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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 902931, 45 pages
http://dx.doi.org/10.1155/2012/902931
Review Article

Nonlinear Random Stability via Fixed-Point Method

1Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea
2Department of Mathematics and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea
3Department of Mathematics, Iran University of Science and Technology, Behshahr, Iran

Received 31 October 2011; Accepted 22 December 2011

Academic Editor: Yeong-ChengΒ Liou

Copyright Β© 2012 Yeol Je Cho 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

We prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation 𝑓(π‘₯+2𝑦)+𝑓(π‘₯βˆ’2𝑦)=4𝑓(π‘₯+𝑦)+4𝑓(π‘₯βˆ’π‘¦)βˆ’6𝑓(π‘₯)+𝑓(2𝑦)+𝑓(βˆ’2𝑦)βˆ’4𝑓(𝑦)βˆ’4𝑓(βˆ’π‘¦) 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(π‘₯+𝑦)+𝑓(π‘₯βˆ’π‘¦)=2𝑓(π‘₯)+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 π‘“βˆΆπ‘‹β†’π‘Œ, where 𝑋 is a normed space and π‘Œ 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(2π‘₯+𝑦)+𝑓(2π‘₯βˆ’π‘¦)=2𝑓(π‘₯+𝑦)+2𝑓(π‘₯βˆ’π‘¦)+12𝑓(π‘₯).(1.2)

It is easy to show that the function 𝑓(π‘₯)=π‘₯3 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(2π‘₯+𝑦)+𝑓(2π‘₯βˆ’π‘¦)=4𝑓(π‘₯+𝑦)+4𝑓(π‘₯βˆ’π‘¦)+24𝑓(π‘₯)βˆ’6𝑓(𝑦).(1.3) It is easy to show that the function 𝑓(π‘₯)=π‘₯4 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 π‘“βˆΆπ‘‹β†’π‘Œ satisfies the additive-quadratic-cubic-quadratic functional equation 𝑓(π‘₯+2𝑦)+𝑓(π‘₯βˆ’2𝑦)=4𝑓(π‘₯+𝑦)+4𝑓(π‘₯βˆ’π‘¦)βˆ’6𝑓(π‘₯)+𝑓(2𝑦)+𝑓(βˆ’2𝑦)βˆ’4𝑓(𝑦)βˆ’4𝑓(βˆ’π‘¦)(1.4) if and only if it is an additive-cubic mapping, that is, 𝑓(π‘₯+2𝑦)+𝑓(π‘₯βˆ’2𝑦)=4𝑓(π‘₯+𝑦)+4𝑓(π‘₯βˆ’π‘¦)βˆ’6𝑓(π‘₯).(1.5)

It was shown in Lemma  2.2 of [14] that 𝑔(π‘₯)∢=𝑓(2π‘₯)βˆ’2𝑓(π‘₯) and β„Ž(π‘₯)∢=𝑓(2π‘₯)βˆ’8𝑓(π‘₯) are cubic and additive, respectively, and that 𝑓(π‘₯)=(1/6)𝑔(π‘₯)βˆ’(1/6)β„Ž(π‘₯).

One can easily show that an even mapping π‘“βˆΆπ‘‹β†’π‘Œ satisfies (1.4) if and only if it is a quadratic-quartic mapping, that is, 𝑓(π‘₯+2𝑦)+𝑓(π‘₯βˆ’2𝑦)=4𝑓(π‘₯+𝑦)+4𝑓(π‘₯βˆ’π‘¦)βˆ’6𝑓(π‘₯)+2𝑓(2𝑦)βˆ’8𝑓(𝑦).(1.6)

Also 𝑔(π‘₯)∢=𝑓(2π‘₯)βˆ’4𝑓(π‘₯) and β„Ž(π‘₯)∢=𝑓(2π‘₯)βˆ’16𝑓(π‘₯) are quartic and quadratic, respectively, and 𝑓(π‘₯)=(1/12)𝑔(π‘₯)βˆ’(1/12)β„Ž(π‘₯).

For a given mapping π‘“βˆΆπ‘‹β†’π‘Œ, we define 𝐷𝑓(π‘₯,𝑦)∢=𝑓(π‘₯+2𝑦)+𝑓(π‘₯βˆ’2𝑦)βˆ’4𝑓(π‘₯+𝑦)βˆ’4𝑓(π‘₯βˆ’π‘¦)+6𝑓(π‘₯)βˆ’π‘“(2𝑦)βˆ’π‘“(βˆ’2𝑦)+4𝑓(𝑦)+4𝑓(βˆ’π‘¦)(1.7)

for all π‘₯,π‘¦βˆˆπ‘‹.

Let 𝑋 be a set. A function π‘‘βˆΆπ‘‹Γ—π‘‹β†’[0,∞] is called a generalized metric on 𝑋 if 𝑑 satisfies(1)𝑑(π‘₯,𝑦)=0 if and only if π‘₯=𝑦,(2)𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯) for all π‘₯,π‘¦βˆˆπ‘‹,(3)𝑑(π‘₯,𝑧)≀𝑑(π‘₯,𝑦)+𝑑(𝑦,𝑧) for all π‘₯,𝑦,π‘§βˆˆπ‘‹.

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 π½βˆΆπ‘‹β†’π‘‹ be a strictly contractive mapping with Lipschitz constant 𝐿<1, then for each given element π‘₯βˆˆπ‘‹, either 𝑑𝐽𝑛π‘₯,𝐽𝑛+1π‘₯ξ€Έ=∞(1.8) for all nonnegative integers 𝑛 or there exists a positive integer 𝑛0 such that(1)𝑑(𝐽𝑛π‘₯,𝐽𝑛+1π‘₯)<∞ for all 𝑛β‰₯𝑛0,(2)the sequence {𝐽𝑛π‘₯} converges to a fixed point π‘¦βˆ— of 𝐽,(3)π‘¦βˆ— is the unique fixed point of 𝐽 in the set π‘Œ={π‘¦βˆˆπ‘‹βˆ£π‘‘(𝐽𝑛0π‘₯,𝑦)<∞},(4)𝑑(𝑦,π‘¦βˆ—)≀(1/(1βˆ’πΏ))𝑑(𝑦,𝐽𝑦) for all π‘¦βˆˆπ‘Œ.

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, Ξ”+ is the space of all probability distribution functions, that is, the space of all mappings πΉβˆΆβ„βˆͺ{βˆ’βˆž,+∞}β†’[0,1], such taht 𝐹 is left continuous, nondecreasing on ℝ, 𝐹(0)=0 and {𝐹(+∞)=1}. 𝐷+ is a subset of Ξ”+ consisting of all functions πΉβˆˆΞ”+ for which π‘™βˆ’πΉ(+∞)=1, where π‘™βˆ’π‘“(π‘₯) denotes the left limit of the function 𝑓 at the point π‘₯, that is, π‘™βˆ’π‘“(π‘₯)=lim𝑑→π‘₯βˆ’π‘“(𝑑). The space Ξ”+ is partially ordered by the usual pointwise ordering of functions, that is, 𝐹≀𝐺 if and only if 𝐹(𝑑)≀𝐺(𝑑) for all 𝑑 in ℝ. The maximal element for Ξ”+ in this order is the distribution function πœ€0 given byπœ€0ξ‚»(𝑑)=0,if𝑑≀0,1,if𝑑>0.(2.1)

A triangular norm (shortly 𝑑-norm) is a binary operation on the unit interval [0,1], that is, a function π‘‡βˆΆ[0,1]Γ—[0,1]β†’[0,1], such that for all π‘Ž,𝑏,π‘βˆˆ[0,1] the following four axioms satisfied:(T1)𝑇(π‘Ž,𝑏)=𝑇(𝑏,π‘Ž) (commutativity),(T2)𝑇(π‘Ž,(𝑇(𝑏,𝑐)))=𝑇(𝑇(π‘Ž,𝑏),𝑐) (associativity),(T3)𝑇(π‘Ž,1)=π‘Ž (boundary condition),(T4)𝑇(π‘Ž,𝑏)≀𝑇(π‘Ž,𝑐) whenever 𝑏≀𝑐 (monotonicity).

Basic examples are the Łukasiewicz  𝑑-norm  𝑇𝐿,𝑇𝐿(π‘Ž,𝑏)=max(π‘Ž+π‘βˆ’1,0) for all π‘Ž,π‘βˆˆ[0,1] and the 𝑑-norms  𝑇𝑃,𝑇𝑀,𝑇𝐷, where   𝑇𝑃(π‘Ž,𝑏)∢=π‘Žπ‘, 𝑇𝑀(π‘Ž,𝑏)∢=min{π‘Ž,𝑏}, 𝑇𝐷(π‘Ž,𝑏)∢=min(π‘Ž,𝑏),ifmax(π‘Ž,𝑏)=1,0,otherwise.(2.2)

If 𝑇 is a 𝑑-norm, then π‘₯𝑇(𝑛) is defined for every π‘₯∈[0,1] and π‘›βˆˆπ‘βˆͺ{0} by 1, if 𝑛=0 and 𝑇(π‘₯𝑇(π‘›βˆ’1),π‘₯) if 𝑛β‰₯1. A 𝑑-norm𝑇 is said to be of HadΕΎiΔ‡ type (we denote by π‘‡βˆˆβ„‹) if the family (π‘₯𝑇(𝑛))π‘›βˆˆπ‘ is equicontinuous at π‘₯=1 (cf. [27]).

Other important triangular norms are the following (see [28]):(1)The Sugeno-Weber family {𝑇SWπœ†}πœ†βˆˆ[βˆ’1,∞] is defined by 𝑇SWβˆ’1=𝑇𝐷, 𝑇SW∞=𝑇𝑃 and 𝑇SWπœ†ξ‚΅(π‘₯,𝑦)=max0,π‘₯+π‘¦βˆ’1+πœ†π‘₯𝑦1+πœ†(2.3) ifβ€‰β€‰πœ†βˆˆ(βˆ’1,∞).(2)The Domby family {π‘‡π·πœ†}πœ†βˆˆ[0,∞] is defined by 𝑇𝐷 if πœ†=0, 𝑇𝑀 if πœ†=∞, and π‘‡π·πœ†1(π‘₯,𝑦)=ξ€·1+((1βˆ’π‘₯)/π‘₯)πœ†+((1βˆ’π‘¦)/𝑦)πœ†ξ€Έ1/πœ†(2.4) ifβ€‰πœ†βˆˆ(0,∞).(3)The Aczel-Alsina family {𝑇AAπœ†}πœ†βˆˆ[0,∞] is defined by 𝑇𝐷 if πœ†=0, 𝑇𝑀 if πœ†=∞ and 𝑇AAπœ†(π‘₯,𝑦)=π‘’βˆ’(|logπ‘₯|πœ†+|log𝑦|πœ†)1/πœ†(2.5) ifβ€‰πœ†βˆˆ(0,∞).

A 𝑑-norm   𝑇 can be extended (by associativity) in a unique way to an 𝑛-array operation taking for (π‘₯1,…,π‘₯𝑛)∈[0,1]𝑛 the value 𝑇(π‘₯1,…,π‘₯𝑛) defined by 𝑇0𝑖=1π‘₯𝑖=1,𝑇𝑛𝑖=1π‘₯𝑖𝑇=π‘‡π‘›βˆ’1𝑖=1π‘₯𝑖,π‘₯𝑛π‘₯=𝑇1,…,π‘₯𝑛.(2.6)

𝑇 can also be extended to a countable operation taking for any sequence (π‘₯𝑛)π‘›βˆˆπ‘ in [0,1] the valueπ‘‡βˆžπ‘–=1π‘₯𝑖=limπ‘›β†’βˆžπ‘‡π‘›π‘–=1π‘₯𝑖.(2.7) The limit on the right side of (6.4) exists since the sequence (𝑇𝑛𝑖=1π‘₯𝑖)π‘›βˆˆβ„• is nonincreasing and bounded from below.

Proposition 2.1 (see [28]). We have the following.(1)For 𝑇β‰₯𝑇𝐿, the following implication holds: limπ‘›β†’βˆžTβˆžπ‘–=1π‘₯𝑛+𝑖=1βŸΊβˆžξ“π‘›=1ξ€·1βˆ’π‘₯𝑛<∞.(2.8)(2)If 𝑇 is of HadΕΎiΔ‡ type, then limπ‘›β†’βˆžTβˆžπ‘–=1π‘₯𝑛+𝑖=1(2.9) for every sequence (π‘₯𝑛)π‘›βˆˆπ‘ in [0,1] such that limπ‘›β†’βˆžπ‘₯𝑛=1.(3)If π‘‡βˆˆ{𝑇AAπœ†}πœ†βˆˆ(0,∞)βˆͺ{π‘‡π·πœ†}πœ†βˆˆ(0,∞), then limπ‘›β†’βˆžTβˆžπ‘–=1π‘₯𝑛+𝑖=1βŸΊβˆžξ“π‘›=1ξ€·1βˆ’π‘₯𝑛𝛼<∞.(2.10)(4)If π‘‡βˆˆ{𝑇SWπœ†}πœ†βˆˆ[βˆ’1,∞), then limπ‘›β†’βˆžTβˆžπ‘–=1π‘₯𝑛+𝑖=1βŸΊβˆžξ“π‘›=1ξ€·1βˆ’π‘₯𝑛<∞.(2.11)

Definition 2.2 (see [26]). A Random normed space (briefly, RN-space) is a triple (𝑋,πœ‡,𝑇), where 𝑋 is a vector space, 𝑇 is a continuous 𝑑-norm, and πœ‡ is a mapping from 𝑋 into 𝐷+ such that, the following conditions hold: (RN1) πœ‡π‘₯(𝑑)=πœ€0(𝑑) for all 𝑑>0 if and only if π‘₯=0,(RN2) πœ‡π›Όπ‘₯(𝑑)=πœ‡π‘₯(𝑑/|𝛼|) for all π‘₯βˆˆπ‘‹, and 𝛼≠0,(RN3) πœ‡π‘₯+𝑦(𝑑+𝑠)β‰₯𝑇(πœ‡π‘₯(𝑑),πœ‡π‘¦(𝑠)) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑,𝑠β‰₯0.

Definition 2.3. Let (𝑋,πœ‡,𝑇) be an RN-space.(1)A sequence {π‘₯𝑛} in 𝑋 is said to be convergent to π‘₯ in 𝑋 if, for every πœ–>0 and πœ†>0, there exists positive integer 𝑁 such that πœ‡π‘₯π‘›βˆ’π‘₯(πœ–)>1βˆ’πœ† whenever 𝑛β‰₯𝑁.(2)A sequence {π‘₯𝑛} in 𝑋 is called a Cauchy sequence if, for every πœ–>0 and πœ†>0, there exists positive integer 𝑁 such that πœ‡π‘₯π‘›βˆ’π‘₯π‘š(πœ–)>1βˆ’πœ† whenever 𝑛β‰₯π‘šβ‰₯𝑁.(3)An RN-space (𝑋,πœ‡,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,πœ‡,𝑇) is an RN-space and {π‘₯𝑛} is a sequence such that π‘₯𝑛→π‘₯, then limπ‘›β†’βˆžπœ‡π‘₯𝑛(𝑑)=πœ‡π‘₯(𝑑) 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) |β‹…| from 𝐾 into [0,∞) such that |π‘Ÿ|=0 if and only if π‘Ÿ=0, |π‘Ÿπ‘ |=|π‘Ÿ||𝑠|, and |π‘Ÿ+𝑠|≀max{|π‘Ÿ|,|𝑠|} for all π‘Ÿ,π‘ βˆˆπ’¦. Clearly, |1|=|βˆ’1|=1 and |𝑛|≀1 for all π‘›βˆˆβ„•. By the trivial valuation, we mean the mapping |β‹…| 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 |β‹…|. A function β€–β‹…β€–βˆΆπ‘‹β†’[0,∞) is called a non-Archimedean norm if it satisfies the following conditions:(NAN1) β€–π‘₯β€–=0 if and only if π‘₯=0,(NAN2) for any π‘Ÿβˆˆπ’¦ and π‘₯βˆˆπ‘‹, β€–π‘Ÿπ‘₯β€–=|π‘Ÿ|β€–π‘₯β€–,(NAN3) the strong triangle inequality (ultrametric), namely, β€–π‘₯+𝑦‖≀max{β€–π‘₯β€–,‖𝑦‖}(π‘₯,π‘¦βˆˆπ‘‹),(3.1)

then (𝑋,β€–β‹…β€–) is called a non-Archimedean normed space. Due to the fact that β€–β€–π‘₯π‘›βˆ’π‘₯π‘šβ€–β€–ξ€½β€–β€–π‘₯≀max𝑗+1βˆ’π‘₯π‘—β€–β€–ξ€ΎβˆΆπ‘šβ‰€π‘—β‰€π‘›βˆ’1(𝑛>π‘š),(3.2)

a sequence {π‘₯𝑛} is a Cauchy sequence if and only if {π‘₯𝑛+1βˆ’π‘₯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 π‘₯, there exists a unique integer 𝑛π‘₯βˆˆβ„€ such that π‘₯=(π‘Ž/𝑏)𝑝𝑛π‘₯, where π‘Ž and 𝑏 are integers not divisible by 𝑝. Then |π‘₯|π‘βˆΆ=π‘βˆ’π‘›π‘₯ defines a non-Archimedean norm on β„š. The completion of β„š with respect to the metric 𝑑(π‘₯,𝑦)=|π‘₯βˆ’π‘¦|𝑝 is denoted by β„šπ‘, which is called the 𝑝-adic number field.

Throughout the paper, we assume that 𝑋 is a vector space and π‘Œ 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,πœ‡,𝑇), where 𝑋 is a linear space over a non-Archimedean field 𝒦, 𝑇 is a continuous 𝑑-norm, and πœ‡ is a mapping from 𝑋 into 𝐷+ such that the following conditions hold:(NA-RN1) πœ‡π‘₯(𝑑)=πœ€0(𝑑) for all 𝑑>0 if and only if π‘₯=0,(NA-RN2) πœ‡π›Όπ‘₯(𝑑)=πœ‡π‘₯(𝑑/|𝛼|) for all π‘₯βˆˆπ‘‹, 𝑑>0, and 𝛼≠0,(NA-RN3) πœ‡π‘₯+𝑦(max{𝑑,𝑠})β‰₯𝑇(πœ‡π‘₯(𝑑),πœ‡π‘¦(𝑠)) for all π‘₯,𝑦,π‘§βˆˆπ‘‹ and 𝑑,𝑠β‰₯0.
It is easy to see that if (NA-RN3) holds, then so is(RN3)πœ‡π‘₯+𝑦(𝑑+𝑠)β‰₯𝑇(πœ‡π‘₯(𝑑),πœ‡π‘¦(𝑠)).

As a classical example, if (𝑋,β€–.β€–) is a non-Archimedean normed linear space, then the triple (𝑋,πœ‡,𝑇𝑀), where πœ‡π‘₯ξ‚»(𝑑)=0,𝑑≀‖π‘₯β€–,1,𝑑>β€–π‘₯β€–,(3.3)

is a non-Archimedean RN-space.

Example 3.2. Let (𝑋,β€–β‹…β€–) be a non-Archimedean normed linear space. Define πœ‡π‘₯(𝑑𝑑)=(𝑑+β€–π‘₯β€–π‘₯βˆˆπ‘‹,𝑑>0),(3.4) then (𝑋,πœ‡,𝑇𝑀) is a non-Archimedean RN-space.

Definition 3.3. Let (𝑋,πœ‡,𝑇) be a non-Archimedean RN-space. Let {π‘₯𝑛} be a sequence in 𝑋, then {π‘₯𝑛} is said to be convergent if there exists π‘₯βˆˆπ‘‹ such that limπ‘›β†’βˆžπœ‡π‘₯π‘›βˆ’π‘₯(𝑑)=1(3.5) for all 𝑑>0. In that case, π‘₯ is called the limit of the sequence {π‘₯𝑛}.

A sequence {π‘₯𝑛} in 𝑋 is called a Cauchy sequence if for each πœ€>0 and each 𝑑>0 there exists 𝑛0 such that for all 𝑛β‰₯𝑛0 and all 𝑝>0, we have πœ‡π‘₯𝑛+π‘βˆ’π‘₯𝑛(𝑑)>1βˆ’πœ€.

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,πœ‡,𝑇𝑀) be a non-Archimedean RN-space, then πœ‡π‘₯𝑛+π‘βˆ’π‘₯π‘›ξ‚†πœ‡(𝑑)β‰₯minπ‘₯𝑛+𝑗+1βˆ’π‘₯𝑛+𝑗(𝑑)βˆΆπ‘—=0,1,2,…,π‘βˆ’1.(3.6) So, the sequence {π‘₯𝑛} is a Cauchy sequence if for each πœ€>0 and 𝑑>0 there exists 𝑛0 such that for all 𝑛β‰₯𝑛0, πœ‡π‘₯𝑛+1βˆ’π‘₯𝑛(𝑑)>1βˆ’πœ€.(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 (π‘Œ,πœ‡,𝑇) be a non-Archimedean random Banach space over 𝒦.

Next, we define a random approximately AQCQ mapping. Let Ξ¨ be a distribution function on 𝑋×𝑋×[0,∞) such that Ξ¨(π‘₯,𝑦,β‹…) is nondecreasing and 𝑑Ψ(𝑐π‘₯,𝑐π‘₯,𝑑)β‰₯Ξ¨π‘₯,π‘₯,ξ‚Ά|𝑐|(π‘₯βˆˆπ‘‹,𝑐≠0).(4.1)

Definition 4.1. A mapping π‘“βˆΆπ‘‹β†’π‘Œ is said to be Ξ¨-approximately AQCQ if πœ‡π·π‘“(π‘₯,𝑦)(𝑑)β‰₯Ξ¨(π‘₯,𝑦,𝑑)(π‘₯,π‘¦βˆˆπ‘‹,𝑑>0).(4.2)

In this section, we assume that 2β‰ 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(π‘₯,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 (π‘Œ,πœ‡,𝑇) be a non-Archimedean random Banach space over 𝒦. Let π‘“βˆΆπ‘‹β†’π‘Œ be an odd mapping and Ξ¨-approximately AQCQ mapping. If for some π›Όβˆˆβ„, 𝛼>0, and some integer π‘˜, π‘˜>3 with |2π‘˜|<𝛼, Ξ¨ξ€·2βˆ’π‘˜π‘₯,2βˆ’π‘˜ξ€Έπ‘¦,𝑑β‰₯Ξ¨(π‘₯,𝑦,𝛼𝑑)(π‘₯βˆˆπ‘‹,𝑑>0),(4.3)limπ‘›β†’βˆžTβˆžπ‘—=𝑛𝑀𝛼2π‘₯,𝑗𝑑||8||π‘˜π‘—ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0),(4.4) then there exists a unique cubic mapping πΆβˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’2𝑓(π‘₯/2)βˆ’πΆ(π‘₯/2)(𝑑)β‰₯π‘‡βˆžπ‘–=1𝑀𝛼π‘₯,𝑖+1𝑑||8||π‘˜π‘–ξƒͺ(4.5) for all π‘₯βˆˆπ‘‹ and 𝑑>0, where 𝑀(π‘₯,𝑑)∢=π‘‡π‘˜βˆ’1ξ‚ΈΞ¨ξ‚΅π‘₯2,π‘₯2,𝑑||4||ξ‚Άξ‚€π‘₯,Ξ¨π‘₯,22,𝑑,…,Ξ¨π‘˜βˆ’1π‘₯2,2π‘˜βˆ’1π‘₯2,𝑑||4||ξ‚Άξ‚΅2,Ξ¨π‘˜βˆ’12π‘₯,π‘˜βˆ’1π‘₯2,𝑑(π‘₯βˆˆπ‘‹,𝑑>0).(4.6)

Proof. Letting π‘₯=𝑦 in (4.2), we get πœ‡π‘“(3𝑦)βˆ’4𝑓(2𝑦)+5𝑓(𝑦)(𝑑)β‰₯Ξ¨(𝑦,𝑦,𝑑)(4.7) for all π‘¦βˆˆπ‘‹ and 𝑑>0. Replacing π‘₯ by 2𝑦 in (4.2), we get πœ‡π‘“(4𝑦)βˆ’4𝑓(3𝑦)+6𝑓(2𝑦)βˆ’4𝑓(𝑦)(𝑑)β‰₯Ξ¨(2𝑦,𝑦,𝑑)(4.8) for all π‘¦βˆˆπ‘‹ and 𝑑>0. By (4.7) and (4.8), we have πœ‡π‘“(4𝑦)βˆ’10𝑓(2𝑦)+16𝑓(𝑦)ξ€·πœ‡(𝑑)β‰₯𝑇4(𝑓(3𝑦)βˆ’4𝑓(2𝑦)+5𝑓(𝑦))(𝑑),πœ‡π‘“(4𝑦)βˆ’4𝑓(3𝑦)+6𝑓(2𝑦)βˆ’4𝑓(𝑦)ξ€Έξ‚΅πœ‡(𝑑)=𝑇𝑓(3𝑦)βˆ’4𝑓(2𝑦)+5𝑓(𝑦)𝑑||4||ξ‚Ά,πœ‡π‘“(4𝑦)βˆ’4𝑓(3𝑦)+6𝑓(2𝑦)βˆ’4𝑓(𝑦)Ψ𝑑(𝑑)β‰₯𝑇𝑦,𝑦,||4||ξ‚Άξ‚Ά,Ξ¨(2𝑦,𝑦,𝑑)(4.9) for all π‘¦βˆˆπ‘‹ and 𝑑>0. Letting π‘¦βˆΆ=π‘₯/2 and 𝑔(π‘₯)∢=𝑓(2π‘₯)βˆ’2𝑓(π‘₯) for all π‘₯βˆˆπ‘‹ in (4.9), we get πœ‡π‘”(π‘₯)βˆ’8𝑔(π‘₯/2)ξ‚΅Ξ¨ξ‚΅π‘₯(𝑑)β‰₯𝑇2,π‘₯2,𝑑||4||ξ‚Άξ‚€π‘₯,Ξ¨π‘₯,2,𝑑(4.10) for all π‘₯βˆˆπ‘‹ and 𝑑>0. Now, we show by induction on 𝑗 that for all π‘₯βˆˆπ‘‹, 𝑑>0 and 𝑗β‰₯1, πœ‡π‘”(2π‘—βˆ’1π‘₯)βˆ’8𝑗𝑔(π‘₯/2)(𝑑)β‰₯𝑀𝑗(π‘₯,𝑑)∢=𝑇2π‘—βˆ’1ξ‚ΈΞ¨ξ‚΅π‘₯2,π‘₯2,𝑑||4||ξ‚Άξ‚€π‘₯,Ξ¨π‘₯,22,𝑑,…,Ξ¨π‘—βˆ’1π‘₯2,2π‘—βˆ’1π‘₯2,𝑑||4||ξ‚Άξ‚΅2,Ξ¨π‘—βˆ’12π‘₯,π‘—βˆ’1π‘₯2.,𝑑(4.11) Putting 𝑗=1 in (4.11), we obtain (4.10). Assume that (4.11) holds for some 𝑗β‰₯1. Replacing π‘₯ by 2𝑗π‘₯ in (4.10), we get πœ‡π‘”(2𝑗π‘₯)βˆ’8𝑔(2π‘—βˆ’1π‘₯)ξ‚΅Ξ¨ξ‚΅2(𝑑)β‰₯π‘‡π‘—βˆ’1π‘₯,2π‘—βˆ’1𝑑π‘₯,||4||ξ‚Άξ€·2,Ψ𝑗π‘₯,2π‘—βˆ’1ξ€Έξ‚Άπ‘₯,𝑑.(4.12) Since |8|≀1, πœ‡π‘”(2𝑗π‘₯)βˆ’8𝑗+1𝑔(π‘₯/2)ξ€·πœ‡(𝑑)β‰₯𝑇𝑔(2𝑗π‘₯)βˆ’8𝑔(2π‘—βˆ’1π‘₯)(𝑑),πœ‡8𝑔(2π‘—βˆ’1π‘₯)βˆ’8𝑗+1𝑔(π‘₯/2)ξ€Έξ‚΅πœ‡(𝑑)=𝑇𝑔(2𝑗π‘₯)βˆ’8𝑔(2π‘—βˆ’1π‘₯)(𝑑),πœ‡π‘”(2π‘—βˆ’1π‘₯)βˆ’8𝑗𝑔(π‘₯/2)𝑑||8||ξ‚Άξ‚Άβ‰₯𝑇2ξ‚΅Ξ¨ξ‚΅2π‘—βˆ’1π‘₯,2π‘—βˆ’1𝑑π‘₯,||4||ξ‚Άξ€·2,Ψ𝑗π‘₯,2π‘—βˆ’1ξ€Έπ‘₯,𝑑,𝑀𝑗(π‘₯,𝑑)=𝑀𝑗+1(π‘₯,𝑑)(4.13)for all π‘₯βˆˆπ‘‹ and 𝑑>0. Thus, (4.11) holds for all 𝑗β‰₯2. In particular, πœ‡π‘”(2π‘˜βˆ’1π‘₯)βˆ’8π‘˜π‘”(π‘₯/2)(𝑑)β‰₯𝑀(π‘₯,𝑑)(π‘₯βˆˆπ‘‹,𝑑>0).(4.14) Replacing π‘₯ by 2βˆ’(π‘˜π‘›+π‘˜βˆ’1)π‘₯ in (4.14) and using inequality (4.3), we obtain πœ‡π‘”(π‘₯/2π‘˜π‘›)βˆ’8π‘˜π‘”(π‘₯/2π‘˜(𝑛+1))ξ‚΅(𝑑)β‰₯𝑀2π‘₯2π‘˜(𝑛+1)ξ‚Ά,𝑑(π‘₯βˆˆπ‘‹,𝑑>0,𝑛=0,1,2,…).(4.15) Then πœ‡8π‘˜π‘›π‘”(π‘₯/2π‘˜π‘›)βˆ’8π‘˜(𝑛+1)𝑔(π‘₯/2π‘˜(𝑛+1))𝛼(𝑑)β‰₯𝑀2π‘₯,𝑛+1||8π‘˜(𝑛+1)||𝑑ξƒͺ(π‘₯βˆˆπ‘‹,𝑑>0,𝑛=0,1,2,…).(4.16) Hence πœ‡8π‘˜π‘›π‘”(π‘₯/2π‘˜π‘›)βˆ’8π‘˜(𝑛+𝑝)𝑔(π‘₯/2π‘˜(𝑛+𝑝))(𝑑)β‰₯𝑇𝑛+𝑝𝑗=π‘›ξ€·πœ‡8π‘˜π‘—π‘”(π‘₯/2π‘˜π‘—)βˆ’8π‘˜(𝑗+𝑝)𝑔(π‘₯/2π‘˜(𝑗+𝑝))(𝑑)β‰₯𝑇𝑛+𝑝𝑗=π‘›π‘€βŽ›βŽœβŽœβŽœβŽπ›Ό2π‘₯,𝑗+1|||ξ€·8π‘˜ξ€Έπ‘—+1|||π‘‘βŽžβŽŸβŽŸβŽŸβŽ β‰₯𝑇𝑛+𝑝𝑗=π‘›π‘€βŽ›βŽœβŽœβŽœβŽπ›Ό2π‘₯,𝑗+1|||ξ€·8π‘˜ξ€Έπ‘—+1|||π‘‘βŽžβŽŸβŽŸβŽŸβŽ (π‘₯βˆˆπ‘‹,𝑑>0,𝑛=0,1,2,…).(4.17) Since limπ‘›β†’βˆžπ‘‡βˆžπ‘—=π‘›π‘€βŽ›βŽœβŽœβŽœβŽπ›Ό2π‘₯,𝑗+1|||ξ€·8π‘˜ξ€Έπ‘—+1|||π‘‘βŽžβŽŸβŽŸβŽŸβŽ =1(π‘₯βˆˆπ‘‹,𝑑>0),(4.18) then ξ‚»8π‘˜π‘›π‘”ξ‚΅π‘₯2π‘˜π‘›ξ‚Άξ‚Όπ‘›βˆˆβ„•(4.19)is a Cauchy sequence in the non-Archimedean random Banach space (π‘Œ,πœ‡,𝑇). Hence we can define a mapping πΆβˆΆπ‘‹β†’π‘Œ such that limπ‘›β†’βˆžπœ‡(88π‘˜)𝑛𝑔(π‘₯/2π‘˜π‘›)βˆ’πΆ(π‘₯)(𝑑)=1(π‘₯βˆˆπ‘‹,𝑑>0).(4.20)
Next for each 𝑛β‰₯1, π‘₯βˆˆπ‘‹ and 𝑑>0, πœ‡π‘”(π‘₯)βˆ’(88π‘˜)𝑛𝑔(π‘₯/2π‘˜π‘›)(𝑑)=πœ‡βˆ‘π‘›βˆ’1𝑖=0(88π‘˜)𝑖𝑔(π‘₯/2π‘˜π‘–)βˆ’(88π‘˜)𝑖+1𝑔(π‘₯/2π‘˜(𝑖+1))(𝑑)β‰₯π‘‡π‘›βˆ’1𝑖=0ξ‚€πœ‡(88π‘˜)𝑖𝑔(π‘₯/2π‘˜π‘–)βˆ’(88π‘˜)𝑖+1𝑔(π‘₯/2π‘˜(𝑖+1))(𝑑)β‰₯π‘‡π‘›βˆ’1𝑖=0𝑀𝛼2π‘₯,𝑖+1𝑑||8π‘˜||𝑖+1ξƒͺ.(4.21)Therefore, πœ‡π‘”(π‘₯)βˆ’πΆ(π‘₯)ξ€·πœ‡(𝑑)β‰₯𝑇𝑔(π‘₯)βˆ’(88π‘˜)𝑛𝑔(π‘₯/2π‘˜π‘›)(𝑑),πœ‡(88π‘˜)𝑛𝑔(π‘₯/2π‘˜π‘›)βˆ’πΆ(π‘₯)𝑇(𝑑)β‰₯π‘‡π‘›βˆ’1𝑖=0𝑀𝛼2π‘₯,𝑖+1𝑑||8π‘˜||𝑖+1ξƒͺ,πœ‡(88π‘˜)𝑛𝑔(π‘₯/2π‘˜π‘›)βˆ’πΆ(π‘₯)ξƒͺ.(𝑑)(4.22) By letting π‘›β†’βˆž, we obtain πœ‡π‘”(π‘₯)βˆ’πΆ(π‘₯)(𝑑)β‰₯π‘‡βˆžπ‘–=1𝑀𝛼2π‘₯,𝑖+1𝑑||8π‘˜||𝑖+1ξƒͺ.(4.23) So, πœ‡π‘“(π‘₯)βˆ’2𝑓(π‘₯/2)βˆ’πΆ(π‘₯/2)(𝑑)β‰₯π‘‡βˆžπ‘–=1𝑀𝛼π‘₯,𝑖+1𝑑||8π‘˜||𝑖+1ξƒͺ.(4.24) This proves (4.5). From 𝐷𝑔(π‘₯,𝑦)=𝐷𝑓(2π‘₯,2𝑦)βˆ’2𝐷𝑓(π‘₯,𝑦), by (4.2), we deduce that πœ‡π·π‘“(2π‘₯,2𝑦)πœ‡(𝑑)β‰₯Ξ¨(2π‘₯,2𝑦,𝑑),βˆ’2𝐷𝑓(π‘₯,𝑦)(𝑑)=πœ‡π·π‘“(π‘₯,𝑦)𝑑||2||ξ‚Άβ‰₯πœ‡π·π‘“(π‘₯,𝑦)(𝑑)β‰₯Ξ¨(π‘₯,𝑦,𝑑),(4.25) and so, by (NA-RN3) and (4.2), we obtain πœ‡π·π‘”(π‘₯,𝑦)ξ€·πœ‡(𝑑)β‰₯𝑇𝐷𝑓(2π‘₯,2𝑦)(𝑑),πœ‡βˆ’2𝐷𝑓(π‘₯,𝑦)ξ€Έ(𝑑)β‰₯𝑇(Ξ¨(2π‘₯,2𝑦,𝑑),Ξ¨(π‘₯,𝑦,𝑑))∢=𝑁(π‘₯,𝑦,𝑑).(4.26) It follows that πœ‡8π‘˜π‘›π·π‘”(π‘₯/2π‘˜π‘›,𝑦/2π‘˜π‘›)(𝑑)=πœ‡π·π‘”(π‘₯/2π‘˜π‘›,𝑦/2π‘˜π‘›)𝑑||8||π‘˜π‘›ξƒͺπ‘₯β‰₯𝑁2π‘˜π‘›,𝑦2π‘˜π‘›,𝑑||8||π‘˜π‘›ξƒͺ𝛼β‰₯β‹―β‰₯𝑁π‘₯,𝑦,π‘›βˆ’1𝑑||8||π‘˜(π‘›βˆ’1)ξƒͺ(4.27)for all π‘₯,π‘¦βˆˆπ‘‹, 𝑑>0, and π‘›βˆˆβ„•. Since limπ‘›β†’βˆžπ‘ξƒ©π›Όπ‘₯,𝑦,π‘›βˆ’1𝑑||8||π‘˜(π‘›βˆ’1)ξƒͺ=1(4.28)for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0, by Theorem 2.4, we deduce that πœ‡π·πΆ(π‘₯,𝑦)(𝑑)=1(4.29)for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0. Thus, the mapping πΆβˆΆπ‘‹β†’π‘Œ satisfies (1.4).
Now, we have 𝐢(2π‘₯)βˆ’8𝐢(π‘₯)=limπ‘›β†’βˆžξ‚Έ8𝑛𝑔π‘₯2π‘›βˆ’1ξ‚Άβˆ’8𝑛+1𝑔π‘₯2𝑛=8limπ‘›β†’βˆžξ‚Έ8π‘›βˆ’1𝑔π‘₯2π‘›βˆ’1ξ‚Άβˆ’8𝑛𝑔π‘₯2𝑛=0(4.30)for all π‘₯βˆˆπ‘‹. Since the mapping π‘₯→𝐢(2π‘₯)βˆ’2𝐢(π‘₯) is cubic (see Lemma  2.2 of [14]), from the equality 𝐢(2π‘₯)=8𝐢(π‘₯), we deduce that the mapping πΆβˆΆπ‘‹β†’π‘Œ is cubic.

Corollary 4.3. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦, and let (π‘Œ,πœ‡,𝑇) be a non-Archimedean random Banach space over 𝒦 under a t-norm π‘‡βˆˆβ„‹. Let π‘“βˆΆπ‘‹β†’π‘Œ be an odd and Ξ¨-approximately AQCQ mapping. If, for some π›Όβˆˆβ„, 𝛼>0, and some integer π‘˜, π‘˜>3, with |2π‘˜|<𝛼, Ξ¨ξ€·2βˆ’π‘˜π‘₯,2βˆ’π‘˜ξ€Έπ‘¦,𝑑β‰₯Ξ¨(π‘₯,𝑦,𝛼𝑑)(π‘₯βˆˆπ‘‹,𝑑>0),(4.31) then there exists a unique cubic mapping πΆβˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’2𝑓(π‘₯/2)βˆ’πΆ(π‘₯/2)(𝑑)β‰₯π‘‡βˆžπ‘–=1𝑀𝛼π‘₯,𝑖+1𝑑||8||π‘˜π‘–ξƒͺ(4.32) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Proof. Since limπ‘›β†’βˆžπ‘€ξƒ©π›Όπ‘₯,𝑗𝑑||8||π‘˜π‘—ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0)(4.33) and 𝑇 is of HadΕΎiΔ‡ type, from Proposition 2.1, it follows that limπ‘›β†’βˆžπ‘‡βˆžπ‘—=𝑛𝑀𝛼π‘₯,𝑗𝑑||8||π‘˜π‘—ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0).(4.34)Now, we can apply Theorem 4.2 to obtain the result.

Example 4.4. Let (𝑋,πœ‡,𝑇𝑀) be non-Archimedean random normed space in which πœ‡π‘₯(𝑑𝑑)=(𝑑+β€–π‘₯β€–π‘₯βˆˆπ‘‹,𝑑>0).(4.35) And let (π‘Œ,πœ‡,𝑇𝑀) be a complete non-Archimedean random normed space (see Example 3.2). Define 𝑑Ψ(π‘₯,𝑦,𝑑)=.1+𝑑(4.36) It is easy to see that (4.3) holds for 𝛼=1. Also, since 𝑑𝑀(π‘₯,𝑑)=,1+𝑑(4.37) we have limπ‘›β†’βˆžπ‘‡βˆžπ‘€,𝑗=𝑛𝑀𝛼π‘₯,𝑗𝑑||8||π‘˜π‘—ξƒͺ=limπ‘›β†’βˆžξƒ©limπ‘šβ†’βˆžπ‘‡π‘šπ‘€,𝑗=𝑛𝑀𝑑π‘₯,||8||π‘˜π‘—ξƒͺξƒͺ=limπ‘›β†’βˆžlimπ‘šβ†’βˆžξƒ©π‘‘||8𝑑+π‘˜||𝑛ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0).(4.38) Let π‘“βˆΆπ‘‹β†’π‘Œ be an odd and Ξ¨-approximately AQCQ mapping. Thus, all the conditions of Theorem 4.2 hold, and so there exists a unique cubic mapping πΆβˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’2𝑓(π‘₯/2)βˆ’πΆ(π‘₯/2)(𝑑𝑑)β‰₯||8𝑑+π‘˜||.(4.39)

Theorem 4.5. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦, and let (π‘Œ,πœ‡,𝑇) be a non-Archimedean random Banach space over 𝒦. Let π‘“βˆΆπ‘‹β†’π‘Œ be an odd mapping and Ξ¨-approximately AQCQ mapping. If for some π›Όβˆˆβ„, 𝛼>0, and some integer π‘˜, π‘˜>1 with |2π‘˜|<𝛼, Ξ¨ξ€·2βˆ’π‘˜π‘₯,2βˆ’π‘˜ξ€Έπ‘¦,𝑑β‰₯Ξ¨(π‘₯,𝑦,𝛼𝑑)(π‘₯βˆˆπ‘‹,𝑑>0),limπ‘›β†’βˆžπ‘‡βˆžπ‘—=𝑛𝑀𝛼2π‘₯,𝑗𝑑||2||π‘˜π‘—ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0),(4.40) then there exists a unique additive mapping π΄βˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’8𝑓(π‘₯/2)βˆ’π΄(π‘₯/2)(𝑑)β‰₯π‘‡βˆžπ‘–=1𝑀𝛼π‘₯,𝑖+1𝑑||2||π‘˜π‘–ξƒͺ(4.41) for all π‘₯βˆˆπ‘‹ and 𝑑>0, where 𝑀(π‘₯,𝑑)∢=π‘‡π‘˜βˆ’1ξ‚ΈΞ¨ξ‚΅π‘₯2,π‘₯2,𝑑||4||ξ‚Άξ‚€π‘₯,Ξ¨π‘₯,22,𝑑,…,Ξ¨π‘˜βˆ’1π‘₯2,2π‘˜βˆ’1π‘₯2,𝑑||4||ξ‚Άξ‚΅2,Ξ¨π‘˜βˆ’12π‘₯,π‘˜βˆ’1π‘₯2,𝑑(π‘₯βˆˆπ‘‹,𝑑>0)(4.42)

Proof. Letting π‘¦βˆΆ=π‘₯/2 and 𝑔(π‘₯)∢=𝑓(2π‘₯)βˆ’8𝑓(π‘₯) for all π‘₯βˆˆπ‘‹ in (4.9), we get πœ‡π‘”(π‘₯)βˆ’2𝑔(π‘₯/2)ξ‚΅Ξ¨ξ‚΅π‘₯(𝑑)β‰₯𝑇2,π‘₯2,𝑑||4||ξ‚Άξ‚€π‘₯,Ξ¨π‘₯,2,𝑑(4.43) for all π‘₯βˆˆπ‘‹ and 𝑑>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 (π‘Œ,πœ‡,𝑇) be a non-Archimedean random Banach space over 𝒦 under a t-norm π‘‡βˆˆβ„‹. Let π‘“βˆΆπ‘‹β†’π‘Œ be an odd and Ξ¨-approximately AQCQ mapping. If, for some π›Όβˆˆβ„,𝛼>0, and some integer π‘˜,π‘˜>1, with |2π‘˜|<𝛼, Ξ¨ξ€·2βˆ’π‘˜π‘₯,2βˆ’π‘˜ξ€Έπ‘¦,𝑑β‰₯Ξ¨(π‘₯,𝑦,𝛼𝑑)(π‘₯βˆˆπ‘‹,𝑑>0),(4.44) then there exists a unique additive mapping π΄βˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’8𝑓(π‘₯/2)βˆ’π΄(π‘₯/2)(𝑑)β‰₯π‘‡βˆžπ‘–=1𝑀𝛼π‘₯,𝑖+1𝑑||2||π‘˜π‘–ξƒͺ(4.45) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Proof. Since limπ‘›β†’βˆžπ‘€ξƒ©π›Όπ‘₯,𝑗𝑑||2||π‘˜π‘—ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0)(4.46) and 𝑇 is of HadΕΎiΔ‡ type, from Proposition 2.1, it follows that limπ‘›β†’βˆžπ‘‡βˆžπ‘—=𝑛𝑀𝛼π‘₯,𝑗𝑑||2||π‘˜π‘—ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0).(4.47)Now, we can apply Theorem 4.5 to obtain the result.

Example 4.7. Let (𝑋,πœ‡,𝑇𝑀) non-Archimedean random normed space in which πœ‡π‘₯=𝑑(𝑑)𝑑+β€–π‘₯β€–(π‘₯βˆˆπ‘‹,𝑑>0),(4.48)and let (π‘Œ,πœ‡,𝑇𝑀) be a complete non-Archimedean random normed space (see Example 3.2). Define 𝑑Ψ(π‘₯,𝑦,𝑑)=.1+𝑑(4.49) It is easy to see that (4.3) holds for 𝛼=1. Also, since 𝑑𝑀(π‘₯,𝑑)=,1+𝑑(4.50) we have limπ‘›β†’βˆžπ‘‡βˆžπ‘€,𝑗=𝑛𝑀𝛼π‘₯,𝑗𝑑||2||π‘˜π‘—ξƒͺ=limπ‘›β†’βˆžξƒ©limπ‘šβ†’βˆžπ‘‡π‘šπ‘€,𝑗=𝑛𝑀𝑑π‘₯,||2||π‘˜π‘—ξƒͺξƒͺ=limπ‘›β†’βˆžlimπ‘šβ†’βˆžξƒ©π‘‘||2𝑑+π‘˜||𝑛ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0).(4.51) Let π‘“βˆΆπ‘‹β†’π‘Œ be an odd and Ξ¨-approximately AQCQ mapping. Thus, all the conditions of Theorem 4.2 hold, and so there exists a unique additive mapping π΄βˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’8𝑓(π‘₯/2)βˆ’π΄(π‘₯/2)(𝑑𝑑)β‰₯||2𝑑+π‘˜||.(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(π‘₯,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 (π‘Œ,πœ‡,𝑇) be a non-Archimedean random Banach space over 𝒦. Let π‘“βˆΆπ‘‹β†’π‘Œ be an even mapping, 𝑓(0)=0, and Ξ¨-approximately AQCQ mapping. If for some π›Όβˆˆβ„, 𝛼>0, and some integer π‘˜, π‘˜>4 with |2π‘˜|<𝛼, Ξ¨ξ€·2βˆ’π‘˜π‘₯,2βˆ’π‘˜ξ€Έπ‘¦,𝑑β‰₯Ξ¨(π‘₯,𝑦,𝛼𝑑)(π‘₯βˆˆπ‘‹,𝑑>0),limπ‘›β†’βˆžTβˆžπ‘—=𝑛𝑀𝛼2π‘₯,𝑗𝑑||||16π‘˜π‘—ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0),(5.1) then there exists a unique quartic mapping π‘„βˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’4𝑓(π‘₯/2)βˆ’π‘„(π‘₯/2)(𝑑)β‰₯Tβˆžπ‘–=1𝑀𝛼π‘₯,𝑖+1𝑑||||16π‘˜π‘–ξƒͺ(5.2) for all π‘₯βˆˆπ‘‹ and 𝑑>0, where 𝑀(π‘₯,𝑑)∢=π‘‡π‘˜βˆ’1ξ‚ΈΞ¨ξ‚΅π‘₯2,π‘₯2,𝑑||4||ξ‚Άξ‚€π‘₯,Ξ¨π‘₯,22,𝑑,…,Ξ¨π‘˜βˆ’1π‘₯2,2π‘˜βˆ’1π‘₯2,𝑑||4||ξ‚Άξ‚΅2,Ξ¨π‘˜βˆ’12π‘₯,π‘˜βˆ’1π‘₯2,𝑑(π‘₯βˆˆπ‘‹,𝑑>0).(5.3)

Proof. Letting π‘₯=𝑦 in (4.2), we get πœ‡π‘“(3𝑦)βˆ’6𝑓(2𝑦)+15𝑓(𝑦)(𝑑)β‰₯Ξ¨(𝑦,𝑦,𝑑)(5.4) for all π‘¦βˆˆπ‘‹ and 𝑑>0. Replacing π‘₯ by 2𝑦 in (4.2), we get πœ‡π‘“(4𝑦)βˆ’4𝑓(3𝑦)+4𝑓(2𝑦)+4𝑓(𝑦)(𝑑)β‰₯Ξ¨(2𝑦,𝑦,𝑑)(5.5) for all π‘¦βˆˆπ‘‹ and 𝑑>0. By (5.4) and (5.5), we have πœ‡π‘“(4𝑦)βˆ’20𝑓(2𝑦)+64𝑓(𝑦)ξ€·πœ‡(𝑑)β‰₯𝑇4(𝑓(3𝑦)βˆ’4𝑓(2𝑦)+5𝑓(𝑦))(𝑑),πœ‡π‘“(4𝑦)βˆ’4𝑓(3𝑦)+6𝑓(2𝑦)βˆ’4𝑓(𝑦)ξ€Έξ‚΅πœ‡(𝑑)=𝑇𝑓(3𝑦)βˆ’4𝑓(2𝑦)+5𝑓(𝑦)𝑑||4||ξ‚Ά,πœ‡π‘“(4𝑦)βˆ’4𝑓(3𝑦)+6𝑓(2𝑦)βˆ’4𝑓(𝑦)Ψ𝑑(𝑑)β‰₯𝑇𝑦,𝑦,||4||ξ‚Άξ‚Ά,Ξ¨(2𝑦,𝑦,𝑑)(5.6) for all π‘¦βˆˆπ‘‹ and 𝑑>0. Letting π‘¦βˆΆ=π‘₯/2 and 𝑔(π‘₯)∢=𝑓(2π‘₯)βˆ’4𝑓(π‘₯) for all π‘₯βˆˆπ‘‹ in (5.6), we get πœ‡π‘”(π‘₯)βˆ’16𝑔(π‘₯/2)ξ‚΅Ξ¨ξ‚΅π‘₯(𝑑)β‰₯𝑇2,π‘₯2,𝑑||4||ξ‚Άξ‚€π‘₯,Ξ¨π‘₯,2,𝑑(5.7) for all π‘₯βˆˆπ‘‹ and 𝑑>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 (π‘Œ,πœ‡,𝑇) be a non-Archimedean random Banach space over 𝒦 under a t-norm π‘‡βˆˆβ„‹. Let π‘“βˆΆπ‘‹β†’π‘Œ be an even, 𝑓(0)=0, and Ξ¨-approximately AQCQ mapping. If,   for some π›Όβˆˆβ„, 𝛼>0, and some integer π‘˜, π‘˜>4, with |2π‘˜|<𝛼, Ξ¨ξ€·2βˆ’π‘˜π‘₯,2βˆ’π‘˜ξ€Έπ‘¦,𝑑β‰₯Ξ¨(π‘₯,𝑦,𝛼𝑑)(π‘₯βˆˆπ‘‹,𝑑>0),(5.8) then there exists a unique quartic mapping π‘„βˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’4𝑓(π‘₯/2)βˆ’π‘„(π‘₯/2)(𝑑)β‰₯Tβˆžπ‘–=1𝑀𝛼π‘₯,𝑖+1𝑑||||16π‘˜π‘–ξƒͺ(5.9) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Proof. Since limπ‘›β†’βˆžπ‘€ξƒ©π›Όπ‘₯,𝑗𝑑||||16π‘˜π‘—ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0)(5.10) and 𝑇 is of HadΕΎiΔ‡ type, from Proposition 2.1, it follows that limπ‘›β†’βˆžπ‘‡βˆžπ‘—=𝑛𝑀𝛼π‘₯,𝑗𝑑||||16π‘˜π‘—ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0).(5.11) Now, we can apply Theorem 5.1 to obtain the result.

Example 5.3. Let (𝑋,πœ‡,𝑇𝑀) be non-Archimedean random normed space in which πœ‡π‘₯(𝑑𝑑)=(𝑑+β€–π‘₯β€–π‘₯βˆˆπ‘‹,𝑑>0).(5.12) And let (π‘Œ,πœ‡,𝑇𝑀) be a complete non-Archimedean random normed space (see Example 3.2). Define 𝑑Ψ(π‘₯,𝑦,𝑑)=.1+𝑑(5.13) It is easy to see that (4.3) holds for 𝛼=1. Also, since 𝑑𝑀(π‘₯,𝑑)=,1+𝑑(5.14) we have limπ‘›β†’βˆžπ‘‡βˆžπ‘€,𝑗=𝑛𝑀𝛼π‘₯,𝑗𝑑||||16π‘˜π‘—ξƒͺ=limπ‘›β†’βˆžξƒ©limπ‘šβ†’βˆžπ‘‡π‘šπ‘€,𝑗=𝑛𝑀𝑑π‘₯,||||16π‘˜π‘—ξƒͺξƒͺ=limπ‘›β†’βˆžlimπ‘šβ†’βˆžξƒ©π‘‘||𝑑+16π‘˜||𝑛ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0).(5.15) Let π‘“βˆΆπ‘‹β†’π‘Œ be an even, 𝑓(0)=0, and Ξ¨-approximately AQCQ mapping. Thus all the conditions of Theorem 5.1 hold, and so there exists a unique quartic mapping π‘„βˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’4𝑓(π‘₯/2)βˆ’π‘„(π‘₯/2)(𝑑𝑑)β‰₯||𝑑+16π‘˜||.(5.16)

Theorem 5.4. Let 𝒦 be a non-Archimedean field, let 𝑋 be a vector space over 𝒦 and let (π‘Œ,πœ‡,𝑇) be a non-Archimedean random Banach space over 𝒦. Let π‘“βˆΆπ‘‹β†’π‘Œ be an even mapping, 𝑓(0)=0 and Ξ¨-approximately AQCQ mapping. If for some π›Όβˆˆβ„, 𝛼>0, and some integer π‘˜, π‘˜>2 with |2π‘˜|<𝛼, Ξ¨ξ€·2βˆ’π‘˜π‘₯,2βˆ’π‘˜ξ€Έπ‘¦,𝑑β‰₯Ξ¨(π‘₯,𝑦,𝛼𝑑)(π‘₯βˆˆπ‘‹,𝑑>0),limπ‘›β†’βˆžTβˆžπ‘—=𝑛𝑀𝛼2π‘₯,𝑗𝑑||4||π‘˜π‘—ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0),(5.17) then there exists a unique quadratic mapping π‘„βˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’16𝑓(π‘₯/2)βˆ’π‘„(π‘₯/2)(𝑑)β‰₯Tβˆžπ‘–=1𝑀𝛼π‘₯,𝑖+1𝑑||4||π‘˜π‘–ξƒͺ(5.18) for all π‘₯βˆˆπ‘‹ and 𝑑>0, where 𝑀(π‘₯,𝑑)∢=π‘‡π‘˜βˆ’1ξ‚ΈΞ¨ξ‚΅π‘₯2,π‘₯2,𝑑||4||ξ‚Άξ‚€π‘₯,Ξ¨π‘₯,22,𝑑,…,Ξ¨π‘˜βˆ’1π‘₯2,2π‘˜βˆ’1π‘₯2,𝑑||4||ξ‚Άξ‚΅2,Ξ¨π‘˜βˆ’12π‘₯,π‘˜βˆ’1π‘₯2,𝑑(π‘₯βˆˆπ‘‹,𝑑>0).(5.19)

Proof. Letting π‘¦βˆΆ=π‘₯/2 and 𝑔(π‘₯)∢=𝑓(2π‘₯)βˆ’16𝑓(π‘₯) for all π‘₯βˆˆπ‘‹ in (5.6), we get πœ‡π‘”(π‘₯)βˆ’4𝑔(π‘₯/2)ξ‚΅Ξ¨ξ‚΅π‘₯(𝑑)β‰₯𝑇2,π‘₯2,𝑑||4||ξ‚Άξ‚€π‘₯,Ξ¨π‘₯,2,𝑑(5.20) for all π‘₯βˆˆπ‘‹ and 𝑑>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 (π‘Œ,πœ‡,𝑇) be a non-Archimedean random Banach space over 𝒦 under a t-norm π‘‡βˆˆβ„‹. Let π‘“βˆΆπ‘‹β†’π‘Œ be an even, 𝑓(0)=0, and Ξ¨-approximately AQCQ mapping. If,   for some π›Όβˆˆβ„, 𝛼>0, and some integer π‘˜, π‘˜>2, with |2π‘˜|<𝛼, Ξ¨ξ€·2βˆ’π‘˜π‘₯,2βˆ’π‘˜ξ€Έπ‘¦,𝑑β‰₯Ξ¨(π‘₯,𝑦,𝛼𝑑)(π‘₯βˆˆπ‘‹,𝑑>0),(5.21) then there exists a unique quadratic mapping π‘„βˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’16𝑓(π‘₯/2)βˆ’π‘„(π‘₯/2)(𝑑)β‰₯Tβˆžπ‘–=1𝑀𝛼π‘₯,𝑖+1𝑑||4||π‘˜π‘–ξƒͺ(5.22) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Proof. Since limπ‘›β†’βˆžπ‘€ξƒ©π›Όπ‘₯,𝑗𝑑||4||π‘˜π‘—ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0)(5.23) and 𝑇 is of HadΕΎiΔ‡ type, from Proposition 2.1, it follows that limπ‘›β†’βˆžπ‘‡βˆžπ‘—=𝑛𝑀𝛼π‘₯,𝑗𝑑||4||π‘˜π‘—ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0).(5.24)Now, we can apply Theorem 5.4 to obtain the result.

Example 5.6. Let (𝑋,πœ‡,𝑇𝑀) be a non-Archimedean random normed space in which πœ‡π‘₯(𝑑𝑑)=(𝑑+β€–π‘₯β€–π‘₯βˆˆπ‘‹,𝑑>0).(5.25) And let (π‘Œ,πœ‡,𝑇𝑀) be a complete non-Archimedean random normed space (see Example 3.2). Define 𝑑Ψ(π‘₯,𝑦,𝑑)=.1+𝑑(5.26) It is easy to see that (4.3) holds for 𝛼=1. Also, since 𝑑𝑀(π‘₯,𝑑)=,1+𝑑(5.27) we have limπ‘›β†’βˆžπ‘‡βˆžπ‘€,𝑗=𝑛𝑀𝛼π‘₯,𝑗𝑑||4||π‘˜π‘—ξƒͺ=limπ‘›β†’βˆžξƒ©limπ‘šβ†’βˆžπ‘‡π‘šπ‘€,𝑗=𝑛𝑀𝑑π‘₯,||4||π‘˜π‘—ξƒͺξƒͺ=limπ‘›β†’βˆžlimπ‘šβ†’βˆžξƒ©π‘‘||4𝑑+π‘˜||𝑛ξƒͺ=1(π‘₯βˆˆπ‘‹,𝑑>0).(5.28) Let π‘“βˆΆπ‘‹β†’π‘Œ be an even, 𝑓(0)=0, and Ξ¨-approximately AQCQ mapping. Thus, all the conditions of Theorem 5.4 hold, and so there exists a unique quadratic mapping π‘„βˆΆπ‘‹β†’π‘Œ such that πœ‡π‘“(π‘₯)βˆ’16𝑓(π‘₯/2)βˆ’π‘„(π‘₯/2)(𝑑𝑑)β‰₯||4𝑑+π‘˜||.(5.29)

6. Latticetic Random Normed Space

Let β„’=(𝐿,β‰₯𝐿) be a complete lattice, that is, a partially ordered set in which every nonempty subset admits supremum and infimum, and 0β„’=inf𝐿, 1β„’=sup𝐿. The space of latticetic random distribution functions, denoted by Ξ”+𝐿, is defined as the set of all mappings πΉβˆΆβ„βˆͺ{βˆ’βˆž,+∞}→𝐿 such that 𝐹 is left continuous and nondecreasing on ℝ, 𝐹(0)=0β„’, 𝐹(+∞)=1β„’.

𝐷+πΏβŠ†Ξ”+𝐿 is defined as 𝐷+𝐿={πΉβˆˆΞ”+πΏβˆΆπ‘™βˆ’πΉ(+∞)=1β„’}, where π‘™βˆ’π‘“(π‘₯) denotes the left limit of the function 𝑓 at the point π‘₯. The space Ξ”+𝐿 is partially ordered by the usual pointwise ordering of functions, that is, 𝐹β‰₯𝐺 if and only if 𝐹(𝑑)β‰₯𝐿𝐺(𝑑) for all 𝑑 in ℝ. The maximal element for Ξ”+𝐿 in this order is the distribution function given byπœ€0ξ‚»0(𝑑)=β„’1,if𝑑≀0,β„’,if𝑑>0.(6.1)

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

Definition 6.1 (see [42]). A triangular norm (𝑑-norm) on 𝐿 is a mapping π’―βˆΆ(𝐿)2→𝐿 satisfying the following conditions:(a)(forallπ‘₯∈𝐿)(𝒯(π‘₯,1β„’)=π‘₯) (boundary condition);(b)(forall(π‘₯,𝑦)∈(𝐿)2)(𝒯(π‘₯,𝑦)=𝒯(𝑦,π‘₯)) (commutativity);(c)(forall(π‘₯,𝑦,𝑧)∈(𝐿)3)(𝒯(π‘₯,𝒯(𝑦,𝑧))=𝒯(𝒯(π‘₯,𝑦),𝑧)) (associativity);(d)(forall(π‘₯,π‘₯ξ…ž,𝑦,π‘¦ξ…ž)∈(𝐿)4)(π‘₯≀𝐿π‘₯ξ…žandπ‘¦β‰€πΏπ‘¦ξ…žβ‡’π’―(π‘₯,𝑦)≀𝐿𝒯(π‘₯ξ…ž,π‘¦ξ…ž)) (monotonicity).

Let {π‘₯𝑛} be a sequence in 𝐿 converges to π‘₯∈𝐿 (equipped order topology). The 𝑑-norm 𝒯 is said to be a continuous 𝑑-norm if limπ‘›β†’βˆžπ’―ξ€·π‘₯𝑛,𝑦=𝒯(π‘₯,𝑦)(6.2) for all π‘¦βˆˆπΏ.

A 𝑑-norm 𝒯 can be extended (by associativity) in a unique way to an 𝑛-array operation taking for (π‘₯1,…,π‘₯𝑛)βˆˆπΏπ‘› the value 𝒯(π‘₯1,…,π‘₯𝑛) defined by 𝒯0𝑖=1π‘₯𝑖=1,𝒯𝑛𝑖=1π‘₯𝑖𝒯=π’―π‘›βˆ’1i=1π‘₯𝑖,π‘₯𝑛π‘₯=𝒯1,…,π‘₯𝑛.(6.3)

𝒯 can also be extended to a countable operation taking for any sequence (π‘₯𝑛)π‘›βˆˆπ‘ in 𝐿 the valueπ’―βˆžπ‘–=1π‘₯𝑖=limπ‘›β†’βˆžπ’―π‘›π‘–=1π‘₯𝑖.(6.4) The limit on the right side of (6.4) exists since the sequence (𝒯𝑛𝑖=1π‘₯𝑖)π‘›βˆˆβ„• is nonincreasing and bounded from below.

Note that we put 𝒯=𝑇 whenever 𝐿=[0,1]. If 𝑇 is a 𝑑-norm, then π‘₯𝑇(𝑛) is defined for every π‘₯∈[0,1] and π‘›βˆˆπ‘βˆͺ{0} by 1 if 𝑛=0 and 𝑇(π‘₯𝑇(π‘›βˆ’1),π‘₯) if 𝑛β‰₯1. A 𝑑-norm 𝑇 is said to be of HadΕΎiΔ‡ type, (we denote by π‘‡βˆˆβ„‹) if the family (π‘₯𝑇(𝑛))π‘›βˆˆπ‘ is equicontinuous at π‘₯=1 (cf. [27]).

Definition 6.2 (see [42]). A continuous 𝑑-norm 𝒯 on 𝐿=[0,1]2 is said to be continuous 𝑑–representable if there exist a continuous 𝑑-norm βˆ— and a continuous 𝑑-conorm ⋄ on [0,1] such that, for all π‘₯=(π‘₯1,π‘₯2), 𝑦=(𝑦1,𝑦2)∈𝐿, 𝒯π‘₯(π‘₯,𝑦)=1βˆ—π‘¦1,π‘₯2⋄𝑦2ξ€Έ.(6.5)

For example, ξ€·π‘Žπ’―(π‘Ž,𝑏)=1𝑏1ξ€½π‘Ž,min2+𝑏2,ξ€·ξ€½π‘Ž,1ξ€Ύξ€ΈπŒ(π‘Ž,𝑏)=min1,𝑏1ξ€Ύξ€½π‘Ž,max2,𝑏2ξ€Ύξ€Έ(6.6) for all π‘Ž=(π‘Ž1,π‘Ž2), 𝑏=(𝑏1,𝑏2)∈[0,1]2 are continuous 𝑑-representable. Define the mapping π’―βˆ§ from 𝐿2 to 𝐿 by π’―βˆ§ξ‚»(π‘₯,𝑦)=π‘₯,if𝑦β‰₯𝐿π‘₯,𝑦,ifπ‘₯β‰₯𝐿𝑦.(6.7)

Recall (see [27, 28]) that if {π‘₯𝑛} is a given sequence in 𝐿, (π’―βˆ§)𝑛𝑖=1π‘₯𝑖 is defined recurrently by (π’―βˆ§)1𝑖=1π‘₯𝑖=π‘₯1 and (π’―βˆ§)𝑛𝑖=1π‘₯𝑖=π’―βˆ§((π’―βˆ§)π‘›βˆ’1𝑖=1π‘₯𝑖,π‘₯𝑛) for all 𝑛β‰₯2.

A negation on β„’ is any decreasing mapping π’©βˆΆπΏβ†’πΏ satisfying 𝒩(0β„’)=1β„’ and 𝒩(1β„’)=0β„’. If 𝒩(𝒩(π‘₯))=π‘₯, for all π‘₯∈𝐿, then 𝒩 is called an involutive negation. In the following, β„’ is endowed with a (fixed) negation 𝒩.

Definition 6.3. A latticetic random normed space (in short LRN-space) is a triple (𝑋,πœ‡,π’―βˆ§), where 𝑋 is a vector space and πœ‡ is a mapping from 𝑋 into 𝐷+𝐿 such that the following conditions hold: (LRN1) πœ‡π‘₯(𝑑)=πœ€0(𝑑) for all 𝑑>0 if and only if π‘₯=0, (LRN2) πœ‡π›Όπ‘₯(𝑑)=πœ‡π‘₯(𝑑/|𝛼|) for all π‘₯ in 𝑋, 𝛼≠0 and 𝑑β‰₯0, (LRN3) πœ‡π‘₯+𝑦(𝑑+𝑠)β‰₯πΏπ’―βˆ§(πœ‡π‘₯(𝑑),πœ‡π‘¦(𝑠)) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑,𝑠β‰₯0.

We note that from (LPN2) it follows that πœ‡βˆ’π‘₯(𝑑)=πœ‡π‘₯(𝑑) for all π‘₯βˆˆπ‘‹ and 𝑑β‰₯0.

Example 6.4. Let 𝐿=[0,1]Γ—[0,1] and operation ≀𝐿 be defined by π‘ŽπΏ=ξ€½ξ€·1,π‘Ž2ξ€ΈβˆΆξ€·π‘Ž1,π‘Ž2ξ€Έβˆˆ[]Γ—[]0,10,1,π‘Ž1+π‘Ž2ξ€Ύ,ξ€·π‘Žβ‰€11,π‘Ž2≀𝐿𝑏1,𝑏2ξ€ΈβŸΊπ‘Ž1≀𝑏1,π‘Ž2β‰₯𝑏2ξ€·π‘Ž,βˆ€π‘Ž=1,π‘Ž2𝑏,𝑏=1,𝑏2ξ€ΈβˆˆπΏ.(6.8) then (𝐿,≀𝐿) is a complete lattice (see [42]). In this complete lattice, we denote its units by 0𝐿=(0,1) and 1𝐿=(1,0). Let (𝑋,β€–β‹…β€–) be a normed space. Let 𝒯(π‘Ž,𝑏)=(min{π‘Ž1,𝑏1},max{π‘Ž2,𝑏2}) for all π‘Ž=(π‘Ž1,π‘Ž2), 𝑏=(𝑏1,𝑏2)∈[0,1]Γ—[0,1] and πœ‡ be a mapping defined by πœ‡π‘₯𝑑(𝑑)=,𝑑+β€–π‘₯β€–β€–π‘₯‖𝑑+β€–π‘₯β€–π‘‘βˆˆβ„+ξ€Έ,(6.9) then (𝑋,πœ‡,𝒯) is a latticetic random normed spaces.
If (𝑋,πœ‡,π’―βˆ§) is a latticetic random normed space, then 𝑉𝒱=(πœ€,πœ†)βˆΆπœ€>𝐿0β„’ξ€½0,πœ†βˆˆπΏβ§΅β„’,1β„’ξ€½ξ€Ύξ€Ύ,𝑉(πœ€,πœ†)=π‘₯βˆˆπ‘‹βˆΆπΉπ‘₯(πœ€)>𝐿𝒩(πœ†),(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,πœ‡,π’―βˆ§) be a latticetic r