Fuzzy stability of the monomial functional equation

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Available online at www.ispacs.com/jnaa Volume 2014, Year 2014 Article ID jnaa-00244, 13 Pages

doi:10.5899/2014/jnaa-00244 Research Article

Fuzzy stability of the monomial functional equation

M. Almahalebi1∗, M. Sirouni1, A. Charifi1, S. Kabbaj1

(1)Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, Kenitra, Morocco.

Copyright 2014 c⃝M. Almahalebi, M. Sirouni, A. Charifi and S. Kabbaj. 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

By using the fixed point method, we prove the generalized Hyers-Ulam-Rassias stability of the monomial functional equation∆n

xf(y) =n!f(x)in fuzzy normed spaces.

Keywords:fuzzy normed spaces, monomial functional equation, fixed point alternative, Hyers-Ulam-Rassias stability.

2010 Mathematics Subject Classification:Primary 39B82; Secondary 46S40; 46S50; 47H10.

1 Introduction

One of the interesting questions in the theory of functional analysis concerning the stability problem of functional equations is as follows:

“When is it true that a mapping satisfying approximately a functional equation must be close to an exact solution of the given functional equation?”

The first stability problem was raised by S. M. Ulam [22],[23] during his talk at the University of Wisconsin in 1940. For very general functional equations, the concept of stability for functional equations arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus the stability question of functional equations is that how do the solutions of the inequality differ from those of the given functional equation? LetXandY be linear spaces andYXbe the vector space of all functions fromXtoY. Following [12], for eachx∈X, define∆x:YX→YX by

∆xf(y) =f(x+y)−f(y) (f ∈YX,y∈X). Inductively, we define

∆x1,...,xnf(y) =∆x1,...,xn−1(∆xnf(y)) for eachy,x1, ...,xn∈Xand all f ∈YX. Ifx1=...=xn=x, we write

∆n

xf(y) =∆x, ...,x | {z }

n times

f(y)

By induction onn, it can be easily verified that

∆nxf(y) = n

∑

r=0

(−1)n−rCr_nf(rx+y) (n∈N,x,y∈X). (1.1)

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The functional equation

∆n

xf(y)−n!f(x) =0 (1.2)

is calledthe monomial functional equationof degreen, since the function f(x) =cxnis a solution of this functional equation. Every solution of the monomial functional equation of degreenis said to bea monomial mappingof degree n. In particular additive, quadratic, cubic and quartic functions are monomials of degree one, two, three and four respectively. The stability of monomial equations was initiated by D. H. Hyers in [12]. The problem has been recently considered by many authors, we refer, for example, to [4], [8]-[11], [14], [17] and [24].

In 2006, Z. Kaiser [14] proved the stability of monomial functional equation where the functions map a normed space over a field with valuation to a Banach space over a field with valuation and the control function is of the form

ε(∥x∥α+∥y∥α). In 2007, L. C˘adariu and V. Radu proved the stability of the monomial functional equation

n

∑

i=0

Ci_n(−1)n−i_f₍_ix₊_y₎₋_n_!_f₍_x_{) =}₀ _(1.3)

In 2008, Y. Lee [17] modified the results of L. C˘adarui and V. Radu for the stability of the monomial functional equa-tion (1.3) in the sense of Th. M. Rassias and he investigated the superstability of this equaequa-tion.

In 2010, A. K. Mirmostafaee [21] proved the Hyers-Ulam-Rassias stability of monomial functional equation of an arbitrary degree in non-Archimedean normed spaces over a field with valuation and the control functionϕ(x,y).

In this paper, we use the fixed point method to prove the Hyers-Ulam-Rassias stability of monomial functional equa-tion of an arbitrary degree in fuzzy normed spaces with the control funcequa-tion is of the formN′(ϕ(x,y),t).

2 Preliminaries and notations

The theory of fuzzy space has much progressed as developing the theory of randomness. Some mathematicians have defined fuzzy norms on a vector space from various points of view [1], [7], [16], [19] and [25]. Following Cheng and Mordeson [5], Bag and Samanta [1] gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [15] and investigated some properties of fuzzy normed spaces [2]. We use the definition of fuzzy normed spaces given in [1], [19] and [20] to investigate a fuzzy version of the Hyers-Ulam stability for the monomial functional equation (1.2) in the fuzzy normed vector space setting.

Definition 2.1. Let X be a real vector space. A function N:X×R_→_[₀_,₁_](so-called fuzzy subset) is said a fuzzy norm on X if

(N1) N(x,t) =0for t≤0;

(N2) x=0if and only if N(x,t) =1for all t>0; (N3) N(cx,t) =N(x,_|_ct_|)if c̸=0;

(N4) N(x+y,s+t)≥min{N(x,s),N(y,t)};

(N5) N(x, .)is a non-decreasing function ofRand lim

t→+∞N(x,t) =1;

(N6) For x̸=0, N(x, .)is a left continuous function onR.

for all x,y∈X and all s,t∈R. The pair (X,N) is called a fuzzy normed vector space (briefly, FNS).

Definition 2.2. Let(X,N)be an FNS. A sequence{xn}in X is said to be convergent if there exists an x∈X such that lim

t→+∞N(xn−x,t) =1 for all t>0. In this case, x is called the limit of the sequence{xn}in X and we denote it by

N− lim t→+∞xn=x.

Definition 2.3. Let(X,N)be an FNS. A sequence{xn}in X is called Cauchy if for eachε>0and each t>0there

exists an n₀∈Nsuch that for all n0and all p>0, we have N(xn+p−xn)>1−ε.

It is well-known that every convergent sequence in an FNS is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to becompleteand the fuzzy normed vector space is calleda fuzzy Banach space. We say that a mapping f :X→Y between FNSX andY is continuous at a pointx∈X if for each sequence{xn}converging to

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said to becontinuousonX(see [2]).

Definition 2.4. Let X be a set. A function d:X×X→[0,∞]is called a generalized metric on X if d satisfies the following:

(i) d(x,y) =0if and only if x=y;

(ii) d(x,y) =d(y,x)for all x,y∈X ;

(iii) d(x,z)≤d(x,y) +d(y,z)for all x,y,z∈X .

Then(X,d)is called a generalized metric space. (X,d)is called complete if every Cauchy sequence in X is d-convergent.

Note that the distance between two points in a generalized metric space is permitted to be infinity.

Definition 2.5.Let(X,d)be a generalized complete metric space. A mapping J:X→X satisfies a Lipschitz condition with a Lipschitz constant L>0if

d(J(x),J(y))≤Ld(x,y) (x,y∈X).

If L<1, then J is called a strictly contractive operator.

Theorem 2.1. [3], [6] Suppose we are given a complete generalized metric space(X,d)and a strictly contractive

mapping J:X→X , with the Lipshitz constant L<1. If there exists a nonnegative integer k such that

d(Jk_x_,_Jk+1_x₎_<_∞_{for some x}_∈_{X , then the following are true:} (1) the sequence Jnx converges to a fixed point x∗of J;

(2) x∗is the unique fixed point of J in the set Y ={y∈X:d(Jkx,y)<∞}; (3) d(y,x∗)≤₁₋1_Ld(y,Jy)for all y∈Y .

In 1996, Isac and Th. M. Rassias [13] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications.

3 Main Results

Throughout this section, we assume thatXis a linear space,(Y,N)is a fuzzy Banach space and(Z,N′)is a fuzzy normed space.

From a result of A. Gil`anyi [8], Z. Kaiser [14] proved the following lemma.

Lemma 3.1. For every mapping f :X −→Y and n,k∈N, there correspond positive integersξ0,ξ1, ...,ξn(k−1) such

that

f(kx)−knf(x) = 1

n! (_n(k₋₁₎

∑

i=0

ξiFi(x)−G(x)

)

for all x∈X , where Fi:X−→Y and G:X−→Y are defined by

Fi(x) =∆n

xf(ix)−n!f(x)

and

G(x) =∆n

kxf(0)−n!f(kx)

for all x∈X and i=0,1, ...,n(k−1).

Theorem 3.1. Assume that a mapping f:X→Y satisfies the inequality

N(∆n

xf(y)−n!f(x),t)≥N′(ϕ(x,y),t) (3.4)

whereϕ:X2→Z is a mapping which there is a constant0<L<1satisfying N′(ϕ(x,y),t)≥N′(knLϕ(x

k, y k )

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for all x,y∈X , all t>0and some k≥2.

Define

ψk(x,t) =min {

N′(ϕ(kx,0),n!t),N′ (

ϕ(x,ix),n!t

ξi

)

i=0,1,...,n(k−1)

}

for each k≥2. Then the limit

M(x):=N− lim s→+∞

f(ksx)

kns (3.6)

exists for each x∈X and defines a monomial mapping M:X→Y of degree n such that

N(f(x)−M(x),t)≥min {

N′ (_ϕ

(kx,0)

kn₍₁₋_L₎,n!t

)

,N′ ( _ϕ

(x,ix)

kn₍₁₋_L₎,

n!t

ξi

)

i=0,1,...,n(k−1)

}

(3.7)

for all x∈X , and all t>0.

Proof. Form Lemma 3.1 and (3.4), we obtain

N(f(kx)−knf(x),t) = N (_n(k₋₁₎

∑

i=0

ξiFi(x)−G(x),n!t

)

≥ N

(_n(k₋₁₎

∑

i=0

ξiFi(x) +G(x),n!t

)

≥ min{N(G(x),n!t),N(ξiFi(x),n!t)i=0,1,...,n(k−1)

}

≥ min

{

N′(ϕ(kx,0),n!t),N′ (

ϕ(x,ix),n!t

ξi

)

i=0,1,...,n(k−1)

}

So, we get

N(f(kx)−knf(x),t)≥ψk(x,t) (3.8) Let us consider the set

S:={g:X→Y}

and introduce the generalized metric onS:

d(g,h) =in f{µ∈[0,∞):N(g(x)−h(x),µt)≥ψk(x,t), ∀x∈X,∀t>0}

where, as usual,in fØ= +∞. It is easy to show that(S,d)is complete (see the proof of [18], Lemma 2.1). Now we consider the linear mappingJ:S−→Ssuch that

J(g(x)):= f(kx)

kn

for allx∈X.

Letg,h∈Sbe given such thatd(g,h) =ε. Then

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for allx∈Xand allt>0. Hence

N(J(g(x))−J(h(x)),Lεt) = N (

1

kn(g(kx)−h(kx)),Lεt

)

= N(g(kx)−h(kx),knLεt)

≥ min

{

N′(ϕ(k2x,0),n!Lknt),N′ (

ϕ(kx,kix),n!Lk

n_t

ξi

)

i=0,1,...,n(k−1)

}

≥ min

{

N′ (

ϕ(kx,0),n!Lk

n_t

Lkn

)

,N′ (

ϕ(x,ix),n!Lk

n_t

Lkn_ξ i

)

i=0,1,...,n(k−1)

}

≥ min

{

N′(ϕ(kx,0),n!t),N′ (

ϕ(x,ix),n!t

ξi

)

i=0,1,...,n(k−1)

}

= ψk(x,t)

for allx∈Xand allt>0. Sod(g,h) =εimplies thatd(Jg,Jh)≤Lε. This means that

d(Jg,Jh)≤Ld(g,h)

for allg,h∈S. It follows from (3.8) that

N (_f₍_kx₎

kn −f(x),

t kn

)

≥ψk(x,t)

for allx∈Xand allt>0. So

d(f,J f)≤ 1

kn<∞.

Therefore, by Theorem 2.1, there exists a mappingM:X−→Y satisfying the following:

(1) Mis fixed point of the operatorJ, that is,

M(kx) =knM(x) (3.9)

for allx∈X.

The mappingMis a unique fixed point ofJin the set

A={g∈S:d(f,g)≤∞};

This implies thatMis a unique mapping satisfying (3.9) such that there exists aµ∈(0,∞)satisfying

N(f(x)−M(x),µt)≥ψk(x,t)

for allx∈Xand allt>0.

(2) We find thatd(Jsf,M)−→0 ass−→∞. This implies the equality

N− lim s→+∞

f(ksx)

ksn =M(x)

for allx∈X;

(3) We have

d(f,M)≤ 1

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Thus we can get

N (

f(x)−M(x), t

kn₋_kn_L

)

≥min

{

N′(ϕ(kx,0),n!t),N′ (

ϕ(x,ix),n!t

ξi

)

i=0,1,...,n(k−1)

}

Then

N′(ϕ(kx,0),n!kn(1−L)t),N′ (

ϕ(x,ix),n!k

n₍₁₋_L₎_t

ξi

)

i=0,1,...,n(k−1)

}

which means that the inequality (3.7) holds true, that is,

N′ (

ϕ(kx,0)

)

,N′ (

ϕ(x,ix)

kn₍₁₋_L₎,

n!t

ξi

)

i=0,1,...,n(k−1)

}

for allx∈Xand allt>0.

In addition, it follows from the condition (3.5) that

N (_∆n

ks_xf(ksy)

kns −

n!f(ksx)

kns ,

t kns

)

≥N′(ϕ(ksx,ksy),t)

for allx,y∈Xand allt>0. So

N (_∆n

ks_xf(ksy)

kns −

n!f(ks_x₎

kns ,t

)

≥N′

(

Lϕ(x,y),ks(n−1)t )

for allx,y∈Xand allt>0. Since

lim s→+∞N

′(_L_ϕ₍_x_,_y₎_,_ks(n−1)_t)₌₁ for allx,y∈Xand allt>0,

N(∆n

xM(y)−n!M(x),t) =1 for allx,y∈Xand allt>0. Thus

∆nxM(y)−n!M(x) =0

for allx,y∈X. It means thatM:X−→Y is a monomial function of degreen, as desired.

We obtain the following corollaries concerning the stability for approximate mappings controlled by a sum of powers of norms then by a product of powers of norms.

Corollary 3.1. Suppose that(X,∥.∥)is a normed vector space, (R,N′)a fuzzy normed space and(Y,N)is a fuzzy Banach space. Letθ≥0and p>0be real numbers with p<n and let f :X→Y be a mapping satisfying

N(∆n

xf(y)−n!f(x),t)≥N′(θ(∥x∥p+∥y∥p),t) (3.10)

for all x,y∈X and all t>0. Then the limit

f(ksx)

kns (3.11)

N′ (

∥x∥p

kn−p₋₁,n!t

)

,N′ (

(1+ip)∥x∥p

kn₋_kp ,

n!t

ξi

)

i=0,1,...,n(k−1)

}

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for all x∈X and all t>0.

Proof. The proof follows from Theorem 3.1 by taking

ϕ(x,y):=θ(∥x∥p₊_∥_y_∥p₎

for allx,y∈X. Then we can chooseL=kp−nand we get the desired result.

Corollary 3.2. Suppose that(X,∥.∥)is a normed vector space, (R,N′)a fuzzy normed space and(Y,N)is a fuzzy Banach space. Letθ≥0, p>0and q>0be real numbers with p+q<n and let f :X→Y be a mapping satisfying

N(∆n

xf(y)−n!f(x),t)≥N′(θ(∥x∥p.∥y∥p),t) (3.13)

f(ks_x₎

kns (3.14)

1,N′ (

iq∥x∥p+q

kn₋_kp+q,

n!t

ξi

)

i=0,1,...,n(k−1)

}

(3.15)

ϕ(x,y):=θ(∥x∥p_._∥_y_∥q₎

for allx,y∈X. Then we can chooseL=k(p+q)−nand we get the desired result.

By the similar method of the proof of Theorem 3.1, we can prove the following theorem and corollaries.

Theorem 3.2. Assume that a mapping f :X→Y satisfies the inequality (3.4) whereϕ:X2→Z is a mapping which there is a constant0<L<1satisfying

N′(ϕ(x,y),t)≥N′ (

L

knϕ(kx,ky),t

)

(3.16)

for all x,y∈X , all t>0and some k≥2. Then the limit

M(x):=N− lim s→+∞k

ns_f(x

ks

)

(3.17)

N′ (

Lϕ(kx,0)

)

,N′ (

Lϕ(x,ix)

kn₍₁₋_L₎,

n!t

ξi

)

i=0,1,...,n(k−1)

}

(3.18)

for all x∈X , and all t>0.

Proof. Let(S,d)be the generalized metric space defined in the proof of Theorem 3.1. Consider the linear mapping J:S→Ssuch that

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for allx∈X. It follows from (3.8) that

N (

f(x)−knf(x k )

,Lt

kn

)

≥ψk(x,t)

for allx∈Xand allt>0. So

d(f,J f)≤ L

kn<∞. The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.3. Suppose that(X,∥.∥)is a normed vector space, (R,N′)a fuzzy normed space and(Y,N)is a fuzzy Banach space. Letθ≥0and p>0be real numbers with p>n and let f :X→Y be a mapping satisfying

N(∆n

xf(y)−n!f(x),t)≥N′(θ(∥x∥p+∥y∥p),t) (3.19)

ns_f(x

ks

)

(3.20)

N′ (

∥x∥p 1−kn−p,n!t

)

,N′ (

(1+ip)∥x∥p

kp₋_kn ,

n!t

ξi

)

i=0,1,...,n(k−1)

}

(3.21)

ϕ(x,y):=θ(∥x∥p+∥y∥p)

for allx,y∈X. Then we can chooseL=kn−pand we get the desired result.

Corollary 3.4. Suppose that(X,∥.∥)is a normed vector space, (R,N′)a fuzzy normed space and(Y,N)is a fuzzy Banach space. Letθ≥0, p>0and q>0be real numbers with p+q>n and let f :X→Y be a mapping satisfying

N(∆n

xf(y)−n!f(x),t)≥N′(θ(∥x∥p.∥y∥p),t) (3.22)

ns_f(x

ks

)

(3.23)

1,N′ (

iq∥x∥p+q

kp+q₋_kn,

n!t

ξi

)

i=0,1,...,n(k−1)

}

(3.24)

ϕ(x,y):=θ(∥x∥p.∥y∥q)

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We obtain the following theorems and corollaries concerning the stability for approximate mappings controlled by a fraction function.

Theorem 3.3. Letϕ:X2−→[0,∞)be a function such that there exists an constant0<L<1with

ϕ(x,y)≤knLϕ(x

k, y k )

(3.25)

for all x,y∈X , for all t>0and for some integers k≥2. Let f:X→Y be a mapping satisfying

N(∆n

xf(y)−n!f(x),t)≥

t

t+ϕ(x,y) (3.26)

f(ksx)

kns (3.27)

exists for each x∈X and defines a monomial function M:X→Y of degree n such that

N(f(x)−M(x),t)≥min {(

n!kn(1−L)t n!kn₍₁₋_L₎_t₊_ϕ₍_kx_,₀₎

)

,

(

n!kn(1−L)t n!kn₍₁₋_L₎_t₊_ξ

iϕ(x,ix)

)

i=0,1,...,n(k−1)

}

(3.28)

N′(ϕ(x,y),t):= t

t+ϕ(x,y) for allx,y∈Xand allt>0.

Corollary 3.5. Suppose that(X,∥.∥)is a normed vector space and(Y,N)is a fuzzy Banach space. Letθ≥0and p>0be real numbers with p<n and let f :X→Y be a mapping satisfying

N(∆n

t

t+θ(∥x∥p₊_∥_y_∥p₎ (3.29)

f(ks_x₎

kns (3.30)

N(f(x)−M(x),t)≥min {(

n!(kn−kp)t n!(kn₋_kp₎_t₊_θ_kp_∥_x_∥p

)

,

(

n!(kn−kp)t n!(kn₋_kp₎_t₊_ξ

iθ(1+ip)∥x∥p

)

i=0,1,...,n(k−1)

}

(3.31)

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Corollary 3.6. Suppose that(X,∥.∥)is a normed vector space and(Y,N)is a fuzzy Banach space. Letθ≥0, p>0

and q>0be real numbers with p+q<n and let f :X→Y be a mapping satisfying

N(∆n

t

t+θ(∥x∥p_._∥_y_∥q₎ (3.32)

f(ksx)

kns (3.33)

N(f(x)−M(x),t)≥min  

 1,

(

n!(kn−k(p+q))t n!(kn₋_k(p+q)₎_t₊_iq_{θ ξ}

i∥x∥p+q

)

i=0,1,...,n(k−1)

 



(3.34)

ϕ(x,y):=θ(∥x∥p.∥y∥q)

for allx,y∈X. Then we can chooseL=k(p+q)−n_{and we get the desired result.}

Theorem 3.4. Letϕ:X2−→[0,∞)be a function such that there exists an constant0<L<1with

ϕ(x,y)≤ L

knϕ(kx,ky) (3.35)

for all x,y∈X , for all t>0and for some integers k≥2. Let f:X→Y be a mapping satisfying

N(∆n

t

t+ϕ(x,y) (3.36)

ns_f(x

ks

)

(3.37)

exists for each x∈X and defines a monomial function M:X→Y of degree n such that

N(f(x)−M(x),t)≥min {(

n!kn(1−L)t n!kn₍₁₋_L₎_t₊_L_ϕ₍_kx_,₀₎

)

,

(

n!kn(1−L)t n!kn₍₁₋_L₎_t₊_L_ξ

iϕ(x,ix)

)

i=0,1,...,n(k−1)

}

(3.38)

N′(ϕ(x,y),t):= t

t+ϕ(x,y) for allx,y∈Xand allt>0.

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p>0be real numbers with p>n and let f :X→Y be a mapping satisfying

N(∆n

t

t+θ(∥x∥p₊_∥_y_∥p₎ (3.39)

f(ksx)

kns (3.40)

N(f(x)−M(x),t)≥ f(k

s_x₎

kns min

{(

n!(1−kn−p)t n!(1−kn−p₎_t₊_θ_∥_x_∥p

)

,

(

n!(kp−kn)t n!(kp₋_kn₎_t₊_ξ

iθ(1+ip)∥x∥p

)

i=0,1,...,n(k−1)

}

(3.41) for all x∈X and all t>0.

ϕ(x,y):=θ(∥x∥p₊_∥_y_∥p₎ for allx,y∈X. Then we can chooseL=kn−pand we get the desired result.

Corollary 3.8. Suppose that(X,∥.∥)is a normed vector space and(Y,N)is a fuzzy Banach space. Letθ≥0, p>0

and q>0be real numbers with p+q>n and let f :X→Y be a mapping satisfying

N(∆n

t

t+θ(∥x∥p_._∥_y_∥q₎ (3.42)

ns_f(x

ks

)

(3.43)

N(f(x)−M(x),t)≥min  

 1,

(

n!(k(p+q)−kn)t n!(k(p+q)−kn₎_t₊_iq_{θ ξ}

i∥x∥p+q

)

i=0,1,...,n(k−1)

 



(3.44)

ϕ(x,y):=θ(∥x∥p_._∥_y_∥q₎

for allx,y∈X. Then we can chooseL=kn−(p+q)and we get the desired result.

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