑥(𝑥)(𝑥),Ψ≤𝜆𝑑𝑥,𝑦,𝑥,𝑦∈𝑋,𝑀∶=sup+𝐺))∶𝑥∈𝐾<∞,)))⊂𝐹+𝐺𝑓𝑜𝑟𝑥∈𝐾. (7)(8) Then there exists a unique function 𝑓∶𝐾→𝑋 such that Ψ∘𝑓∘𝑎=𝑓 and 𝑑𝑓(𝑥),𝐹(𝑥)≤𝑀,𝑥∈𝐾. (9)

Proof. Let us fix 𝑥∈𝐾. Replacing 𝑥 by 𝑎^𝑛(𝑥) in (8), we get Ψ𝐹𝑎^𝑛+1(𝑥)⊂𝐹𝑎^𝑛(𝑥)+𝐺𝑎^𝑛(𝑥), (10) and as 0∈𝐺(𝑥), we have 𝐹𝑎^𝑛(𝑥)⊂𝐹𝑎^𝑛(𝑥)+𝐺𝑎^𝑛(𝑥) (11) for 𝑛∈ℕ₀. Thus, ℎΨ^𝑛+1𝐹𝑎^𝑛+1(𝑥),Ψ^𝑛𝐹𝑎^𝑛(𝑥)≤𝜆^𝑛ℎΨ𝐹𝑎^𝑛+1(𝑥),𝐹𝑎^𝑛(𝑥)≤𝜆^𝑛𝛿𝐹𝑎^𝑛(𝑥)+𝐺𝑎^𝑛(𝑥)≤𝜆^𝑛𝑀 (12) for all 𝑛∈ℕ₀. According to Lemma 1 and lim_𝑛→∞𝜆^𝑛𝑀=0, the sequence (Ψ^𝑛(𝐹(𝑎^𝑛(𝑥))))_𝑛∈ℕ0 is a Cauchy sequence. Since (𝑏cl(𝑌),ℎ) is complete, there exists the limit lim_𝑛→∞clΨ^𝑛(𝐹(𝑎^𝑛(𝑥))). Moreover, 𝛿clΨ^𝑛𝐹𝑎^𝑛(𝑥)≤𝜆^𝑛𝛿𝐹𝑎^𝑛(𝑥), (13) and the right side of the last inequality converges to 0 with 𝑛→∞. Therefore, lim_𝑛→∞clΨ^𝑛𝐹𝑎^𝑛(𝑥)=∶𝑓(𝑥) (14) is a singleton and as Ψ is continuous, Ψ𝑓(𝑎(𝑥))=Ψlim_𝑛→∞clΨ^𝑛𝐹𝑎^𝑛(𝑎(𝑥))⊂lim_𝑛→∞clΨ^𝑛+1𝐹𝑎^𝑛+1(𝑥)=𝑓(𝑥), (15) so Ψ∘𝑓∘𝑎=𝑓. Notice that ℎΨ^𝑛𝐹𝑎^𝑛(𝑥(𝑥ℎΨ),𝐹)≤max^𝑛𝐹𝑎^𝑛(𝑥),Ψ^𝑛−1𝐹𝑎^𝑛−1(𝑥)ℎ(Ψ(𝐹(𝑎(𝑥(𝑥𝜆,…,))),𝐹))≤max^𝑛−1𝛿𝐹𝑎^𝑛−1(𝑥)+𝐺𝑎^𝑛−1(𝑥𝛿(𝐹(𝑥)(𝑥𝜆),…,+𝐺))≤max^𝑛−1𝑀,…,𝑀≤𝑀. (16) Consequently, 𝑑𝑓(𝑥)(𝑥)𝑓(𝑥)(𝑥),𝐹≤𝑀,∈𝐹+𝑀𝑆, (17) where 𝑆 is a closed unit ball.
It remains to prove the uniqueness of 𝑓. Let 𝑓,𝑔 be such that Ψ∘𝑓∘𝑎=𝑓, Ψ∘𝑔∘𝑎=𝑔, 𝑑(𝑓(𝑥),𝐹(𝑥))≤𝑀, 𝑑(𝑔(𝑥),𝐹(𝑥))≤𝑀 for 𝑥∈𝐾. By induction we get Ψ^𝑛∘𝑓∘𝑎^𝑛=𝑓 and Ψ^𝑛∘𝑔∘𝑎^𝑛=𝑔 for 𝑛∈ℕ₀. Hence, 𝑑𝑓(𝑥),𝑔(𝑥)=𝑑Ψ^𝑛∘𝑓∘𝑎^𝑛(𝑥),Ψ^𝑛∘𝑔∘𝑎^𝑛(𝑥)≤𝜆^𝑛𝑑𝑓𝑎^𝑛(𝑥),𝑔𝑎^𝑛(𝑥)≤𝜆^𝑛𝛿𝐹𝑎^𝑛(𝑥)+𝑀𝑆,𝑛∈ℕ₀. (18) Since lim_𝑛→∞𝜆^𝑛𝛿(𝐹(𝑎^𝑛(𝑥))+𝑀𝑆)=0, we have 𝑓=𝑔.

Theorem 3. Assume that 𝐹,𝐺∶𝐾→𝑏𝑑(𝑋), 0∈𝐺(𝑥) for all 𝑥∈𝐾, 𝑘∈ℕ, Ψ∶𝐾×𝑋^𝑘→𝑋, 𝛼₁,…,𝛼_𝑘∶𝐾→𝐾, 𝜆₁,…,𝜆_𝑘∶𝐾→[0,∞) are such that 𝑑Ψ𝑥,𝑤₁,…,𝑤_𝑘,Ψ𝑥,𝑧₁,…,𝑧_𝑘≤max_{{}𝑖∈1,…,𝑘}𝜆_𝑖(𝑥)𝑑𝑤_𝑖,𝑧_𝑖 (19) for 𝑥∈𝐾, 𝑤₁,…,𝑤_𝑘,𝑧₁,…,𝑧_𝑘∈𝑋, lim_𝑛→∞max_{𝑖1=1,…,𝑘}𝜆_𝑖1(𝑥)max_{𝑖2=1,…,𝑘}𝜆_𝑖2∘𝛼_𝑖1(𝑥)⋯max_{𝑖𝑛=1,…,𝑘}𝜆_𝑖𝑛∘𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥)×𝛿𝐹𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1𝑥)𝛼(+𝐺_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥Ψ𝛼)=0,𝑥∈𝐾,𝑥,𝐹₁(𝑥),…,𝐹𝛼_𝑘(𝑥)⊂𝐹(𝑥)+𝐺(𝑥)𝑓𝑜𝑟𝑥∈𝐾. (20)(21) Then there exists a unique function 𝑓∶𝐾→𝑋 such that Ψ(𝑥,𝑓(𝛼₁(𝑥)),…,𝑓(𝛼_𝑘(𝑥)))=𝑓(𝑥) and 𝑑(𝑓(𝑥),𝐹(𝑥))≤𝑘(𝑥) for 𝑥∈𝐾, where 𝑘(𝑥)=sup_𝑛∈ℕ𝛿(𝐹(𝑥)+𝐺(𝑥)),max_{𝑖1=1,…,𝑘}𝜆_𝑖1(𝑥)max_{𝑖2=1,…,𝑘}𝜆_𝑖2∘𝛼_𝑖1(𝑥)⋯max_{𝑖𝑛=1,…,𝑘}𝜆_𝑖𝑛∘𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥)×𝛿𝐹𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥)+𝐺𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥). (22)

Proof. Let us fix 𝑥∈𝐾. Replacing 𝑥 by 𝛼_𝑖(𝑥), 𝑖=1,…,𝑘, in (21), we obtain Ψ𝛼_𝑖(𝑥),𝐹𝛼₁𝛼_𝑖(𝑥),…,𝐹𝛼_𝑘𝛼_𝑖(𝑥)⊂𝐹𝛼_𝑖(𝑥)+𝐺𝛼_𝑖(𝑥). (23) Since 0∈𝐺(𝑥), we have 𝐹(𝛼_𝑖(𝑥))⊂𝐹(𝛼_𝑖(𝑥))+𝐺(𝛼_𝑖(𝑥)) for 𝑖=1,…,𝑘. Hence, Ψ𝑥,Ψ𝛼₁(𝑥),𝐹𝛼₁𝛼₁(𝑥),…,𝐹𝛼_𝑘𝛼₁(𝑥)Ψ𝛼,…,_𝑘(𝑥),𝐹𝛼₁𝛼_𝑘(𝑥),…,𝐹𝛼_𝑘𝛼_𝑘(𝑥)⊂Ψ𝑥,𝐹𝛼₁(𝑥)+𝐺𝛼₁(𝑥),…,𝐹𝛼_𝑘(𝑥)+𝐺𝛼_𝑘(𝑥). (24) We define a sequence (𝐴_𝑛(𝑥))_𝑛∈ℕ0 by the following recurrence relation: 𝐴₀(𝑥)(𝑥),𝐴=𝐹_𝑛+1(𝑥)=Ψ𝑥,𝐴_𝑛𝛼₁(𝑥),…,𝐴_𝑛𝛼_𝑘(𝑥). (25) It is easy to see that ℎ𝐴₁(𝑥),𝐴₀(𝑥)≤ℎΨ𝑥,𝐹𝛼₁(𝑥),…,𝐹𝛼_𝑘(𝑥)(𝑥)(𝐹(𝑥)(𝑥,ℎ𝐴,𝐹≤𝛿+𝐺))_𝑛+1(𝑥),𝐴_𝑛(𝑥)≤max_{𝑖1=1,…,𝑘}𝜆_𝑖1(𝑥)max_{𝑖2=1,…,𝑘}𝜆_𝑖2∘𝛼_𝑖1(𝑥)⋯max_{𝑖𝑛=1,…,𝑘}𝜆_𝑖𝑛∘𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥)×𝛿𝐹𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥)+𝐺𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥) (26) for 𝑛∈ℕ. In virtue of (20), the sequence (𝐴_𝑛(𝑥))_𝑛∈ℕ is a Cauchy sequence. As (𝑏cl(𝑌),ℎ) is a complete metric space, there exists the limit lim_𝑛→∞cl𝐴_𝑛(𝑥). Moreover, 𝛿cl𝐴_𝑛(𝑥)≤max_{𝑖1=1,…,𝑘}𝜆_𝑖1(𝑥)max_{𝑖2=1,…,𝑘}𝜆_𝑖2∘𝛼_𝑖1(𝑥)⋯max_{𝑖𝑛=1,…,𝑘}𝜆_𝑖𝑛∘𝛼_𝑖𝑙−1∘⋯∘𝛼_𝑖1(𝑥)×𝛿𝐹𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥), (27) and the right side of the last inequality converges to 0 as 𝑛→∞. Therefore, lim_𝑛→∞cl𝐴_𝑛(𝑥)=∶𝑓(𝑥) (28) is a singleton, and as Ψ satisfies (19), Ψ𝑥,𝑓𝛼₁(𝑥),…,𝑓𝛼_𝑘(𝑥)=Ψ𝑥,lim_𝑛→∞cl𝐴_𝑛𝛼₁(𝑥),…,lim_𝑛→∞cl𝐴_𝑛𝛼_𝑘(𝑥)⊂lim_𝑛→∞clΨ𝑥,𝐴_𝑛𝛼₁(𝑥),…,𝐴_𝑛𝛼_𝑘(𝑥)=lim_𝑛→∞cl𝐴_𝑛+1(𝑥)=𝑓(𝑥). (29) By (26), we get ℎ𝐴_𝑛(𝑥)(𝑥)𝛿(𝐹(𝑥)(𝑥,𝐹≤max+𝐺)),…,max_{𝑖1=1,…,𝑘}𝜆_𝑖1(𝑥)max_{𝑖2=1,…,𝑘}𝜆_𝑖2∘𝛼_𝑖1(𝑥)⋯max_{𝑖𝑛−1=1,…,𝑘}𝜆_𝑖𝑛−1∘𝛼_𝑖𝑛−2∘⋯∘𝛼_𝑖1(𝑥)×𝛿𝐹𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥)+𝐺𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥) (30) which, with 𝑛→∞, yields 𝑑𝑓(𝑥),𝐹(𝑥)≤sup_𝑛∈ℕ𝛿(𝐹(𝑥)+𝐺(𝑥)),max_{𝑖1=1,…,𝑘}𝜆_𝑖1(𝑥)max_{𝑖2=1,…,𝑘}𝜆_𝑖2∘𝛼_𝑖1(𝑥)⋯max_{𝑖𝑛=1,…,𝑘}𝜆_𝑖𝑛∘𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥)×𝛿𝐹𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥)+𝐺𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥)=𝑘(𝑥). (31)
It remains to prove the uniqueness of 𝑓. Suppose that 𝑓,𝑔 are such that Ψ(𝑥,𝑓(𝛼₁(𝑥)),…,𝑓(𝛼_𝑘(𝑥)))=𝑓(𝑥), Ψ(𝑥,𝑔(𝛼₁(𝑥)),…,𝑔(𝛼_𝑘(𝑥)))=𝑔(𝑥), 𝑑(𝑓(𝑥),𝐹(𝑥))≤𝑘(𝑥), and 𝑑(𝑔(𝑥),𝐹(𝑥))≤𝑘(𝑥). Replacing 𝑥 by 𝛼_𝑖(𝑥), 𝑖=1,…,𝑘, in the penultimate equality, we get Ψ𝛼_𝑖(𝑥),𝑓𝛼₁𝛼_𝑖(𝑥),…,𝑓𝛼_𝑘𝛼_𝑖(𝑥)=𝑓𝛼_𝑖(𝑥). (32) Thus, Ψ𝑥,Ψ𝛼₁(𝑥),𝑓𝛼₁𝛼₁(𝑥),…,𝑓𝛼_𝑘𝛼₁(𝑥)Ψ𝛼,…,_𝑘(𝑥),𝑓𝛼₁𝛼_𝑘(𝑥),…,𝑓𝛼_𝑘𝛼_𝑘(𝑥)=Ψ𝑥,𝑓𝛼₁(𝑥),…,𝑓𝛼_𝑘(𝑥)=𝑓(𝑥), (33) and we get a constant sequence 𝑓₀(𝑥)(𝑥),𝑓=𝑓_𝑛+1(𝑥)=Ψ𝑥,𝑓_𝑛𝛼₁(𝑥),…,𝑓_𝑛𝛼_𝑘(𝑥). (34) In the same way, we get a constant sequence 𝑔₀(𝑥)(𝑥),𝑔=𝑔_𝑛+1(𝑥)=Ψ𝑥,𝑔_𝑛𝛼₁(𝑥),…,𝑔_𝑛𝛼_𝑘(𝑥). (35) Hence, 𝑑𝑓(𝑥),𝑔(𝑥)=𝑑𝑓_𝑛(𝑥),𝑔_𝑛(𝑥)≤max_{𝑖1=1,…,𝑘}𝜆_𝑖1(𝑥)max_{𝑖2=1,…,𝑘}𝜆_𝑖2∘𝛼_𝑖1(𝑥)⋯max_{𝑖𝑛=1,…,𝑘}𝜆_𝑖𝑛∘𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥)×𝑑𝑓𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥),𝑔𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥)≤max_{𝑖1=1,…,𝑘}𝜆_𝑖1(𝑥)max_{𝑖2=1,…,𝑘}𝜆_𝑖2∘𝛼_𝑖1(𝑥)⋯max_{𝑖𝑛=1,…,𝑘}𝜆_𝑖𝑛∘𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥)×𝑑𝑓𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥),𝐹𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥)+𝑑𝐹𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥),𝑔𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥)≤2max_{𝑖1=1,…,𝑘}𝜆_𝑖1(𝑥)max_{𝑖2=1,…,𝑘}𝜆_𝑖2∘𝛼_𝑖1(𝑥)⋯max_{𝑖𝑛=1,…,𝑘}𝜆_𝑖𝑛∘𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥)×𝑘𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥). (36) Using the definition of 𝑘, we get 𝑑𝑓(𝑥),𝑔(𝑥)≤2max_{𝑖1=1,…,𝑘}𝜆_𝑖1(𝑥)max_{𝑖2=1,…,𝑘}𝜆_𝑖2∘𝛼_𝑖1(𝑥)⋯max_{𝑖𝑛=1,…,𝑘}𝜆_𝑖𝑛∘𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥)×𝛿𝐹𝛼_𝑖𝑛∘⋯∘𝛼_𝑖1𝑥)𝛼(+𝐺_𝑖𝑛∘⋯∘𝛼_𝑖1(𝑥)+sup_{𝑙=1,…,∞}max_{𝑗1=1,…,𝑘}𝜆_𝑗1∘𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1×(𝑥)⋯max_{𝑗𝑙=1,…,𝑘}𝜆_𝑗𝑙∘𝛼_𝑗𝑙−1∘⋯∘𝛼_𝑗1∘𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥)×𝛿𝐹𝛼_𝑗𝑙∘⋯∘𝛼_𝑗1∘𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥)+𝐺𝛼_𝑗𝑛∘⋯∘𝛼_𝑗1∘𝛼_𝑖𝑛−1∘⋯∘𝛼_𝑖1(𝑥)=sup_{𝑙=𝑛,…,∞}max_{𝑖1=1,…,𝑘}𝜆_𝑖1(𝑥)max_{𝑖2=1,…,𝑘}𝜆_𝑖2∘𝛼_𝑖1×(𝑥)⋯max_{𝑖𝑙=1,…,𝑘}𝜆_𝑖𝑙∘𝛼_𝑖𝑙−1∘⋯∘𝛼_𝑖1(𝑥)×𝛿𝐹𝛼_𝑖𝑙∘⋯∘𝛼_𝑖1(𝑥)+𝐺𝛼_𝑖𝑙∘⋯∘𝛼_𝑖1(𝑥). (37) It follows that 𝑓=𝑔 with 𝑛→0, and the proof is completed.

3. Stability Results

We present the applications of the above theorems to the results concerning the stability of functional equations.

Corollary 4.  Let 𝑓∶𝐾→𝑋, Ψ∶𝑋→𝑋, 𝑎∶𝐾→𝐾, 𝜆∈(0,1), 𝜖>0 satisfy 𝑑Ψ(𝑥)𝑦𝑑Ψ𝑓(𝑎(𝑥(𝑥),Ψ≤𝜆𝑑𝑥,𝑦,𝑥,𝑦∈𝑋,)),𝑓≤𝜖,𝑥∈𝐾. (38) Then there exists a unique function 𝑔∶𝐾→𝑋 such that Ψ∘𝑔∘𝑎=𝑔 and 𝑑𝑓(𝑥),𝑔(𝑥)≤2𝜖,𝑥∈𝐾. (39)

Proof. Let 𝐹(𝑥)={𝑓(𝑥)} for 𝑥∈𝐾. Then Ψ(𝐹(𝑎(𝑥)))⊂𝐹(𝑥)+𝜖𝑆, (40) where 𝑆 is a closed unit ball and 𝛿(𝐹(𝑥)+𝜖𝑆)=2𝜖. (41) According to Theorem 2 there exists a unique function 𝑔 such that Ψ∘𝑔∘𝑎=𝑔 and 𝑑(𝑓(𝑥),𝑔(𝑥))≤2𝜖 for 𝑥∈𝐾.

Corollary 5. Let 𝑓∶𝐾→𝑋, 𝑘∈ℕ, 𝛼₁,…,𝛼_𝑘∶𝐾→𝐾, 𝜆₁,…,𝜆_𝑘∶𝐾→[0,1], 𝜖>0, Ψ∶𝐾×𝑋^𝑘→𝑋 satisfy (19) and 𝑑Ψ𝑥,𝑓𝛼₁(𝑥),…,𝑓𝛼_𝑘(𝑥),𝑓(𝑥)≤𝜖𝑓𝑜𝑟𝑥∈𝐾. (42) Then there exists a unique function 𝑔 such that Ψ𝑥,𝑔𝛼₁(𝑥),…,𝑔𝛼_𝑘(𝑥)(𝑥),𝑑𝑓(𝑥)(𝑥)=𝑔,𝑔≤2𝜖. (43)

Proof. Let 𝐹(𝑥)={𝑓(𝑥)} for 𝑥∈𝐾. Then Ψ𝑥,𝐹𝛼₁(𝑥),…,𝐹𝛼_𝑘(𝑥)⊂𝐹(𝑥)+𝜖𝑆, (44) where 𝑆 is a closed unit ball and sup_{𝑙=1,…,∞}𝛿(𝐹(𝑥)+𝜖𝑆),max_{𝑖1=1,…,𝑘}𝜆_𝑖1(𝑥)max_{𝑖2=1,…,𝑘}𝜆_𝑖2∘𝛼_𝑖1(𝑥)⋯max_{𝑖𝑙=1,…,𝑘}𝜆_𝑖𝑙∘𝛼_𝑖𝑙−1∘⋯∘𝛼_𝑖1(𝑥)×𝛿𝐹𝛼_𝑖𝑙∘…∘𝛼_𝑖1(𝑥)+𝜖𝑆=2𝜖. (45) By Theorem 3, we get the assertion.

As it was observed in [9, 10] from the stability results concerning the equation Ψ∘𝑓∘𝑎=𝑓, we can easily derive the stability of functional equations in several variables, for example, the Cauchy equation, the Jensen equation, or the quadratic equation. The equation Ψ(𝑥,𝑓(𝛼₁(𝑥)),…,𝑓(𝛼_𝑘(𝑥)))=𝑓(𝑥) is a generalization of the gamma-type equations or the linear equations (see [11, 12]).