Báo cáo hóa học: "Lattictic non-archimedean random stability of ACQ functional equation" doc

Iran Full list of author information is available at the end of the article Abstract In this paper, we prove the generalized Hyers-Ulam stability of the following additive-cubic-quartic

R E S E A R C H Open Access

Lattictic non-archimedean random stability of

ACQ functional equation

Yeol Je Cho1and Reza Saadati2*

* Correspondence: rsaadati@eml.cc

2

Department of Mathematics,

Science and Research Branch,

Islamic Azad University, Tehran, I.R.

Iran

Full list of author information is

available at the end of the article

Abstract

In this paper, we prove the generalized Hyers-Ulam stability of the following additive-cubic-quartic functional equation

in various complete lattictic random normed spaces

Mathematics Subject Classification (2000) Primary 54E40; Secondary 39B82, 46S50, 46S40

Keywords: Stability, Random normed space, Fixed point, Generalized Hyers-Ulam sta-bility, Additive-cubic-quartic functional equation, Lattice, non-Archimedean normed spaces

1 Introduction

Probability theory is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering It has also very useful applications in various fields, e.g., population dynamics, chaos control, computer pro-gramming, nonlinear dynamical systems, nonlinear operators, statistical convergence and others The random topology proves to be a very useful tool to deal with such situations where the use of classical theories breaks down The usual uncertainty prin-ciple of Werner Heisenberg leads to a generalized uncertainty prinprin-ciple, which has been motivated by string theory and non-commutative geometry In strong quantum gravity, regime space-time points are determined in a random manner Thus, impossi-bility of determining the position of particles gives the space-time a random structure Because of this random structure, position space representation of quantum mechanics breaks down and so a generalized normed space of quasi-position eigenfunction is required Hence one needs to discuss on a new family of random norms There are many situations where the norm of a vector is not possible to be found and the con-cept of random norm seems to be more suitable in such cases, i.e., we can deal with such situations by modeling the inexactness through the random norm

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 gener-alized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by

© 2011 Cho and Saadati; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in

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 Găvruta [5] by replacing the unbounded Cauchy difference

by a general control function in the spirit of Rassias approach

The stability problems of several functional equations have been extensively investi-gated by a number of authors and there are many interesting results concerning this

problem (see [4,6-27])

In [28,29], Jun and Kim considered the following cubic functional equation

f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x). (2)

It is easy to show that the function f(x) = x3 satisfies the functional equation (2), which is called a cubic functional equation and every solution of the cubic functional

equation is said to be a cubic mapping

In [8], Lee et al considered the following quartic functional equation

f (2x + y) + f (2x − y) = 4f (x + y) + 4f (x − y) + 24f (x) − 6f (y). (3)

It is easy to show that the function f(x) = x4 satisfies the functional equation (3), which is called a quartic functional equation and every solution of the quartic

func-tional equation is said to be a quartic mapping

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

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

(2) d(x, y) = d(y, x) for all x, yÎ X;

(3) d(x, z)≤ d(x, y) + d(y, z) for all x, y, z Î X

We recall a fundamental result in fixed point theory

Theorem 1.1 [30,31]Let (X, d) be a complete generalized metric space and J : X ® X

be a strictly contractive mapping with Lipschitz constant L < 1 Then, for any x Î X,

either

d(J n x, J n+1 x) =∞

for all nonnegative integers n or there exists a positive integer n0such 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 setY = {y ∈ X|d(J n0x, y) < ∞}; (4)d(y, y∗)≤ 1

1−Ld(y, Jy)for all yÎ Y

In 1996, Isac and Rassias [32] were the first to provide applications of stability theory

of functional equations for the proof of new fixed point theorems with applications

Using fixed point methods, the stability problems of several functional equations have

been extensively investigated by a number of authors (see [33-38])

2 Preliminaries

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 the appropriate tools to study the

geo-metry 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 by

Alsina [39], Mirmostafaee, Mirzavaziri and Moslehian [40,35], Miheţ, and Radu [41],

Miheţ, et al [42,43], Baktash et al [44], Najati [45] and Saadati et al [24]

LetL = (L, ≥ L)be a complete lattice, i.e., a partially ordered set in which every none-mpty subset admits supremum and infimum and0 = infL,1 = supL The space of

latticetic random distribution functions, denoted by+

L, is defined as the set of all map-pings F : ℝ ∪ {-∞, +∞} ® L such that F is left continuous, non-decreasing on ℝ and

The subspace D+

L ⊆ +

L is defined as D+L ={F ∈ +

denotes the left limit of the function f at the point x The space+

Lis partially ordered

by the usual point-wise ordering of functions, i.e., F ≥ G if and only if F(t) ≥LG(t) for

all t in ℝ The maximal element for+

Lin this order is the distribution function given by

ε0(t) =



1 , if t > 0.

Definition 2.1 [46] A triangular norm (t-norm) on L is a mapping T : (L)2→ L

satisfying the following conditions:

(3)(∀(x, y, z) ∈ (L)3)(T (x, T (y, z)) = T (T (x, y), z))(: associativity);

(4) (∀(x, x’, y, y’) Î (L)4

)(x≤Lx’ andy≤Ly ⇒T (x, y)≤ L T (x , y))(: monotonicity)

Let {xn} be a sequence in L converges to xÎ L (equipped the order topology) The t-normT is called a continuous t-norm if

lim

n→∞T (x n, y) = T (x, y),

for any yÎ L

A t-norm T can be extended (by associativity) in a unique way to an n-array opera-tion taking for (x1, , xn)Î Ln

the valueT (x1, , x n)defined by

T0

i=1 x i= 1, T n

i=1 x i=T (T n−1

i=1 x i, xn) = T (x1, , x n).

The t-norm T can also be extended to a countable operation taking, for any sequence {xn} in L, the value

T∞

i=1 x i= lim

n→∞T n

The limit on the right side of (4) exists since the sequence(T n

i=1 x i)n∈Nis non-increas-ing and bounded from below

Note that we putT = T whenever L = [0, 1] If T is a t-norm then, for all xÎ [0, 1]

and n Î N ∪ {0}, x (n) T is defined by 1 if n = 0 andT(x (n T−1), x)if n≥ 1 A t-norm T is

said to be of Hadžić-type (we denote byT∈H) if the family(x (n) T )n∈Nis

equicontinu-ous at x = 1 (see [47])

Definition 2.2 [46] A continuous t-normT on L = [0, 1]2is said to be continuous t-representable if there exist a continuous t-norm * and a continuous t-conorm ◇ on

[0, 1] such that, for all x = (x1, x2), y = (y1, y2)Î L,

T (x, y) = (x1∗ y1, x2♦y2)

For example,

T (a, b) = (a1b1, min{a2+ b2, 1})

and

M(a, b) = (min {a1, b1}, max{a2, b2})

for all a = (a1, a2), b = (b1, b2)Î [0, 1]2

are continuous t-representable

Define the mappingT∧from L2 to L by

T∧(x, y) =



x, if y≥Lx,

y, if x≥Ly.

Recall (see [47,48]) that, if {xn} is a given sequence in L, then(T∧)n i=1 x iis defined recurrently by(T∧)1

i=1 x i = x1and(T∧)n i=1 x i=T∧((T∧)n i=1−1x i, xn)for all n≥ 2

A negation onLis any decreasing mapping N : L → Lsatisfying N (0 L) = 1Land

the following,Lis endowed with a (fixed) negation N

Definition 2.3 A latticetic random normed space is a triple(X, μ, T∧), where X is a vector space andμ is a mapping from X intoD+

Lsatisfying the following conditions:

(LRN1)μx(t) =ε0(t) for all t > 0 if and only if x = 0;

(LRN2)μ αx (t) = μ x



t

|α|



for all x in X,a ≠ 0 and t ≥ 0;

(LRN3)μ x+y(t + s)≥L T∧(μ x(t), μ y(s))for all x, yÎ X and t, s ≥ 0

We note that, from (LPN2), it follows μ-x(t) =μx(t) for all xÎ X and t ≥ 0

Example 2.4 Let L = [0, 1] × [0, 1] and an operation ≤Lbe defined by

L = {(a1, a2) : (a1, a2)∈ [0, 1] × [0, 1] and a1+ a2≤ 1},

(a1, a2)≤L(b1, b2)⇔ a1≤ b1, a2≥ b2, ∀a = (a1, a2), b = (b1, b2)∈ L.

Then (L, ≤L) is a complete lattice (see [46]) In this complete lattice, we denote its units by 0L = (0, 1) and 1L = (1, 0) Let (X, ||·||) be a normed space Let

T (a, b) = (min{a1, b1}, max{a2, b2})for all a = (a1, a2), b = (b1, b2)Î [0, 1] × [0, 1]

and μ be a mapping defined by

μ x(t) =



t

t + ||x||,

||x||

t + ||x||



Then,(X, μ, T )is a latticetic random normed spaces

If(X, μ, T∧)is a latticetic random normed space, then we have

V = {V(ε, λ) : ε> L0 ,λ ∈ L\{0 L, 1L}

is a complete system of neighborhoods of null vector for a linear topology on X gen-erated by the norm F, where

V( ε, λ) = {x ∈ X : F x(ε) > L N (λ)}.

Definition 2.5 Let(X, μ, T∧)be a latticetic random normed spaces

(1) A sequence {xn} in X is said to be convergent to a point x Î X if, for any t > 0 andε ∈ L\{0 L}, there exists a positive integer N such thatμ x n −x (t) > L N (ε)for all n≥

N

(2) A sequence {xn} in X is called a Cauchy sequence if, for any t > 0 andε ∈ L\{0 L}, there exists a positive integer N such thatμ x n −x m (t) > L N (ε)for all n≥ m ≥ N

(3) A latticetic random normed space(X, μ, T∧)is said to be complete if every Cau-chy sequence in X is convergent to a point in X

Theorem 2.6 If(X, μ, T∧)is a latticetic random normed space and{xn} is a sequence such that xn® x, thenlimn→∞μ x n (t) = μ x (t)

Proof The proof is the same as classical random normed spaces (see [49]).□ Lemma 2.7 Let(X, μ, T∧)be a latticetic random normed space and xÎ X If

μ x(t) = C, ∀t > 0,

thenC= 1 Land x= 0

Proof Letμx(t) = C for all t > 0 Since Ran(μ) ⊆ D+

L, we haveC= 1 Land, by (LRN1),

we conclude that x = 0.□

3 Non-Archimedean Lattictic random normed space

By a non-Archimedean field, we mean a fieldKequipped with a function (valuation) | ·

| from K into [0, ∞) such that |r| = 0 if and only if r = 0, |rs| = |r| |s| and |r + s| ≤

max{|r|, |s|} for allr, s∈K Clearly, |1| = | - 1| = 1 and |n| ≤ 1 for all n Î N By the

trivial valuationwe mean the mapping | · | taking everything but 0 into 1 and |0| = 0

LetX be a vector space over a fieldKwith a non-Archimedean non-trivial valuation

| · | A function|| · || :X → [0, ∞)is called a non-Archimedean norm, if it satisfies the

following conditions:

(1) ||x|| = 0 if and only if x = 0;

(2) for anyr∈K,x∈X, ||rx|| = |r| ||x||;

(3) the strong triangle inequality (ultrametric), i.e.,

Then(X , || · ||)is called a non-Archimedean normed space

Due to the fact that

a sequence {xn} is a Cauchy sequence if and only if {xn+1- xn} 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 [50] discovered the p-adic numbers as a number theoretical analo-gue of power series in complex analysis Fix a prime number p For any nonzero

rational number x, there exists a unique integer nx Î ℤ such that x = a b p n x, where a

and b are integers not divisible by p Then,|x|p := p −n x defines a non-Archimedean

norm on Q The completion ofQwith respect to the metric d(x, y) = |x - y|p is

denoted byQp, which is called the p-adic number field

Throughout the paper, we assume thatX is a vector space andYis a complete non-Archimedean normed space

Definition 3.1 A Archimedean lattictic random normed space (briefly, non-Archimedean LRN-space) is a triple(X , μ, T ), where X is a linear space over a

non-Archimedean fieldK,T is a continuous t-norm and is μ is a mapping fromX intoD+

L

satisfying the following conditions hold:

(NA-LRN1)μx(t) =ε0(t) for all t > 0 if and only if x = 0;

(NA-LRN2)μ αx (t) = μ x



t

|α|



for allx∈X, t > 0,a ≠ 0;

(NA-LRN3)μ x+y(max {t, s})≥L T (μ x(t), μ y(s))for allx, y, z∈X and t, s ≥ 0

It is easy to see that, if (NA-LRN3) holds, then we have (RN3)μ x+y(t + s)≥LT (μ x(t), μ y(s))

As a classical example, if(X , ||.||)is a non-Archimedean normed linear space, then the triple(X , μ, T ), where L = [0, 1],T = minand

μ x(t) =



1, t > ||x||,

is a non-Archimedean LRN-space

Example 3.2 Let(X , ||.||)be is a non-Archimedean normed linear space in which L

= [0, 1] Define

t + ||x||, ∀x ∈ X , t > 0.

Then(X , μ, min)is a non-Archimedean RN-space

Definition 3.3 Let(X , μ, T )be a non-Archimedean LRN-space and {xn} be a sequence inX

(1) The sequence {xn} is said to be convergent if there exists x∈X such that

lim

n→∞μ x n −x (t) = 1 L

for all t > 0 In that case, x is called the limit of the sequence {xn}

(2) The sequence {xn} inX is called a Cauchy sequence if, for anyε ∈ L\{0 L}and t >

0, there exists a poisitve integer n0 such that, for all n ≥ n0 and p > 0,

μ x n+p −x n (t) > L N (ε)

(3) If every Cauchy sequence is convergent, then the random norm is said to be com-plete and the non-Archimedean RN-space is called a non-Archimedean random

Banach space

Remark 3.4 [51] Let(X , μ, T∧)be a non-Archimedean LRN-space Then, we have

μ x n+p −x n (t)≥LT∧{μx n+j+1 −x n+j (t) : j = 0, 1, 2, , p− 1}

Thus the sequence {xn} is Cauchy sequence if, for anyε ∈ L\{0 L}and t > 0, there exists a positive integer n0such that, for all n≥ n0,

μ x n+1 −x n (t) > L N (ε).

4 Generalized Ulam-Hyers stability for functional equation (1): an odd case

in non-Archimedean LRN-spaces

Let Kbe a non-Archimedean field,X be a vector space overKand(Y, μ, T)be a

non-Archimedean random Banach space over KIn this section, we investigate the stability

of the functional equation (1): an odd case where f is a mapping fromKtoY

LetΨ be a distribution function onX × X toD+

L (Ψ(x, y, t) denoted by Ψx,y(t) such that

 cx,cx(t)≥L x,x



t

|c|

 , ∀x ∈ X , c = 0.

Definition 4.1 A mapping f : X → Y is said to beΨ-approximately mixed ACQ if

We assume that 2 ≠ 0 inK(i.e., the characteristic ofKis not 2) Our main result, in this section, is as follows:

Theorem 4.2 Let Kbe a non-Archimedean field, Xbe a vector space over Kand

(Y, μ, T)be a non-Archimedean complete LRN-space overKLet f : X → Ybe an odd

andΨ-approximately mixed ACQ mapping If, for some a Î ℝ, a > 0, and some integer

k, k > 3 with |2k| <a,

and

lim

n→∞T

∞

j=n M



x, α j t

|2|kj



then there exists a unique cubic mappingC : X → Ysuch that

μ f (x) −C(x) (t)≥LT∞

i=1 M



x, α i+1 t

|2|ki



where

M(x, t) := T(  x,0(t),  2x,0 (t), , 2k−1x,0 (t)), ∀x ∈ X , t > 0.

Proof First, by induction on j, we show that for anyx∈X, t > 0 and j≥ 2,

μ f (4 j

x)−256j f (x) (t) ≥ Mj(x, t) := T( (x, 0, t), , (4 j−1x, 0, t)). (9) Putting y = 0 in (5), we obtain

μ f (4x) −256f (x) (t) ≥ (x, 0, t), ∀x ∈ X , t > 0.

This proves (9) for j = 2 Assume that (9) holds for some j≥ 2 Replacing y by 0 and

xby 4jxin (5), we get

μ f (4 j+1 x) −256f (4 j x) (t) ≥ (4 j x, 0, t), ∀x ∈ X , t > 0.

Since |256|≤ 1, we have

μ f (4 j+1 x)−256j+1 f (x) (t) ≥ T(μf (4 j+1 x) −256f (4 j x) (t), μ 256f (4 j x)−256j+1 f (x) (t))

= T



μ f (4 j+1 x) −256f (4 j x) (t), μ f (4 j x)−256j f (x)



t

|256|



≥ T(μ f (4 j+1

x) −256f (4 j

x) (t), μ f (4 j

x)−256j f (x) (t))

≥ T((4 j x, 0, t), M j (x, t))

Thus (9) holds for all j≥ 2 In particular,

Replacing x by 4-(kn+k)xin (10) and using inequality (6), we obtain

μ f x

4kn



−256k f x

4kn+k

(t) ≥ M  x

4kn+k , t

(11)

Then, we have

μ

(44k)n f



x

(4k)n



−(44k)n+1 f



x

(4k)n+1

(t) ≥ M



x, α n+1

|(44k)n|t

 , ∀x ∈ X , t > 0, n ≥ 0,

and so

μ

(44k)n f



x

(4k)n



−(44k)n+p f



x

(4k)n+p

(t)

j=n

⎛

⎜

⎝μ

(44k)j f



x

(4k)j



−(44k

)j+p f



x

(4k)j+p

(t)

⎞

⎟

⎠

j=n M



x, α j+1

|(44k

)j|t



j=n M



x, α j+1

|(4k

)j|t

 , ∀x ∈ X , t > 0, n ≥ 0.

Sincelimn→∞T∞j=n M



x, α j+1

|(4k)j|t



= 1for allx∈X and t > 0,

 (44k)n f

x

(4k)n



is a Cau-chy sequence in the non-Archimedean random Banach space(Y, μ, T) Hence we can

define a mappingQ : X → Ysuch that

lim

n→∞μ

(44k)n f



x

(4k)n



−Q(x) (t) = 1, ∀x ∈ X , t > 0. (12)

Next, for all n≥ 1,x∈X and t > 0, we have

μ

f (x)−(44k

)n f



x

(4k)n

(t) = μn−1

i=0 (44k)i f



x

(4k)i



−(44k

)i+1 f



x

(4k)i+1

(t)

≥ Tn−1

i=0



μ

(44k)i f



x

(4k)i



−(44k

)i+1 f



x

(4k)i+1

(t)



≥ Tn−1



x, α i+1 t

|44k|i



Therefore, it follows that

μ f (x) −Q(x) (t) ≥ T



μ

f (x)−(44k)n f



x

(4k)n

(t), μ

(44k)n f



x

(4k)n



−Q(x) (t)



≥ T



Tn i=0−1M



x, α i+1 t

|44k|i

 ,μ

(44k)n f



x

(4k)n



−Q(x) (t)



By letting n® ∞, we obtain

μ f (x) −Q(x) (t)≥ T∞

i=1 M



x, α i+1 t

|4k|i

 ,

which proves (8) Since T is continuous, from a well-known result in probabilistic metric space (see [49], Chapter 12), it follows that

lim

n→∞μ 1(x,y,k) (t) = μ 2(x,y) (t), ∀x, y ∈ X , t > 0,

for almost all t > 0., where

1(x, y, k) =(4 k)n · 16f (4 −kn (x + 4y)) + (4 k)n f (4 −kn (4x − y))

− 306[(4k)n · 9f (4 −kn (x + y

k)n f (4 −kn (x + 2y))]

− 136(4k)n f (4 −kn (x − y)) + 1394(4 k)n f (4 −kn (x + y))

− 425(4k)n f (4 −kn y) + 1530(4 k)n f (4 −kn x)

and

x + y

3



− 136Q(x − y) + 1394Q(x + y) − 425Q(y) + 1530Q(x).

On the other hand, replacing x, y by 4-knx, 4-kny, respectively, in (5) and using (NA-RN2) and (6), we get

μ 1(x,y,k) (t) ≥ 



4−kn x, 4 −kn y, t

|4k|n



≥ 



x, y, α n t

|4k|n



Sincelimn→∞x, y, α n t

|4k|n



= 1, it follows that Q is a quartic mapping

IfQ :X → Y is another quartic mapping such that μQ ’(x)-f(x)(t) ≥ M(x, t) for all

x∈X and t > 0, then, for all nÎ N,x∈X and t > 0,

μ Q(x) −Q (x) (t) ≥ T



μ Q(x)−(44k)n f



x

(4k)n

(t), μ

(44k)n f



x

(4k)n



−Q (x) (t), t)



Therefore, by (12), we conclude that Q = Q’ This completes the proof □ Corollary 4.3 Let Kbe a non-Archimedean field, Xbe a vector space over Kand

(Y, μ, T)be a non-Archimedean random Banach space over Kunder a t-normT∈H

Let f : X → Ybe aΨ-approximately quartic mapping If, for some a Î ℝ, a > 0, and

some integer k, k> 3, with |4k| <a

(4 −k x, 4 −k y, t) ≥ (x, y, αt), ∀x ∈ X , t > 0,

then there exists a unique quartic mappingQ : X → Ysuch that

μ f (x) −Q(x) (t)≥ T∞

i=1 M



x, α i+1 t

|4|ki

 , ∀x ∈ X , t > 0,

where

M(x, t) := T((x, 0, t), (4x, 0, t), , (4 k−1x, 0, t)), ∀x ∈ X , t > 0.

Proof Since

lim



x, α j t

|4|kj



= 1, ∀x ∈ X , t > 0,

and T is of Hadžić type, it follows that

lim

n→∞T∞j=n M



x, α j t

|4|kj



= 1, ∀x ∈ X , t > 0.

Now, if we apply Theorem 4.2, we get the conclusion.□ Example 4.4 Let(X , μ, T M)non-Archimedean random normed space in which

t + ||x||, ∀x ∈ X , t > 0,

and(Y, μ, T M)a complete non-Archimedean random normed space (see Example 3.2) Define

(x, y, t) = t

1 + t.

It is easy to see that (6) holds for a = 1 Also, since

M(x, t) = t

1 + t,

we have

lim

n→∞T

∞

M,j=n M



x, α j t

|4|kj



= lim

n→∞

 lim

m M,j=n M



x, t

|4|kj



= lim

n→∞mlim→∞



t

t +|4k|n



= 1, ∀x ∈ X , t > 0.