Hyers-Ulam-Rassias RNS Approximation of Euler-Lagrange-Type Additive Mappings

Abstract

Recently the generalized Hyers-Ulam (or Hyers-Ulam-Rassias) stability of the following functional equation ∑^𝑚_𝑗=1𝑓(−𝑟_𝑗𝑥_𝑗+∑_{1≤𝑖≤𝑚,𝑖≠𝑗}𝑟_𝑖𝑥_𝑖)+2∑^𝑚_𝑖=1𝑟_𝑖𝑓(𝑥_𝑖)=𝑚𝑓(∑^𝑚_𝑖=1𝑟_𝑖𝑥_𝑖) where 𝑟₁,…,𝑟_𝑚∈ℝ, proved in Banach modules over a unital 𝐶^∗-algebra. It was shown that if ∑^𝑚_𝑖=1𝑟_𝑖≠0, 𝑟_𝑖,𝑟_𝑗≠0 for some 1≤𝑖<𝑗≤𝑚 and a mapping 𝑓∶𝑋→𝑌 satisfies the above mentioned functional equation then the mapping 𝑓∶𝑋→𝑌 is Cauchy additive. In this paper we prove the Hyers-Ulam-Rassias stability of the above mentioned functional equation in random normed spaces (briefly RNS).

1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference.

The paper of Rassias has provided a lot of influence in the development of what we call the generalized Hyers-Ulam stability of functional equations. In 1994, a generalization of Rassias’ theorem was obtained by Găvruţa [5] by replacing the bound 𝜖(‖𝑥‖^𝑝+‖𝑦‖^𝑝) by a general control function 𝜑(𝑥,𝑦).

The functional equation:𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+2𝑓𝑦, (1.1) is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [6] for mappings 𝑓∶𝑋→𝑌, where 𝑋 is a normed space and 𝑌 is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain 𝑋 is replaced by an Abelian group. Czerwik [8] proved the generalized Hyers-Ulam stability of the quadratic functional equation.

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [2, 4, 5, 9–28]).

In the sequel, we will adopt the usual terminology, notions, and conventions of the theory of random normed spaces as in [29]. Throughout this paper, the spaces of all probability distribution functions are denoted by Δ⁺. Elements of Δ⁺ are functions 𝐹∶ℝ∪{−∞,+∞}→[0,1], such that 𝐹 is left continuous and nondecreasing on ℝ, 𝐹(0)=0 and 𝐹(+∞)=1. It’s clear that the subset 𝐷⁺={𝐹∈Δ⁺∶𝑙⁻𝐹(+∞)=1}, where 𝑙⁻𝑓(𝑥)=lim_{𝑡→𝑥−}𝑓(𝑡), is a subset of Δ⁺. The space Δ⁺ is partially ordered by the usual point-wise ordering of functions, that is, for all 𝑡∈𝑅, 𝐹≤𝐺 if and only if 𝐹(𝑡)≤𝐺(𝑡). For every 𝑎≥0, 𝐻_𝑎(𝑡) is the element of 𝐷⁺ defined by𝐻_𝑎(𝑡)=0if𝑡≤𝑎1if𝑡>𝑎. (1.2) One can easily show that the maximal element for Δ⁺ in this order is the distribution function 𝐻₀(𝑡).

Definition 1.1. A function 𝑇∶[0,1]²→[0,1] is a continuous triangular norm (briefly a 𝑡-norm) if 𝑇 satisfies the following conditions:(i)𝑇 is commutative and associative;(ii)𝑇 is continuous;(iii)𝑇(𝑥,1)=𝑥 for all 𝑥∈[0,1];(iv)𝑇(𝑥,𝑦)≤𝑇(𝑧,𝑤) whenever 𝑥≤𝑧 and 𝑦≤𝑤 for all 𝑥,𝑦,𝑧,𝑤∈[0,1].

Three typical examples of continuous 𝑡-norms are 𝑇_𝑃(𝑥,𝑦)=𝑥𝑦,𝑇_max(𝑥,𝑦)=max{𝑎+𝑏−1,0}, and 𝑇_𝑀(𝑥,𝑦)=min(𝑎,𝑏). Recall that, if 𝑇 is a 𝑡-norm and {𝑥_𝑛} is a given of numbers in [0,1], 𝑇^𝑛_𝑖=1𝑥_𝑖 is defined recursively by 𝑇¹_𝑖=1𝑥₁ and 𝑇^𝑛_𝑖=1𝑥_𝑖=𝑇(𝑇^𝑛−1_𝑖=1𝑥_𝑖,𝑥_𝑛) for 𝑛≥2.

Definition 1.2. A random normed space (briefly 𝑅𝑁𝑆) is a triple (𝑋,𝜇^′,𝑇), where 𝑋 is a vector space, 𝑇 is a continuous 𝑡-norm, and 𝜇′∶𝑋→𝐷⁺ is a mapping such that the following conditions hold.(i)𝜇^′_𝑥(𝑡)=𝐻₀(𝑡) for all 𝑡>0 if and only if 𝑥=0.(ii)𝜇^′_𝛼𝑥(𝑡)=𝜇^′_𝑥(𝑡/|𝛼|) for all 𝛼∈ℝ, 𝛼≠0, 𝑥∈𝑋 and 𝑡≥0.(iii)𝜇^′_𝑥+𝑦(𝑡+𝑠)≥𝑇(𝜇^′_𝑥(𝑡),𝜇^′_𝑦(𝑠)), for all 𝑥,𝑦∈𝑋 and 𝑡,𝑠≥0.

Definition 1.3. Let (𝑋,𝜇^′,𝑇) be an RNS.
(i)A sequence {𝑥_𝑛} in 𝑋 is said to be convergent to 𝑥∈𝑋 in 𝑋 if for all 𝑡>0, lim_𝑛→∞𝜇^′_{𝑥𝑛−𝑥}(𝑡)=1(ii)A sequence {𝑥_𝑛} in 𝑋 is said to be Cauchy sequence in 𝑋 if for all 𝑡>0,lim_𝑛→∞𝜇^′_{𝑥𝑛−𝑥𝑚}(𝑡)=1.(iii)The 𝑅𝑁-space (𝑋,𝜇^′,𝑇) is said to be complete if every Cauchy sequence in 𝑋 is convergent.

Theorem 1.4. If (𝑋,𝜇^′,𝑇) is RNS and {𝑥_𝑛} is a sequence such that 𝑥_𝑛→𝑥, then lim_𝑛→∞𝜇^′_𝑥𝑛(𝑡)=𝜇^′_𝑥(𝑡).

In this paper, we investigate the generalized Hyers-Ulam stability of the following additive functional equation of Euler-Lagrange type:^𝑚_𝑗=1𝑓−𝑟_𝑗𝑥_𝑗+_{1≤𝑖≤𝑚,𝑖≠𝑗}𝑟_𝑖𝑥_𝑖+2^𝑚_𝑖=1𝑟_𝑖𝑓𝑥_𝑖=𝑚𝑓^𝑚_𝑖=1𝑟_𝑖𝑥_𝑖, (1.3) where 𝑟₁,…,𝑟_𝑛∈ℝ, ∑^𝑚_𝑖=1𝑟_𝑖≠0, and 𝑟_𝑖,𝑟_𝑗≠0 for some 1≤𝑖<𝑗≤𝑚, in random normed spaces.

Every solution of the functional equation (1.3) is said to be a generalized Euler-Lagrange type additive mapping.

2. RNS Approximation of Functional Equation (1.3)

Remark 2.1. Throughout this paper, 𝑟₁,…,𝑟_𝑚 will be real numbers such that 𝑟_𝑖,𝑟_𝑗≠0 for fixed 1≤𝑖<𝑗≤𝑚.

Theorem 2.2. Let 𝑋 be a real linear space, (𝑍,𝜇′,min) be an RN space, 𝜑∶𝑋^𝑛→𝑍 be a function such that for some 0<𝛼<2, 𝜇^′_{𝜑2𝑥1,…,2𝑥𝑚}(𝑡)≥𝜇^′_{𝑥𝛼𝜑1,…,𝑥𝑚}(𝑡)∀𝑥_𝑖∈𝑋,𝑡>0. (2.1)𝑓(0)=0 and for all 𝑥_𝑖∈𝑋 and 𝑡>0lim_𝑛→∞𝜇^′_{𝜑2𝑛𝑥1,…,2𝑛𝑥𝑚/2𝑛}(𝑡)=1. (2.2)
Let (𝑌,𝜇,min) be a complete RN space. If 𝑓∶𝑥→𝑌 is a mapping such that for all 𝑥_𝑖,𝑥_𝑗∈𝑋 and 𝑡>0𝜇_∑^𝑚_{𝑗=1𝑓(−𝑟𝑗𝑥𝑗+∑1≤𝑖≤𝑚,𝑖≠𝑗𝑟𝑖𝑥𝑖)+2∑}^𝑚_{𝑖=1𝑟𝑖𝑓(𝑥𝑖)−𝑚𝑓(∑}^𝑚_{𝑖=1𝑟𝑖𝑥𝑖)}(𝑡)≥𝜇^′_{𝜑(𝑥1,…,𝑥𝑚)}(𝑡) (2.3) then there is a unique generalized Euler-Lagrange-type additive mapping EL∶𝑋→𝑌 such that, for all 𝑥∈𝑋 and all 𝑡>0𝜇_{EL(𝑥)−𝑓(𝑥)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}(2−𝛼)𝑡6,𝜇′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}(2−𝛼)𝑡6,𝜇′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}(2−𝛼)𝑡6,𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}(2−𝛼)𝑡3,𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}(2−𝛼)𝑡3,𝜇′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}(2−𝛼)𝑡3. (2.4)

Proof. For each 1≤𝑘≤𝑚 with 𝑘≠𝑖,𝑗, let 𝑥_𝑘=0 in (2.3). Then we get the following inequality: 𝜇_{𝜆(𝑥𝑖,𝑥𝑗)}(𝑡)≥𝜇^′_{𝜑𝑖,𝑗(𝑥𝑖,𝑥𝑗)}(𝑡), (2.5) for all 𝑥_𝑖,𝑥_𝑗∈𝑋, where 𝜑_𝑖,𝑗𝑥,𝑦∶=𝜑0,…,0,𝑥_}𝑖th,0,…,0,𝑦^}_𝑗th,0,…,0, (2.6) for all 𝑥,𝑦∈𝑋 and all 1≤𝑖<𝑗≤𝑚, and 𝜆𝑥_𝑖,𝑥_𝑗=𝑓−𝑟_𝑖𝑥_𝑖+𝑟_𝑗𝑥_𝑗+𝑓𝑟_𝑖𝑥_𝑖−𝑟_𝑗𝑥_𝑗−2𝑓𝑟_𝑖𝑥_𝑖+𝑟_𝑗𝑥_𝑗+2𝑟_𝑖𝑓𝑥_𝑖+2𝑟_𝑗𝑓𝑥_𝑗. (2.7)
Letting 𝑥_𝑖=0 in (2.5), we get 𝜇_{𝑓(−𝑟𝑗𝑥𝑗)−𝑓(𝑟𝑗𝑥𝑗)+2𝑟𝑗𝑓(𝑥𝑗)}(𝑡)≥𝜇^′_{𝜑𝑖,𝑗(0,𝑥𝑗)}(𝑡), (2.8) for all 𝑥_𝑗∈𝑋. Similarly, letting 𝑥_𝑗=0 in (2.5), we get 𝜇_{𝑓(−𝑟𝑖𝑥𝑖)−𝑓(𝑟𝑖𝑥𝑖)+2𝑟𝑖𝑓(𝑥𝑖)}(𝑡)≥𝜇^′_{𝜑𝑖,𝑗(𝑥𝑖,0)}(𝑡), (2.9) for all 𝑥_𝑖∈𝑋. It follows from (2.5), (2.8), and (2.9) that for all 𝑥_𝑖,𝑥_𝑗∈𝑋𝜇_{𝜆(𝑥𝑖,𝑥𝑗)−(𝑓(−𝑟𝑖𝑥𝑖)−𝑓(𝑟𝑖𝑥𝑖)+2𝑟𝑖𝑓(𝑥𝑖))−(𝑓(−𝑟𝑗𝑥𝑗)−𝑓(𝑟𝑗𝑥𝑗)+2𝑟𝑗𝑓(𝑥𝑗))}(𝑡)≥𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥𝑖,𝑥𝑗)}𝑡3,𝜇^′𝜑_𝑖,𝑗𝑥_𝑖𝑡,03,𝜇^′𝜑_𝑖,𝑗0,𝑥_𝑗𝑡3. (2.10)
Replacing 𝑥_𝑖 and 𝑥_𝑗 by (𝑥/𝑟_𝑖) and (𝑦/𝑟_𝑗) in (2.10), we get that 𝜇_{𝑓(−𝑥+𝑦)+𝑓(𝑥−𝑦)−2𝑓(𝑥+𝑦)+𝑓(𝑥)+𝑓(𝑦)−𝑓(−𝑥)−𝑓(−𝑦)}(𝑡)≥𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑦/𝑟𝑗)}𝑡3,𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}𝑡3,𝜇′_{𝜑𝑖,𝑗(0,𝑦/𝑟𝑗)}𝑡3, (2.11) for all 𝑥,𝑦∈𝑋. Putting 𝑦=𝑥 in (2.11), we get 𝜇_{2𝑓(𝑥)−2𝑓(−𝑥)−2𝑓(2𝑥)}(𝑡)≥𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}𝑡3,𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}𝑡3,𝜇′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}𝑡3, (2.12) for all 𝑥∈𝑋. Replacing 𝑥 and 𝑦 by (𝑥/2) and −(𝑥/2) in (2.11), respectively, we get 𝜇_{𝑓(𝑥)+𝑓(−𝑥)}(𝑡)≥𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}𝑡3,𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}𝑡3,𝜇^′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}𝑡3, (2.13) for all 𝑥∈𝑋. It follows from (2.12) and (2.13) that 𝜇_{𝑓(2𝑥)−2𝑓(𝑥)}(𝑡)=𝜇_{𝑓(𝑥)+𝑓(−𝑥)+((2𝑓(𝑥)−2𝑓(−𝑥)−2𝑓(2𝑥))/2)}(𝑡)≥𝑇_𝑀𝜇_{𝑓(𝑥)+𝑓(−x)}𝑡2,𝜇_{2𝑓(𝑥)−2𝑓(−𝑥)−2𝑓(2𝑥)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}𝑡6,𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}𝑡6,𝜇^′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}𝑡6,𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}𝑡3,𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}𝑡3,𝜇^′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}𝑡3, (2.14) for all 𝑥∈𝑋. So 𝜇_{(𝑓(2𝑥)/2)−𝑓(𝑥)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}𝑡3,𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}𝑡3,𝜇^′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}𝑡3,𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}2𝑡3,𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}2𝑡3,𝜇^′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}2𝑡3. (2.15)
Replacing 𝑥 by 2^𝑛𝑥 in (2.15) and using (2.1), we get 𝜇_{(𝑓(2𝑛+1𝑥)/2𝑛+1)−(𝑓(2𝑛𝑥)/2𝑛)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗((2𝑛𝑥/2𝑟𝑖),−(2𝑛𝑥/2𝑟𝑗))}2^𝑛𝑡3,𝜇^′_{𝜑𝑖,𝑗((2𝑛𝑥/2𝑟𝑖),0)}2^𝑛𝑡3,𝜇^′_{𝜑𝑖,𝑗(0,−(2𝑛𝑥/2𝑟𝑗))}2^𝑛𝑡3,𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗((2𝑛𝑥/𝑟𝑖),(2𝑛𝑥/𝑟𝑗))}2^𝑛+1𝑡3,𝜇^′_{𝜑𝑖,𝑗((2𝑛𝑥/𝑟𝑖),0)}2^𝑛+1𝑡3,𝜇^′_{𝜑𝑖,𝑗(0,(2𝑛𝑥/𝑟𝑗))}2^𝑛+1𝑡3≥𝑇_𝑀𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}2^𝑛𝑡3𝛼^𝑛,𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}2^𝑛𝑡3𝛼^𝑛,𝜇^′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}2^𝑛𝑡3𝛼^𝑛,𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}2^𝑛+1𝑡3𝛼^𝑛,𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}2^𝑛+1𝑡3𝛼^𝑛,𝜇^′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}2^𝑛+1𝑡3𝛼^𝑛, (2.16) for all 𝑥∈𝑋 and all 𝑛∈ℕ. Therefore, we have 𝜇_{(𝑓(2𝑛𝑥)/2𝑛)−𝑓(𝑥)}^𝑛−1_𝑘=0𝛼^𝑘𝑡2^𝑘=𝜇_∑^𝑛−1_{𝑘=0((𝑓(2𝑘+1𝑥)/2𝑘+1)−(𝑓(2𝑘𝑥)/2𝑘))}^𝑛−1_𝑘=0𝛼^𝑘𝑡2^𝑘≥𝑇^𝑛−1_𝑘=0𝜇_{((𝑓(2𝑘+1𝑥)/2𝑘+1)−(𝑓(2𝑘𝑥)/2𝑘))}𝛼^𝑘𝑡2^𝑘≥𝑇^𝑛−1_𝑘=0𝑇_𝑀𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}𝑡3,𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}𝑡3,𝜇^′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}𝑡3,𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}2𝑡3,𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}2𝑡3,𝜇^′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}2𝑡3=𝑇_𝑀𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}𝑡3,𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}𝑡3,𝜇^′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}𝑡3,𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}2𝑡3,𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}2𝑡3,𝜇^′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}2𝑡3, (2.17) for all 𝑥∈𝑋. This implies that 𝜇_{(𝑓(2𝑛𝑥)/2𝑛)−𝑓(𝑥)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝜇′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}2𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}2𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝜇′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}2𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘. (2.18)
Replacing 𝑥 by 2^𝑝𝑥 in (2.18), we obtain 𝜇_{(𝑓(2𝑛+𝑝𝑥)/2𝑛+𝑝)−(𝑓(2𝑝𝑥)/2𝑝)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}𝑡3∑^{𝑝+𝑛−1}_𝑘=𝑝𝛼^𝑘/2^𝑘,𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}𝑡3∑^{𝑝+𝑛−1}_𝑘=𝑝𝛼^𝑘/2^𝑘,𝜇′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}𝑡3∑^{𝑝+𝑛−1}_𝑘=𝑝𝛼^𝑘/2^𝑘,𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}2𝑡3∑^{𝑝+𝑛−1}_𝑘=𝑝𝛼^𝑘/2^𝑘,𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}2𝑡3∑^{𝑝+𝑛−1}_𝑘=𝑝𝛼^𝑘/2^𝑘𝜇′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}2𝑡3∑^{𝑝+𝑛−1}_𝑘=𝑝𝛼^𝑘/2^𝑘 (2.19)
Since the right-hand side of the above inequality tends to 1, when 𝑝,𝑛→∞, then the sequence {𝑓(2^𝑘𝑥)/2^𝑘}^+∞_𝑛=1 is a Cauchy sequence in complete RN space (𝑌,𝜇,min), so there exists some point EL(𝑥)∈𝑌 such that EL(𝑥)=lim_𝑛→∞𝑓2^𝑘𝑥2^𝑘, (2.20) for all 𝑥∈𝑋.
Fix 𝑥∈𝑋 and put 𝑃=0 in (2.19). Then we obtain 𝜇_{(𝑓(2𝑛𝑥)/2𝑛)−𝑓(𝑥)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝜇′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝜇′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}2𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}2𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝜇′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}2𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘, (2.21) and so, for every 𝜖>0, we have 𝜇_{EL(𝑥)−𝑓(𝑥)}(𝑡+𝜖)≥𝑇𝜇_{EL(𝑥)−(𝑓(2𝑛𝑥)/2𝑛)}(𝜖),𝜇_{(𝑓(2𝑛𝑥)/2𝑛)−𝑓(𝑥)}(𝑡)≥𝑇𝜇_{EL(𝑥)−(𝑓(2𝑛𝑥)/2𝑛)}(𝜖),𝑇_𝑀𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝜇′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘𝜇′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}2𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}2𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘,𝜇′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}2𝑡3∑^𝑛−1_𝑘=0𝛼^𝑘/2^𝑘. (2.22)
Taking the limit as 𝑛→∞ and using (2.22), we get 𝜇_{EL(𝑥)−𝑓(𝑥)}(𝑡+𝜖)≥𝑇_M𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}(2−𝛼)𝑡6,𝜇′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}(2−𝛼)𝑡6,𝜇′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}(2−𝛼)𝑡6,𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}(2−𝛼)𝑡3,𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}(2−𝛼)𝑡3,𝜇′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}(2−𝛼)𝑡3. (2.23)
Since 𝜖 was arbitrary by taking 𝜖→0 in (2.23), we get 𝜇_{EL(𝑥)−𝑓(𝑥)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}(2−𝛼)𝑡6,𝜇′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}(2−𝛼)𝑡6,𝜇′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}(2−𝛼)𝑡6,𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}(2−𝛼)𝑡3,𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}(2−𝛼)𝑡3,𝜇′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}(2−𝛼)𝑡3. (2.24)
Replacing 𝑥_𝑖 by 2^𝑛𝑥_𝑖 for all 1≤𝑖≤𝑚, in (2.3), we get for all 𝑥_𝑖,𝑥_𝑗∈𝑋 and for all 𝑡>0, 𝜇_∑^𝑚_{𝑗=1𝑓(−2𝑛𝑟𝑗𝑥𝑗+∑1≤𝑖≤𝑚,𝑖≠𝑗2𝑛𝑟𝑖𝑥𝑖)+2∑}^𝑚_{𝑖=1𝑟𝑖𝑓(2𝑛𝑥𝑖)−𝑚𝑓(∑}^𝑚_{𝑖=12𝑛𝑟𝑖𝑥𝑖)/2𝑛}(𝑡)≥𝜇^′_{𝜑(2𝑛𝑥1,…,2𝑛𝑥𝑚)/2𝑛}(𝑡). (2.25)
since lim_𝑛→∞𝜇^′_{𝜑2𝑛𝑥1,…,2𝑛𝑥𝑚/2𝑛}(𝑡)=1, (2.26) We conclude that ^𝑚_𝑗=1EL−𝑟_𝑗𝑥_𝑗+_{1≤𝑖≤𝑚,𝑖≠𝑗}𝑟_𝑖𝑥_𝑖+2^𝑚_𝑖=1𝑟_𝑖EL𝑥_𝑖−𝑚EL^𝑚_𝑖=1𝑟_𝑖𝑥_𝑖=0. (2.27)
To prove the uniqueness of mapping EL, assume that there exists another mapping 𝐴∶𝑋→𝑌 which satisfies (2.4). Fix 𝑥∈𝑋, clearly EL(2^𝑛𝑥)=2^𝑛EL(𝑥) and 𝐴(2^𝑛𝑥)=2^𝑛𝐴(𝑥), for all 𝑛∈𝑁. Since 𝜇_{EL(𝑥)−𝐴(𝑥)}(𝑡)=lim_𝑛→∞𝜇_{(EL(2𝑛𝑥)/2𝑛)−(𝐴(2𝑛𝑥)/2𝑛)}(𝑡), so 𝜇_{(EL(2𝑛𝑥)/2𝑛)−(𝐴(2𝑛𝑥)/2𝑛)}(𝑡)≥min𝜇_{(EL(2𝑛𝑥)/2𝑛)−(𝑓(2𝑛𝑥)/2𝑛)}(𝑡)𝑡2,𝜇_{(𝑓(2𝑛𝑥)/2𝑛)−(𝐴(2𝑛𝑥)/2𝑛)}(𝑡)𝑡2≥𝑇_𝑀𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}2^𝑛(2−𝛼)𝑡12𝛼^𝑛,𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}2^𝑛(2−𝛼)𝑡12𝛼^𝑛,𝜇^′_{𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}2^𝑛(2−𝛼)𝑡12𝛼^𝑛,𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}2^𝑛(2−𝛼)𝑡6𝛼^𝑛,𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}2^𝑛(2−𝛼)𝑡6𝛼^𝑛,𝜇^′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}2^𝑛(2−𝛼)𝑡6𝛼^𝑛. (2.28)
Since the right-hand side of the above inequality tends to 1, when 𝑛→∞, therefore, it follows that for all 𝑡>0, 𝜇_{EL(𝑥)−𝐴(𝑥)}(𝑡)=1 and so EL(𝑥)=𝐴(𝑥). This completes the proof.

Corollary 2.3. Let 𝑋 be a real linear space, (𝑍,𝜇′,min) be an RN space, and (𝑌,𝜇,min) a complete RN space. Let 0<𝑝<1, 𝑧₀∈𝑍 and 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying 𝜇_∑^𝑚_{𝑗=1𝑓(−𝑟𝑗𝑥𝑗+∑1≤𝑖≤𝑚,𝑖≠𝑗𝑟𝑖𝑥𝑖)+2∑}^𝑚_{𝑖=1𝑟𝑖𝑓(𝑥𝑖)−𝑚𝑓(∑}^𝑚_{𝑖=1𝑟𝑖𝑥𝑖)}(𝑡)≥𝜇^′_(∑^𝑚_{𝑘=1‖𝑥𝑘‖}^𝑝_)𝑧0(𝑡), (2.29) for all 𝑥_𝑖,𝑥_𝑗∈𝑋 and 𝑡>0. Then the limit EL(𝑥)=lim_𝑛→∞𝑓(2^𝑛𝑥)/2^𝑛 exists for all 𝑥∈𝑋 and defines a unique Euler-Lagrange additive mapping EL∶𝑋→𝑌 such that 𝜇_{EL(𝑥)−𝑓(𝑥)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇′_‖𝑥‖^𝑝_𝑧02^𝑝𝑟_𝑖𝑟_𝑗^𝑝2−2^𝑝𝑡6𝑟_𝑖^𝑝+𝑟_𝑗^𝑝,𝜇′_‖𝑥‖^𝑝_𝑧02𝑟_𝑖^𝑝2−2^𝑝𝑡6,𝜇′_‖𝑥‖^𝑝_𝑧02𝑟_𝑗^𝑝2−2^𝑝𝑡6,𝑇_𝑀𝜇′_‖𝑥‖^𝑝_𝑧0𝑟_𝑖𝑟_𝑗^𝑝2−2^𝑝𝑡3𝑟_𝑖^𝑝+𝑟_𝑗^𝑝,𝜇′_‖𝑥‖^𝑝_𝑧0𝑟_𝑖^𝑝2−2^𝑝𝑡3,𝜇′_‖𝑥‖^𝑝_𝑧0𝑟_𝑗^𝑝2−2^𝑝𝑡3, (2.30) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let 𝛼=2^𝑝 and 𝜑∶𝑋^𝑚→𝑍 be defined as 𝜑(𝑥₁,…,𝑥_𝑚)=(∑^𝑚_𝑘=1‖𝑥_𝑖‖^𝑝)𝑧₀.

Corollary 2.4. Let 𝑋 be a real linear space, (𝑍,𝜇′,min) be an RN space, and (𝑌,𝜇,min) a complete RN space. Let 𝑧₀∈𝑍 and 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying 𝜇_∑^𝑚_{𝑗=1𝑓(−𝑟𝑗𝑥𝑗+∑1≤𝑖≤𝑚,𝑖≠𝑗𝑟𝑖𝑥𝑖)+2∑}^𝑚_{𝑖=1𝑟𝑖𝑓(𝑥𝑖)−𝑚𝑓(∑}^𝑚_{𝑖=1𝑟𝑖𝑥𝑖)}(𝑡)≥𝜇^′_𝛿𝑧0(𝑡), (2.31) for all 𝑥_𝑖∈𝑋 for all 1≤𝑖≤𝑚 and all 𝑡>0. Then, the limit 𝐶(𝑥)=lim_𝑛→∞(𝑓(2^𝑛𝑥)/2^𝑛) exists for all 𝑥∈𝑋 and defines a unique Euler-Lagrange additive mapping EL∶𝑋→𝑌 such that 𝜇_{EL(𝑥)−𝑓(𝑥)}(𝑡)≥𝑇_𝑀𝜇′_𝛿𝑧0𝑡6,𝜇′_𝛿𝑧0𝑡3, (2.32) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let 𝛼=1 and 𝜑∶𝑋^𝑚→𝑍 be defined as 𝜑(𝑥₁,…,𝑥_𝑚)=𝛿𝑧₀.

Theorem 2.5. Let 𝑋 be a real linear space, (𝑍,𝜇′,min) be an RN space, 𝜑∶𝑋^𝑚→𝑍 be a function such that for some 0<𝛼<1/2, 𝜇^′_{𝜑(𝑥1/2,…,𝑥𝑚/2)}(𝑡)≥𝜇^′_{𝛼𝜑(𝑥1,…,𝑥𝑚)}(𝑡)∀𝑥_𝑖∈𝑋,𝑡>0, (2.33)𝑓(0)=0 and for all 𝑥_𝑖∈𝑋 and 𝑡>0, lim_𝑛→∞𝜇^′_{2𝑛𝜑(𝑥1/2𝑛,…,𝑥𝑚/2𝑛)}(𝑡)=1. Let (𝑌,𝜇,min) be a complete RN space. If 𝑓∶𝑋→𝑌 is a mapping satisfying (2.3), then there is a unique generalized Euler-Lagrange-type additive mapping EL∶𝑋→𝑌 such that, for all 𝑥∈𝑋𝜇_{EL(𝑥)−𝑓(𝑥)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,−𝑥/𝑟𝑗)}(1−2𝛼)𝑡6𝛼,𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑗,0)}(1−2𝛼)𝑡,6𝛼𝜇′_{𝜑𝑖,𝑗(0,−(𝑥/𝑟𝑗))}(1−2𝛼)𝑡,𝑇6𝛼_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}(1−2𝛼)𝑡,3𝛼𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}(1−2𝛼)𝑡3𝛼,𝜇′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}(1−2𝛼)𝑡,3𝛼 (2.34) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. Replacing 𝑥 by 𝑥/2^𝑛+1 in (2.14) and using (2.33), we obtain 𝜇_{2𝑛𝑓(𝑥/2𝑛)−2𝑛+1𝑓(𝑥/2𝑛+1)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑛+2𝑟𝑖,−(𝑥/2𝑛+2𝑟𝑗))}𝑡2^𝑛⋅6,𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑛+2𝑟𝑖,0)}𝑡2^𝑛,𝜇⋅6_{′𝜑𝑖,𝑗(0,−(𝑥/2𝑛+2𝑟𝑗))}𝑡2^𝑛,𝑇⋅6_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑛+1𝑟𝑖,𝑥/2𝑛+1𝑟𝑗)}𝑡2^𝑛⋅3,𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑛+1𝑟𝑖,0)}𝑡2^𝑛⋅3,𝜇^′_{𝜑𝑖,𝑗(0,𝑥/2𝑛+1𝑟𝑗)}𝑡2^𝑛⋅3≥𝑇_𝑀𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,−(𝑥/2𝑟𝑗))}𝑡𝛼^𝑛+12^𝑛⋅6,𝜇^′_{𝜑𝑖,𝑗(𝑥/2𝑟𝑖,0)}𝑡𝛼^𝑛+12^𝑛,𝜇⋅6_{′𝜑𝑖,𝑗(0,−(𝑥/2𝑟𝑗))}𝑡𝛼^𝑛+12^𝑛,𝑇⋅6_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑗)}𝑡𝛼^𝑛+12^𝑛⋅3,𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}𝑡𝛼^𝑛+12^𝑛⋅3,𝜇^′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}𝑡𝛼^𝑛+12^𝑛.⋅3 (2.35)
So 𝜇_{2𝑛𝑓(𝑥/2𝑛)−𝑓(𝑥)}^𝑛−1_𝑖=12^𝑘𝛼^𝑘+1𝑡≥𝑇_𝑀𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,−(𝑥/𝑟𝑗))}𝑡6,𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}𝑡6,𝜇^′_{𝜑𝑖,𝑗(0,−(𝑥/𝑟𝑗))}𝑡6,𝑇_𝑀𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}𝑡3,𝜇^′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}𝑡3,𝜇^′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}𝑡3, (2.36) for all 𝑥∈𝑋. This implies that 𝜇_{2𝑛𝑓(𝑥/2𝑛)−𝑓(𝑥)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,−(𝑥/𝑟𝑗))}𝑡6𝛼∑^𝑛−1_𝑘=02^𝑘𝛼^𝑘,𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}𝑡6𝛼∑^𝑛−1_𝑘=02^𝑘𝛼^𝑘,𝜇′_{𝜑𝑖,𝑗(0,−(𝑥/𝑟𝑗))}𝑡6𝛼∑^𝑛−1_𝑘=02^𝑘𝛼^𝑘,𝑇_M𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,𝑥/𝑟𝑗)}𝑡3𝛼∑^𝑛−1_𝑘=02^𝑘𝛼^𝑘,𝜇′_{𝜑𝑖,𝑗(𝑥/𝑟𝑖,0)}𝑡3𝛼∑^𝑛−1_𝑘=02^𝑘𝛼^𝑘,𝜇′_{𝜑𝑖,𝑗(0,𝑥/𝑟𝑗)}𝑡3𝛼∑^𝑛−1_𝑘=02^𝑘𝛼^𝑘. (2.37)
Proceeding as in the proof of Theorem 2.2, one can easily show that the sequence {2^𝑛𝑓(𝑥/2^𝑛)}^+∞_𝑛=1 is a Cauchy sequence in complete RN space (𝑌,𝜇,min), so there exists some point EL(𝑥)∈𝑌 such that EL(𝑥)=lim_𝑛→∞2^𝑛𝑓𝑥2^𝑛, (2.38) for all 𝑥∈𝑋.
Taking the limit 𝑛→∞ from both sides of the above inequality, we obtain (2.34).
The rest of the proof is similar to the proof of Theorem 2.2.

Corollary 2.6. Let 𝑋 be a real linear space, (𝑍,𝜇′,min) be an RN space and (𝑌,𝜇,min) a complete RN space. Let 𝑝>1, 𝑧₀∈𝑍 and 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying 𝜇_∑^𝑚_{𝑗=1𝑓(−𝑟𝑗𝑥𝑗+∑1≤𝑖≤𝑚,𝑖≠𝑗𝑟𝑖𝑥𝑖)+2∑}^𝑚_{𝑖=1𝑟𝑖𝑓(𝑥𝑖)−𝑚𝑓(∑}^𝑚_{𝑖=1𝑟𝑖𝑥𝑖)}(𝑡)≥𝜇^′_(∑^𝑚_{𝑘=1‖𝑥𝑖‖}^𝑝_)𝑧0(𝑡), (2.39) for all 𝑥_𝑖∈𝑋 for all 1≤𝑖≤𝑚 and all 𝑡>0. Then the limit EL(𝑥)=lim_𝑛→∞2^𝑛𝑓(𝑥/2^𝑛) exists for all 𝑥∈𝑋 and defines a unique Euler-Lagrange additive mapping EL∶𝑋→𝑌 such that 𝜇_{EL(𝑥)−𝑓(𝑥)}(𝑡)≥𝑇_𝑀𝑇_𝑀𝜇′_‖𝑥‖^𝑝_𝑧02^𝑝𝑟_𝑖𝑟_𝑗^𝑝2^𝑝−2𝑡6𝑟_𝑖^𝑝+𝑟_𝑗^𝑝,𝜇′_‖𝑥‖^𝑝_𝑧02𝑟_𝑖^𝑝2^p−2𝑡6,𝜇′_‖𝑥‖^𝑝_𝑧02𝑟_𝑗^𝑝2^𝑝−2𝑡6,𝑇_𝑀𝜇^′_‖𝑥‖^𝑝_𝑧0𝑟_𝑖𝑟_𝑗^𝑝2^𝑝−2𝑡3𝑟_𝑖^𝑝+𝑟_𝑗^𝑝,𝜇′_‖𝑥‖^𝑝_𝑧0𝑟_𝑖^𝑝2^𝑝−2𝑡3,𝜇′_‖𝑥‖^𝑝_𝑧0𝑟_𝑗^𝑝2^𝑝−2𝑡3, (2.40) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let 𝛼=2^−𝑝 and 𝜑∶𝑋^𝑚→𝑍 be defined as 𝜑(𝑥₁,…,𝑥_𝑚)=(∑^𝑚_𝑘=1‖𝑥_𝑖‖^𝑝)𝑧₀.

Corollary 2.7. Let 𝑋 be a real linear space, (𝑍,𝜇′,min) be an RN space and (𝑌,𝜇,min) a complete RN space. Let 𝑧₀∈𝑍 and 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying 𝜇_∑^𝑚_{𝑗=1𝑓(−𝑟𝑗𝑥𝑗+∑1≤𝑖≤𝑚,𝑖≠𝑗𝑟𝑖𝑥𝑖)+2∑}^𝑚_{𝑖=1𝑟𝑖𝑓(𝑥𝑖)−𝑚𝑓(∑}^𝑚_{𝑖=1𝑟𝑖𝑥𝑖)}(𝑡)≥𝜇^′_𝛿𝑧0(𝑡), (2.41) for all 𝑥_𝑖,𝑥_𝑗∈𝑋 and 𝑡>0. Then, the limit EL(𝑥)=lim_𝑛→∞2^𝑛𝑓(𝑥/2^𝑛) exists for all 𝑥∈𝑋 and defines a unique Euler-Lagrange additive mapping EL∶𝑋→𝑌 such that 𝜇_{EL(𝑥)−𝑓(𝑥)}(𝑡)≥𝑇_𝑀𝜇′_𝛿𝑧04𝑡3,𝜇′_𝛿𝑧02𝑡3, (2.42) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let 𝛼=1/4 and 𝜑∶𝑋^𝑚→𝑍 be defined as 𝜑(𝑥₁,…,𝑥_𝑚)=𝛿𝑧₀.