# A General System of  Euler–Lagrange-Type Quadratic Functional Equations in Menger Probabilistic Non-Archimedean 2-Normed Spaces

Authors: M. Eshaghi Gordji; M. B. Ghaemi; Y. J. Cho; H. Majani
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We prove the generalized Hyers-Ulam-Rassias stability of a general system of Euler-Lagrange-type quadratic functional equations in non-Archimedean 2-normed spaces and Menger probabilistic non-Archimedean-normed spaces.


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## Body

## 1. Introduction and Preliminaries

Gähler [1, 2] introduced the concept of linear 2-normed spaces and Gähler and White [3–5] introduced the concept of 2-Banach spaces. In 1999–2003, Lewanwdoska published a series of some papers on 2-normed sets and generalized 2-normed spaces [6, 7]. For more details on linear 2-normed spaces, see the books written by Freese and Cho [8] and Cho et al. [9].Recently, Park [10] has investigated approximate additive mappings, approximate Jensen mappings and approximate quadratic mappings in 2-Banach spaces. We recall and apply notions and notes which are given in [10].Definition 1.1.
LetX be a linear space over ℝ with dim X>1 and ∥·,·∥:X×X→ℝ be a function satisfying the following properties: for all x,y,z∈X and α∈ℝ,(1)
∥x,y∥=0, if and only if x,y are linearly dependent;(2)
∥x,y∥=∥y,x∥;(3)
∥x,αy∥=|α|∥x,y∥;(4)
∥x,y+z∥≤∥x,y∥+∥x,z∥.Then the function ∥·,·∥ is called a 2-norm on X and the pair (X,∥·,·∥) is called a linear 2-normed space.Lemma 1.2.
Let(X,∥·,·∥) be a linear 2-normed space. If x∈X and ∥x,y∥=0 for all y∈X, then x=0.Remark 1.3.
Let(X,∥·,·∥) be a linear 2-normed space. One can show that the conditions (2) and (4) in Definition 1.1 imply that
(1.1)|‖x,z‖-‖y,z‖|≤‖x-y,z‖
for all x,y,z∈X. Hence the function x→∥x,y∥ is continuous function of X into ℝ for any fixed y∈X.Definition 1.4.
A sequence{xn} in a linear 2-normed space X is called a Cauchy sequence if there are two linearly independent points y,z∈X such that
(1.2)lim⁡m,n→+∞‖xn-xm,y‖=0,lim⁡m,n→+∞‖xn-xm,z‖=0.Definition 1.5.
A sequence{xn} in a linear 2-normed space X is called a convergent sequence if there exists x∈X such that
(1.3)lim⁡n→∞‖xn-x,y‖=0,
for all y∈X. If {xn} converges to x, write xn→x as n→∞, and x is called the limit of {xn}. In this case, we also write limn→∞xn=x.Lemma 1.6.
For any convergent sequence{xn} in a linear 2-normed space X,
(1.4)lim⁡n→∞‖xn,y‖=‖lim⁡n→∞xn,y‖
for all y∈X.Definition 1.7.
A linear 2-normed space in which every Cauchy sequence is a convergent sequence is called a 2-Banach space.Hensel [11] has introduced a normed space which does not have the Archimedean property. During the last three decades theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, p-adic strings, and superstrings [12]. Although many results in the classical normed space theory have a non-Archimedean counterpart, their proofs are different and require a rather new kind of intuition [13–17]. One may note that |n|≤1 in each valuation field, every triangle is isosceles, and there may be no unit vector in a non-Archimedean normed space [15]. These facts show that the non-Archimedean framework is of special interest.Definition 1.8.
Let𝕂 be a field. A valuation mapping on 𝕂 is a function |·|:𝕂→ℝ such that, for any a,b∈𝕂,(a)
|a|≥0 and equality holds if and only if a=0;(b)
|ab|=|a||b|;(c)
|a+b|≤|a|+|b|.A field endowed with a valuation mapping is called a valued field.If the condition(c) in the definition of a valuation mapping is replaced with the following condition:(c)'
|a+b|≤max{|a|,|b|},then the valuation|·| is said to be non-Archimedean.The condition(c)′ is called the strict triangle inequality. By (b), we have |1|=|-1|=1. Thus, by induction, it follows from (c)′ that |n|≤1 for each integer n≥1. We always assume in addition that |·| is nontrivial, that is, that there exists a0∈𝕂 such that |a0|∉{0,1}. The most important examples of non-Archimedean spaces are p–adic numbers.Example 1.9.
Letp be a prime number. For any nonzero rational number a=pr(m/n) such that m and n are coprime to the prime number p, define the p-adic absolute value |a|p=p-r. Then |·| is a non-Archimedean norm on ℚ. The completion of ℚ with respect to |·| is denoted by ℚp and is called the p-adic number field.Definition 1.10.
LetX be a linear space over a scalar field 𝕂 with a non-Archimedean nontrivial valuation |·|. A function ∥·∥:X→ℝ is a non-Archimedean norm (valuation) if it satisfies the following conditions: for all x,y∈X and r∈𝕂,(NA1)
∥x∥=0 if and only if x=0;(NA2)
∥rx∥=|r|∥x∥;(NA3)
∥x+y∥≤max{∥x∥,∥y∥} (the strong triangle inequality (ultrametric)). Then (X,∥·∥) is called a non–Archimedean normed space.Definition 1.11 (Freese and Cho [8]).
LetX be a linear space with dim X>1 over a scalar field 𝕂 with a non-Archimedean non-trivial valuation |·|. A function ∥·,·∥:X→ℝ is a non-Archimedean 2-norm (valuation) if it satisfies the following conditions: for any x,y,z∈X and α∈𝕂,(NA1)
∥x,y∥=0 if and only if x,y are linearly dependent;(NA2)
∥x,y∥=∥y,x∥;(NA3)
∥x,αy∥=|α|∥x,y∥;(NA4)
∥x+y,z∥≤max{∥x,z∥,∥y,z∥}. Then (X,∥·,·∥) is called a non-Archimedean 2-normed space.It follows from (NA4) that(1.5)‖xm-xl,y‖≤max⁡{‖xj+1-xj,y‖:l≤j≤m-1}(m>l),and so a sequence{xm} is a Cauchy sequence in X if and only if {xm+1-xm} converges to zero in a non-Archimedean 2-normed space.Theory of probabilistic normed spaces were first defined by Šerstnev in 1962 (see [18]), which was generalized in [19]. We recall and apply the definition of Menger probabilistic normed spaces briefly as given in [20–22].Definition 1.12.
A distance distribution function (briefly, a d.d.f.) is a nondecreasing functionF from [0,+∞] into [0,1] satisfying F(0)=0, F(+∞)=1, and F is left-continuous on (0,+∞). The space of d.d.f.'s is denoted by Δ+ and the set of all F in Δ+ for which limt→+∞-F(t)=1 by D+.The spaceΔ+ is partially ordered by the usual pointwise ordering of functions, that is, F≤G if and only if F(x)≤G(x) for all x in [0,+∞]. For any a≥0, ɛa is the d.d.f. given by
(1.6)ɛa(t)={0,ift≤a,1,ift>a.Definition 1.13.
A triangular norm (briefly,t-norm) is a binary operation T:[0,1]×[0,1]→[0,1] which is commutative, associative, and non-decreasing in each variable and has 1 as the unit element.Basic examples are the Łukasiewiczt-norm TL,  TL(a,b)=max(a+b-1,0), the product t-norm TP, TP(a,b)=ab, and the strongest triangular norm TM,  TM(a,b)=min(a,b).Definition 1.14.
A Menger probabilistic normed space is a triple(X,ν,T), where X is a real vector space, T is continuous t-norm, and ν is a mapping the probabilistic norm from X into Δ+, such that, for any choice of p,q∈X and a,s,t∈(0,+∞), the following hold:(PN1)
ν(p)=ɛ0 if and only if p=θ (θ is the null vector in X);(PN2)
ν(ap)(t)=ν(p)(t/(|a|));(PN3)
ν(p+q)(s+t)≥T(ν(p)(s),ν(q)(t)).Mirmostafaee and Moslehian [23] have introduced a notion of a non-Archimedean fuzzy normed space and, recently, Mihet [24] has restated the definition of them.Now, we introduce the definition of a Menger probabilistic non-Archimedean 2-normed space by the definitions which is given in [24] and [23].Definition 1.15.
A Menger Probabilistic non-Archimedean 2-normed space is a triple(X,ν,T), where X is a linear space with dim X>1 over a non-Archimedean field 𝕂, T is continuous t-norm, and ν is a mapping the probabilistic norm from X2 into Δ+, such that, for any choice of p,q,r∈X, α∈𝕂, and s,t∈(0,+∞), the following hold:(PNA1)
ν(p,q)=ɛ0 if and only if p,q are linearly dependent;(PNA2)
ν(p,q)=ν(q,p);(PNA3)
ν(αp,q)(t)=ν(p,q)(t/(|α|));(PNA4)
ν(p+q,r)(max{s,t})≥T(ν(p,r)(s),ν(q,r)(t)).It follows fromν(p,q)∈Δ+ that ν(p,q) is nondecreasing for any fixed p,q∈X. So one can show that the condition (PNA4) is equivalent to the following condition:(PNA4)'
ν(p+q,r)(t)≥T(ν(p,r)(t),ν(q,r)(t)).Definition 1.16.
Let(X,ν,T) be a Menger probabilistic non-Archimedean 2-normed space and {xn} be a sequence in X. Then(1)
the sequence{xn} is said to be convergent if there exists x∈X such that
(1.7)lim⁡n→∞ν(xn-x,y)(t)=1,
for all y∈X and t>0. In this case, x is called the limit of the sequence {xn};(2)
the sequence{xn} in X is called a Cauchy sequence if, for any ɛ>0 and t>0, there exists n0 and linearly independent elements y,z∈X such that, for all n≥n0 and p>0,
(1.8)ν(xn+p-xn,y)(t)>1-ɛ,ν(xn+p-xn,z)(t)>1-ɛ.LetT be a given t-norm. Then, by associativity, a family of mappings Tn:[0,1]→[0,1] for all n∈ℕ is defined as follows: (1.9)T1(x)=T(x,x),Tn(x)=T(Tn-1(x),x),for allx∈[0,1]. For three important t-norms TM, TP, and TL, we have (1.10)TMn(x)=x,TPn(x)=xn,TLn(x)=max⁡{(n+1)x-n,0},for alln∈ℕ.Definition 1.17 (Hadžić [25]).
At-norm T is said to be of H-type if a family of functions {Tn(t)} for all n∈ℕ is equicontinuous at t=1, that is, for all ɛ∈(0,1), there exists δ∈(0,1) such that
(1.11)t>1-δ⟹Tn(t)>1-ɛ
for alln∈ℕ.Thet-norm TM is a trivial example of t-norm of H-type, but there are t-norms of H-type with T≠TM (see Hadžić [26]).Lemma 1.18.
Assume thatT is a t-norm of H-type. Then the sequence {xn} is a Cauchy sequence if, for any ɛ>0 and t>0, there exists n0∈N and linearly independent elements y,z∈X such that, for all n≥n0,
(1.12)ν(xn+1-xn,y)(t)>1-ɛ,ν(xn+1-xn,z)(t)>1-ɛ.Proof.
For anyy∈X, we have
(1.13)ν(xn+p-xn,y)(t)≥T(ν(xn+p-xn+p-1,y)(t),ν(xn+p-1-xn,y)(t))≥T(ν(xn+p-xn+p-1,y)(t),T(ν(xn+p-1-xn+p-2,y)(t),ν(xn+p-2-xn,y)(t)))≥⋯≥T(ν(xn+p-xn+p-1,y)(t),T(ν(xn+p-1-xn+p-2,y)(t),…,T(ν(xn+2-xn+1,y)(t),ν(xn+1-xn,y)(t)))⋯).
For anyz∈X, we have the same inequalities to (1.13). Thus, from (1.13), the t-norm T of H-type and the assumption, it follows that the sequence {xn} is a Cauchy sequence in X. This completes the proof.It is easy to see that every convergent sequence in a (Menger probabilistic) non-Archimedean 2-normed spaceX is a Cauchy sequence. If every Cauchy sequence is convergent in X, then the (Menger probabilistic) non-Archimedean 2-normed space X is said to be complete and, further, the space X is called (Menger probabilistic) non-Archimedean 2-Banach space.The first stability problem concerning group homomorphisms was raised by Ulam [27] in 1940 and solved in the next year by Hyers [28]. Hyers' theorem was generalized by Aoki [29] for additive mappings and by Rassias [30] for linear mappings by considering an unbounded Cauchy difference. In 1994, a generalization of the Rassias theorem was obtained by Găvruta [31] by replacing the unbounded Cauchy difference by a general control function.The functional equation(1.14)f(x+y)+f(x-y)=2f(x)+2f(y)
is related to a symmetric biadditive function [32, 33]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.14) is said to be a quadratic function. It is well known that a function f:X→Y, where X and Y are real vector spaces, is quadratic if and only if there exists a unique symmetric bi-additive function B such that f(x)=B(x,x). The bi-additive function B is given by (1.15)B(x,y)=14(f(x+y)-f(x-y)).The Hyers-Ulam stability problem for the quadratic functional equation was solved by Skof [34]. In [35], Czerwik proved the Hyers-Ulam-Rassias stability of (1.14). Later, Jung [36] has generalized the results obtained by Skof and Czerwik. Rassias [37, 38] has solved the stability problem of Ulam for the generalized Euler-Lagrange type quadratic functional equation of
(1.16)f(ax+by)+f(bx-ay)=(a2+b2)(f(x)+f(y)).Lee [39] proved the stability of (1.16) in the spirit of Hyers, Ulam, Rassias and Găvruta in Banach spaces.In recent years, many authors proved the stability of various functional equations in various spaces (see, for instance, [10, 40, 41]). Using the method of our paper, one can investigate the stability of many general systems of various functional equations with n functional equations and n variables (n∈N) and our paper notably generalizes previous papers in this area.We assume thatf:Xn→Y is a mapping and consider a general system of Euler-Lagrange type quadratic functional equations as follows:
(1.17)f(a1x1+b1y1,x2,…,xn)+f(b1x1-a1y1,x2,…,xn)=(a12+b12)(f(x1,…,xn)+f(y1,x2,…,xn))…f(x1,…,xn-1,anxn+bnyn)+f(x1,…,xn-1,bnxn-anyn)=(an2+bn2)(f(x1,…,xn)+f(x1,…,xn-1,yn)).In Section2, we establish the generalized Hyers-Ulam–Rassias stability of the system (1.17) in non-Archimedean 2-Banach spaces. In Section 3, we prove the generalized Hyers-Ulam-Rassias stability of the system (1.17) in Menger probabilistic non-Archimedean 2-Banach spaces.


## 2. Stability of the System (1.17) in Non-Archimedean 2-Banach Spaces

In this section, we prove the generalized Hyers-Ulam-Rassias stability of system (1.17) in non-Archimedean 2-Banach spaces. Throughout this section, we assume that i,k,n,p∈ℕ∪{0}, 𝕂 is a non–Archimedean field, Y is a non-Archimedean 2-Banach space over 𝕂, and X is a vector space over 𝕂. Also assume that f:Xn→Y is a mapping.Theorem 2.1.
Letφi:Xn+1→[0,∞) for i∈{1,…,n} be a function such that
(2.1)lim⁡k→∞|1m12k⋯mn2k|×max⁡{|1m12⋯mi-12mi|φi(m1k+1x1,…,mi-1k+1xi-1,mikxi,0,mi+1kxi+1,…,mnkxn),|1m12⋯mi2|φi(m1k+1x1,…,mi-1k+1xi-1,aimikxi,bimikxi,mi+1kxi+1,…,mnkxn),‖1m12⋯mi2f(m1k+1x1,…,mi-1k+1xi-1,0,mi+1kxi+1,…,mnkxn),wi‖,‖1m12⋯mi-12f(m1k+1x1,…,mi-1k+1xi-1,0,mi+1kxi+1,…,mnkxn),wi‖:i=1,…,n}=0,(2.2)Φ=Φ(x1,…,xi,0,xi+1,…,xn)=lim⁡p→∞⁡max⁡⁡{|1m12k⋯mn2k|×max⁡{|1m12⋯mi-12mi|φi(m1k+1x1,…,mi-1k+1xi-1,mikxi,0,mi+1kxi+1,…,mnkxn),|1m12⋯mi2|φi(m1k+1x1,…,mi-1k+1xi-1,aimikxi,bimikxi,mi+1kxi+1,…,mnkxn),‖1m12⋯mi2f(m1k+1x1,…,mi-1k+1xi-1,0,mi+1kxi+1,…,mnkxn),wi‖,‖1m12⋯mi-12f(m1k+1x1,…,mi-1k+1xi-1,0,mi+1kxi+1,…,mnkxn),wi‖:i=1,…,n|1m12⋯mi-12mi|}:k=0,1,…,p}<∞,(2.3)lim⁡k→∞|1m12k…mn2k|φi(m1kx1,…,mikxi,mikyi,…,mnkxn)=0,
for all xi,yi,wi∈X,  and  ai,bi∈𝕂  with mi=ai2+bi2≠0,1, i=1,…,n. Let f:Xn→Y be a mapping satisfying
(2.4)‖f(a1x1+b1y1,x2,…,xn)+f(b1x1-a1y1,x2,…,xn)-(a12+b12)(f(x1,…,xn)+f(y1,x2,…,xn)),w1‖≤φ1(x1,y1,x2,…,xn),⋮‖f(x1,…,xn-1,anxn+bnyn)+f(x1,…,xn-1,bnxn-anyn)-(an2+bn2)(f(x1,…,xn)+f(x1,…,xn-1,yn)),wn‖≤φn(x1,…,xn-1,xn,yn),
for all xi,yi,wi∈X, and  ai,bi∈𝕂 with mi=ai2+bi2≠0,1, i=1,…,n. Then there exists a unique mapping T:Xn→Y satisfying (1.17) and
(2.5)‖f(x1,…,xn)-T(x1,…,xn),wi‖≤Φ,
for all xi,wi∈X,  i=1,…,n.Proof.
Fixi∈{1,2,…,n} and consider the following inequality. 							(2.6)‖f(x1,…,aixi+biyi,…,xn)+f(x1,…,bixi-aiyi,…,xn)-(ai2+bi2)(f(x1,…,xn)+f(x1,…,yi,…,xn)),wi‖≤φi(x1,…,xi,yi,…,xn).
Putting yi=0 and dividing by mi in (2.6), we get
(2.7)‖1mi(f(x1,…,aixi,…,xn)+f(x1,…,bixi,…,xn))-f(x1,…,xn),wi‖≤max⁡{|1mi|φi(x1,…,xi,0,xi+1,…,xn),‖f(x1,…,xi-1,0,xi+1,…,xn),wi‖}.
Replacing xi by aixi and yi by bixi and dividing by mi2 in (2.6), we obtain
(2.8)‖1mi2f(x1,…,mixi,…,xn)-1mi(f(x1,…,aixi,…,xn)+f(x1,…,bixi,…,xn)),wi‖≤max⁡{|1mi2|φi(x1,…,xi-1,aixi,bixi,xi+1,…,xn),‖1mi2f(x1,…,xi-1,0,xi+1,…,xn),wi‖}.
By (2.7) and (2.8), we have
(2.9)‖f(x1,…,xn)-1mi2f(x1,…,mixi,…,xn),wi‖≤max⁡{|1mi|φi(x1,…,xi,0,xi+1,…,xn),|1mi2|φi(x1,…,xi-1,aixi,bixi,xi+1,…,xn),‖1mi2f(x1,…,xi-1,0,xi+1,…,xn),wi‖,‖f(x1,…,xi-1,0,xi+1,…,xn),wi‖}.
Therefore one can obtain
(2.10)‖1m12⋯mi-12f(m1x1,…,mi-1xi-1,xi,…,xn)-1m12⋯mi2f(m1x1,…,mixi,xi+1,…,xn),wi‖≤max⁡{|1m12⋯mi-12mi|φi(m1x1,…,mi-1xi-1,xi,0,xi+1,…,xn),|1m12⋯mi2|φi(m1x1,…,mi-1xi-1,aixi,bixi,xi+1,…,xn),‖1m12⋯mi2f(m1x1,…,mi-1xi-1,0,xi+1,…,xn),wi‖,‖1m12⋯mi-12f(m1x1,…,mi-1xi-1,0,xi+1,…,xn),wi‖}.
So we conclude
(2.11)‖f(x1,…,xn)-1m12⋯mn2f(m1x1,…,mnxn),wi‖≤max⁡{|1m12⋯mi-12mi|φi(m1x1,…,mi-1xi-1,xi,0,xi+1,…,xn),|1m12⋯mi2|φi(m1x1,…,mi-1xi-1,aixi,bixi,xi+1,…,xn),‖1m12⋯mi2f(m1x1,…,mi-1xi-1,0,xi+1,…,xn),wi‖,‖1m12…mi-12f(m1x1,…,mi-1xi-1,0,xi+1,…,xn),wi‖:i=1,…,n|1m12⋯mi-12mi|}.
Therefore we get
(2.12)‖1m12k⋯mn2kf(m1kx1,…,mnkxn)-1m12(k+1)⋯mn2(k+1)f(m1k+1x1,…,mnk+1xn),wi‖≤|1m12k⋯mn2k|max⁡{|1m12…mi-12mi|φi(m1k+1x1,…,mi-1k+1xi-1,mikxi,0,mi+1kxi+1,…,mnkxn),|1m12⋯mi2|φi(m1k+1x1,…,mi-1k+1xi-1,aimikxi,bimikxi,mi+1kxi+1,…,mnkxn),‖1m12⋯mi2f(m1k+1x1,…,mi-1k+1xi-1,0,mi+1kxi+1,…,mnkxn),wi‖,‖1m12…mi-12f(m1k+1x1,…,mi-1k+1xi-1,0,mi+1kxi+1,…,mnkxn),wi‖,:i=1,…,n|1m12…mi-12mi|}
for all k∈ℕ∪{0}. It follows from (2.12) and (2.1) that the sequence
(2.13){1m12k⋯mn2kf(m1kx1,…,mnkxn)}
is Cauchy. Since the space Y is complete, this sequence is convergent. Therefore we can define T:Xn→Y by
(2.14)T(x1,…,xn):=lim⁡k→∞1m12k⋯mn2kf(m1kx1,…,mnkxn),
for all xi,wi∈X  and  ai,bi∈𝕂  with mi=ai2+bi2≠0,1, i=1,…,n. Using induction with (2.12) one can show that
(2.15)‖f(x1,…,xn)-1m12p⋯mn2pf(m1px1,…,mnpxn),wi‖≤max⁡{|1m12k⋯mn2k|×max⁡{|1m12⋯mi-12mi|φi(m1k+1x1,…,mi-1k+1xi-1,mikxi,0,mi+1kxi+1,…,mnkxn),|1m12⋯mi2|φi(m1k+1x1,…,mi-1k+1xi-1,aimikxi,bimikxi,mi+1kxi+1,…,mnkxn),‖1m12⋯mi2f(m1k+1x1,…,mi-1k+1xi-1,0,mi+1kxi+1,…,mnkxn),wi‖,‖1m12⋯mi-12f(m1k+1x1,…,mi-1k+1xi-1,0,mi+1kxi+1,…,mnkxn),wi‖:i=1,…,n|1m12⋯mi-12mi|}:k=0,1,…,p},
for all xi,wi∈X,i=1,…,n and p∈ℕ∪{0}. By taking p to approach infinity in (2.15) and using (2.2) one obtains (2.5).
For fixedi∈{1,2,…,n} and by (2.6) and (2.14), we get
(2.16)‖T(x1,…,aixi+biyi,…,xn)+T(x1,…,bixi-aiyi,…,xn)-(ai2+bi2)(T(x1,…,xn)+T(x1,…,yi,…,xn)),wi‖,=lim⁡k→∞⁡‖1m12k⋯mn2k{f(m1kx1,…,mik(aixi+biyi),…,mnkxn)+f(m1kx1,…,mik(bixi-aiyi),…,mnkxn)-(ai2+bi2)(f(m1kx1,nkxn)+f(m1kx1,…,mikyi,…,mnkxn))},wi1m12k…mn2k‖≤lim⁡k→∞|1m12k⋯mn2k|φi(m1kx1,…,mikxi,mikyi,…,mnkxn)=0.
By (2.16) and (2.3), we conclude that T satisfies (1.17).
Suppose that there exists another mappingT′:Xn→Y which satisfies (1.17) and (2.5). So we have
(2.17)‖T(x1,…,xn)-T′(x1,…,xn),wi‖≤|1m12k⋯mn2k|max⁡{‖T(m1kx1,…,mnkxn)-f(m1kx1,…,mnkxn),wi‖,‖f(m1kx1,…,mnkxn)-T′(m1kx1,…,mnkxn),wi‖}≤|1m12k⋯mn2k|max⁡{Φ(m1kx1,…,mnkxn),Φ(m1kx1,…,mnkxn)}
which tends to zero as k→∞ by (2.2). Therefore T=T′. This completes the proof.In the manner of proof of Theorem2.1, one can prove the following corollary.Corollary 2.2.
Letφi:Xn+1→[0,∞) for i∈{1,…,n} be a function such that
(2.18)lim⁡k→∞|1m12k⋯mn2k|max⁡{|1m12⋯mi-12mi|×φi(m1k+1x1,…,mi-1k+1xi-1,mikxi,0,mi+1kxi+1,…,mnkxn),|1m12…mi2|φi(m1k+1x1,…,mi-1k+1xi-1,aimikxi,bimikxi,mi+1kxi+1,…,mnkxn):i=1,…,n|1m12⋯mi-12mi|}=0,(2.19)Φ=Φ(x1,…,xi,0,xi+1,…,xn)=lim⁡p→∞max⁡⁡{|1m12k⋯mn2k|×max⁡⁡{|1m12…mi-12mi|φi(m1k+1x1,…,mi-1k+1xi-1,mikxi,0,mi+1kxi+1,…,mnkxn),|1m12…mi2|φi(m1k+1x1,…,mi-1k+1xi-1,aimikxi,bimikxi,mi+1kxi+1,…,mnkxn):i=1,…,n|1m12…mi2|}:k=0,1,…,p}<∞,(2.20)lim⁡k→∞|1m12k⋯mn2k|φi(m1kx1,…,mikxi,mikyi,…,mnkxn)=0,
for all xi,yi,wi∈X  and  ai,bi∈𝕂  with mi=ai2+bi2≠0,1,i=1,…,n. Let f:Xn→Y be a mapping satisfying
(2.21)‖f(a1x1+b1y1,x2,…,xn)+f(b1x1-a1y1,x2,…,xn)-(a12+b12)(f(x1,…,xn)+f(y1,x2,…,xn)),w1‖≤φ1(x1,y1,x2,…,xn),⋮‖f(x1,…,xn-1,anxn+bnyn)+f(x1,…,xn-1,bnxn-anyn)-(an2+bn2)(f(x1,…,xn)+f(x1,…,xn-1,yn)),wn‖≤φn(x1,…,xn-1,xn,yn),
for all xi,yi,wi∈X,  and  ai,bi∈𝕂  with mi=ai2+bi2≠0,1,i=1,…,n. Then there exists a unique mapping T:Xn→Y satisfying (1.17) and
(2.22)‖f(x1,…,xn)-T(x1,…,xn),wi‖≤Φ
for all xi,wi∈X, i=1,…,n.


## 3. System (1.17) Stability in Menger Probabilistic Non-Archimedean 2-Banach Spaces

In this section, we prove the generalized Hyers-Ulam-Rassias stability of system (1.17) in Menger probabilistic non-Archimedean 2–Banach spaces. Throughout this section, we assume that u∈ℝ,i,k,n∈ℕ∪{0}, 𝕂 is a non–Archimedean field, T is a continuous t–norm of H-type, (Y,ν,T) is a Menger probabilistic non-Archimedean 2-Banach space over 𝕂, (Z,ω,T) is a Menger probabilistic non-Archimedean 2-normed space over 𝕂, and X is a vector space over 𝕂.  Also assume that f:Xn→Y is a mapping.Theorem 3.1.
Letφi:Xn+1→Z for i∈{1,…,n} be a mapping such that
(3.1)φ̃i=φ̃i(x1,…,xi,xi,xi+1,…,xn,u)=T{T(ω(1m12⋯mi-12miφi(m1x1,…,mi-1xi-1,xi,0,…,xn),wi)(u),ν(1m12⋯mi-12f(m1x1,…,mi-1xi-1,0,xi+1,…,xn),wi)(u)),T(ω(1m12⋯mi2φi(m1x1,…,mi-1xi-1,aixi,bixi,xi+1,…,xn),wi)(u),ν(1m12⋯mi2f(m1x1,…,mi-1xi-1,0,xi+1,…,xn),wi)(u))},Φ1=Φ1(x1,x1,x2,…,xn,u)=φ̃1(x1,y1,x2,…,xn,u),Φi=Φi(x1,…,xi,xi,xi+1,…,xn,u)=T(φ̃i(x1,…,xi,xi,xi+1,…,xn,u),Φi-1(x1,…,xi-1,xi-1,xi,…,xn,u)),lim⁡k→∞Φn(m1kx1,…,mnkxn,mnkxn,|m12k⋯mn2k|u)=1,(3.2)lim⁡k→∞ω(1m12k⋯mn2kφi(m1kx1,…,mikxi,mikyi,…,mnkxn),wi)(u)=1,(3.3)Φk*=Φk*(x1,…,xn,xn,u)=Φn(m1kx1,…,mnkxn,mnkxn,|m12k⋯mn2k|u),Ψ0=Φ0*(x1,…,xn,xn,u)=Φn(x1,…,xn,xn,u),Ψk=Ψk(x1,…,xn,xn,u)=T(Φk*(x1,…,xn,xn,u),Ψk-1(x1,…,xn,xn,u)),Ψ=Ψ(x1,…,xn,xn,u)=lim⁡k→∞Ψk=1,
for all u>0, xi,wi∈X,andai,bi∈𝕂withmi=ai2+bi2≠0,1,i=1,…,n. Let f:Xn→Y be a mapping satisfying(3.4)ν((a12+b12)f(a1x1+b1y1,x2,…,xn)+f(b1x1-a1y1,x2,…,xn)-(a12+b12)(f(x1,…,xn)+f(y1,x2,…,xn)),w1)(u)≥ω(φ1(x1,y1,x2,…,xn),w1)(u),⋮ν(an2+bn2f(x1,…,xn-1,anxn+bnyn)+f(x1,…,xn-1,bnxn-anyn)-(an2+bn2)(f(x1,…,xn)+f(x1,…,xn-1,yn)),wn)(u)≥ω(φn(x1,…,xn,yn),wn)(u),
for all u>0,   xi,yi,wi∈Xandai,bi∈𝕂 withmi=ai2+bi2≠0,1,i=1,…,n. Then there exists a unique mapping F:Xn→Y satisfying (1.17) and
(3.5)ν(f(x1,…,xn)-F(x1,…,xn),wi)(u)≥Ψ,
for all u>0 and xi,wi∈X,i=1,…,n.Proof.
Fixi∈{1,2,…,n} and consider the following inequality:
(3.6)ν((ai2+bi2)f(x1,…,aixi+biyi,…,xn)+f(x1,…,bixi-aiyi,…,xn)-(ai2+bi2)(f(x1,…,xn)+f(x1,…,yi,…,xn)),wi)(u)≥ω(φi(x1,…,xi,yi,…,xn),wi)(u).
Putting yi=0 and dividing by mi in (3.6), we get
(3.7)ν(1mi(f(x1,…,aixi,…,xn)+f(x1,…,bixi,…,xn))-f(x1,…,xn),wi)(u)≥T(ω(1miφi(x1,…,xi,0,…,xn),wi)(u),ν(f(x1,…,0,…,xn),wi)(u)).
Replacing xi by aixi and yi by bixi and dividing by mi2 in (3.6), we obtain
(3.8)ν(1mi2f(x1,…,mixi,…,xn)-1mi(f(x1,…,aixi,…,xn)+f(x1,…,bixi,…,xn)),wi)(u)≥T(ω(1mi2φi(x1,…,xi-1,aixi,bixi,xi+1,…,xn),wi)(u),ν(1mi2f(x1,…,xi-1,0,xi+1,…,xn),wi)(u)).
By (3.7) and (3.8), we have
(3.9)ν(f(x1,…,xn)-1mi2f(x1,…,mixi,…,xn),wi)(u)≥T{T(ω(1miφi(x1,…,xi,0,…,xn),wi)(u),ν(f(x1,…,,xi-1,0,xi+1,…,xn),wi)(u)),T(ω(1mi2φi(x1,…,xi-1,aixi,bixi,xi+1,…,xn),wi)(u),ν(1mi2f(x1,…,xi-1,0,xi+1,…,xn),wi)(u))}.
Therefore one can obtain
(3.10)ν(1m12⋯mi-12f(m1x1,…,mi-1xi-1,xi,…,xn)-1m12⋯mi2f(m1x1,…,mixi,…,xn),wi)(u)≥T{T(ω(1m12⋯mi-12miφi(m1x1,…,mi-1xi-1,xi,0,…,xn),wi)(u),ν(1m12⋯mi-12f(m1x1,…,mi-1xi-1,0,xi+1,…,xn),wi)(u)),T(ω(1m12⋯mi2φi(m1x1,…,mi-1xi-1,aixi,bixi,xi+1,…,xn),wi)(u),ν(1m12⋯mi2f(m1x1,…,mi-1xi-1,0,xi+1,…,xn),wi)(u))}=φ̃i.
So we conclude
(3.11)ν(f(x1,…,xn)-1m12⋯mn2f(m1x1,…,mnxn),wi)(u)≥Φn.
Therefore we get
(3.12)ν(1m12k⋯mn2kf(m1kx1,…,mnkxn)-1m12(k+1)⋯mn2(k+1)f(m1kx1,…,mnkxn),wi)(u)≥Φn(m1kx1,…,mnkxn,mnkxn,|m12k⋯mn2k|u),
for all xi,wi∈X and k∈ℕ∪{0}. It follows from (3.12) and (3.1) that the sequence
(3.13){1m12k⋯mn2kf(m1kx1,…,mnkxn)}
is Cauchy. Since the space Y is complete, this sequence is convergent. Therefore we can define F:Xn→Y by
(3.14)lim⁡k→∞ν(F(x1,…,xn)-1m12k⋯mn2kf(m1kx1,…,mnkxn),wi)(u)=1,
for all u>0, xi,wi∈X and  ai,bi∈𝕂 with  mi=ai2+bi2≠0,1, i=1,…,n. Using induction with (3.12) one can show that
(3.15)ν(f(x1,…,xn)-1m12k⋯mn2kf(m1kx1,…,mnkxn),wi)(u)≥Ψk,
for all xi,wi∈X,i=1,…,n and k∈ℕ∪{0}. By taking k to approach infinity in (3.15) and using (3.3) one obtains (3.5).
For fixedi∈{1,2,…,n} and by (3.6) and (3.14), we get
(3.16)ν(ai2+bi2F(x1,…,aixi+biyi,…,xn)+F(x1,…,bixi-aiyi,…,xn)-(ai2+bi2)(F(x1,…,xn)+F(x1,…,yi,…,xn)),wi)(u)=lim⁡n→∞ν(1m12k⋯mn2k{f(m1kx1,…,mik(aixi+biyi),…,mnkxn)+f(m1kx1,…,mik(bixi-aiyi),…,mnkxn)-(ai2+bi2)(f(m1kx1,nkxn)+f(m1kx1,…,mikyi,…,mnkxn))},wi)(u)≥lim⁡k→∞ω(1m12k⋯mn2kφi(m1kx1,…,mikxi,mikyi,…,mnkxn),wi)(u)=1.
By  (3.2) and (3.16), we conclude that F satisfies (1.17).
Suppose that there exists another mappingF′:Xn→Y which satisfies (1.17) and (3.5). So we have
(3.17)ν(F(x1,…,xn)-F′(x1,…,xn),w)(u)=ν(1m12k…mn2k{F(m1kx1,…,mnkxn)-f(m1kx1,…,mnkxn)+f(m1kx1,…,mnkxn)-F′(m1kx1,…,mnkxn)},w)(u)≥T{Ψ(m1kx1,…,mnkxn,mnkxn,|m12k…mn2k|u),Ψ(m1kx1,…,mnkxn,mnkxn,|m12k…mn2k|u)},
which tends to 1 as k→∞ by (3.3). Therefore F=F′. This completes the proof.In the manner of proof of Theorem3.1, one can investigate the following corollary.Corollary 3.2.
Letφi:Xn+1→Z for i∈{1,…,n} be a mapping such that
(3.18)φ̃i=φ̃i(x1,…,xi,xi,xi+1,…,xn,u)=T{ω(1m12⋯mi-12miφi(m1x1,…,mi-1xi-1,xi,0,…,xn),wi)(u),ω(1m12⋯mi2φi(m1x1,…,mi-1xi-1,aixi,bixi,xi+1,…,xn),wi)(u)},Φ1=Φ1(x1,x1,x2,…,xn,u)=φ̃1(x1,y1,x2,…,xn,u),Φi=Φi(x1,…,xi,xi,xi+1,…,xn,u)=T(φ̃i(x1,…,xi,xi,xi+1,…,xn,u),Φi-1(x1,…,xi-1,xi-1,xi,…,xn,u)),lim⁡k→∞Φn(m1kx1,…,mnkxn,mnkxn,|m12k…mn2k|u)=1,lim⁡k→∞ω(1m12k⋯mn2kφi(m1kx1,…,mikxi,mikyi,…,mnkxn),wi)(u)=1,Φk*=Φk*(x1,…,xn,xn,u)=Φn(m1kx1,…,mnkxn,mnkxn,|m12k⋯mn2k|u),Ψ0=Φ0*(x1,…,xn,xn,u)=Φn(x1,…,xn,xn,u),Ψk=Ψk(x1,…,xn,xn,u)=T(Φk*(x1,…,xn,xn,u),Ψk-1(x1,…,xn,xn,u)),Ψ=Ψ(x1,…,xn,xn,u)=lim⁡k→∞Ψk=1,
for all u>0, xi,wi∈X  and  ai,bi∈𝕂 "with"  mi=ai2+bi2≠0,1, i=1,…,n. Let f:Xn→Y be a mapping satisfying
(3.19)ν(a12+b12f(a1x1+b1y1,x2,…,xn)+f(b1x1-a1y1,x2,…,xn)-(a12+b12)(f(x1,…,xn)+f(y1,x2,…,xn)),w1)(u),≥ω(φ1(x1,y1,x2,…,xn),w1)(u);⋮ν(an2+bn2f(x1,…,xn-1,anxn+bnyn)+f(x1,…,xn-1,bnxn-anyn)-(an2+bn2)(f(x1,…,xn)+f(x1,…,xn-1,yn)),wn)(u),≥ω(φn(x1,…,xn,yn),w1)(u);
for all u>0,xi,yi,wi∈Xandai,bi∈𝕂 with mi=ai2+bi2≠0,1,i=1,…,n. Assume that f(x1,x2,…,xn)=0 if xi=0 for some i=1,…,n. Then there exists a unique mapping F:Xn→Y satisfying (1.17) and
(3.20)ν(f(x1,…,xn)-F(x1,…,xn),wi)(u)≥Ψ,
for all u>0 and xi,wi∈X,i=1,…,n.



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*Source: 208163-2011-10-19.xml*