Generalized Hyers-Ulam-Rassias Theorem in Menger Probabilistic Normed Spaces

Abstract

We introduce two reasonable versions of approximately additive functions in a Šerstnev probabilistic normed space endowed with Π_𝑀 triangle function. More precisely, we show under some suitable conditions that an approximately additive function can be approximated by an additive mapping in above mentioned spaces.

1. Introduction and Preliminaries

Menger proposed transferring the probabilistic notions of quantum mechanic from physics to the underlying geometry. The theory of probabilistic normed spaces (briefly, PN spaces) is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The theory of probabilistic metric spaces introduced in 1942 by Menger [1], as well as by the authors in [2, 3]. The notion of a probabilistic normed space was introduced by Šerstnev [4]. Alsina, Schweizer and Skalar gave a general definition of probabilistic normed space based on the definition of Menger for probabilistic metric spaces in [5, 6].

It can be defined, in some way, the class of approximate solutions of the given functional equation one can ask whether each mapping from this class can be somehow approximated by an exact solution of the considered equation. Such a problem was formulated by Ulam in 1940 (cf., [7]) and solved the next year for the Cauchy functional equation by Hyers [8]. In 1950, Aoki [9] and in 1978, Rassias [10] proved a generalization of Hyers' theorem for additive and linear mappings, respectively.

Theorem 1.1. Let 𝑓 be an approximately additive mapping from a normed vector space 𝐸 into a Banach space 𝐹, that is, 𝑓 satisfies the inequality ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜀(‖𝑥‖^𝑝+‖𝑦‖^𝑝) (1.1) for all 𝑥,𝑦∈𝐸, where 𝜀 and 𝑝 are constants with 𝜀>0 and 0≤𝑝<1. Then the mapping 𝐿∶𝐸→𝐹 defined by 𝐿(𝑥)=lim_𝑛→∞2^−𝑛𝑓(2^𝑛𝑥) is the unique additive mapping which satisfies ‖𝑓(𝑥)−𝐿(𝑥)‖≤2𝜀2−2^𝑝‖𝑥‖^𝑝 (1.2) for all 𝑥∈𝐸.

The result of Rassias has influenced the development of what is now called the Hyers-Ulam-Rassias stability theory for functional equations. In 1994, a generalization of Rassias' theorem was obtained by Găvruţa [11] by replacing the bound 𝜀(‖𝑥‖^𝑝+‖𝑦‖^𝑝) by a general control function 𝜑(𝑥,𝑦). Several stability results have been recently obtained for various equations, also for mapping with more general domains and ranges (see [12–18]).

PN spaces were first defined by Šerstnev in 1962 (see [4]). Their definition was generalized in [5]. We recall and apply the definition of probabilistic space briefly as given in [2], together with the notation that will be needed (see [2]). A distance distribution function (briefly, a d.d.f.) is a nondecreasing function 𝐹 from ℝ⁺ into [0,1] that satisfies 𝐹(0)=0 and 𝐹(+∞)=1, and is left-continuous on (0,+∞); here as usual, ℝ⁺∶=[0,+∞]. The space of d.d.f.'s will be denoted by Δ⁺; and the set of all 𝐹 in Δ⁺ for which 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,if𝑡≤𝑎,1,if𝑡>𝑎. (1.3) The space Δ⁺ can be metrized in several ways [2], but we will here adopt the Sibley metric 𝑑_𝑆. If 𝐹,𝐺 are d.f.'s and ℎ is in ]0,1[, let (𝐹,𝐺;ℎ) denote the condition:

1𝐺(𝑥)≤𝐹(𝑥+ℎ)+ℎ∀𝑥∈]0,ℎ[. (1.4) Then the Sibley metric 𝑑_𝑆 is defined by

𝑑_𝑆(𝐹,𝐺)∶=inf{ℎ∈]0,1[∶both(𝐹,𝐺;ℎ)and(𝐺,𝐹;ℎ)hold}. (1.5) In particular, under the usual pointwise ordering of functions, 𝜀₀ is the maximal element of Δ⁺. A triangle function is a binary operation on Δ⁺, namely a function 𝜏∶Δ⁺×Δ⁺→Δ⁺ that is associative, commutative, nondecreasing in each place and has 𝜀₀ as 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.

Moreover, a triangle function is continuous if it is continuous in the metric space (Δ⁺,𝑑_𝑆).

Typical continuous triangle functions are Π_𝑇(𝐹,𝐺)(𝑥)=sup_{𝑠+𝑡=𝑥}𝑇(𝐹(𝑠),𝐺(𝑡)), and Π_𝑇^∗(𝐹,𝐺)=inf_{𝑠+𝑡=𝑥}𝑇^∗(𝐹(𝑠),𝐺(𝑡)). Here 𝑇 is a continuous 𝑡-norm, that is, a continuous binary operation on [0,1] that is commutative, associative, nondecreasing in each variable and has 1 as identity; 𝑇^∗ is a continuous 𝑡-conorm, namely a continuous binary operation on [0,1] which is related to the continuous 𝑡-norm 𝑇 through 𝑇^∗(𝑥,𝑦)=1−𝑇(1−𝑥,1−𝑦). For example 𝑇(𝑥,𝑦)=min(𝑥,𝑦)=𝑀(𝑥,𝑦) and 𝑇^∗(𝑥,𝑦)=max(𝑥,𝑦) or 𝑇(𝑥,𝑦)=𝜋(𝑥,𝑦)=𝑥𝑦 and 𝑇^∗(𝑥,𝑦)=𝜋^∗(𝑥,𝑦)=𝑥+𝑦−𝑥𝑦.

Definition 1.2. A Probabilistic Normed space (briefly, PN space) is a quadruple (𝑋,𝜈,𝜏,𝜏^∗), where 𝑋 is a real vector space, 𝜏 and 𝜏^∗ are continuous triangle functions with 𝜏≤𝜏^∗ and 𝜈 is a mapping (the probabilistic norm) from 𝑉 into Δ⁺, such that for every choice of 𝑝 and 𝑞 in 𝑉 the following hold:(N1)𝜈_𝑝=𝜀₀ if and only if 𝑝=𝜃 (𝜃 is the null vector in 𝑋);(N2)𝜈_−𝑝=𝜈_𝑝;(N3)𝜈_𝑝+𝑞≥𝜏(𝜈_𝑝,𝜈_𝑞);(N4)𝜈_𝑝≤𝜏^∗(𝜈_𝜆𝑝,𝜈_{(1−𝜆)𝑝}) for every 𝜆∈[0,1].

A PN space is called a Šerstnev space if it satisfies (N1), (N3) and the following condition:

𝜈_𝛼𝑝(𝑥)=𝜈_𝑝(𝑥|𝛼|), (1.6) holds for every 𝛼≠0∈ℝ and 𝑥>0. When here is a continuous 𝑡-norm 𝑇 such that 𝜏=Π_𝑇 and 𝜏^∗=Π_𝑇^∗, the PN space (𝑋,𝜈,𝜏,𝜏^∗) is called Menger PN space (briefly, MPN space), and is denoted by (𝑋,𝜈,𝜏).

Let (𝑋,𝜈,𝜏) be an MPN space let {𝑥_𝑛} be 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 (1.7) for all 𝑡>0. In this case 𝑥 is called the limit of {𝑥_𝑛}.

The sequence 𝑥_𝑛 in MPN space (𝑋,𝜈,𝜏) is called Cauchy if for each 𝜀>0 and 𝛿>0, there exists some 𝑛₀ such that 𝜈(𝑥_𝑛−𝑥_𝑚)(𝛿)>1−𝜀 for all 𝑚,𝑛≥𝑛₀.

Clearly, every convergent sequence in a MPN space is Cauchy. If each Cauchy sequence is convergent in a MPN space (𝑋,𝜈,𝜏), then (𝑋,𝜈,𝜏) is called Menger probabilistic Banach space (briefly, MPB space).

Recently, the stability of functional equations on PN spaces and MPN spaces have been investigated by some authors; see [19–23] and references therein. In this paper, we investigate the stability of additive functional equations on Šerstnev probabilistic normed space endowed with Π_𝑀 triangle function.

2. Main Results

We begin our work with uniform version of the Hyers-Ulam-Rassias stability in a Šerstnev PN space in which we uniformly approximate a uniform approximate additive mapping.

Theorem 2.1. Let 𝑋 be a linear space and (𝑌,𝜈,Π_𝑀) be a Šerstnev PB space. Let 𝜑∶𝑋×𝑋→[0,∞) be a control function such that 𝜑_𝑛(𝑥,𝑦)={2^−𝑛−1𝜑(2^𝑛𝑥,2^𝑛𝑦)}(𝑥,𝑦∈𝑋) (2.1) converges to zero. Let 𝑓∶𝑋→𝑌 be a uniformly approximately additive function with respect to 𝜑 in the sense that lim_𝑡→∞𝜈(𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦))(𝑡𝜑(𝑥,𝑦))=1 (2.2) uniformly on 𝑋×𝑋. Then 𝑇(𝑥)∶=lim_𝑛→∞2^−𝑛𝑓(2^𝑛𝑥) for any 𝑥∈𝑋 exists and defines an additive mapping 𝑇∶𝑋→𝑌 such that if for some 𝛿>0,𝛼>0𝜈(𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦))(𝛿𝜑(𝑥,𝑦))>𝛼(𝑥,𝑦∈𝑋) (2.3) then 𝜈(𝑇(𝑥)−𝑓(𝑥))(𝛿𝜑_𝑛(𝑥,𝑥))>𝛼(𝑥∈𝑋). (2.4)

Proof. Given 𝜀>0, by (2.2), we can choose some 𝑡₀ such that 𝜈(𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦))(𝑡𝜑(𝑥,𝑦))≥1−𝜀 (2.5) for all 𝑥,𝑦∈𝑋 and all 𝑡≥𝑡₀. Putting 𝑦=𝑥 in (2.5) we get 𝜈(𝑓(2𝑥)−2𝑓(𝑥))(𝑡𝜑(𝑥,𝑥))≥1−𝜀 (2.6) and by replacing 𝑥 by 2^𝑛𝑥, we obtain 𝜈(2^−𝑛−1𝑓(2^𝑛+1𝑥)−2^−𝑛𝑓(2^𝑛𝑥))(𝑡2^−𝑛−1𝜑(2^𝑛𝑥,2^𝑛𝑥))≥1−𝜀. (2.7) By passing to a nonincreasing subsequence, if necessary, we may assume that {2^−𝑛−1𝜑(2^𝑛𝑥,2^𝑛𝑦)} is nonincreasing.
Thus for each 𝑛>𝑚 we have 𝜈(2^−𝑚𝑓(2^𝑚𝑥)−2^−𝑛𝑓(2^𝑛𝑥))(𝑡2^−𝑚−1𝜑(2^𝑚𝑥,2^𝑚𝑥))=𝜈(^𝑛−1_𝑘=𝑚(2^−𝑘𝑓(2^𝑘𝑥)−2^−𝑘−1𝑓(2^𝑘+1𝑥)))(𝑡2^−𝑚−1𝜑(2^𝑚𝑥,2^𝑚𝑥))≥Π_𝑀{𝜈(2^−𝑚𝑓(2^𝑚𝑥)−2^−𝑚−1𝑓(2^𝑚+1𝑥)),𝜈(^𝑛−1_𝑘=𝑚+1(2^−𝑘𝑓(2^𝑘𝑥)−2^−𝑘−1𝑓(2^𝑘+1𝑥)))}(𝑡2^−𝑚−1𝜑(2^𝑚𝑥,2^𝑚𝑥))≥Π_𝑀{1−𝜀,Π_𝑀{𝜈(2^−𝑚−1𝑓(2^𝑚+1𝑥)−2^−𝑚−2𝑓(2^𝑚+2𝑥)),𝜈(^𝑛−1_𝑘=𝑚+2(2^−𝑘𝑓(2^𝑘𝑥)−2^−𝑘−1𝑓(2^𝑘+1𝑥)))}(𝑡2^−𝑚−2𝜑(2^𝑚+1𝑥,2^𝑚+1𝑥))}≥1−𝜀. (2.8) The convergence of (2.1) implies that for given 𝛿>0 there is 𝑛₀∈ℕ such that 𝑡₀2^−𝑛−1𝜑(2^𝑛𝑥,2^𝑛𝑥)<𝛿∀𝑛≥𝑛₀. (2.9) Thus by (2.8) we deduce that 𝜈(2^−𝑚𝑓(2^𝑚𝑥)−2^−𝑛𝑓(2^𝑛𝑥))(𝛿)≥𝜈(2^−𝑚𝑓(2^𝑚𝑥)−2^−𝑛𝑓(2^𝑛𝑥))(𝑡₀2^−𝑚−1𝜑(2^𝑚𝑥,2^𝑚𝑥))≥1−𝜀 (2.10) for each 𝑛≥𝑛₀. Hence 2^−𝑛𝑓(2^𝑛𝑥) is a Cauchy sequence in 𝑌. Since (𝑌,𝜈,Π_𝑀) is complete, this sequence converges to some 𝑇(𝑥)∈𝑌. Therefore, we can define a mapping 𝑇∶𝑋→𝑌 by 𝑇(𝑥)∶=lim_𝑛→∞2^−𝑛𝑓(2^𝑛𝑥), namely, for each 𝑡>0, and 𝑥∈𝑋, 𝜈(𝑇(𝑥)−2^−𝑛𝑓(2^𝑛𝑥))(𝑡)=1. (2.11) Next, let 𝑥,𝑦∈𝑋. Temporarily fix 𝑡>0 and 0<𝜀<1. Since 2^−𝑛−1𝜑(2^𝑛𝑥,2^𝑛𝑦) converges to zero, there is some 𝑛₁>𝑛₀ such that 𝑡₀𝜑(2^𝑛𝑥,2^𝑛𝑦)<𝑡2^𝑛+1 for all 𝑛≥𝑛₁. Hence for each 𝑛≥𝑛₁, we have 𝜈(𝑇(𝑥+𝑦)−𝑇(𝑥)−𝑇(𝑦))(𝑡)≥Π_𝑀{Π_𝑀{𝜈(𝑇(𝑥+𝑦)−2^−𝑛−1𝑓(2^𝑛+1(𝑥+𝑦)))(𝑡),𝜈(𝑇(𝑥)−2^−𝑛−1𝑓(2^𝑛+1Π𝑥))(𝑡)},_𝑀{𝜈(𝑇(𝑦)−2^−𝑛−1𝑓(2^𝑛+1𝑦))(𝑡),𝜈(𝑓(2^𝑛+1(𝑥+𝑦))−𝑓(2^𝑛+1𝑥)−𝑓(2^𝑛+1𝑦))(2^𝑛+1𝑡)}} (2.12) but we have lim_𝑛→∞𝜈(𝑇(𝑥+𝑦)−2^−𝑛−1𝑓(2^𝑛+1))(𝑡)=1,lim_𝑛→∞𝜈(𝑇(𝑥)−2^−𝑛−1𝑓(2^𝑛+1𝑥))(𝑡)=1,lim_𝑛→∞𝜈(𝑇(𝑦)−2^−𝑛−1𝑓(2^𝑛+1𝑦))(𝑡)=1, (2.13) and by (2.5) and for large enough 𝑛, we have 𝜈(𝑓(2^𝑛+1(𝑥+𝑦))−𝑓(2^𝑛+1𝑥)−𝑓(2^𝑛+1𝑦))(2^𝑛+1𝑡)≥𝜈(𝑓(2^𝑛+1(𝑥+𝑦))−𝑓(2^𝑛+1𝑥)−𝑓(2^𝑛+1𝑦))(𝑡₀𝜑(2^𝑛𝑥,2^𝑛𝑦))≥1−𝜀. (2.14) Thus 𝜈(𝑇(𝑥+𝑦)−𝑇(𝑥)−𝑇(𝑦))(𝑡)≥1−𝜀∀𝑡>0,0<𝜀<1. (2.15) It follows that 𝜈(𝑇(𝑥+𝑦)−𝑇(𝑥)−𝑇(𝑦))(𝑡)=1 for all 𝑡>0 and by (N1), we have 𝑇(𝑥+𝑦)=𝑇(𝑥)+𝑇(𝑦).
To end the proof, let, for some positive 𝛿 and 𝛼, (2.3) hold. Let 𝑥∈𝑋. Putting 𝑚=0 and 𝛼=1−𝜀 in (2.10), we get 𝜈(𝑓(2^𝑛𝑥)−2^𝑛𝑓(𝑥))(𝛿)≥𝛼 (2.16) for all positive integers 𝑛≥𝑛₀. Thus for large enough 𝑛, we have 𝜈(𝑓(𝑥)−𝑇(𝑥))(𝛿2^−𝑛−1𝜑(2^𝑛𝑥,2^𝑛𝑥))≥Π_𝑀{𝜈(𝑓(𝑥)−2^−𝑛𝑓(2^𝑛𝑥)),𝜈(2^−𝑛𝑓(2^𝑛𝑥)−𝑇(𝑥))}(𝛿2^−𝑛−1𝜑(2^𝑛𝑥,2^𝑛𝑥))≥𝛼 (2.17) therefore 𝜈(𝑇(𝑥)−𝑓(𝑥))(𝛿𝜑_𝑛(𝑥,𝑥))>𝛼. (2.18)

Corollary 2.2. Let 𝑋 be a linear space and (𝑌,𝜈,Π_𝑀) be a Šerstnev PB space. Let 𝜑∶𝑋×𝑋→[0,∞) be a control function satisfying (2.2). Let 𝑓∶𝑋→𝑌 be a uniformly approximately additive function with respect to 𝜑. Then there is a unique additive mapping 𝑇∶𝑋→𝑌 such that lim_𝑛→∞𝜈(𝑓(𝑥)−𝑇(𝑥))(𝑡𝜑_𝑛(𝑥,𝑥))=1 (2.19) uniformly on 𝑋.

Proof. The existence of uniform limit (2.19) immediately follows from Theorem 2.1. It remains to prove the uniqueness assertion.
Let 𝑆 be another additive mapping satisfying (2.19). Fix 𝑐>0. Given 𝜀>0, by (2.19) for 𝑇 and 𝑆, we can find some 𝑡₀>0 such that 𝜈(𝑓(𝑥)−𝑇(𝑥))(𝑡𝜑_𝑛𝜑(𝑥,𝑥))≥1−𝜀,𝜈(𝑓(𝑥)−𝑆(𝑥))(𝑡_𝑛(𝑥,𝑥))≥1−𝜀 (2.20) for all 𝑥∈𝑋 and 𝑡≥𝑡₀. Fix some 𝑥∈𝑋 and find some integer 𝑛₀ such that 𝑡₀2^−𝑛𝜑(2^𝑛+1𝑥,2^𝑛+1)<𝑐∀𝑛≥𝑛₀. (2.21) Then we have 𝜈(𝑆(𝑥)−𝑇(𝑥))(𝑐)≥Π_𝑀{𝜈(2^−𝑛𝑓(2^𝑛𝑥)−𝑇(𝑥)),𝜈(𝑆(𝑥)−2^−𝑛𝑓(2^𝑛𝑥))}(𝑐)=Π_𝑀{𝜈(𝑓(2^𝑛𝑥)−𝑇(2^𝑛𝑥)),𝜈(𝑆(2^𝑛𝑥)−𝑓(2^𝑛𝑥))}(2^𝑛𝑐)≥Π_𝑀{𝜈(𝑓(2^𝑛𝑥)−𝑇(2^𝑛𝑥)),𝜈(𝑆(2^𝑛𝑥)−𝑓(2^𝑛𝑥))}(𝑡₀𝜑(2^𝑛+1𝑥,2^𝑛+1𝑥))≥1−𝜀. (2.22) It follows that 𝜈(𝑆(𝑥)−𝑇(𝑥))(𝑐)=1 for all 𝑐>0. Thus 𝑇(𝑥)=𝑆(𝑥) for all 𝑥∈𝑋.

Considering the control function 𝜑(𝑥,𝑦)=𝜃(‖𝑥‖^𝑞+‖𝑦‖^𝑞) for some 𝜃>0, we obtain the following.

Corollary 2.3. Let 𝑋 be a normed linear space. Let (𝑌,𝜈,Π_𝑀) be a Šerstnev PB space. Let 𝜃≥0 and 0≤𝑞<1. Suppose that 𝑓∶𝑋→𝑌 is a function such that lim_𝑡→∞𝜈(𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦))(𝑡𝜃(‖𝑥‖^𝑞+‖𝑦‖^𝑞))=1 (2.23) uniformly on 𝑋×𝑋. Then there is a unique additive mapping 𝑇∶𝑋→𝑌 such that lim_𝑡→∞𝜈(𝑇(𝑥)−𝑓(𝑥))(𝑡2^{𝑛(𝑞−1)}𝜃‖𝑥‖^𝑞)=1 (2.24) uniformly on 𝑋.

We are ready to give our nonuniform version of the Hyers-Ulam-Rassias theorem in Šerstnev PN spaces.

Theorem 2.4. Let 𝑋 be a linear space. Let (𝑍,𝜔,Π_𝑀) be a Šerstnev MPN space. Let 𝜓∶𝑋×𝑋→𝑍 be a function such that for some 0<𝛼<2, 𝜔(𝜓(2𝑥,2𝑦))(𝑡)≥𝜔(𝛼𝜓(𝑥,𝑦))(𝑡) (2.25) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Let (𝑌,𝜈,Π_𝑀) be a Šerstnev PB space and let 𝑓∶𝑋→𝑌 be a 𝜓-approximately additive mapping in the sense that 𝜈(𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦))(𝑡)≥𝜔(𝜓(𝑥,𝑦))(𝑡) (2.26) for each 𝑡>0 and 𝑥,𝑦∈𝑋. Then there exists unique additive mapping 𝑇∶𝑋→𝑌 such that 1𝜈(𝑓(𝑥)−𝑇(𝑥))(𝑡)≥𝜔(2𝜓(𝑥,𝑥)(𝑡)) (2.27) where 𝑥∈𝑋 and 𝑡>0.

Proof. Put 𝑦=𝑥 in (2.26) to obtain 𝜈(𝑓(2𝑥)−2𝑓(𝑥))(𝑡)≥𝜔(𝜓(𝑥,𝑥))(𝑡)(𝑥∈𝑋,𝑡>0). (2.28) Using (2.25) and induction on 𝑛, one can verify that 𝜔(𝜓(2^𝑛𝑥,2^𝑛)𝑥)(𝑡)≥𝜔(𝛼^𝑛𝜓(𝑥,𝑥))(𝑡) (2.29) for all 𝑥∈𝑋 and 𝑡>0. Replacing 𝑥 by 2^𝑛−1𝑥 in (2.28) and using (2.29) we get 𝜈(𝑓(2^𝑛𝑥)−2𝑓(2^𝑛−1𝑥))(𝑡)≥𝜔(𝛼^𝑛−1𝜓(𝑥,𝑥))(𝑡) (2.30) for all 𝑥∈𝑋 and 𝑡>0. It follows from (2.30) that 𝜈(2^−𝑛𝑓(2^𝑛𝑥)−2^−𝑛+1𝑓(2^𝑛−1𝑥))(2^−𝑛1𝑡)≥𝜔((𝛼)𝜓(𝑥,𝑥))(𝛼^−𝑛𝑡) (2.31) whence 𝜈(2^−𝑛𝑓(2^𝑛𝑥)−2^−𝑛+1𝑓(2^𝑛−1𝛼𝑥))((^𝑛2^𝑛1)𝑡)≥𝜔((𝛼)𝜓(𝑥,𝑥))(𝑡) (2.32) for all 𝑛>𝑚≥0, 𝑥∈𝑋 and 𝑡>0. So 𝜈(2^−𝑛𝑓(2^𝑛𝑥)−2^−𝑚𝑓(2^𝑚𝛼𝑥))((^𝑚+12^𝑚+1)𝑡)=𝜈(^𝑛_𝑘=𝑚+12^−𝑘𝑓(2^𝑘𝑥)−2^−𝑘+1𝑓(2^𝑘−1𝛼𝑥))((^𝑚+12^𝑚+11)𝑡)≥𝜔((𝛼)𝜓(𝑥,𝑥))(𝑡) (2.33) whence 𝜈(2^−𝑛𝑓(2^𝑛𝑥)−2^−𝑚𝑓(2^𝑚1𝑥))(𝑡)≥𝜔(𝛼𝛼𝜓(𝑥,𝑥))((^𝑚+12^𝑚+1)𝑡) (2.34) for all 𝑛>𝑚≥0, 𝑥∈𝑋 and 𝑡>0. Fix 𝑥∈𝑋. By lim_𝑠→∞1𝜔(𝛼𝜓(𝑥,𝑥))(𝑠)=1 (2.35) we deduce that {2^−𝑛𝑓(2^𝑛𝑥)} is a Cauchy sequence in (𝑌,𝜈,Π_𝑀). Since (𝑌,𝜈,Π_𝑀) is complete, this sequence converges to some point 𝑇(𝑥)∈𝑌. It follows from (2.26) that 𝜈(𝑓(2^𝑛(𝑥+𝑦))−𝑓(2^𝑛𝑥)−𝑓(2^𝑛𝑦))(𝑡)≥𝜔(𝜓(2^𝑛𝑥,2^𝑛𝑦))(𝑡)≥𝜔(𝛼^𝑛𝜓(𝑥,𝑦))(𝑡)≥𝜔(𝜓(𝑥,𝑦))(𝛼^−𝑛𝑡) (2.36) whence 𝜈(2^−𝑛𝑓(2^𝑛(𝑥+𝑦))−2^−𝑛𝑓(2^𝑛𝑥)−2^−𝑛𝑓(2^𝑛2𝑦))(𝑡)≥𝜔(𝜓(𝑥,𝑦))((𝛼)^𝑛𝑡). (2.37) We have 𝜈(𝑇(𝑥+𝑦)−𝑇(𝑥)−𝑇(𝑦))(𝑡)≥Π_𝑀{Π_𝑀{𝜈(𝑇(𝑥+𝑦)−2^−𝑛𝑓(2^𝑛(𝑥+𝑦))),𝜈(𝑇(𝑥)−2^−𝑛𝑓(2^𝑛Π𝑥))}(𝑡),_𝑀{𝜈(𝑇(𝑦)−2^−𝑛𝑓(2^𝑛𝑦)),𝜈(2^−𝑛𝑓(2^𝑛(𝑥+𝑦))−2^−𝑛𝑓(2^𝑛𝑥)−2^−𝑛𝑓(2^𝑛𝑦))}(𝑡)}. (2.38) By (2.37) and the fact that lim_𝑛→∞𝜈(𝑇(𝑧)−2^−𝑛𝑓(2^𝑛𝑧))=1 (2.39) for all 𝑧∈𝑋 and 𝑟>0, each term on the right-hand side tends to 1 as 𝑛→∞. Hence 𝜈(𝑇(𝑥+𝑦)−𝑇(𝑥)−𝑇(𝑦))(𝑡)=1. (2.40) By (N1), it means that 𝑇(𝑥+𝑦)=𝑇(𝑥)+𝑇(𝑦). (2.41) Furthermore, let 𝑥∈𝑋 and 𝑡>0. Using (2.34) with 𝑚=0 we obtain 𝜈(𝑇(𝑥)−𝑓(𝑥))(𝑡)≥Π_𝑀{𝜈(𝑇(𝑥)−2^−𝑛𝑓(2^𝑛𝑥)),𝜈(2^−𝑛𝑓(2^𝑛𝑥)−𝑓(𝑥))}(𝑡)≥Π_𝑀{𝜈(𝑇(𝑥)−2^−𝑛𝑓(2^𝑛1𝑥)),𝜔((2)𝜓(𝑥,𝑥))}(𝑡). (2.42) Hence 𝜈(𝑇(𝑥)−𝑓(𝑥))(𝑡)≥Π_𝑀{lim_𝑛→∞𝜈(𝑇(𝑥)−2^−𝑛𝑓(2^𝑛1𝑥)),𝜔((21)𝜓(𝑥,𝑥))}(𝑡)=𝜔((2)𝜓(𝑥,𝑥))(𝑡). (2.43) The uniqueness of 𝑇 can be proved in a similar fashion as in the proof of Corollary 2.2.