E extracta mathematicae Vol. 33, Núm. 1, 1 – 10 (2018)
A Generalization of the Hyers -Ulam-Aoki Type Stability
of Some Banach Lattice -Valued Functional Equation
Nutefe Kwami Agbeko, Patŕıcia Szokol
Institute of Mathematics, University of Miskolc, 3515 Miskolc-Egyetemváros, Hungary
matagbek@uni-miskolc.hu
Institute of Mathematics, MTA-DE Research Group “Equations, Functions and Curves”,
Hungarian Academy of Sciences and University of Debrecen,
P. O. Box 12, 4010 Debrecen, Hungary
szokol.patricia@inf.unideb.hu
Presented by Pier L. Papini Received April 30, 2017
Abstract: We obtained a generalization of the stability of some Banach lattice-valued func-
tional equation with the addition replaced in the Cauchy functional equation by lattice
operations and their combinations.
Key words: Banach lattices, Hyers-Ulam-Aoki type of stability.
AMS Subject Class. (2010): 39B82, 46B30, 46B42.
1. Introduction
All along (X , ∧X , ∨X ) will stand for a normed Riesz space and (Y, ∧Y, ∨Y)
for a Banach lattice with X + and Y+ their respective positive cones. Let us
pose the following problem.
Problem 1. Given three numbers ε, p, q ∈ (0, ∞), two Riesz spaces G1
and G2 with G2 being endowed with a metric d(·, ·), four lattice operations
∆∗G1, ∆
∗∗
G1
∈ {∧G1, ∨G1} and ∆
∗
G2
, ∆∗∗G2 ∈ {∧G2, ∨G2}, does there exist some
real number δ > 0 such that, if a mapping F : G1 → G2 satisfies
d
((
F
(
(τq|xt|)∆∗G1(η
q|y|)
))
∆∗G2
(
F
(
(τq|x|)∆∗∗G1(η
q|y|)
))
,
(
τpF(|x|)
)
∆∗∗G2
(
ηpF(|y|)
))
≤ δ
(1.1)
for all x, y ∈ G1 and all τ, η ∈ [0, ∞), then an operation-preserving functional
T : G1 → G2 exists with the property that
d
(
T(x), F(x)
)
≤ ε
1
2 n.k. agbeko, p. szokol
for all x ∈ G1 and all τ, η ∈ [0, ∞)?
If in (1.1) we let τ = η = 1, then the above problem reduces to the problem
posed and treated in [5].
The study of functional equations and inequalities in lattice environments
is motivated by the fact that many addition-related results or theorems can
be extended and can be proved mutatis mutandis. For more references about
the earliest extensions of the kind, we would refer the reader to the papers
[1, 2, 3, 4].
The main goal of this paper is to show how Ulam-Hyers-Aoki styled version
of perturbation (1.1) leads to the unique solution of the functional equation(
T
(
(τq|x|)∆∗X (η
q|y|)
))
∆∗Y
(
T
(
(τq|x|)∆∗∗X (η
q|y|)
))
=
(
τpT(|x|)
)
∆∗∗Y
(
ηpT(|y|)
) (1.2)
for all x, y ∈ X and all τ, η ∈ [0, ∞), where ∆∗X , ∆
∗∗
X ∈ {∧X , ∨X } and
∆∗Y, ∆
∗∗
Y ∈ {∧Y, ∨Y} are fixed lattice operations.
Remark 1.1. If we let η = τ and y = x in equation (1.2), then we observe
that
T (τq |x|) = τpT (|x|) (1.3)
for all x ∈ X and all τ ∈ [0, ∞).
The results we obtained are straightforward generalizations of Agbeko [3,
5, 6] and Salahi et al [16]. For an additional reference we would like to mention
the paper [7] where we proved separation and stability results for operators
mapping a semi-group with values in a Riesz lattice.
We recall that a functional H : X → Y is cone-related if H (X +) =
{H (|x|) : x ∈ X} ⊂ Y+ (see more about this notion in [3, 4]).
Some few references about Hyers-Ulam stability problems and solutions
can be found, e.g. in [8, 11, 13, 14, 15].
Our theorems will be deduced from the following Forti’s result [10].
Theorem 1.1. (Forti) Let (X, d) be a complete metric space and S an
appropriate set. Assume some functions f : S → X, G : S → S, H : X → X
and δ : S → [0, ∞) satisfy the inequality
d
(
H
(
f(G(x))
)
, f(x)
)
≤ δ(x), (1.4)
stability of a functional equation on Banach lattices 3
for all x ∈ S. If function H both is continuous and satisfies the inequality
d
(
H (u) , H (v)
)
≤ φ
(
d (u, v)
)
, u, v ∈ X, (1.5)
for a certain non-decreasing subadditive function φ : [0, ∞) → [0, ∞) and the
series
∞∑
j=0
φj
(
δ
(
Gj (x)
))
(1.6)
is convergent for every x ∈ S, then there exists a unique function F : S → X
solution of the functional equation
H
(
F (G (x))
)
= F (x) , x ∈ S, (1.7)
and satisfying the following inequality:
d
(
F (x) , f (x)
)
≤
∞∑
j=0
φj
(
δ
(
Gj (x)
))
. (1.8)
The function F is given by
F (x) = lim
n→∞
Hn
(
f(Gn(x))
)
. (1.9)
2. The main results
Theorem 2.1. Given a pair of real numbers (p, q) ∈ (0, ∞) × (0, ∞),
consider a cone-related functional F : X → Y for which there are numbers
ϑ > 0 and α with qα ∈ (p, ∞) such that∥∥∥F((τq|x|)∆∗X (ηq|y|))∆∗YF((τq|x|)∆∗∗X (ηq|y|))
−
(
τpF(|x|)
)
∆∗∗Y
(
ηpF(|y|)
)∥∥∥ ≤ 2(p−1)ϑ(∥x∥α + ∥y∥α) (2.1)
for all x, y ∈ X and all τ, η ∈ [0, ∞). Then the sequence (2npF (2−nq |x|))n∈N
is a Cauchy sequence for every x ∈ X . Let the functional T : X + → Y+ be
defined by
T (|x|) = lim
n→∞
2npF
(
2−nq |x|
)
(2.2)
for all x ∈ X . Then T both is the unique solution of (1.2) and satisfies
inequality ∥∥T (|x|) − F (|x|) ∥∥ ≤ 2pϑ
2qα − 2p
∥x∥α (2.3)
for every x ∈ X .
4 n.k. agbeko, p. szokol
Theorem 2.2. Given a pair of real numbers (p, q) ∈ (0, ∞) × (0, ∞),
consider a cone-related functional F : X → Y for which there are numbers
β ∈ [0, ∞), ϑ > 0 and α with qα ∈ (0, p) such that∥∥∥F((τq|x|)∆∗X (ηq|y|))∆∗YF((τq|x|)∆∗∗X (ηq|y|))
−
(
τpF(|x|)
)
∆∗∗Y
(
ηpF(|y|)
)∥∥∥ ≤ β + ϑ2−(p+1)(∥x∥α + ∥y∥α) (2.4)
for all x, y ∈ X and all τ, η ∈ [0, ∞). Then the sequence (2−npF (2nq |x|))n∈N
is a Cauchy sequence for every fixed x ∈ X . Let the functional T : X + → Y+
be defined by
T (|x|) = lim
n→∞
2−npF
(
2nq|x|
)
(2.5)
for all x ∈ X . Then T both is the unique solution of (1.2) and satisfies
inequality ∥∥T(|x|) − F(|x|)∥∥ ≤ β2p
2p − 1
+
ϑ∥x∥α2qα
2p − 2qα
(2.6)
for every x ∈ X .
Remark 2.1. If the conditions of Theorem 2.1 or Theorem 2.2 hold true,
then F (0) = 0.
Proof. The proof is similar to its counterpart in [5, 6] under the conditions
of Theorem 2.1 or Theorem 2.2 when β = 0. Under the condition of Theorem
2.2 with β > 0, we need to prove that F (0) = 0. Suppose in the contrary that
F (0) > 0 were true. Then by letting x = y = 0 and η = τ in (2.4), inequality∥∥F(0) − τpF(0)∥∥ ≤ β
follows or, equivalently
|τp − 1| ≤
β
∥F (0)∥
< ∞,
which, as τ tends to infinity, would lead to an absurdity, indeed. Hence the
relation F (0) = 0 must be true.
Remark 2.2. Let Z be a set closed under the scalar multiplication, i.e.
bz ∈ Z whenever b ∈ R and z ∈ Z. Given a number c ∈ R let the function
γ : Z → Z be defined by γ (z) = cz. Then γj : Z → Z the j-th iteration of γ
is given by γj (z) = cjz for every natural number j ≥ 2.
stability of a functional equation on Banach lattices 5
3. Proof of the main results
We shall use the technique in [5] to prove the main theorems.
Proof of Theorem 2.1. First, if we choose τ = η = 2, y = x and replace x
by 2−qx in inequality (2.1) then we obviously have∥∥2pF(2−q|x|) − F(|x|)∥∥ ≤ ϑ2p−qα∥x∥α. (3.1)
Next, let us define the following functions:
G : X + → X +, G (|x|) = 2−q |x| , for all x ∈ X ,
δ : X + → [0, ∞) , δ (|x|) = ϑ2p−qα ∥x∥α , for all x ∈ X ,
φ : [0, ∞) → [0, ∞) , φ (t) = 2pt,
H : Y+ → Y+, H (|y|) = 2p |y| , for all y ∈ Y,
d(·, ·) : Y+ × Y+ → [0, ∞), d
(
|y1|, |y2|
)
=
∥∥|y1| − |y2|∥∥, for all y1, y2 ∈ Y.
We shall verify the fulfilment of all the three conditions of the Forti’s theorem
as follows.
(I) From inequality (3.1) we obviously have
d
(
H
(
F(G(|xt|))
)
, F(|x|)
)
=
∥∥H(F(G(|x|))) − F(|x|)∥∥
=
∥∥2pF(2−q|x|) − F(|x|)∥
≤ ϑ2p−qα
∥∥x∥α = δ(|x|).
(II) d
(
H(|y1|), H(|y2|)
)
= 2p
∥∥|y1|−|y2|∥∥ = φ(d(|y1| , |y2|)) for all y1, y2 ∈ Y.
(III) Clearly, on the one hand φ is a non-decreasing subadditive function on
the positive half line, and on the other hand by applying Remark 2.2 on
both the iterations Gj and φj of G and φ respectively, one can observe
that
∞∑
j=0
φj
(
δ(Gj(|x|))
)
= ϑ2p−qα∥x∥α
∞∑
j=0
2(p−qα)j = ϑ∥x∥α
2p
2qα − 2p
< ∞.
Then in view of Forti’s theorem, sequence (Hn(F(Gn|x|)))n∈N is a Cauchy
sequence for every x ∈ X and thus so is (2npF(2−nq|x|))n∈N. Furthermore,
the mapping (2.2) satisfies inequality (2.3).
6 n.k. agbeko, p. szokol
Next, we prove that T solves (1.2). In fact, in (2.1) substitute x with
2−nqx also y with 2−nqy, and fix arbitrarily τ, η ∈ [0, ∞). Then∥∥∥∥∥F
(
(τq|x|)∆∗X (η
q|y|)
2nq
)
∆∗YF
(
(τq|x|)∆∗∗X (η
q|y|)
2nq
)
−
(
τpF
(
|x|
2nq
))
∆∗∗Y
(
ηpF
(
|y|
2nq
))∥∥∥∥∥ ≤ 2(p−1)ϑ
(∥∥∥ x
2nq
∥∥∥α + ∥∥∥ y
2nq
∥∥∥α).
Multiplying both sides of this last inequality by 2np yields
2np
∥∥∥∥∥F
(
(τq|x|)∆∗X (η
q|y|)
2nq
)
∆∗YF
(
(τq|x|)∆∗∗X (η
q|y|)
2nq
)
−
(
τpF
(
|x|
2nq
))
∆∗∗Y
(
ηpF
(
|y|
2nq
))∥∥∥∥∥ ≤ ϑ2(1−p) ∥x∥
α + ∥y∥α
2n(qα−p)
.
(3.2)
Taking the limit in (3.2) we have via (2.2) that∥∥T((τq|x|)∆∗X (ηq|y|))∆∗YT((τq|x|)∆∗∗X (ηq|y|))−(τpT(|x|))∆∗∗Y (ηpT(|y|))∥∥ = 0
for all τ, η ∈ [0, ∞) and all x, y ∈ X , which is equivalent to (1.2). Thus T
also satisfies (1.3) in Remark 1.1. Finally we show the uniqueness, using a
technique in [16]. In fact, assume that there is another functional S : X → Y
which satisfies (1.2) and the inequality ∥S(|x|) − F(|x|)∥ ≤ δ2∥x∥α2 for some
numbers α2, δ2 ∈ (0, ∞) with qα2 > p, and for all x ∈ X . In (2.3) let
δ1 :=
2pϑ
2qα−2p , α1 := α and by choosing τ = 2
−n in Remark 1.1 one can
observe that for all x ∈ X∥∥S(|x|) − T(|x|)∥∥ = 2np∥∥S(2−nq|x|) − T(2−nq|x|)∥∥
≤ 2np
∥∥F(2−nq|x|) − T(2−nq|x|)∥∥
+ 2np
∥∥S(2−nq|x|) − F(2−nq|x|)∥∥
≤ 2npδ1∥2−nqx∥α1 + 2npδ2∥2−nqx∥α2
= 2(p−qα1)nδ1∥x∥α1 + 2(p−qα2)nδ2∥x∥α2.
Hence ∥∥S(|x|) − T(|x|)∥∥ ≤ 2(p−qα1)nδ1∥x∥α1 + 2(p−qα2)nδ2∥x∥α2
which, in the limit, yields ∥S(|x|)−T(|x|)∥ = 0 or equivalently S(|x|) = T(|x|)
for all x ∈ X .
stability of a functional equation on Banach lattices 7
This was to be proven.
Proof of Theorem 2.2. First, if we choose τ = η = 2−1, y = x and replace
x by 2qx in inequality (2.4) then we obviously have∥∥2−pF(2q|x|) − F(|x|)∥∥ ≤ β + ϑ2qα−p∥x∥α. (3.3)
Next, let us define the following functions:
G : X + → X +, G(|x|) = 2q|x|, for all x ∈ X ,
δ : X + → [0, ∞), δ(|x|) = β + ϑ2qα−p∥x∥α, for all x ∈ X ,
φ : [0, ∞) → [0, ∞), φ(t) = 2−pt,
H : Y+ → Y+, H(|y|) = 2−p|y|, for all y ∈ Y,
d(· , ·) : Y+× Y+ → [0, ∞), d
(
|y1|, |y2|
)
=
∥∥|y1| − |y2|∥∥, for all y1, y2 ∈ Y.
We shall verify the fulfilment of all the three conditions of the Forti’s theorem
as follows.
(I) From inequality (3.3) we obviously have
d
(
H
(
F(G(|x|))
)
, F(|x|)
)
=
∥∥H(F(G(|x|))) − F(|x|)∥∥
=
∥∥2−pF(2q|x|) − F(|x|)∥∥
≤ β + ϑ2qα−p∥x∥α = δ(|x|).
(II) d
(
H(|y1|), H(|y2|)
)
= 2−p
∥∥|y1|−|y2|∥∥= φ(d(|y1|, |y2|)) for all y1, y2 ∈ Y.
(III) Clearly, on the one hand φ is a non-decreasing subadditive function on
the positive half line, and on the other hand by applying Remark 2.2 on
both the iterations Gj and φj of G and φ respectively, one can observe
that
∞∑
j=0
φj
(
δ(Gj(|x|))
)
= β
∞∑
j=0
2−pj + ϑ2qα−p∥x∥α
∞∑
j=0
2(qα−p)j
=
β2p
2p − 1
+
ϑ∥x∥α2qα
2p − 2qα
< ∞.
Then in view of Forti’s theorem, sequence (Hn
(
F(Gn|x|))
)
n∈N is a Cauchy
sequence for every x ∈ X and thus so is
(
2−npF(2nq|x|)
)
n∈N. Furthermore,
the mapping (2.5) satisfies inequality (2.6).
8 n.k. agbeko, p. szokol
Next, we prove that T solves (1.2). In fact, in (2.4) substitute x with 2nqx
also y with 2nqy, and fix arbitrarily τ, η ∈ [0, ∞). Then∥∥∥F(2nq((τq|x|)∆∗X (ηq|y|)))∆∗YF(2nq((τq|x|)∆∗∗X (ηq|y|)))
−
(
τpF(2nq|x|)
)
∆∗∗Y
(
ηpF(2nq|y|)
)∥∥∥
≤ β + 2−(p+1)ϑ
(
∥2nqx∥α + ∥2nqy∥α
)
.
Dividing both sides of this last inequality by 2np yields∥∥∥∥∥
F
(
2nq
(
(τq|x|)∆∗X (η
q|y|)
))
∆∗YF
(
2nq
(
(τq|x|)∆∗∗X (η
q|y|)
))
2np
−
(
τpF(2nq|x|)
)
∆∗∗Y
(
ηpF(2nq|y|)
)
2np
∥∥∥∥∥
≤ β2−np + 2−(p+1)ϑ(∥x∥α + ∥y∥α)2(qα−p)n.
(3.4)
Taking the limit in (3.4) we have via (2.5) that∥∥∥T((τq|x|)∆∗X (ηq|y|))∆∗YT((τq|x|)∆∗∗X (ηq|y|))−(τpT(|x|))∆∗∗Y (ηpT(|y|))∥∥∥ = 0
for all τ, η ∈ [0, ∞) and all x, y ∈ X , which is equivalent to (1.2). Thus
T satisfies (1.3) in Remark 1.1. Finally we show the uniqueness, using a
technique in [16]. In fact, assume that there is another functional S : X → Y
which satisfies (1.2) and the inequality ∥S(|x|) − F(|x|)∥ ≤ β2 + δ2∥x∥α2 for
some numbers α2, δ2 ∈ (0, ∞), β2 ∈ [0, ∞) with qα2 < p, and for all x ∈ X .
In (2.6) let β1 :=
β2p
2p−1, δ1 :=
ϑ2qα
2p−2qα , α1 := α and by choosing τ = 2
n in
Remark 1.1 one can observe that for all x ∈ X∥∥S(|x|) − T(|x|)∥∥ = 2−np∥∥S(2nq|x|) − T(2nq|x|)∥∥
≤ 2−np
∥∥F(2nq|x|) − T(2nq|x|)∥∥
+ 2−np
∥∥S(2nq|x|) − F(2nq|x|)∥∥
≤ 2−np
(
β1 + δ1∥2nqx∥α1
)
+ 2−np
(
β2 + δ2∥2nqx∥α2
)
= 2−np(β1 + β2) + δ12
(qα1−p)n∥x∥α1 + δ22(qα2−p)n∥x∥α2.
Hence∥∥S(|x|) − T(|x|)∥∥ ≤ 2−np(β1 + β2) + δ12(qα1−p)n∥x∥α1 + δ22(qα2−p)n∥x∥α2
stability of a functional equation on Banach lattices 9
which, in the limit, yields ∥S(|x|)−T(|x|)∥ = 0 or equivalently S(|x|) = T(|x|)
for all x ∈ X . This completes the proof.
To end our paper we give an example showing that stability fails to occur
in general.
Example 1. Fix arbitrarily τ, η ∈ (0, 2) and consider the function
F : [0, ∞) → [0, ∞) , F (x) = xα+1, α =
p
q
.
Since F is increasing the first equality in the chain below is valid, entailing
the subsequent relations:∣∣∣F((τqx) ∨ (ηqy)) − (τpF(x)) ∧ (ηpF(y))∣∣∣
=
∣∣(τqx)α+1 ∨ (ηqy)α+1 − (τpxα+1) ∧ (ηpyα+1)∣∣
≤ (τqx)α+1 ∨ (ηqy)α+1 + (τpxα+1) ∧ (ηpyα+1)
≤ (2qx)α+1 ∨ (2qy)α+1 + (2pxα+1) ∧ (2pyα+1)
≤ 2p+q(xα+1 ∨ yα+1) + 2p+q(xα+1 ∧ yα+1)
= 2p+q(xα+1 + yα+1)
for all x, y ∈ [0, ∞). Now, let T : [0, ∞) → [0, ∞) be a function such that
T (µqx) = µpT (x) for all x ∈ [0, ∞) and all µ ∈ [0, ∞). Since x =
(
x1/q
)q
,
and α is the ratio of p and q, we can then note that T (x) = xαT (1) for every
x ∈ [0, ∞). Now,
sup
x∈(0, ∞)
|F (x) − T (x)|
2p+qxα+1
= sup
x∈(0, ∞)
∣∣∣xα+1 − T((x1q )q)∣∣∣
2p+qxα+1
= sup
x∈(0, ∞)
∣∣xα+1 − xαT (1)∣∣
2p+qxα+1
=
1
2p+q
sup
x∈(0, ∞)
∣∣∣∣1 − T (1)x
∣∣∣∣ = ∞.
The above example about the lack of stability on the real line in lattice en-
vironments is the counterpart of the example given by S. Czerwik [9] in the
addition environments for quadratic mappings.
10 n.k. agbeko, p. szokol
Acknowledgements
The second author is supported by the Hungarian Academy of Sci-
ences, and OTKA Grant PD124875.
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