Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 902931, 45 pages
doi:10.1155/2012/902931
Review Article
Nonlinear Random Stability via
Fixed-Point Method
Yeol Je Cho,1 Shin Min Kang,2 and Reza Saadati3
1
Department of Mathematics Education and the RINS, Gyeongsang National University,
Chinju 660-701, Republic of Korea
2
Department of Mathematics and the RINS, Gyeongsang National University,
Chinju 660-701, Republic of Korea
3
Department of Mathematics, Iran University of Science and Technology, Behshahr, Iran
Correspondence should be addressed to Shin Min Kang, smkang@gnu.ac.kr
and Reza Saadati, rsaadati@eml.cc
Received 31 October 2011; Accepted 22 December 2011
Academic Editor: Yeong-Cheng Liou
Copyright q 2012 Yeol Je Cho 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.
We prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic
functional equation fx2yfx−2y 4fxy4fx−y−6fxf2yf−2y−4fy−4f−y
in various complete random normed spaces.
1. Introduction
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 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 Găvruţa 5 by replacing the
unbounded Cauchy difference by a general control function in the spirit of Rassias approach.
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. A generalized Hyers-Ulam stability problem for
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the quadratic functional equation was proved by Cholewa 6 for mappings f : X → Y ,
where X is a normed space and Y is a Banach space. Czerwik 7 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 8–12.
In 13, Jun and Kim consider the following cubic functional equation:
f 2x y f 2x − y 2f x y 2f x − y 12fx.
1.2
It is easy to show that the function fx x3 satisfies the functional equation 1.2, which is
called a cubic functional equation, and every solution of the cubic functional equation is said
to be a cubic mapping.
Considered the following quartic functional equation
f 2x y f 2x − y 4f x y 4f x − y 24fx − 6f y .
1.3
It is easy to show that the function fx x4 satisfies the functional equation, which is called
a quartic functional equation, and every solution of the quartic functional equation is said to
be a quartic mapping. One can easily show that an odd mapping f : X → Y satisfies the
additive-quadratic-cubic-quadratic functional equation
f x 2y f x − 2y 4f x y 4f x − y − 6fx
f 2y f −2y − 4f y − 4f −y
1.4
if and only if it is an additive-cubic mapping, that is,
f x 2y f x − 2y 4f x y 4f x − y − 6fx.
1.5
It was shown in Lemma 2.2 of 14 that gx : f2x−2fx and hx : f2x−8fx
are cubic and additive, respectively, and that fx 1/6gx − 1/6hx.
One can easily show that an even mapping f : X → Y satisfies 1.4 if and only if it is
a quadratic-quartic mapping, that is,
f x 2y f x − 2y 4f x y 4f x − y − 6fx 2f 2y − 8f y .
1.6
Also gx : f2x − 4fx and hx : f2x − 16fx are quartic and quadratic, respectively,
and fx 1/12gx − 1/12hx.
For a given mapping f : X → Y , we define
Df x, y : f x 2y f x − 2y − 4f x y − 4f x − y 6fx
− f 2y − f −2y 4f y 4f −y
for all x, y ∈ X.
1.7
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Let X be a set. A function d : X × X → 0, ∞ is called a generalized metric on X if d
satisfies
1 dx, y 0 if and only if x y,
2 dx, y dy, x for all x, y ∈ X,
3 dx, z ≤ dx, y dy, z for all x, y, z ∈ X.
We recall the fixed-point alternative of Diaz and Margolis.
Theorem 1.1 see 15, 16. Let X, d be a complete generalized metric space and let J : X → X be
a strictly contractive mapping with Lipschitz constant L < 1, then for each given element x ∈ X, either
d J n x, J n1 x ∞
1.8
for all nonnegative integers n or there exists a positive integer n0 such that
1 dJ n x, J n1 x < ∞ for all n ≥ n0 ,
2 the sequence {J n x} converges to a fixed point y∗ of J,
3 y∗ is the unique fixed point of J in the set Y {y ∈ X | dJ n0 x, y < ∞},
4 dy, y∗ ≤ 1/1 − Ldy, Jy for all y ∈ Y .
In 1996, Isac and Rassias 17 were the first to provide applications of stability theory
of functional equations for the proof of new fixed-point theorems with applications. By
using fixed point methods, the stability problems of several functional equations have been
extensively investigated by a number of authors see 18–21.
2. Preliminaries
In the sequel, we adopt the usual terminology, notations, and conventions of the theory of
random normed spaces, as in 22–26. Throughout this paper, Δ is the space of all probability
distribution functions, that is, the space of all mappings F : R ∪ {−∞, ∞} → 0, 1, such taht
F is left continuous, nondecreasing on R, F0 0 and {F∞ 1}. D is a subset of Δ
consisting of all functions F ∈ Δ for which l− F∞ 1, where l− fx denotes the left limit
of the function f at the point x, that is, l− fx limt → x− ft. The space Δ is partially ordered
by the usual pointwise ordering of functions, that is, F ≤ G if and only if Ft ≤ Gt for all t
in R. The maximal element for Δ in this order is the distribution function ε0 given by
0,
ε0 t 1,
if t ≤ 0,
if t > 0.
2.1
A triangular norm shortly t-norm is a binary operation on the unit interval 0, 1, that
is, a function T : 0, 1 × 0, 1 → 0, 1, such that for all a, b, c ∈ 0, 1 the following four
axioms satisfied:
T1 T a, b T b, a commutativity,
T2 T a, T b, c T T a, b, c associativity,
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T3 T a, 1 a boundary condition,
T4 T a, b ≤ T a, c whenever b ≤ c monotonicity.
Basic examples are the Łukasiewicz t-norm TL , TL a, b maxa b − 1, 0 for all a, b ∈
0, 1 and the t-norms TP , TM , TD , where TP a, b : ab, TM a, b : min{a, b},
TD a, b :
mina, b,
if maxa, b 1,
0,
otherwise.
2.2
n
If T is a t-norm, then xT is defined for every x ∈ 0, 1 and n ∈ N ∪ {0} by 1, if n 0
n−1
and T xT , x if n ≥ 1. A t-norm T is said to be of Hadžić type we denote by T ∈ H if the
n
family xT n∈N is equicontinuous at x 1 cf. 27.
Other important triangular norms are the following see 28:
SW
SW
1 The Sugeno-Weber family {TλSW }λ∈−1,∞ is defined by T−1
TD , T∞
TP and
x y − 1 λxy
TλSW x, y max 0,
1λ
2.3
if λ ∈ −1, ∞.
2 The Domby family {TλD }λ∈0,∞ is defined by TD if λ 0, TM if λ ∞, and
TλD x, y 1
λ 1/λ
1 1 − x/xλ 1 − y/y
2.4
if λ ∈ 0, ∞.
3 The Aczel-Alsina family {TλAA }λ∈0,∞ is defined by TD if λ 0, TM if λ ∞ and
λ
λ 1/λ
TλAA x, y e−| log x| | log y| 2.5
if λ ∈ 0, ∞.
A t-norm T can be extended by associativity in a unique way to an n-array operation
taking for x1 , . . . , xn ∈ 0, 1n the value T x1 , . . . , xn defined by
0
Ti1
xi 1,
n
n−1
Ti1
xi T Ti1
xi , xn T x1 , . . . , xn .
2.6
T can also be extended to a countable operation taking for any sequence xn n∈N in
0, 1 the value
∞
n
xi lim Ti1
xi .
Ti1
n→∞
2.7
n
The limit on the right side of 6.4 exists since the sequence Ti1
xi n∈N is nonincreasing and
bounded from below.
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Proposition 2.1 see 28. We have the following.
1 For T ≥ TL , the following implication holds:
lim T∞
i1 xni 1 ⇐⇒
n→∞
∞
1 − xn < ∞.
2.8
n1
2 If T is of Hadžić type, then
lim T∞ xni
n → ∞ i1
1
2.9
for every sequence xn n∈N in 0, 1 such that limn → ∞ xn 1.
3 If T ∈ {TλAA }λ∈0,∞ ∪ {TλD }λ∈0,∞ , then
lim T∞
i1 xni 1 ⇐⇒
n→∞
∞
1 − xn α < ∞.
2.10
n1
4 If T ∈ {TλSW }λ∈−1,∞ , then
lim T∞
i1 xni 1 ⇐⇒
n→∞
∞
1 − xn < ∞.
2.11
n1
Definition 2.2 see 26. A Random normed space briefly, RN-space is a triple X, μ, T , where
X is a vector space, T is a continuous t-norm, and μ is a mapping from X into D such that,
the following conditions hold:
RN1 μx t ε0 t for all t > 0 if and only if x 0,
RN2 μαx t μx t/|α| for all x ∈ X, and α /
0,
RN3 μxy t s ≥ T μx t, μy s for all x, y ∈ X and t, s ≥ 0.
Definition 2.3. Let X, μ, T be an RN-space.
1 A sequence {xn } in X is said to be convergent to x in X if, for every > 0 and λ > 0,
there exists positive integer N such that μxn −x > 1 − λ whenever n ≥ N.
2 A sequence {xn } in X is called a Cauchy sequence if, for every > 0 and λ > 0, there
exists positive integer N such that μxn −xm > 1 − λ whenever n ≥ m ≥ N.
3 An RN-space X, μ, 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 random Banach
space.
Theorem 2.4 see 25. If X, μ, T is an RN-space and {xn } is a sequence such that xn → x, then
limn → ∞ μxn t μx t almost everywhere.
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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 with 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 20, 24, 29–39.
3. Non-Archimedean Random Normed Space
By a non-Archimedean field, we mean a field K equipped 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 all
r, s ∈ K. Clearly, |1| | − 1| 1 and |n| ≤ 1 for all n ∈ N. By the trivial valuation, we mean the
mapping | · | taking everything but 0 into 1 and |0| 0. Let X be a vector space over a field
K with a non-Archimedean nontrivial valuation | · |. A function · : X → 0, ∞ is called a
non-Archimedean norm if it satisfies the following conditions:
NAN1 x 0 if and only if x 0,
NAN2 for any r ∈ K and x ∈ X, rx |r|x,
NAN3 the strong triangle inequality ultrametric, namely,
x y
≤ max x, y
x, y ∈ X ,
3.1
then X, · is called a non-Archimedean normed space. Due to the fact that
xn − xm ≤ max xj1 − xj : m ≤ j ≤ n − 1
n > m,
3.2
a sequence {xn } is a Cauchy sequence if and only if {xn1 − xn } converges to zero in a nonArchimedean normed space. By a complete non-Archimedean normed space, we mean one
in which every Cauchy sequence is convergent.
In 1897, Hensel 40 discovered the p-adic numbers of as a number theoretical
analogues of power series in complex analysis. Fix a prime number p. For any nonzero
rational number x, there exists a unique integer nx ∈ Z such that x a/bpnx , where a
and b are integers not divisible by p. Then |x|p : p−nx defines a non-Archimedean norm on
Q. The completion of Q with respect to the metric dx, y |x − y|p is denoted by Qp , which
is called the p-adic number field.
Throughout the paper, we assume that X is a vector space and Y is a complete nonArchimedean normed space.
Definition 3.1. A non-Archimedean random normed space briefly, non-Archimedean RN-space
is a triple X, μ, T , where X is a linear space over a non-Archimedean field K, T is a
continuous t-norm, and μ is a mapping from X into D such that the following conditions
hold:
NA-RN1 μx t ε0 t for all t > 0 if and only if x 0,
0,
NA-RN2 μαx t μx t/|α| for all x ∈ X, t > 0, and α /
NA-RN3 μxy max{t, s} ≥ T μx t, μy s for all x, y, z ∈ X and t, s ≥ 0.
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7
It is easy to see that if NA-RN3 holds, then so is
RN3 μxy t s ≥ T μx t, μy s.
As a classical example, if X, . is a non-Archimedean normed linear space, then the
triple X, μ, TM , where
0,
μx t 1,
t ≤ x,
t > x,
3.3
is a non-Archimedean RN-space.
Example 3.2. Let X, · be a non-Archimedean normed linear space. Define
μx t t
t x
x ∈ X, t > 0,
3.4
then X, μ, TM is a non-Archimedean RN-space.
Definition 3.3. Let X, μ, T be a non-Archimedean RN-space. Let {xn } be a sequence in X,
then {xn } is said to be convergent if there exists x ∈ X such that
lim μxn −x t 1
n→∞
3.5
for all t > 0. In that case, x is called the limit of the sequence {xn }.
A sequence {xn } in X is called a Cauchy sequence if for each ε > 0 and each t > 0 there
exists n0 such that for all n ≥ n0 and all p > 0, we have μxnp −xn t > 1 − ε.
If each Cauchy sequence is convergent, then the random norm is said to be complete
and the non-Archimedean RN-space is called a non-Archimedean random Banach space.
Remark 3.4 see 41. Let X, μ, TM be a non-Archimedean RN-space, then
μxnp −xn t ≥ min μxnj1 −xnj t : j 0, 1, 2, . . . , p − 1 .
3.6
So, the sequence {xn } is a Cauchy sequence if for each ε > 0 and t > 0 there exists n0 such that
for all n ≥ n0 ,
μxn1 −xn t > 1 − ε.
3.7
4. Generalized Ulam-Hyers Stability for a Quartic
Functional Equation in Non-Archimedean RN-Spaces of
Functional Equation 1.4: An Odd Case
Let K be a non-Archimedean field, let X be a vector space over K, and let Y, μ, T be a nonArchimedean random Banach space over K.
8
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Next, we define a random approximately AQCQ mapping. Let Ψ be a distribution
function on X × X × 0, ∞ such that Ψx, y, · is nondecreasing and
t
Ψcx, cx, t ≥ Ψ x, x,
|c|
0.
x ∈ X, c /
4.1
Definition 4.1. A mapping f : X → Y is said to be Ψ-approximately AQCQ if
μDfx,y t ≥ Ψ x, y, t
x, y ∈ X, t > 0 .
4.2
In this section, we assume that 2 / 0 in K i.e., characteristic of K is not 2. Our main
result, in this section, is the following.
We prove the generalized Hyers-Ulam stability of the functional equation Dfx, y 0
in non-Archimedean random spaces, an odd case.
Theorem 4.2. Let K be a non-Archimedean field, let X be a vector space over K and let Y, μ, T be a non-Archimedean random Banach space over K. Let f : X → Y be an odd mapping and Ψapproximately AQCQ mapping. If for some α ∈ R, α > 0, and some integer k, k > 3 with |2k | < α,
Ψ 2−k x, 2−k y, t ≥ Ψ x, y, αt
lim T∞ M
n → ∞ jn
2x,
αj t
x ∈ X, t > 0,
4.3
|8|kj
1 x ∈ X, t > 0,
4.4
then there exists a unique cubic mapping C : X → Y such that
μfx−2fx/2−Cx/2 t ≥
∞
M
Ti1
x,
αi1 t
|8|ki
4.5
for all x ∈ X and t > 0, where
Mx, t : T
k−1
x x t
2k−1 x 2k−1 x t
2k−1 x
x k−1
Ψ
, ,
,
,
,t
, Ψ x, , t , . . . , Ψ
, Ψ 2 x,
2 2 |4|
2
2
2
2
|4|
x ∈ X, t > 0.
4.6
Proof. Letting x y in 4.2, we get
μf3y−4f2y5fy t ≥ Ψ y, y, t
4.7
for all y ∈ X and t > 0. Replacing x by 2y in 4.2, we get
μf4y−4f3y6f2y−4fy t ≥ Ψ 2y, y, t
4.8
Journal of Applied Mathematics
9
for all y ∈ X and t > 0. By 4.7 and 4.8, we have
μf4y−10f2y16fy t ≥ T μ4f3y−4f2y5fy t, μf4y−4f3y6f2y−4fy t
t
T μf3y−4f2y5fy
, μf4y−4f3y6f2y−4fy t
|4|
t
, Ψ 2y, y, t
≥ T Ψ y, y,
|4|
4.9
for all y ∈ X and t > 0. Letting y : x/2 and gx : f2x − 2fx for all x ∈ X in 4.9, we
get
x x t
x , ,
, Ψ x, , t
μgx−8gx/2 t ≥ T Ψ
2 2 |4|
2
4.10
for all x ∈ X and t > 0. Now, we show by induction on j that for all x ∈ X, t > 0 and j ≥ 1,
μg2j−1 x−8j gx/2 t
≥ Mj x, t
: T
2j−1
x x t
2j−1 x 2j−1 x t
2j−1 x
x j−1
Ψ
, ,
,
,
,t .
, Ψ x, , t , . . . , Ψ
, Ψ 2 x,
2 2 |4|
2
2
2
2
|4|
4.11
Putting j 1 in 4.11, we obtain 4.10. Assume that 4.11 holds for some j ≥ 1. Replacing x
by 2j x in 4.10, we get
t
μg2j x−8g2j−1 x t ≥ T Ψ 2j−1 x, 2j−1 x,
, Ψ 2j x, 2j−1 x, t .
|4|
4.12
Since |8| ≤ 1,
μg2j x−8j1 gx/2 t ≥ T μg2j x−8g2j−1 x t, μ8g2j−1 x−8j1 gx/2 t
t
T μg2j x−8g2j−1 x t, μg2j−1 x−8j gx/2
|8|
t
≥ T 2 Ψ 2j−1 x, 2j−1 x,
, Ψ 2j x, 2j−1 x, t , Mj x, t
|4|
4.13
Mj1 x, t
for all x ∈ X and t > 0. Thus, 4.11 holds for all j ≥ 2. In particular,
μg2k−1 x−8k gx/2 t ≥ Mx, t
x ∈ X, t > 0.
4.14
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Replacing x by 2−knk−1 x in 4.14 and using inequality 4.3, we obtain
μgx/2kn −8k gx/2kn1 t ≥ M
2x
2kn1
,t
x ∈ X, t > 0, n 0, 1, 2, . . ..
4.15
Then
αn1
μ8kn gx/2kn −8kn1 gx/2kn1 t ≥ M 2x, kn1 t
8
x ∈ X, t > 0, n 0, 1, 2, . . ..
4.16
Hence
np
μ8kn gx/2kn −8knp gx/2knp t ≥ Tjn μ8kj gx/2kj −8kjp gx/2kjp t
⎛
≥
np
Tjn M⎝2x,
≥
np
Tjn M⎝2x,
⎛
⎞
αj1
t⎠
k j1 8
⎞
αj1
t⎠
k j1 8
x ∈ X, t > 0, n 0, 1, 2, . . ..
4.17
Since
⎞
j1
α
lim T ∞ M⎝2x, j1 t⎠ 1
k
n → ∞ jn
8
⎛
x ∈ X, t > 0,
4.18
then
x
8kn g kn
2
n∈N
4.19
is a Cauchy sequence in the non-Archimedean random Banach space Y, μ, T . Hence we can
define a mapping C : X → Y such that
lim μ88k n gx/2kn −Cx t 1 x ∈ X, t > 0.
n→∞
4.20
Next for each n ≥ 1, x ∈ X and t > 0,
μgx−88k n gx/2kn t μn−1 88k i gx/2ki −88k i1 gx/2ki1 t
i0
n−1
≥ Ti0 μ88k i gx/2ki −88k i1 gx/2ki1 t
αi1 t
n−1
≥ Ti0 M 2x, i1 .
8k 4.21
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11
Therefore,
μgx−Cx t ≥ T μgx−88k n gx/2kn t, μ88k n gx/2kn −Cx t
≥T
αi1 t
2x, i1
8k n−1
Ti0
M
,μ
n
88k gx/2kn −Cx
4.22
t .
By letting n → ∞, we obtain
μgx−Cx t ≥
αi1 t
2x, i1
8k ∞
M
Ti1
4.23
.
So,
μfx−2fx/2−Cx/2 t ≥
αi1 t
x, i1
8k ∞
M
Ti1
4.24
.
This proves 4.5. From Dgx, y Df2x, 2y − 2Dfx, y, by 4.2, we deduce that
μDf2x,2y t ≥ Ψ 2x, 2y, t ,
t
≥ μDfx,y t ≥ Ψ x, y, t ,
μ−2Dfx,y t μDfx,y
|2|
4.25
and so, by NA-RN3 and 4.2, we obtain
μDgx,y t ≥ T μDf2x,2y t, μ−2Dfx,y t ≥ T Ψ 2x, 2y, t , Ψ x, y, t : N x, y, t .
4.26
It follows that
μ8kn Dgx/2kn ,y/2kn t μDgx/2kn ,y/2kn ≥N
t
|8|kn
t
x y
,
,
2kn 2kn |8|kn
≥ · · · ≥ N x, y,
αn−1 t
4.27
|8|kn−1
for all x, y ∈ X, t > 0, and n ∈ N. Since
lim N x, y,
n→∞
αn−1 t
|8|kn−1
1
4.28
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Journal of Applied Mathematics
for all x, y ∈ X and t > 0, by Theorem 2.4, we deduce that
μDCx,y t 1
4.29
for all x, y ∈ X and t > 0. Thus, the mapping C : X → Y satisfies 1.4.
Now, we have
x x
C2x − 8Cx lim 8n g n−1 − 8n1 g n
n→∞
2
2
x
x
8 lim 8n−1 g n−1 − 8n g n
0
n→∞
2
2
4.30
for all x ∈ X. Since the mapping x → C2x − 2Cx is cubic see Lemma 2.2 of 14, from
the equality C2x 8Cx, we deduce that the mapping C : X → Y is cubic.
Corollary 4.3. Let K be a non-Archimedean field, let X be a vector space over K, and let Y, μ, T be a
non-Archimedean random Banach space over K under a t-norm T ∈ H. Let f : X → Y be an odd and
Ψ-approximately AQCQ mapping. If, for some α ∈ R, α > 0, and some integer k, k > 3, with |2k | < α,
Ψ 2−k x, 2−k y, t ≥ Ψ x, y, αt
x ∈ X, t > 0,
4.31
then there exists a unique cubic mapping C : X → Y such that
μfx−2fx/2−Cx/2 t ≥
∞
M
Ti1
x,
αi1 t
|8|ki
4.32
for all x ∈ X and t > 0.
Proof. Since
lim M x,
n→∞
αj t
1 x ∈ X, t > 0
|8|kj
4.33
and T is of Hadžić type, from Proposition 2.1, it follows that
lim T ∞ M
n → ∞ jn
x,
αj t
|8|kj
1
x ∈ X, t > 0.
4.34
Now, we can apply Theorem 4.2 to obtain the result.
Example 4.4. Let X, μ, TM be non-Archimedean random normed space in which
μx t t
t x
x ∈ X, t > 0.
4.35
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13
And let Y, μ, TM be a complete non-Archimedean random normed space see Example 3.2.
Define
Ψ x, y, t t
.
1t
4.36
It is easy to see that 4.3 holds for α 1. Also, since
Mx, t t
,
1t
4.37
we have
lim T ∞ M
n → ∞ M,jn
x,
αj t
|8|kj
lim
n→∞
lim T m M
m → ∞ M,jn
lim lim
n→∞m→∞
1
t
n
t 8k x,
t
|8|kj
4.38
x ∈ X, t > 0.
Let f : X → Y be an odd and Ψ-approximately AQCQ mapping. Thus, all the conditions of
Theorem 4.2 hold, and so there exists a unique cubic mapping C : X → Y such that
μfx−2fx/2−Cx/2 t ≥
t
.
t 8k 4.39
Theorem 4.5. Let K be a non-Archimedean field, let X be a vector space over K, and let Y, μ, T be a non-Archimedean random Banach space over K. Let f : X → Y be an odd mapping and Ψapproximately AQCQ mapping. If for some α ∈ R, α > 0, and some integer k, k > 1 with |2k | < α,
Ψ 2−k x, 2−k y, t ≥ Ψ x, y, αt
lim T ∞ M
n → ∞ jn
2x,
αj t
x ∈ X, t > 0,
|2|kj
4.40
1
x ∈ X, t > 0,
then there exists a unique additive mapping A : X → Y such that
μfx−8fx/2−Ax/2 t ≥
∞
M
Ti1
x,
αi1 t
|2|ki
4.41
14
Journal of Applied Mathematics
for all x ∈ X and t > 0, where
Mx, t : T
k−1
x x t
2k−1 x 2k−1 x t
x 2k−1 x
k−1
Ψ
, ,
,
,
,t
, Ψ x, , t , . . . , Ψ
, Ψ 2 x,
2 2 |4|
2
2
2
2
|4|
x ∈ X, t > 0
4.42
Proof. Letting y : x/2 and gx : f2x − 8fx for all x ∈ X in 4.9, we get
x x x t
μgx−2gx/2 t ≥ T Ψ
, ,
, Ψ x, , t
2 2 |4|
2
4.43
for all x ∈ X and t > 0.
The rest of the proof is similar to the proof of Theorem 4.2.
Corollary 4.6. Let K be a non-Archimedean field, let X be a vector space over K, and let Y, μ, T be
a non-Archimedean random Banach space over K under a t-norm T ∈ H. Let f : X → Y be an odd
and Ψ-approximately AQCQ mapping. If, for some α ∈ R, α > 0, and some integer k, k > 1, with
|2k | < α,
Ψ 2−k x, 2−k y, t ≥ Ψ x, y, αt
x ∈ X, t > 0,
4.44
then there exists a unique additive mapping A : X → Y such that
μfx−8fx/2−Ax/2 t ≥
∞
M
Ti1
x,
αi1 t
|2|ki
4.45
for all x ∈ X and t > 0.
Proof. Since
lim M x,
n→∞
αj t
|2|kj
1 x ∈ X, t > 0
4.46
and T is of Hadžić type, from Proposition 2.1, it follows that
lim T ∞ M
n → ∞ jn
x,
αj t
|2|kj
1
Now, we can apply Theorem 4.5 to obtain the result.
x ∈ X, t > 0.
4.47
Journal of Applied Mathematics
15
Example 4.7. Let X, μ, TM non-Archimedean random normed space in which
t
t x
μx t x ∈ X, t > 0,
4.48
and let Y, μ, TM be a complete non-Archimedean random normed space see Example 3.2.
Define
Ψ x, y, t t
.
1t
4.49
It is easy to see that 4.3 holds for α 1. Also, since
Mx, t t
,
1t
4.50
we have
lim T ∞ M
n → ∞ M,jn
x,
αj t
|2|kj
lim
n→∞
lim T m M
m → ∞ M,jn
lim lim
n→∞m→∞
1
t
n
t 2k x,
t
|2|kj
4.51
x ∈ X, t > 0.
Let f : X → Y be an odd and Ψ-approximately AQCQ mapping. Thus, all the conditions of
Theorem 4.2 hold, and so there exists a unique additive mapping A : X → Y such that
μfx−8fx/2−Ax/2 t ≥
t
.
t 2k 4.52
5. Generalized Hyers-Ulam Stability of the Functional Equation 1.4
in Non-Archimedean Random Normed Spaces: An Even Case
Now, we prove the generalized Hyers-Ulam stability of the functional equation Dfx, y 0
in non-Archimedean Banach spaces, an even case.
Theorem 5.1. Let K be a non-Archimedean field, let X be a vector space over K, and let Y, μ, T be
a non-Archimedean random Banach space over K. Let f : X → Y be an even mapping, f0 0, and
Ψ-approximately AQCQ mapping. If for some α ∈ R, α > 0, and some integer k, k > 4 with |2k | < α,
Ψ 2−k x, 2−k y, t ≥ Ψ x, y, αt
lim T∞ M
n → ∞ jn
2x,
αj t
|16|kj
x ∈ X, t > 0,
5.1
1 x ∈ X, t > 0,
16
Journal of Applied Mathematics
then there exists a unique quartic mapping Q : X → Y such that
μfx−4fx/2−Qx/2 t ≥
T∞
i1 M
x,
αi1 t
5.2
|16|ki
for all x ∈ X and t > 0, where
Mx, t : T
k−1
2k−1 x 2k−1 x t
x x t
x 2k−1 x
k−1
Ψ
, ,
,
,
,t
, Ψ x, , t , . . . , Ψ
, Ψ 2 x,
2 2 |4|
2
2
2
2
|4|
x ∈ X, t > 0.
5.3
Proof. Letting x y in 4.2, we get
μf3y−6f2y15fy t ≥ Ψ y, y, t
5.4
for all y ∈ X and t > 0. Replacing x by 2y in 4.2, we get
μf4y−4f3y4f2y4fy t ≥ Ψ 2y, y, t
5.5
for all y ∈ X and t > 0. By 5.4 and 5.5, we have
μf4y−20f2y64fy t ≥ T μ4f3y−4f2y5fy t, μf4y−4f3y6f2y−4fy t
t
T μf3y−4f2y5fy
, μf4y−4f3y6f2y−4fy t
|4|
t
, Ψ 2y, y, t
≥ T Ψ y, y,
|4|
5.6
for all y ∈ X and t > 0. Letting y : x/2 and gx : f2x − 4fx for all x ∈ X in 5.6, we
get
x x x t
μgx−16gx/2 t ≥ T Ψ
, ,
, Ψ x, , t
2 2 |4|
2
5.7
for all x ∈ X and t > 0.
The rest of the proof is similar to the proof of Theorem 4.2.
Corollary 5.2. Let K be a non-Archimedean field, let X be a vector space over K, and let Y, μ, T be a non-Archimedean random Banach space over K under a t-norm T ∈ H. Let f : X → Y be an
even, f0 0, and Ψ-approximately AQCQ mapping. If, for some α ∈ R, α > 0, and some integer
k, k > 4, with |2k | < α,
Ψ 2−k x, 2−k y, t ≥ Ψ x, y, αt
x ∈ X, t > 0,
5.8
Journal of Applied Mathematics
17
then there exists a unique quartic mapping Q : X → Y such that
μfx−4fx/2−Qx/2 t ≥
T∞
i1 M
x,
αi1 t
5.9
|16|ki
for all x ∈ X and t > 0.
Proof. Since
lim M x,
n→∞
αj t
|16|kj
1
x ∈ X, t > 0
5.10
and T is of Hadžić type, from Proposition 2.1, it follows that
lim T ∞ M
n → ∞ jn
x,
αj t
1
|16|kj
x ∈ X, t > 0.
5.11
Now, we can apply Theorem 5.1 to obtain the result.
Example 5.3. Let X, μ, TM be non-Archimedean random normed space in which
μx t t
t x
x ∈ X, t > 0.
5.12
And let Y, μ, TM be a complete non-Archimedean random normed space see Example 3.2.
Define
Ψ x, y, t t
.
1t
5.13
It is easy to see that 4.3 holds for α 1. Also, since
Mx, t t
,
1t
5.14
we have
lim T ∞ M
n → ∞ M,jn
x,
αj t
|16|kj
lim
n→∞
lim T m M
m → ∞ M,jn
lim lim
n→∞m→∞
1
t
n
t 16k x ∈ X, t > 0.
x,
t
|16|kj
5.15
18
Journal of Applied Mathematics
Let f : X → Y be an even, f0 0, and Ψ-approximately AQCQ mapping. Thus all the
conditions of Theorem 5.1 hold, and so there exists a unique quartic mapping Q : X → Y
such that
μfx−4fx/2−Qx/2 t ≥
t
.
t 16k 5.16
Theorem 5.4. Let K be a non-Archimedean field, let X be a vector space over K and let Y, μ, T be
a non-Archimedean random Banach space over K. Let f : X → Y be an even mapping, f0 0 and
Ψ-approximately AQCQ mapping. If for some α ∈ R, α > 0, and some integer k, k > 2 with |2k | < α,
Ψ 2−k x, 2−k y, t ≥ Ψ x, y, αt
lim T∞ M
n → ∞ jn
2x,
αj t
|4|kj
x ∈ X, t > 0,
5.17
1 x ∈ X, t > 0,
then there exists a unique quadratic mapping Q : X → Y such that
μfx−16fx/2−Qx/2 t ≥
T∞
i1 M
x,
αi1 t
5.18
|4|ki
for all x ∈ X and t > 0, where
Mx, t : T
k−1
x x t
2k−1 x 2k−1 x t
2k−1 x
x k−1
Ψ
, ,
,
,
,t
, Ψ x, , t , . . . , Ψ
, Ψ 2 x,
2 2 |4|
2
2
2
2
|4|
x ∈ X, t > 0.
5.19
Proof. Letting y : x/2 and gx : f2x − 16fx for all x ∈ X in 5.6, we get
x x x t
μgx−4gx/2 t ≥ T Ψ
, ,
, Ψ x, , t
2 2 |4|
2
5.20
for all x ∈ X and t > 0.
The rest of the proof is similar to the proof of Theorem 5.1.
Corollary 5.5. Let K be a non-Archimedean field, let X be a vector space over K, and let Y, μ, T be a non-Archimedean random Banach space over K under a t-norm T ∈ H. Let f : X → Y be an
even, f0 0, and Ψ-approximately AQCQ mapping. If, for some α ∈ R, α > 0, and some integer
k, k > 2, with |2k | < α,
Ψ 2−k x, 2−k y, t ≥ Ψ x, y, αt
x ∈ X, t > 0,
5.21
Journal of Applied Mathematics
19
then there exists a unique quadratic mapping Q : X → Y such that
T∞
i1 M
μfx−16fx/2−Qx/2 t ≥
x,
αi1 t
5.22
|4|ki
for all x ∈ X and t > 0.
Proof. Since
lim M x,
n→∞
αj t
1 x ∈ X, t > 0
|4|kj
5.23
and T is of Hadžić type, from Proposition 2.1, it follows that
lim T ∞ M
n → ∞ jn
x,
αj t
1
|4|kj
x ∈ X, t > 0.
5.24
Now, we can apply Theorem 5.4 to obtain the result.
Example 5.6. Let X, μ, TM be a non-Archimedean random normed space in which
μx t t
t x
x ∈ X, t > 0.
5.25
And let Y, μ, TM be a complete non-Archimedean random normed space see Example 3.2.
Define
Ψ x, y, t t
.
1t
5.26
It is easy to see that 4.3 holds for α 1. Also, since
Mx, t t
,
1t
5.27
we have
lim T ∞ M
n → ∞ M,jn
x,
αj t
|4|kj
lim
n→∞
lim T m M
m → ∞ M,jn
lim lim
n→∞m→∞
1
t
n
t 4k x ∈ X, t > 0.
x,
t
|4|kj
5.28
20
Journal of Applied Mathematics
Let f : X → Y be an even, f0 0, and Ψ-approximately AQCQ mapping. Thus, all the
conditions of Theorem 5.4 hold, and so there exists a unique quadratic mapping Q : X → Y
such that
μfx−16fx/2−Qx/2 t ≥
t
.
t 4k 5.29
6. Latticetic Random Normed Space
Let L L, ≥L be a complete lattice, that is, a partially ordered set in which every nonempty
subset admits supremum and infimum, and 0L inf L, 1L sup L. The space of latticetic
random distribution functions, denoted by ΔL , is defined as the set of all mappings F : R ∪
{−∞, ∞} → L such that F is left continuous and nondecreasing on R, F0 0L , F∞ 1L .
DL ⊆ ΔL is defined as DL {F ∈ ΔL : l− F∞ 1L }, where l− fx denotes the left
limit of the function f at the point x. The space ΔL is partially ordered by the usual pointwise
ordering of functions, that is, F ≥ G if and only if Ft ≥L Gt for all t in R. The maximal
element for ΔL in this order is the distribution function given by
ε0 t 0L ,
if t ≤ 0,
1L ,
if t > 0.
6.1
In Section 2, we defined t-norms on 0, 1, and now we extend t-norms on a complete
lattice.
Definition 6.1 see 42. A triangular norm t-norm on L is a mapping T : L2 → L satisfying
the following conditions:
a for all x ∈ LTx, 1L x boundary condition;
b for all x, y ∈ L2 Tx, y Ty, x commutativity;
c for all x, y, z ∈ L3 Tx, Ty, z TTx, y, z associativity;
d for all x, x , y, y ∈ L4 x≤L x and y≤L y ⇒ Tx, y≤L Tx , y monotonicity.
Let {xn } be a sequence in L converges to x ∈ L equipped order topology. The t-norm
T is said to be a continuous t-norm if
lim T xn , y T x, y
n→∞
6.2
for all y ∈ L.
A t-norm T can be extended by associativity in a unique way to an n-array operation
taking for x1 , . . . , xn ∈ Ln the value Tx1 , . . . , xn defined by
T0i1 xi 1,
Tni1 xi T Tn−1
i1 xi , xn Tx1 , . . . , xn .
6.3
Journal of Applied Mathematics
21
T can also be extended to a countable operation taking for any sequence xn n∈N in L
the value
n
T∞
i1 xi lim Ti1 xi .
n→∞
6.4
The limit on the right side of 6.4 exists since the sequence Tni1 xi n∈N is nonincreasing and
bounded from below.
n
Note that we put T T whenever L 0, 1. If T is a t-norm, then xT is defined for
n−1
every x ∈ 0, 1 and n ∈ N ∪ {0} by 1 if n 0 and T xT , x if n ≥ 1. A t-norm T is said to
n
be of Hadžić type, we denote by T ∈ H if the family xT n∈N is equicontinuous at x 1 cf.
27.
Definition 6.2 see 42. A continuous t-norm T on L 0, 12 is 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 ∈ L,
T x, y x1 ∗ y1 , x2 y2 .
6.5
For example,
Ta, b a1 b1 , min{a2 b2 , 1},
Ma, b min{a1 , b1 }, max{a2 , b2 }
6.6
for all a a1 , a2 , b b1 , b2 ∈ 0, 12 are continuous t-representable. Define the mapping
T∧ from L2 to L by
T∧ x, y ⎧
⎨x,
if y≥L x,
⎩y,
if x≥L y.
6.7
Recall see 27, 28 that if {xn } is a given sequence in L, T∧ ni1 xi is defined
recurrently by T∧ 1i1 xi x1 and T∧ ni1 xi T∧ T∧ n−1
i1 xi , xn for all n ≥ 2.
A negation on L is any decreasing mapping N : L → L satisfying N0L 1L and
N1L 0L . If NNx x, for all x ∈ L, then N is called an involutive negation. In the
following, L is endowed with a fixed negation N.
Definition 6.3. A latticetic random normed space in short LRN-space is a triple X, μ, T∧ , where
X is a vector space and μ is a mapping from X into DL such that the following conditions
hold:
LRN1 μx t ε0 t for all t > 0 if and only if x 0,
LRN2 μαx t μx t/|α| for all x in X, α /
0 and t ≥ 0,
LRN3 μxy t s ≥L T∧ μx t, μy s for all x, y ∈ X and t, s ≥ 0.
We note that from LPN2 it follows that μ−x t μx t for all x ∈ X and t ≥ 0.
22
Journal of Applied Mathematics
Example 6.4. Let L 0, 1 × 0, 1 and operation ≤L be defined by
L {a1 , a2 : a1 , a2 ∈ 0, 1 × 0, 1, a1 a2 ≤ 1},
a1 , a2 ≤L b1 , b2 ⇐⇒ a1 ≤ b1 , a2 ≥ b2 ,
∀a a1 , a2 , b b1 , b2 ∈ L.
6.8
then L, ≤L is a complete lattice see 42. In this complete lattice, we denote its units by 0L 0, 1 and 1L 1, 0. Let X, · be a normed space. Let Ta, b min{a1 , b1 }, max{a2 , b2 }
for all a a1 , a2 , b b1 , b2 ∈ 0, 1 × 0, 1 and μ be a mapping defined by
μx t t
x
,
t x t x
t ∈ R ,
6.9
then X, μ, T is a latticetic random normed spaces.
If X, μ, T∧ is a latticetic random normed space, then
V {V ε, λ : ε >L 0L , λ ∈ L \ {0L , 1L }},
V ε, λ {x ∈ X : Fx ε >L Nλ},
6.10
is a complete system of neighborhoods of null vector for a linear topology on X generated by
the norm F.
Definition 6.5. Let X, μ, T∧ be a latticetic random normed spaces.
1 A sequence {xn } in X is said to be convergent to x in X if, for every t > 0 and
ε ∈ L \ {0L }, there exists a positive integer N such that μxn −x t >L Nε whenever
n ≥ N.
2 A sequence {xn } in X is called a Cauchy sequence if, for every t > 0 and ε ∈ L \ {0L },
there exists a positive integer N such that μxn −xm t >L Nε whenever n ≥ m ≥ N.
3 A latticetic random normed spaces X, μ, T∧ is said to be complete if and only if
every Cauchy sequence in X is convergent to a point in X.
Theorem 6.6. If X, μ, T∧ is a latticetic random normed space and {xn } is a sequence such that
xn → x, then limn → ∞ μxn t μx t.
Proof. The proof is the same as classical random normed spaces, see 25.
7. Generalized Hyers-Ulam Stability of the Functional Equation 1.4:
An Odd Case via Fixed-Point Method
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the
functional equation Dfx, y 0 in random Banach spaces: an odd case.
Theorem 7.1. Let X be a linear space, let Y, μ, T∧ be a complete LRN-space, and Φ let be a mapping
from X 2 to DL (Φx, y is denoted by Φx,y ) such that, for some 0 < α < 1/8,
Φ2x,2y t ≤L Φx,y αt
x, y ∈ X, t > 0 .
7.1
Journal of Applied Mathematics
23
Let f : X → Y be an odd mapping satisfying
μDfx,y t≥L Φx,y t
7.2
x x
Cx : lim 8 f n−1 − 2f n
n→∞
2
2
7.3
for all x, y ∈ X and t > 0. Then
n
exists for each x ∈ X and defines a cubic mapping C : X → Y such that
μf2x−2fx−Cx t ≥L T∧ Φx,x
1 − 8α
1 − 8α
t , Φ2x,x
t
5α
5α
7.4
for all x ∈ X and t > 0.
Proof. Letting x y in 7.2, we get
μf3y−4f2y5fy t ≥L Φy,y t
7.5
for all y ∈ X and t > 0. Replacing x by 2y in 7.2, we get
μf4y−4f3y6f2y−4fy t ≥L Φ2y,y t
7.6
for all y ∈ X and t > 0. By 7.5 and 7.6,
μf4y−10f2y16fy 5t ≥L T∧ μ4f3y−4f2y5fy 4t, μf4y−4f3y6f2y−4fy t
T∧ μf3y−4f2y5fy t, μf4y−4f3y6f2y−4fy t
≥L T∧ Φy,y t, Φ2y,y t
7.7
for all y ∈ X and t > 0. Letting y : x/2 and gx : f2x − 2fx for all x ∈ X, we get
μgx−8gx/2 5t ≥L T∧ Φx/2,x/2 t, Φx,x/2 t
7.8
S : {h : X −→ Y, h0 0}
7.9
for all x ∈ X and t > 0.
Consider the set
and introduce the generalized metric on S:
dh, k inf u ∈ R : μhx−kx ut ≥L T∧ Φx,x t, Φ2x,x t, ∀x ∈ X, ∀t > 0
7.10
24
Journal of Applied Mathematics
where, as usual, inf ∅ ∞. It is easy to show that S, d is complete see the proof of
Lemma 2.1 of 24.
Now, we consider the linear mapping J : S → S such that
x
Jhx : 8h
2
7.11
for all x ∈ X, and we prove that J is a strictly contractive mapping with the Lipschitz constant
8α.
Let h, k ∈ S be given such that dh, k < ε. Then
μhx−kx εt ≥L T∧ Φx,x t, Φ2x,x t
7.12
for all x ∈ X and t > 0. Hence
μJhx−Jkx 8αεt μ8hx/2−8kx/2 8αεt
μhx/2−kx/2 αεt
≥ T∧ Φx/2,x/2 αt, Φx,x/2 αt
7.13
≥L T∧ Φx,x t, Φ2x,x t
for all x ∈ X and t > 0. So, dh, k < ε implies that
α
ε.
8
7.14
α
dh, k
8
7.15
dJh, Jk ≤
This means that
dJh, Jk ≤
for all h, k ∈ S. It follows from 7.8 that
μgx−8gx/2 5αt ≥L T∧ Φx,x t, Φ2x,x t
7.16
for all x ∈ X and t > 0. So, dg, Jg ≤ 5α ≤ 5/8.
By Theorem 1.1, there exists a mapping C : X → Y satisfying the following:
1 C is a fixed point of J, that is,
C
x
2
1
Cx
8
7.17
for all x ∈ X. Since g : X → Y is odd, C : X → Y is an odd mapping. The mapping
C is a unique fixed point of J in the set
M h ∈ S : d h, g < ∞ .
7.18
Journal of Applied Mathematics
25
This implies that C is a unique mapping satisfying 7.17 such that there exists a
u ∈ 0, ∞ satisfying
μgx−Cx ut ≥L T∧ Φx,x t, Φ2x,x t
7.19
for all x ∈ X and t > 0.
2 dJ n g, C → 0 as n → ∞. This implies the equality
lim 8n g
n→∞
x
Cx
2n
7.20
for all x ∈ X.
3 dh, C ≤ 1/1 − 8αdh, Jh with h ∈ M, which implies the inequality
d g, C ≤
5α
,
1 − 8α
7.21
5α
t ≥L T∧ Φx,x t, Φ2x,x t.
1 − 8α
7.22
from which it follows that
μgx−Cx
This implies that the inequality 7.4 holds. From Dgx, y Df2x, 2y −
2Dfx, y, by 7.2, we deduce that
μDf2x,2y t ≥L Φ2x,2y t,
t
t
≥L Φx,y
μ−2Dfx,y t μDfx,y
2
2
7.23
and so, by LRN3 and 7.1, we obtain
μDgx,y 3t ≥L T∧ μDf2x,2y t, μ−2Dfx,y 2t
≥L T∧ Φ2x,2y t, Φx,y t ≥L Φ2x,2y t.
7.24
It follows that
t
μ8n Dgx/2n ,y/2n 3t μDgx/2n ,y/2n 3 n
8
1
t
t
≥ Φx/2n−1 ,y/2n−1
≥L · · · ≥L Φx,y
8n
8 8αn−1
for all x, y ∈ X, t > 0 and n ∈ N.
7.25
26
Journal of Applied Mathematics
Since limn → ∞ Φx,y 3/8t/8αn−1 1 for all x, y ∈ X and t > 0, by Theorem 2.4, we
deduce that
7.26
μDCx,y 3t 1L
for all x, y ∈ X and t > 0. Thus the mapping C : X → Y satisfies 1.4.
Now, we have
x C2x − 8Cx lim 8 g n−1 − 8 g n
n→∞
2
2
x x
n−1
n
8 lim 8 g n−1 − 8 g n
0
n→∞
2
2
x
n
n1
7.27
for all x ∈ X. Since the mapping x → C2x − 2Cx is cubic see Lemma 2.2 of 14, from
the equality C2x 8Cx, we deduce that the mapping C : X → Y is cubic.
Corollary 7.2. Let θ ≥ 0 and let p be a real number with p > 3. Let X be a normed vector space with
norm · . Let f : X → Y be an odd mapping satisfying
μDfx,y t ≥
t
p t θ xp y
7.28
for all x, y ∈ X and t > 0. Note that X, μ, TM is a complete LRN-space, in which L 0, 1, then
x x
Cx : lim 8n f n−1 − 2f n
n→∞
2
2
7.29
exists for each x ∈ X and defines a cubic mapping C : X → Y such that
μf2x−2fx−Cx t ≥
2p
2p − 8t
− 8t 51 2p θxp
7.30
for all x ∈ X and t > 0.
Proof. The proof follows from Theorem 7.1 by taking
Φx,y t :
t
p t θ xp y
7.31
Journal of Applied Mathematics
27
for all x, y ∈ X and t > 0. Then we can choose α 2−p , and we get
1 − 23−p t
μf2x−2fx−Cx t ≥ min , 1 − 23−p t 5 · 2−p θ 2xp
1 − 23−p t 5 · 2−p θ 2xp xp
1 − 23−p t
≥
1 − 23−p t 5 · 2−p θ 2xp xp
1 − 23−p t
2p − 8t
,
2p − 8t 5 · 2p 1θxp
7.32
which is the desired result.
Theorem 7.3. Let X be a linear space, let Y, μ, T∧ be a complete LRN-space, and let Φ be a mapping
from X 2 to DL (Φx, y is denoted by Φx,y ) such that, for some 0 < α < 8,
Φx/2,y/2 t ≤L Φx,y αt
x, y ∈ X, t > 0 .
7.33
Let f : X → Y be an odd mapping satisfying 7.2, then
1 n1 n
f
2
x
−
2f2
x
n → ∞ 8n
Cx : lim
7.34
exists for each x ∈ X and defines a cubic mapping C : X → Y such that
8−α
8−α
μf2x−2fx−Cx t ≥L T∧ Φx,x
t , Φ2x,x
t
5
5
7.35
for all x ∈ X and t > 0.
Proof. Let S, d be the generalized metric space defined in the proof of Theorem 7.1.
Consider the linear mapping J : S → S such that
Jhx :
1
h2x
8
7.36
for all x ∈ X, and we prove that J is a strictly contractive mapping with the Lipschitz constant
α/8.
Let h, k ∈ S be given such that dh, k < ε, then
μhx−kx εt ≥L T∧ Φx,x t, Φ2x,x t
7.37
28
Journal of Applied Mathematics
for all x ∈ X and t > 0. Hence
μJhx−Jkx
α
εt
8
μ1/8h2x−1/8k2x
α
εt
8
μh2x−k2x αεt
7.38
≥L T∧ Φ2x,2x αt, Φ4x,2x αt
≥ T∧ Φx,x t, Φ2x,x t
for all x ∈ X and t > 0. So, dh, k < ε implies that
α
ε.
8
7.39
α
dh, k
8
7.40
dJh, Jk ≤
This means that
dJh, Jk ≤
for all g, h ∈ S. Letting gx : f2x − 2fx for all x ∈ X, from 7.8, we get that
5
μgx−1/8g2x
t ≥L T∧ Φx,x t, Φ2x,x t
8
7.41
for all x ∈ X and t > 0. So, dg, Jg ≤ 5/8.
By Theorem 1.1, there exists a mapping C : X → Y satisfying the following:
1 C is a fixed point of J, that is,
C2x 8Cx
7.42
for all x ∈ X. Since g : X → Y is odd, C : X → Y is an odd mapping. The mapping
C is a unique fixed point of J in the set
M h ∈ S : d h, g < ∞ .
7.43
This implies that C is a unique mapping satisfying 7.42 such that there exists a
u ∈ 0, ∞ satisfying
μgx−Cx ut ≥L T∧ Φx,x t, Φ2x,x t
7.44
for all x ∈ X and t > 0.
2 dJ n g, C → 0 as n → ∞. This implies the equalit
lim
1
n → ∞ 8n
for all x ∈ X.
g2n x Cx
7.45
Journal of Applied Mathematics
29
3 dh, C ≤ 1/1 − α/8dh, Jh for every h ∈ M, which implies the inequality
d g, C ≤
5
,
8−α
7.46
5
t ≥L T∧ Φx,x t, Φ2x,x t
8−α
7.47
from which it follows that
μgx−Cx
for all x ∈ X and t > 0. This implies that the inequality 7.35 holds.
From
t
μDgx,y 3t ≥L T∧ Φ2x,2y t, Φx,y t ≥L T∧ Φ2x,2y t, Φx,y
,
8
7.48
by 7.33, we deduce that
μ
8−n Dg2n x,2n y
3t μ
Dg2n x,2n y
3 · 8 t≥L Φ
n
2n x,2n y
n−1
8
n−1 t
8
t ≥L · · · ≥ Φx,y
α
α
7.49
for all x, y ∈ X, t > 0, and n ∈ N. As n → ∞, we deduce that
μDCx,y 3t 1L
7.50
for all x, y ∈ X and t > 0. Thus the mapping C : X → Y satisfies 1.4.
Now, we have
1
1 n1 n
C2x − 8Cx lim n g 2 x − n−1 g2 x
n→∞ 8
8
1
1
n1
n
8 lim n1 g 2 x − n g2 x 0
n→∞ 8
8
7.51
for all x ∈ X. Since the mapping x → C2x − 2Cx is cubic see Lemma 2.2 of 14, from
the equality C2x 8Cx, we deduce that the mapping C : X → Y is cubic.
Corollary 7.4. Let θ ≥ 0 and let p be a real number with 0 < p < 3. Let X be a normed vector space
with norm · . Let f : X → Y be an odd mapping satisfying 7.28, then
1 n1 f 2 x − 2f2n x
n
n→∞8
Cx : lim
7.52
30
Journal of Applied Mathematics
exists for each x ∈ X and defines a cubic mapping C : X → Y such that
μf2x−2fx−Cx t ≥
8 −
2p t
8 − 2p t
51 2p θxp
7.53
for all x ∈ X and t > 0. Note that X, μ, TM is a complete LRN-space, in which L 0, 1.
Proof. The proof follows from Theorem 7.3 by taking
μDfx,y t ≥
t
p t θ xp y
7.54
for all x, y ∈ X and t > 0. Then we can choose α 2p , and we get the desired result.
Theorem 7.5. Let X be a linear space, let Y, μ, T∧ be a complete LRN-space, and let Φ be a mapping
from X 2 to DL (Φx, y is denoted by Φx,y ) such that, for some 0 < α < 1/2,
Φ2x,2y t ≤L Φx,y αt
x, y ∈ X, t > 0 .
7.55
Let f : X → Y be an odd mapping satisfying 7.2, then
x x
Ax : lim 2 f n−1 − 8f n
n→∞
2
2
n
7.56
exists for each x ∈ X and defines an additive mapping A : X → Y such that
1 − 2α
1 − 2α
t , Φ2x,x
t
μf2x−8fx−Ax t ≥L T∧ Φx,x
5α
5α
7.57
for all x ∈ X and t > 0.
Proof. Let S, d be the generalized metric space defined in the proof of Theorem 7.1.
Letting y : x/2 and gx : f2x − 8fx for all x ∈ X in 7.7, we get
μgx−2gx/2 5t ≥L T∧ Φx/2,x/2 t, Φx,x/2 t
7.58
for all x ∈ X and t > 0.
Now, we consider the linear mapping J : S → S such that
x
Jhx : 2h
2
7.59
for all x ∈ X. It is easy to see that J is a strictly contractive self-mapping on S with the
Lipschitz constant 2α.
Journal of Applied Mathematics
31
It follows from 7.58 and 7.55 that
μgx−2gx/2 5αt ≥ TM Φx,x t, Φ2x,x t
7.60
for all x ∈ X and t > 0. So, dg, Jg ≤ 5α < ∞.
By Theorem 1.1, there exists a mapping A : X → Y satisfying the following:
1 A is a fixed point of J, that is,
x
A
2
1
Ax
2
7.61
for all x ∈ X. Since g : X → Y is odd, A : X → Y is an odd mapping. The
mapping A is a unique fixed point of J in the set
M h ∈ S : d h, g < ∞ .
7.62
This implies that A is a unique mapping satisfying 7.61 such that there exists a
u ∈ 0, ∞ satisfying
μgx−Ax ut ≥L T∧ Φx,x t, Φ2x,x t
7.63
for all x ∈ X and t > 0.
2 dJ n g, A → 0 as n → ∞. This implies the equality
lim 2n g
n→∞
x
Ax
2n
7.64
for all x ∈ X.
3 dh, A ≤ 1/1 − 2αdh, Jh for each h ∈ M, which implies the inequality
d g, A ≤
5α
.
1 − 2α
7.65
This implies that the inequality 7.57 holds. Since μDgx,y 3t≥L Φ2x,2y t, it follows
that
t
μ2n Dgx/2n ,y/2n 3t μDgx/2n ,y/2n 3 n
2
1
t
t
≥ Φx/2n−1 ,y/2n−1
≥L · · · ≥L Φx,y
2n
2 2αn−1
7.66
32
Journal of Applied Mathematics
for all x, y ∈ X, t > 0, and n ∈ N. As n → ∞, we deduce that
μDAx,y 3t 1L
7.67
for all x, y ∈ X and t > 0. Thus, the mapping A : X → Y satisfies 1.4.
Now, we have
x x
A2x − 2Ax lim 2n g n−1 − 2n1 g n
n→∞
2
2
x
x
2 lim 2n−1 g n−1 − 2n g n
0
n→∞
2
2
7.68
for all x ∈ X. Since the mapping x → A2x − 8Ax is additive see Lemma 2.2 of 14,
from the equality A2x 2Ax, we deduce that the mapping A : X → Y is additive.
Corollary 7.6. Let θ ≥ 0 and let p be a real number with p > 1. Let X be a normed vector space with
norm · . Let f : X → Y be an odd mapping satisfying 7.28, then
x x
Ax : lim 2n f n−1 − 8f n
n→∞
2
2
7.69
exists for each x ∈ X and defines an additive mapping A : X → Y such that
μf2x−8fx−Ax t ≥
2p − 2t
2p − 2t 51 2p θxp
7.70
for all x ∈ X and t > 0, where X, μ, TM is a complete LRN-space in which L 0, 1.
Proof. The proof follows from Theorem 7.5 by taking
μDfx,y t ≥
t
p t θ xp y
7.71
for all x, y ∈ X and t > 0. Then we can choose α 2−p , and we get the desired result.
Theorem 7.7. Let X be a linear space, let Y, μ, T∧ be a complete LRN-space, and let Φ be a mapping
from X 2 to DL (Φx, y is denoted by Φx,y ) such that, for some 0 < α < 2,
Φx,y αt≥L Φx/2,y/2 t
x, y ∈ X, t > 0 .
7.72
Let f : X → Y be an odd mapping satisfying 7.2, then
1 n1 f 2 x − 8f2n x
n
n→∞2
Ax : lim
7.73
Journal of Applied Mathematics
33
exists for each x ∈ X and defines an additive mapping A : X → Y such that
2−α
2−α
t , Φ2x,x
t
μf2x−8fx−Ax t ≥L T∧ Φx,x
5α
5α
7.74
for all x ∈ X and t > 0.
Proof. Let S, d be the generalized metric space defined in the proof of Theorem 7.1.
Consider the linear mapping J : S → S such that
Jhx :
1
h2x
2
7.75
for all x ∈ X. It is easy to see that J is a strictly contractive self-mapping on S with the
Lipschitz constant α/2. Let gx f2x − 8fx, from 7.58, it follows that
μgx−1/2g2x
5
t ≥L T∧ Φx,x t, Φ2x,x t
2
7.76
for all x ∈ X and t > 0. So, dg, Jg ≤ 5/2. By Theorem 1.1, there exists a mapping A : X → Y
satisfying the following:
1 A is a fixed point of J, that is,
A2x 2Ax
7.77
for all x ∈ X. Since h : X → Y is odd, A : X → Y is an odd mapping. The mapping
A is a unique fixed point of J in the set
M h ∈ S : d h, g < ∞ .
7.78
This implies that A is a unique mapping satisfying 7.77 such that there exists a
u ∈ 0, ∞ satisfying
μgx−Ax ut ≥L T∧ Φx,x t, Φ2x,x t
7.79
for all x ∈ X and t > 0.
2 dJ n g, A → 0 as n → ∞. This implies the equality
1
g2n x Ax
n → ∞ 2n
lim
for all x ∈ X.
7.80
34
Journal of Applied Mathematics
3 dh, A ≤ 1/1 − α/2dh, Jh, which implies the inequality
d g, A ≤
5
.
2−α
7.81
This implies that the inequality 7.74 holds.
Proceeding as in the proof of Theorem 7.5, we obtain that the mapping A : X → Y
satisfies 1.4. Now, we have
1 n1 1
n
A2x − 2Ax lim n g 2 x − n−1 g2 x
n→∞ 2
2
1 n1 1
n
2 lim n1 g 2 x − n g2 x 0
n→∞ 2
2
7.82
for all x ∈ X. Since the mapping x → A2x − 8Ax is additive see Lemma 2.2 of 14,
from the equality A2x 2Ax, we deduce that the mapping A : X → Y is additive.
Corollary 7.8. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space
with norm · . Let f : X → Y be an odd mapping satisfying 7.28, then
1 n1 n
f
2
x
−
8f2
x
n → ∞ 2n
Ax : lim
7.83
exists for each x ∈ X and defines an additive mapping A : X → Y such that
μf2x−8fx−Ax t ≥
2 − 2p t
2 − 2p t 51 2p θxp
7.84
for all x ∈ X and t > 0, where X, μ, TM is a complete LRN-space in which L 0, 1.
Proof. The proof follows from Theorem 7.7 by taking
μDfx,y t ≥
t
p t θ xp y
7.85
for all x, y ∈ X and t > 0. Then we can choose α 2p , and we get the desired result.
8. Generalized Hyers-Ulam Stability of the Functional Equation 1.4:
An Even Case via Fixed-Point Method
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the
functional equation Dfx, y 0 in random Banach spaces, an even case.
Journal of Applied Mathematics
35
Theorem 8.1. Let X be a linear space, let Y, μ, T∧ be a complete LRN-space, and let Φ be a mapping
from X 2 to DL (Φx, y is denoted by Φx,y ) such that, for some 0 < α < 1/16,
Φx,y αt ≥L Φ2x,2y t
x, y ∈ X, t > 0 .
8.1
Let f : X → Y be an even mapping satisfying f0 0 and 7.2, then
x x
Qx : lim 16n f n−1 − 4f n
n→∞
2
2
8.2
exists for each x ∈ X and defines a quartic mapping Q : X → Y such that
1 − 16α
1 − 16α
μf2x−4fx−Qx t≥L T∧ Φx,x
t , Φ2x,x
t
5α
5α
8.3
for all x ∈ X and t > 0.
Proof. Letting x y in 7.2, we get
μf3y−6f2y15fy t ≥L Φy,y t
8.4
for all y ∈ X and t > 0. Replacing x by 2y in 7.2, we get
μf4y−4f3y4f2y4fy t ≥L Φ2y,y t
8.5
for all y ∈ X and t > 0. By 8.4 and 8.5,
μf4x−20f2x64fx 5t≥L T∧ μ4f3x−6f2x15fx 4t, μf4x−4f3x4f2x4fx t
≥L T∧ Φx,x t, Φ2x,x t
8.6
for all x ∈ X and t > 0. Letting gx : f2x − 4fx for all x ∈ X, we get
μgx−16gx/2 5t ≥L T∧ Φx/2,x/2 t, Φx,x/2 t
8.7
for all x ∈ X and t > 0. Let S, d be the generalized metric space defined in the proof of
Theorem 7.1.
Now we consider the linear mapping J : S → S such that Jhx : 16hx/2 for all
x ∈ X. It is easy to see that J is a strictly contractive self-mapping on S with the Lipschitz
constant 16α. It follows from 8.7 that
μgx−16gx/2 5αt ≥L T∧ Φx,x t, Φ2x,x t
8.8
5
< ∞.
d g, Jg ≤ 5α ≤
16
8.9
for all x ∈ X and t > 0. So,
36
Journal of Applied Mathematics
By Theorem 1.1, there exists a mapping Q : X → Y satisfying the following:
1 Q is a fixed point of J, that is,
Q
x
2
1
Qx
16
8.10
for all x ∈ X. Since g : X → Y is even with g0 0, Q : X → Y is an even
mapping with Q0 0. The mapping Q is a unique fixed point of J in the set
M h ∈ S : d h, g < ∞ .
8.11
This implies that Q is a unique mapping satisfying 8.10 such that there exists a
u ∈ 0, ∞ satisfying
μgx−Qx ut ≥L T∧ Φx,x t, Φ2x,x t
8.12
for all x ∈ X and t > 0.
2 dJ n g, Q → 0 as n → ∞. This implies the equality
lim 16n g
n→∞
x
Qx
2n
8.13
for all x ∈ X.
3 dh, Q ≤ 1/1 − 16αdh, Jh for every h ∈ M, which implies the inequality
d g, Q ≤
5α
.
1 − 16α
8.14
This implies that the inequality 8.3 holds.
Proceeding as in the proof of Theorem 7.1, we obtain that the mapping Q : X → Y
satisfies 1.4. Now, we have
x x
Q2x − 16Qx lim 16n g n−1 − 16n1 g n
n→∞
2
2
x x
16 lim 16n−1 g n−1 − 16n g n
0
n→∞
2
2
8.15
for all x ∈ X. Since the mapping x → Q2x − 4Qx is quartic, we get that the mapping
Q : X → Y is quartic.
Corollary 8.2. Let θ ≥ 0 and let p be a real number with p > 4. Let X be a normed vector space with
norm · . Let f : X → Y be an even mapping satisfying f0 0 and 7.28, then
x x
Qx : lim 16 f n−1 − 4f n
n→∞
2
2
n
8.16
Journal of Applied Mathematics
37
exists for each x ∈ X and defines a quartic mapping Q : X → Y such that
μf2x−4fx−Qx t ≥
2p
2p − 16t
− 16t 51 2p θxp
8.17
for all x ∈ X and t > 0, where X, μ, TM is a complete LRN-space in which L 0, 1.
Proof. The proof follows from Theorem 8.1 by taking
μDfx,y t ≥
t
p t θ xp y
8.18
for all x, y ∈ X and t > 0. Then we can choose α 2−p , and we get the desired result.
Theorem 8.3. Let X be a linear space, let Y, μ, T∧ be a complete LRN-space, and let Φ be a mapping
from X 2 to DL (Φx, y is denoted by Φx,y ) such that, for some 0 < α < 16,
Φx,y αt ≥ Φx/2,y/2 t
x, y ∈ X, t > 0 .
8.19
Let f : X → Y be an even mapping satisfying f0 0 and 7.2, then
1 n1 f 2 x − 4f2n x
n
n → ∞ 16
Qx : lim
8.20
exists for each x ∈ X and defines a quartic mapping Q : X → Y such that
16 − α
16 − α
μf2x−4fx−Qx t ≥L T∧ Φx,x
t , Φ2x,x
t
5
5
8.21
for all x ∈ X and t > 0.
Proof. In the generalized metric space S, d defined in the proof of Theorem 7.1, we consider
the linear mapping J : S → S such that
Jhx :
1
h2x
16
8.22
for all x ∈ X. It is easy to see that J is a strictly contractive self-mapping on S with the
Lipschitz constant α/16.
Letting gx : f2x − 4fx for all x ∈ X, by 8.7, we get
μgx−1/16g2x
5
t ≥L T∧ Φx,x t, Φ2x,x t
16
for all x ∈ X and t > 0. So, dg, Jg ≤ 5/16.
8.23
38
Journal of Applied Mathematics
By Theorem 1.1, there exists a mapping Q : X → Y satisfying the following:
1 Q is a fixed point of J, that is,
Q2x 16Qx
8.24
for all x ∈ X. Since g : X → Y is even with g0 0, Q : X → Y is an even
mapping with Q0 0. The mapping Q is a unique fixed point of J in the set
M h ∈ S : d h, g < ∞ .
8.25
This implies that Q is a unique mapping satisfying 8.24 such that there exists a
u ∈ 0, ∞ satisfying
μgx−Qx ut ≥L T∧ Φx,x t, Φ2x,x t
8.26
for all x ∈ X and t > 0.
2 dJ n g, Q → 0 as n → ∞. This implies the equality
lim
1
n → ∞ 16n
g2n x Qx
8.27
for all x ∈ X.
3 dg, Q ≤ 16/16 − αdg, Jg for each h ∈ M, which implies the inequality
d g, Q ≤ 5/16 − α.
8.28
This implies that the inequality 8.21 holds.
Proceeding as in the proof of Theorem 7.3, we obtain that the mapping Q : X → Y
satisfies 1.4. Now, we have
1 n1 1
n
Q2x − 16Qx lim
g 2 x − n−1 g2 x
n → ∞ 16n
16
1
1
n1
n
16 lim
g
2
x
−
g2
x
0
n → ∞ 16n1
16n
8.29
for all x ∈ X. Since the mapping x → Q2x − 4Qx is quartic, we get that the mapping
Q : X → Y is quartic.
Corollary 8.4. Let θ ≥ 0 and let p be a real number with 0 < p < 4. Let X be a normed vector space
with norm · . Let f : X → Y be an even mapping satisfying f0 0 and 7.28, then
1 n1 f 2 x − 4f2n x
n
n → ∞ 16
Qx : lim
8.30
Journal of Applied Mathematics
39
exists for each x ∈ X and defines a quartic mapping Q : X → Y such that
μf2x−4fx−Qx t ≥
16 − 2p t
16 − 2p t 51 2p θxp
8.31
for all x ∈ X and t > 0, where X, μ, TM is a complete LRN-space in which L 0, 1.
Proof. The proof follows from Theorem 8.3 by taking
μDfx,y t ≥
t
p t θ xp y
8.32
for all x, y ∈ X and t > 0. Then we can choose α 2p , and we get the desired result.
Theorem 8.5. Let X be a linear space, let Y, μ, T∧ be a complete LRN-space, and let Φ be a mapping
from X 2 to DL (Φx, y is by denoted Φx,y ) such that, for some 0 < α < 1/4,
x, y ∈ X, t > 0 .
Φx,y αt≥L Φ2x,2y t
8.33
Let f : X → Y be an even mapping satisfying f0 0 and 7.2, then
x x
T x : lim 4n f n−1 − 16f n
n→∞
2
2
8.34
exists for each x ∈ X and defines a quadratic mapping T : X → Y such that
μf2x−16fx−T x t ≥L T∧ Φx,x
1 − 4α
1 − 4α
t , Φ2x,x
t
5α
5α
8.35
for all x ∈ X and t > 0.
Proof. Let S, d be the generalized metric space defined in the proof of Theorem 7.1.
Letting gx : f2x − 16fx for all x ∈ X in 8.6, we get
μgx−4gx/2 5t ≥L T∧ Φx/2,x/2 t, Φx,x/2 t
8.36
for all x ∈ X and t > 0. It is easy to see that the linear mapping J : S → S such that
x
Jhx : 4h
2
8.37
for all x ∈ X, is a strictly contractive self-mapping with the Lipschitz constant 4α.
It follows from 8.36 that
μgx−4gx/2 5αt ≥L T∧ Φx,x t, Φ2x,x t
for all x ∈ X and t > 0. So, dg, Jg ≤ 5α < ∞.
8.38
40
Journal of Applied Mathematics
By Theorem 1.1, there exists a mapping T : X → Y satisfying the following:
1 T is a fixed point of J, that is,
T
x
2
1
T x
4
8.39
for all x ∈ X. Since g : X → Y is even with g0 0, T : X → Y is an even
mapping with T 0 0. The mapping T is a unique fixed point of J in the set
M {h ∈ S : dh, g < ∞}. This implies that T is a unique mapping satisfying
8.39 such that there exists a u ∈ 0, ∞ satisfying
μgx−T x ut ≥L T∧ Φx,x t, Φ2x,x t
8.40
for all x ∈ X and t > 0.
2 dJ n g, T → 0 as n → ∞. This implies the equality
lim 4n g
n→∞
x
T x
2n
8.41
for all x ∈ X.
3 dh, T ≤ 1/1 − 4αdh, Jh for each h ∈ M, which implies the inequality
d g, T ≤
5α
.
1 − 4α
8.42
This implies that the inequality 8.35 holds.
Proceeding as in the proof of Theorem 7.1, we obtain that the mapping T : X → Y
satisfies 1.4. Now, we have
x x
T 2x − 4T x lim 4n g n−1 − 4n1 g n
n→∞
2
2
x x
4 lim 4n−1 g n−1 − 4n g n
0
n→∞
2
2
8.43
for all x ∈ X. Since the mapping x → T 2x − 16T x is quadratic, we get that the mapping
T : X → Y is quadratic.
Corollary 8.6. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with
norm · . Let f : X → Y be an even mapping satisfying f0 0 and 7.28, then
x x
T x : lim 4 f n−1 − 16f n
n→∞
2
2
n
8.44
Journal of Applied Mathematics
41
exists for each x ∈ X and defines a quadratic mapping T : X → Y such that
μf2x−16fx−T x t ≥
2p − 4t
2p − 4t 51 2p θxp
8.45
for all x ∈ X and t > 0.
Proof. The proof follows from Theorem 8.5 by taking
Φx,y t :
t
p t θ xp y
8.46
for all x, y ∈ X. Then we can choose α 2−p , and we get the desired result.
Theorem 8.7. Let X be a linear space, let Y, μ, TM be a complete RN-space, and let Φ be a mapping
from X 2 to D (Φx, y is denoted by Φx,y ) such that, for some 0 < α < 4,
Φx,y αt ≥ Φx/2,y/2 t
x, y ∈ X, t > 0 .
8.47
Let f : X → Y be an even mapping satisfying f0 0 and 7.2, then
1 n1 n
f
2
x
−
16f2
x
n → ∞ 4n
T x : lim
8.48
exists for each x ∈ X and defines a quadratic mapping T : X → Y such that
4−α
4−α
μf2x−16fx−T x t ≥ TM Φx,x
t , Φ2x,x
t
5
5
8.49
for all x ∈ X and t > 0.
Proof. Let S, d be the generalized metric space defined in the proof of Theorem 7.1.
It is easy to see that the linear mapping J : S → S such that
Jhx :
1
h2x
4
8.50
for all x ∈ X is a strictly contractive self-mapping with the Lipschitz constant α/4.
Letting gx : f2x − 16fx for all x ∈ X, from 8.36, we get
μgx−1/4g2x
5
t
4
for all x ∈ X and t > 0. So, dg, Jg ≤ 5/4.
≥ TM Φx,x t, Φ2x,x t
8.51
42
Journal of Applied Mathematics
By Theorem 1.1, there exists a mapping T : X → Y satisfying the following:
1 T is a fixed point of J, that is,
T 2x 4T x
8.52
for all x ∈ X. Since g : X → Y is even with g0 0, T : X → Y is an even mapping
with T 0 0. The mapping T is a unique fixed point of J in the set
M h ∈ S : d h, g < ∞ .
8.53
This implies that T is a unique mapping satisfying 8.52 such that there exists a
u ∈ 0, ∞ satisfying
μgx−T x ut ≥ TM Φx,x t, Φ2x,x t
8.54
for all x ∈ X and t > 0.
2 dJ n g, T → 0 as n → ∞. This implies the equality
lim
1
n → ∞ 4n
g2n x T x
8.55
for all x ∈ X.
3 dh, T ≤ 1/1 − α/4dh, Jh for each h ∈ M, which implies the inequality
d g, T ≤ 5/4 − α.
8.56
This implies that the inequality 8.49 holds.
Proceeding as in the proof of Theorem 2.3, we obtain that the mapping Q : X → Y
satisfies 1.4. Now, we have
1
1 n1 n
T 2x − 4T x lim n g 2 x − n−1 g2 x
n→∞ 4
4
1
1 4 lim n1 g 2n1 x − n g2n x 0
n→∞ 4
4
8.57
for all x ∈ X. Since the mapping x → T 2x − 16T x is quadratic, we get that the mapping
T : X → Y is quadratic.
Corollary 8.8. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed vector space
with norm · . Let f : X → Y be an even mapping satisfying f0 0 and 7.28. Then
1 n1 f 2 x − 16f2n x
n
n→∞4
T x : lim
8.58
Journal of Applied Mathematics
43
exists for each x ∈ X and defines a quadratic mapping T : X → Y such that
μf2x−16fx−T x t ≥
4 − 2p t
4 − 2p t 51 2p θxp
8.59
for all x ∈ X and t > 0, where X, μ, TM is a complete LRN-space in which L 0, 1.
Proof. The proof follows from Theorem 8.5 by taking
Φx,y t :
t
p t θ xp y
8.60
for all x, y ∈ X and t > 0. Then we can choose α 2p , and we get the desired result.
Acknowledgments
The authors are grateful to the area Editor Professor Yeong-Cheng Liou and the reviewer
for their valuable comments and suggestions. Y. J. Cho was supported by the Basic Science
Research Program through the National Research Foundation of Korea NRF funded by the
Ministry of Education, Science and Technology Grant no. 2011-0021821.
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