Hindawi Publishing Corporation
Mathematical Problems in Engineering
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
H. Azadi Kenary,1 H. Rezaei,1 A. Ebadian,2 and A. R. Zohdi3
1
Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran
Department of Mathematics, Payame Noor University, Tehran, Iran
3
Department of Mathematics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran
2
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
Copyright q 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
generalizedHyers-Ulam or Hyers-Ulam-Rassias
m
mstability of the following functional
equation m
j ri xi 2
j1 f−rj xj 1≤i≤m,i /
i1 ri fxi mf i1 ri xi where r1 , . . . , rm ∈ R, proved
0, ri , rj /
0 for some
in Banach modules over a unital C∗ -algebra. It was shown that if m
i1 ri /
1 ≤ i < j ≤ m and a mapping f : X → Y satisfies the above mentioned functional equation
then the mapping f : X → Y is Cauchy additive. In this paper we prove the Hyers-Ulam-Rassias
stability of the above mentioned functional equation in random normed spaces briefly RNS.
1. Introduction and Preliminaries
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 generalized by Aoki 3 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 Găvruţa 5 by replacing the bound xp yp by a
general control function ϕx, y.
The functional equation:
f x y f x − y 2fx 2f y ,
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
2
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the quadratic functional equation was proved by Skof 6 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 domain X 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 investigated 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 all t ∈ R, F ≤ G if and only if Ft ≤ Gt. For every a ≥ 0,
Ha t is the element of D defined by
Ha t ⎧
⎨0
if t ≤ a
⎩1
if t > a.
1.2
One can easily show that the maximal element for Δ in this order is the distribution function
H0 t.
Definition 1.1. A function T : 0, 12 → 0, 1 is a continuous triangular norm briefly a tnorm if T satisfies the following conditions:
i T is commutative and associative;
ii T is continuous;
iii T x, 1 x for all x ∈ 0, 1;
iv T x, y ≤ T z, w whenever x ≤ z and y ≤ w for all x, y, z, w ∈ 0, 1.
Three typical examples of continuous t-norms are TP x, y xy, Tmax x, y max{a b − 1, 0}, and TM x, y mina, b. Recall that, if T is a t-norm and {xn } is a given of numbers
n−1
n
n
1
xi is defined recursively by Ti1
x1 and Ti1
xi T Ti1
xi , xn for n ≥ 2.
in 0, 1, Ti1
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 H0 t for all t > 0 if and only if x 0.
0, x ∈ X and t ≥ 0.
ii μ
αx t μ
x t/|α| for all α ∈ R, α /
iii μ
xy t s ≥ T μ
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 {xn } in X is said to be convergent to x ∈ X in X if for all t > 0,
limn → ∞ μ
xn −x t 1
Mathematical Problems in Engineering
3
ii A sequence {xn } in X is said to be Cauchy sequence in X if for all t > 0,
limn → ∞ μ
xn −xm 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 {xn } is a sequence such that xn → x, then limn → ∞ 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
f ⎝−rj xj j1
1≤i≤m, i /
j
⎞
m
ri xi ⎠ 2 ri fxi mf
i1
m
ri xi ,
1.3
i1
0, and ri , rj /
0 for some 1 ≤ i < j ≤ m, in random normed
where r1 , . . . , rn ∈ R, m
i1 ri /
spaces.
Every solution of the functional equation 1.3 is said to be a generalized Euler-Lagrange
type additive mapping.
2. RNS Approximation of Functional Equation 1.3
Remark 2.1. Throughout this paper, r1 , . . . , rm will be real numbers such that ri , rj /
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 ,...,2xm t ≥ μ
αϕx1 ,...,xm t ∀xi ∈ X, t > 0.
2.1
f0 0 and for all xi ∈ X and t > 0
lim μ
ϕ2n x ,...,2n x /2n t 1.
1
m
n→∞
2.2
Let Y, μ, min be a complete RN space. If f : x → Y is a mapping such that for all xi , xj ∈ X
and t > 0
μmj1 f−rj xj 1≤i≤m, i / j ri xi 2 mi1 ri fxi −mfmi1 ri xi t ≥ μ
ϕx1 ,...,xm t
2.3
4
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then there is a unique generalized Euler-Lagrange-type additive mapping EL : X → Y such that, for
all x ∈ X and all t > 0
2−αt
2−αt
μELx−fx t ≥ TM TM μ
ϕi,j x/2ri ,−x/2rj , μ
ϕi,j x/2ri ,0
,
6
6
2−αt
2−αt
, TM μ
ϕi,j x/ri , x/rj ,
μ
ϕi,j 0,−x/2rj 6
3
2−αt
μ ϕi,j x/ri ,0
,
3
2 − αt
μ ϕi,j 0,x/rj .
3
2.4
Proof. For each 1 ≤ k ≤ m with k /
i, j, let xk 0 in 2.3. Then we get the following inequality:
μλxi ,xj t ≥ μ
ϕi,j xi ,xj t,
2.5
for all xi , xj ∈ X, where
⎞
⎛
⎜
ϕi,j x, y : ϕ⎜
x , 0, . . . , 0,
⎝0, . . . , 0, ith
⎟
y , 0, . . . , 0⎟
⎠,
2.6
jth
for all x, y ∈ X and all 1 ≤ i < j ≤ m, and
λ xi , xj f −ri xi rj xj f ri xi − rj xj − 2f ri xi rj xj 2ri fxi 2rj f xj .
2.7
Letting xi 0 in 2.5, we get
μf−rj xj −frj xj 2rj fxj t ≥ μ
ϕi,j 0,xj t,
2.8
for all xj ∈ X. Similarly, letting xj 0 in 2.5, we get
μf−ri xi −fri xi 2ri fxi t ≥ μ
ϕi,j xi ,0 t,
2.9
for all xi ∈ X. It follows from 2.5, 2.8, and 2.9 that for all xi , xj ∈ X
μλxi ,xj −f−ri xi −fri xi 2ri fxi −f−rj xj −frj xj 2rj fxj t
t
t
t
, μ ϕi,j xi , 0
, μ ϕi,j 0, xj
.
≥ TM μ ϕi,j xi ,xj 3
3
3
2.10
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5
Replacing xi and xj by x/ri and y/rj in 2.10, we get that
μf−xyfx−y−2fxyfxfy−f−x−f−y t
t
t
t
≥ TM μ ϕi,j x/ri ,y/rj , μ ϕi,j x/ri ,0
, μ ϕi,j 0,y/rj ,
3
3
3
2.11
for all x, y ∈ X. Putting y x in 2.11, we get
t
t
t
μ2fx−2f−x−2f2x t ≥ TM μ
ϕi,j x/ri , x/rj , μ
ϕi,j x/ri ,0
, μ
ϕi,j 0,x/rj ,
3
3
3
2.12
for all x ∈ X. Replacing x and y by x/2 and −x/2 in 2.11, respectively, we get
μfxf−x t ≥ TM
t
t
t
, μ ϕi,j x/2ri ,0
, μ ϕi,j 0,−x/2rj ,
μ ϕi,j x/2ri ,−x/2rj 3
3
3
2.13
for all x ∈ X. It follows from 2.12 and 2.13 that
μf2x−2fx t μfxf−x2fx−2f−x−2f2x/2 t
t
≥ TM μfxf−x
, μ2fx−2f−x−2f2x t
2
t
t
t
, μ
ϕi,j x/2ri ,0
, μ
ϕi,j 0,−x/2rj ,
≥ TM TM μ
ϕi,j x/2ri ,−x/2rj 6
6
6
t
t
t
TM μ
ϕi,j x/ri ,x/rj , μ
ϕi,j x/ri ,0
, μ
ϕi,j 0,x/rj ,
3
3
3
2.14
for all x ∈ X. So
t
t
μf2x/2−fx t ≥ TM TM μ
ϕi,j x/2ri ,−x/2rj , μ
ϕi,j x/2ri ,0
,
3
3
t
,
3
2t
2t
2t
, μ
ϕi,j x/ri ,0
, μ
ϕi,j 0,x/rj .
TM μ
ϕi,j x/ri ,x/rj 3
3
3
2.15
μ
ϕi,j 0,−x/2rj 6
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Replacing x by 2n x in 2.15 and using 2.1, we get
μf2n1 x/2n1 −f2n x/2n t
n n n 2 t
2 t
2 t
≥ TM TM μ ϕi,j 2n x/2ri ,−2n x/2rj , μ ϕi,j 2n x/2ri ,0
, μ ϕi,j 0,−2n x/2rj ,
3
3
3
2n1 t
2n1 t
, μ ϕi,j 2n x/ri ,0
,
T M μ ϕi,j 2n x/ri ,2n x/rj 3
3
2n1 t
μ ϕi,j 0,2n x/rj 3
n n n 2 t
2 t
2 t
≥ TM TM μ
ϕi,j x/2ri ,−x/2rj ,
μ
,
μ
,
ϕi,j x/2ri ,0
ϕi,j 0,−x/2rj n
n
3α
3α
3αn
2n1 t
2n1 t
2n1 t
, μ ϕi,j x/ri ,0
, μ ϕi,j 0,x/rj ,
TM μ ϕi,j x/ri ,x/rj 3αn
3αn
3αn
2.16
for all x ∈ X and all n ∈ N. Therefore, we have
μf2n x/2n −fx
n−1 k
α t
k0
μn−1
2k
k1 x/2k1 −f2k x/2k k0 f2
n−1 k
α t
k0
2k
αk t
μf2k1 x/2k1 −f2k x/2k ≥
2k
t
t
t
n−1
≥ Tk0
TM TM μ
ϕi,j x/2ri ,−x/2rj , μ
ϕi,j x/2ri ,0
, μ
ϕi,j 0,−x/2rj ,
3
3
3
2t
2t
2t
, μ
ϕi,j x/ri ,0
, μ
ϕi,j 0,x/rj TM μ
ϕi,j x/ri ,x/rj 3
3
3
t
t
t
TM TM μ ϕi,j x/2ri ,−x/2rj , μ ϕi,j x/2ri ,0
, μ ϕi,j 0,−x/2rj ,
3
3
3
2t
2t
2t
, μ ϕi,j x/ri ,0
, μ ϕi,j 0,x/rj ,
TM μ ϕi,j x/ri ,x/rj 3
3
3
n−1
Tk0
2.17
Mathematical Problems in Engineering
7
for all x ∈ X. This implies that
μf2n x/2n −fx t
t
t
,
≥ TM TM μ ϕi,j x/2ri ,−x/2rj k k , μ ϕi,j x/2ri ,0
k
k
3 n−1
3 n−1
k0 α /2
k0 α /2
t
μ ϕi,j 0,−x/2rj ,
k k
3 n−1
k0 α /2
2t
2t
TM μ ϕi,j x/ri ,x/rj k k , μ ϕi,j x/ri ,0
k k ,
3 n−1
3 n−1
k0 α /2
k0 α /2
2t
.
μ
ϕi,j 0,x/rj n−1 k k 3 k0 α /2
2.18
Replacing x by 2p x in 2.18, we obtain
μf2np x/2np −f2p x/2p t
⎛ ⎛
⎞
⎞
⎛
t
t
≥ TM ⎝TM ⎝μ
ϕi,j x/2ri ,−x/2rj ⎝ pn−1 ⎠, μ ϕi,j x/2ri ,0 ⎝ pn−1 k k ⎠,
3 kp αk /2k
3 kp α /2
⎛
⎞⎞
t
μ
ϕi,j 0,−x/2rj ⎝ pn−1 ⎠⎠,
3 kp αk /2k
⎛
⎞
2t
T M ⎝μ
ϕi,j x/ri ,x/rj ⎝ pn−1 ⎠,
3 kp αk /2k
⎛
⎛
⎞
⎞⎞⎞
⎛
2t
2t
μ
ϕi,j x/ri ,0 ⎝ pn−1 ⎠ μ ϕi,j 0,x/rj ⎝ pn−1 k k ⎠⎠⎠
3 kp αk /2k
3 kp α /2
⎛
2.19
Since the right-hand side of the above inequality tends to 1, when p, n → ∞, then
∞
the sequence {f2k x/2k }n1 is a Cauchy sequence in complete RN space Y, μ, min, so there
exists some point ELx ∈ Y such that
ELx lim
n→∞
for all x ∈ X.
f 2k x
2k
,
2.20
8
Mathematical Problems in Engineering
Fix x ∈ X and put P 0 in 2.19. Then we obtain
μf2n x/2n −fx t
≥ TM
t
t
TM μ ϕi,j x/2ri ,−x/2rj k k , μ ϕi,j x/2ri ,0
k k ,
3 n−1
3 n−1
k0 α /2
k0 α /2
t
μ
ϕi,j 0,−x/2rj ,
n−1 k k 3 k0 α /2
2t
2t
TM μ ϕi,j x/ri ,x/rj k k , μ ϕi,j x/ri ,0
k k ,
3 n−1
3 n−1
k0 α /2
k0 α /2
2t
μ
ϕi,j 0,x/rj ,
n−1 k k 3 k0 α /2
2.21
and so, for every > 0, we have
μELx−fx t ≥ T μELx−f2n x/2n , μf2n x/2n −fx t
≥T
μELx−f2n x/2n , TM
t
TM μ ϕi,j x/2ri ,−x/2rj k k ,
3 n−1
k0 α /2
t
μ ϕi,j x/2ri ,0
k k
3 n−1
k0 α /2
t
,
μ ϕi,j 0,−x/2rj k k
3 n−1
k0 α /2
2t
TM μ ϕi,j x/ri ,x/rj k k ,
3 n−1
k0 α /2
2t
μ ϕi,j x/ri ,0
k k ,
3 n−1
k0 α /2
2t
μ ϕi,j 0,x/rj .
k k
3 n−1
k0 α /2
2.22
Mathematical Problems in Engineering
9
Taking the limit as n → ∞ and using 2.22, we get
μELx−fx t 2 − αt
2 − αt
, μ ϕi,j x/2ri ,0
,
≥ TM TM μ ϕi,j x/2ri ,−x/2rj 6
6
2 − αt
2 − αt
μ ϕi,j 0,−x/2rj , TM μ ϕi,j x/ri ,x/rj ,
6
3
2 − αt
,
μ
ϕi,j x/ri ,0
3
2 − αt
μ
ϕi,j 0,x/rj .
3
2.23
Since was arbitrary by taking → 0 in 2.23, we get
μELx−fx t ≥ TM
2 − αt
2 − αt
, μ ϕi,j x/2ri ,0
,
TM μ ϕi,j x/2ri ,−x/2rj 6
6
2 − αt
μ ϕi,j 0,−x/2rj ,
6
2 − αt
2 − αt
TM μ
ϕi,j x/ri ,x/rj , μ
ϕi,j x/ri ,0
,
3
3
2 − αt
μ
ϕi,j 0,x/rj .
3
2.24
Replacing xi by 2n xi for all 1 ≤ i ≤ m, in 2.3, we get for all xi , xj ∈ X and for all t > 0,
μmj1 f−2n rj xj 1≤i≤m,i / j 2n ri xi 2 mi1 ri f2n xi −mfmi1 2n ri xi /2n t ≥ μ
ϕ2n x1 ,...,2n xm /2n t.
2.25
since
lim μ
n
n
n t
n → ∞ ϕ2 x1 ,...,2 xm /2
1,
2.26
We conclude that
⎛
m
EL⎝−rj xj j1
1≤i≤m,i /
j
⎞
m
m
ri xi ⎠ 2 ri ELxi − mEL
ri xi 0.
i1
i1
2.27
10
Mathematical Problems in Engineering
To prove the uniqueness of mapping EL, assume that there exists another mapping
A : X → Y which satisfies 2.4. Fix x ∈ X, clearly EL2n x 2n ELx and A2n x 2n Ax,
for all n ∈ N. Since μELx−Ax t limn → ∞ μEL2n x/2n −A2n x/2n t, so
t
t
n
n
n
n
, μf2 x/2 −A2 x/2 t
t ≥ min μ
t
2
2
n
n
2 2 − αt
2 2 − αt
≥ TM TM μ ϕi,j x/2ri ,−x/2rj , μ ϕi,j x/2ri ,0
,
12αn
12αn
n
2 2 − αt
μ ϕi,j 0,−x/2rj ,
12αn
n
n
2 2 − αt
2 2 − αt
T M μ
ϕi,j x/ri ,x/rj ,
μ
,
ϕi,j x/ri ,0
6αn
6αn
n
2 2 − αt
μ
ϕi,j 0,x/rj .
6αn
2.28
μ
EL2n x/2n −A2n x/2n EL2n x/2n −f2n x/2n Since the right-hand side of the above inequality tends to 1, when n → ∞, therefore, it
follows that for all t > 0, μELx−Ax t 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
μmj1 f−rj xj 1≤i≤m, i / j ri xi 2 mi1 ri fxi −mfmi1 ri xi t ≥ μ
m
k1
xk p z0 t,
2.29
for all xi , xj ∈ X and t > 0. Then the limit ELx limn → ∞ f2n x/2n exists for all x ∈ X and
defines a unique Euler-Lagrange additive mapping EL : X → Y such that
p
2p ri rj 2 − 2p t
|2ri |p 2 − 2p t
, μ xp z0
TM μ xp z0
,
p 6
6 |ri |p rj p
2rj 2 − 2p t
,
μ xp z0
6
p
p
ri rj 2 − 2p t
|ri | 2 − 2p t
TM μ xp z0
,
p p , μ
xp z0
3
3 |ri | rj μELx−fx t ≥ TM
μ xp z0
p
rj 2 − 2p t
3
,
for all x ∈ X and t > 0.
Proof. Let α 2p and ϕ : X m → Z be defined as ϕx1 , . . . , xm m
k1
xi p z0 .
2.30
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11
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
μmj1 f−rj xj 1≤i≤m, i / j ri xi 2 mi1 ri fxi −mfmi1 ri xi t ≥ μ
δz0 t,
2.31
for all xi ∈ X for all 1 ≤ i ≤ m and all t > 0. Then, the limit Cx limn → ∞ f2n x/2n exists for
all x ∈ X and defines a unique Euler-Lagrange additive mapping EL : X → Y such that
t
t
, μ
δz0
,
μELx−fx t ≥ TM μ
δz0
6
3
2.32
for all x ∈ X and t > 0.
Proof. Let α 1 and ϕ : X m → Z be defined as ϕx1 , . . . , xm 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 ,...,xm t
∀xi ∈ X, t > 0,
2.33
f0 0 and for all xi ∈ X and t > 0, limn → ∞ μ
2n ϕx1 /2n ,...,xm /2n t 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
1−2αt
1−2αt
, μ
ϕi,j x/rj ,0
,
μELx−fx t ≥ TM TM μ
ϕi,j x/ri ,−x/rj 6α
6α
1−2αt
μ
ϕi,j 0,−x/rj ,
6α
1−2αt
,
TM μ
ϕi,j x/ri ,x/rj 3α
1−2αt
1−2αt
μ
ϕi,j x/ri ,0
, μ
ϕi,j 0,x/rj ,
3α
3α
for all x ∈ X and all t > 0.
2.34
12
Mathematical Problems in Engineering
Proof. Replacing x by x/2n1 in 2.14 and using 2.33, we obtain
μ2n fx/2n −2n1 fx/2n1 t
≥ TM TM μ
ϕi,j x/2n2 ri ,−x/2n2 rj t
t
,
μ
,
ϕi,j x/2n2 ri ,0
2n · 6
2n · 6
t
,
μ
ϕi,j 0,−x/2n2 rj n
2 ·6
t
t
t
TM μ
ϕi,j x/2n1 ri ,x/2n1 rj n
, μ
ϕi,j x/2n1 ri ,0 n
, μ
ϕi,j 0,x/2n1 rj n
2 ·3
2 ·3
2 ·3
t
t
≥ TM TM μ
ϕi,j x/2ri ,−x/2rj , μ
ϕi,j x/2ri ,0
,
n1
n
n1
α 2 ·6
α 2n · 6
t
,
μ
ϕi,j 0,−x/2rj αn1 2n · 6
t
t
t
TM μ
ϕi,j x/ri ,x/j
,
μ
,
μ
.
ϕi,j x/ri ,0
ϕi,j 0,x/rj αn1 2n · 3
αn1 2n · 3
αn1 2n · 3
2.35
So
μ2n fx/2n −fx
n−1
k k1
2 α
t
i1
t
t
t
≥ TM TM μ
ϕi,j x/ri ,−x/rj , μ
ϕi,j x/ri ,0
, μ
ϕi,j 0,−x/rj ,
6
6
6
t
t
t
, μ
ϕi,j x/ri ,0
, μ
ϕi,j 0,x/rj ,
TM μ
ϕi,j x/ri ,x/rj 3
3
3
2.36
for all x ∈ X. This implies that
μ2n fx/2n −fx t
≥ TM TM μ ϕi,j x/ri ,−x/rj 6α
μ ϕi,j 0,−x/rj 6α
3α
k0
t
n−1
k0
n−1
k0
3α
n−1
k0
2k αk
2k αk
6α
n−1
k0
2k αk
,
,
2.37
,
, μ ϕi,j 0,x/rj t
2k αk
t
, μ ϕi,j x/ri ,0
2k αk
t
μ ϕi,j x/ri ,0
n−1
t
TM μ ϕi,j x/ri ,x/rj t
3α
n−1
k0
2k αk
.
Mathematical Problems in Engineering
13
Proceeding as in the proof of Theorem 2.2, one can easily show that the sequence
{2n fx/2n }∞
n1 is a Cauchy sequence in complete RN space Y, μ, min, so there exists some
point ELx ∈ Y such that
ELx lim 2n f
n→∞
x
,
2n
2.38
for all x ∈ X.
Taking the limit n → ∞ from both sides of the above inequality, we obtain 2.34.
The rest of the proof is similar to the proof of Theorem 2.2.
Corollary 2.6. Let X be a real linear space, Z, μ
, min be an RN space and Y, μ, min a complete
RN space. Let p > 1, z0 ∈ Z and f : X → Y be a mapping with f0 0 and satisfying
μmj1 f−rj xj 1≤i≤m, i / j ri xi 2 mi1 ri fxi −mfmi1 ri xi t ≥ μ
m
k1
xi p z0 t,
2.39
for all xi ∈ X for all 1 ≤ i ≤ m and all t > 0. Then the limit ELx limn → ∞ 2n fx/2n exists for all
x ∈ X and defines a unique Euler-Lagrange additive mapping EL : X → Y such that
p
2p ri rj 2p − 2t
|2ri |p 2p − 2t
, μ
xp z0
TM μ xp z0
,
p p 6
6 |ri | rj p p
2rj 2 − 2t
,
μ xp z0
6
p p
p p
ri rj 2 − 2t
|ri | 2 − 2t
TM μ xp z0
,
p , μ xp z0
3
3 |ri |p rj μELx−fx t ≥ TM
μ xp z0
2.40
p p
rj 2 − 2t
3
,
for all x ∈ X and t > 0.
Proof. Let α 2−p and ϕ : X m → Z be defined as ϕx1 , . . . , xm m
k1
xi p z0 .
Corollary 2.7. 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
μmj1 f−rj xj 1≤i≤m, i / j ri xi 2 mi1 ri fxi −mfmi1 ri xi t ≥ μ
δz0 t,
2.41
for all xi , xj ∈ X and t > 0. Then, the limit ELx limn → ∞ 2n fx/2n exists for all x ∈ X and
defines a unique Euler-Lagrange additive mapping EL : X → Y such that
4t
2t
μELx−fx t ≥ TM μ
δz0
, μ
δz0
,
3
3
for all x ∈ X and t > 0.
2.42
14
Mathematical Problems in Engineering
Proof. Let α 1/4 and ϕ : X m → Z be defined as ϕx1 , . . . , xm δz0 .
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