Nonlinear Random Stability via Fixed-Point Method

Abstract

We prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation 𝑓(𝑥+2𝑦)+𝑓(𝑥−2𝑦)=4𝑓(𝑥+𝑦)+4𝑓(𝑥−𝑦)−6𝑓(𝑥)+𝑓(2𝑦)+𝑓(−2𝑦)−4𝑓(𝑦)−4𝑓(−𝑦) in various complete random normed spaces.

1. Introduction

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 [4] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations. A generalization of the Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias approach.

The functional equation 𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦)=2𝑓(𝑥)+2𝑓(𝑦)(1.1)

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Cholewa [6] for mappings 𝑓∶𝑋→𝑌, where 𝑋 is a normed space and 𝑌 is a Banach space. Czerwik [7] 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 [8–12]).

In [13], Jun and Kim consider the following cubic functional equation:𝑓(2𝑥+𝑦)+𝑓(2𝑥−𝑦)=2𝑓(𝑥+𝑦)+2𝑓(𝑥−𝑦)+12𝑓(𝑥).(1.2)

It is easy to show that the function 𝑓(𝑥)=𝑥³ satisfies the functional equation (1.2), which is called a cubic functional equation, and every solution of the cubic functional equation is said to be a cubic mapping.

Considered the following quartic functional equation𝑓(2𝑥+𝑦)+𝑓(2𝑥−𝑦)=4𝑓(𝑥+𝑦)+4𝑓(𝑥−𝑦)+24𝑓(𝑥)−6𝑓(𝑦).(1.3) It is easy to show that the function 𝑓(𝑥)=𝑥⁴ satisfies the functional equation, which is called a quartic functional equation, and every solution of the quartic functional equation is said to be a quartic mapping. One can easily show that an odd mapping 𝑓∶𝑋→𝑌 satisfies the additive-quadratic-cubic-quadratic functional equation 𝑓(𝑥+2𝑦)+𝑓(𝑥−2𝑦)=+4𝑓𝑓((𝑥2𝑦+)𝑦+)𝑓+(4−𝑓2(𝑦𝑥)−−4𝑦𝑓)−(𝑦6)𝑓−(4𝑥𝑓)(−𝑦)(1.4) if and only if it is an additive-cubic mapping, that is, 𝑓(𝑥+2𝑦)+𝑓(𝑥−2𝑦)=4𝑓(𝑥+𝑦)+4𝑓(𝑥−𝑦)−6𝑓(𝑥).(1.5)

It was shown in Lemma  2.2 of [14] that 𝑔(𝑥)∶=𝑓(2𝑥)−2𝑓(𝑥) and ℎ(𝑥)∶=𝑓(2𝑥)−8𝑓(𝑥) are cubic and additive, respectively, and that 𝑓(𝑥)=(1/6)𝑔(𝑥)−(1/6)ℎ(𝑥).

One can easily show that an even mapping 𝑓∶𝑋→𝑌 satisfies (1.4) if and only if it is a quadratic-quartic mapping, that is, 𝑓(𝑥+2𝑦)+𝑓(𝑥−2𝑦)=4𝑓(𝑥+𝑦)+4𝑓(𝑥−𝑦)−6𝑓(𝑥)+2𝑓(2𝑦)−8𝑓(𝑦).(1.6)

Also 𝑔(𝑥)∶=𝑓(2𝑥)−4𝑓(𝑥) and ℎ(𝑥)∶=𝑓(2𝑥)−16𝑓(𝑥) are quartic and quadratic, respectively, and 𝑓(𝑥)=(1/12)𝑔(𝑥)−(1/12)ℎ(𝑥).

For a given mapping 𝑓∶𝑋→𝑌, we define 𝐷𝑓(𝑥,𝑦)∶=−𝑓𝑓((𝑥2𝑦+)2−𝑦𝑓)+(−𝑓2(𝑦𝑥)+−42𝑓𝑦)(−𝑦)4+𝑓4(𝑓𝑥+(−𝑦𝑦))−4𝑓(𝑥−𝑦)+6𝑓(𝑥)(1.7)

for all 𝑥,𝑦∈𝑋.

Let 𝑋 be a set. A function 𝑑∶𝑋×𝑋→[0,∞] is called a generalized metric on 𝑋 if 𝑑 satisfies(1)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦,(2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋,(3)𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋.

We recall the fixed-point alternative of Diaz and Margolis.

Theorem 1.1 (see [15, 16]). Let (𝑋,𝑑) be a complete generalized metric space and let 𝐽∶𝑋→𝑋 be a strictly contractive mapping with Lipschitz constant 𝐿<1, then for each given element 𝑥∈𝑋, either 𝑑𝐽^𝑛𝑥,𝐽^𝑛+1𝑥=∞(1.8) 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 𝑌={𝑦∈𝑋∣𝑑(𝐽^𝑛0𝑥,𝑦)<∞},(4)𝑑(𝑦,𝑦^∗)≤(1/(1−𝐿))𝑑(𝑦,𝐽𝑦) for all 𝑦∈𝑌.

In 1996, Isac and Rassias [17] were the first to provide applications of stability theory of functional equations for the proof of new fixed-point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [18–21]).

2. Preliminaries

In the sequel, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [22–26]. Throughout this paper, Δ⁺ is the space of all probability distribution functions, that is, the space of all mappings 𝐹∶ℝ∪{−∞,+∞}→[0,1], such taht 𝐹 is left continuous, nondecreasing on ℝ, 𝐹(0)=0 and {𝐹(+∞)=1}. 𝐷⁺ is a subset of Δ⁺ consisting of all functions 𝐹∈Δ⁺ for which 𝑙⁻𝐹(+∞)=1, where 𝑙⁻𝑓(𝑥) denotes the left limit of the function 𝑓 at the point 𝑥, that is, 𝑙⁻𝑓(𝑥)=lim_{𝑡→𝑥−}𝑓(𝑡). 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 in ℝ. The maximal element for Δ⁺ in this order is the distribution function 𝜀₀ given by𝜀₀(𝑡)=01,,iiff𝑡𝑡≤>00,.(2.1)

A triangular norm (shortly 𝑡-norm) is a binary operation on the unit interval [0,1], that is, a function 𝑇∶[0,1]×[0,1]→[0,1], such that for all 𝑎,𝑏,𝑐∈[0,1] the following four axioms satisfied:(T1)𝑇(𝑎,𝑏)=𝑇(𝑏,𝑎) (commutativity),(T2)𝑇(𝑎,(𝑇(𝑏,𝑐)))=𝑇(𝑇(𝑎,𝑏),𝑐) (associativity),(T3)𝑇(𝑎,1)=𝑎 (boundary condition),(T4)𝑇(𝑎,𝑏)≤𝑇(𝑎,𝑐) whenever 𝑏≤𝑐 (monotonicity).

Basic examples are the Łukasiewicz  𝑡-norm  𝑇_𝐿,𝑇_𝐿(𝑎,𝑏)=max(𝑎+𝑏−1,0) for all 𝑎,𝑏∈[0,1] and the 𝑡-norms  𝑇_𝑃,𝑇_𝑀,𝑇_𝐷, where   𝑇_𝑃(𝑎,𝑏)∶=𝑎𝑏, 𝑇_𝑀(𝑎,𝑏)∶=min{𝑎,𝑏}, 𝑇_𝐷(𝑎,𝑏)∶=m0,in(𝑎,𝑏),iofthmerawxis(e𝑎.,𝑏)=1,(2.2)

If 𝑇 is a 𝑡-norm, then 𝑥^𝑇(𝑛) is defined for every 𝑥∈[0,1] and 𝑛∈𝑁∪{0} by 1, if 𝑛=0 and 𝑇(𝑥^{𝑇(𝑛−1)},𝑥) if 𝑛≥1. A 𝑡-norm𝑇 is said to be of Hadžić type (we denote by 𝑇∈ℋ) if the family (𝑥^𝑇(𝑛))_𝑛∈𝑁 is equicontinuous at 𝑥=1 (cf. [27]).

Other important triangular norms are the following (see [28]):(1)The Sugeno-Weber family {𝑇^SW_𝜆}_{𝜆∈[−1,∞]} is defined by 𝑇^SW₋₁=𝑇_𝐷, 𝑇^SW_∞=𝑇_𝑃 and 𝑇^SW_𝜆(𝑥,𝑦)=max0,𝑥+𝑦−1+𝜆𝑥𝑦1+𝜆(2.3) if  𝜆∈(−1,∞).(2)The Domby family {𝑇^𝐷𝜆}_{𝜆∈[0,∞]} is defined by 𝑇_𝐷 if 𝜆=0, 𝑇_𝑀 if 𝜆=∞, and 𝑇^𝐷𝜆(𝑥,𝑦)=11+((1−𝑥)/𝑥)^𝜆+((1−𝑦)/𝑦)^𝜆^1/𝜆(2.4) if 𝜆∈(0,∞).(3)The Aczel-Alsina family {𝑇^AA_𝜆}_{𝜆∈[0,∞]} is defined by 𝑇_𝐷 if 𝜆=0, 𝑇_𝑀 if 𝜆=∞ and 𝑇^AA_𝜆(𝑥,𝑦)=𝑒^{−(|log𝑥|𝜆+|log𝑦|𝜆)1/𝜆}(2.5) if 𝜆∈(0,∞).

A 𝑡-norm   𝑇 can be extended (by associativity) in a unique way to an 𝑛-array operation taking for (𝑥₁,…,𝑥_𝑛)∈[0,1]^𝑛 the value 𝑇(𝑥₁,…,𝑥_𝑛) defined by 𝑇^0𝑖=1𝑥_𝑖=1,𝑇^𝑛𝑖=1𝑥_𝑖=𝑇𝑇^{𝑛−1𝑖=1}𝑥_𝑖,𝑥_𝑛=𝑇𝑥₁,…,𝑥_𝑛.(2.6)

𝑇 can also be extended to a countable operation taking for any sequence (𝑥_𝑛)_𝑛∈𝑁 in [0,1] the value𝑇^∞𝑖=1𝑥_𝑖=lim_𝑛→∞𝑇^𝑛𝑖=1𝑥_𝑖.(2.7) The limit on the right side of (6.4) exists since the sequence (𝑇^𝑛𝑖=1𝑥_𝑖)_𝑛∈ℕ is nonincreasing and bounded from below.

Proposition 2.1 (see [28]). We have the following.(1)For 𝑇≥𝑇_𝐿, the following implication holds: lim_𝑛→∞T^∞𝑖=1𝑥_𝑛+𝑖=1⟺^∞_𝑛=11−𝑥_𝑛<∞.(2.8)(2)If 𝑇 is of Hadžić type, then lim_𝑛→∞T^∞𝑖=1𝑥_𝑛+𝑖=1(2.9) for every sequence (𝑥_𝑛)_𝑛∈𝑁 in [0,1] such that lim_𝑛→∞𝑥_𝑛=1.(3)If 𝑇∈{𝑇^AA_𝜆}_{𝜆∈(0,∞)}∪{𝑇^𝐷𝜆}_{𝜆∈(0,∞)}, then lim_𝑛→∞T^∞𝑖=1𝑥_𝑛+𝑖=1⟺^∞_𝑛=11−𝑥_𝑛^𝛼<∞.(2.10)(4)If 𝑇∈{𝑇^SW_𝜆}_{𝜆∈[−1,∞)}, then lim_𝑛→∞T^∞𝑖=1𝑥_𝑛+𝑖=1⟺^∞_𝑛=11−𝑥_𝑛<∞.(2.11)

Definition 2.2 (see [26]). A Random normed space (briefly, RN-space) is a triple (𝑋,𝜇,𝑇), where 𝑋 is a vector space, 𝑇 is a continuous 𝑡-norm, and 𝜇 is a mapping from 𝑋 into 𝐷⁺ such that, the following conditions hold: (RN1) 𝜇_𝑥(𝑡)=𝜀₀(𝑡) for all 𝑡>0 if and only if 𝑥=0,(RN2) 𝜇_𝛼𝑥(𝑡)=𝜇_𝑥(𝑡/|𝛼|) for all 𝑥∈𝑋, and 𝛼≠0,(RN3) 𝜇_𝑥+𝑦(𝑡+𝑠)≥𝑇(𝜇_𝑥(𝑡),𝜇_𝑦(𝑠)) for all 𝑥,𝑦∈𝑋 and 𝑡,𝑠≥0.

Definition 2.3. Let (𝑋,𝜇,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 𝜇_{𝑥𝑛−𝑥}(𝜖)>1−𝜆 whenever 𝑛≥𝑁.(2)A sequence {𝑥_𝑛} in 𝑋 is called a Cauchy sequence if, for every 𝜖>0 and 𝜆>0, there exists positive integer 𝑁 such that 𝜇_{𝑥𝑛−𝑥𝑚}(𝜖)>1−𝜆 whenever 𝑛≥𝑚≥𝑁.(3)An RN-space (𝑋,𝜇,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 random Banach space.

Theorem 2.4 (see [25]). If (𝑋,𝜇,𝑇) is an RN-space and {𝑥_𝑛} is a sequence such that 𝑥_𝑛→𝑥, then lim_𝑛→∞𝜇_𝑥𝑛(𝑡)=𝜇_𝑥(𝑡) almost everywhere.

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 RN-spaces may also provide us with 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 [20, 24, 29–39].

3. Non-Archimedean Random Normed Space

By a non-Archimedean field, we mean a field 𝒦 equipped with a function (valuation) |⋅| from 𝐾 into [0,∞) such that |𝑟|=0 if and only if 𝑟=0, |𝑟𝑠|=|𝑟||𝑠|, and |𝑟+𝑠|≤max{|𝑟|,|𝑠|} for all 𝑟,𝑠∈𝒦. Clearly, |1|=|−1|=1 and |𝑛|≤1 for all 𝑛∈ℕ. By the trivial valuation, we mean the mapping |⋅| taking everything but 0 into 1 and |0|=0. Let 𝑋 be a vector space over a field 𝒦 with a non-Archimedean nontrivial valuation |⋅|. A function ‖⋅‖∶𝑋→[0,∞) is called a non-Archimedean norm if it satisfies the following conditions:(NAN1) ‖𝑥‖=0 if and only if 𝑥=0,(NAN2) for any 𝑟∈𝒦 and 𝑥∈𝑋, ‖𝑟𝑥‖=|𝑟|‖𝑥‖,(NAN3) the strong triangle inequality (ultrametric), namely, ‖𝑥+𝑦‖≤max{‖𝑥‖,‖𝑦‖}(𝑥,𝑦∈𝑋),(3.1)

then (𝑋,‖⋅‖) is called a non-Archimedean normed space. Due to the fact that ‖‖𝑥_𝑛−𝑥_𝑚‖‖≤max‖‖𝑥_𝑗+1−𝑥_𝑗‖‖∶𝑚≤𝑗≤𝑛−1(𝑛>𝑚),(3.2)

a sequence {𝑥_𝑛} is a Cauchy sequence if and only if {𝑥_𝑛+1−𝑥_𝑛} converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space, we mean one in which every Cauchy sequence is convergent.

In 1897, Hensel [40] discovered the 𝑝-adic numbers of as a number theoretical analogues of power series in complex analysis. Fix a prime number 𝑝. For any nonzero rational number 𝑥, there exists a unique integer 𝑛_𝑥∈ℤ such that 𝑥=(𝑎/𝑏)𝑝^𝑛𝑥, where 𝑎 and 𝑏 are integers not divisible by 𝑝. Then |𝑥|_𝑝∶=𝑝^−𝑛𝑥 defines a non-Archimedean norm on ℚ. The completion of ℚ with respect to the metric 𝑑(𝑥,𝑦)=|𝑥−𝑦|_𝑝 is denoted by ℚ_𝑝, which is called the 𝑝-adic number field.

Throughout the paper, we assume that 𝑋 is a vector space and 𝑌 is a complete non-Archimedean normed space.

Definition 3.1. A non-Archimedean random normed space (briefly, non-Archimedean RN-space) is a triple (𝑋,𝜇,𝑇), where 𝑋 is a linear space over a non-Archimedean field 𝒦, 𝑇 is a continuous 𝑡-norm, and 𝜇 is a mapping from 𝑋 into 𝐷⁺ such that the following conditions hold:(NA-RN1) 𝜇_𝑥(𝑡)=𝜀₀(𝑡) for all 𝑡>0 if and only if 𝑥=0,(NA-RN2) 𝜇_𝛼𝑥(𝑡)=𝜇_𝑥(𝑡/|𝛼|) for all 𝑥∈𝑋, 𝑡>0, and 𝛼≠0,(NA-RN3) 𝜇_𝑥+𝑦(max{𝑡,𝑠})≥𝑇(𝜇_𝑥(𝑡),𝜇_𝑦(𝑠)) for all 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠≥0.
It is easy to see that if (NA-RN3) holds, then so is(RN3)𝜇_𝑥+𝑦(𝑡+𝑠)≥𝑇(𝜇_𝑥(𝑡),𝜇_𝑦(𝑠)).

As a classical example, if (𝑋,‖.‖) is a non-Archimedean normed linear space, then the triple (𝑋,𝜇,𝑇_𝑀), where 𝜇_𝑥(𝑡)=01,,𝑡𝑡≤>‖‖𝑥𝑥‖‖,,(3.3)

is a non-Archimedean RN-space.

Example 3.2. Let (𝑋,‖⋅‖) be a non-Archimedean normed linear space. Define 𝜇_𝑥(𝑡)=𝑡𝑡+‖𝑥‖(𝑥∈𝑋,𝑡>0),(3.4) then (𝑋,𝜇,𝑇_𝑀) is a non-Archimedean RN-space.

Definition 3.3. Let (𝑋,𝜇,𝑇) be a non-Archimedean RN-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→∞𝜇_{𝑥𝑛−𝑥}(𝑡)=1(3.5) for all 𝑡>0. In that case, 𝑥 is called the limit of the sequence {𝑥_𝑛}.

A sequence {𝑥_𝑛} in 𝑋 is called a Cauchy sequence if for each 𝜀>0 and each 𝑡>0 there exists 𝑛₀ such that for all 𝑛≥𝑛₀ and all 𝑝>0, we have 𝜇_{𝑥𝑛+𝑝−𝑥𝑛}(𝑡)>1−𝜀.

If each Cauchy sequence is convergent, then the random norm is said to be complete and the non-Archimedean RN-space is called a non-Archimedean random Banach space.

Remark 3.4 (see [41]). Let (𝑋,𝜇,𝑇_𝑀) be a non-Archimedean RN-space, then 𝜇_{𝑥𝑛+𝑝−𝑥𝑛}(𝑡)≥min𝜇_{𝑥𝑛+𝑗+1−𝑥𝑛+𝑗}(𝑡)∶𝑗=0,1,2,…,𝑝−1.(3.6) So, the sequence {𝑥_𝑛} is a Cauchy sequence if for each 𝜀>0 and 𝑡>0 there exists 𝑛₀ such that for all 𝑛≥𝑛₀, 𝜇_{𝑥𝑛+1−𝑥𝑛}(𝑡)>1−𝜀.(3.7)

4. Generalized Ulam-Hyers Stability for a Quartic Functional Equation in Non-Archimedean RN-Spaces of Functional Equation (1.4): An Odd Case

Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦, and let (𝑌,𝜇,𝑇) be a non-Archimedean random Banach space over 𝒦.

Next, we define a random approximately AQCQ mapping. Let Ψ be a distribution function on 𝑋×𝑋×[0,∞) such that Ψ(𝑥,𝑦,⋅) is nondecreasing and Ψ(𝑐𝑥,𝑐𝑥,𝑡)≥Ψ𝑥,𝑥,𝑡|𝑐|(𝑥∈𝑋,𝑐≠0).(4.1)

Definition 4.1. A mapping 𝑓∶𝑋→𝑌 is said to be Ψ-approximately AQCQ if 𝜇_{𝐷𝑓(𝑥,𝑦)}(𝑡)≥Ψ(𝑥,𝑦,𝑡)(𝑥,𝑦∈𝑋,𝑡>0).(4.2)

In this section, we assume that 2≠0 in 𝒦 (i.e., characteristic of 𝒦 is not 2). Our main result, in this section, is the following.

We prove the generalized Hyers-Ulam stability of the functional equation 𝐷𝑓(𝑥,𝑦)=0 in non-Archimedean random spaces, an odd case.

Theorem 4.2. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦 and let (𝑌,𝜇,𝑇) be a non-Archimedean random Banach space over 𝒦. Let 𝑓∶𝑋→𝑌 be an odd mapping and Ψ-approximately AQCQ mapping. If for some 𝛼∈ℝ, 𝛼>0, and some integer 𝑘, 𝑘>3 with |2^𝑘|<𝛼, Ψlim2^−𝑘_𝑛→∞𝑥,2^−𝑘𝑦,𝑡≥Ψ(𝑥,𝑦,𝛼𝑡)(𝑥∈𝑋,𝑡>0),(4.3)T^∞𝑗=𝑛𝑀2𝑥,𝛼^𝑗𝑡||8||^𝑘𝑗=1(𝑥∈𝑋,𝑡>0),(4.4) then there exists a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−2𝑓(𝑥/2)−𝐶(𝑥/2)}(𝑡)≥𝑇^∞𝑖=1𝑀𝑥,𝛼^𝑖+1𝑡||8||^𝑘𝑖(4.5) for all 𝑥∈𝑋 and 𝑡>0, where 𝑀(𝑥,𝑡)∶=𝑇^𝑘−1Ψ𝑥2,𝑥2,||4𝑡||,Ψ𝑥,𝑥2,𝑡,…,Ψ2^𝑘−1𝑥2,2^𝑘−1𝑥2,||4𝑡||,Ψ2^𝑘−1𝑥,2^𝑘−1𝑥(𝑥∈𝑋,𝑡>0).2,𝑡(4.6)

Proof. Letting 𝑥=𝑦 in (4.2), we get 𝜇_{𝑓(3𝑦)−4𝑓(2𝑦)+5𝑓(𝑦)}(𝑡)≥Ψ(𝑦,𝑦,𝑡)(4.7) for all 𝑦∈𝑋 and 𝑡>0. Replacing 𝑥 by 2𝑦 in (4.2), we get 𝜇_{𝑓(4𝑦)−4𝑓(3𝑦)+6𝑓(2𝑦)−4𝑓(𝑦)}(𝑡)≥Ψ(2𝑦,𝑦,𝑡)(4.8) for all 𝑦∈𝑋 and 𝑡>0. By (4.7) and (4.8), we have 𝜇_{𝑓(4𝑦)−10𝑓(2𝑦)+16𝑓(𝑦)}(𝑡)≥=𝑇𝑇𝜇𝜇_{4(𝑓(3𝑦)−4𝑓(2𝑦)+5𝑓(𝑦))𝑓(3𝑦)−4𝑓(2𝑦)+5𝑓(𝑦)}(𝑡),𝜇_{𝑓(4𝑦)−4𝑓(3𝑦)+6𝑓(2𝑦)−4𝑓(𝑦)}(𝑡)||4𝑡||,𝜇^{𝑓(4𝑦)−4𝑓(3𝑦)+6𝑓(2𝑦)−4𝑓(𝑦)}≥𝑇Ψ𝑦,𝑦,𝑡(𝑡)||4||,Ψ(2𝑦,𝑦,𝑡)(4.9) for all 𝑦∈𝑋 and 𝑡>0. Letting 𝑦∶=𝑥/2 and 𝑔(𝑥)∶=𝑓(2𝑥)−2𝑓(𝑥) for all 𝑥∈𝑋 in (4.9), we get 𝜇_{𝑔(𝑥)−8𝑔(𝑥/2)}(𝑡)≥𝑇Ψ𝑥2,𝑥2,||4𝑡||,Ψ𝑥,𝑥2,𝑡(4.10) for all 𝑥∈𝑋 and 𝑡>0. Now, we show by induction on 𝑗 that for all 𝑥∈𝑋, 𝑡>0 and 𝑗≥1, 𝜇≥𝑀_𝑔(2^𝑗−1_𝑥)−8^𝑗_{𝑔(𝑥/2)𝑗}(𝑡)∶=𝑇(𝑥_2𝑗−1,𝑡)Ψ𝑥2,𝑥2,||4𝑡||,Ψ𝑥,𝑥2,𝑡,…,Ψ2^𝑗−1𝑥2,2^𝑗−1𝑥2,||4𝑡||,Ψ2^𝑗−1𝑥,2^𝑗−1𝑥2,𝑡.(4.11) Putting 𝑗=1 in (4.11), we obtain (4.10). Assume that (4.11) holds for some 𝑗≥1. Replacing 𝑥 by 2^𝑗𝑥 in (4.10), we get 𝜇_𝑔(2^𝑗_{𝑥)−8𝑔(2}^𝑗−1_𝑥)(𝑡)≥𝑇Ψ2^𝑗−1𝑥,2^𝑗−1𝑥,𝑡||4||,Ψ2^𝑗𝑥,2^𝑗−1𝑥,𝑡.(4.12) Since |8|≤1, 𝜇_𝑔(2^𝑗_𝑥)−8^𝑗+1_𝑔(𝑥/2)(𝑡)≥=𝑇𝑇𝜇𝜇_𝑔(2^𝑗_{𝑥)−8𝑔(2}^𝑗−1_{𝑥)𝑔(2𝑗𝑥)−8𝑔(2𝑗−1𝑥)}(𝑡),𝜇_8𝑔(2^𝑗−1_𝑥)−8^𝑗+1_𝑔(𝑥/2)(𝑡)(𝑡),𝜇_𝑔(2^𝑗−1_𝑥)−8^𝑗_𝑔(𝑥/2)≥𝑇₂||8𝑡||Ψ2^𝑗−1𝑥,2^𝑗−1𝑥,𝑡||4||,Ψ2^𝑗=𝑀_𝑗+1𝑥,2^𝑗−1𝑥,𝑡,𝑀_𝑗(𝑥,𝑡)(𝑥,𝑡)(4.13)for all 𝑥∈𝑋 and 𝑡>0. Thus, (4.11) holds for all 𝑗≥2. In particular, 𝜇_𝑔(2^𝑘−1_𝑥)−8^𝑘_𝑔(𝑥/2)(𝑡)≥𝑀(𝑥,𝑡)(𝑥∈𝑋,𝑡>0).(4.14) Replacing 𝑥 by 2^{−(𝑘𝑛+𝑘−1)}𝑥 in (4.14) and using inequality (4.3), we obtain 𝜇_𝑔(𝑥/2^𝑘𝑛₎₋₈^𝑘_𝑔(𝑥/2^𝑘(𝑛+1)₎(𝑡)≥𝑀2𝑥2^𝑘(𝑛+1),𝑡(𝑥∈𝑋,𝑡>0,𝑛=0,1,2,…).(4.15) Then 𝜇₈^𝑘𝑛_𝑔(𝑥/2^𝑘𝑛₎₋₈^𝑘(𝑛+1)_𝑔(𝑥/2^𝑘(𝑛+1)₎(𝑡)≥𝑀2𝑥,𝛼^𝑛+1||8^𝑘(𝑛+1)||𝑡(𝑥∈𝑋,𝑡>0,𝑛=0,1,2,…).(4.16) Hence 𝜇₈^𝑘𝑛_𝑔(𝑥/2^𝑘𝑛₎₋₈^{𝑘(𝑛+𝑝)}_𝑔(𝑥/2^{𝑘(𝑛+𝑝)}₎(𝑡)≥≥𝑇𝑇^{𝑛+𝑝𝑗=𝑛}_{𝑛+𝑝𝑗=𝑛}𝜇₈^𝑘𝑗_𝑔(𝑥/2^𝑘𝑗₎₋₈^{𝑘(𝑗+𝑝)}_𝑔(𝑥/2^{𝑘(𝑗+𝑝)}₎(𝑡)𝑀⎛⎜⎜⎜⎝2𝑥,𝛼^𝑗+1|||8_𝑘^𝑗+1≥𝑇_{𝑛+𝑝𝑗=𝑛}|||𝑡⎞⎟⎟⎟⎠𝑀⎛⎜⎜⎜⎝2𝑥,𝛼^𝑗+1|||8_𝑘^𝑗+1|||𝑡⎞⎟⎟⎟⎠(𝑥∈𝑋,𝑡>0,𝑛=0,1,2,…).(4.17) Since lim_𝑛→∞𝑇^∞𝑗=𝑛𝑀⎛⎜⎜⎜⎝2𝑥,𝛼^𝑗+1|||8_𝑘^𝑗+1|||𝑡⎞⎟⎟⎟⎠=1(𝑥∈𝑋,𝑡>0),(4.18) then 8^𝑘𝑛𝑔𝑥2^𝑘𝑛_𝑛∈ℕ(4.19)is a Cauchy sequence in the non-Archimedean random Banach space (𝑌,𝜇,𝑇). Hence we can define a mapping 𝐶∶𝑋→𝑌 such that lim_𝑛→∞𝜇_(88𝑘)^𝑛_{𝑔(𝑥/2𝑘𝑛)−𝐶(𝑥)}(𝑡)=1(𝑥∈𝑋,𝑡>0).(4.20)
Next for each 𝑛≥1, 𝑥∈𝑋 and 𝑡>0, 𝜇_{𝑔(𝑥)−(88𝑘)}^𝑛_{𝑔(𝑥/2𝑘𝑛)}(𝑡)=≥𝜇𝑇_{∑𝑛−1𝑖=0(88𝑘)}^𝑖_{𝑔(𝑥/2𝑘𝑖)−(88𝑘)}^𝑖+1_{𝑔(𝑥/2𝑘(𝑖+1))𝑛−1𝑖=0}(𝑡)≥𝑇_{𝑛−1𝑖=0}𝜇₍₈^8𝑘₎^𝑖_𝑔(𝑥/2^𝑘𝑖₎₋₍₈^8𝑘₎^𝑖+1_𝑔(𝑥/2^𝑘(𝑖+1)₎(𝑡)𝑀2𝑥,𝛼^𝑖+1𝑡||8^𝑘||^𝑖+1.(4.21)Therefore, 𝜇_{𝑔(𝑥)−𝐶(𝑥)}(𝑡)≥≥𝑇𝑇𝜇𝑇_{𝑔(𝑥)−(88𝑘)}^𝑛_{𝑔(𝑥/2𝑘𝑛)𝑛−1𝑖=0}(𝑡),𝜇_(88𝑘)^𝑛_{𝑔(𝑥/2𝑘𝑛)−𝐶(𝑥)}(𝑡)𝑀2𝑥,𝛼^𝑖+1𝑡||8^𝑘||^𝑖+1,𝜇^{(88𝑘)𝑛𝑔(𝑥/2𝑘𝑛)−𝐶(𝑥)}(𝑡).(4.22) By letting 𝑛→∞, we obtain 𝜇_{𝑔(𝑥)−𝐶(𝑥)}(𝑡)≥𝑇^∞𝑖=1𝑀2𝑥,𝛼^𝑖+1𝑡||8^𝑘||^𝑖+1.(4.23) So, 𝜇_{𝑓(𝑥)−2𝑓(𝑥/2)−𝐶(𝑥/2)}(𝑡)≥𝑇^∞𝑖=1𝑀𝑥,𝛼^𝑖+1𝑡||8^𝑘||^𝑖+1.(4.24) This proves (4.5). From 𝐷𝑔(𝑥,𝑦)=𝐷𝑓(2𝑥,2𝑦)−2𝐷𝑓(𝑥,𝑦), by (4.2), we deduce that 𝜇𝜇_{𝐷𝑓(2𝑥,2𝑦)}(𝑡)≥Ψ(2𝑥,2𝑦,𝑡),_{−2𝐷𝑓(𝑥,𝑦)}(𝑡)=𝜇_{𝐷𝑓(𝑥,𝑦)}||2𝑡||≥𝜇^{𝐷𝑓(𝑥,𝑦)}(𝑡)≥Ψ(𝑥,𝑦,𝑡),(4.25) and so, by (NA-RN3) and (4.2), we obtain 𝜇_{𝐷𝑔(𝑥,𝑦)}(𝑡)≥𝑇𝜇_{𝐷𝑓(2𝑥,2𝑦)}(𝑡),𝜇_{−2𝐷𝑓(𝑥,𝑦)}(𝑡)≥𝑇(Ψ(2𝑥,2𝑦,𝑡),Ψ(𝑥,𝑦,𝑡))∶=𝑁(𝑥,𝑦,𝑡).(4.26) It follows that 𝜇₈^𝑘𝑛_{𝐷𝑔(𝑥/2}^𝑘𝑛_,𝑦/2^𝑘𝑛₎(𝑡)=𝜇_{𝐷𝑔(𝑥/2}^𝑘𝑛_,𝑦/2^𝑘𝑛₎≥𝑁𝑥||8𝑡||^𝑘𝑛2^𝑘𝑛,𝑦2^𝑘𝑛,𝑡||8||^𝑘𝑛≥⋯≥𝑁𝑥,𝑦,𝛼^𝑛−1𝑡||8||^{𝑘(𝑛−1)}(4.27)for all 𝑥,𝑦∈𝑋, 𝑡>0, and 𝑛∈ℕ. Since lim_𝑛→∞𝑁𝑥,𝑦,𝛼^𝑛−1𝑡||8||^{𝑘(𝑛−1)}=1(4.28)for all 𝑥,𝑦∈𝑋 and 𝑡>0, by Theorem 2.4, we deduce that 𝜇_{𝐷𝐶(𝑥,𝑦)}(𝑡)=1(4.29)for all 𝑥,𝑦∈𝑋 and 𝑡>0. Thus, the mapping 𝐶∶𝑋→𝑌 satisfies (1.4).
Now, we have 𝐶(2𝑥)−8𝐶(𝑥)=lim_𝑛→∞8^𝑛𝑔𝑥2^𝑛−1−8^𝑛+1𝑔𝑥=8lim_𝑛→∞2^𝑛8^𝑛−1𝑔𝑥2^𝑛−1−8^𝑛𝑔𝑥2^𝑛=0(4.30)for all 𝑥∈𝑋. Since the mapping 𝑥→𝐶(2𝑥)−2𝐶(𝑥) is cubic (see Lemma  2.2 of [14]), from the equality 𝐶(2𝑥)=8𝐶(𝑥), we deduce that the mapping 𝐶∶𝑋→𝑌 is cubic.

Corollary 4.3. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦, and let (𝑌,𝜇,𝑇) be a non-Archimedean random Banach space over 𝒦 under a t-norm 𝑇∈ℋ. Let 𝑓∶𝑋→𝑌 be an odd and Ψ-approximately AQCQ mapping. If, for some 𝛼∈ℝ, 𝛼>0, and some integer 𝑘, 𝑘>3, with |2^𝑘|<𝛼, Ψ2^−𝑘𝑥,2^−𝑘𝑦,𝑡≥Ψ(𝑥,𝑦,𝛼𝑡)(𝑥∈𝑋,𝑡>0),(4.31) then there exists a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−2𝑓(𝑥/2)−𝐶(𝑥/2)}(𝑡)≥𝑇^∞𝑖=1𝑀𝑥,𝛼^𝑖+1𝑡||8||^𝑘𝑖(4.32) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Since lim_𝑛→∞𝑀𝑥,𝛼^𝑗𝑡||8||^𝑘𝑗=1(𝑥∈𝑋,𝑡>0)(4.33) and 𝑇 is of Hadžić type, from Proposition 2.1, it follows that lim_𝑛→∞𝑇^∞𝑗=𝑛𝑀𝑥,𝛼^𝑗𝑡||8||^𝑘𝑗=1(𝑥∈𝑋,𝑡>0).(4.34)Now, we can apply Theorem 4.2 to obtain the result.

Example 4.4. Let (𝑋,𝜇,𝑇_𝑀) be non-Archimedean random normed space in which 𝜇_𝑥(𝑡)=𝑡𝑡+‖𝑥‖(𝑥∈𝑋,𝑡>0).(4.35) And let (𝑌,𝜇,𝑇_𝑀) be a complete non-Archimedean random normed space (see Example 3.2). Define Ψ(𝑥,𝑦,𝑡)=𝑡1+𝑡.(4.36) It is easy to see that (4.3) holds for 𝛼=1. Also, since 𝑀(𝑥,𝑡)=𝑡1+𝑡,(4.37) we have lim_𝑛→∞𝑇^{∞𝑀,𝑗=𝑛}𝑀𝑥,𝛼^𝑗𝑡||8||^𝑘𝑗=lim^𝑛→∞lim^𝑚→∞𝑇^{𝑚𝑀,𝑗=𝑛}𝑀𝑥,𝑡=lim_𝑛→∞||8||^𝑘𝑗lim_𝑚→∞=1(𝑥∈𝑋,𝑡𝑡+>𝑡||08)^𝑘||.^𝑛(4.38) Let 𝑓∶𝑋→𝑌 be an odd and Ψ-approximately AQCQ mapping. Thus, all the conditions of Theorem 4.2 hold, and so there exists a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−2𝑓(𝑥/2)−𝐶(𝑥/2)}(𝑡)≥𝑡𝑡+||8^𝑘||.(4.39)

Theorem 4.5. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦, and let (𝑌,𝜇,𝑇) be a non-Archimedean random Banach space over 𝒦. Let 𝑓∶𝑋→𝑌 be an odd mapping and Ψ-approximately AQCQ mapping. If for some 𝛼∈ℝ, 𝛼>0, and some integer 𝑘, 𝑘>1 with |2^𝑘|<𝛼, Ψlim2^−𝑘_𝑛→∞𝑥,2^−𝑘𝑦,𝑡≥Ψ(𝑥,𝑦,𝛼𝑡)(𝑥∈𝑋,𝑡>0),𝑇^∞𝑗=𝑛𝑀2𝑥,𝛼^𝑗𝑡||2||^𝑘𝑗=1(𝑥∈𝑋,𝑡>0),(4.40) then there exists a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−8𝑓(𝑥/2)−𝐴(𝑥/2)}(𝑡)≥𝑇^∞𝑖=1𝑀𝑥,𝛼^𝑖+1𝑡||2||^𝑘𝑖(4.41) for all 𝑥∈𝑋 and 𝑡>0, where 𝑀(𝑥,𝑡)∶=𝑇^𝑘−1Ψ𝑥2,𝑥2,||4𝑡||,Ψ𝑥,𝑥2,𝑡,…,Ψ2^𝑘−1𝑥2,2^𝑘−1𝑥2,||4𝑡||,Ψ2^𝑘−1𝑥,2^𝑘−1𝑥(𝑥∈𝑋,𝑡>0)2,𝑡(4.42)

Proof. Letting 𝑦∶=𝑥/2 and 𝑔(𝑥)∶=𝑓(2𝑥)−8𝑓(𝑥) for all 𝑥∈𝑋 in (4.9), we get 𝜇_{𝑔(𝑥)−2𝑔(𝑥/2)}(𝑡)≥𝑇Ψ𝑥2,𝑥2,||4𝑡||,Ψ𝑥,𝑥2,𝑡(4.43) for all 𝑥∈𝑋 and 𝑡>0.
The rest of the proof is similar to the proof of Theorem 4.2.

Corollary 4.6. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦, and let (𝑌,𝜇,𝑇) be a non-Archimedean random Banach space over 𝒦 under a t-norm 𝑇∈ℋ. Let 𝑓∶𝑋→𝑌 be an odd and Ψ-approximately AQCQ mapping. If, for some 𝛼∈ℝ,𝛼>0, and some integer 𝑘,𝑘>1, with |2^𝑘|<𝛼, Ψ2^−𝑘𝑥,2^−𝑘𝑦,𝑡≥Ψ(𝑥,𝑦,𝛼𝑡)(𝑥∈𝑋,𝑡>0),(4.44) then there exists a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−8𝑓(𝑥/2)−𝐴(𝑥/2)}(𝑡)≥𝑇^∞𝑖=1𝑀𝑥,𝛼^𝑖+1𝑡||2||^𝑘𝑖(4.45) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Since lim_𝑛→∞𝑀𝑥,𝛼^𝑗𝑡||2||^𝑘𝑗=1(𝑥∈𝑋,𝑡>0)(4.46) and 𝑇 is of Hadžić type, from Proposition 2.1, it follows that lim_𝑛→∞𝑇^∞𝑗=𝑛𝑀𝑥,𝛼^𝑗𝑡||2||^𝑘𝑗=1(𝑥∈𝑋,𝑡>0).(4.47)Now, we can apply Theorem 4.5 to obtain the result.

Example 4.7. Let (𝑋,𝜇,𝑇_𝑀) non-Archimedean random normed space in which 𝜇_𝑥(𝑡)=𝑡𝑡+‖𝑥‖(𝑥∈𝑋,𝑡>0),(4.48)and let (𝑌,𝜇,𝑇_𝑀) be a complete non-Archimedean random normed space (see Example 3.2). Define Ψ(𝑥,𝑦,𝑡)=𝑡1+𝑡.(4.49) It is easy to see that (4.3) holds for 𝛼=1. Also, since 𝑀(𝑥,𝑡)=𝑡1+𝑡,(4.50) we have lim_𝑛→∞𝑇^{∞𝑀,𝑗=𝑛}𝑀𝑥,𝛼^𝑗𝑡||2||^𝑘𝑗=lim^𝑛→∞lim^𝑚→∞𝑇^{𝑚𝑀,𝑗=𝑛}𝑀𝑥,𝑡=lim_𝑛→∞||2||^𝑘𝑗lim_𝑚→∞=1(𝑥∈𝑋,𝑡𝑡+>𝑡||02)^𝑘||.^𝑛(4.51) Let 𝑓∶𝑋→𝑌 be an odd and Ψ-approximately AQCQ mapping. Thus, all the conditions of Theorem 4.2 hold, and so there exists a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−8𝑓(𝑥/2)−𝐴(𝑥/2)}(𝑡)≥𝑡𝑡+||2^𝑘||.(4.52)

5. Generalized Hyers-Ulam Stability of the Functional Equation (1.4) in Non-Archimedean Random Normed Spaces: An Even Case

Now, we prove the generalized Hyers-Ulam stability of the functional equation 𝐷𝑓(𝑥,𝑦)=0 in non-Archimedean Banach spaces, an even case.

Theorem 5.1. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦, and let (𝑌,𝜇,𝑇) be a non-Archimedean random Banach space over 𝒦. Let 𝑓∶𝑋→𝑌 be an even mapping, 𝑓(0)=0, and Ψ-approximately AQCQ mapping. If for some 𝛼∈ℝ, 𝛼>0, and some integer 𝑘, 𝑘>4 with |2^𝑘|<𝛼, Ψlim2^−𝑘_𝑛→∞𝑥,2^−𝑘𝑦,𝑡≥Ψ(𝑥,𝑦,𝛼𝑡)(𝑥∈𝑋,𝑡>0),T^∞𝑗=𝑛𝑀2𝑥,𝛼^𝑗𝑡||16||^𝑘𝑗=1(𝑥∈𝑋,𝑡>0),(5.1) then there exists a unique quartic mapping 𝑄∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−4𝑓(𝑥/2)−𝑄(𝑥/2)}(𝑡)≥T^∞𝑖=1𝑀𝑥,𝛼^𝑖+1𝑡||16||^𝑘𝑖(5.2) for all 𝑥∈𝑋 and 𝑡>0, where 𝑀(𝑥,𝑡)∶=𝑇^𝑘−1Ψ𝑥2,𝑥2,||4𝑡||,Ψ𝑥,𝑥2,𝑡,…,Ψ2^𝑘−1𝑥2,2^𝑘−1𝑥2,||4𝑡||,Ψ2^𝑘−1𝑥,2^𝑘−1𝑥(𝑥∈𝑋,𝑡>0).2,𝑡(5.3)

Proof. Letting 𝑥=𝑦 in (4.2), we get 𝜇_{𝑓(3𝑦)−6𝑓(2𝑦)+15𝑓(𝑦)}(𝑡)≥Ψ(𝑦,𝑦,𝑡)(5.4) for all 𝑦∈𝑋 and 𝑡>0. Replacing 𝑥 by 2𝑦 in (4.2), we get 𝜇_{𝑓(4𝑦)−4𝑓(3𝑦)+4𝑓(2𝑦)+4𝑓(𝑦)}(𝑡)≥Ψ(2𝑦,𝑦,𝑡)(5.5) for all 𝑦∈𝑋 and 𝑡>0. By (5.4) and (5.5), we have 𝜇_{𝑓(4𝑦)−20𝑓(2𝑦)+64𝑓(𝑦)}(𝑡)≥=𝑇𝑇𝜇𝜇_{4(𝑓(3𝑦)−4𝑓(2𝑦)+5𝑓(𝑦))𝑓(3𝑦)−4𝑓(2𝑦)+5𝑓(𝑦)}(𝑡),𝜇_{𝑓(4𝑦)−4𝑓(3𝑦)+6𝑓(2𝑦)−4𝑓(𝑦)}(𝑡)||4𝑡||,𝜇^{𝑓(4𝑦)−4𝑓(3𝑦)+6𝑓(2𝑦)−4𝑓(𝑦)}≥𝑇Ψ𝑦,𝑦,𝑡(𝑡)||4||,Ψ(2𝑦,𝑦,𝑡)(5.6) for all 𝑦∈𝑋 and 𝑡>0. Letting 𝑦∶=𝑥/2 and 𝑔(𝑥)∶=𝑓(2𝑥)−4𝑓(𝑥) for all 𝑥∈𝑋 in (5.6), we get 𝜇_{𝑔(𝑥)−16𝑔(𝑥/2)}(𝑡)≥𝑇Ψ𝑥2,𝑥2,||4𝑡||,Ψ𝑥,𝑥2,𝑡(5.7) for all 𝑥∈𝑋 and 𝑡>0.
The rest of the proof is similar to the proof of Theorem 4.2.

Corollary 5.2. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦, and let (𝑌,𝜇,𝑇) be a non-Archimedean random Banach space over 𝒦 under a t-norm 𝑇∈ℋ. Let 𝑓∶𝑋→𝑌 be an even, 𝑓(0)=0, and Ψ-approximately AQCQ mapping. If,   for some 𝛼∈ℝ, 𝛼>0, and some integer 𝑘, 𝑘>4, with |2^𝑘|<𝛼, Ψ2^−𝑘𝑥,2^−𝑘𝑦,𝑡≥Ψ(𝑥,𝑦,𝛼𝑡)(𝑥∈𝑋,𝑡>0),(5.8) then there exists a unique quartic mapping 𝑄∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−4𝑓(𝑥/2)−𝑄(𝑥/2)}(𝑡)≥T^∞𝑖=1𝑀𝑥,𝛼^𝑖+1𝑡||16||^𝑘𝑖(5.9) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Since lim_𝑛→∞𝑀𝑥,𝛼^𝑗𝑡||16||^𝑘𝑗=1(𝑥∈𝑋,𝑡>0)(5.10) and 𝑇 is of Hadžić type, from Proposition 2.1, it follows that lim_𝑛→∞𝑇^∞𝑗=𝑛𝑀𝑥,𝛼^𝑗𝑡||16||^𝑘𝑗=1(𝑥∈𝑋,𝑡>0).(5.11) Now, we can apply Theorem 5.1 to obtain the result.

Example 5.3. Let (𝑋,𝜇,𝑇_𝑀) be non-Archimedean random normed space in which 𝜇_𝑥(𝑡)=𝑡𝑡+‖𝑥‖(𝑥∈𝑋,𝑡>0).(5.12) And let (𝑌,𝜇,𝑇_𝑀) be a complete non-Archimedean random normed space (see Example 3.2). Define Ψ(𝑥,𝑦,𝑡)=𝑡1+𝑡.(5.13) It is easy to see that (4.3) holds for 𝛼=1. Also, since 𝑀(𝑥,𝑡)=𝑡1+𝑡,(5.14) we have lim_𝑛→∞𝑇^{∞𝑀,𝑗=𝑛}𝑀𝑥,𝛼^𝑗𝑡||16||^𝑘𝑗=lim^𝑛→∞lim^𝑚→∞𝑇^{𝑚𝑀,𝑗=𝑛}𝑀𝑥,𝑡=lim_𝑛→∞||16||^𝑘𝑗lim_𝑚→∞=1(𝑥∈𝑋,𝑡𝑡+>||01𝑡)6.^𝑘||^𝑛(5.15) Let 𝑓∶𝑋→𝑌 be an even, 𝑓(0)=0, and Ψ-approximately AQCQ mapping. Thus all the conditions of Theorem 5.1 hold, and so there exists a unique quartic mapping 𝑄∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−4𝑓(𝑥/2)−𝑄(𝑥/2)}(𝑡)≥𝑡𝑡+||16^𝑘||.(5.16)

Theorem 5.4. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦 and let (𝑌,𝜇,𝑇) be a non-Archimedean random Banach space over 𝒦. Let 𝑓∶𝑋→𝑌 be an even mapping, 𝑓(0)=0 and Ψ-approximately AQCQ mapping. If for some 𝛼∈ℝ, 𝛼>0, and some integer 𝑘, 𝑘>2 with |2^𝑘|<𝛼, Ψlim2^−𝑘_𝑛→∞𝑥,2^−𝑘𝑦,𝑡≥Ψ(𝑥,𝑦,𝛼𝑡)(𝑥∈𝑋,𝑡>0),T^∞𝑗=𝑛𝑀2𝑥,𝛼^𝑗𝑡||4||^𝑘𝑗=1(𝑥∈𝑋,𝑡>0),(5.17) then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−16𝑓(𝑥/2)−𝑄(𝑥/2)}(𝑡)≥T^∞𝑖=1𝑀𝑥,𝛼^𝑖+1𝑡||4||^𝑘𝑖(5.18) for all 𝑥∈𝑋 and 𝑡>0, where 𝑀(𝑥,𝑡)∶=𝑇^𝑘−1Ψ𝑥2,𝑥2,||4𝑡||,Ψ𝑥,𝑥2,𝑡,…,Ψ2^𝑘−1𝑥2,2^𝑘−1𝑥2,||4𝑡||,Ψ2^𝑘−1𝑥,2^𝑘−1𝑥(𝑥∈𝑋,𝑡>0).2,𝑡(5.19)

Proof. Letting 𝑦∶=𝑥/2 and 𝑔(𝑥)∶=𝑓(2𝑥)−16𝑓(𝑥) for all 𝑥∈𝑋 in (5.6), we get 𝜇_{𝑔(𝑥)−4𝑔(𝑥/2)}(𝑡)≥𝑇Ψ𝑥2,𝑥2,||4𝑡||,Ψ𝑥,𝑥2,𝑡(5.20) for all 𝑥∈𝑋 and 𝑡>0.
The rest of the proof is similar to the proof of Theorem 5.1.

Corollary 5.5. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦, and let (𝑌,𝜇,𝑇) be a non-Archimedean random Banach space over 𝒦 under a t-norm 𝑇∈ℋ. Let 𝑓∶𝑋→𝑌 be an even, 𝑓(0)=0, and Ψ-approximately AQCQ mapping. If,   for some 𝛼∈ℝ, 𝛼>0, and some integer 𝑘, 𝑘>2, with |2^𝑘|<𝛼, Ψ2^−𝑘𝑥,2^−𝑘𝑦,𝑡≥Ψ(𝑥,𝑦,𝛼𝑡)(𝑥∈𝑋,𝑡>0),(5.21) then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−16𝑓(𝑥/2)−𝑄(𝑥/2)}(𝑡)≥T^∞𝑖=1𝑀𝑥,𝛼^𝑖+1𝑡||4||^𝑘𝑖(5.22) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Since lim_𝑛→∞𝑀𝑥,𝛼^𝑗𝑡||4||^𝑘𝑗=1(𝑥∈𝑋,𝑡>0)(5.23) and 𝑇 is of Hadžić type, from Proposition 2.1, it follows that lim_𝑛→∞𝑇^∞𝑗=𝑛𝑀𝑥,𝛼^𝑗𝑡||4||^𝑘𝑗=1(𝑥∈𝑋,𝑡>0).(5.24)Now, we can apply Theorem 5.4 to obtain the result.

Example 5.6. Let (𝑋,𝜇,𝑇_𝑀) be a non-Archimedean random normed space in which 𝜇_𝑥(𝑡)=𝑡𝑡+‖𝑥‖(𝑥∈𝑋,𝑡>0).(5.25) And let (𝑌,𝜇,𝑇_𝑀) be a complete non-Archimedean random normed space (see Example 3.2). Define Ψ(𝑥,𝑦,𝑡)=𝑡1+𝑡.(5.26) It is easy to see that (4.3) holds for 𝛼=1. Also, since 𝑀(𝑥,𝑡)=𝑡1+𝑡,(5.27) we have lim_𝑛→∞𝑇^{∞𝑀,𝑗=𝑛}𝑀𝑥,𝛼^𝑗𝑡||4||^𝑘𝑗=lim^𝑛→∞lim^𝑚→∞𝑇^{𝑚𝑀,𝑗=𝑛}𝑀𝑥,𝑡=lim_𝑛→∞||4||^𝑘𝑗lim_𝑚→∞=1(𝑥∈𝑋,𝑡𝑡+>𝑡||04)^𝑘||.^𝑛(5.28) Let 𝑓∶𝑋→𝑌 be an even, 𝑓(0)=0, and Ψ-approximately AQCQ mapping. Thus, all the conditions of Theorem 5.4 hold, and so there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−16𝑓(𝑥/2)−𝑄(𝑥/2)}(𝑡)≥𝑡𝑡+||4^𝑘||.(5.29)

6. Latticetic Random Normed Space

Let ℒ=(𝐿,≥_𝐿) be a complete lattice, that is, a partially ordered set in which every nonempty subset admits supremum and infimum, and 0_ℒ=inf𝐿, 1_ℒ=sup𝐿. The space of latticetic random distribution functions, denoted by Δ^+𝐿, is defined as the set of all mappings 𝐹∶ℝ∪{−∞,+∞}→𝐿 such that 𝐹 is left continuous and nondecreasing on ℝ, 𝐹(0)=0_ℒ, 𝐹(+∞)=1_ℒ.

𝐷^+𝐿⊆Δ^+𝐿 is defined as 𝐷^+𝐿={𝐹∈Δ^+𝐿∶𝑙⁻𝐹(+∞)=1_ℒ}, where 𝑙⁻𝑓(𝑥) denotes the left limit of the function 𝑓 at the point 𝑥. The space Δ^+𝐿 is partially ordered by the usual pointwise ordering of functions, that is, 𝐹≥𝐺 if and only if 𝐹(𝑡)≥_𝐿𝐺(𝑡) for all 𝑡 in ℝ. The maximal element for Δ^+𝐿 in this order is the distribution function given by𝜀₀(𝑡)=0^ℒ1,if𝑡≤0,_ℒ,if𝑡>0.(6.1)

In Section 2, we defined 𝑡-norms on [0,1], and now we extend 𝑡-norms on a complete lattice.

Definition 6.1 (see [42]). A triangular norm (𝑡-norm) on 𝐿 is a mapping 𝒯∶(𝐿)²→𝐿 satisfying the following conditions:(a)(forall𝑥∈𝐿)(𝒯(𝑥,1_ℒ)=𝑥) (boundary condition);(b)(forall(𝑥,𝑦)∈(𝐿)²)(𝒯(𝑥,𝑦)=𝒯(𝑦,𝑥)) (commutativity);(c)(forall(𝑥,𝑦,𝑧)∈(𝐿)³)(𝒯(𝑥,𝒯(𝑦,𝑧))=𝒯(𝒯(𝑥,𝑦),𝑧)) (associativity);(d)(forall(𝑥,𝑥^,𝑦,𝑦^)∈(𝐿)⁴)(𝑥≤_𝐿𝑥^and𝑦≤_𝐿𝑦^⇒𝒯(𝑥,𝑦)≤_𝐿𝒯(𝑥^,𝑦^)) (monotonicity).

Let {𝑥_𝑛} be a sequence in 𝐿 converges to 𝑥∈𝐿 (equipped order topology). The 𝑡-norm 𝒯 is said to be a continuous 𝑡-norm if lim_𝑛→∞𝒯𝑥_𝑛,𝑦=𝒯(𝑥,𝑦)(6.2) for all 𝑦∈𝐿.

A 𝑡-norm 𝒯 can be extended (by associativity) in a unique way to an 𝑛-array operation taking for (𝑥₁,…,𝑥_𝑛)∈𝐿^𝑛 the value 𝒯(𝑥₁,…,𝑥_𝑛) defined by 𝒯^0𝑖=1𝑥_𝑖=1,𝒯^𝑛𝑖=1𝑥_𝑖=𝒯𝒯^𝑛−1_i=1𝑥_𝑖,𝑥_𝑛=𝒯𝑥₁,…,𝑥_𝑛.(6.3)

𝒯 can also be extended to a countable operation taking for any sequence (𝑥_𝑛)_𝑛∈𝑁 in 𝐿 the value𝒯^∞𝑖=1𝑥_𝑖=lim_𝑛→∞𝒯^𝑛𝑖=1𝑥_𝑖.(6.4) The limit on the right side of (6.4) exists since the sequence (𝒯^𝑛𝑖=1𝑥_𝑖)_𝑛∈ℕ is nonincreasing and bounded from below.

Note that we put 𝒯=𝑇 whenever 𝐿=[0,1]. If 𝑇 is a 𝑡-norm, then 𝑥^𝑇(𝑛) is defined for every 𝑥∈[0,1] and 𝑛∈𝑁∪{0} by 1 if 𝑛=0 and 𝑇(𝑥^{𝑇(𝑛−1)},𝑥) if 𝑛≥1. A 𝑡-norm 𝑇 is said to be of Hadžić type, (we denote by 𝑇∈ℋ) if the family (𝑥^𝑇(𝑛))_𝑛∈𝑁 is equicontinuous at 𝑥=1 (cf. [27]).

Definition 6.2 (see [42]). A continuous 𝑡-norm 𝒯 on 𝐿=[0,1]² is said to be continuous 𝑡–representable if there exist a continuous 𝑡-norm ∗ and a continuous 𝑡-conorm ⋄ on [0,1] such that, for all 𝑥=(𝑥₁,𝑥₂), 𝑦=(𝑦₁,𝑦₂)∈𝐿, 𝒯(𝑥,𝑦)=𝑥₁∗𝑦₁,𝑥₂⋄𝑦₂.(6.5)

For example, 𝐌𝒯(𝑎,(𝑎𝑏,)𝑏=)=mi𝑎n₁𝑏₁𝑎,₁min𝑎₂+𝑏₂,1,,𝑏₁,max𝑎₂,𝑏₂(6.6) for all 𝑎=(𝑎₁,𝑎₂), 𝑏=(𝑏₁,𝑏₂)∈[0,1]² are continuous 𝑡-representable. Define the mapping 𝒯_∧ from 𝐿² to 𝐿 by 𝒯_∧(𝑥,𝑦)=𝑥,if𝑦≥^𝐿𝑦,if𝑥≥_𝐿𝑥,𝑦.(6.7)

Recall (see [27, 28]) that if {𝑥_𝑛} is a given sequence in 𝐿, (𝒯_∧)^𝑛𝑖=1𝑥_𝑖 is defined recurrently by (𝒯_∧)^1𝑖=1𝑥_𝑖=𝑥₁ and (𝒯_∧)^𝑛𝑖=1𝑥_𝑖=𝒯_∧((𝒯_∧)^{𝑛−1𝑖=1}𝑥_𝑖,𝑥_𝑛) for all 𝑛≥2.

A negation on ℒ is any decreasing mapping 𝒩∶𝐿→𝐿 satisfying 𝒩(0_ℒ)=1_ℒ and 𝒩(1_ℒ)=0_ℒ. If 𝒩(𝒩(𝑥))=𝑥, for all 𝑥∈𝐿, then 𝒩 is called an involutive negation. In the following, ℒ is endowed with a (fixed) negation 𝒩.

Definition 6.3. A latticetic random normed space (in short LRN-space) is a triple (𝑋,𝜇,𝒯_∧), where 𝑋 is a vector space and 𝜇 is a mapping from 𝑋 into 𝐷^+𝐿 such that the following conditions hold: (LRN1) 𝜇_𝑥(𝑡)=𝜀₀(𝑡) for all 𝑡>0 if and only if 𝑥=0, (LRN2) 𝜇_𝛼𝑥(𝑡)=𝜇_𝑥(𝑡/|𝛼|) for all 𝑥 in 𝑋, 𝛼≠0 and 𝑡≥0, (LRN3) 𝜇_𝑥+𝑦(𝑡+𝑠)≥_𝐿𝒯_∧(𝜇_𝑥(𝑡),𝜇_𝑦(𝑠)) for all 𝑥,𝑦∈𝑋 and 𝑡,𝑠≥0.

We note that from (LPN2) it follows that 𝜇_−𝑥(𝑡)=𝜇_𝑥(𝑡) for all 𝑥∈𝑋 and 𝑡≥0.

Example 6.4. Let 𝐿=[0,1]×[0,1] and operation ≤_𝐿 be defined by 𝑎𝐿=𝑎₁,𝑎₂∶𝑎₁,𝑎₂∈[0,1]×[0,1],𝑎₁+𝑎₂≤1₁,,𝑎₂≤_𝐿𝑏₁,𝑏₂⟺𝑎₁≤𝑏₁,𝑎₂≥𝑏₂,∀𝑎=𝑎₁,𝑎₂,𝑏=𝑏₁,𝑏₂∈𝐿.(6.8) then (𝐿,≤_𝐿) is a complete lattice (see [42]). In this complete lattice, we denote its units by 0_𝐿=(0,1) and 1_𝐿=(1,0). Let (𝑋,‖⋅‖) be a normed space. Let 𝒯(𝑎,𝑏)=(min{𝑎₁,𝑏₁},max{𝑎₂,𝑏₂}) for all 𝑎=(𝑎₁,𝑎₂), 𝑏=(𝑏₁,𝑏₂)∈[0,1]×[0,1] and 𝜇 be a mapping defined by 𝜇_𝑥(𝑡)=𝑡𝑡+‖𝑥‖,𝑡+‖𝑥‖‖𝑥‖𝑡∈ℝ⁺,(6.9) then (𝑋,𝜇,𝒯) is a latticetic random normed spaces.
If (𝑋,𝜇,𝒯_∧) is a latticetic random normed space, then 𝒱=𝑉(𝜀,𝜆)∶𝜀>_𝐿0_ℒ,𝜆∈𝐿⧵0_ℒ,1_ℒ,𝑉(𝜀,𝜆)=𝑥∈𝑋∶𝐹_𝑥(𝜀)>_𝐿𝒩(𝜆),(6.10) is a complete system of neighborhoods of null vector for a linear topology on 𝑋 generated by the norm 𝐹.

Definition 6.5. Let (𝑋,𝜇,𝒯_∧) be a latticetic random normed spaces.(1)A sequence {𝑥_𝑛} in 𝑋 is said to be convergent to 𝑥 in 𝑋 if, for every 𝑡>0 and 𝜀∈𝐿⧵{0_ℒ}, there exists a positive integer 𝑁 such that 𝜇_{𝑥𝑛−𝑥}(𝑡)>_𝐿𝒩(𝜀) whenever 𝑛≥𝑁.(2)A sequence {𝑥_𝑛} in 𝑋 is called a Cauchy sequence if, for every 𝑡>0 and 𝜀∈𝐿⧵{0_ℒ}, there exists a positive integer 𝑁 such that 𝜇_{𝑥𝑛−𝑥𝑚}(𝑡)>_𝐿𝒩(𝜀) whenever 𝑛≥𝑚≥𝑁.(3)A latticetic random normed spaces (𝑋,𝜇,𝒯_∧) is said to be complete if and only if every Cauchy sequence in 𝑋 is convergent to a point in 𝑋.

Theorem 6.6. If (𝑋,𝜇,𝒯_∧) is a latticetic random normed space and {𝑥_𝑛} is a sequence such that 𝑥_𝑛→𝑥, then lim_𝑛→∞𝜇_𝑥𝑛(𝑡)=𝜇_𝑥(𝑡).

Proof. The proof is the same as classical random normed spaces, see [25].

7. Generalized Hyers-Ulam Stability of the Functional Equation (1.4): An Odd Case via Fixed-Point Method

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation 𝐷𝑓(𝑥,𝑦)=0 in random Banach spaces: an odd case.

Theorem 7.1. Let 𝑋 be a linear space, let (𝑌,𝜇,𝒯_∧) be a complete LRN-space, and Φ let be a mapping from 𝑋² to 𝐷^+𝐿 (Φ(𝑥,𝑦) is denoted by Φ_𝑥,𝑦) such that, for some 0<𝛼<1/8, Φ_2𝑥,2𝑦(𝑡)≤_𝐿Φ_𝑥,𝑦(𝛼𝑡)(𝑥,𝑦∈𝑋,𝑡>0).(7.1) Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying 𝜇_{𝐷𝑓(𝑥,𝑦)}(𝑡)≥_𝐿Φ_𝑥,𝑦(𝑡)(7.2) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then 𝐶(𝑥)∶=lim_𝑛→∞8^𝑛𝑓2𝑥^𝑛−1−2𝑓2𝑥^𝑛(7.3) exists for each 𝑥∈𝑋 and defines a cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−2𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥_𝐿𝒯_∧Φ^𝑥,𝑥1−8𝛼5𝛼𝑡,Φ^2𝑥,𝑥1−5𝛼8𝛼𝑡(7.4) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Letting 𝑥=𝑦 in (7.2), we get 𝜇_{𝑓(3𝑦)−4𝑓(2𝑦)+5𝑓(𝑦)}(𝑡)≥_𝐿Φ_𝑦,𝑦(𝑡)(7.5) for all 𝑦∈𝑋 and 𝑡>0. Replacing 𝑥 by 2𝑦 in (7.2), we get 𝜇_{𝑓(4𝑦)−4𝑓(3𝑦)+6𝑓(2𝑦)−4𝑓(𝑦)}(𝑡)≥_𝐿Φ_2𝑦,𝑦(𝑡)(7.6) for all 𝑦∈𝑋 and 𝑡>0. By (7.5) and (7.6), 𝜇_{𝑓(4𝑦)−10𝑓(2𝑦)+16𝑓(𝑦)}(5𝑡)≥=_𝐿𝒯𝒯_∧∧𝜇_{4(𝑓(3𝑦)−4𝑓(2𝑦)+5𝑓(𝑦))}(4𝑡),𝜇_{𝑓(4𝑦)−4𝑓(3𝑦)+6𝑓(2𝑦)−4𝑓(𝑦)}(𝑡)≥𝜇_{𝑓(3𝑦)−4𝑓(2𝑦)+5𝑓(𝑦)}(𝑡),𝜇_{𝑓(4𝑦)−4𝑓(3𝑦)+6𝑓(2𝑦)−4𝑓(𝑦)}(𝑡)_𝐿𝒯_∧Φ_𝑦,𝑦(𝑡),Φ_2𝑦,𝑦(𝑡)(7.7) for all 𝑦∈𝑋 and 𝑡>0. Letting 𝑦∶=𝑥/2 and 𝑔(𝑥)∶=𝑓(2𝑥)−2𝑓(𝑥) for all 𝑥∈𝑋, we get 𝜇_{𝑔(𝑥)−8𝑔(𝑥/2)}(5𝑡)≥_𝐿𝒯_∧Φ_{𝑥/2,𝑥/2}(𝑡),Φ_𝑥,𝑥/2(𝑡)(7.8) for all 𝑥∈𝑋 and 𝑡>0.  
Consider the set 𝑆∶={ℎ∶𝑋⟶𝑌,ℎ(0)=0}(7.9) and introduce the generalized metric on 𝑆: 𝑑(ℎ,𝑘)=inf𝑢∈ℝ⁺∶𝜇_{ℎ(𝑥)−𝑘(𝑥)}(𝑢𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡),∀𝑥∈𝑋,∀𝑡>0(7.10) where, as usual, inf∅=+∞. It is easy to show that (𝑆,𝑑) is complete (see the proof of Lemma  2.1 of [24]).
Now, we consider the linear mapping 𝐽∶𝑆→𝑆 such that 𝐽ℎ(𝑥)∶=8ℎ𝑥2(7.11) for all 𝑥∈𝑋, and we prove that 𝐽 is a strictly contractive mapping with the Lipschitz constant 8𝛼.
Let ℎ,𝑘∈𝑆 be given such that 𝑑(ℎ,𝑘)<𝜀. Then 𝜇_{ℎ(𝑥)−𝑘(𝑥)}(𝜀𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.12) for all 𝑥∈𝑋 and 𝑡>0. Hence 𝜇_{𝐽ℎ(𝑥)−𝐽𝑘(𝑥)}(8𝛼𝜀𝑡)==𝜇𝜇_{8ℎ(𝑥/2)−8𝑘(𝑥/2)ℎ(𝑥/2)−𝑘(𝑥/2)}(8𝛼𝜀𝑡)≥𝒯_∧(𝛼𝜀𝑡)≥Φ_{𝑥/2,𝑥/2}(𝛼𝑡),Φ_𝑥,𝑥/2(𝛼𝑡)_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.13) for all 𝑥∈𝑋 and 𝑡>0. So, 𝑑(ℎ,𝑘)<𝜀 implies that 𝑑(𝐽ℎ,𝐽𝑘)≤𝛼8𝜀.(7.14) This means that 𝑑(𝐽ℎ,𝐽𝑘)≤𝛼8𝑑(ℎ,𝑘)(7.15) for all ℎ,𝑘∈𝑆. It follows from (7.8) that 𝜇_{𝑔(𝑥)−8𝑔(𝑥/2)}(5𝛼𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.16) for all 𝑥∈𝑋 and 𝑡>0. So, 𝑑(𝑔,𝐽𝑔)≤5𝛼≤5/8.
By Theorem 1.1, there exists a mapping 𝐶∶𝑋→𝑌 satisfying the following:(1)𝐶 is a fixed point of 𝐽, that is, 𝐶𝑥2=18𝐶(𝑥)(7.17) for all 𝑥∈𝑋. Since 𝑔∶𝑋→𝑌 is odd, 𝐶∶𝑋→𝑌 is an odd mapping. The mapping 𝐶 is a unique fixed point of 𝐽 in the set 𝑀={ℎ∈𝑆∶𝑑(ℎ,𝑔)<∞}.(7.18) This implies that 𝐶 is a unique mapping satisfying (7.17) such that there exists a 𝑢∈(0,∞) satisfying 𝜇_{𝑔(𝑥)−𝐶(𝑥)}(𝑢𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.19) for all 𝑥∈𝑋 and 𝑡>0.(2)𝑑(𝐽^𝑛𝑔,𝐶)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞8^𝑛𝑔𝑥2^𝑛=𝐶(𝑥)(7.20) for all 𝑥∈𝑋.(3)𝑑(ℎ,𝐶)≤(1/(1−8𝛼))𝑑(ℎ,𝐽ℎ) with ℎ∈𝑀, which implies the inequality 𝑑(𝑔,𝐶)≤5𝛼1−8𝛼,(7.21) from which it follows that 𝜇_{𝑔(𝑥)−𝐶(𝑥)}5𝛼1−8𝛼𝑡≥^𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡).(7.22) This implies that the inequality (7.4) holds. From 𝐷𝑔(𝑥,𝑦)=𝐷𝑓(2𝑥,2𝑦)−2𝐷𝑓(𝑥,𝑦), by (7.2), we deduce that 𝜇𝜇_{𝐷𝑓(2𝑥,2𝑦)}(𝑡)≥_𝐿Φ_2𝑥,2𝑦(𝑡),_{−2𝐷𝑓(𝑥,𝑦)}(𝑡)=𝜇_{𝐷𝑓(𝑥,𝑦)}𝑡2≥^𝐿Φ_𝑥,𝑦𝑡2(7.23) and so, by (LRN3) and (7.1), we obtain 𝜇_{𝐷𝑔(𝑥,𝑦)}(3𝑡)≥≥_𝐿𝒯_∧𝜇_{𝐷𝑓2𝑥,2𝑦}(𝑡),𝜇_{−2𝐷𝑓(𝑥,𝑦)}(2𝑡)_𝐿𝒯_∧Φ_2𝑥,2𝑦(𝑡),Φ_𝑥,𝑦(𝑡)≥_𝐿Φ_2𝑥,2𝑦(𝑡).(7.24) It follows that 𝜇₈^𝑛_{𝐷𝑔(𝑥/2}^𝑛_,𝑦/2^𝑛₎(3𝑡)=𝜇_{𝐷𝑔(𝑥/2}^𝑛_,𝑦/2^𝑛₎3𝑡≥Φ_{𝑥/2𝑛−1,𝑦/2𝑛−1}8^𝑛8𝑡^𝑛≥^𝐿⋯≥_𝐿Φ_𝑥,𝑦18(8𝛼𝑡)^𝑛−1(7.25) for all 𝑥,𝑦∈𝑋, 𝑡>0 and 𝑛∈ℕ.
Since lim_𝑛→∞Φ_𝑥,𝑦((3/8)(𝑡/(8𝛼)^𝑛−1))=1 for all 𝑥,𝑦∈𝑋 and 𝑡>0, by Theorem 2.4, we deduce that 𝜇_{𝐷𝐶(𝑥,𝑦)}(3𝑡)=1_ℒ(7.26) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Thus the mapping 𝐶∶𝑋→𝑌 satisfies (1.4).
Now, we have 𝐶(2𝑥)−8𝐶(𝑥)=lim_𝑛→∞8^𝑛𝑔𝑥2^𝑛−1−8^𝑛+1𝑔𝑥=8lim_𝑛→∞2^𝑛8^𝑛−1𝑔𝑥2^𝑛−1−8^𝑛𝑔𝑥2^𝑛=0(7.27) for all 𝑥∈𝑋. Since the mapping 𝑥→𝐶(2𝑥)−2𝐶(𝑥) is cubic (see Lemma  2.2 of [14]), from the equality 𝐶(2𝑥)=8𝐶(𝑥), we deduce that the mapping 𝐶∶𝑋→𝑌 is cubic.

Corollary 7.2. Let 𝜃≥0 and let 𝑝 be a real number with 𝑝>3. Let 𝑋 be a normed vector space with norm ‖⋅‖. Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying 𝜇_{𝐷𝑓(𝑥,𝑦)}(𝑡)≥𝑡𝑡+𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(7.28) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Note that (𝑋,𝜇,𝑇_𝑀) is a complete LRN-space, in which 𝐿=[0,1], then 𝐶(𝑥)∶=lim_𝑛→∞8^𝑛𝑓2𝑥^𝑛−1−2𝑓2𝑥^𝑛(7.29) exists for each 𝑥∈𝑋 and defines a cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−2𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥(2^𝑝−8)𝑡(2^𝑝−8)𝑡+5(1+2^𝑝)𝜃‖𝑥‖^𝑝(7.30) for all 𝑥∈𝑋 and 𝑡>0.

Proof. The proof follows from Theorem 7.1 by taking Φ_𝑥,𝑦(𝑡)∶=𝑡𝑡+𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(7.31) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then we can choose 𝛼=2^−𝑝, and we get 𝜇_{𝑓(2𝑥)−2𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥min1−2^3−𝑝𝑡1−2^3−𝑝𝑡+5⋅2^−𝑝𝜃2‖𝑥‖^𝑝,1−2^3−𝑝𝑡≥1−2_3−𝑝1−2^3−𝑝𝑡+5⋅2^−𝑝𝜃‖2𝑥‖^𝑝+‖𝑥‖^𝑝𝑡=1−2^3−𝑝(𝑡2+_𝑝5⋅2^−𝑝𝜃‖2𝑥‖^𝑝+‖𝑥‖^𝑝−8)𝑡(2^𝑝−8)𝑡+5⋅(2^𝑝+1)𝜃‖𝑥‖^𝑝,(7.32) which is the desired result.

Theorem 7.3. Let 𝑋 be a linear space, let (𝑌,𝜇,𝒯_∧) be a complete LRN-space, and let Φ be a mapping from 𝑋² to 𝐷^+𝐿 (Φ(𝑥,𝑦) is denoted by Φ_𝑥,𝑦) such that, for some 0<𝛼<8, Φ_{𝑥/2,𝑦/2}(𝑡)≤_𝐿Φ_𝑥,𝑦(𝛼𝑡)(𝑥,𝑦∈𝑋,𝑡>0).(7.33) Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying (7.2), then 𝐶(𝑥)∶=lim_𝑛→∞18^𝑛𝑓2^𝑛+1𝑥−2𝑓(2^𝑛𝑥)(7.34) exists for each 𝑥∈𝑋 and defines a cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−2𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥_𝐿𝒯_∧Φ^𝑥,𝑥8−𝛼5𝑡,Φ^2𝑥,𝑥8−5𝛼𝑡(7.35) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let (𝑆,𝑑) be the generalized metric space defined in the proof of Theorem 7.1.  
Consider the linear mapping 𝐽∶𝑆→𝑆 such that 𝐽ℎ(𝑥)∶=18ℎ(2𝑥)(7.36) for all 𝑥∈𝑋, and we prove that 𝐽 is a strictly contractive mapping with the Lipschitz constant 𝛼/8.
Let ℎ,𝑘∈𝑆 be given such that 𝑑(ℎ,𝑘)<𝜀, then 𝜇_{ℎ(𝑥)−𝑘(𝑥)}(𝜀𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.37) for all 𝑥∈𝑋 and 𝑡>0. Hence 𝜇_{𝐽ℎ(𝑥)−𝐽𝑘(𝑥)}𝛼8𝜀𝑡==𝜇𝜇^{(1/8)ℎ(2𝑥)−(1/8)𝑘(2𝑥)}_{ℎ(2𝑥)−𝑘(2𝑥)}𝛼8𝜀𝑡≥(𝛼𝜀𝑡)_𝐿≥𝒯𝒯_∧∧Φ_2𝑥,2𝑥(𝛼𝑡),Φ_4𝑥,2𝑥(𝛼𝑡)Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.38) for all 𝑥∈𝑋 and 𝑡>0. So, 𝑑(ℎ,𝑘)<𝜀 implies that 𝑑(𝐽ℎ,𝐽𝑘)≤𝛼8𝜀.(7.39)
This means that 𝑑(𝐽ℎ,𝐽𝑘)≤𝛼8𝑑(ℎ,𝑘)(7.40) for all 𝑔,ℎ∈𝑆. Letting 𝑔(𝑥)∶=𝑓(2𝑥)−2𝑓(𝑥) for all 𝑥∈𝑋, from (7.8), we get that 𝜇_{𝑔(𝑥)−(1/8)𝑔(2𝑥)}58𝑡≥^𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.41) for all 𝑥∈𝑋 and 𝑡>0. So, 𝑑(𝑔,𝐽𝑔)≤5/8.
By Theorem 1.1, there exists a mapping 𝐶∶𝑋→𝑌 satisfying the following:(1)𝐶 is a fixed point of 𝐽, that is, 𝐶(2𝑥)=8𝐶(𝑥)(7.42) for all 𝑥∈𝑋. Since 𝑔∶𝑋→𝑌 is odd, 𝐶∶𝑋→𝑌 is an odd mapping. The mapping 𝐶 is a unique fixed point of 𝐽 in the set 𝑀={ℎ∈𝑆∶𝑑(ℎ,𝑔)<∞}.(7.43) This implies that 𝐶 is a unique mapping satisfying (7.42) such that there exists a 𝑢∈(0,∞) satisfying 𝜇_{𝑔(𝑥)−𝐶(𝑥)}(𝑢𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.44) for all 𝑥∈𝑋 and 𝑡>0.(2)𝑑(𝐽^𝑛𝑔,𝐶)→0 as 𝑛→∞. This implies the equalit lim_𝑛→∞18^𝑛𝑔(2^𝑛𝑥)=𝐶(𝑥)(7.45) for all 𝑥∈𝑋.(3)𝑑(ℎ,𝐶)≤(1/(1−𝛼/8))𝑑(ℎ,𝐽ℎ) for every ℎ∈𝑀, which implies the inequality 𝑑(𝑔,𝐶)≤58−𝛼,(7.46) from which it follows that 𝜇_{𝑔(𝑥)−𝐶(𝑥)}58−𝛼𝑡≥^𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.47) for all 𝑥∈𝑋 and 𝑡>0. This implies that the inequality (7.35) holds.
From 𝜇_{𝐷𝑔(𝑥,𝑦)}(3𝑡)≥_𝐿𝒯_∧Φ_2𝑥,2𝑦(𝑡),Φ_𝑥,𝑦(𝑡)≥_𝐿𝒯_∧Φ^2𝑥,2𝑦(𝑡),Φ_𝑥,𝑦𝑡8,(7.48) by (7.33), we deduce that 𝜇₈^−𝑛_𝐷𝑔(2^𝑛_𝑥,2^𝑛_𝑦)(3𝑡)=𝜇_𝐷𝑔(2^𝑛_𝑥,2^𝑛_𝑦)(3⋅8^𝑛𝑡)≥_𝐿Φ₂^𝑛_𝑥,2^𝑛_𝑦8^𝑛−1𝑡≥_𝐿⋯≥Φ_𝑥,𝑦𝛼8^𝑛−1𝑡𝛼(7.49) for all 𝑥,𝑦∈𝑋, 𝑡>0, and 𝑛∈ℕ. As 𝑛→∞, we deduce that 𝜇_{𝐷𝐶(𝑥,y)}(3𝑡)=1_ℒ(7.50) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Thus the mapping 𝐶∶𝑋→𝑌 satisfies (1.4).
Now, we have 𝐶(2𝑥)−8𝐶(𝑥)=lim_𝑛→∞81^𝑛𝑔2^𝑛+1𝑥−18^𝑛−1=8lim_𝑛→∞𝑔(2^𝑛𝑥)81^𝑛+1𝑔2^𝑛+1𝑥−18^𝑛𝑔(2^𝑛𝑥)=0(7.51) for all 𝑥∈𝑋. Since the mapping 𝑥→𝐶(2𝑥)−2𝐶(𝑥) is cubic (see Lemma  2.2 of [14]), from the equality 𝐶(2𝑥)=8𝐶(𝑥), we deduce that the mapping 𝐶∶𝑋→𝑌 is cubic.

Corollary 7.4. Let 𝜃≥0 and let 𝑝 be a real number with 0<𝑝<3. Let 𝑋 be a normed vector space with norm ‖⋅‖. Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying (7.28), then 𝐶(𝑥)∶=lim_𝑛→∞18^𝑛𝑓2^𝑛+1𝑥−2𝑓(2^𝑛𝑥)(7.52) exists for each 𝑥∈𝑋 and defines a cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−2𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥(8−2^𝑝)𝑡(8−2^𝑝)𝑡+5(1+2^𝑝)𝜃‖𝑥‖^𝑝(7.53) for all 𝑥∈𝑋 and 𝑡>0. Note that (𝑋,𝜇,𝑇_𝑀) is a complete LRN-space, in which 𝐿=[0,1].

Proof. The proof follows from Theorem 7.3 by taking 𝜇_{𝐷𝑓(𝑥,𝑦)}(𝑡)≥𝑡𝑡+𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(7.54) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then we can choose 𝛼=2^𝑝, and we get the desired result.

Theorem 7.5. Let 𝑋 be a linear space, let (𝑌,𝜇,𝒯_∧) be a complete LRN-space, and let Φ be a mapping from 𝑋² to 𝐷^+𝐿 (Φ(𝑥,𝑦) is denoted by Φ_𝑥,𝑦) such that, for some 0<𝛼<1/2, Φ_2𝑥,2𝑦(𝑡)≤_𝐿Φ_𝑥,𝑦(𝛼𝑡)(𝑥,𝑦∈𝑋,𝑡>0).(7.55) Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying (7.2), then 𝐴(𝑥)∶=lim_𝑛→∞2^𝑛𝑓2𝑥^𝑛−1−8𝑓2𝑥^𝑛(7.56) exists for each 𝑥∈𝑋 and defines an additive mapping 𝐴∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−8𝑓(𝑥)−𝐴(𝑥)}(𝑡)≥_𝐿𝒯_∧Φ^𝑥,𝑥1−2𝛼5𝛼𝑡,Φ^2𝑥,𝑥1−5𝛼2𝛼𝑡(7.57) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let (𝑆,𝑑) be the generalized metric space defined in the proof of Theorem 7.1.
Letting 𝑦∶=𝑥/2 and 𝑔(𝑥)∶=𝑓(2𝑥)−8𝑓(𝑥) for all 𝑥∈𝑋 in (7.7), we get 𝜇_{𝑔(𝑥)−2𝑔(𝑥/2)}(5𝑡)≥_𝐿𝒯_∧Φ_{𝑥/2,𝑥/2}(𝑡),Φ_𝑥,𝑥/2(𝑡)(7.58) for all 𝑥∈𝑋 and 𝑡>0.
Now, we consider the linear mapping 𝐽∶𝑆→𝑆 such that 𝐽ℎ(𝑥)∶=2ℎ𝑥2(7.59) for all 𝑥∈𝑋. It is easy to see that 𝐽 is a strictly contractive self-mapping on 𝑆 with the Lipschitz constant 2𝛼.  
It follows from (7.58) and (7.55) that 𝜇_{𝑔(𝑥)−2𝑔(𝑥/2)}(5𝛼𝑡)≥𝑇_𝑀Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.60) for all 𝑥∈𝑋 and 𝑡>0. So, 𝑑(𝑔,𝐽𝑔)≤5𝛼<∞.
By Theorem 1.1, there exists a mapping 𝐴∶𝑋→𝑌 satisfying the following:(1)𝐴 is a fixed point of 𝐽, that is, 𝐴𝑥2=12𝐴(𝑥)(7.61) for all𝑥∈𝑋. Since𝑔∶𝑋→𝑌 is odd,𝐴∶𝑋→𝑌 is an odd mapping. The mapping𝐴 is a unique fixed point of𝐽 in the set 𝑀={ℎ∈𝑆∶𝑑(ℎ,𝑔)<∞}.(7.62) This implies that 𝐴 is a unique mapping satisfying (7.61) such that there exists a 𝑢∈(0,∞) satisfying 𝜇_{𝑔(𝑥)−𝐴(𝑥)}(𝑢𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.63) for all 𝑥∈𝑋 and 𝑡>0.(2)𝑑(𝐽^𝑛𝑔,𝐴)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞2^𝑛𝑔𝑥2^𝑛=𝐴(𝑥)(7.64) for all 𝑥∈𝑋.(3)𝑑(ℎ,𝐴)≤(1/(1−2𝛼))𝑑(ℎ,𝐽ℎ) for each ℎ∈𝑀, which implies the inequality 𝑑(𝑔,𝐴)≤5𝛼1−2𝛼.(7.65) This implies that the inequality (7.57) holds. Since 𝜇_{𝐷𝑔(𝑥,𝑦)}(3𝑡)≥_𝐿Φ_2𝑥,2𝑦(𝑡), it follows that 𝜇₂^𝑛_{𝐷𝑔(𝑥/2}^𝑛_,𝑦/2^𝑛₎(3𝑡)=𝜇_{𝐷𝑔(𝑥/2}^𝑛_,𝑦/2^𝑛₎3𝑡≥Φ_{𝑥/2𝑛−1,𝑦/2𝑛−1}2^𝑛2𝑡^𝑛≥^𝐿⋯≥_𝐿Φ_𝑥,𝑦12(2𝛼𝑡)^𝑛−1(7.66) for all 𝑥,𝑦∈𝑋, 𝑡>0, and 𝑛∈ℕ. As 𝑛→∞, we deduce that 𝜇_{𝐷𝐴(𝑥,𝑦)}(3𝑡)=1_ℒ(7.67) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Thus, the mapping 𝐴∶𝑋→𝑌 satisfies (1.4).
Now, we have 𝐴(2𝑥)−2𝐴(𝑥)=lim_𝑛→∞2^𝑛𝑔𝑥2^𝑛−1−2^𝑛+1𝑔𝑥=2lim_𝑛→∞2^𝑛2^𝑛−1𝑔𝑥2^𝑛−1−2^𝑛𝑔𝑥2^𝑛=0(7.68) for all 𝑥∈𝑋. Since the mapping 𝑥→𝐴(2𝑥)−8𝐴(𝑥) is additive (see Lemma  2.2 of [14]), from the equality 𝐴(2𝑥)=2𝐴(𝑥), we deduce that the mapping 𝐴∶𝑋→𝑌 is additive.

Corollary 7.6. Let 𝜃≥0 and let 𝑝 be a real number with 𝑝>1. Let 𝑋 be a normed vector space with norm ‖⋅‖. Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying (7.28), then 𝐴(𝑥)∶=lim_𝑛→∞2^𝑛𝑓2𝑥^𝑛−1−8𝑓2𝑥^𝑛(7.69) exists for each 𝑥∈𝑋 and defines an additive mapping 𝐴∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−8𝑓(𝑥)−𝐴(𝑥)}(𝑡)≥(2^𝑝−2)𝑡(2^𝑝−2)𝑡+5(1+2^𝑝)𝜃‖𝑥‖^𝑝(7.70) for all 𝑥∈𝑋 and 𝑡>0, where (𝑋,𝜇,𝑇_𝑀) is a complete LRN-space in which 𝐿=[0,1].

Proof. The proof follows from Theorem 7.5 by taking 𝜇_{𝐷𝑓(𝑥,𝑦)}(𝑡)≥𝑡𝑡+𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(7.71) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then we can choose 𝛼=2^−𝑝, and we get the desired result.

Theorem 7.7. Let 𝑋 be a linear space, let (𝑌,𝜇,𝒯_∧) be a complete LRN-space, and let Φ be a mapping from 𝑋² to 𝐷^+𝐿 (Φ(𝑥,𝑦) is denoted by Φ_𝑥,𝑦) such that, for some 0<𝛼<2, Φ_𝑥,𝑦(𝛼𝑡)≥_𝐿Φ_{𝑥/2,𝑦/2}(𝑡)(𝑥,𝑦∈𝑋,𝑡>0).(7.72) Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying (7.2), then 𝐴(𝑥)∶=lim_𝑛→∞12^𝑛𝑓2^𝑛+1𝑥−8𝑓(2^𝑛𝑥)(7.73) exists for each 𝑥∈𝑋 and defines an additive mapping 𝐴∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−8𝑓(𝑥)−𝐴(𝑥)}(𝑡)≥_𝐿𝒯_∧Φ^𝑥,𝑥2−𝛼5𝛼𝑡,Φ^2𝑥,𝑥25−𝛼𝛼𝑡(7.74) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let (𝑆,𝑑) be the generalized metric space defined in the proof of Theorem 7.1.
Consider the linear mapping 𝐽∶𝑆→𝑆 such that 𝐽ℎ(𝑥)∶=12ℎ(2𝑥)(7.75) for all 𝑥∈𝑋. It is easy to see that 𝐽 is a strictly contractive self-mapping on 𝑆 with the Lipschitz constant 𝛼/2. Let 𝑔(𝑥)=𝑓(2𝑥)−8𝑓(𝑥), from (7.58), it follows that 𝜇_{𝑔(𝑥)−1/2𝑔(2𝑥)}52𝑡≥^𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.76) for all 𝑥∈𝑋 and 𝑡>0. So, 𝑑(𝑔,𝐽𝑔)≤5/2. By Theorem 1.1, there exists a mapping 𝐴∶𝑋→𝑌 satisfying the following:(1)𝐴 is a fixed point of 𝐽, that is, 𝐴(2𝑥)=2𝐴(𝑥)(7.77) for all 𝑥∈𝑋. Since ℎ∶𝑋→𝑌 is odd, 𝐴∶𝑋→𝑌 is an odd mapping. The mapping 𝐴 is a unique fixed point of 𝐽 in the set 𝑀={ℎ∈𝑆∶𝑑(ℎ,𝑔)<∞}.(7.78) This implies that 𝐴 is a unique mapping satisfying (7.77) such that there exists a 𝑢∈(0,∞) satisfying 𝜇_{𝑔(𝑥)−𝐴(𝑥)}(𝑢𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(7.79) for all 𝑥∈𝑋 and 𝑡>0.(2)𝑑(𝐽^𝑛𝑔,𝐴)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞12^𝑛𝑔(2^𝑛𝑥)=𝐴(𝑥)(7.80) for all 𝑥∈𝑋.(3)𝑑(ℎ,𝐴)≤(1/(1−𝛼/2))𝑑(ℎ,𝐽ℎ), which implies the inequality 𝑑(𝑔,𝐴)≤52−𝛼.(7.81) This implies that the inequality (7.74) holds.
Proceeding as in the proof of Theorem 7.5, we obtain that the mapping 𝐴∶𝑋→𝑌 satisfies (1.4). Now, we have 𝐴(2𝑥)−2𝐴(𝑥)=lim_𝑛→∞21^𝑛𝑔2^𝑛+1𝑥−12^𝑛−1=2lim_𝑛→∞𝑔(2^𝑛𝑥)21^𝑛+1𝑔2^𝑛+1𝑥−12^𝑛𝑔(2^𝑛𝑥)=0(7.82) for all 𝑥∈𝑋. Since the mapping 𝑥→𝐴(2𝑥)−8𝐴(𝑥) is additive (see Lemma  2.2 of [14]), from the equality 𝐴(2𝑥)=2𝐴(𝑥), we deduce that the mapping 𝐴∶𝑋→𝑌 is additive.

Corollary 7.8. Let 𝜃≥0 and let 𝑝 be a real number with 0<𝑝<1. Let 𝑋 be a normed vector space with norm ‖⋅‖. Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying (7.28), then 𝐴(𝑥)∶=lim_𝑛→∞12^𝑛𝑓2^𝑛+1𝑥−8𝑓(2^𝑛𝑥)(7.83) exists for each 𝑥∈𝑋 and defines an additive mapping 𝐴∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−8𝑓(𝑥)−𝐴(𝑥)}(𝑡)≥(2−2^𝑝)𝑡(2−2^𝑝)𝑡+5(1+2^𝑝)𝜃‖𝑥‖^𝑝(7.84) for all 𝑥∈𝑋 and 𝑡>0, where (𝑋,𝜇,𝑇_𝑀) is a complete LRN-space in which 𝐿=[0,1].

Proof. The proof follows from Theorem 7.7 by taking 𝜇_{𝐷𝑓(𝑥,𝑦)}(𝑡)≥𝑡𝑡+𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(7.85) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then we can choose 𝛼=2^𝑝, and we get the desired result.

8. Generalized Hyers-Ulam Stability of the Functional Equation (1.4): An Even Case via Fixed-Point Method

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation 𝐷𝑓(𝑥,𝑦)=0 in random Banach spaces, an even case.

Theorem 8.1. Let 𝑋 be a linear space, let (𝑌,𝜇,𝒯_∧) be a complete LRN-space, and let Φ be a mapping from 𝑋² to 𝐷^+𝐿 (Φ(𝑥,𝑦) is denoted by Φ_𝑥,𝑦) such that, for some 0<𝛼<1/16, Φ_𝑥,𝑦(𝛼𝑡)≥_𝐿Φ_2𝑥,2𝑦(𝑡)(𝑥,𝑦∈𝑋,𝑡>0).(8.1) Let 𝑓∶𝑋→𝑌 be an even mapping satisfying 𝑓(0)=0 and (7.2), then 𝑄(𝑥)∶=lim_𝑛→∞16^𝑛𝑓2𝑥^𝑛−1−4𝑓2𝑥^𝑛(8.2) exists for each 𝑥∈𝑋 and defines a quartic mapping 𝑄∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−4𝑓(𝑥)−𝑄(𝑥)}(𝑡)≥_𝐿𝒯_∧Φ^𝑥,𝑥1−16𝛼5𝛼𝑡,Φ^2𝑥,𝑥1−5𝛼16𝛼𝑡(8.3) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Letting 𝑥=𝑦 in (7.2), we get 𝜇_{𝑓(3𝑦)−6𝑓(2𝑦)+15𝑓(𝑦)}(𝑡)≥_𝐿Φ_𝑦,𝑦(𝑡)(8.4) for all 𝑦∈𝑋 and 𝑡>0. Replacing 𝑥 by 2𝑦 in (7.2), we get 𝜇_{𝑓(4𝑦)−4𝑓(3𝑦)+4𝑓(2𝑦)+4𝑓(𝑦)}(𝑡)≥_𝐿Φ_2𝑦,𝑦(𝑡)(8.5) for all 𝑦∈𝑋 and 𝑡>0. By (8.4) and (8.5), 𝜇_{𝑓(4𝑥)−20𝑓(2𝑥)+64𝑓(𝑥)}(5𝑡)≥≥_𝐿𝒯_∧𝜇_{4(𝑓(3𝑥)−6𝑓(2𝑥)+15𝑓(𝑥))}(4𝑡),𝜇_{𝑓(4𝑥)−4𝑓(3𝑥)+4𝑓(2𝑥)+4𝑓(𝑥)}(𝑡)_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(8.6) for all 𝑥∈𝑋 and 𝑡>0. Letting 𝑔(𝑥)∶=𝑓(2𝑥)−4𝑓(𝑥) for all 𝑥∈𝑋, we get 𝜇_{𝑔(𝑥)−16𝑔(𝑥/2)}(5𝑡)≥_𝐿𝒯_∧Φ_{𝑥/2,𝑥/2}(𝑡),Φ_𝑥,𝑥/2(𝑡)(8.7) for all 𝑥∈𝑋 and 𝑡>0. Let (𝑆,𝑑) be the generalized metric space defined in the proof of Theorem 7.1.  
Now we consider the linear mapping 𝐽∶𝑆→𝑆 such that 𝐽ℎ(𝑥)∶=16ℎ(𝑥/2) for all 𝑥∈𝑋. It is easy to see that 𝐽 is a strictly contractive self-mapping on 𝑆 with the Lipschitz constant 16𝛼. It follows from (8.7) that 𝜇_{𝑔(𝑥)−16𝑔(𝑥/2)}(5𝛼𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(8.8) for all 𝑥∈𝑋 and 𝑡>0. So, 𝑑(𝑔,𝐽𝑔)≤5𝛼≤516<∞.(8.9)
By Theorem 1.1, there exists a mapping 𝑄∶𝑋→𝑌 satisfying the following:(1)𝑄 is a fixed point of 𝐽, that is, 𝑄𝑥2=116𝑄(𝑥)(8.10) for all𝑥∈𝑋. Since𝑔∶𝑋→𝑌 is even with𝑔(0)=0,𝑄∶𝑋→𝑌 is an even mapping with𝑄(0)=0. The mapping𝑄 is a unique fixed point of𝐽 in the set 𝑀={ℎ∈𝑆∶𝑑(ℎ,𝑔)<∞}.(8.11) This implies that 𝑄 is a unique mapping satisfying (8.10) such that there exists a 𝑢∈(0,∞) satisfying 𝜇_{𝑔(𝑥)−𝑄(𝑥)}(𝑢𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(8.12) for all 𝑥∈𝑋 and 𝑡>0.(2)𝑑(𝐽^𝑛𝑔,𝑄)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞16^𝑛𝑔𝑥2^𝑛=𝑄(𝑥)(8.13) for all 𝑥∈𝑋.(3)𝑑(ℎ,𝑄)≤(1/(1−16𝛼))𝑑(ℎ,𝐽ℎ) for every ℎ∈𝑀, which implies the inequality 𝑑(𝑔,𝑄)≤5𝛼1−16𝛼.(8.14) This implies that the inequality (8.3) holds.
Proceeding as in the proof of Theorem 7.1, we obtain that the mapping 𝑄∶𝑋→𝑌 satisfies (1.4). Now, we have 𝑄(2𝑥)−16𝑄(𝑥)=lim_𝑛→∞16^𝑛𝑔𝑥2^𝑛−1−16^𝑛+1𝑔𝑥=16lim_𝑛→∞2^𝑛16^𝑛−1𝑔𝑥2^𝑛−1−16^𝑛𝑔𝑥2^𝑛=0(8.15) for all 𝑥∈𝑋.  Since the mapping 𝑥→𝑄(2𝑥)−4𝑄(𝑥) is quartic, we get that the mapping 𝑄∶𝑋→𝑌 is quartic.

Corollary 8.2. Let 𝜃≥0 and let 𝑝 be a real number with 𝑝>4. Let 𝑋 be a normed vector space with norm ‖⋅‖.  Let 𝑓∶𝑋→𝑌 be an even mapping satisfying 𝑓(0)=0 and (7.28), then 𝑄(𝑥)∶=lim_𝑛→∞16^𝑛𝑓2𝑥^𝑛−1−4𝑓2𝑥^𝑛(8.16) exists for each 𝑥∈𝑋 and defines a quartic mapping 𝑄∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−4𝑓(𝑥)−𝑄(𝑥)}(𝑡)≥(2^𝑝−16)𝑡(2^𝑝−16)𝑡+5(1+2^𝑝)𝜃‖𝑥‖^𝑝(8.17) for all 𝑥∈𝑋 and 𝑡>0, where (𝑋,𝜇,𝑇_𝑀) is a complete LRN-space in which 𝐿=[0,1].

Proof. The proof follows from Theorem 8.1 by taking 𝜇_{𝐷𝑓(𝑥,𝑦)}(𝑡)≥𝑡𝑡+𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(8.18) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then we can choose 𝛼=2^−𝑝, and we get the desired result.

Theorem 8.3. Let 𝑋 be a linear space, let (𝑌,𝜇,𝒯_∧) be a complete LRN-space, and let Φ be a mapping from 𝑋² to 𝐷^+𝐿 (Φ(𝑥,𝑦) is denoted by Φ_𝑥,𝑦) such that, for some 0<𝛼<16, Φ_𝑥,𝑦(𝛼𝑡)≥Φ_{𝑥/2,𝑦/2}(𝑡)(𝑥,𝑦∈𝑋,𝑡>0).(8.19) Let 𝑓∶𝑋→𝑌 be an even mapping satisfying 𝑓(0)=0 and (7.2), then 𝑄(𝑥)∶=lim_𝑛→∞116^𝑛𝑓2^𝑛+1𝑥−4𝑓(2^𝑛𝑥)(8.20) exists for each 𝑥∈𝑋 and defines a quartic mapping 𝑄∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−4𝑓(𝑥)−𝑄(𝑥)}(𝑡)≥_𝐿𝒯_∧Φ^𝑥,𝑥16−𝛼5𝑡,Φ^2𝑥,𝑥165−𝛼𝑡(8.21) for all 𝑥∈𝑋 and 𝑡>0.

Proof. In the generalized metric space (𝑆,𝑑) defined in the proof of Theorem 7.1, we consider the linear mapping 𝐽∶𝑆→𝑆 such that 𝐽ℎ(𝑥)∶=116ℎ(2𝑥)(8.22) for all 𝑥∈𝑋. It is easy to see that 𝐽 is a strictly contractive self-mapping on 𝑆 with the Lipschitz constant 𝛼/16.  
Letting 𝑔(𝑥)∶=𝑓(2𝑥)−4𝑓(𝑥) for all 𝑥∈𝑋, by (8.7), we get 𝜇_{𝑔(𝑥)−(1/16)𝑔(2𝑥)}516𝑡≥^𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(8.23) for all 𝑥∈𝑋 and 𝑡>0. So, 𝑑(𝑔,𝐽𝑔)≤5/16.
By Theorem 1.1, there exists a mapping 𝑄∶𝑋→𝑌 satisfying the following:(1)𝑄 is a fixed point of 𝐽, that is, 𝑄(2𝑥)=16𝑄(𝑥)(8.24) for all 𝑥∈𝑋. Since 𝑔∶𝑋→𝑌 is even with 𝑔(0)=0, 𝑄∶𝑋→𝑌 is an even mapping with 𝑄(0)=0. The mapping 𝑄 is a unique fixed point of 𝐽 in the set 𝑀={ℎ∈𝑆∶𝑑(ℎ,𝑔)<∞}.(8.25) This implies that 𝑄 is a unique mapping satisfying (8.24) such that there exists a 𝑢∈(0,∞) satisfying 𝜇_{𝑔(𝑥)−𝑄(𝑥)}(𝑢𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(8.26) for all 𝑥∈𝑋 and 𝑡>0.(2)d(𝐽^𝑛𝑔,𝑄)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞116^𝑛𝑔(2^𝑛𝑥)=𝑄(𝑥)(8.27) for all 𝑥∈𝑋.(3)𝑑(𝑔,𝑄)≤(16/(16−𝛼))𝑑(𝑔,𝐽𝑔) for each ℎ∈𝑀, which implies the inequality 𝑑(𝑔,𝑄)≤5/(16−𝛼).(8.28) This implies that the inequality (8.21) holds.
Proceeding as in the proof of Theorem 7.3, we obtain that the mapping 𝑄∶𝑋→𝑌 satisfies (1.4). Now, we have 𝑄(2𝑥)−16𝑄(𝑥)=lim_𝑛→∞116^𝑛𝑔2^𝑛+1𝑥−116^𝑛−1=16lim_𝑛→∞𝑔(2^𝑛𝑥)161^𝑛+1𝑔2^𝑛+1𝑥−116^𝑛𝑔(2^𝑛𝑥)=0(8.29) for all 𝑥∈𝑋. Since the mapping 𝑥→𝑄(2𝑥)−4𝑄(𝑥) is quartic, we get that the mapping 𝑄∶𝑋→𝑌 is quartic.

Corollary 8.4. Let 𝜃≥0 and let 𝑝 be a real number with 0<𝑝<4. Let 𝑋 be a normed vector space with norm ‖⋅‖. Let 𝑓∶𝑋→𝑌 be an even mapping satisfying 𝑓(0)=0 and (7.28), then 𝑄(𝑥)∶=lim_𝑛→∞116^𝑛𝑓2^𝑛+1𝑥−4𝑓(2^𝑛𝑥)(8.30)exists for each 𝑥∈𝑋 and defines a quartic mapping 𝑄∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−4𝑓(𝑥)−𝑄(𝑥)}(𝑡)≥(16−2^𝑝)𝑡(16−2^𝑝)𝑡+5(1+2^𝑝)𝜃‖𝑥‖^𝑝(8.31) for all 𝑥∈𝑋 and 𝑡>0, where (𝑋,𝜇,𝑇_𝑀) is a complete LRN-space in which 𝐿=[0,1].

Proof. The proof follows from Theorem 8.3 by taking 𝜇_{𝐷𝑓(𝑥,𝑦)}(𝑡)≥𝑡𝑡+𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(8.32) for all 𝑥,𝑦∈𝑋 and 𝑡>0.  Then we can choose 𝛼=2^𝑝, and we get the desired result.

Theorem 8.5. Let 𝑋be a linear space, let (𝑌,𝜇,𝒯_∧) be a complete LRN-space, and let Φ be a mapping from 𝑋² to 𝐷^+𝐿 (Φ(𝑥,𝑦) is by denoted Φ_𝑥,𝑦) such that, for some 0<𝛼<1/4, Φ_𝑥,𝑦(𝛼𝑡)≥_𝐿Φ_2𝑥,2𝑦(𝑡)(𝑥,𝑦∈𝑋,𝑡>0).(8.33) Let 𝑓∶𝑋→𝑌 be an even mapping satisfying 𝑓(0)=0 and (7.2), then 𝑇(𝑥)∶=lim_𝑛→∞4^𝑛𝑓2𝑥^𝑛−1−16𝑓2𝑥^𝑛(8.34) exists for each 𝑥∈𝑋 and defines a quadratic mapping 𝑇∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−16𝑓(𝑥)−𝑇(𝑥)}(𝑡)≥_𝐿𝒯_∧Φ^𝑥,𝑥1−4𝛼5𝛼𝑡,Φ^2𝑥,𝑥1−5𝛼4𝛼𝑡(8.35) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let (𝑆,𝑑) be the generalized metric space defined in the proof of Theorem 7.1.
Letting 𝑔(𝑥)∶=𝑓(2𝑥)−16𝑓(𝑥) for all 𝑥∈𝑋 in (8.6), we get 𝜇_{𝑔(𝑥)−4𝑔(𝑥/2)}(5𝑡)≥_𝐿𝒯_∧Φ_{𝑥/2,𝑥/2}(𝑡),Φ_𝑥,𝑥/2(𝑡)(8.36) for all 𝑥∈𝑋 and 𝑡>0. It is easy to see that the linear mapping 𝐽∶𝑆→𝑆 such that 𝐽ℎ(𝑥)∶=4ℎ𝑥2(8.37) for all 𝑥∈𝑋, is a strictly contractive self-mapping with the Lipschitz constant 4𝛼.
It follows from (8.36) that 𝜇_{𝑔(𝑥)−4𝑔(𝑥/2)}(5𝛼𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(8.38) for all 𝑥∈𝑋 and 𝑡>0. So, 𝑑(𝑔,𝐽𝑔)≤5𝛼<∞.
By Theorem 1.1, there exists a mapping 𝑇∶𝑋→𝑌 satisfying the following:(1)𝑇 is a fixed point of 𝐽, that is, 𝑇𝑥2=14𝑇(𝑥)(8.39) for all 𝑥∈𝑋. Since 𝑔∶𝑋→𝑌 is even with 𝑔(0)=0, 𝑇∶𝑋→𝑌 is an even mapping with 𝑇(0)=0. The mapping 𝑇 is a unique fixed point of 𝐽 in the set 𝑀={ℎ∈𝑆∶𝑑(ℎ,𝑔)<∞}. This implies that 𝑇 is a unique mapping satisfying (8.39) such that there exists a 𝑢∈(0,∞) satisfying 𝜇_{𝑔(𝑥)−𝑇(𝑥)}(𝑢𝑡)≥_𝐿𝒯_∧Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(8.40) for all 𝑥∈𝑋 and 𝑡>0.(2)𝑑(𝐽^𝑛𝑔,𝑇)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞4^𝑛𝑔𝑥2^𝑛=𝑇(𝑥)(8.41) for all 𝑥∈𝑋.(3)𝑑(ℎ,𝑇)≤(1/(1−4𝛼))𝑑(ℎ,𝐽ℎ) for each ℎ∈𝑀, which implies the inequality 𝑑(𝑔,𝑇)≤5𝛼1−4𝛼.(8.42) This implies that the inequality (8.35) holds.
Proceeding as in the proof of Theorem 7.1, we obtain that the mapping 𝑇∶𝑋→𝑌 satisfies (1.4). Now, we have 𝑇(2𝑥)−4𝑇(𝑥)=lim_𝑛→∞4^𝑛𝑔𝑥2^𝑛−1−4^𝑛+1𝑔𝑥=4lim_𝑛→∞2^𝑛4^𝑛−1𝑔𝑥2^𝑛−1−4^𝑛𝑔𝑥2^𝑛=0(8.43) for all 𝑥∈𝑋. Since the mapping 𝑥→𝑇(2𝑥)−16𝑇(𝑥) is quadratic, we get that the mapping 𝑇∶𝑋→𝑌 is quadratic.

Corollary 8.6. Let 𝜃≥0 and let 𝑝 be a real number with 𝑝>2. Let 𝑋 be a normed vector space with norm ‖⋅‖. Let 𝑓∶𝑋→𝑌 be an even mapping satisfying 𝑓(0)=0 and (7.28), then 𝑇(𝑥)∶=lim_𝑛→∞4^𝑛𝑓2𝑥^𝑛−1−16𝑓2𝑥^𝑛(8.44) exists for each 𝑥∈𝑋 and defines a quadratic mapping 𝑇∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−16𝑓(𝑥)−𝑇(𝑥)}(𝑡)≥(2^𝑝−4)𝑡(2^𝑝−4)𝑡+5(1+2^𝑝)𝜃‖𝑥‖^𝑝(8.45) for all 𝑥∈𝑋 and 𝑡>0.

Proof. The proof follows from Theorem 8.5 by taking Φ_𝑥,𝑦(𝑡)∶=𝑡𝑡+𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(8.46) for all 𝑥,𝑦∈𝑋. Then we can choose 𝛼=2^−𝑝, and we get the desired result.

Theorem 8.7. Let 𝑋 be a linear space, let (𝑌,𝜇,𝑇_𝑀) be a complete RN-space, and let Φ be a mapping from 𝑋² to 𝐷⁺ (Φ(𝑥,𝑦) is denoted by Φ_𝑥,𝑦) such that, for some 0<𝛼<4, Φ_𝑥,𝑦(𝛼𝑡)≥Φ_{𝑥/2,𝑦/2}(𝑡)(𝑥,𝑦∈𝑋,𝑡>0).(8.47) Let 𝑓∶𝑋→𝑌 be an even mapping satisfying 𝑓(0)=0 and (7.2), then 𝑇(𝑥)∶=lim_𝑛→∞14^𝑛𝑓2^𝑛+1𝑥−16𝑓(2^𝑛𝑥)(8.48) exists for each 𝑥∈𝑋 and defines a quadratic mapping 𝑇∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−16𝑓(𝑥)−𝑇(𝑥)}(𝑡)≥𝑇_𝑀Φ^𝑥,𝑥4−𝛼5𝑡,Φ^2𝑥,𝑥4−5𝛼𝑡(8.49) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let (𝑆,𝑑) be the generalized metric space defined in the proof of Theorem 7.1.
It is easy to see that the linear mapping 𝐽∶𝑆→𝑆 such that 𝐽ℎ(𝑥)∶=14ℎ(2𝑥)(8.50) for all 𝑥∈𝑋 is a strictly contractive self-mapping with the Lipschitz constant 𝛼/4.
Letting 𝑔(𝑥)∶=𝑓(2𝑥)−16𝑓(𝑥) for all 𝑥∈𝑋, from (8.36), we get 𝜇_{𝑔(𝑥)−1/4𝑔(2𝑥)}54𝑡≥𝑇^𝑀Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(8.51) for all 𝑥∈𝑋 and 𝑡>0. So, 𝑑(𝑔,𝐽𝑔)≤5/4.  
By Theorem 1.1, there exists a mapping 𝑇∶𝑋→𝑌 satisfying the following:(1) 𝑇 is a fixed point of 𝐽, that is, 𝑇(2𝑥)=4𝑇(𝑥)(8.52) for all 𝑥∈𝑋. Since 𝑔∶𝑋→𝑌 is even with 𝑔(0)=0, 𝑇∶𝑋→𝑌 is an even mapping with 𝑇(0)=0. The mapping 𝑇 is a unique fixed point of 𝐽 in the set 𝑀={ℎ∈𝑆∶𝑑(ℎ,𝑔)<∞}.(8.53) This implies that 𝑇 is a unique mapping satisfying (8.52) such that there exists a 𝑢∈(0,∞) satisfying 𝜇_{𝑔(𝑥)−𝑇(𝑥)}(𝑢t)≥𝑇_𝑀Φ_𝑥,𝑥(𝑡),Φ_2𝑥,𝑥(𝑡)(8.54) for all 𝑥∈𝑋 and 𝑡>0.(2) 𝑑(𝐽^𝑛𝑔,𝑇)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞14^𝑛𝑔(2^𝑛𝑥)=𝑇(𝑥)(8.55) for all 𝑥∈𝑋.(3) 𝑑(ℎ,𝑇)≤(1/(1−𝛼/4))𝑑(ℎ,𝐽ℎ) for each ℎ∈𝑀, which implies the inequality 𝑑(𝑔,𝑇)≤5/(4−𝛼).(8.56) This implies that the inequality (8.49) holds.
Proceeding as in the proof of Theorem  2.3, we obtain that the mapping 𝑄∶𝑋→𝑌 satisfies (1.4). Now, we have 𝑇(2𝑥)−4𝑇(𝑥)=lim_𝑛→∞41^𝑛𝑔2^𝑛+1𝑥−14^𝑛−1=4lim_𝑛→∞𝑔(2^𝑛𝑥)41^𝑛+1𝑔2^𝑛+1𝑥−14^𝑛𝑔(2^𝑛𝑥)=0(8.57) for all 𝑥∈𝑋. Since the mapping 𝑥→𝑇(2𝑥)−16𝑇(𝑥) is quadratic, we get that the mapping 𝑇∶𝑋→𝑌 is quadratic.

Corollary 8.8. Let 𝜃≥0 and let 𝑝 be a real number with 0<𝑝<2. Let 𝑋 be a normed vector space with norm ‖⋅‖. Let 𝑓∶𝑋→𝑌 be an even mapping satisfying 𝑓(0)=0 and (7.28). Then 𝑇(𝑥)∶=lim_𝑛→∞14^𝑛𝑓2^𝑛+1𝑥−16𝑓(2^𝑛𝑥)(8.58) exists for each 𝑥∈𝑋 and defines a quadratic mapping 𝑇∶𝑋→𝑌 such that 𝜇_{𝑓(2𝑥)−16𝑓(𝑥)−𝑇(𝑥)}(𝑡)≥(4−2^𝑝)𝑡(4−2^𝑝)𝑡+5(1+2^𝑝)𝜃‖𝑥‖^𝑝(8.59) for all 𝑥∈𝑋 and 𝑡>0, where (𝑋,𝜇,𝑇_𝑀) is a complete LRN-space in which 𝐿=[0,1].

Proof. The proof follows from Theorem 8.5 by taking Φ_𝑥,𝑦(𝑡)∶=𝑡𝑡+𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(8.60) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then we can choose 𝛼=2^𝑝, and we get the desired result.

Acknowledgments

The authors are grateful to the area Editor Professor Yeong-Cheng Liou and the reviewer for their valuable comments and suggestions. Y. J. Cho was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021821).