hyers ulam rassias rns approximation of euler lagrange type additive mappings

Volume 2012, Article ID 672531, 15 pagesdoi:10.1155/2012/672531 Research Article Hyers-Ulam-Rassias RNS Approximation of Euler-Lagrange-Type Additive Mappings 1 Department of Mathematics

Volume 2012, Article ID 672531, 15 pages

doi:10.1155/2012/672531

Research Article

Hyers-Ulam-Rassias RNS Approximation of

Euler-Lagrange-Type Additive Mappings

1 Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran

2 Department of Mathematics, Payame Noor University, Tehran, Iran

3 Department of Mathematics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran

Correspondence should be addressed to A Ebadian,ebadian.ali@gmail.com

Received 24 December 2011; Revised 5 March 2012; Accepted 19 March 2012

Academic Editor: Tadeusz Kaczorek

Copyrightq 2012 H Azadi Kenary 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

Recently the generalized Hyers-Ulamor Hyers-Ulam-Rassias stability of the following functional equationm

j1 f−r j x j1≤i≤m,i / j r i x i2m

i1 r i fx i   mfm

i1 r i x i  where r1, , r m∈R, proved

in Banach modules over a unitalC∗-algebra It was shown that ifm

i1 r i / 0, r i , r j / 0 for some

1 ≤ i < j ≤ m and a mapping f : X → Y satisfies the above mentioned functional equation

then the mappingf : X → Y is Cauchy additive In this paper we prove the Hyers-Ulam-Rassias

stability of the above mentioned functional equation in random normed spacesbriefly RNS

1 Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam1 concern-ing the stability of group homomorphisms Hyers2 gave a first affirmative partial answer to the question of Ulam for Banach spaces Hyers’ Theorem was generalized by Aoki3 for additive mappings and by Rassias 4 for linear mappings by considering an unbounded Cauchy difference

The paper of Rassias has provided a lot of influence in the development of what we

call the generalized Hyers-Ulam stability of functional equations In 1994, a generalization of

Rassias’ theorem was obtained by G˘avrut¸a5 by replacing the bound x p  y p by a general control functionϕx, y.

The functional equation:

fx  y fx − y 2fx  2fy, 1.1

is called a quadratic functional equation In particular, every solution of the quadratic functional equation is said to be a quadratic mapping The generalized Hyers-Ulam stability problem for

the quadratic functional equation was proved by Skof6 for mappings f : X → Y, where

X is a normed space and Y is a Banach space Cholewa 7 noticed that the theorem of Skof

is still true if the relevant domainX is replaced by an Abelian group Czerwik 8 proved the generalized Hyers-Ulam stability of the quadratic functional equation

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 2,4,5,9 28

In the sequel, we will adopt the usual terminology, notions, and conventions of the theory of random normed spaces as in 29 Throughout this paper, the spaces of all probability distribution functions are denoted by Δ Elements of Δ are functions F :

R ∪ {−∞, ∞} → 0, 1, such that F is left continuous and nondecreasing on R, F0  0

and F∞  1 It’s clear that the subset D  {F ∈ Δ : l−F∞  1}, where l−fx 

limt → x−ft, is a subset of Δ The space Δ is partially ordered by the usual point-wise ordering of functions, that is, for allt ∈ R, F ≤ G if and only if Ft ≤ Gt For every a ≥ 0,

H a t is the element of Ddefined by

H a t 

⎧

⎨

⎩

0 if t ≤ a

1 if t > a. 1.2

One can easily show that the maximal element forΔin this order is the distribution function

H0t.

Definition 1.1 A function T : 0, 12 → 0, 1 is a continuous triangular norm briefly a

t-norm if T satisfies the following conditions:

i T is commutative and associative;

ii T is continuous;

iii Tx, 1  x for all x ∈ 0, 1;

iv Tx, y ≤ Tz, w whenever x ≤ z and y ≤ w for all x, y, z, w ∈ 0, 1.

Three typical examples of continuoust-norms are T P x, y  xy, Tmaxx, y  max{a

b − 1, 0}, and T M x, y  mina, b Recall that, if T is a t-norm and {x n} is a given of numbers

in0, 1, T n

i1 x iis defined recursively byT1

i1 x1andT n

i1 x i  TT n−1

i1 x i , x n  for n ≥ 2.

Definition 1.2 A random normed space briefly RNS is a triple X, μ , T, where X is a vector

space, T is a continuous t-norm, and μ : X → D is a mapping such that the following conditions hold

i μ x t  H0t for all t > 0 if and only if x  0.

ii μ αx t  μ x t/|α| for all α ∈ R, α / 0, x ∈ X and t ≥ 0.

iii μ xy t  s ≥ Tμ x t, μ y s, for all x, y ∈ X and t, s ≥ 0.

Definition 1.3 Let X, μ , T be an RNS.

i A sequence {x n } in X is said to be convergent to x ∈ X in X if for all t > 0,

limn → ∞ μ xn −x t  1

ii A sequence {x n } in X is said to be Cauchy sequence in X if for all t > 0,

limn → ∞ μ xn −x m t  1.

iii The RN-space X, μ , T is said to be complete if every Cauchy sequence in X is

convergent

Theorem 1.4 If X, μ , T is RNS and {x n } is a sequence such that x n → x, then lim n → ∞ μ xn t 

μ x t.

In this paper, we investigate the generalized Hyers-Ulam stability of the following additive functional equation of Euler-Lagrange type:

m



j1

f

⎛

⎝−r j x j 

1≤i≤m, i / j

r i x i

⎞

⎠  2m

i1

r i fx i   mf m

i1

r i x i



where r1, , r n ∈ R,m i1 r i / 0, and r i , r j / 0 for some 1 ≤ i < j ≤ m, in random normed

spaces

Every solution of the functional equation1.3 is said to be a generalized Euler-Lagrange

type additive mapping.

Remark 2.1 Throughout this paper, r1, , r mwill be real numbers such thatr i , r j / 0 for fixed

1≤ i < j ≤ m.

Theorem 2.2 Let X be a real linear space, Z, μ , min be an RN space, ϕ : X n → Z be a function

such that for some 0 < α < 2,

μ ϕ2x1, ,2xmt ≥ μ αϕx1, ,xmt ∀x i ∈ X, t > 0. 2.1

f0  0 and for all x i ∈ X and t > 0

lim

n → ∞ μϕ2 n x

Let Y, μ, min be a complete RN space If f : x → Y is a mapping such that for all x i , x j ∈ X

and t > 0

μm

j1 f−rjxj1≤i≤m, i / jrixi2m

i1 rifxi −mfm

i1 rixit ≥ μ ϕx , ,xmt 2.3

then there is a unique generalized Euler-Lagrange-type additive mapping EL : X → Y such that, for all x ∈ X and all t > 0

μELx−fxt ≥ T M



T M



μ ϕi,j x/2r i,−x/2rj

2−αt 6



, μ ϕi,j x/2r i,0

2−αt 6



,

μ ϕi,j 0,−x/2r j

2−αt 6



, T M



μ ϕi,j x/r i, x/rj

2−αt 3



,

μ ϕi,j x/r i,0

2−αt 3



,

μ ϕi,j 0,x/r j

2 − αt 3



.

2.4

Proof For each 1 ≤ k ≤ m with k / i, j, let x k 0 in 2.3 Then we get the following inequality:

for allx i , x j ∈ X, where

ϕ i,j

x, y: ϕ

⎛

⎜

⎝0, , 0, x

ith

, 0, , 0, y

jth

, 0, , 0

⎞

⎟

for allx, y ∈ X and all 1 ≤ i < j ≤ m, and

λx i , x j f−r i x i  r j x j fr i x i − r j x j− 2fr i x i  r j x j 2r i fx i   2r j fx j. 2.7 Lettingx i 0 in 2.5, we get

μ f−rjxj −fr jxj 2r jfxjt ≥ μ ϕi,j 0,x jt, 2.8 for allx j ∈ X Similarly, letting x j 0 in 2.5, we get

μ f−rixi −fr ixi 2r ifxit ≥ μ ϕi,j x i,0 t, 2.9

for allx i ∈ X It follows from 2.5, 2.8, and 2.9 that for all x i , x j ∈ X

μ λxi,xj −f−r ixi −fr ixi 2r ifxi −f−r jxj −fr jxj 2r jfxjt

≥ T M



μ ϕi,j x i,xj

t 3



, μ ϕ i,j x i , 0

t 3



, μ ϕ i,j

0, x j  t

3



Replacingx iandx jbyx/r i  and y/r j in 2.10, we get that

μ f−xyfx−y−2fxyfxfy−f−x−f−y t

≥ T Mμ ϕi,j x/r i,y/rjt

3



, μ ϕi,j x/r i,0t

3



, μ ϕi,j 0,y/r jt

3



for allx, y ∈ X Putting y  x in 2.11, we get

μ2fx−2f−x−2f2x t ≥ T M



μ ϕi,j x/r i, x/rj

t 3



, μ ϕi,j x/r i,0

t 3



, μ ϕi,j 0,x/r j

t 3



,

2.12 for allx ∈ X Replacing x and y by x/2 and −x/2 in 2.11, respectively, we get

μ fxf−x t ≥ T M



μ ϕi,j x/2r i ,−x/2rj

t 3



, μ ϕi,j x/2r i,0

t 3



, μ ϕi,j 0,−x/2r j

t 3



,

2.13 for allx ∈ X It follows from 2.12 and 2.13 that

μ f2x−2fx t  μ fxf−x2fx−2f−x−2f2x/2 t

≥ T Mμ fxf−xt

2



, μ2fx−2f−x−2f2x t



≥ T M



T M



μ ϕi,j x/2r i,−x/2rj

t 6



, μ ϕi,j x/2r i,0

t 6



, μ ϕi,j 0,−x/2r j

t 6



,

T M



μ ϕi,j x/r i,x/rj

t 3



, μ ϕi,j x/r i,0

t 3



, μ ϕi,j 0,x/r j

t 3



,

2.14 for allx ∈ X So

μ f2x/2−fx t ≥ T M



T M



μ ϕi,j x/2r i,−x/2rj

t 3



, μ ϕi,j x/2r i,0

t 3



,

μ ϕi,j 0,−x/2r j

t 3



,

T M



μ ϕi,j x/r i,x/rj



2t

3



, μ ϕi,j x/r i,0



2t

3



, μ ϕi,j 0,x/r j



2t

3



.

2.15

Replacingx by 2 n x in 2.15 and using 2.1, we get

μ f2 n1 x/2 n1 −f2 n x/2 nt

≥ T M



T M



μ ϕi,j2n x/2ri ,−2 n x/2rj



2n t

3



, μ ϕi,j2n x/2ri ,0



2n t

3



, μ ϕi,j 0,−2 n x/2rj



2n t

3



,

T M μ ϕi,j2n x/ri ,2 n x/rj

2n1 t

3



, μ ϕi,j2n x/ri ,0

2n1 t

3



,

μ ϕi,j 0,2 n x/rj

2n1 t

3



≥ T M



T M



μ ϕi,j x/2r i,−x/2rj



2n t

3α n



, μ ϕi,j x/2r i,0



2n t

3α n



, μ ϕi,j 0,−x/2r j



2n t

3α n



,

T M μ ϕi,j x/r i,x/rj 2n1 t

3α n



, μ ϕi,j x/r i,0 2n1 t

3α n



, μ ϕi,j 0,x/r j 2n1 t

3α n



,

2.16

for allx ∈ X and all n ∈ N Therefore, we have

μ f2 n x/2 n −fx

n−1



k0

α k t

2k



 μn−1 f2 k1 x/2 k1 −f2 k x/2 k n−1

k0

α k t

2k



≥ T n−1

k0 μ f2 k1 x/2 k1 −f2 k x/2 k α k t

2k



≥ T n−1

k0



T M



T M



μ ϕi,j x/2r i,−x/2rj

t 3



, μ ϕi,j x/2r i,0

t 3



, μ ϕi,j 0,−x/2r j

t 3



,

T M



μ ϕi,j x/r i,x/rj



2t

3



, μ ϕi,j x/r i,0



2t

3



, μ ϕi,j 0,x/r j



2t

3



 T M



T M



μ ϕi,j x/2r i,−x/2rj

t 3



, μ ϕi,j x/2r i,0

t 3



, μ ϕi,j 0,−x/2r j

t 3



,

T M



μ ϕi,j x/r i,x/rj



2t

3



, μ ϕi,j x/r i,0



2t

3



, μ ϕi,j 0,x/r j



2t

3



,

2.17

for allx ∈ X This implies that

μ f2 n x/2 n −fx t

≥ T M T M μ ϕi,j x/2r i,−x/2rj t

3n−1

k0



α k /2 k



, μ ϕi,j x/2r i,0 t

3n−1

k0 α k /2 k



,

μ ϕi,j 0,−x/2r j t

3n−1

k0



α k /2 k



,

3n−1

k0



α k /2 k



, μ ϕi,j x/r i,0 2t

3n−1

k0



α k /2 k



,

μ ϕi,j 0,x/r j 2t

3n−1

k0



α k /2 k



.

2.18

Replacingx by 2 p x in 2.18, we obtain

μ f2 np x/2 np −f2 p x/2 pt

≥ T M

⎛

⎝T M

⎛

⎝μ ϕi,j x/2r i,−x/2rj

⎛

3pn−1 kp



α k /2 k

⎞

⎠, μ ϕi,j x/2r i,0

⎛

3pn−1 kp



α k /2 k

⎞

⎠,

μ ϕi,j 0,−x/2r j

⎛

3pn−1

kp



α k /2 k

⎞

⎠

⎞

⎠,

T M

⎛

⎝μ ϕi,j x/r i,x/rj

⎛

3pn−1

kp



α k /2 k

⎞

⎠,

μ ϕi,j x/r i,0

⎛

3pn−1 kp



α k /2 k

⎞

⎠ μ ϕi,j 0,x/r j

⎛

3pn−1 kp



α k /2 k

⎞

⎠

⎞

⎠

⎞

⎠

2.19

Since the right-hand side of the above inequality tends to 1, whenp, n → ∞, then

the sequence{f2 k x/2 k}∞n1is a Cauchy sequence in complete RN spaceY, μ, min, so there

exists some point ELx ∈ Y such that

ELx  lim

n → ∞

f2k x

for allx ∈ X.

Fixx ∈ X and put P  0 in 2.19 Then we obtain

μ f2 n x/2 n −fx t

≥ T M T M μ ϕi,j x/2r i,−x/2rj t

3n−1

k0



α k /2 k



, μ ϕi,j x/2r i,0 t

3n−1

k0



α k /2 k



,

μ ϕi,j 0,−x/2r j t

3n−1

k0



α k /2 k



,

3n−1

k0



α k /2 k



, μ ϕi,j x/r i,0 2t

3n−1

k0



α k /2 k



,

μ ϕi,j 0,x/r j 2t

3n−1

k0



α k /2 k



,

2.21

and so, for every > 0, we have

μELx−fxt   ≥ TμELx−f2n x/2 n, μ f2 n x/2 n −fx t

≥ T μELx−f2n x/2 n, T M T M μ ϕi,j x/2r i,−x/2rj t

3n−1

k0



α k /2 k



,

μ ϕi,j x/2r i,0 t

3n−1

k0



α k /2 k



μ ϕi,j 0,−x/2r j t

3n−1

k0



α k /2 k



,

3n−1

k0



α k /2 k



,

μ ϕi,j x/r i,0 2t

3n−1

k0



α k /2 k



,

μ ϕi,j 0,x/r j 2t

3n−1

k0



α k /2 k



.

2.22

Taking the limit asn → ∞ and using 2.22, we get

μELx−fxt  

≥ TM



T M



μ ϕi,j x/2r i,−x/2rj

2 − αt 6



, μ ϕi,j x/2r i,0

2 − αt 6



,

μ ϕi,j 0,−x/2r j

2 − αt 6



, T M



μ ϕi,j x/r i,x/rj

2 − αt 3



,

μ ϕi,j x/r i,0

2 − αt 3



,

μ ϕi,j 0,x/r j

2 − αt 3



.

2.23

Since was arbitrary by taking  → 0 in 2.23, we get

μELx−fxt ≥ T M



T M



μ ϕi,j x/2r i,−x/2rj

2 − αt 6



, μ ϕi,j x/2r i,0

2 − αt 6



,

μ ϕi,j 0,−x/2r j

2 − αt 6



,

T M



μ ϕi,j x/r i,x/rj

2 − αt 3



, μ ϕi,j x/r i,0

2 − αt 3



,

μ ϕi,j 0,x/r j2 − αt

3



.

2.24

Replacingx iby 2n x ifor all 1≤ i ≤ m, in 2.3, we get for all x i , x j ∈ X and for all t > 0,

μm

j1 f−2 n rjxj1≤i≤m,i / j2n rixi2m

i1 rif2 n xi −mfm

i12n rixi /2 n t ≥ μ ϕ2 n x1, ,2 n xm /2 n t. 2.25 since

lim

We conclude that

m



j1

EL

⎛

⎝−r j x j 

1≤i≤m,i / j

r i x i

⎞

⎠  2m

i1

r iELxi  − mEL m

i1

r i x i



 0. 2.27

To prove the uniqueness of mapping EL, assume that there exists another mapping

A : X → Y which satisfies 2.4 Fix x ∈ X, clearly EL2 n x  2 n ELx and A2n x  2 n Ax,

for alln ∈ N Since μELx−Axt  lim n → ∞ μEL2n x/2 n −A2 n x/2 nt, so

μEL2n x/2 n −A2 n x/2 nt ≥ min



μEL2n x/2 n −f2 n x/2 nt

t 2



, μ f2 n x/2 n −A2 n x/2 nt

t 2



≥ T M



T M



μ ϕi,j x/2r i,−x/2rj



2n 2 − αt

12α n



, μ ϕi,j x/2r i,0



2n 2 − αt

12α n



,

μ ϕi,j 0,−x/2r j



2n 2 − αt

12α n



,

T M



μ ϕi,j x/r i,x/rj



2n 2 − αt

6α n



, μ ϕi,j x/r i,0



2n 2 − αt

6α n



,

μ ϕi,j 0,x/r j



2n 2 − αt

6α n



.

2.28

Since the right-hand side of the above inequality tends to 1, whenn → ∞, therefore, it

follows that for allt > 0, μELx−Axt  1 and so ELx  Ax This completes the proof.

Corollary 2.3 Let X be a real linear space, Z, μ , min be an RN space, and Y, μ, min a complete

RN space Let 0 < p < 1, z0∈ Z and f : X → Y be a mapping with f0  0 and satisfying

μm

j1 f−rjxj1≤i≤m, i / j rixi2m

i1 rifxi −mfm

i1 rixit ≥ μm

k1 x kp z0t, 2.29

for all x i , x j ∈ X and t > 0 Then the limit ELx  lim n → ∞ f2 n x/2 n exists for all x ∈ X and defines a unique Euler-Lagrange additive mapping EL : X → Y such that

μELx−fxt ≥ T M T M μ x p z0

2p r i r jp2 − 2p t

6

|r i|pr jp



, μ x p z0

|2r

i|p2 − 2p t

6



,

μ x p z0

2r jp2 − 2p t

6



,

T M μ x p z0

i r jp2 − 2p t

3

|r i|pr jp



, μ x p z0

|r

i|p2 − 2p t

3



,

μ x p z0

jp2 − 2p t

3



,

2.30

for all x ∈ X and t > 0.

Proof Let α  2 pandϕ : X m → Z be defined as ϕx1, , x m  m

k1 x ip z0

Corollary 2.4 Let X be a real linear space, Z, μ , min be an RN space, and Y, μ, min a complete

RN space Let z0∈ Z and f : X → Y be a mapping with f0  0 and satisfying

μm

j1 f−rjxj1≤i≤m, i / jri xi2m

i1 rifxi −mfm

i1 rixit ≥ μ δz0t, 2.31

for all x i ∈ X for all 1 ≤ i ≤ m and all t > 0 Then, the limit Cx  lim n → ∞ f2 n x/2 n  exists for

all x ∈ X and defines a unique Euler-Lagrange additive mapping EL : X → Y such that

μELx−fxt ≥ T M



μ δz0

t 6



, μ δz0

t 3



for all x ∈ X and t > 0.

Proof Let α  1 and ϕ : X m → Z be defined as ϕx1, , x m   δz0

Theorem 2.5 Let X be a real linear space, Z, μ , min be an RN space, ϕ : X m → Z be a function

such that for some 0 < α < 1/2,

μ ϕx1/2, ,xm/2 t ≥ μ αϕx1, ,xmt ∀x i ∈ X, t > 0, 2.33

f0  0 and for all x i ∈ X and t > 0, lim n → ∞ μ2n ϕx1/2 n , ,xm/2 nt  1 Let Y, μ, min be a

complete RN space If f : X → Y is a mapping satisfying 2.3, then there is a unique generalized

Euler-Lagrange-type additive mapping EL : X → Y such that, for all x ∈ X

μELx−fxt ≥ T M



T M



μ ϕi,j x/r i,−x/rj

1−2αt

6α



, μ ϕi,j x/r j,0

1−2αt

6α



,

μ ϕi,j 0,−x/r j

1−2αt

6α



,

T M



μ ϕi,j x/r i,x/rj

1−2αt

3α



,

μ ϕi,j x/r i ,01−2αt

3α



, μ ϕi,j 0,x/r j1−2αt

3α



,

2.34

for all x ∈ X and all t > 0.