On the Stability of an 𝑚-Variables Functional Equation in Random Normed Spaces via Fixed Point Method

Abstract

At first we find the solution of the functional equation 𝐷_𝑓(𝑥₁,…,𝑥_𝑚)∶=∑^𝑚_𝑘=2(∑^𝑘_𝑖1=2∑_𝑖^𝑘+1_2=𝑖1+1⋯∑^𝑚_{𝑖𝑚−𝑘+1=𝑖𝑚−𝑘+1})𝑓(∑^𝑚_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑚−𝑘+1}𝑥_𝑖−∑^{𝑚−𝑘+1}_𝑟=1𝑥_𝑖𝑟)+𝑓(∑^𝑚_𝑖=1𝑥_𝑖)−2^𝑚−1𝑓(𝑥₁)=0, where 𝑚≥2 is an integer number. Then, we obtain the generalized Hyers-Ulam-Rassias stability in random normed spaces via the fixed point method for the above functional equation.

1. Introduction and Preliminaries

A basic question in the theory of functional equations is as follows: “when is it true that a function that approximately satisfies a functional equation must be close to an exact solution of the equation?”

If the problem accepts a solution, we say the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940 and affirmatively solved by Hyers [2]. The result of Hyers was generalized by Aoki [3] for approximate additive function and by Rassias [4] for approximate linear functions by allowing the difference Cauchy equation ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖ to be controlled by 𝜀(‖𝑥‖^𝑝+‖𝑦‖^𝑝). Taking into consideration a lot of influence of Ulam, Hyers, and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias is called the Hyers-Ulam-Rassias stability. In 1994, a generalization of Rassias theorem was obtained by Găvruţa [5], who replaced 𝜀(‖𝑥‖^𝑝+‖𝑦‖^𝑝) by a general control function 𝜑(𝑥,𝑦) (see also [6–24]).

In the sequel we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [25–29]. Throughout this paper, let Δ⁺ be the space of distribution functions, that is,Δ⁺∶={𝐹∶ℝ∪{−∞,∞}⟶[0,1]∶𝐹isleft-continuous,nondecreasingonℝ,𝐹(0)=0and𝐹(+∞)=1 (1.1) and the subset 𝐷⁺⊆Δ⁺ is the set𝐷⁺=𝐹∈Δ⁺∶𝑙⁻𝐹(+∞)=1, (1.2) where 𝑙⁻𝑓(𝑥) denotes the left limit of function 𝑓 at the point 𝑥. The space Δ⁺ is partially ordered by the usual pointwise ordering of functions, that is, 𝐹≤𝐺 if and only if 𝐹(𝑡)≤𝐺(𝑡) for all 𝑡∈ℝ. The maximal element for Δ⁺ in this order is the distribution function given by𝜀₀(𝑡)=0if1𝑡≤0,if𝑡>0. (1.3)

Definition 1.1 (see [28]). A mapping 𝑇∶[0,1]×[0,1]→[0,1] is a continuous triangular norm (briefly, a 𝑡-norm) if 𝑇 satisfies the following conditions:(a)𝑇 is commutative and associative,(b)𝑇 is continuous,(c)𝑇(𝑎,1)=𝑎 for all 𝑎∈[0,1];(d)𝑇(𝑎,𝑏)≤𝑇(𝑐,𝑑) whenever 𝑎≤𝑐 and 𝑏≤𝑑 for all 𝑎,𝑏,𝑐,𝑑∈[0,1].

Typical examples of continuous 𝑡-norms are 𝑇_𝑃(𝑎,𝑏)=𝑎𝑏, 𝑇_𝑀(𝑎,𝑏)=min(𝑎,𝑏), and 𝑇_𝐿(𝑎,𝑏)=max(𝑎+𝑏−1,0) (the Łukasiewicz 𝑡-norm).

Recall (see [30, 31]) that if 𝑇 is a 𝑡-norm and {𝑥_𝑛} is a given sequence of numbers in [0,1], 𝑇^𝑛_𝑖=1𝑥_𝑖 is defined recurrently by𝑇^𝑛_𝑖=1𝑥_𝑖=𝑥₁if𝑇𝑇𝑛=1,_{𝑛−1𝑖=1}𝑥_𝑖,𝑥_𝑛if𝑛≥2. (1.4)𝑇^∞_𝑖=𝑛𝑥_𝑖 is defined as 𝑇^∞_𝑖=1𝑥_𝑛+𝑖.

It is known [31] that for the Łukasiewicz 𝑡-norm the following implication holds:lim_𝑛→∞𝑇_𝐿^∞_𝑖=1𝑥_𝑛+𝑖=1⟺^∞_𝑛=11−𝑥_𝑛<∞. (1.5)

Definition 1.2 (see [29]). A random normed space (briefly, RN space) is a triple (𝑋,Λ,𝑇), where 𝑋 is a vector space, 𝑇 is a continuous 𝑡-norm, and Λ is a mapping from 𝑋 into 𝐷⁺ such that the following conditions hold:(RN1)Λ_𝑥(𝑡)=𝜀₀(𝑡) for all 𝑡>0 if and only if 𝑥=0,(RN2)Λ_𝛼𝑥(𝑡)=Λ_𝑥(𝑡/|𝛼|) for all 𝑥∈𝑋, 𝛼≠0,(RN3)Λ_𝑥+𝑦(𝑡+𝑠)≥𝑇(Λ_𝑥(𝑡),Λ_𝑦(𝑠)) for all 𝑥,𝑦∈𝑋 and 𝑡,𝑠≥0.

Definition 1.3. Let (X,Λ,T) be an RN space.(1)A sequence {𝑥_𝑛} in 𝑋 is said to be convergent to 𝑥 in 𝑋 if, for every 𝜖>0 and 𝜆>0, there exists positive integer 𝑁 such that Λ_{𝑥𝑛−𝑥}(𝜖)>1−𝜆 whenever 𝑛≥𝑁.(2)A sequence {𝑥_𝑛} in 𝑋 is called Cauchy if, for every 𝜖>0 and 𝜆>0, there exists positive integer 𝑁 such that Λ_{𝑥𝑛−𝑥𝑚}(𝜖)>1−𝜆 whenever 𝑛≥𝑚≥𝑁.(3)An RN space (𝑋,Λ,𝑇) is said to be complete if and only if every Cauchy sequence in 𝑋 is convergent to a point in 𝑋. A complete RN space is said to be a random Banach space.

Theorem 1.4 (see [28]). If (𝑋,Λ,𝑇) is an RN space and {𝑥_𝑛} is a sequence such that 𝑥_𝑛→𝑥, then lim_𝑛→∞Λ_𝑥𝑛(𝑡)=Λ_𝑥(𝑡) almost everywhere.

Theorem 1.5 (see [32, 33]). Let (𝑆,𝑑) be a complete generalized metric space, and let 𝐽∶𝑆→𝑆 be a strictly contractive mapping with Lipschitz constant 𝐿<1. Then, for each given element 𝑥∈𝑆, either 𝑑𝐽^𝑛𝑥,𝐽^𝑛+1𝑥=∞ (1.6) for all nonnegative integers 𝑛 or there exists a positive integer 𝑛₀ such that(1)𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)<∞, for all 𝑛≥𝑛₀,(2)the sequence {𝐽^𝑛𝑥} converges to a fixed point 𝑦^∗ of 𝐽,(3)𝑦^∗ is the unique fixed point of 𝐽 in the set Ω={𝑦∈𝑆∣𝑑(𝐽^𝑛0𝑥,𝑦)<∞},(4)𝑑(𝑦,𝑦^∗)≤(1/(1−𝐿))𝑑(𝑦,𝐽𝑦) for all 𝑦∈Ω.

The theory of random normed spaces (RN spaces) is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The notion of an RN space corresponds to the situations when we do not know exactly the norm of point and we know only probabilities of possible values of this norm. The RN spaces may also provide us the appropriate tools to study the geometry of nuclear physics and have important application in quantum particle physics. The generalized Hyers-Ulam stability of different functional equations in random normed spaces (RN spaces) and fuzzy normed spaces has been recently studied in, Alsina [34], Mirmostafaee et al. [35–38], Miheţ and Radu [26, 27, 39, 40], Miheţ et al. [41, 42], Baktash et al. [43] and Saadati et al. [44].

In this paper, we consider the 𝑚-dimensional additive functional equation^𝑚_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑚_{𝑖𝑚−𝑘+1=𝑖𝑚−𝑘+1}𝑓^𝑚_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑚−𝑘+1}𝑥_𝑖−^{𝑚−𝑘+1}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑚_𝑖=1𝑥_𝑖=2^𝑚−1𝑓𝑥₁, (1.7) where 𝑚≥2 is an integer number. It is easy to see that the function 𝑓(𝑥)=𝑎𝑥 is a solution of the functional equation (1.7).

As a special case, if 𝑚=2 in (1.7), then the functional equation (1.7) reduces to𝑓𝑥₁+𝑥₂+𝑓𝑥₁−𝑥₂=2𝑓𝑥₁. (1.8) Also by putting 𝑚=3 in (1.7), we obtain²_𝑖1³_{=2𝑖2=𝑖1+1}𝑓³_{𝑖=1,𝑖≠𝑖1,𝑖2}𝑥_𝑖−²_𝑟=1𝑥_𝑖𝑟+³_𝑖1=2𝑓³_{𝑖=1,𝑖≠𝑖1}𝑥_𝑖−𝑥_𝑖1+𝑓³_𝑖=1𝑥_𝑖=2²𝑓𝑥₁, (1.9) that is,𝑓𝑥₁−𝑥₂−𝑥₃+𝑓𝑥₁−𝑥₂+𝑥₃+𝑓𝑥₁+𝑥₂−𝑥₃+𝑓𝑥₁+𝑥₂+𝑥₃=4𝑓𝑥₁. (1.10)

The main purpose of this paper is to prove the stability of (1.7) in random normed spaces via the fixed point method.

2. Results in RN spaces via Fixed Point Method

Lemma 2.1. Let 𝑋 and 𝑌 be real vector spaces. A function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies (1.7) if and only if 𝑓∶𝑋→𝑌 is additive.

Proof. Let 𝑓 satisfy the functional equation (1.7). Hence, according to (1.7), we get ²_𝑖1³_{=2𝑖2=𝑖1+1}⋯^𝑚_{𝑖𝑚−1=𝑖𝑚−2+1}𝑓^𝑚_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑚−1}𝑥_𝑖−^𝑚−1_𝑟=1𝑥_𝑖𝑟+³_𝑖1⁴_{=2𝑖2=𝑖1+1}⋯^𝑚_{𝑖𝑚−2=𝑖𝑚−3+1}𝑓^𝑚_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑚−2}𝑥_𝑖−^𝑚−2_𝑟=1𝑥_𝑖𝑟+⋯+^𝑚_𝑖1=2𝑓^𝑚_{𝑖=1,𝑖≠𝑖1}𝑥_𝑖−𝑥_𝑖1+𝑓^𝑚_𝑖=1𝑥_𝑖=2^𝑚−1𝑓𝑥₁ (2.1) for all 𝑥₁,…,𝑥_𝑚∈𝑋. Setting 𝑥_𝑖=0(𝑖=2,…,𝑚−1) in (2.1), we have 𝑓𝑥₁−𝑥_𝑚+1𝑓𝑥𝑚−2₁−𝑥_𝑚+𝑓𝑥𝑚−2𝑚−2¹+𝑥_𝑚+⋯+𝑓𝑥𝑚−2𝑚−3¹−𝑥_𝑚+2𝑓𝑥𝑚−2₁+𝑥_𝑚+𝑓𝑥𝑚−2𝑚−2¹−𝑥_𝑚+1𝑓𝑥𝑚−2₁+𝑥_𝑚+𝑓𝑥₁+𝑥_𝑚=2^𝑚−1𝑓𝑥₁, (2.2) that is, 1+^𝑚−2_ℓ=1ℓ𝑓𝑥𝑚−2₁+𝑥_𝑚+𝑓𝑥₁−𝑥_𝑚=2^𝑚−1𝑓𝑥₁ (2.3) for all 𝑥₁,𝑥_𝑚∈𝑋. On the other hand, we have the relation 1+^𝑚−𝑗_ℓ=1ℓ=𝑚−𝑗^𝑚−𝑗_ℓ=0ℓ𝑚−𝑗=2^𝑚−𝑗 (2.4) for all 𝑚>𝐽. Hence, we obtain from (2.3) and (2.4) that 𝑓𝑥₁+𝑥_𝑚+𝑓𝑥₁−𝑥_𝑚=2𝑓𝑥₁ (2.5) for all 𝑥₁,𝑥_𝑚∈𝑋. Setting 𝑥_𝑚=𝑥₁ in (2.5) we get 𝑓(2𝑥₁)=2𝑓(𝑥₁) for all 𝑥₁∈𝑋. Replacing 𝑥₁ and 𝑥_𝑚 by 𝑥₁+𝑥_𝑚 and 𝑥₁−𝑥_𝑚 in (2.5), respectively, and then using 𝑓(2𝑥₁)=2𝑓(𝑥₁), we obtain that 𝑓𝑥₁+𝑥_𝑚=𝑓𝑥₁+𝑓𝑥_𝑚 (2.6) for all 𝑥₁,𝑥_𝑚∈𝑋, which implies that 𝑓 is additive.
Conversely, suppose that 𝑓 is additive, and thus 𝑓 satisfies the equation 𝑓(𝑥₁+𝑥₂)=𝑓(𝑥₁)+𝑓(𝑥₂). Hence we have 𝑓(0)=0 and 𝑓(2𝑥₁)=2𝑓(𝑥₁) for all 𝑥₁∈𝑋. Replacing 𝑥₁ and 𝑥₂ by 𝑥₁+𝑥₂ and 𝑥₁−𝑥₂ in the additive equation and then using 𝑓(2𝑥₁)=2𝑓(𝑥₁) lead to 𝑓𝑥₁+𝑥₂+𝑓𝑥₁−𝑥₂=2𝑓𝑥₁ (2.7) for all 𝑥₁,𝑥₂∈𝑋.
Now, we are going to prove our assumption by induction on 𝑚≥2. It holds for 𝑚=2; see (2.7). Assume that (1.7) holds for the case, where 𝑚=𝑝; that is, we have ^𝑝_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑝_{𝑖𝑝−𝑘+1=𝑖𝑝−𝑘+1}𝑓^𝑝_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑝−𝑘+1}𝑥_𝑖−^{𝑝−𝑘+1}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑝_𝑖=1𝑥_𝑖=2^𝑝−1𝑓𝑥₁ (2.8) for all 𝑥₁,…,𝑥_𝑝∈𝑋. Replacing 𝑥₁ by 𝑥₁+𝑥_𝑝+1 in (2.8), we obtain ^𝑝_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑝_{𝑖𝑝−𝑘+1=𝑖𝑝−𝑘+1}𝑓𝑥₁+𝑥_𝑝+1+^𝑝_{𝑖=2,𝑖≠𝑖1,…,𝑖𝑝−𝑘+1}𝑥_𝑖−^{𝑝−𝑘+1}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑝+1_𝑖=1𝑥_𝑖=2^𝑝−1𝑓𝑥₁+𝑥_𝑝+1 (2.9) for all 𝑥₁,…,𝑥_𝑝∈𝑋. Replacing 𝑥_𝑝+1 by −𝑥_𝑝+1 in (2.9), we obtain ^𝑝_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑝_{𝑖𝑝−𝑘+1=𝑖𝑝−𝑘+1}𝑓𝑥₁−𝑥_𝑝+1+^𝑝_{𝑖=2,𝑖≠𝑖1,…,𝑖𝑝−𝑘+1}𝑥_𝑖−^{𝑝−𝑘+1}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑝_𝑖=1𝑥_𝑖−𝑥_𝑝+1=2^𝑝−1𝑓𝑥₁−𝑥_𝑝+1 (2.10) for all 𝑥₁,…,𝑥_𝑝+1∈𝑋. Adding (2.9) to (2.10), one gets ^𝑝+1_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑝+1_{𝑖𝑝−𝑘+2=𝑖𝑝−𝑘+1+1}𝑓^𝑝+1_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑝−𝑘+2}𝑥_𝑖−^{𝑝−𝑘+2}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑝+1_𝑖=1𝑥_𝑖=2^𝑝−1𝑓𝑥₁+𝑥_𝑝+1+𝑓𝑥₁−𝑥_𝑝+1 (2.11) for all 𝑥₁,…,𝑥_𝑝+1∈𝑋. Therefore, it follows from (2.7) and (2.11) that (1.7) holds for 𝑚=𝑝+1. This completes the proof of the theorem.

From now on, let 𝑋 be a linear space and (𝑌,Λ,𝑇_𝑀) a complete RN space. For convenience, we use the following abbreviation for a given function 𝑓∶𝑋→𝑌:𝐷_𝑓𝑥₁,…,𝑥_𝑚=^𝑚_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}…^𝑚_{𝑖𝑚−𝑘+1=𝑖𝑚−𝑘+1}𝑓^𝑚_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑚−𝑘+1}𝑥_𝑖−^{𝑚−𝑘+1}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑚_𝑖=1𝑥_𝑖−2^𝑚−1𝑓𝑥₁ (2.12) for all 𝑥₁,…,𝑥_𝑚∈𝑋, where 𝑚≥2 is an integer number.

Theorem 2.2. Let Φ∶𝑋×𝑋×⋯×𝑋_𝑚-terms→𝐷⁺ be a function (Φ(𝑥₁,…,𝑥_𝑚) is denoted by Φ_{𝑥1,…,𝑥𝑚}) such that, for some 0<𝛼<2, Φ_{2𝑥1,…,2𝑥𝑚}(𝛼𝑡)≥Φ_{𝑥1,…,𝑥𝑚}(𝑡) (2.13) for all 𝑥₁,…,𝑥_𝑚∈𝑋 and all 𝑡>0. Suppose that a function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality Λ_{𝐷𝑓(𝑥1,…,𝑥𝑚)}(𝑡)≥Φ_{𝑥1,…,𝑥𝑚}(𝑡) (2.14) for all 𝑥₁,…,𝑥_𝑚∈𝑋 and all 𝑡>0. Then, there exists a unique additive function 𝐴∶𝑋→𝑌 such that Λ_{𝑓(𝑥)−𝐴(𝑥)}(𝑡)≥Φ_𝑥,𝑥,0,…,0_𝑚−22^𝑚−2(2−𝛼)𝑡 (2.15) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. Letting 𝑥_𝑖=0(𝑖=3,…,𝑚) in (2.14), we get Λ_{(1+∑𝑚−2ℓ=1ℓ𝑚−2)(𝑓(𝑥1+𝑥2)+𝑓(𝑥1−𝑥2))−2𝑚−1𝑓(𝑥1)}(𝑡)≥Φ_{𝑥1,𝑥2,0,…,0}(𝑡) (2.16) for all 𝑥₁,𝑥₂∈𝑋 and all 𝑡>0. Setting 𝑥₁=𝑥₂=𝑥 in (2.16), we obtain from (2.4) and 𝑓(0)=0 that Λ₂^𝑚−2_{𝑓(2𝑥)−2}^𝑚−1_𝑓(𝑥)(𝑡)≥Φ_{𝑥,𝑥,0,…,0}(𝑡) (2.17) for all 𝑥∈𝑋 and all 𝑡>0, or Λ_{𝑓(2𝑥)/2−𝑓(𝑥)}(𝑡)≥Φ_{𝑥,𝑥,0,…,0}2^𝑚−1𝑡 (2.18) for all 𝑥∈𝑋 and all 𝑡>0. Let 𝑆 be the set of all functions ℎ∶𝑋→𝑌 with ℎ(0)=0 and introduce a generalized metric on 𝑆 as follows: 𝑑(ℎ,𝑘)=inf𝑢∈ℝ⁺∶Λ_{ℎ(𝑥)−𝑘(𝑥)}(𝑢𝑡)≥Φ_{𝑥,𝑥,0,…,0}(𝑡),∀𝑥∈𝑋,∀𝑡>0, (2.19) where, as usual, inf∅=+∞. It is easy to show that (𝑆,𝑑) is a generalized complete metric space [26, 45].
Now we consider the function 𝐽∶𝑆→𝑆 defined by (𝑥)ℎ()𝐽ℎ∶=2𝑥2 (2.20) for all ℎ∈𝑆 and 𝑥∈𝑋.
Now let 𝑔,𝑓∈𝑆 such that 𝑑(𝑓,𝑔)<𝜀. Then, Λ_{𝐽𝑔(𝑥)−𝐽𝑓(𝑥)}𝛼𝜀2𝑡=Λ_{𝑔(2𝑥)−𝑓(2𝑥)}(𝛼𝜀𝑡)≥Φ_{2𝑥,2𝑥,0,…,0}(𝛼𝑡)≥Φ_{𝑥,𝑥,0,…,0}(𝑡), (2.21) that is, if 𝑑(𝑓,𝑔)<𝜀, we have 𝑑(𝐽𝑓,𝐽𝑔)<(𝛼/2)𝜀. This means that 𝑑≤𝛼𝐽𝑓,𝐽𝑔2𝑑𝑓,𝑔 (2.22) for all 𝑓,𝑔∈𝑆, that is, 𝐽 is a strictly contractive self-function on 𝑆 with the Lipschitz constant 𝛼/2.
It follows from (2.18) that Λ_{𝐽𝑓(𝑥)−𝑓(𝑥)}𝑡2^𝑚−1≥Φ_{𝑥,𝑥,0,…,0}(𝑡) (2.23) for all 𝑥∈𝑋 and all 𝑡>0, which implies that 𝑑(𝐽𝑓,𝑓)≤1/2^𝑚−1.
Due to Theorem 1.5, there exists a function 𝐴∶𝑋→𝑌 such that 𝐴 is a fixed point of 𝐽, that is, 𝐴(2𝑥)=2𝐴(𝑥) for all 𝑥∈𝑋.
Also, 𝑑(𝐽^𝑛𝑓,𝐴)→0 as 𝑛→∞, implies the equality lim_𝑛→∞𝑓2^𝑛𝑥2^𝑛=𝐴(𝑥) (2.24) for all 𝑥∈𝑋. If we replace 𝑥₁,…,𝑥_𝑚 with 2^𝑛𝑥₁,…,2^𝑛𝑥_𝑚 in (2.14), respectively, and divide by 2^𝑛, then it follows from (2.13) that Λ_𝐷𝑓(2^𝑛_𝑥1,…,2^𝑛_𝑥𝑚)/2^𝑛(𝑡)≥Φ₂^𝑛_𝑥1,…,2^𝑛_𝑥𝑚2^𝑛𝑡=Φ₂^𝑛_𝑥1,…,2^𝑛_𝑥𝑚𝛼^𝑛2𝛼^𝑛𝑡≥Φ_{𝑥1,…,𝑥𝑚}2𝛼^𝑛𝑡 (2.25) for all 𝑥₁,…,𝑥_𝑚∈𝑋 and all 𝑡>0. By letting 𝑛→∞ in (2.25), we find that Λ_{𝐷𝐴(𝑥1,…,𝑥𝑚)}(𝑡)=1 for all 𝑡>0, which implies 𝐷_𝐴(𝑥₁,…,𝑥_𝑚)=0, and thus 𝐴 satisfies (1.7). Hence by Lemma 2.1, the function 𝐴∶𝑋→𝑌 is additive.
According to the fixed point alterative, since 𝐴 is the unique fixed point of 𝐽 in the set Ω={𝑔∈𝑆∶𝑑(𝑓,𝑔)<∞}, 𝐴 is the unique function such that Λ_{𝑓(𝑥)−𝐴(𝑥)}(𝑢𝑡)≥Φ_{𝑥,𝑥,0,…,0}(𝑡) (2.26) for all 𝑥∈𝑋 and all 𝑡>0. Again using the fixed point alterative gives 𝑑≤1𝑓,𝐴𝑑≤11−𝐿𝑓,𝐽𝑓2^𝑚−1()=11−𝐿2^𝑚−1(),1−𝛼/2 (2.27) which implies the inequality Λ_{𝑓(𝑥)−𝐴(𝑥)}𝑡2^𝑚−2(2−𝛼)≥Φ_{𝑥,𝑥,0,…,0}(𝑡) (2.28) for all 𝑥∈𝑋 and all 𝑡>0. So, Λ_{𝑓(𝑥)−𝐴(𝑥)}(𝑡)≥Φ_{𝑥,𝑥,0,…,0}2^𝑚−2(2−𝛼)𝑡 (2.29) for all 𝑥∈𝑋 and all 𝑡>0. This completes the proof.

Now, we present a corollary that is an application of the last theorem in the classical case.

Corollary 2.3. Let (𝑋,‖⋅‖) and (𝑌,‖⋅‖), normed linear spaces, define Λ_𝑥(𝑡)=𝑡𝑡+‖𝑥‖ (2.30) for 𝑥∈𝑋 and 𝑡>0. Define Φ_{𝑥1,…,𝑥𝑚}(𝑡)=𝑡𝑡+∑^𝑚_𝑖=1𝑥_𝑖^𝑝 (2.31) for all 𝑥₁,…,𝑥_𝑚∈𝑋 and all 𝑡>0 in which 𝑝<1. Now, for 𝛼=2^𝑝, (2.13) holds for all 𝑥₁,…,𝑥_𝑚∈𝑋 and all 𝑡>0. Suppose that an odd function 𝑓∶𝑋→𝑌 satisfies (2.14) for all 𝑥₁,…,𝑥_𝑛∈𝑋 and all 𝑡>0. Then, by the last theorem there exists a unique additive function 𝐴∶𝑋→𝑌 such that 𝑡𝑓(𝑥)(𝑥≥𝑡𝑡+−𝐴)𝑡+1/2^𝑚−21−2^𝑝−1‖𝑥‖^𝑝 (2.32) for all 𝑥∈𝑋 and all 𝑡>0. Hence, 𝑓(𝑥)(𝑥≤1−𝐴)2^𝑚−21−2^𝑝−1‖𝑥‖^𝑝 (2.33) for all 𝑥∈𝑋.

Theorem 2.4. Let Φ∶𝑋×𝑋×⋯×𝑋_{𝑚-𝑡𝑒𝑟𝑚𝑠}→𝐷⁺ be a function such that, for some 0<𝛼<3, Φ_{3𝑥1,…,3𝑥𝑚}(𝛼𝑡)≥Φ_{𝑥1,…,𝑥𝑚}(𝑡) (2.34) for all 𝑥₁,…,𝑥_𝑚∈𝑋 and all 𝑡>0. Suppose that an odd function 𝑓∶𝑋→𝑌 satisfies (2.14) for all 𝑥₁,…,𝑥_𝑛∈𝑋 and all 𝑡>0. Then, there exists a unique additive function 𝐴∶𝑋→𝑌 such that Λ_{𝑓(𝑥)−𝐴(𝑥)}(𝑡)≥Φ_{𝑥,𝑥,𝑥,}0,…,0_𝑚−32^𝑚−3(3−𝛼)𝑡 (2.35) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. Letting 𝑥_𝑖=0(𝑖=4,…,𝑚) in (2.14), we get Λ_{(1+∑𝑚−3ℓ=1ℓ𝑛−3)(𝑓(𝑥1+𝑥2+𝑥3)+𝑓(𝑥1−𝑥2+𝑥3)+𝑓(𝑥1+𝑥2−𝑥3)+𝑓(𝑥1−𝑥2−𝑥3))−2𝑚−1𝑓(𝑥1)}(𝑡)≥Φ_{𝑥1,𝑥2,𝑥3,0,…,0}(𝑡) (2.36) for all 𝑥₁,𝑥₂,𝑥₃∈𝑋 and all 𝑡>0. Setting 𝑥₁=𝑥₂=𝑥₃=𝑥 in the last inequality, we obtain by using oddness of 𝑓 and (2.4) that Λ₂^𝑚−3_{𝑓(3𝑥)+𝑓(𝑥)−2}^𝑚−1_𝑓(𝑥)(𝑡)≥Φ_{𝑥,𝑥,𝑥,0,…,0}(𝑡) (2.37) for all 𝑥∈𝑋 and all 𝑡>0, or Λ_{(𝑓(3𝑥)/3)−𝑓(𝑥)}(𝑡)≥Φ_{𝑥,𝑥,𝑥,0,…,0}3.2^𝑚−3𝑡 (2.38) for all 𝑥∈𝑋 and all 𝑡>0. Let 𝑆 be the set of all odd functions ℎ∶𝑋→𝑌, and introduce a generalized metric on 𝑆 as follows: 𝑑(ℎ,𝑘)=inf𝑢∈ℝ⁺∶Λ_{ℎ(𝑥)−𝑘(𝑥)}(𝑢𝑡)≥Φ_{𝑥,𝑥,𝑥,0,…,0}(𝑡),∀𝑥∈𝑋,∀𝑡>0. (2.39) It is easy to show that (𝑆,𝑑) is a generalized complete metric space [26, 45]. Let 𝐽∶𝑆→𝑆 be the function defined by (𝑥)ℎ()𝐽ℎ∶=3𝑥3 (2.40) for all ℎ∈𝑆 and 𝑥∈𝑋. One can show that 𝑑≤𝛼𝐽𝑓,𝐽𝑔3𝑑𝑓,𝑔 (2.41) for all 𝑓,𝑔∈𝑆, that is, 𝐽 is a strictly contractive self-function on 𝑆 with the Lipschitz constant 𝛼/3.
It follows from (2.38) that Λ_{𝐽𝑓(𝑥)−𝑓(𝑥)}𝑡3.2^𝑚−3≥Φ_{𝑥,𝑥,𝑥,0,…,0}(𝑡) (2.42) for all 𝑥∈𝑋 and all 𝑡>0, which implies that 𝑑(𝐽𝑓,𝑓)≤(1/3.2^𝑚−3).
Due to Theorem 1.5, the sequence {𝐽^𝑛} converges to a fixed point 𝐴 of 𝐽, that is, 𝐴∶𝑋⟶𝑌,𝐴(𝑥)=lim_𝑛→∞𝐽^𝑛𝑓(𝑥)=lim_𝑛→∞𝑓3^𝑛𝑥3^𝑛, (2.43) and 𝐴(3𝑥)=3𝐴(𝑥) for all 𝑥∈𝑋.
Also, 𝐴 is the unique fixed point of 𝐽 in the set Ω={𝑔∈𝑆∶𝑑(𝑓,𝑔)<∞}, and 𝑑≤1𝑓,𝐴𝑑≤11−𝐿𝑓,𝐽𝑓3.2^𝑚−3()=11−𝐿3.2^𝑚−3(1−𝛼/3) (2.44) implies the inequality Λ_{𝑓(𝑥)−𝐴(𝑥)}𝑡2^𝑚−3(3−𝛼)≥Φ_{𝑥,𝑥,𝑥,0,…,0}(𝑡) (2.45) for all 𝑥∈𝑋 and all 𝑡>0. This implies that inequality (2.35) holds. Furthermore, we can obtain that the function 𝐴∶𝑋→𝑌 satisfies (1.7). Hence by Lemma 2.1, we get that the function 𝐴∶𝑋→𝑌 is additive.

Now, we present a corollary that is an application of the last theorem in the classical case.

Corollary 2.5. Let (𝑋,‖⋅‖) and (𝑌,‖⋅‖), normed linear spaces, define Λ_𝑥(𝑡)=𝑡𝑡+‖𝑥‖ (2.46) for 𝑥∈𝑋 and 𝑡>0. Define Φ_{𝑥1,…,𝑥𝑚}(𝑡)=𝑡𝑡+∑^𝑚_𝑖=1𝑥_𝑖^𝑝 (2.47) for all 𝑥₁,…,𝑥_𝑚∈𝑋 and all 𝑡>0 in which 𝑝<1. Now, for 𝛼=3^𝑝, (2.34) holds for all 𝑥₁,…,𝑥_𝑚∈𝑋 and all 𝑡>0. Suppose that an odd function 𝑓∶𝑋→𝑌 satisfies (2.14) for all 𝑥₁,…,𝑥_𝑛∈𝑋 and all 𝑡>0. Then, by the last theorem there exists a unique additive function 𝐴∶𝑋→𝑌 such that 𝑡𝑓(𝑥)(𝑥≥𝑡𝑡+−𝐴)𝑡+1/2^𝑚−31−3^𝑝−1‖𝑥‖^𝑝 (2.48) for all 𝑥∈𝑋 and all 𝑡>0. Hence, 𝑓(𝑥)(𝑥≤1−𝐴)2^𝑚−31−3^𝑝−1‖𝑥‖^𝑝 (2.49) for all 𝑥∈𝑋.

Acknowledgments

The authors would like to thank the referees and the Editor Professor Seenith Sivasundaram for giving useful suggestions for the improvement of this paper.