1. Introduction
Let be the field of real or complex numbers. Throughout this paper, denotes a Banach space over the field while denotes the linear space of all n times differentiable functions with continuous n-th derivatives, defined on with values in X.
Let A and B be two linear spaces over the field
Definition 1. The function satisfying the following properties:
- (i)
if and only if ;
- (ii)
for all
is called a gauge on
For the function
, we define
Then,
is a gauge on
We suppose that both linear spaces
and
are endowed with the same gauge
Let be two gauges on the linear spaces A and respectively, and let be a linear operator.
We denote by the kernel of D and by the range of the operator D, respectively.
Definition 2. We say that the operator D is Ulam-stable if there exists such that, for every and every with the property that , there exists such that
The number
K in the above definition is called an
Ulam constant of the operator
Further, we denote by
the infimum of all Ulam constants of the operator
In general, the infimum of all Ulam constants of the operator
D is not necessarily an Ulam constant of
D (see [
1]). However, for the case where
is also an Ulam constant of the operator
D, we will call it
the best Ulam constant of
D.
The stability problem was initially raised by Ulam [
2] in the fall of 1940 and partially answered a year later by Hyers [
3], and it has developed ever since, growing as such into a vast area of research. Nowadays, Ulam stability follows various directions of research, from the stability of operators to the stability of different types of equations. For a complete approach to this topic, we refer the reader to [
4,
5].
For the sake of completeness, we will first present a brief historical background of the problem of finding the best Ulam constant of differential equations and operators. Consequently, we will mention here only some results in the field connected with the stability of operators that also serve the purpose of the present paper. As far as we know, the first Ulam stability result for differential equations was obtained by M.Obłoza in [
6]. Hereafter, the topic was deeply investigated by many mathematicians. We can mention here T. Miura, S. Miyajima, S.E. Takahasi [
7,
8,
9], and S. M Jung [
10], who obtained stability results for various differential equations and partial differential equations. Representations of the best Ulam constants of linear and bounded operators acting on normed spaces were given in [
1,
11].
The study of Ulam stability has also been developed for the higher-order differential operators with constant coefficients. In [
12,
13], sharp estimates for the Ulam constant of the first-order and the higher-order linear differential operators were given. Later, the work was improved, and the best Ulam constant was obtained for the case of the first-order linear differential operator in [
7]. Shortly after, in [
14], A.R. Baias and D. Popa extended the study of Ulam stability to the case of the second-order linear differential operator with constant coefficients
and obtained its best Ulam constant as
where
p and
q are the characteristic roots of the equation; namely,
and
The stability of the second-order linear differential equations with variable coefficients was treated by M. Onitsuka in [
15]. For linear differential equations with periodic coefficients, we can mention the stability results obtained in [
16], while for second-order linear dynamic equations on time scales, we refer the reader to the paper by D.R. Anderson and M. Onitsuka [
17]. The Ulam stability of some integro-differential equations was studied in [
18,
19].
In [
8], it was proved that the
n-order linear differential operator with constant coefficients is Ulam stable if and only if its characteristic equation has no pure imaginary roots. In this case, the Ulam constant is given by
where
denote the roots of the characteristic equation. Important steps in finding the best Ulam constant of the same operator were made in [
20], where the best Ulam constant was obtained only for the case of distinct roots in the characteristic equation.
The results in the next section streamline those given by [
14] and extend them from the case of the second-order differential operator to the case of the third-order linear differential operator. In this paper, we first obtain a stability result for the third-order linear differential operator acting in a Banach space. However, the main purpose of the present paper is to give a complete answer to the problem of the best Ulam constant for this operator by obtaining an expression of the best Ulam constant in all possible cases. This research was motivated by the fact that the best Ulam constant of an equation or operator offers the best measure of the error between the approximate and the exact solution of the corresponding equation or operator.
2. Main Results
Let
and let
be defined by
If
, and
r are the roots of the characteristic polynomial
, then, as is well known, the kernel of
D takes one of the below forms, depending on the order of multiplicity of the roots of the characteristic equation:
for the case of distinct roots;
for the case
;
for the case
respectively.
The operator D is surjective; so, for every , one can find a particular solution of the equation using, for example, the method of variation of constants. Next, we will determine the form of the particular solutions, taking into account the order of multiplicity of the roots of the characteristic equation.
For the case of
distinct roots in the characteristic equation, the form of a particular solution to the equation
is
where
, and
are functions of class
that satisfy
Consequently, we obtain
For the sake of simplicity, we denote this by
; hence, a particular solution of the equation
is given by
Analogously, for the case
, a particular solution is given by
while, for the case
, the form of the particular solution is
The main results concerning the Ulam stability of the operator D are given in the next theorems.
Theorem 1. Suppose that and r are distinct roots of the characteristic equation with and , and let Thenm for every satisfyingthere exists a unique such thatwhere Proof. Existence.
Suppose that
satisfies Equation (
10) and let
Then,
and
for some
where
is a particular solution of the equation
given by Equation (
7).
- (i)
Let
and
. Define
by the relation
where
The integrals in the definition of the constants
are convergent since
,
, and
Then,
Now, letting
in the above integral, we obtain
- (ii)
Let
and
. Define
by the relation
where
Letting
and, respectively,
in the above integrals, it follows that
- (iii)
Let
and
. Define
by the relation
where
Letting
and, respectively,
in the above integrals, it follows that
- (iv)
Let
and
. Define
by the relation
where
Now, letting
in the above integral, we obtain
Uniqueness. Suppose that, for some
satisfying Equation (
10), there exist
such that
Then,
However,
; thus,
belongs to the set given by Equation (3). If
then
which contradicts the boundedness of
We conclude that
; therefore,
The theorem is proved. □
Theorem 2. Suppose that with and and let Then, for every satisfyingthere exists a unique such thatwhere Proof. Existence.
Let
satisfying Equation (
13) and let
with
Then,
- (i)
Let
and
Define
by the relation
where
The proof of the convergence of the improper integrals is analogous to that given in Theorem 1.
Now, letting
in the above integral, we obtain
- (ii)
Let
and
The proof follows analogously, defining
by the relation
with
Letting
and, respectively,
in the above integrals, it follows that
- (iii)
Let
and
Then, we define
by the relation
with
Letting
in the above integrals, it follows that
Uniqueness. Suppose that, for some
satisfying Equation (
13), there exist
such that
Then,
However,
; hence, there exist
such that
If
then
which contradicts the boundedness of
We conclude that
; therefore,
The theorem is proved. □
Theorem 3. Suppose that with and let Then, for every satisfyingthere exists a unique such that Proof. Existence. Let
satisfying Equation (
17) and let
with
Then,
where
is given by Equation (9).
- (i)
Let
. Define
by the relation
where
Now, letting
in the above integral, we obtain
- (ii)
Let
. Define
by the relation
where
Now, letting
in the above integral, we obtain
The existence is proved.
Uniqueness. Suppose that, for some
satisfying Equation (
17), there exist
such that
However,
; hence, there exist
such that
If
then
which contradicts the boundedness of
We conclude that
; therefore,
and the result holds. □
Theorem 4. Suppose that and r aredistinct rootsof the characteristic equation with and Then, the best Ulam constant of D is given by Proof. Suppose that D admits an Ulam constant
- (i)
First, let
and
Then,
Let
Take
and
as arbitrarily chosen and consider
defined by
where
denotes the conjugate of
Obviously, the function
f is continuous on
and
for all
Let
be the solution to
given by
Since
f is bounded and
, and
, it follows that
is bounded on
. Furthermore,
and the Ulam stability of
D for
with the constant
K leads to the existence of
which is given by Equation (3) such that
If
, we get, in view of the boundedness of
which contradicts relation (
23). Therefore,
and relation (
23) becomes
Taking
in Equation (
25) we get
; i.e.,
or, equivalently,
Consequently, letting in Equation (26), we get which contradicts the supposition
- (ii)
The case where
and
follows analogously for
for
, and
where
h is defined by
- (iii)
Consider
, and
Let
Take an arbitrary
and define the function
by
It can be seen that f is continuous and
Let
be the solution to
given by
Since f is bounded, taking account of the signs of the roots and r, it follows that is bounded.
On the other hand, all the elements of
are unbounded, except
We conclude that the relation
takes place only for
; therefore,
For
, it follows that
, which is equivalent to
Now, letting in Equation (29), it follows that which is a contradiction.
- (iv)
The case where
and
follows analogously for
for
, and
where
and
are defined by
respectively.
□
Theorem 5. Suppose that p is a double root and r a simple root of the characteristic equation with and Then, the best Ulam constant of D is given by Proof. First, let
and
Let
and
Consider
and
and define
The function
f is continuous on
and
for all
Suppose that
D admits an Ulam constant
Let
given by
be the solution to
where
f is given by Equation (
32). The relation
leads to the existence of a unique
such that
in view of the Ulam stability of
D with the constant
The function
is bounded since
f is bounded and
and
On the other hand, all the elements of
are unbounded except
We conclude that Equation (
34) applies only for
; therefore,
For
, it follows that
, which is equivalent to
We prove that
Indeed,
Now, letting
in Equation (35), it follows that
which is a contradiction.
The proof of the other cases is obtained similarly by converting the signs of and respectively. □
Theorem 6. Suppose that p is a triple root of the characteristic equation with Then, the best Ulam constant of D is given by Proof. Consider first the case where
Suppose that
D is stable with an Ulam constant
Take
and let
be given by
where
denotes the imaginary part of the root
p.
The function of the right-hand side of Equation (
19) is a solution to the equation
and, in the following, we denote it by
. Consequently,
Replacing
f given by Equation (
36) in Equation (
37), it follows that
The substitution
leads to
Since
it follows in view of the Ulam stability of
D, for
that there exists a solution
to
such that
If
, we have
which contradicts Equation (
39), since
is bounded. If
then Equation (
39) becomes
which contradicts the supposition
The case where
follows analogously. The theorem is proved. □