Stabilities of Cubic Mappings in Various Normed Spaces: Direct and Fixed Point Methods

Abstract

In 1940 and 1964, Ulam proposed the general problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?”. In 1941, Hyers solved this stability problem for linear mappings. According to Gruber (1978) this kind of stability problems are of the particular interest in probability theory and in the case of functional equations of different types. In 1981, Skof was the first author to solve the Ulam problem for quadratic mappings. In 1982–2011, J. M. Rassias solved the above Ulam problem for linear and nonlinear mappings and established analogous stability problems even on restricted domains. The purpose of this paper is the generalized Hyers-Ulam stability for the following cubic functional equation: 𝑓(𝑚𝑥+𝑦)+𝑓(𝑚𝑥−𝑦)=𝑚𝑓(𝑥+𝑦)+𝑚𝑓(𝑥−𝑦)+2(𝑚³−𝑚)𝑓(𝑥),𝑚≥2 in various normed spaces.

1. Introduction

A classical question in the theory of functional equations is the following: “When is it true that a function which approximately satisfies a functional equation 𝐷 must be close to an exact solution of 𝐷?”

If the problem accepts a solution, we say that the equation 𝐷 is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1964.

In the next year, Hyers [2] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces.

In 1978, Th. M. Rassias [3] proved a generalization of Hyers’ theorem for additive mappings.

Theorem 1.1 (Th. M. Rassias). Let 𝑓∶𝐸→𝐸^ be a mapping from a normed vector space 𝐸 into a Banach space 𝐸^ subject to 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 𝑝<1. Then, the limit 𝐿(𝑥)=lim_𝑛→∞𝑓(2^𝑛𝑥)2^𝑛(1.2) exists for all 𝑥∈𝐸 and 𝐿∶𝐸→𝐸′ is the unique additive mapping which satisfies ‖𝑓(𝑥)−𝐿(𝑥)‖≤2𝜖2−2^𝑝‖𝑥‖^𝑝(1.3) for all 𝑥∈𝐸. If 𝑝<0, then inequality (1.1) holds for 𝑥,𝑦≠0 and (1.3) for 𝑥≠0. Also, if for each 𝑥∈𝐸 the mapping 𝑡↦𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ, then 𝐿 is ℝ-linear.

The result of Th. M. 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 [4] by replacing the bound 𝜖(‖𝑥‖^𝑝+‖𝑦‖^𝑝) by a general control function 𝜑(𝑥,𝑦).

The functional equation𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦)=2𝑓(𝑥)+2𝑓(𝑦)(1.4) is called a quadratic functional equation. In particular, every solution of the above quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [5] for mappings 𝑓∶𝑋→𝑌, where 𝑋 is a normed space and 𝑌 is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain 𝑋 is replaced by an Abelian group. 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 [2–4, 8–48]).

On the other hand, J. M. Rassias [38] considered the Cauchy difference controlled by a product of different powers of norm.

Theorem 1.2 (J. M. Rassias). Let 𝑓∶𝐸→𝐸^ be a mapping from a real normed vector space 𝐸 into a Banach space 𝐸^ subject to the inequality ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜖‖𝑥‖^𝑝‖𝑦‖^𝑞(1.5) for all 𝑥,𝑦∈𝐸, where 𝜖 and 𝑟=𝑝+𝑞 are constants with 𝜖>0 and 𝑟≠1. Then, 𝐿∶𝐸→𝐸^ is the unique additive mapping which satisfies ‖𝑓(𝑥)−𝐿(𝑥)‖≤𝜖2−2^𝑟‖𝑥‖^𝑟(1.6) for all 𝑥∈𝐸.

However, there was a singular case, for this singularity a counterexample was given by Găvruţa [19]. This stability phenomenon is called the Ulam-Gavruta-Rassias product stability (see also [13–17, 49]). In addition, J. M. Rassias considered the mixed product-sum of powers of norms control function. This stability is called JMRassias mixed product-sum stability (see also [44, 50–53]).

Jun and Kim [22] introduced the functional equation𝑓(2𝑥+𝑦)+𝑓(2𝑥−𝑦)=2𝑓(𝑥+𝑦)+2𝑓(𝑥−𝑦)+12𝑓(𝑥),(1.7) and they established the general solution and the generalized Hyers-Ulam-Rassias stability problem for the functional equation (1.7) in Banach spaces.

Park and Jung [35] introduced the functional equation𝑓(3𝑥+𝑦)+𝑓(3𝑥−𝑦)=3𝑓(𝑥+𝑦)+3𝑓(𝑥−𝑦)+48𝑓(𝑥),(1.8) and they established the general solution and the generalized Hyers-Ulam-Rassias stability problem for the functional equation (1.8) in Banach spaces.

It is easy to see that the function 𝑓(𝑥)=𝑥³ is a solution of the functional equations (1.7) and (1.8). Thus, it is natural that functional equations (1.7) and (1.8) are called cubic functional equations and every solution of these cubic functional equations is said to be a cubic mapping.

In this paper, we prove the generalized Hyers-Ulam stability of the following functional equation:𝑓(𝑚𝑥+𝑦)+𝑓(𝑚𝑥−𝑦)=𝑚𝑓(𝑥+𝑦)+𝑚𝑓(𝑥−𝑦)+2𝑚³−𝑚𝑓(𝑥).(1.9) where 𝑚 is a positive integer greater than 2, in various normed spaces.

2. Preliminaries

In the sequel, we will adopt the usual terminology, notions, and conventions of the theory of random normed spaces as in [54]. Throughout this paper, the space of all probability distribution functions is 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 is 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 pointwise 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)=01,,iiff𝑡𝑡≤>𝑎𝑎,.(2.1) One can easily show that the maximal element for Δ⁺ in this order is the distribution function 𝐻₀(𝑡).

Definition 2.1. A function  𝑇∶[0,1]²→[0,1]  is a continuous triangular norm (briefly  a 𝑡-norm) if 𝑇satisfies 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(𝑥,𝑦)=𝑥𝑦,𝑇(𝑥,𝑦)=max{𝑎+𝑏−1,0}, and 𝑇(𝑥,𝑦)=min(𝑎,𝑏). Recall that, if 𝑇 is a 𝑡-norm and {𝑥_𝑛} is a given group of numbers in [0,1], 𝑇^𝑛𝑖=1𝑥_𝑖 is defined recursively by 𝑇^1𝑖=1𝑥₁ and 𝑇^𝑛𝑖=1𝑥_𝑖=𝑇(𝑇^{𝑛−1𝑖=1}𝑥_𝑖,𝑥_𝑛) for 𝑛≥2.

Definition 2.2. 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^∶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.

Every normed space (𝑋,||⋅||) defines a random normed space (𝑋,𝜇^,𝑇_𝑀) where, for every 𝑡>0, 𝜇^𝑢(𝑡)=𝑡𝑡+‖𝑢‖(2.2) and 𝑇_𝑀 is the minimum 𝑡-norm. This space is called the induced random normed space.

If the 𝑡-norm 𝑇 is such that sup_0<𝑎<1𝑇(𝑎,𝑎)=1, then every RN-space (𝑋,𝜇′,𝑇) is a metrizable linear topological space with the topology 𝜏 (called the 𝜇′-topology or the (𝜖,𝛿)-topology) induced by the base of neighborhoods of 𝜃, {𝑈(𝜖,𝜆)∣𝜖>0,𝜆∈(0,1)}, where𝑈(𝜖,𝜆)=𝑥∈𝑋∣𝜇^𝑥(𝜖)>1−𝜆.(2.3)

Definition 2.3. Let  (𝑋,𝜇^,𝑇)   be an RN-space. (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^𝑥_𝑛−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^𝑥_{𝑛−𝑥𝑚}(𝑡)=1.(iii) The RN-space (𝑋,𝜇^,𝑇) is said to be complete if every Cauchy sequence in 𝑋 is convergent.

Theorem 2.4. If (𝑋,𝜇^,𝑇) is RN-space and {𝑥_𝑛} is a sequence such that 𝑥_𝑛→𝑥, then lim_𝑛→∞𝜇^𝑥_𝑛(𝑡)=𝜇^𝑥(𝑡).

A valuation is a function |⋅| from a field 𝕂 into [0,∞) such that 0 is the unique element having the 0 valuation, |𝑟𝑠|=|𝑟||𝑠|, and the triangle inequality holds, that is, |𝑟+𝑠|≤max{|𝑟|,|𝑠|}.(2.4) A field 𝕂 is called a valued field if 𝕂 carries a valuation. The usual absolute values of ℝ and ℂ are examples of valuations.

Let us consider a valuation which satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by|𝑟+𝑠|≤max{|𝑟|,|𝑠|}(2.5) for all 𝑟,𝑠∈𝕂, then the function |⋅| is called a non-Archimedean valuation and the field is called a non-Archimedean field. Clearly, |1|=|−1|=1 and |𝑛|≤1 for all 𝑛≥1. A trivial example of a non-Archimedean valuation is the function |⋅| taking everything except for 0 into 1 and |0|=0.

Definition 2.5. Let  𝑋  be a vector space over a field  𝕂  with a non-Archimedean valuation  |⋅|. A function  ‖⋅‖∶𝑋→[0,∞) is called a non-Archimedean norm if the following conditions hold:(a)‖𝑥‖=0 if and only if 𝑥=0 for all 𝑥∈𝑋;(b)‖𝑟𝑥‖=|𝑟|‖𝑥‖ for all 𝑟∈𝕂 and 𝑥∈𝑋;(c)the strong triangle inequality holds: ‖𝑥+𝑦‖≤max{‖𝑥‖,‖𝑦‖}(2.6) for all 𝑥,𝑦∈𝑋. Then (𝑋,‖⋅‖) is called a non-Archimedean normed space.

Definition 2.6. Let  {𝑥_𝑛} be a sequence in a non-Archimedean normed space  𝑋. (a)A sequence {𝑥_𝑛}^∞𝑛=1 in a non-Archimedean space is a Cauchy sequence if and only if, the sequence {𝑥_𝑛+1−𝑥_𝑛}^∞𝑛=1 converges to zero.(b)The sequence {𝑥_𝑛} is said to be convergent if, for any 𝜀>0, there are a positive integer 𝑁 and 𝑥∈𝑋 such that ‖‖𝑥_𝑛−𝑥‖‖≤𝜀(2.7) for all 𝑛≥𝑁. Then, the point 𝑥∈𝑋 is called the limit of the sequence {𝑥_𝑛}, which is denoted by lim_𝑛→∞𝑥_𝑛=𝑥.(c)If every Cauchy sequence in 𝑋 converges, then the non-Archimedean normed space 𝑋 is called a non-Archimedean Banach space.

Definition 2.7. Let 𝑋 be a set. A function 𝑑∶𝑋×𝑋→[0,∞] is called a generalized metric on 𝑋 if 𝑑 satisfies the following conditions:(a)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦 for all 𝑥,𝑦∈𝑋;(b)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋;(c)𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋.

Theorem 2.8. Let (𝑋,𝑑) be a complete generalized metric space and 𝐽∶𝑋→𝑋 a strictly contractive mapping with Lipschitz constant 𝐿<1. Then, for all 𝑥∈𝑋, either 𝑑𝐽^𝑛𝑥,𝐽^𝑛+1𝑥=∞(2.8) for all nonnegative integers 𝑛 or there exists a positive integer 𝑛₀ such that(a)𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)<∞ for all 𝑛₀≥𝑛₀;(b)the sequence {𝐽^𝑛𝑥} converges to a fixed point 𝑦^∗ of 𝐽;(c)𝑦^∗ is the unique fixed point of 𝐽 in the set 𝑌={𝑦∈𝑋∶𝑑(𝐽^𝑛0𝑥,𝑦)<∞};(d)𝑑(𝑦,𝑦^∗)≤(1/(1−𝐿))𝑑(𝑦,𝐽𝑦) for all 𝑦∈𝑌.

3. Random Stability of Functional Equation (1.9): A Direct Method

In this section, using direct method, we prove the generalized Hyers-Ulam stability of cubic functional equation (1.9) in random normed spaces.

Lemma 3.1. Let 𝐸₁ and 𝐸₂ be real vector spaces. A function 𝑓∶𝐸₁→𝐸₂ satisfies the functional equation (1.7) if and only if 𝑓∶𝐸₁→𝐸₂ satisfies the functional equation (1.9). Therefore, every solution of functional equation (1.9) is also cubic function.

Proof. Let 𝑓∶𝐸₁→𝐸₂ satisfy the equation (1.7). Putting 𝑥=𝑦=0 in (1.7), we get 𝑓(0)=0. Set 𝑦=0 in (1.7) to get 𝑓(−𝑦)=−𝑓(𝑦). By induction, we lead to 𝑓(𝑘𝑥)=𝑘³𝑓(𝑥) for all positive integer 𝑘. Replacing 𝑦 by 𝑥+𝑦 in (1.7), we have 𝑓(3𝑥+𝑦)+𝑓(𝑥−𝑦)=2𝑓(2𝑥+𝑦)−2𝑓(−𝑦)+12𝑓(𝑥),(3.1) for all 𝑥,𝑦∈𝐸₁. Once again replacing 𝑦 by 𝑦−𝑥 in (1.7), we have 𝑓(𝑥+𝑦)+𝑓(3𝑥−𝑦)=2𝑓(𝑦)+2𝑓(2𝑥−𝑦)+12𝑓(𝑥),(3.2) for all 𝑥,𝑦∈𝐸₁. Adding (3.1) to (3.2) and using (1.7), we obtain 𝑓(3𝑥+𝑦)+𝑓(3𝑥−𝑦)=3𝑓(𝑥+𝑦)+3𝑓(𝑥−𝑦)+48𝑓(𝑥),(3.3) for all 𝑥,𝑦∈𝐸₁. By using the previous method, by induction, we infer that 𝑓(𝑚𝑥+𝑦)+𝑓(𝑚𝑥−𝑦)=𝑚𝑓(𝑥+𝑦)+𝑚𝑓(𝑥−𝑦)+2𝑚³−𝑚𝑓(𝑥),(3.4) for all 𝑥,𝑦∈𝐸₁ and each positive integer 𝑚≥3.
Let 𝑓∶𝐸₁→𝐸₂ satisfy the functional equation (1.9) with the positive integer 𝑚≥3. Putting 𝑥=𝑦=0 in (1.9), we get 𝑓(0)=0. Setting 𝑥=0, we get 𝑓(−𝑦)=−𝑓(𝑦). Let 𝑘 be a positive integer. Replacing 𝑦 by 𝑘𝑥+𝑦 in (1.9), we have 𝑓((𝑚+𝑘)𝑥+𝑦)+𝑓((𝑚−𝑘)𝑥−𝑦)=−+𝑚𝑚2𝑓𝑓(𝑚(𝑘(₃(𝑘+−1)1𝑥)𝑥++𝑦)𝑦)−𝑚𝑓(𝑥),(3.5) for all 𝑥,𝑦∈𝐸₁. Replacing 𝑦 by 𝑦−𝑘𝑥 in (1.9), we have 𝑓((𝑚−𝑘)𝑥+𝑦)+𝑓((𝑚+𝑘)𝑥−𝑦)=−+𝑚𝑚2𝑓𝑓(𝑚(𝑘(₃(𝑘+−1)1𝑥)𝑥−−𝑦)𝑦)−𝑚𝑓(𝑥),(3.6) for all 𝑥,𝑦∈𝐸₁. Adding (3.5) to (3.6), we obtain 𝑓((𝑚=+−𝑚𝑘𝑚𝑓)(𝑥((𝑓𝑘+(+(𝑦𝑘)1−+)𝑥1𝑓)+(𝑥(𝑦𝑚+)+−𝑦)𝑚𝑘+)𝑓𝑥𝑓(−((𝑘(𝑘𝑦+)−+11)𝑓)𝑥𝑥(−(−𝑚𝑦𝑦)−))𝑘+)𝑥2+𝑚𝑦₃)+𝑓((𝑚+𝑘)𝑥−𝑦)−𝑚𝑓(𝑥),(3.7) for all 𝑥,𝑦∈𝐸₁ and for all integer 𝑘≥1. Let 𝜓_𝑚(𝑥,𝑦)=𝑓(𝑚𝑥+𝑦)+𝑓(𝑚𝑥−𝑦) for each integer 𝑚≥0. Then, (3.7) means that 𝜓_𝑚+𝑘(𝑥,𝑦)+𝜓_𝑚−𝑘(𝑥,𝑦)=𝑚𝜓_𝑘+1(𝑥,𝑦)−𝑚𝜓_𝑘−1(𝑥,𝑦)+4𝑚³−𝑚𝑓(𝑥),(3.8) for all 𝑥,𝑦∈𝐸₁ and for all integer 𝑘≥1. For 𝑘=1 and 𝑘=𝑚 in (3.8), we obtain 𝜓𝜓_𝑚+1(𝑥,𝑦)+𝜓_𝑚−1(𝑥,𝑦)=𝑚𝜓₂(𝑥,𝑦)+4𝑚³−𝑚𝑓(𝑥),_2𝑚(𝑥,𝑦)=𝑚𝜓_𝑚+1(𝑥,𝑦)−𝑚𝜓_𝑚−1(𝑥,𝑦)+4𝑚³−𝑚𝑓(𝑥),(3.9) for all 𝑥,𝑦∈𝐸₁. By the proof of the first part, since 𝑓∶𝐸₁→𝐸₂ satisfies the functional equation (1.9) with the positive integer 𝑚≥3, then 𝑓 satisfies the functional equation (1.9) with the positive integer 𝑘≥𝑚. It follows from (3.9) that 𝑓 satisfies the functional equation (1.7) and 𝑓((𝑚−1)𝑥+𝑦)+𝑓((𝑚−1)𝑥−𝑦)=++(𝑚2(𝑚−(−𝑚1)1−𝑓)𝑓1(𝑥)(₃𝑥+−𝑦)𝑦)−(𝑚−1)𝑓(𝑥).(3.10)

Theorem 3.2. Let 𝑋 be a real linear space, (𝑍,𝜇^,min) an RN-space, and 𝜓∶𝑋²→𝑍 a function such that, for some 0<𝛼<𝑚³, 𝜇^{𝜓(𝑚𝑥,0)}(𝑡)≥𝜇^{𝛼𝜓(𝑥,0)}(𝑡)∀𝑥∈𝑋,𝑡>0,(3.11) and, for all 𝑥,𝑦∈𝑋 and 𝑡>0, lim_𝑛→∞𝜇^𝜓(𝑚_{𝑛𝑥,𝑚𝑛𝑦)/𝑚3𝑛}(𝑡)=1. Let (𝑌,𝜇,min) be a complete RN-space. If 𝑓∶𝑋→𝑌 is a mapping with 𝑓(0)=0 such that for all 𝑥∈𝑋 and 𝑡>0𝜇_{𝑓(𝑚𝑥±𝑦)−𝑚𝑓(𝑥±𝑦)−2(𝑚3−𝑚)𝑓(𝑥)}(𝑡)≥𝜇^{𝜓(𝑥,𝑦)}(𝑡),(3.12) then the limit 𝐶(𝑥)=lim_𝑛→∞(𝑓(𝑚^𝑛𝑥)/𝑚^3𝑛) exists for all 𝑥∈𝑋 and defines a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥𝜇^{𝜓(𝑥,0)/2(𝑚}_3−𝛼)(𝑡).(3.13)

Proof. Putting 𝑦=0 in (3.12) we see that, for all 𝑥∈𝑋, 𝜇_{𝑓(𝑚𝑥)/𝑚3−𝑓(𝑥)}(𝑡)≥𝜇^{𝜓(𝑥,0)/2𝑚}₃(𝑡).(3.14) Replacing 𝑥 by 𝑚^𝑛𝑥 in (3.14) and using (3.11), we obtain 𝜇_{𝑓(𝑚𝑛+1𝑥)/𝑚3𝑛+3−𝑓(𝑚𝑛𝑥)/𝑚3𝑛}(𝑡)≥𝜇^𝜓(𝑚_{𝑛𝑥,0)/2𝑚3𝑛+3}(𝑡)≥𝜇^𝛼_{𝑛𝜓(𝑥,0)/2𝑚3𝑛+3}(𝑡).(3.15) So 𝜇_{(𝑓(𝑚𝑛𝑥)/𝑚3𝑛∑)−𝑓(𝑥)/(}^{𝑛−1𝑘=0}_{(𝛼𝑘/2𝑚3𝑘+3))}(𝑡)=≥𝜇𝑇_∑((^{𝑛−1𝑘=0}_{(𝑓(𝑚𝑘+1𝑥)/𝑚3𝑘+3))−(𝑓(𝑚𝑘𝑥)/𝑚3𝑘∑))/(}^{𝑛−1𝑘=0}_{(𝛼𝑘/2𝑚3𝑘+3))𝑛−1𝑘=0}(𝑡)=𝑇_{𝑛−1𝑘=0}𝜇_((𝑓(𝑚^𝑘+1_𝑥)/𝑚^3𝑘+3_{)−(𝑓(𝑚}^𝑘_𝑥)/𝑚^3𝑘_))/(𝛼^𝑘_/2𝑚^3𝑘+3₎(𝑡)=𝜇_{𝜓(𝑥,0)}𝜇^{𝜓(𝑥,0)}(𝑡)(𝑡).(3.16) This implies that 𝜇_{𝑓(𝑚𝑛𝑥)/𝑚3𝑛−𝑓(𝑥)}(𝑡)≥𝜇^∑_{𝑛−1𝑘=0(𝛼𝑘𝜓(𝑥,0)/2𝑚3𝑘+3)}(𝑡).(3.17) Replacing 𝑥 by 𝑚^𝑝𝑥 in (3.17), we obtain 𝜇_{𝑓(𝑚𝑛+𝑝𝑥)/𝑚3(𝑛+𝑝)−𝑓(𝑚𝑝𝑥)/𝑚3𝑝}(𝑡)≥𝜇^∑_{𝑛−1𝑘=0(𝛼𝑘𝜓(𝑚𝑝𝑥,0)/2𝑚3(𝑘+𝑝)+3)}(𝑡)≥=𝜇𝜇^∑_{𝑛−1𝑘=0(𝛼𝑘+𝑝𝜓(𝑥,0)/2𝑚3(𝑘+𝑝)+3)∑𝑛+𝑝−1𝑘=𝑝(𝛼𝑘𝜓(𝑥,0)/2𝑚3𝑘+3)}(𝑡)(𝑡).(3.18) As lim_{𝑝,𝑛→∞}𝜇^∑_{𝑛+𝑝−1𝑘=𝑝(𝛼𝑘𝜓(𝑥,0)/2𝑚3𝑘+3)}(𝑡)=1,(3.19){𝑓(𝑚^𝑛𝑥)/𝑚^3𝑛} is a Cauchy sequence in complete RN-space (𝑌,𝜇,min), so there exists some point 𝐶(𝑥)∈𝑌 such that lim_𝑛→∞(𝑓(𝑚^𝑛𝑥)/𝑚^3𝑛)=𝐶(𝑥). Fix 𝑥∈𝑋 and put 𝑝=0 in (3.18). Then, we obtain 𝜇_{𝑓(𝑚𝑛𝑥)/𝑚3𝑛−𝑓(𝑥)}(𝑡)≥𝜇^∑_{𝑛−1𝑘=0(𝛼𝑘𝜓(𝑥,0)/2𝑚3𝑘+3)}(𝑡),(3.20) and so, for every 𝜖>0, we have 𝜇_{𝐶(𝑥)−𝑓(𝑥)}(𝑡+𝜖)≥≥𝑇𝑇𝜇𝜇_{𝐶(𝑥)−𝑓(𝑚𝑛𝑥)/𝑚3𝑛𝐶(𝑥)−𝑓(𝑚𝑛𝑥)/𝑚3𝑛}(𝜖),𝜇_{𝑓(𝑚𝑛𝑥)/𝑚3𝑛−𝑓(𝑥)}(𝑡)(𝜖),𝜇^∑_{𝑛−1𝑘=0(𝛼𝑘𝜓(𝑥,0)/2𝑚3𝑘+3)}(𝑡).(3.21) Taking the limit as 𝑛→∞ and using (3.21), we get 𝜇_{𝐶(𝑥)−𝑓(𝑥)}(𝑡+𝜖)≥𝜇^{𝜓(𝑥,0)/2(𝑚}_3−𝛼)(𝑡).(3.22) Since 𝜖 was arbitrary by taking 𝜖→0 in (3.22), we get 𝜇_{𝐶(𝑥)−𝑓(𝑥)}(𝑡)≥𝜇^{𝜓(𝑥,0)/2(𝑚}_3−𝛼)(𝑡).(3.23) Replacing 𝑥 and 𝑦 by 𝑚^𝑛𝑥 and 𝑚^𝑛𝑦 in (3.12), respectively, we get, for all 𝑥,𝑦∈𝑋 and for all 𝑡>0, 𝜇_{(𝑓(𝑚𝑛+1𝑥±𝑚𝑛𝑦)−𝑚𝑓(𝑚𝑛𝑥±𝑚𝑛𝑦)−2(𝑚3−𝑚)𝑓(𝑚𝑛𝑥))/𝑚3𝑛}(𝑡)≥𝜇^𝜓(𝑚_{𝑛𝑥,𝑚𝑛𝑦)/𝑚3𝑛}(𝑡).(3.24) Since lim_𝑛→∞𝜇^𝜓(𝑚_{𝑛𝑥,𝑚𝑛𝑦)/𝑚3𝑛}(𝑡)=1, we conclude that 𝐶(𝑚𝑥±𝑦)=𝐶(𝑥±𝑦)+2(𝑚³−𝑚)𝐶(𝑥). To prove the uniqueness of the cubic mapping 𝐶, assume that there exists another cubic mapping 𝐿∶𝑋→𝑌 which satisfies (3.13).
By induction one can easily see that, since 𝑓 is a cubic functional equation, so, for all 𝑛∈ℕ and every 𝑥∈𝑋, 𝐶(𝑚^𝑛𝑥)=𝑚^3𝑛𝐶(𝑥), and 𝐿(𝑚^𝑛𝑥)=𝑚^3𝑛𝐿(𝑥), we have 𝜇_{𝐶(𝑥)−𝐿(𝑥)}(𝑡)=lim_𝑛→∞𝜇_{(𝐶(𝑚𝑛𝑥)/𝑚3𝑛)−(𝐿(𝑚𝑛𝑥)/𝑚3𝑛)}(𝑡),(3.25) so 𝜇_{𝐶(𝑚𝑛𝑥)/𝑚3𝑛−𝐿(𝑚3𝑛𝑥)/𝑚3𝑛}(𝑡)≥min𝜇_{𝐶(𝑚𝑛𝑥)/𝑚3𝑛−𝑓(𝑚𝑛𝑥)/𝑚3𝑛}𝑡≥𝜇_{𝜓(𝑚𝑛𝑥,0)/𝑚3𝑛(𝑚3−𝛼)}2,𝜇^{𝐿(𝑚𝑛𝑥)/𝑚3𝑛−𝑓(𝑚𝑛𝑥)/𝑚3𝑛}2𝑡≥𝜇_{𝛼𝑛𝜓(𝑥,0)/𝑚3𝑛(𝑚3−𝛼)}(𝑡)(𝑡).(3.26) Since lim_𝑛→∞𝜇^𝛼_{𝑛𝜓(𝑥,0)/𝑚3𝑛(𝑚3−𝛼)}(𝑡)=1, it follows that, for all 𝑡>0, 𝜇_{𝐶(𝑥)−𝐿(𝑥)}(𝑡)=1 and so 𝐶(𝑥)=𝐿(𝑥). This completes the proof.

Corollary 3.3. Let 𝑋 be a real linear space, (𝑍,𝜇^,min) an RN-space, and (𝑌,𝜇,min) a complete RN-space. Let 0<𝑟<1 and 𝑧₀∈𝑍, and let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying 𝜇_{𝑓(𝑚𝑥±𝑦)−𝑚𝑓(𝑥±𝑦)−2(𝑚3−𝑚)𝑓(𝑥)}(𝑡)≥𝜇^(||𝑥||_{𝑟+||𝑦||𝑟)𝑧0}(𝑡),(3.27) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then, the limit 𝐶(𝑥)=lim_𝑛→∞(𝑓(𝑚^𝑛𝑥)/𝑚^3𝑛) exists for all 𝑥∈𝑋 and defines a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥𝜇^||𝑥||_{𝑟𝑧0/2(𝑚3−𝑚3𝑟)}(𝑡)(3.28) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let 𝛼=𝑚^3𝑟, and let 𝜓∶𝑋²→𝑍 be defined as 𝜓(𝑥,𝑦)=(||𝑥||^𝑟+||𝑦||^𝑟)𝑧₀.

Remark 3.4. In Corollary 3.3, if we assume that 𝜓(𝑥,𝑦)=(‖𝑥‖^𝑟.‖𝑦‖^𝑟)𝑧₀ or 𝜓(𝑥,𝑦)=(‖𝑥‖^𝑟+𝑠+‖𝑦‖^𝑟+𝑠+‖𝑥‖^𝑟‖𝑦‖^𝑠)𝑧₀, then we get Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability, respectively. But, since we put  𝑦=0  in this functional equation, the Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability corollaries will be obvious. Meanwhile, the JMRassias mixed product-sum stability when 𝑟+𝑠=3  is an open question.

Corollary 3.5. Let 𝑋 be a real linear space, (𝑍,𝜇^,min) an RN-space, and (𝑌,𝜇,min) a complete RN-space. Let 𝑧₀∈𝑍, and let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying 𝜇_{𝑓(𝑚𝑥±𝑦)−𝑚𝑓(𝑥±𝑦)−2(𝑚3−𝑚)𝑓(𝑥)}(𝑡)≥𝜇^𝛿𝑧₀(𝑡),(3.29) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then, the limit 𝐶(𝑥)=lim_𝑛→∞(𝑓(𝑚^𝑛𝑥)/𝑚^3𝑛) exists for all 𝑥∈𝑋 and defines a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥𝜇^𝛿𝑧_{0/2(𝑚3−1)}(𝑡)(3.30) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let 𝛼=1, and let 𝜓∶𝑋²→𝑍 be defined by 𝜓(𝑥,𝑦)=𝛿𝑧₀.

Theorem 3.6. Let 𝑋 be a real linear space, (𝑍,𝜇^,min) an RN-space, and 𝜓∶𝑋²→𝑍 a function such that for some 0<𝛼<1/𝑚³𝜇^{𝜓(𝑥/𝑚,0)}(𝑡)≥𝜇^{𝛼𝜓(𝑥,0)}(𝑡)∀𝑥∈𝑋,𝑡>0,(3.31) and, for all 𝑥,𝑦∈𝑋 and 𝑡>0, lim_𝑛→∞𝜇^𝑚_{3𝑛𝜓(𝑥/𝑚𝑛,𝑦/𝑚𝑛)}(𝑡)=1.(3.32) Let (𝑌,𝜇,min) be a complete RN-space. If 𝑓∶𝑋→𝑌 is a mapping with 𝑓(0)=0 and satisfying (3.12), then the limit 𝐶(𝑥)=lim_𝑛→∞𝑚^3𝑛𝑓(𝑥/𝑚^𝑛) exists for all 𝑥∈𝑋 and defines a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥𝜇^{𝜓(𝑥,0)/2(1−𝑚}_3𝛼)(𝑡).(3.33)

Proof. Putting 𝑦=0 in (3.12) and replacing 𝑥 by 𝑥/𝑚, we obtain that for all 𝑥∈𝑋𝜇_{𝑓(𝑥)−𝑚3𝑓(𝑥/𝑚)}(𝑡)≥𝜇^{𝜓(𝑥,0)/2}(𝑡).(3.34) Replacing 𝑥 by 𝑥/𝑚^𝑛 in (3.34) and using (3.31), we obtain 𝜇_{𝑚3𝑛𝑓(𝑥/𝑚𝑛)−𝑚3𝑛+3𝑓(𝑥/𝑚𝑛+1)}(𝑡)≥𝜇^𝑚_{3𝑛𝜓(𝑥/𝑚𝑛,0)/2}(𝑡)≥𝜇^𝛼_{𝑛𝑚3𝑛𝜓(𝑥,0)/2}(𝑡).(3.35) So 𝜇_{(𝑚3𝑛𝑓(𝑥/𝑚𝑛∑)−𝑓(𝑥))/}^{𝑛−1𝑘=0}_{(𝛼𝑘𝑚3𝑘/2)}(𝑡)=≥𝜇𝑇_(∑^{𝑛−1𝑘=0}_{𝑚3𝑘+3𝑓(𝑥/𝑚𝑘+1)−𝑚3𝑘𝑓(𝑥/𝑚𝑘∑))/}^{𝑛−1𝑘=0}_{(𝛼𝑘𝑚3𝑘/2)𝑛−1𝑘=0}(𝑡)=𝜇′_𝜓(𝑥,0)𝜇^{𝜓(𝑥,0)}(𝑡)(𝑡).(3.36) This implies that 𝜇_{𝑚3𝑛𝑓(𝑥/𝑚𝑛)−𝑓(𝑥)}(𝑡)≥𝜇^∑_{𝑛−1𝑘=0𝛼𝑘𝑚3𝑘𝜓(𝑥,0)/2}(𝑡).(3.37) The rest of the proof is similar to the proof of Theorem 3.2.

Corollary 3.7. Let 𝑋 be a real linear space, (𝑍,𝜇^,min) an RN-space, and (𝑌,𝜇,min) a complete RN-space. Let 𝑟>1 and 𝑧₀∈𝑍, and let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying (3.27). Then, the limit 𝐶(𝑥)=lim_𝑛→∞𝑚^3𝑛𝑓(𝑥/𝑚^𝑛) exists for all 𝑥∈𝑋 and defines a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥𝜇^𝑚_{3𝑟||𝑥||𝑟𝑧0/2(𝑚3𝑟−𝑚3)}(𝑡)(3.38) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let 𝛼=𝑚^−3𝑟, and let 𝜓∶𝑋²→𝑍 be defined as 𝜓(𝑥,𝑦)=(||𝑥||^𝑟+||𝑦||^𝑟)𝑧₀.

Corollary 3.8. 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 let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying (3.29). Then, the limit 𝐶(𝑥)=lim_𝑛→∞𝑚^3𝑛𝑓(𝑥/𝑚^𝑛) exists for all 𝑥∈𝑋 and defines a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝜇_{𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥𝜇^{𝑚𝛿𝑧}_{0/2(𝑚−1)}(𝑡)(3.39) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let 𝛼=1/𝑚⁴, and let 𝜓∶𝑋²→𝑍 be defined by 𝜓(𝑥,𝑦)=𝛿𝑧₀.

4. Random Stability of the Functional Equation (1.9): A Fixed Point Approach

In this section, using the fixed point alternative approach, we prove the generalized Hyers-Ulam stability of functional equation (1.9) in random normed spaces.

Theorem 4.1. Let 𝑋 be a linear space, (𝑌,𝜇,𝑇_𝑀) a complete RN-space, and Φ a mapping from 𝑋² to 𝐷⁺(Φ(𝑥,𝑦) is denoted by Φ_𝑥,𝑦) such that there exists 0<𝛼<1/𝑚³ such that Φ_𝑥,𝑦(𝑡)≤Φ_{𝑥/𝑚,𝑦/𝑚}(𝛼𝑡)(4.1) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying 𝜇_{𝑓(𝑚𝑥±𝑦)−𝑚𝑓(𝑥±𝑦)−2(𝑚3−𝑚)𝑓(𝑥)}(𝑡)≥Φ_𝑥,𝑦(𝑡)(4.2) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then, for all 𝑥∈𝑋𝐶(𝑥)∶=lim_𝑛→∞𝑚^3𝑛𝑓𝑥𝑚^𝑛(4.3) exists and 𝐶∶𝑋→𝑌 is a unique cubic mapping such that 𝜇_{𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥Φ_𝑥,02−2𝑚³𝛼𝑡𝛼(4.4) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Putting 𝑦=0 in (4.2) and replacing 𝑥 by 𝑥/𝑚, we have 𝜇_{𝑓(𝑥)−𝑚3𝑓(𝑥/𝑚)}(𝑡)≥Φ_𝑥/𝑚,0(2𝑡)≥Φ_𝑥,02𝑡𝛼(4.5) for all 𝑥∈𝑋 and 𝑡>0. Consider the set 𝑆∶={𝑔∶𝑋→𝑌;𝑔(0)=0}(4.6) and the generalized metric 𝑑 in 𝑆 defined by 𝑑(𝑓,𝑔)=inf𝑢∈ℝ⁺∶𝜇_{𝑔(𝑥)−ℎ(𝑥)}(𝑢𝑡)≥Φ_𝑥,0(𝑡),∀𝑥∈𝑋,𝑡>0,(4.7) where inf∅=+∞. It is easy to show that (𝑆,𝑑) is complete (see [26], Lemma 2.1). Now, we consider a linear mapping 𝐽∶𝑆→𝑆 such that 𝐽ℎ(𝑥)∶=𝑚³ℎ𝑥𝑚(4.8) for all 𝑥∈𝑋. First, we prove that 𝐽 is a strictly contractive mapping with the Lipschitz constant 𝑚³𝛼.
In fact, let 𝑔,ℎ∈𝑆 be such that 𝑑(𝑔,ℎ)<𝜖. Then, we have 𝜇_{𝑔(𝑥)−ℎ(𝑥)}(𝜖𝑡)≥Φ_𝑥,0(𝑡)(4.9) for all 𝑥∈𝑋 and 𝑡>0, and so 𝜇_{𝐽𝑔(𝑥)−𝐽ℎ(𝑥)}𝑚³𝛼𝜖𝑡=𝜇_{𝑚3𝑔(𝑥/𝑚)−𝑚3ℎ(𝑥/𝑚)}𝑚³𝛼𝜖𝑡=≥𝜇Φ_{𝑔(𝑥/𝑚)−ℎ(𝑥/𝑚)𝑥/𝑚,0}(𝛼𝜖𝑡)≥Φ_𝑥,0(𝛼𝑡)(𝑡)(4.10) for all 𝑥∈𝑋 and 𝑡>0. Thus, 𝑑(𝑔,ℎ)<𝜖 implies that 𝑑(𝐽𝑔,𝐽ℎ)=𝑑𝑚³𝑔𝑥𝑚,𝑚³ℎ𝑥𝑚<𝑚³𝛼𝜖.(4.11) This means that 𝑑(𝐽𝑔,𝐽ℎ)=𝑑𝑚³𝑔𝑥𝑚,𝑚³ℎ𝑥𝑚≤𝑚³𝛼𝑑(𝑔,ℎ)(4.12) for all 𝑔,ℎ∈𝑆. It follows from (4.5) that 𝑑(𝑓,𝐽𝑓)=𝑑𝑓,𝑚³𝑓𝑥𝑚≤𝛼2.(4.13) By Theorem 2.8, there exists a mapping 𝐶∶𝑋→𝑌 satisfying the following.
(1) 𝐶 is a fixed point of 𝐽, that is, 𝐶𝑥𝑚=𝑚1³𝐶(𝑥)(4.14) for all 𝑥∈𝑋.
The mapping 𝐶 is a unique fixed point of 𝐽 in the set Ω={ℎ∈𝑆∶𝑑(𝑔,ℎ)<∞}.(4.15) This implies that 𝐶 is a unique mapping satisfying (4.14) such that there exists 𝑢∈(0,∞) satisfying 𝜇_{𝑓(𝑥)−𝐶(𝑥)}(𝑢𝑡)≥Φ_𝑥,0(𝑡)(4.16) for all 𝑥∈𝑋 and 𝑡>0.
(2) 𝑑(𝐽^𝑛𝑓,𝐶)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞𝑚^3𝑛𝑓𝑥𝑚^𝑛=𝐶(𝑥)(4.17) for all 𝑥∈𝑋.
(3) 𝑑(𝑓,𝐶)≤𝑑(𝑓,𝐽𝑓)/(1−𝑚³𝛼) with 𝑓∈Ω, which implies the inequality 𝑑(𝑓,𝐶)≤𝛼/(2−2𝑚³𝛼)  and so 𝜇_{𝑓(𝑥)−𝐶(𝑥)}𝛼𝑡2−2𝑚³𝛼≥Φ^𝑥,0(𝑡)(4.18) for all 𝑥∈𝑋 and 𝑡>0. This implies that inequality (4.4) holds. Now, we have 𝜇_{𝑚3𝑛𝑓((𝑚𝑥±𝑦)/𝑚𝑛)−𝑚3𝑛+1𝑓((𝑥±𝑦)/𝑚𝑛)−2𝑚3𝑛(𝑚3−𝑚)𝑓(𝑥/𝑚𝑛)}(𝑡)≥Φ_{𝑥/𝑚𝑛,𝑦/𝑚𝑛}𝑡𝑚^3𝑛(4.19) for all 𝑥,𝑦∈𝑋, 𝑡>0, and 𝑛≥1, and so, from (4.1), it follows that Φ_{𝑥/𝑚𝑛,𝑦/𝑚𝑛}𝑡𝑚^3𝑛≥Φ^𝑥,𝑦𝑚𝑡_3𝑛𝛼^𝑛,(4.20) Since lim_𝑛→∞Φ_𝑥,𝑦𝑡𝑚^3𝑛𝛼^𝑛=1,(4.21) for all 𝑥,𝑦∈𝑋 and 𝑡>0, we have 𝜇_{𝐶(𝑚𝑥±𝑦)−𝑚𝐶(𝑥±𝑦)−2(𝑚3−𝑚)𝐶(𝑥)}(𝑡)=1(4.22) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Thus, the mapping 𝐶∶𝑋→𝑌 is cubic. This completes the proof.

Corollary 4.2. Let 𝑋 be a real normed space, 𝜃≥0, and 𝑝 a real number with 𝑝∈(1,+∞). Let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying 𝜇_{𝑓(𝑚𝑥±𝑦)−𝑚𝑓(𝑥±𝑦)−2(𝑚3−𝑚)𝑓(𝑥)}(𝑡)≥𝑡𝑡+𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(4.23) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then, for all 𝑥∈𝑋, the limit 𝐶(𝑥)=lim_𝑛→∞𝑚^3𝑛𝑓(𝑥/𝑚^𝑛) exists and 𝐶∶𝑋→𝑌 is a unique cubic mapping such that 𝜇_{𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥2𝑚^3𝑝−𝑚³𝑡2𝑚^3𝑝−𝑚³𝑡+𝜃‖𝑥‖^𝑝(4.24) for all 𝑥∈𝑋 and 𝑡>0.

Proof. The proof follows from Theorem 4.1 if we take Φ_𝑥,𝑦(𝑡)=𝑡𝑡+𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(4.25) for all 𝑥,𝑦∈𝑋 and 𝑡>0. In fact, if we choose 𝛼=𝑚^−3𝑝, then we get the desired result.

Theorem 4.3. Let 𝑋 be a linear space, (𝑌,𝜇,𝑇_𝑀) a complete RN-space, and Φ a mapping from 𝑋² to 𝐷⁺ ( Φ(𝑥,𝑦) is denoted by Φ_𝑥,𝑦) such that for some 0<𝛼<𝑚³Φ_{𝑥/𝑚,𝑦/𝑚}(𝑡)≤Φ_𝑥,𝑦(𝛼𝑡)(4.26) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying (4.2). Then, for all 𝑥∈𝑋, the limit 𝐶(𝑥)∶=lim_𝑛→∞𝑓(𝑚^𝑛𝑥)/𝑚^3𝑛 exists and 𝐶∶𝑋→𝑌 is a unique cubic mapping such that 𝜇_{𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥Φ_𝑥,02𝑚³−2𝛼𝑡(4.27) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let (𝑆,𝑑) be the generalized metric space defined in the proof of Theorem 4.1. Now, we consider a linear mapping 𝐽∶𝑆→𝑆 such that 𝐽ℎ(𝑥)∶=1𝑚³ℎ(𝑚𝑥)(4.28) for all 𝑥∈𝑋.
Let 𝑔,ℎ∈𝑆 be such that 𝑑(𝑔,ℎ)<𝜖. Then, we have 𝜇_{𝑔(𝑥)−ℎ(𝑥)}(𝜖𝑡)≥Φ_𝑥,0(𝑡)(4.29) for all 𝑥∈𝑋 and 𝑡>0 and so 𝜇_{𝐽𝑔(𝑥)−𝐽ℎ(𝑥)}𝛼𝜖𝑡𝑚³=𝜇^{(1/𝑚3)𝑔(𝑚𝑥)−(1/𝑚3)ℎ(𝑚𝑥)}𝛼𝑚𝜖₃𝑡=𝜇^{𝑔(𝑚𝑥)−ℎ(𝑚𝑥)}≥Φ_𝑚𝑥,0(𝛼𝜖𝑡)≥Φ_𝑥,0(𝛼𝑡)(𝑡)(4.30) for all 𝑥∈𝑋 and 𝑡>0. Thus, 𝑑(𝑔,ℎ)<𝜖 implies that 𝑑(𝐽𝑔,𝐽ℎ)=𝑑𝑔(𝑚𝑥)𝑚³,ℎ(𝑚𝑥)𝑚³<𝑚𝛼𝜖³.(4.31) This means that 𝑑(𝐽𝑔,𝐽ℎ)=𝑑𝑔(𝑚𝑥)𝑚³,ℎ(𝑚𝑥)𝑚³≤𝑚𝛼³𝑑(𝑔,ℎ)(4.32) for all 𝑔,ℎ∈𝑆.
Putting 𝑦=0 in (4.2), we see that, for all 𝑥∈𝑋, 𝜇_{𝑓(𝑚𝑥)/𝑚3−𝑓(𝑥)}𝑡2𝑚³≥Φ^𝑥,0(𝑡).(4.33) It follows from (4.33) that 𝑑(𝑓,𝐽𝑓)=𝑑𝑓,𝑓(𝑚𝑥)𝑚³≤2𝑚1³.(4.34) By Theorem 2.8, there exists a mapping 𝐶∶𝑋→𝑌 satisfying the following.
(1) 𝐶 is a fixed point of 𝐽, that is, 𝐶(𝑚𝑥)=𝑚³𝐶(𝑥)(4.35) for all 𝑥∈𝑋.
The mapping 𝐶 is a unique fixed point of 𝐽 in the set Ω={ℎ∈𝑆∶𝑑(𝑔,ℎ)<∞}.(4.36) This implies that 𝐶 is a unique mapping satisfying (4.35) such that there exists 𝑢∈(0,∞) satisfying 𝜇_{𝑓(𝑥)−𝐶(𝑥)}(𝑢𝑡)≥Φ_𝑥,0(𝑡)(4.37) for all 𝑥∈𝑋 and 𝑡>0.
(2) 𝑑(𝐽^𝑛𝑓,𝐶)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞𝑓(𝑚^𝑛𝑥)𝑚^3𝑛=𝐶(𝑥)(4.38) for all 𝑥∈𝑋.
(3) 𝑑(𝑓,𝐶)≤𝑑(𝑓,𝐽𝑓)/(1−𝛼/𝑚³) with 𝑓∈Ω, which implies the inequality 𝑑(𝑓,𝐶)≤1/(2𝑚³−2𝛼), and so 𝜇_{𝑓(𝑥)−𝐶(𝑥)}𝑡2𝑚³−2𝛼≥Φ^𝑥,0(𝑡)(4.39) for all 𝑥∈𝑋 and 𝑡>0. The rest of the proof is similar to the proof of Theorem 4.1.

Corollary 4.4. Let 𝑋 be a real normed space, 𝜃≥0, and 𝑝 a real number with 𝑝∈(0,1). Let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying (4.23). Then, for all 𝑥∈𝑋, the limit 𝐶(𝑥)=lim_𝑛→∞𝑓(𝑚^𝑛𝑥)/𝑚^3𝑛 exists and 𝐶∶𝑋→𝑌 is a unique cubic mapping such that 𝜇_{𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥2𝑚³−𝑚^3𝑝𝑡2𝑚³−𝑚^3𝑝𝑡+𝜃‖𝑥‖^𝑝(4.40) for all 𝑥∈𝑋 and 𝑡>0.

Proof. The proof follows from Theorem 6.3 if we take Φ_𝑥,𝑦(𝑡)=𝑡𝑡+𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(4.41) for all 𝑥,𝑦∈𝑋 and 𝑡>0. In fact, if we choose 𝛼=𝑚^3𝑝, then we get the desired result.

Remark 4.5. In Corollaries 4.2 and 4.4, if we assume that Φ_𝑥,𝑦(𝑡)=𝑡/(𝑡+𝜃(‖𝑥‖^𝑝⋅‖𝑦‖^𝑝)) or Φ_𝑥,𝑦(𝑡)=𝑡/(𝑡+𝜃(‖𝑥‖^𝑝+𝑞+‖𝑦‖^𝑝+𝑞+‖𝑥‖^𝑝‖𝑦‖^𝑞)), then we get Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability, respectively. But, since we put 𝑦=0 in this functional equation, the Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability corollaries will be obvious. Meanwhile, the JMRassias mixed product-sum stability when 𝑝+𝑞=3 is an open question.

5. Non-Archimedean Stability of Functional Equation (1.9): A Direct Method

In this section, using direct method, we prove the generalized Hyers-Ulam stability of cubic functional equation (1.9) in non-Archimedean normed spaces. Throughout this section, we assume that 𝐺 is an additive semigroup and 𝑋 is a complete non-Archimedean space.

Theorem 5.1. Let 𝜁∶𝐺²→[0,+∞) be a function such that lim_𝑛→∞|𝑚|^3𝑛𝜁𝑥𝑚^𝑛,𝑦𝑚^𝑛=0(5.1) for all 𝑥,𝑦∈𝐺 and let for each 𝑥∈𝐺 the limit Θ(𝑥)=lim_𝑛→∞max|𝑚|^3(𝑘+1)𝜁𝑥𝑚^𝑘+1,0;0≤𝑘<𝑛(5.2) exist. Suppose that 𝑓∶𝐺→𝑋 a mapping with 𝑓(0)=0 and satisfying the following inequality: ‖‖𝑓(𝑚𝑥±𝑦)−𝑚𝑓(𝑥±𝑦)−2𝑚³−𝑚𝑓(𝑥)‖‖≤𝜁(𝑥,𝑦,𝑧).(5.3) Then, the limit 𝐶(𝑥)∶=lim_𝑛→∞𝑚^3𝑛𝑄(𝑥/𝑚^𝑛) exists for all 𝑥∈𝐺 and defines a cubic mapping 𝐶∶𝐺→𝑋 such that ‖𝑓(𝑥)−𝐶(𝑥)‖≤Θ(𝑥)||2𝑚³||.(5.4) Moreover, if lim_𝑗→∞lim_𝑛→∞max|𝑚|^3(𝑘+1)𝜁𝑥𝑚^𝑘+1,0;𝑗≤𝑘<𝑛+𝑗=0,(5.5) then 𝐶 is the unique cubic mapping satisfying (5.4).

Proof. Putting 𝑦=0 in (5.3), we get ‖‖𝑓(𝑚𝑥)−𝑚³𝑓(𝑥)‖‖≤𝜁(𝑥,0)||2||(5.6) for all 𝑥∈𝐺. Replacing 𝑥 by 𝑥/𝑚^𝑛+1 in (5.6), we obtain ‖‖‖𝑚_3𝑛+3𝑓𝑥𝑚^𝑛+1−𝑚^3𝑛𝑓𝑥𝑚^𝑛‖‖‖≤|𝑚||2|||^3𝑛𝜁𝑚𝑥^𝑛+1,0.(5.7) It follows from (5.1) and (5.7) that the sequence {𝑚^3𝑛𝑓(𝑥/𝑚^𝑛)}_𝑛≥1 is a Cauchy sequence. Since 𝑋 is complete, {𝑚^3𝑛𝑓(𝑥/𝑚^𝑛)}_𝑛≥1 is convergent. Set 𝐶(𝑥)∶=lim_𝑛→∞𝑚^3𝑛𝑓(𝑥/𝑚^𝑛).
Using induction, one can show that ‖‖‖𝑚_3𝑛𝑓𝑥𝑚^𝑛−𝑓(𝑥)‖‖‖≤||2𝑚1₃||max|𝑚|^3(𝑘+1)𝜁𝑥𝑚^𝑘+1,0;0≤𝑘<𝑛(5.8) for all 𝑛∈ℕ and all 𝑥∈𝐺. By taking 𝑛 to approach infinity in (5.8) and using (5.2), one obtains (5.4). By (5.1) and (5.3), we get ‖‖𝐶(=𝑚𝑥li±m_𝑛→∞𝑦)−𝑚𝐶(𝑥±𝑦)−2𝑚³−𝑚𝐶(𝑥)‖‖‖‖‖𝑚_3𝑛𝑓𝑚𝑥±𝑦𝑚^𝑛−𝑚^3𝑛+1𝑓𝑥±𝑦𝑚^𝑛−2𝑚^3𝑛𝑚³−𝑚𝑓𝑥≤lim_𝑛→∞𝑚^𝑛‖‖‖|𝑚|^3𝑛𝜁𝑥𝑚^𝑛,𝑦=0𝑚^𝑛(5.9) for all 𝑥,𝑦∈𝐺. Therefore, the function 𝐶∶𝐺→𝑋 satisfies (1.9). To prove the uniqueness property of 𝐶, let 𝐿∶𝐺→𝑋 be another function satisfying (5.4). Then, ‖𝐶(𝑥)−𝐿(𝑥)‖=lim_𝑗→∞|𝑚|^3𝑗‖‖‖𝐶𝑥≤lim_𝑗→∞𝑚^𝑗−𝐿𝑚𝑥_𝑗‖‖‖|𝑚|^3𝑗max‖‖‖𝐶𝑥≤lim_𝑗→∞𝑚^𝑗−𝑓𝑚𝑥_𝑗‖‖‖,‖‖‖𝑓𝑚𝑥_𝑗−𝐿𝑚𝑥_𝑗‖‖‖lim_𝑛→∞max|𝑚|^3(𝑘+1)𝜁𝑥𝑚^𝑘+1=0,0;𝑗≤𝑘<𝑛+𝑗(5.10) for all 𝑥∈𝐺. Therefore, 𝐶=𝐿, and the proof is complete.

Corollary 5.2. Let 𝜉∶[0,∞)→[0,∞) be a function satisfying 𝜉𝑡|𝑚|≤𝜉|𝑚1|𝜉(𝑡)(𝑡≥0)𝜉|𝑚1|<|𝑚|⁻³.(5.11) Let 𝜅>0, and let 𝑓∶𝐺→𝑋 be a mapping with 𝑓(0)=0 and satisfying the following inequality: ‖‖𝑓(𝑚𝑥±𝑦)−𝑚𝑓(𝑥±𝑦)−2𝑚³−𝑚𝑓(𝑥)‖‖≤𝜅𝜉(|𝑥|)+𝜉||𝑦||(5.12) for all 𝑥,𝑦∈𝐺. Then there exists a unique cubic mapping 𝐶∶𝐺→𝑋 such that ‖𝑓(𝑥)−𝐶(𝑥)‖≤𝜅𝜉(|𝑥|)||2𝑚³||.(5.13)

Proof. Defining 𝜁∶𝐺²→[0,∞) by 𝜁(𝑥,𝑦)∶=𝜅(𝜉(|𝑥|)+𝜉(|𝑦|)), we have lim_𝑛→∞|𝑚|^3𝑛𝜁𝑥𝑚^𝑛,𝑦𝑚^𝑛≤lim_𝑛→∞|𝑚|³𝜉1|𝑚|^𝑛𝜁(𝑥,𝑦)=0(5.14) for all 𝑥,𝑦∈𝐺. The last equality comes from the fact that |𝑚|³𝜉(1/|𝑚|)<1. On the other hand, Θ(𝑥)=lim_𝑛→∞max|𝑚|^3𝑘+3𝜁𝑥𝑚^𝑘+1,0;0≤𝑘<𝑛=|𝑚|³𝜁𝑥𝑚,0=𝜅𝜉(|𝑥|),(5.15) for all 𝑥∈𝐺, exists. Also, lim_𝑗→∞lim_𝑛→∞max|𝑚|^3𝑘+3𝜁𝑥𝑚^𝑘+1,0;𝑗≤𝑘<𝑛+𝑗=lim_𝑗→∞|𝑚|^3𝑗+3𝜁𝑥𝑚^𝑗+1,0=0.(5.16) Applying Theorem 5.1, we get the desired result.

Theorem 5.3. Let 𝜁∶𝐺³→[0,+∞) be a function such that lim_𝑛→∞𝜁(𝑚^𝑛𝑥,𝑚^𝑛𝑦)|𝑚|^3𝑛=0(5.17) for all 𝑥,𝑦∈𝐺, and let for each 𝑥∈𝐺 the limit Θ(𝑥)=lim_𝑛→∞max𝜁𝑚^𝑘𝑥,0|𝑚|^3𝑘+3;0≤𝑘<𝑛(5.18) exist. Suppose that 𝑓∶𝐺→𝑋 a mapping with 𝑓(0)=0 and satisfying (5.3). Then, the limit 𝐶(𝑥)∶=lim_𝑛→∞𝑓(𝑚^𝑛𝑥)/𝑚^3𝑛 exists for all 𝑥∈𝐺 and defines a cubic mapping 𝐶∶𝐺→𝑋 such that ‖𝑓(𝑥)−𝐶(𝑥)‖≤1||2||Θ(𝑥).(5.19) Moreover, if lim_𝑗→∞lim_𝑛→∞max𝜁𝑚^𝑘𝑥,0|𝑚|^3𝑘+3;𝑗≤𝑘<𝑛+𝑗=0,(5.20) then 𝐶 is the unique cubic mapping satisfying (5.19).

Proof. Putting 𝑦=0 in (5.3), we get ‖‖‖𝑓(𝑥)−𝑓(𝑚𝑥)𝑚³‖‖‖≤𝜁(𝑥,0)||2𝑚³||(5.21) for all 𝑥∈𝐺. Replacing 𝑥 by 𝑚^𝑛𝑥 in (5.21), we obtain ‖‖‖‖𝑓(𝑚^𝑛𝑥)𝑚^3𝑛−𝑓𝑚^𝑛+1𝑥𝑚^3𝑛+3‖‖‖‖≤𝜁(𝑚^𝑛𝑥,0)||2|||𝑚|^3𝑛+3.(5.22) It follows from (5.17) and (5.22) that the sequence {𝑓(𝑚^𝑛𝑥)/𝑚^3𝑛}_𝑛≥1 is convergent. Set 𝐶(𝑥)∶=lim_𝑛→∞𝑓(𝑚^𝑛𝑥)/𝑚^3𝑛. On the other hand, it follows from (5.22) that ‖‖‖𝑓(𝑚^𝑝𝑥)𝑚^3𝑝−𝑓(𝑚^𝑞𝑥)𝑚^3𝑞‖‖‖=‖‖‖‖^𝑞−1_𝑘=𝑝𝑓𝑚^𝑘+1𝑥𝑚^3𝑘+3−𝑓𝑚^𝑘𝑥𝑚^3𝑘≤max‖‖‖‖𝑓𝑚_𝑘+1‖‖‖‖𝑥𝑚^3𝑘+3−𝑓𝑚^𝑘𝑥𝑚^3𝑘≤1‖‖‖‖;𝑝≤𝑘<𝑞−1||2||max𝜁𝑚^𝑘𝑥,0|𝑚|^3𝑘+3;𝑝≤𝑘<𝑞(5.23) for all 𝑥∈𝐺 and all nonnegative integers 𝑝,𝑞 with 𝑞>𝑝≥0. Letting 𝑝=0, passing the limit 𝑞→∞ in the last inequality, and using (5.18), we obtain (5.19). The rest of the proof is similar to the proof of Theorem 5.1.

6. Non-Archimedean Stability of Functional Equation (1.9): A Fixed Point Method

In this section, using the fixed point alternative approach, we prove the generalized Hyers-Ulam stability of cubic functional equation (1.9) in non-Archimedean normed spaces. Throughout this section, let 𝑋 be a non-Archimedean normed space that 𝑌 a complete non-Archimedean normed space. Also, |2𝑚³|≠1.

Theorem 6.1. Let 𝜁∶𝑋²→[0,∞) be a function such that there exists an 𝐿<1 with 𝜁𝑥𝑚,𝑚𝑦≤𝐿𝜁||𝑚(𝑥³,||𝑦)(6.1) for all 𝑥,𝑦∈𝑋. Let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 and satisfying the following inequality: ‖‖𝑓(𝑚𝑥±𝑦)−𝑚𝑓(𝑚