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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 546819, 28 pages
http://dx.doi.org/10.1155/2012/546819
Research Article

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

1Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran
2Department of Mathematics, Islamic Azad University, Firoozabad Branch, Firoozabad, Iran
3Department of Mathematics, Daejin University, Kyeonggi 487-711, Republic of Korea

Received 13 September 2011; Revised 5 November 2011; Accepted 6 November 2011

Academic Editor: Hui-ShenΒ Shen

Copyright Β© 2012 H. Azadi Kenary et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In 1940 and 1964, Ulam proposed the general problem: β€œWhen 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(π‘šπ‘₯+𝑦)+𝑓(π‘šπ‘₯βˆ’π‘¦)=π‘šπ‘“(π‘₯+𝑦)+π‘šπ‘“(π‘₯βˆ’π‘¦)+2(π‘š3βˆ’π‘š)𝑓(π‘₯),π‘šβ‰₯2 in various normed spaces.

1. Introduction

A classical question in the theory of functional equations is the following: β€œWhen 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 π‘“βˆΆπΈβ†’πΈξ…ž be a mapping from a normed vector space 𝐸 into a Banach space πΈξ…ž subject to the inequality ‖𝑓(π‘₯+𝑦)βˆ’π‘“(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœ–β€–π‘₯‖𝑝+‖𝑦‖𝑝(1.1) for all π‘₯,π‘¦βˆˆπΈ, where πœ– and 𝑝 are constants with πœ–>0 and 𝑝<1. Then, the limit 𝐿(π‘₯)=limπ‘›β†’βˆžπ‘“(2𝑛π‘₯)2𝑛(1.2) exists for all π‘₯∈𝐸 and πΏβˆΆπΈβ†’πΈβ€² is the unique additive mapping which satisfies ‖𝑓(π‘₯)βˆ’πΏ(π‘₯)‖≀2πœ–2βˆ’2𝑝‖π‘₯‖𝑝(1.3) for all π‘₯∈𝐸. If 𝑝<0, then inequality (1.1) holds for π‘₯,𝑦≠0 and (1.3) for π‘₯β‰ 0. Also, if for each π‘₯∈𝐸 the mapping 𝑑↦𝑓(𝑑π‘₯) is continuous in π‘‘βˆˆβ„, 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 GΔƒvruΕ£a [4] by replacing the bound πœ–(β€–π‘₯‖𝑝+‖𝑦‖𝑝) by a general control function πœ‘(π‘₯,𝑦).

The functional equation𝑓(π‘₯+𝑦)+𝑓(π‘₯βˆ’π‘¦)=2𝑓(π‘₯)+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 π‘“βˆΆπ‘‹β†’π‘Œ, where 𝑋 is a normed space and π‘Œ 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 π‘“βˆΆπΈβ†’πΈξ…ž be a mapping from a real normed vector space 𝐸 into a Banach space πΈξ…ž subject to the inequality ‖𝑓(π‘₯+𝑦)βˆ’π‘“(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœ–β€–π‘₯β€–π‘β€–π‘¦β€–π‘ž(1.5) for all π‘₯,π‘¦βˆˆπΈ, where πœ– and π‘Ÿ=𝑝+π‘ž are constants with πœ–>0 and π‘Ÿβ‰ 1. Then, πΏβˆΆπΈβ†’πΈξ…ž is the unique additive mapping which satisfies πœ–β€–π‘“(π‘₯)βˆ’πΏ(π‘₯)‖≀2βˆ’2π‘Ÿβ€–π‘₯β€–π‘Ÿ(1.6) for all π‘₯∈𝐸.

However, there was a singular case, for this singularity a counterexample was given by GΔƒvruΕ£a [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(2π‘₯+𝑦)+𝑓(2π‘₯βˆ’π‘¦)=2𝑓(π‘₯+𝑦)+2𝑓(π‘₯βˆ’π‘¦)+12𝑓(π‘₯),(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(3π‘₯+𝑦)+𝑓(3π‘₯βˆ’π‘¦)=3𝑓(π‘₯+𝑦)+3𝑓(π‘₯βˆ’π‘¦)+48𝑓(π‘₯),(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(π‘₯)=π‘₯3 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:ξ€·π‘šπ‘“(π‘šπ‘₯+𝑦)+𝑓(π‘šπ‘₯βˆ’π‘¦)=π‘šπ‘“(π‘₯+𝑦)+π‘šπ‘“(π‘₯βˆ’π‘¦)+23ξ€Έβˆ’π‘šπ‘“(π‘₯).(1.9) where π‘š 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 Ξ”+. Elements of Ξ”+ are functions πΉβˆΆβ„βˆͺ{βˆ’βˆž,+∞}β†’[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+={πΉβˆˆΞ”+βˆΆπ‘™βˆ’πΉ(+∞)=1}, where π‘™βˆ’π‘“(π‘₯)=lim𝑑→π‘₯βˆ’π‘“(𝑑), is a subset of Ξ”+. The space Ξ”+ is partially ordered by the usual pointwise ordering of functions, that is, for all π‘‘βˆˆπ‘…, 𝐹≀𝐺 if and only if 𝐹(𝑑)≀𝐺(𝑑). For every π‘Žβ‰₯0, π»π‘Ž(𝑑) is the element of 𝐷+ defined byπ»π‘Žξ‚»(𝑑)=0,ifπ‘‘β‰€π‘Ž,1,if𝑑>π‘Ž.(2.1) One can easily show that the maximal element for Ξ”+ in this order is the distribution function 𝐻0(𝑑).

Definition 2.1. A functionβ€‰β€‰π‘‡βˆΆ[0,1]2β†’[0,1]  is a continuous triangular norm (briefly  a 𝑑-norm) if 𝑇satisfies the following conditions:(i)𝑇 is commutative and associative;(ii)𝑇 is continuous;(iii)𝑇(π‘₯,1)=π‘₯ for all π‘₯∈[0,1];(iv)𝑇(π‘₯,𝑦)≀𝑇(𝑧,𝑀) whenever π‘₯≀𝑧 and 𝑦≀𝑀 for all π‘₯,𝑦,𝑧,π‘€βˆˆ[0,1].

Three typical examples of continuous 𝑑-norms are 𝑇(π‘₯,𝑦)=π‘₯𝑦,𝑇(π‘₯,𝑦)=max{π‘Ž+π‘βˆ’1,0}, and 𝑇(π‘₯,𝑦)=min(π‘Ž,𝑏). Recall that, if 𝑇 is a 𝑑-norm and {π‘₯𝑛} is a given group of numbers in [0,1], 𝑇𝑛𝑖=1π‘₯𝑖 is defined recursively by 𝑇1𝑖=1π‘₯1 and 𝑇𝑛𝑖=1π‘₯𝑖=𝑇(π‘‡π‘›βˆ’1𝑖=1π‘₯𝑖,π‘₯𝑛) for 𝑛β‰₯2.

Definition 2.2. A random normed space (briefly  RN-space) is a triple (𝑋,πœ‡ξ…ž,𝑇), where  𝑋  is a vector space,  𝑇  is a continuous 𝑑-norm andβ€‰β€‰πœ‡ξ…žβˆΆπ‘‹β†’π·+  is a mapping such that the following conditions hold:(i)β€‰πœ‡ξ…žπ‘₯(𝑑)=𝐻0(𝑑) for all 𝑑>0 if and only if π‘₯=0;(ii)πœ‡ξ…žπ›Όπ‘₯(𝑑)=πœ‡ξ…žπ‘₯(𝑑/|𝛼|) for all π›Όβˆˆβ„, 𝛼≠0, π‘₯βˆˆπ‘‹ and 𝑑β‰₯0;(iii)πœ‡ξ…žπ‘₯+𝑦(𝑑+𝑠)β‰₯𝑇(πœ‡ξ…žπ‘₯(𝑑),πœ‡ξ…žπ‘¦(𝑠)), for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑,𝑠β‰₯0.

Every normed space (𝑋,||β‹…||) defines a random normed space (𝑋,πœ‡ξ…ž,𝑇𝑀) where, for every 𝑑>0, πœ‡ξ…žπ‘’(𝑑𝑑)=𝑑+‖𝑒‖(2.2) and 𝑇𝑀 is the minimum 𝑑-norm. This space is called the induced random normed space.

If the 𝑑-norm 𝑇 is such that sup0<π‘Ž<1𝑇(π‘Ž,π‘Ž)=1, then every RN-space (𝑋,πœ‡β€²,𝑇) is a metrizable linear topological space with the topology 𝜏 (called the πœ‡β€²-topology or the (πœ–,𝛿)-topology) induced by the base of neighborhoods of πœƒ, {π‘ˆ(πœ–,πœ†)βˆ£πœ–>0,πœ†βˆˆ(0,1)}, whereπ‘ˆξ€½(πœ–,πœ†)=π‘₯βˆˆπ‘‹βˆ£πœ‡ξ…žπ‘₯ξ€Ύ(πœ–)>1βˆ’πœ†.(2.3)

Definition 2.3. Let  (𝑋,πœ‡ξ…ž,𝑇)   be an RN-space. (i)A sequence {π‘₯𝑛} in 𝑋 is said to be convergent to π‘₯βˆˆπ‘‹ in 𝑋 if, for all 𝑑>0, limπ‘›β†’βˆžπœ‡ξ…žπ‘₯π‘›βˆ’π‘₯(𝑑)=1.(ii)A sequence {π‘₯𝑛} in 𝑋 is said to be Cauchy sequence in 𝑋 if, for all 𝑑>0, limπ‘›β†’βˆžπœ‡ξ…žπ‘₯π‘›βˆ’π‘₯π‘š(𝑑)=1.(iii) The RN-space (𝑋,πœ‡ξ…ž,𝑇) is said to be complete if every Cauchy sequence in 𝑋 is convergent.

Theorem 2.4. If (𝑋,πœ‡ξ…ž,𝑇) is RN-space and {π‘₯𝑛} is a sequence such that π‘₯𝑛→π‘₯, then limπ‘›β†’βˆžπœ‡ξ…žπ‘₯𝑛(𝑑)=πœ‡ξ…žπ‘₯(𝑑).

A valuation is a function |β‹…| from a field 𝕂 into [0,∞) such that 0 is the unique element having the 0 valuation, |π‘Ÿπ‘ |=|π‘Ÿ||𝑠|, and the triangle inequality holds, that is, |π‘Ÿ+𝑠|≀max{|π‘Ÿ|,|𝑠|}.(2.4) A field 𝕂 is called a valued field if 𝕂 carries a valuation. The usual absolute values of ℝ and β„‚ 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|π‘Ÿ+𝑠|≀max{|π‘Ÿ|,|𝑠|}(2.5) for all π‘Ÿ,π‘ βˆˆπ•‚, then the function |β‹…| is called a non-Archimedean valuation and the field is called a non-Archimedean field. Clearly, |1|=|βˆ’1|=1 and |𝑛|≀1 for all 𝑛β‰₯1. A trivial example of a non-Archimedean valuation is the function |β‹…| 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  |β‹…|. A functionβ€‰β€‰β€–β‹…β€–βˆΆπ‘‹β†’[0,∞) is called a non-Archimedean norm if the following conditions hold:(a)β€–π‘₯β€–=0 if and only if π‘₯=0 for all π‘₯βˆˆπ‘‹;(b)β€–π‘Ÿπ‘₯β€–=|π‘Ÿ|β€–π‘₯β€– for all π‘Ÿβˆˆπ•‚ and π‘₯βˆˆπ‘‹;(c)the strong triangle inequality holds: β€–π‘₯+𝑦‖≀max{β€–π‘₯β€–,‖𝑦‖}(2.6) for all π‘₯,π‘¦βˆˆπ‘‹. Then (𝑋,β€–β‹…β€–) is called a non-Archimedean normed space.

Definition 2.6. Let  {π‘₯𝑛} be a sequence in a non-Archimedean normed space  𝑋. (a)A sequence {π‘₯𝑛}βˆžπ‘›=1 in a non-Archimedean space is a Cauchy sequence if and only if, the sequence {π‘₯𝑛+1βˆ’π‘₯𝑛}βˆžπ‘›=1 converges to zero.(b)The sequence {π‘₯𝑛} is said to be convergent if, for any πœ€>0, there are a positive integer 𝑁 and π‘₯βˆˆπ‘‹ such that β€–β€–π‘₯π‘›β€–β€–βˆ’π‘₯β‰€πœ€(2.7) for all 𝑛β‰₯𝑁. Then, the point π‘₯βˆˆπ‘‹ is called the limit of the sequence {π‘₯𝑛}, which is denoted by limπ‘›β†’βˆžπ‘₯𝑛=π‘₯.(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 π‘‘βˆΆπ‘‹Γ—π‘‹β†’[0,∞] is called a generalized metric on 𝑋 if 𝑑 satisfies the following conditions:(a)𝑑(π‘₯,𝑦)=0 if and only if π‘₯=𝑦 for all π‘₯,π‘¦βˆˆπ‘‹;(b)𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯) for all π‘₯,π‘¦βˆˆπ‘‹;(c)𝑑(π‘₯,𝑧)≀𝑑(π‘₯,𝑦)+𝑑(𝑦,𝑧) for all π‘₯,𝑦,π‘§βˆˆπ‘‹.

Theorem 2.8. Let (𝑋,𝑑) be a complete generalized metric space and π½βˆΆπ‘‹β†’π‘‹ a strictly contractive mapping with Lipschitz constant 𝐿<1. Then, for all π‘₯βˆˆπ‘‹, either 𝑑𝐽𝑛π‘₯,𝐽𝑛+1π‘₯ξ€Έ=∞(2.8) for all nonnegative integers 𝑛 or there exists a positive integer 𝑛0 such that(a)𝑑(𝐽𝑛π‘₯,𝐽𝑛+1π‘₯)<∞ for all 𝑛0β‰₯𝑛0;(b)the sequence {𝐽𝑛π‘₯} converges to a fixed point π‘¦βˆ— of 𝐽;(c)π‘¦βˆ— is the unique fixed point of 𝐽 in the set π‘Œ={π‘¦βˆˆπ‘‹βˆΆπ‘‘(𝐽𝑛0π‘₯,𝑦)<∞};(d)𝑑(𝑦,π‘¦βˆ—)≀(1/(1βˆ’πΏ))𝑑(𝑦,𝐽𝑦) for all π‘¦βˆˆπ‘Œ.

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 π‘“βˆΆπΈ1→𝐸2 satisfies the functional equation (1.7) if and only if π‘“βˆΆπΈ1→𝐸2 satisfies the functional equation (1.9). Therefore, every solution of functional equation (1.9) is also cubic function.

Proof. Let π‘“βˆΆπΈ1→𝐸2 satisfy the equation (1.7). Putting π‘₯=𝑦=0 in (1.7), we get 𝑓(0)=0. Set 𝑦=0 in (1.7) to get 𝑓(βˆ’π‘¦)=βˆ’π‘“(𝑦). By induction, we lead to 𝑓(π‘˜π‘₯)=π‘˜3𝑓(π‘₯) for all positive integer π‘˜. Replacing 𝑦 by π‘₯+𝑦 in (1.7), we have 𝑓(3π‘₯+𝑦)+𝑓(π‘₯βˆ’π‘¦)=2𝑓(2π‘₯+𝑦)βˆ’2𝑓(βˆ’π‘¦)+12𝑓(π‘₯),(3.1) for all π‘₯,π‘¦βˆˆπΈ1. Once again replacing 𝑦 by π‘¦βˆ’π‘₯ in (1.7), we have 𝑓(π‘₯+𝑦)+𝑓(3π‘₯βˆ’π‘¦)=2𝑓(𝑦)+2𝑓(2π‘₯βˆ’π‘¦)+12𝑓(π‘₯),(3.2) for all π‘₯,π‘¦βˆˆπΈ1. Adding (3.1) to (3.2) and using (1.7), we obtain 𝑓(3π‘₯+𝑦)+𝑓(3π‘₯βˆ’π‘¦)=3𝑓(π‘₯+𝑦)+3𝑓(π‘₯βˆ’π‘¦)+48𝑓(π‘₯),(3.3) for all π‘₯,π‘¦βˆˆπΈ1. By using the previous method, by induction, we infer that ξ€·π‘šπ‘“(π‘šπ‘₯+𝑦)+𝑓(π‘šπ‘₯βˆ’π‘¦)=π‘šπ‘“(π‘₯+𝑦)+π‘šπ‘“(π‘₯βˆ’π‘¦)+23ξ€Έβˆ’π‘šπ‘“(π‘₯),(3.4) for all π‘₯,π‘¦βˆˆπΈ1 and each positive integer π‘šβ‰₯3.
Let π‘“βˆΆπΈ1→𝐸2 satisfy the functional equation (1.9) with the positive integer π‘šβ‰₯3. Putting π‘₯=𝑦=0 in (1.9), we get 𝑓(0)=0. Setting π‘₯=0, we get 𝑓(βˆ’π‘¦)=βˆ’π‘“(𝑦). Let π‘˜ be a positive integer. Replacing 𝑦 by π‘˜π‘₯+𝑦 in (1.9), we have ξ€·π‘šπ‘“((π‘š+π‘˜)π‘₯+𝑦)+𝑓((π‘šβˆ’π‘˜)π‘₯βˆ’π‘¦)=π‘šπ‘“((π‘˜+1)π‘₯+𝑦)βˆ’π‘šπ‘“((π‘˜βˆ’1)π‘₯+𝑦)+23ξ€Έβˆ’π‘šπ‘“(π‘₯),(3.5) for all π‘₯,π‘¦βˆˆπΈ1. Replacing 𝑦 by π‘¦βˆ’π‘˜π‘₯ in (1.9), we have ξ€·π‘šπ‘“((π‘šβˆ’π‘˜)π‘₯+𝑦)+𝑓((π‘š+π‘˜)π‘₯βˆ’π‘¦)=π‘šπ‘“((π‘˜+1)π‘₯βˆ’π‘¦)βˆ’π‘šπ‘“((π‘˜βˆ’1)π‘₯βˆ’π‘¦)+23ξ€Έβˆ’π‘šπ‘“(π‘₯),(3.6) for all π‘₯,π‘¦βˆˆπΈ1. Adding (3.5) to (3.6), we obtain ξ€·π‘šπ‘“((π‘š+π‘˜)π‘₯+𝑦)+𝑓((π‘šβˆ’π‘˜)π‘₯βˆ’π‘¦)+𝑓((π‘šβˆ’π‘˜)π‘₯+𝑦)+𝑓((π‘š+π‘˜)π‘₯βˆ’π‘¦)=π‘šπ‘“((π‘˜+1)π‘₯+𝑦)+π‘šπ‘“((π‘˜+1)π‘₯βˆ’π‘¦)βˆ’π‘š(𝑓((π‘˜βˆ’1)π‘₯+𝑦)+𝑓((π‘˜βˆ’1)π‘₯βˆ’π‘¦))+23ξ€Έβˆ’π‘šπ‘“(π‘₯),(3.7) for all π‘₯,π‘¦βˆˆπΈ1 and for all integer π‘˜β‰₯1. Let πœ“π‘š(π‘₯,𝑦)=𝑓(π‘šπ‘₯+𝑦)+𝑓(π‘šπ‘₯βˆ’π‘¦) for each integer π‘šβ‰₯0. Then, (3.7) means that πœ“π‘š+π‘˜(π‘₯,𝑦)+πœ“π‘šβˆ’π‘˜(π‘₯,𝑦)=π‘šπœ“π‘˜+1(π‘₯,𝑦)βˆ’π‘šπœ“π‘˜βˆ’1ξ€·π‘š(π‘₯,𝑦)+43ξ€Έβˆ’π‘šπ‘“(π‘₯),(3.8) for all π‘₯,π‘¦βˆˆπΈ1 and for all integer π‘˜β‰₯1. For π‘˜=1 and π‘˜=π‘š in (3.8), we obtain πœ“π‘š+1(π‘₯,𝑦)+πœ“π‘šβˆ’1(π‘₯,𝑦)=π‘šπœ“2ξ€·π‘š(π‘₯,𝑦)+43ξ€Έπœ“βˆ’π‘šπ‘“(π‘₯),2π‘š(π‘₯,𝑦)=π‘šπœ“π‘š+1(π‘₯,𝑦)βˆ’π‘šπœ“π‘šβˆ’1ξ€·π‘š(π‘₯,𝑦)+43ξ€Έπ‘“βˆ’π‘š(π‘₯),(3.9) for all π‘₯,π‘¦βˆˆπΈ1. By the proof of the first part, since π‘“βˆΆπΈ1→𝐸2 satisfies the functional equation (1.9) with the positive integer π‘šβ‰₯3, then 𝑓 satisfies the functional equation (1.9) with the positive integer π‘˜β‰₯π‘š. It follows from (3.9) that 𝑓 satisfies the functional equation (1.7) and +𝑓((π‘šβˆ’1)π‘₯+𝑦)+𝑓((π‘šβˆ’1)π‘₯βˆ’π‘¦)=(π‘šβˆ’1)𝑓(π‘₯+𝑦)(π‘šβˆ’1)𝑓(π‘₯βˆ’π‘¦)+2(π‘šβˆ’1)3βˆ’ξ€Έπ‘“(π‘šβˆ’1)(π‘₯).(3.10)

Theorem 3.2. Let 𝑋 be a real linear space, (𝑍,πœ‡ξ…ž,min) an RN-space, and πœ“βˆΆπ‘‹2→𝑍 a function such that, for some 0<𝛼<π‘š3, πœ‡ξ…žπœ“(π‘šπ‘₯,0)(𝑑)β‰₯πœ‡ξ…žπ›Όπœ“(π‘₯,0)(𝑑)βˆ€π‘₯βˆˆπ‘‹,𝑑>0,(3.11) and, for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0, limπ‘›β†’βˆžπœ‡ξ…žπœ“(π‘šπ‘›π‘₯,π‘šπ‘›π‘¦)/π‘š3𝑛(𝑑)=1. Let (π‘Œ,πœ‡,min) be a complete RN-space. If π‘“βˆΆπ‘‹β†’π‘Œ is a mapping with 𝑓(0)=0 such that for all π‘₯βˆˆπ‘‹ and 𝑑>0πœ‡π‘“(π‘šπ‘₯±𝑦)βˆ’π‘šπ‘“(π‘₯±𝑦)βˆ’2(π‘š3βˆ’π‘š)𝑓(π‘₯)(𝑑)β‰₯πœ‡ξ…žπœ“(π‘₯,𝑦)(𝑑),(3.12) then the limit 𝐢(π‘₯)=limπ‘›β†’βˆž(𝑓(π‘šπ‘›π‘₯)/π‘š3𝑛) exists for all π‘₯βˆˆπ‘‹ and defines a unique cubic mapping πΆβˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)(𝑑)β‰₯πœ‡ξ…žπœ“(π‘₯,0)/2(π‘š3βˆ’π›Ό)(𝑑).(3.13)

Proof. Putting 𝑦=0 in (3.12) we see that, for all π‘₯βˆˆπ‘‹, πœ‡π‘“(π‘šπ‘₯)/π‘š3βˆ’π‘“(π‘₯)(𝑑)β‰₯πœ‡ξ…žπœ“(π‘₯,0)/2π‘š3(𝑑).(3.14) Replacing π‘₯ by π‘šπ‘›π‘₯ in (3.14) and using (3.11), we obtain πœ‡π‘“(π‘šπ‘›+1π‘₯)/π‘š3𝑛+3βˆ’π‘“(π‘šπ‘›π‘₯)/π‘š3𝑛(𝑑)β‰₯πœ‡ξ…žπœ“(π‘šπ‘›π‘₯,0)/2π‘š3𝑛+3(𝑑)β‰₯πœ‡ξ…žπ›Όπ‘›πœ“(π‘₯,0)/2π‘š3𝑛+3(𝑑).(3.15) So πœ‡(𝑓(π‘šπ‘›π‘₯)/π‘š3π‘›βˆ‘)βˆ’π‘“(π‘₯)/(π‘›βˆ’1π‘˜=0(π›Όπ‘˜/2π‘š3π‘˜+3))(𝑑)=πœ‡βˆ‘((π‘›βˆ’1π‘˜=0(𝑓(π‘šπ‘˜+1π‘₯)/π‘š3π‘˜+3))βˆ’(𝑓(π‘šπ‘˜π‘₯)/π‘š3π‘˜βˆ‘))/(π‘›βˆ’1π‘˜=0(π›Όπ‘˜/2π‘š3π‘˜+3))(𝑑)β‰₯π‘‡π‘›βˆ’1π‘˜=0ξ€·πœ‡((𝑓(π‘šπ‘˜+1π‘₯)/π‘š3π‘˜+3)βˆ’(𝑓(π‘šπ‘˜π‘₯)/π‘š3π‘˜))/(π›Όπ‘˜/2π‘š3π‘˜+3)ξ€Έ(𝑑)=π‘‡π‘›βˆ’1π‘˜=0ξ‚€πœ‡ξ…žπœ“(π‘₯,0)(𝑑)=πœ‡ξ…žπœ“(π‘₯,0)(𝑑).(3.16) This implies that πœ‡π‘“(π‘šπ‘›π‘₯)/π‘š3π‘›βˆ’π‘“(π‘₯)(𝑑)β‰₯πœ‡ξ…žβˆ‘π‘›βˆ’1π‘˜=0(π›Όπ‘˜πœ“(π‘₯,0)/2π‘š3π‘˜+3)(𝑑).(3.17) Replacing π‘₯ by π‘šπ‘π‘₯ in (3.17), we obtain πœ‡π‘“(π‘šπ‘›+𝑝π‘₯)/π‘š3(𝑛+𝑝)βˆ’π‘“(π‘šπ‘π‘₯)/π‘š3𝑝(𝑑)β‰₯πœ‡ξ…žβˆ‘π‘›βˆ’1π‘˜=0(π›Όπ‘˜πœ“(π‘šπ‘π‘₯,0)/2π‘š3(π‘˜+𝑝)+3)(𝑑)β‰₯πœ‡ξ…žβˆ‘π‘›βˆ’1π‘˜=0(π›Όπ‘˜+π‘πœ“(π‘₯,0)/2π‘š3(π‘˜+𝑝)+3)(𝑑)=πœ‡ξ…žβˆ‘π‘›+π‘βˆ’1π‘˜=𝑝(π›Όπ‘˜πœ“(π‘₯,0)/2π‘š3π‘˜+3)(𝑑).(3.18) As lim𝑝,π‘›β†’βˆžπœ‡ξ…žβˆ‘π‘›+π‘βˆ’1π‘˜=𝑝(π›Όπ‘˜πœ“(π‘₯,0)/2π‘š3π‘˜+3)(𝑑)=1,(3.19){𝑓(π‘šπ‘›π‘₯)/π‘š3𝑛} is a Cauchy sequence in complete RN-space (π‘Œ,πœ‡,min), so there exists some point 𝐢(π‘₯)βˆˆπ‘Œ such that limπ‘›β†’βˆž(𝑓(π‘šπ‘›π‘₯)/π‘š3𝑛)=𝐢(π‘₯). Fix π‘₯βˆˆπ‘‹ and put 𝑝=0 in (3.18). Then, we obtain πœ‡π‘“(π‘šπ‘›π‘₯)/π‘š3π‘›βˆ’π‘“(π‘₯)(𝑑)β‰₯πœ‡ξ…žβˆ‘π‘›βˆ’1π‘˜=0(π›Όπ‘˜πœ“(π‘₯,0)/2π‘š3π‘˜+3)(𝑑),(3.20) and so, for every πœ–>0, we have πœ‡πΆ(π‘₯)βˆ’π‘“(π‘₯)ξ€·πœ‡(𝑑+πœ–)β‰₯𝑇𝐢(π‘₯)βˆ’π‘“(π‘šπ‘›π‘₯)/π‘š3𝑛(πœ–),πœ‡π‘“(π‘šπ‘›π‘₯)/π‘š3π‘›βˆ’π‘“(π‘₯)ξ€Έξ‚€πœ‡(𝑑)β‰₯𝑇𝐢(π‘₯)βˆ’π‘“(π‘šπ‘›π‘₯)/π‘š3𝑛(πœ–),πœ‡ξ…žβˆ‘π‘›βˆ’1π‘˜=0(π›Όπ‘˜πœ“(π‘₯,0)/2π‘š3π‘˜+3).(𝑑)(3.21) Taking the limit as π‘›β†’βˆž and using (3.21), we get πœ‡πΆ(π‘₯)βˆ’π‘“(π‘₯)(𝑑+πœ–)β‰₯πœ‡ξ…žπœ“(π‘₯,0)/2(π‘š3βˆ’π›Ό)(𝑑).(3.22) Since πœ– was arbitrary by taking πœ–β†’0 in (3.22), we get πœ‡πΆ(π‘₯)βˆ’π‘“(π‘₯)(𝑑)β‰₯πœ‡ξ…žπœ“(π‘₯,0)/2(π‘š3βˆ’π›Ό)(𝑑).(3.23) Replacing π‘₯ and 𝑦 by π‘šπ‘›π‘₯ and π‘šπ‘›π‘¦ in (3.12), respectively, we get, for all π‘₯,π‘¦βˆˆπ‘‹ and for all 𝑑>0, πœ‡(𝑓(π‘šπ‘›+1π‘₯Β±π‘šπ‘›π‘¦)βˆ’π‘šπ‘“(π‘šπ‘›π‘₯Β±π‘šπ‘›π‘¦)βˆ’2(π‘š3βˆ’π‘š)𝑓(π‘šπ‘›π‘₯))/π‘š3𝑛(𝑑)β‰₯πœ‡ξ…žπœ“(π‘šπ‘›π‘₯,π‘šπ‘›π‘¦)/π‘š3𝑛(𝑑).(3.24) Since limπ‘›β†’βˆžπœ‡ξ…žπœ“(π‘šπ‘›π‘₯,π‘šπ‘›π‘¦)/π‘š3𝑛(𝑑)=1, we conclude that 𝐢(π‘šπ‘₯±𝑦)=𝐢(π‘₯±𝑦)+2(π‘š3βˆ’π‘š)𝐢(π‘₯). To prove the uniqueness of the cubic mapping 𝐢, assume that there exists another cubic mapping πΏβˆΆπ‘‹β†’π‘Œ which satisfies (3.13).
By induction one can easily see that, since 𝑓 is a cubic functional equation, so, for all π‘›βˆˆβ„• and every π‘₯βˆˆπ‘‹, 𝐢(π‘šπ‘›π‘₯)=π‘š3𝑛𝐢(π‘₯), and 𝐿(π‘šπ‘›π‘₯)=π‘š3𝑛𝐿(π‘₯), we have πœ‡πΆ(π‘₯)βˆ’πΏ(π‘₯)(𝑑)=limπ‘›β†’βˆžπœ‡(𝐢(π‘šπ‘›π‘₯)/π‘š3𝑛)βˆ’(𝐿(π‘šπ‘›π‘₯)/π‘š3𝑛)(𝑑),(3.25) so πœ‡πΆ(π‘šπ‘›π‘₯)/π‘š3π‘›βˆ’πΏ(π‘š3𝑛π‘₯)/π‘š3π‘›ξ‚†πœ‡(𝑑)β‰₯min𝐢(π‘šπ‘›π‘₯)/π‘š3π‘›βˆ’π‘“(π‘šπ‘›π‘₯)/π‘š3𝑛𝑑2,πœ‡πΏ(π‘šπ‘›π‘₯)/π‘š3π‘›βˆ’π‘“(π‘šπ‘›π‘₯)/π‘š3𝑛𝑑2β‰₯πœ‡ξ…žπœ“(π‘šπ‘›π‘₯,0)/π‘š3𝑛(π‘š3βˆ’π›Ό)(𝑑)β‰₯πœ‡ξ…žπ›Όπ‘›πœ“(π‘₯,0)/π‘š3𝑛(π‘š3βˆ’π›Ό)(𝑑).(3.26) Since limπ‘›β†’βˆžπœ‡ξ…žπ›Όπ‘›πœ“(π‘₯,0)/π‘š3𝑛(π‘š3βˆ’π›Ό)(𝑑)=1, it follows that, for all 𝑑>0, πœ‡πΆ(π‘₯)βˆ’πΏ(π‘₯)(𝑑)=1 and so 𝐢(π‘₯)=𝐿(π‘₯). This completes the proof.

Corollary 3.3. Let 𝑋 be a real linear space, (𝑍,πœ‡ξ…ž,min) an RN-space, and (π‘Œ,πœ‡,min) a complete RN-space. Let 0<π‘Ÿ<1 and 𝑧0βˆˆπ‘, and let π‘“βˆΆπ‘‹β†’π‘Œ be a mapping with 𝑓(0)=0 and satisfying πœ‡π‘“(π‘šπ‘₯±𝑦)βˆ’π‘šπ‘“(π‘₯±𝑦)βˆ’2(π‘š3βˆ’π‘š)𝑓(π‘₯)(𝑑)β‰₯πœ‡ξ…ž(||π‘₯||π‘Ÿ+||𝑦||π‘Ÿ)𝑧0(𝑑),(3.27) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0. Then, the limit 𝐢(π‘₯)=limπ‘›β†’βˆž(𝑓(π‘šπ‘›π‘₯)/π‘š3𝑛) exists for all π‘₯βˆˆπ‘‹ and defines a unique cubic mapping πΆβˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)(𝑑)β‰₯πœ‡ξ…ž||π‘₯||π‘Ÿπ‘§0/2(π‘š3βˆ’π‘š3π‘Ÿ)(𝑑)(3.28) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Proof. Let 𝛼=π‘š3π‘Ÿ, and let πœ“βˆΆπ‘‹2→𝑍 be defined as πœ“(π‘₯,𝑦)=(||π‘₯||π‘Ÿ+||𝑦||π‘Ÿ)𝑧0.

Remark 3.4. In Corollary 3.3, if we assume that πœ“(π‘₯,𝑦)=(β€–π‘₯β€–π‘Ÿ.β€–π‘¦β€–π‘Ÿ)𝑧0 or πœ“(π‘₯,𝑦)=(β€–π‘₯β€–π‘Ÿ+𝑠+β€–π‘¦β€–π‘Ÿ+𝑠+β€–π‘₯β€–π‘Ÿβ€–π‘¦β€–π‘ )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 π‘Ÿ+𝑠=3  is an open question.

Corollary 3.5. Let 𝑋 be a real linear space, (𝑍,πœ‡ξ…ž,min) an RN-space, and (π‘Œ,πœ‡,min) a complete RN-space. Let 𝑧0βˆˆπ‘, and let π‘“βˆΆπ‘‹β†’π‘Œ be a mapping with 𝑓(0)=0 and satisfying πœ‡π‘“(π‘šπ‘₯±𝑦)βˆ’π‘šπ‘“(π‘₯±𝑦)βˆ’2(π‘š3βˆ’π‘š)𝑓(π‘₯)(𝑑)β‰₯πœ‡ξ…žπ›Ώπ‘§0(𝑑),(3.29) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0. Then, the limit 𝐢(π‘₯)=limπ‘›β†’βˆž(𝑓(π‘šπ‘›π‘₯)/π‘š3𝑛) exists for all π‘₯βˆˆπ‘‹ and defines a unique cubic mapping πΆβˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)(𝑑)β‰₯πœ‡ξ…žπ›Ώπ‘§0/2(π‘š3βˆ’1)(𝑑)(3.30) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Proof. Let 𝛼=1, and let πœ“βˆΆπ‘‹2→𝑍 be defined by πœ“(π‘₯,𝑦)=𝛿𝑧0.

Theorem 3.6. Let 𝑋 be a real linear space, (𝑍,πœ‡ξ…ž,min) an RN-space, and πœ“βˆΆπ‘‹2→𝑍 a function such that for some 0<𝛼<1/π‘š3πœ‡ξ…žπœ“(π‘₯/π‘š,0)(𝑑)β‰₯πœ‡ξ…žπ›Όπœ“(π‘₯,0)(𝑑)βˆ€π‘₯βˆˆπ‘‹,𝑑>0,(3.31) and, for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0, limπ‘›β†’βˆžπœ‡ξ…žπ‘š3π‘›πœ“(π‘₯/π‘šπ‘›,𝑦/π‘šπ‘›)(𝑑)=1.(3.32) Let (π‘Œ,πœ‡,min) be a complete RN-space. If π‘“βˆΆπ‘‹β†’π‘Œ is a mapping with 𝑓(0)=0 and satisfying (3.12), then the limit 𝐢(π‘₯)=limπ‘›β†’βˆžπ‘š3𝑛𝑓(π‘₯/π‘šπ‘›) exists for all π‘₯βˆˆπ‘‹ and defines a unique cubic mapping πΆβˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)(𝑑)β‰₯πœ‡ξ…žπœ“(π‘₯,0)/2(1βˆ’π‘š3𝛼)(𝑑).(3.33)

Proof. Putting 𝑦=0 in (3.12) and replacing π‘₯ by π‘₯/π‘š, we obtain that for all π‘₯βˆˆπ‘‹πœ‡π‘“(π‘₯)βˆ’π‘š3𝑓(π‘₯/π‘š)(𝑑)β‰₯πœ‡ξ…žπœ“(π‘₯,0)/2(𝑑).(3.34) Replacing π‘₯ by π‘₯/π‘šπ‘› in (3.34) and using (3.31), we obtain πœ‡π‘š3𝑛𝑓(π‘₯/π‘šπ‘›)βˆ’π‘š3𝑛+3𝑓(π‘₯/π‘šπ‘›+1)(𝑑)β‰₯πœ‡ξ…žπ‘š3π‘›πœ“(π‘₯/π‘šπ‘›,0)/2(𝑑)β‰₯πœ‡ξ…žπ›Όπ‘›π‘š3π‘›πœ“(π‘₯,0)/2(𝑑).(3.35) So πœ‡(π‘š3𝑛𝑓(π‘₯/π‘šπ‘›βˆ‘)βˆ’π‘“(π‘₯))/π‘›βˆ’1π‘˜=0(π›Όπ‘˜π‘š3π‘˜/2)(𝑑)=πœ‡(βˆ‘π‘›βˆ’1π‘˜=0π‘š3π‘˜+3𝑓(π‘₯/π‘šπ‘˜+1)βˆ’π‘š3π‘˜π‘“(π‘₯/π‘šπ‘˜βˆ‘))/π‘›βˆ’1π‘˜=0(π›Όπ‘˜π‘š3π‘˜/2)(𝑑)β‰₯π‘‡π‘›βˆ’1π‘˜=0ξ‚€πœ‡ξ…žπœ“(π‘₯,0)(𝑑)=πœ‡β€²πœ“(π‘₯,0)(𝑑).(3.36) This implies that πœ‡π‘š3𝑛𝑓(π‘₯/π‘šπ‘›)βˆ’π‘“(π‘₯)(𝑑)β‰₯πœ‡ξ…žβˆ‘π‘›βˆ’1π‘˜=0π›Όπ‘˜π‘š3π‘˜πœ“(π‘₯,0)/2(𝑑).(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,πœ‡ξ…ž,min) an RN-space, and (π‘Œ,πœ‡,min) a complete RN-space. Let π‘Ÿ>1 and 𝑧0βˆˆπ‘, and let π‘“βˆΆπ‘‹β†’π‘Œ be a mapping with 𝑓(0)=0 and satisfying (3.27). Then, the limit 𝐢(π‘₯)=limπ‘›β†’βˆžπ‘š3𝑛𝑓(π‘₯/π‘šπ‘›) exists for all π‘₯βˆˆπ‘‹ and defines a unique cubic mapping πΆβˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)(𝑑)β‰₯πœ‡ξ…žπ‘š3π‘Ÿ||π‘₯||π‘Ÿπ‘§0/2(π‘š3π‘Ÿβˆ’π‘š3)(𝑑)(3.38) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Proof. Let 𝛼=π‘šβˆ’3π‘Ÿ, and let πœ“βˆΆπ‘‹2→𝑍 be defined as πœ“(π‘₯,𝑦)=(||π‘₯||π‘Ÿ+||𝑦||π‘Ÿ)𝑧0.

Corollary 3.8. Let 𝑋 be a real linear space, (𝑍,πœ‡ξ…ž,min) be an RN-space, and (π‘Œ,πœ‡,min) a complete RN-space. Let 𝑧0βˆˆπ‘, and let π‘“βˆΆπ‘‹β†’π‘Œ be a mapping with 𝑓(0)=0 and satisfying (3.29). Then, the limit 𝐢(π‘₯)=limπ‘›β†’βˆžπ‘š3𝑛𝑓(π‘₯/π‘šπ‘›) exists for all π‘₯βˆˆπ‘‹ and defines a unique cubic mapping πΆβˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)(𝑑)β‰₯πœ‡ξ…žπ‘šπ›Ώπ‘§0/2(π‘šβˆ’1)(𝑑)(3.39) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Proof. Let 𝛼=1/π‘š4, and let πœ“βˆΆπ‘‹2→𝑍 be defined by πœ“(π‘₯,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, (π‘Œ,πœ‡,𝑇𝑀) a complete RN-space, and Ξ¦ a mapping from 𝑋2 to 𝐷+(Ξ¦(π‘₯,𝑦) is denoted by Ξ¦π‘₯,𝑦) such that there exists 0<𝛼<1/π‘š3 such that Ξ¦π‘₯,𝑦(𝑑)≀Φπ‘₯/π‘š,𝑦/π‘š(𝛼𝑑)(4.1) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0. Let π‘“βˆΆπ‘‹β†’π‘Œ be a mapping with 𝑓(0)=0 and satisfying πœ‡π‘“(π‘šπ‘₯±𝑦)βˆ’π‘šπ‘“(π‘₯±𝑦)βˆ’2(π‘š3βˆ’π‘š)𝑓(π‘₯)(𝑑)β‰₯Ξ¦π‘₯,𝑦(𝑑)(4.2) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0. Then, for all π‘₯βˆˆπ‘‹πΆ(π‘₯)∢=limπ‘›β†’βˆžπ‘š3𝑛𝑓π‘₯π‘šπ‘›ξ‚(4.3) exists and πΆβˆΆπ‘‹β†’π‘Œ is a unique cubic mapping such that πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)(𝑑)β‰₯Ξ¦π‘₯,02βˆ’2π‘š3𝛼𝑑𝛼ξƒͺ(4.4) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Proof. Putting 𝑦=0 in (4.2) and replacing π‘₯ by π‘₯/π‘š, we have πœ‡π‘“(π‘₯)βˆ’π‘š3𝑓(π‘₯/π‘š)(𝑑)β‰₯Ξ¦π‘₯/π‘š,0(2𝑑)β‰₯Ξ¦π‘₯,0ξ‚€2𝑑𝛼(4.5) for all π‘₯βˆˆπ‘‹ and 𝑑>0. Consider the set π‘†βˆΆ={π‘”βˆΆπ‘‹β†’π‘Œ;𝑔(0)=0}(4.6) and the generalized metric 𝑑 in 𝑆 defined by 𝑑(𝑓,𝑔)=infπ‘’βˆˆβ„+βˆΆπœ‡π‘”(π‘₯)βˆ’β„Ž(π‘₯)(𝑒𝑑)β‰₯Ξ¦π‘₯,0ξ€Ύ(𝑑),βˆ€π‘₯βˆˆπ‘‹,𝑑>0,(4.7) where infβˆ…=+∞. It is easy to show that (𝑆,𝑑) is complete (see [26], Lemma 2.1). Now, we consider a linear mapping π½βˆΆπ‘†β†’π‘† such that π½β„Ž(π‘₯)∢=π‘š3β„Žξ‚€π‘₯π‘šξ‚(4.8) for all π‘₯βˆˆπ‘‹. First, we prove that 𝐽 is a strictly contractive mapping with the Lipschitz constant π‘š3𝛼.
In fact, let 𝑔,β„Žβˆˆπ‘† be such that 𝑑(𝑔,β„Ž)<πœ–. Then, we have πœ‡π‘”(π‘₯)βˆ’β„Ž(π‘₯)(πœ–π‘‘)β‰₯Ξ¦π‘₯,0(𝑑)(4.9) for all π‘₯βˆˆπ‘‹ and 𝑑>0, and so πœ‡π½π‘”(π‘₯)βˆ’π½β„Ž(π‘₯)ξ€·π‘š3ξ€Έπ›Όπœ–π‘‘=πœ‡π‘š3𝑔(π‘₯/π‘š)βˆ’π‘š3β„Ž(π‘₯/π‘š)ξ€·π‘š3ξ€Έπ›Όπœ–π‘‘=πœ‡π‘”(π‘₯/π‘š)βˆ’β„Ž(π‘₯/π‘š)(π›Όπœ–π‘‘)β‰₯Ξ¦π‘₯/π‘š,0(𝛼𝑑)β‰₯Ξ¦π‘₯,0(𝑑)(4.10) for all π‘₯βˆˆπ‘‹ and 𝑑>0. Thus, 𝑑(𝑔,β„Ž)<πœ– implies that ξ‚€π‘šπ‘‘(𝐽𝑔,π½β„Ž)=𝑑3𝑔π‘₯π‘šξ‚,π‘š3β„Žξ‚€π‘₯π‘šξ‚ξ‚<π‘š3π›Όπœ–.(4.11) This means that ξ‚€π‘šπ‘‘(𝐽𝑔,π½β„Ž)=𝑑3𝑔π‘₯π‘šξ‚,π‘š3β„Žξ‚€π‘₯π‘šξ‚ξ‚β‰€π‘š3𝛼𝑑(𝑔,β„Ž)(4.12) for all 𝑔,β„Žβˆˆπ‘†. It follows from (4.5) that 𝑑(𝑓,𝐽𝑓)=𝑑𝑓,π‘š3𝑓π‘₯π‘šβ‰€π›Όξ‚ξ‚2.(4.13) By Theorem 2.8, there exists a mapping πΆβˆΆπ‘‹β†’π‘Œ satisfying the following.
(1) 𝐢 is a fixed point of 𝐽, that is, 𝐢π‘₯π‘šξ‚=1π‘š3𝐢(π‘₯)(4.14) for all π‘₯βˆˆπ‘‹.
The mapping 𝐢 is a unique fixed point of 𝐽 in the set Ξ©={β„Žβˆˆπ‘†βˆΆπ‘‘(𝑔,β„Ž)<∞}.(4.15) This implies that 𝐢 is a unique mapping satisfying (4.14) such that there exists π‘’βˆˆ(0,∞) satisfying πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)(𝑒𝑑)β‰₯Ξ¦π‘₯,0(𝑑)(4.16) for all π‘₯βˆˆπ‘‹ and 𝑑>0.
(2) 𝑑(𝐽𝑛𝑓,𝐢)β†’0 as π‘›β†’βˆž. This implies the equality limπ‘›β†’βˆžπ‘š3𝑛𝑓π‘₯π‘šπ‘›ξ‚=𝐢(π‘₯)(4.17) for all π‘₯βˆˆπ‘‹.
(3) 𝑑(𝑓,𝐢)≀𝑑(𝑓,𝐽𝑓)/(1βˆ’π‘š3𝛼) with π‘“βˆˆΞ©, which implies the inequality 𝑑(𝑓,𝐢)≀𝛼/(2βˆ’2π‘š3𝛼)  and so πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)𝛼𝑑2βˆ’2π‘š3𝛼β‰₯Ξ¦π‘₯,0(𝑑)(4.18) for all π‘₯βˆˆπ‘‹ and 𝑑>0. This implies that inequality (4.4) holds. Now, we have πœ‡π‘š3𝑛𝑓((π‘šπ‘₯±𝑦)/π‘šπ‘›)βˆ’π‘š3𝑛+1𝑓((π‘₯±𝑦)/π‘šπ‘›)βˆ’2π‘š3𝑛(π‘š3βˆ’π‘š)𝑓(π‘₯/π‘šπ‘›)(𝑑)β‰₯Ξ¦π‘₯/π‘šπ‘›,𝑦/π‘šπ‘›ξ‚€π‘‘π‘š3𝑛(4.19) for all π‘₯,π‘¦βˆˆπ‘‹, 𝑑>0, and 𝑛β‰₯1, and so, from (4.1), it follows that Ξ¦π‘₯/π‘šπ‘›,𝑦/π‘šπ‘›ξ‚€π‘‘π‘š3𝑛β‰₯Ξ¦π‘₯,π‘¦ξ‚€π‘‘π‘š3𝑛𝛼𝑛,(4.20) Since limπ‘›β†’βˆžΞ¦π‘₯,π‘¦ξ‚€π‘‘π‘š3𝑛𝛼𝑛=1,(4.21) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0, we have πœ‡πΆ(π‘šπ‘₯±𝑦)βˆ’π‘šπΆ(π‘₯±𝑦)βˆ’2(π‘š3βˆ’π‘š)𝐢(π‘₯)(𝑑)=1(4.22) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0. Thus, the mapping πΆβˆΆπ‘‹β†’π‘Œ is cubic. This completes the proof.

Corollary 4.2. Let 𝑋 be a real normed space, πœƒβ‰₯0, and 𝑝 a real number with π‘βˆˆ(1,+∞). Let π‘“βˆΆπ‘‹β†’π‘Œ be a mapping with 𝑓(0)=0 and satisfying πœ‡π‘“(π‘šπ‘₯±𝑦)βˆ’π‘šπ‘“(π‘₯±𝑦)βˆ’2(π‘š3βˆ’π‘š)𝑓(π‘₯)(𝑑𝑑)β‰₯𝑑+πœƒβ€–π‘₯‖𝑝+‖𝑦‖𝑝(4.23) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0. Then, for all π‘₯βˆˆπ‘‹, the limit 𝐢(π‘₯)=limπ‘›β†’βˆžπ‘š3𝑛𝑓(π‘₯/π‘šπ‘›) exists and πΆβˆΆπ‘‹β†’π‘Œ is a unique cubic mapping such that πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)2ξ€·π‘š(𝑑)β‰₯3π‘βˆ’π‘š3𝑑2ξ€·π‘š3π‘βˆ’π‘š3𝑑+πœƒβ€–π‘₯‖𝑝(4.24) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Proof. The proof follows from Theorem 4.1 if we take Ξ¦π‘₯,𝑦(𝑑𝑑)=𝑑+πœƒβ€–π‘₯‖𝑝+‖𝑦‖𝑝(4.25) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0. In fact, if we choose 𝛼=π‘šβˆ’3𝑝, then we get the desired result.

Theorem 4.3. Let 𝑋 be a linear space, (π‘Œ,πœ‡,𝑇𝑀) a complete RN-space, and Ξ¦ a mapping from 𝑋2 to 𝐷+ ( Ξ¦(π‘₯,𝑦) is denoted by Ξ¦π‘₯,𝑦) such that for some 0<𝛼<π‘š3Ξ¦π‘₯/π‘š,𝑦/π‘š(𝑑)≀Φπ‘₯,𝑦(𝛼𝑑)(4.26) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0. Let π‘“βˆΆπ‘‹β†’π‘Œ be a mapping with 𝑓(0)=0 and satisfying (4.2). Then, for all π‘₯βˆˆπ‘‹, the limit 𝐢(π‘₯)∢=limπ‘›β†’βˆžπ‘“(π‘šπ‘›π‘₯)/π‘š3𝑛 exists and πΆβˆΆπ‘‹β†’π‘Œ is a unique cubic mapping such that πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)(𝑑)β‰₯Ξ¦π‘₯,0ξ€·ξ€·2π‘š3ξ€Έπ‘‘ξ€Έβˆ’2𝛼(4.27) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Proof. Let (𝑆,𝑑) be the generalized metric space defined in the proof of Theorem 4.1. Now, we consider a linear mapping π½βˆΆπ‘†β†’π‘† such that 1π½β„Ž(π‘₯)∢=π‘š3β„Ž(π‘šπ‘₯)(4.28) for all π‘₯βˆˆπ‘‹.
Let 𝑔,β„Žβˆˆπ‘† be such that 𝑑(𝑔,β„Ž)<πœ–. Then, we have πœ‡π‘”(π‘₯)βˆ’β„Ž(π‘₯)(πœ–π‘‘)β‰₯Ξ¦π‘₯,0(𝑑)(4.29) for all π‘₯βˆˆπ‘‹ and 𝑑>0 and so πœ‡π½π‘”(π‘₯)βˆ’π½β„Ž(π‘₯)ξ‚€π›Όπœ–π‘‘π‘š3=πœ‡(1/π‘š3)𝑔(π‘šπ‘₯)βˆ’(1/π‘š3)β„Ž(π‘šπ‘₯)ξ‚€π›Όπœ–π‘‘π‘š3=πœ‡π‘”(π‘šπ‘₯)βˆ’β„Ž(π‘šπ‘₯)(π›Όπœ–π‘‘)β‰₯Ξ¦π‘šπ‘₯,0(𝛼𝑑)β‰₯Ξ¦π‘₯,0(𝑑)(4.30) for all π‘₯βˆˆπ‘‹ and 𝑑>0. Thus, 𝑑(𝑔,β„Ž)<πœ– implies that 𝑑(𝐽𝑔,π½β„Ž)=𝑑𝑔(π‘šπ‘₯)π‘š3,β„Ž(π‘šπ‘₯)π‘š3ξ‚Ά<π›Όπœ–π‘š3.(4.31) This means that 𝑑(𝐽𝑔,π½β„Ž)=𝑑𝑔(π‘šπ‘₯)π‘š3,β„Ž(π‘šπ‘₯)π‘š3ξ‚Άβ‰€π›Όπ‘š3𝑑(𝑔,β„Ž)(4.32) for all 𝑔,β„Žβˆˆπ‘†.
Putting 𝑦=0 in (4.2), we see that, for all π‘₯βˆˆπ‘‹, πœ‡π‘“(π‘šπ‘₯)/π‘š3βˆ’π‘“(π‘₯)𝑑2π‘š3β‰₯Ξ¦π‘₯,0(𝑑).(4.33) It follows from (4.33) that 𝑑(𝑓,𝐽𝑓)=𝑑𝑓,𝑓(π‘šπ‘₯)π‘š3≀12π‘š3.(4.34) By Theorem 2.8, there exists a mapping πΆβˆΆπ‘‹β†’π‘Œ satisfying the following.
(1) 𝐢 is a fixed point of 𝐽, that is, 𝐢(π‘šπ‘₯)=π‘š3𝐢(π‘₯)(4.35) for all π‘₯βˆˆπ‘‹.
The mapping 𝐢 is a unique fixed point of 𝐽 in the set Ξ©={β„Žβˆˆπ‘†βˆΆπ‘‘(𝑔,β„Ž)<∞}.(4.36) This implies that 𝐢 is a unique mapping satisfying (4.35) such that there exists π‘’βˆˆ(0,∞) satisfying πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)(𝑒𝑑)β‰₯Ξ¦π‘₯,0(𝑑)(4.37) for all π‘₯βˆˆπ‘‹ and 𝑑>0.
(2) 𝑑(𝐽𝑛𝑓,𝐢)β†’0 as π‘›β†’βˆž. This implies the equality limπ‘›β†’βˆžπ‘“(π‘šπ‘›π‘₯)π‘š3𝑛=𝐢(π‘₯)(4.38) for all π‘₯βˆˆπ‘‹.
(3) 𝑑(𝑓,𝐢)≀𝑑(𝑓,𝐽𝑓)/(1βˆ’π›Ό/π‘š3) with π‘“βˆˆΞ©, which implies the inequality 𝑑(𝑓,𝐢)≀1/(2π‘š3βˆ’2𝛼), and so πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)𝑑2π‘š3ξ‚βˆ’2𝛼β‰₯Ξ¦π‘₯,0(𝑑)(4.39) for all π‘₯βˆˆπ‘‹ and 𝑑>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, πœƒβ‰₯0, and 𝑝 a real number with π‘βˆˆ(0,1). Let π‘“βˆΆπ‘‹β†’π‘Œ be a mapping with 𝑓(0)=0 and satisfying (4.23). Then, for all π‘₯βˆˆπ‘‹, the limit 𝐢(π‘₯)=limπ‘›β†’βˆžπ‘“(π‘šπ‘›π‘₯)/π‘š3𝑛 exists and πΆβˆΆπ‘‹β†’π‘Œ is a unique cubic mapping such that πœ‡π‘“(π‘₯)βˆ’πΆ(π‘₯)2ξ€·π‘š(𝑑)β‰₯3βˆ’π‘š3𝑝𝑑2ξ€·π‘š3βˆ’π‘š3𝑝𝑑+πœƒβ€–π‘₯‖𝑝(4.40) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Proof. The proof follows from Theorem 6.3 if we take Ξ¦π‘₯,𝑦(𝑑𝑑)=𝑑+πœƒβ€–π‘₯‖𝑝+‖𝑦‖𝑝(4.41) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0. In fact, if we choose 𝛼=π‘š3𝑝, then we get the desired result.

Remark 4.5. In Corollaries 4.2 and 4.4, if we assume that Ξ¦π‘₯,𝑦(𝑑)=𝑑/(𝑑+πœƒ(β€–π‘₯‖𝑝⋅‖𝑦‖𝑝)) or Ξ¦π‘₯,𝑦(𝑑)=𝑑/(𝑑+πœƒ(β€–π‘₯‖𝑝+π‘ž+‖𝑦‖𝑝+π‘ž+β€–π‘₯β€–π‘β€–π‘¦β€–π‘ž)), 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+π‘ž=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∢𝐺2β†’[0,+∞) be a function such that limπ‘›β†’βˆž|π‘š|3π‘›πœξ‚€π‘₯π‘šπ‘›,π‘¦π‘šπ‘›ξ‚=0(5.1) for all π‘₯,π‘¦βˆˆπΊ and let for each π‘₯∈𝐺 the limit Θ(π‘₯)=limπ‘›β†’βˆžξ‚†max|π‘š|3(π‘˜+1)πœξ‚€π‘₯π‘šπ‘˜+1,0;0β‰€π‘˜<𝑛(5.2) exist. Suppose that π‘“βˆΆπΊβ†’π‘‹ a mapping with 𝑓(0)=0 and satisfying the following inequality: β€–β€–ξ€·π‘šπ‘“(π‘šπ‘₯±𝑦)βˆ’π‘šπ‘“(π‘₯±𝑦)βˆ’23ξ€Έβ€–β€–βˆ’π‘šπ‘“(π‘₯)β‰€πœ(π‘₯,𝑦,𝑧).(5.3) Then, the limit 𝐢(π‘₯)∢=limπ‘›β†’βˆžπ‘š3𝑛𝑄(π‘₯/π‘šπ‘›) exists for all π‘₯∈𝐺 and defines a cubic mapping πΆβˆΆπΊβ†’π‘‹ such that ‖𝑓(π‘₯)βˆ’πΆ(π‘₯)β€–β‰€Ξ˜(π‘₯)||2π‘š3||.(5.4) Moreover, if limπ‘—β†’βˆžlimπ‘›β†’βˆžξ‚†max|π‘š|3(π‘˜+1)πœξ‚€π‘₯π‘šπ‘˜+1,0;π‘—β‰€π‘˜<𝑛+𝑗=0,(5.5) then 𝐢 is the unique cubic mapping satisfying (5.4).

Proof. Putting 𝑦=0 in (5.3), we get ‖‖𝑓(π‘šπ‘₯)βˆ’π‘š3‖‖≀𝑓(π‘₯)𝜁(π‘₯,0)||2||(5.6) for all π‘₯∈𝐺. Replacing π‘₯ by π‘₯/π‘šπ‘›+1 in (5.6), we obtain β€–β€–β€–π‘š3𝑛+3𝑓π‘₯π‘šπ‘›+1ξ‚βˆ’π‘š3𝑛𝑓π‘₯π‘šπ‘›ξ‚β€–β€–β€–β‰€|π‘š|3𝑛||2||πœξ‚€π‘₯π‘šπ‘›+1,0.(5.7) It follows from (5.1) and (5.7) that the sequence {π‘š3𝑛𝑓(π‘₯/π‘šπ‘›)}𝑛β‰₯1 is a Cauchy sequence. Since 𝑋 is complete, {π‘š3𝑛𝑓(π‘₯/π‘šπ‘›)}𝑛β‰₯1 is convergent. Set 𝐢(π‘₯)∢=limπ‘›β†’βˆžπ‘š3𝑛𝑓(π‘₯/π‘šπ‘›).
Using induction, one can show that β€–β€–β€–π‘š3𝑛𝑓π‘₯π‘šπ‘›ξ‚β€–β€–β€–β‰€1βˆ’π‘“(π‘₯)||2π‘š3||max|π‘š|3(π‘˜+1)πœξ‚€π‘₯π‘šπ‘˜+1,0;0β‰€π‘˜<𝑛(5.8) for all π‘›βˆˆβ„• and all π‘₯∈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 β€–β€–ξ€·π‘šπΆ(π‘šπ‘₯±𝑦)βˆ’π‘šπΆ(π‘₯±𝑦)βˆ’23ξ€Έβ€–β€–βˆ’π‘šπΆ(π‘₯)=limπ‘›β†’βˆžβ€–β€–β€–π‘š3π‘›π‘“ξ‚€π‘šπ‘₯Β±π‘¦π‘šπ‘›ξ‚βˆ’π‘š3𝑛+1𝑓π‘₯Β±π‘¦π‘šπ‘›ξ‚βˆ’2π‘š3π‘›ξ€·π‘š3𝑓π‘₯βˆ’π‘šπ‘šπ‘›ξ‚β€–β€–β€–β‰€limπ‘›β†’βˆž|π‘š|3π‘›πœξ‚€π‘₯π‘šπ‘›,π‘¦π‘šπ‘›ξ‚=0(5.9) for all π‘₯,π‘¦βˆˆπΊ. Therefore, the function πΆβˆΆπΊβ†’π‘‹ satisfies (1.9). To prove the uniqueness property of 𝐢, let πΏβˆΆπΊβ†’π‘‹ be another function satisfying (5.4). Then, ‖𝐢(π‘₯)βˆ’πΏ(π‘₯)β€–=limπ‘—β†’βˆž|π‘š|3𝑗‖‖‖𝐢π‘₯π‘šπ‘—ξ‚ξ‚€π‘₯βˆ’πΏπ‘šπ‘—ξ‚β€–β€–β€–β‰€limπ‘—β†’βˆž|π‘š|3𝑗‖‖‖𝐢π‘₯maxπ‘šπ‘—ξ‚ξ‚€π‘₯βˆ’π‘“π‘šπ‘—ξ‚β€–β€–β€–,‖‖‖𝑓π‘₯π‘šπ‘—ξ‚ξ‚€π‘₯βˆ’πΏπ‘šπ‘—ξ‚β€–β€–β€–ξ‚‡β‰€limπ‘—β†’βˆžlimπ‘›β†’βˆžξ‚†|maxπ‘š|3(π‘˜+1)πœξ‚€π‘₯π‘šπ‘˜+1,0;π‘—β‰€π‘˜<𝑛+𝑗=0(5.10) for all π‘₯∈𝐺. Therefore, 𝐢=𝐿, and the proof is complete.

Corollary 5.2. Let πœ‰βˆΆ[0,∞)β†’[0,∞) be a function satisfying πœ‰ξ‚΅π‘‘ξ‚Άξ‚΅1|π‘š|β‰€πœ‰ξ‚Άξ‚΅1|π‘š|πœ‰(𝑑)(𝑑β‰₯0)πœ‰ξ‚Ά|π‘š|<|π‘š|βˆ’3.(5.11) Let πœ…>0, and let π‘“βˆΆπΊβ†’π‘‹ be a mapping with 𝑓(0)=0 and satisfying the following inequality: β€–β€–ξ€·π‘šπ‘“(π‘šπ‘₯±𝑦)βˆ’π‘šπ‘“(π‘₯±𝑦)βˆ’23ξ€Έβ€–β€–ξ€·ξ€·||𝑦||βˆ’π‘šπ‘“(π‘₯)β‰€πœ…πœ‰(|π‘₯|)+πœ‰ξ€Έξ€Έ(5.12) for all π‘₯,π‘¦βˆˆπΊ. Then there exists a unique cubic mapping πΆβˆΆπΊβ†’π‘‹ such that (‖𝑓π‘₯)βˆ’πΆ(π‘₯)β€–β‰€πœ…πœ‰(|π‘₯|)||2π‘š3||.(5.13)

Proof. Defining 𝜁∢𝐺2β†’[0,∞) by 𝜁(π‘₯,𝑦)∢=πœ…(πœ‰(|π‘₯|)+πœ‰(|𝑦|)), we have limπ‘›β†’βˆž|π‘š|3π‘›πœξ‚€π‘₯π‘šπ‘›,π‘¦π‘šπ‘›ξ‚β‰€limπ‘›β†’βˆžξ‚΅|π‘š|3πœ‰ξ‚΅1|π‘š|ξ‚Άξ‚Άπ‘›πœ(π‘₯,𝑦)=0(5.14) for all π‘₯,π‘¦βˆˆπΊ. The last equality comes from the fact that |π‘š|3πœ‰(1/|π‘š|)<1. On the other hand, Θ(π‘₯)=limπ‘›β†’βˆžξ‚†max|π‘š|3π‘˜+3πœξ‚€π‘₯π‘šπ‘˜+1,0;0β‰€π‘˜<𝑛=|π‘š|3πœξ‚€π‘₯π‘šξ‚,0=πœ…πœ‰(|π‘₯|),(5.15) for all π‘₯∈𝐺, exists. Also, limπ‘—β†’βˆžlimπ‘›β†’βˆžξ‚†max|π‘š|3π‘˜+3πœξ‚€π‘₯π‘šπ‘˜+1,0;π‘—β‰€π‘˜<𝑛+𝑗=limπ‘—β†’βˆž|π‘š|3𝑗+3πœξ‚€π‘₯π‘šπ‘—+1,0=0.(5.16) Applying Theorem 5.1, we get the desired result.

Theorem 5.3. Let 𝜁∢𝐺3β†’[0,+∞) be a function such that limπ‘›β†’βˆžπœ(π‘šπ‘›π‘₯,π‘šπ‘›π‘¦)|π‘š|3𝑛=0(5.17) for all π‘₯,π‘¦βˆˆπΊ, and let for each π‘₯∈𝐺 the limit Θ(π‘₯)=limπ‘›β†’βˆžξƒ―πœξ€·π‘šmaxπ‘˜ξ€Έπ‘₯,0|π‘š|3π‘˜+3ξƒ°;0β‰€π‘˜<𝑛(5.18) exist. Suppose that π‘“βˆΆπΊβ†’π‘‹ a mapping with 𝑓(0)=0 and satisfying (5.3). Then, the limit 𝐢(π‘₯)∢=limπ‘›β†’βˆžπ‘“(π‘šπ‘›π‘₯)/π‘š3𝑛 exists for all π‘₯∈𝐺 and defines a cubic mapping πΆβˆΆπΊβ†’π‘‹ such that 1‖𝑓(π‘₯)βˆ’πΆ(π‘₯)‖≀||2||Θ(π‘₯).(5.19) Moreover, if limπ‘—β†’βˆžlimπ‘›β†’βˆžξƒ―πœξ€·π‘šmaxπ‘˜ξ€Έπ‘₯,0|π‘š|3π‘˜+3ξƒ°;π‘—β‰€π‘˜<𝑛+𝑗=0,(5.20) then 𝐢 is the unique cubic mapping satisfying (5.19).

Proof. Putting 𝑦=0 in (5.3), we get ‖‖‖𝑓(π‘₯)βˆ’π‘“(π‘šπ‘₯)π‘š3β€–β€–β€–β‰€πœ(π‘₯,0)||2π‘š3||(5.21) for all π‘₯∈𝐺. Replacing π‘₯ by π‘šπ‘›π‘₯ in (5.21), we obtain ‖‖‖‖𝑓(π‘šπ‘›π‘₯)π‘š3π‘›βˆ’π‘“ξ€·π‘šπ‘›+1π‘₯ξ€Έπ‘š3𝑛+3β€–β€–β€–β€–β‰€πœ(π‘šπ‘›π‘₯,0)||2|||π‘š|3𝑛+3.(5.22) It follows from (5.17) and (5.22) that the sequence {𝑓(π‘šπ‘›π‘₯)/π‘š3𝑛}𝑛β‰₯1 is convergent. Set 𝐢(π‘₯)∢=limπ‘›β†’βˆžπ‘“(π‘šπ‘›π‘₯)/π‘š3𝑛. On the other hand, it follows from (5.22) that ‖‖‖𝑓(π‘šπ‘π‘₯)π‘š3π‘βˆ’π‘“(π‘šπ‘žπ‘₯)π‘š3π‘žβ€–β€–β€–=β€–β€–β€–β€–π‘žβˆ’1ξ“π‘˜=π‘π‘“ξ€·π‘šπ‘˜+1π‘₯ξ€Έπ‘š3π‘˜+3βˆ’π‘“ξ€·π‘šπ‘˜π‘₯ξ€Έπ‘š3π‘˜β€–β€–β€–β€–ξƒ―β€–β€–β€–β€–π‘“ξ€·π‘šβ‰€maxπ‘˜+1π‘₯ξ€Έπ‘š3π‘˜+3βˆ’π‘“ξ€·π‘šπ‘˜π‘₯ξ€Έπ‘š3π‘˜β€–β€–β€–β€–ξƒ°β‰€1;π‘β‰€π‘˜<π‘žβˆ’1||2||ξƒ―πœξ€·π‘šmaxπ‘˜ξ€Έπ‘₯,0|π‘š|3π‘˜+3ξƒ°;π‘β‰€π‘˜<π‘ž(5.23) for all π‘₯∈𝐺 and all nonnegative integers 𝑝,π‘ž with π‘ž>𝑝β‰₯0. Letting 𝑝=0, passing the limit π‘žβ†’βˆž 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 π‘Œ a complete non-Archimedean normed space. Also, |2π‘š3|β‰ 1.

Theorem 6.1. Let πœβˆΆπ‘‹2β†’[0,∞) be a function such that there exists an 𝐿<1 with πœξ‚€π‘₯π‘š,π‘¦π‘šξ‚β‰€πΏπœ(π‘₯,𝑦)||π‘š3||(6.1) for all π‘₯,π‘¦βˆˆπ‘‹. Let π‘“βˆΆπ‘‹β†’π‘Œ be a mapping with 𝑓(0)=0 and satisfying the following inequality: β€–β€–ξ€·π‘šπ‘“(π‘šπ‘₯±𝑦)βˆ’π‘šπ‘“(π‘₯±𝑦)βˆ’23ξ€Έβ€–β€–βˆ’π‘šπ‘“(π‘₯)β‰€πœ(π‘₯,𝑦)(6.2) for all π‘₯,π‘¦βˆˆπ‘‹. Then, there is a unique cubic mapping πΆβˆΆπ‘‹β†’π‘Œ such that (‖𝑓π‘₯)βˆ’πΆ(π‘₯)β€–β‰€πΏπœ(π‘₯,0)||2π‘š3||βˆ’||2π‘š3||𝐿.(6.3) for all π‘₯βˆˆπ‘‹.

Proof. Putting 𝑦=0 in (6.2) and replacing π‘₯ by π‘₯/π‘š, we have β€–β€–β€–π‘š3𝑓π‘₯π‘šξ‚β€–β€–β€–β‰€1βˆ’π‘“(π‘₯)||2||πœξ‚€π‘₯π‘šξ‚,0(6.4) for all π‘₯βˆˆπ‘‹. Consider the set π‘†βˆΆ={π‘”βˆΆπ‘‹β†’π‘Œ;𝑔(0)=0}(6.5) and the generalized metric 𝑑 in 𝑆 defined by 𝑑(𝑓,𝑔)=infπœ‡βˆˆβ„+{‖𝑔(π‘₯)βˆ’β„Ž(π‘₯)β€–β‰€πœ‡πœ(π‘₯,0),βˆ€π‘₯βˆˆπ‘‹},(6.6) where infβˆ…=+∞. It is easy to show that (𝑆,𝑑) is complete (see [26], Lemma 2.1).
Now, we consider a linear mapping π½βˆΆπ‘†β†’π‘† such that π½β„Ž(π‘₯)∢=π‘š3β„Žξ‚€π‘₯π‘šξ‚(6.7) for all π‘₯βˆˆπ‘‹. Let 𝑔,β„Žβˆˆπ‘† be such that 𝑑(𝑔,β„Ž)=πœ–. Then, ‖𝑔(π‘₯)βˆ’β„Ž(π‘₯)β€–β‰€πœ–πœ(π‘₯,0)(6.8) for all π‘₯βˆˆπ‘‹, and so β€–β€–β€–β€–π‘šπ½π‘”(π‘₯)βˆ’π½β„Ž(π‘₯)β€–=3𝑔π‘₯π‘šξ‚βˆ’π‘š3β„Žξ‚€π‘₯π‘šξ‚β€–β€–