# On the Stability of anm-Variables Functional Equation in Random Normed Spaces via Fixed Point Method

Authors: A. Ebadian; M. Eshaghi Gordji; H. Khodaei; R. Saadati; Gh. Sadeghi
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

At first we find the solution of the functional equationDf(x1,…,xm):=∑k=2m(∑i1=2k∑i2=i1+1k+1⋯∑im-k+1=im-k+1m)f(∑i=1,i≠i1,…,im-k+1mxi-∑r=1m-k+1xir)+f(∑i=1mxi)-2m-1f(x1)=0, where m≥2 is an integer number. Then, we obtain the generalized Hyers-Ulam-Rassias stability in random normed spaces via the fixed point method for the above functional equation.


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## Body

## 1. Introduction and Preliminaries

A basic question in the theory of functional equations is as follows: “when is it true that a function that approximately satisfies a functional equation must be close to an exact solution of the equation?”If the problem accepts a solution, we say the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940 and affirmatively solved by Hyers [2]. The result of Hyers was generalized by Aoki [3] for approximate additive function and by Rassias [4] for approximate linear functions by allowing the difference Cauchy equation ∥f(x+y)-f(x)-f(y)∥ to be controlled by ε(∥x∥p+∥y∥p). Taking into consideration a lot of influence of Ulam, Hyers, and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias is called the Hyers-Ulam-Rassias stability. In 1994, a generalization of Rassias theorem was obtained by Găvruţa [5], who replaced ε(∥x∥p+∥y∥p) by a general control function φ(x,y) (see also [6–24]).In the sequel we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [25–29]. Throughout this paper, let Δ+ be the space of distribution functions, that is,(1.1)Δ+∶={F:R∪{-∞,∞}⟶[0,1]:Fisleft-continuous,nondecreasingon⁡R,F(0)=0andF(+∞)=1}
and the subset D+⊆Δ+ is the set(1.2)D+={F∈Δ+:l-F(+∞)=1},
where l-f(x) denotes the left limit of function f at the point x. The space Δ+ is partially ordered by the usual pointwise ordering of functions, that is, F≤G if and only if F(t)≤G(t) for all t∈ℝ. The maximal element for Δ+ in this order is the distribution function given by(1.3)ε0(t)={0ift≤0,1ift>0.Definition 1.1 (see [28]).
A mappingT:[0,1]×[0,1]→[0,1] is a continuous triangular norm (briefly, a t-norm) if T satisfies the following conditions:(a)
T is commutative and associative,(b)
T is continuous,(c)
T(a,1)=a for all a∈[0,1];(d)
T(a,b)≤T(c,d) whenever a≤c and b≤d for all a,b,c,d∈[0,1].Typical examples of continuoust-norms are TP(a,b)=ab, TM(a,b)=min(a,b), and TL(a,b)=max(a+b-1,0) (the Łukasiewicz t-norm).Recall (see [30, 31]) that if T is a t-norm and {xn} is a given sequence of numbers in [0,1], Ti=1nxi is defined recurrently by(1.4)Ti=1nxi={x1ifn=1,T(Ti=1n-1xi,xn)ifn≥2.Ti=n∞xi is defined as Ti=1∞xn+i.It is known [31] that for the Łukasiewicz t-norm the following implication holds:(1.5)lim⁡n→∞(TL)i=1∞xn+i=1⟺∑n=1∞(1-xn)<∞.Definition 1.2 (see [29]).
A random normed space (briefly, RN space) is a triple(X,Λ,T), where X is a vector space, T is a continuous t-norm, and Λ is a mapping from X into D+ such that the following conditions hold:(RN1)
Λx(t)=ε0(t) for all t>0 if and only if x=0,(RN2)
Λαx(t)=Λx(t/|α|) for all x∈X, α≠0,(RN3)
Λx+y(t+s)≥T(Λx(t),Λy(s)) for all x,y∈X and t,s≥0.Definition 1.3.
Let(X,Λ,T) be an RN space.(1)
A sequence{xn} in X is said to be convergent to x in X if, for every ϵ>0 and λ>0, there exists positive integer N such that Λxn-x(ϵ)>1-λ whenever n≥N.(2)
A sequence{xn} in X is called Cauchyif, for every ϵ>0 and λ>0, there exists positive integer N such that Λxn-xm(ϵ)>1-λ whenever n≥m≥N.(3)
An RN space(X,Λ,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 a random Banach space.Theorem 1.4 (see [28]).
If(X,Λ,T) is an RN space and  {xn} is a sequence such that  xn→x, then  limn→∞Λxn(t)=Λx(t) almost everywhere.Theorem 1.5 (see [32, 33]).
Let(S,d) be a complete generalized metric space, and let J:S→S be a strictly contractive mapping with Lipschitz constant L<1. Then, for each given element x∈S, either
(1.6)d(Jnx,Jn+1x)=∞
for all nonnegative integers n or there exists a positive integer n0 such that(1)
d(Jnx,Jn+1x)<∞, for all n≥n0,(2)
the sequence{Jnx} converges to a fixed point y* of J,(3)
y* is the unique fixed point of J in the set Ω={y∈S∣d(Jn0x,y)<∞},(4)
d(y,y*)≤(1/(1-L))d(y,Jy) for all y∈Ω.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 notion of an RN space corresponds to the situations when we do not know exactly the norm of point and we know only probabilities of possible values of this norm. The RN spaces may also provide us 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 in, Alsina [34], Mirmostafaee et al. [35–38], Miheţ and Radu [26, 27, 39, 40], Miheţ et al. [41, 42], Baktash et al. [43] and Saadati et al. [44].In this paper, we consider them-dimensional additive functional equation(1.7)∑k=2m(∑i1=2k∑i2=i1+1k+1⋯∑im-k+1=im-k+1m)f(∑i=1,i≠i1,…,im-k+1mxi-∑r=1m-k+1xir)+f(∑i=1mxi)=2m-1f(x1),
where m≥2 is an integer number. It is easy to see that the function f(x)=ax is a solution of the functional equation (1.7).As a special case, ifm=2 in (1.7), then the functional equation (1.7) reduces to(1.8)f(x1+x2)+f(x1-x2)=2f(x1).
Also by putting m=3 in (1.7), we obtain(1.9)∑i1=22∑i2=i1+13f(∑i=1,i≠i1,i23xi-∑r=12xir)+∑i1=23f(∑i=1,i≠i13xi-xi1)+f(∑i=13xi)=22f(x1),
that is,(1.10)f(x1-x2-x3)+f(x1-x2+x3)+f(x1+x2-x3)+f(x1+x2+x3)=4f(x1).The main purpose of this paper is to prove the stability of (1.7) in random normed spaces via the fixed point method.


## 2. Results in RN spaces via Fixed Point Method

Lemma 2.1.
LetX and  Y be real vector spaces. A function  f:X→Y with  f(0)=0 satisfies  (1.7) if and only if  f:X→Y is additive.Proof.
Letf satisfy the functional equation (1.7). Hence, according to (1.7), we get
(2.1)∑i1=22∑i2=i1+13⋯∑im-1=im-2+1mf(∑i=1,i≠i1,…,im-1mxi-∑r=1m-1xir)+∑i1=23∑i2=i1+14⋯∑im-2=im-3+1mf(∑i=1,i≠i1,…,im-2mxi-∑r=1m-2xir)+⋯+∑i1=2mf(∑i=1,i≠i1mxi-xi1)+f(∑i=1mxi)=2m-1f(x1)
for all x1,…,xm∈X. Setting xi=0(i=2,…,m-1) in (2.1), we have
(2.2)f(x1-xm)+((m-21)f(x1-xm)+(m-2m-2)f(x1+xm))+⋯+((m-2m-3)f(x1-xm)+(m-22)f(x1+xm))+((m-2m-2)f(x1-xm)+(m-21)f(x1+xm))+f(x1+xm)=2m-1f(x1),
that is,
(2.3)(1+∑l=1m-2(m-2l))(f(x1+xm)+f(x1-xm))=2m-1f(x1)
for all x1,xm∈X. On the other hand, we have the relation
(2.4)1+∑l=1m-j(m-jl)=∑l=0m-j(m-jl)=2m-j
for all m>J. Hence, we obtain from (2.3) and (2.4) that
(2.5)f(x1+xm)+f(x1-xm)=2f(x1)
for all x1,xm∈X. Setting xm=x1 in (2.5) we get f(2x1)=2f(x1) for all x1∈X. Replacing x1 and xm by x1+xm and x1-xm in (2.5), respectively, and then using f(2x1)=2f(x1), we obtain that
(2.6)f(x1+xm)=f(x1)+f(xm)
for all x1,xm∈X, which implies that f is additive.
Conversely, suppose thatf is additive, and thus f satisfies the equation f(x1+x2)=f(x1)+f(x2). Hence we have f(0)=0 and f(2x1)=2f(x1) for all x1∈X. Replacing x1 and x2 by x1+x2 and x1-x2 in the additive equation and then using f(2x1)=2f(x1) lead to
(2.7)f(x1+x2)+f(x1-x2)=2f(x1)
for all x1,x2∈X.
Now, we are going to prove our assumption by induction onm≥2. It holds for m=2; see (2.7). Assume that (1.7) holds for the case, where m=p; that is, we have
(2.8)∑k=2p(∑i1=2k∑i2=i1+1k+1⋯∑ip-k+1=ip-k+1p)f(∑i=1,i≠i1,…,ip-k+1pxi-∑r=1p-k+1xir)+f(∑i=1pxi)=2p-1f(x1)
for all x1,…,xp∈X. Replacing x1 by x1+xp+1 in (2.8), we obtain
(2.9)∑k=2p(∑i1=2k∑i2=i1+1k+1⋯∑ip-k+1=ip-k+1p)f(x1+xp+1+∑i=2,i≠i1,…,ip-k+1pxi-∑r=1p-k+1xir)+f(∑i=1p+1xi)=2p-1f(x1+xp+1)
for all x1,…,xp∈X. Replacing xp+1 by -xp+1 in (2.9), we obtain
(2.10)∑k=2p(∑i1=2k∑i2=i1+1k+1⋯∑ip-k+1=ip-k+1p)f(x1-xp+1+∑i=2,i≠i1,…,ip-k+1pxi-∑r=1p-k+1xir)+f(∑i=1pxi-xp+1)=2p-1f(x1-xp+1)
for all x1,…,xp+1∈X. Adding (2.9) to (2.10), one gets
(2.11)∑k=2p+1(∑i1=2k∑i2=i1+1k+1⋯∑ip-k+2=ip-k+1+1p+1)f(∑i=1,i≠i1,…,ip-k+2p+1xi-∑r=1p-k+2xir)+f(∑i=1p+1xi)=2p-1(f(x1+xp+1)+f(x1-xp+1))
for all x1,…,xp+1∈X. Therefore, it follows from (2.7) and (2.11) that (1.7) holds for m=p+1. This completes the proof of the theorem.From now on, letX be a linear space and (Y,Λ,TM) a complete RN space. For convenience, we use the following abbreviation for a given function f:X→Y:(2.12)Df(x1,…,xm)=∑k=2m(∑i1=2k∑i2=i1+1k+1…∑im-k+1=im-k+1m)f(∑i=1,i≠i1,…,im-k+1mxi-∑r=1m-k+1xir)+f(∑i=1mxi)-2m-1f(x1)
for all x1,…,xm∈X, where m≥2 is an integer number.Theorem 2.2.
LetΦ:X×X×⋯×X︸m-terms→D+ be a function (Φ(x1,…,xm) is denoted by Φx1,…,xm) such that, for some 0<α<2,
(2.13)Φ2x1,…,2xm(αt)≥Φx1,…,xm(t)
for all x1,…,xm∈X and all t>0. Suppose that a function f:X→Y with f(0)=0 satisfies the inequality
(2.14)ΛDf(x1,…,xm)(t)≥Φx1,…,xm(t)
for all x1,…,xm∈X and all t>0. Then, there exists a unique additive function A:X→Y such that
(2.15)Λf(x)-A(x)(t)≥Φx,x,0,…,0︸m-2(2m-2(2-α)t)
for all x∈X and all t>0.Proof.
Lettingxi=0(i=3,…,m) in (2.14), we get
(2.16)Λ(1+∑l=1m-2(m-2l))(f(x1+x2)+f(x1-x2))-2m-1f(x1)(t)≥Φx1,x2,0,…,0(t)
for all x1,x2∈X and all t>0. Setting x1=x2=x in (2.16), we obtain from (2.4) and f(0)=0 that
(2.17)Λ2m-2f(2x)-2m-1f(x)(t)≥Φx,x,0,…,0(t)
for all x∈X and all t>0, or
(2.18)Λf(2x)/2-f(x)(t)≥Φx,x,0,…,0(2m-1t)
for all x∈X and all t>0. Let S be the set of all functions h:X→Y with h(0)=0 and introduce a generalized metric on S as follows:
(2.19)d(h,k)=inf⁡{u∈R+:Λh(x)-k(x)(ut)≥Φx,x,0,…,0(t),∀x∈X,∀t>0},
where, as usual, inf∅=+∞. It is easy to show that (S,d) is a generalized complete metric space [26, 45].
Now we consider the functionJ:S→S defined by
(2.20)Jh(x)∶=h(2x)2
for all h∈S and x∈X.
Now letg,f∈S such that d(f,g)<ε. Then,
(2.21)ΛJg(x)-Jf(x)(αε2t)=Λg(2x)-f(2x)(αεt)≥Φ2x,2x,0,…,0(αt)≥Φx,x,0,…,0(t),
that is, if d(f,g)<ε, we have d(Jf,Jg)<(α/2)ε. This means that
(2.22)d(Jf,Jg)≤α2d(f,g)
for all f,g∈S, that is, J is a strictly contractive self-function on S with the Lipschitz constant α/2.
It follows from (2.18) that
(2.23)ΛJf(x)-f(x)(t2m-1)≥Φx,x,0,…,0(t)
for all x∈X and all t>0, which implies that d(Jf,f)≤1/2m-1.
Due to Theorem1.5, there exists a function A:X→Y such that A is a fixed point of J, that is, A(2x)=2A(x) for all x∈X.
Also,d(Jnf,A)→0 as n→∞, implies the equality
(2.24)lim⁡n→∞f(2nx)2n=A(x)
for all x∈X. If we replace x1,…,xm with 2nx1,…,2nxm in (2.14), respectively, and divide by 2n, then it follows from (2.13) that
(2.25)ΛDf(2nx1,…,2nxm)/2n(t)≥Φ2nx1,…,2nxm(2nt)=Φ2nx1,…,2nxm(αn(2α)nt)≥Φx1,…,xm((2α)nt)
for all x1,…,xm∈X and all t>0. By letting n→∞ in (2.25), we find that ΛDA(x1,…,xm)(t)=1 for all t>0, which implies DA(x1,…,xm)=0, and thus A satisfies (1.7). Hence by Lemma 2.1, the function A:X→Y is additive.
According to the fixed point alterative, sinceA is the unique fixed point of J in the set Ω={g∈S:d(f,g)<∞}, A is the unique function such that
(2.26)Λf(x)-A(x)(ut)≥Φx,x,0,…,0(t)
for all x∈X and all t>0. Again using the fixed point alterative gives
(2.27)d(f,A)≤11-Ld(f,Jf)≤12m-1(1-L)=12m-1(1-α/2),
which implies the inequality
(2.28)Λf(x)-A(x)(t2m-2(2-α))≥Φx,x,0,…,0(t)
for all x∈X and all t>0. So,
(2.29)Λf(x)-A(x)(t)≥Φx,x,0,…,0(2m-2(2-α)t)
for all x∈X and all t>0. This completes the proof.Now, we present a corollary that is an application of the last theorem in the classical case.Corollary 2.3.
Let(X,∥·∥) and  (Y,∥·∥), normed linear spaces, define
(2.30)Λx(t)=tt+‖x‖
for x∈X and t>0. Define
(2.31)Φx1,…,xm(t)=tt+∑i=1m‖xi‖p
for all x1,…,xm∈X and all t>0 in which p<1. Now, for α=2p, (2.13) holds for all x1,…,xm∈X and all t>0. Suppose that an odd function f:X→Y satisfies (2.14) for all x1,…,xn∈X and all t>0. Then, by the last theorem there exists a unique additive function A:X→Y such that
(2.32)tt+‖f(x)-A(x)‖≥tt+1/2m-2(1-2p-1)‖x‖p
for all x∈X and all t>0. Hence,
(2.33)‖f(x)-A(x)‖≤12m-2(1-2p-1)‖x‖p
for all x∈X.Theorem 2.4.
LetΦ:X×X×⋯×X︸m-terms→D+ be a function such that, for some  0<α<3,
(2.34)Φ3x1,…,3xm(αt)≥Φx1,…,xm(t)
for all x1,…,xm∈X and all t>0. Suppose that an odd function f:X→Y satisfies (2.14) for all x1,…,xn∈X and all t>0. Then, there exists a unique additive function A:X→Y such that
(2.35)Λf(x)-A(x)(t)≥Φx,x,x,0,…,0︸m-3(2m-3(3-α)t)
for all x∈X and all t>0.Proof.
Lettingxi=0(i=4,…,m) in (2.14), we get
(2.36)Λ(1+∑l=1m-3(n-3l))(f(x1+x2+x3)+f(x1-x2+x3)+f(x1+x2-x3)+f(x1-x2-x3))-2m-1f(x1)(t)≥Φx1,x2,x3,0,…,0(t)
for all x1,x2,x3∈X and all t>0. Setting x1=x2=x3=x in the last inequality, we obtain by using oddness of f and (2.4) that
(2.37)Λ2m-3f(3x)+f(x)-2m-1f(x)(t)≥Φx,x,x,0,…,0(t)
for all x∈X and all t>0, or
(2.38)Λ(f(3x)/3)-f(x)(t)≥Φx,x,x,0,…,0(3.2m-3t)
for all x∈X and all t>0. Let S be the set of all odd functions h:X→Y, and introduce a generalized metric on S as follows:
(2.39)d(h,k)=inf⁡{u∈R+:Λh(x)-k(x)(ut)≥Φx,x,x,0,…,0(t),∀x∈X,∀t>0}.
It is easy to show that (S,d) is a generalized complete metric space [26, 45]. Let J:S→S be the function defined by
(2.40)Jh(x):=h(3x)3
for all h∈S and x∈X. One can show that
(2.41)d(Jf,Jg)≤α3d(f,g)
for all f,g∈S, that is, J is a strictly contractive self-function on S with the Lipschitz constant α/3.
It follows from (2.38) that
(2.42)ΛJf(x)-f(x)(t3.2m-3)≥Φx,x,x,0,…,0(t)
for all x∈X and all t>0, which implies that d(Jf,f)≤(1/3.2m-3).
Due to Theorem1.5, the sequence {Jn} converges to a fixed point A of J, that is,
(2.43)A:X⟶Y,A(x)=lim⁡n→∞Jnf(x)=lim⁡n→∞f(3nx)3n,
and A(3x)=3A(x) for all x∈X.
Also,A is the unique fixed point of J in the set Ω={g∈S:d(f,g)<∞}, and
(2.44)d(f,A)≤11-Ld(f,Jf)≤13.2m-3(1-L)=13.2m-3(1-α/3)
implies the inequality
(2.45)Λf(x)-A(x)(t2m-3(3-α))≥Φx,x,x,0,…,0(t)
for all x∈X and all t>0. This implies that inequality (2.35) holds. Furthermore, we can obtain that the function A:X→Y satisfies (1.7). Hence by Lemma 2.1, we get that the function A:X→Y is additive.Now, we present a corollary that is an application of the last theorem in the classical case.Corollary 2.5.
Let(X,∥·∥) and  (Y,∥·∥), normed linear spaces, define
(2.46)Λx(t)=tt+‖x‖
for x∈X and t>0. Define
(2.47)Φx1,…,xm(t)=tt+∑i=1m‖xi‖p
for all x1,…,xm∈X and all t>0 in which p<1. Now, for α=3p, (2.34) holds for all x1,…,xm∈X and all t>0. Suppose that an odd function f:X→Y satisfies (2.14) for all x1,…,xn∈X and all t>0. Then, by the last theorem there exists a unique additive function A:X→Y such that
(2.48)tt+‖f(x)-A(x)‖≥tt+1/2m-3(1-3p-1)‖x‖p
for all x∈X and all t>0. Hence,
(2.49)‖f(x)-A(x)‖≤12m-3(1-3p-1)‖x‖p
for all x∈X.



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