1. Introduction
Ulam stability is one of the main topics in functional equation theory. The starting point of Ulam stability was the question formulated by S.M. Ulam during a talk given at Madison University, Wisconsin, and concerns the approximate homomorphisms of a metric group (see [
1]). The topic was intensively developed and studied in the last 50 years. For some results, methods, extensions and generalizations of the notion we refer the reader to [
1,
2].
The first results on Ulam stability for differential equations were obtained by M. Obloza [
3] and C. Alsina and R. Ger [
4]. T. Miura, S. Miyajima and S. Takahasi proved that the linear differential operator of
n-th order with constant coefficients is stable in Ulam sense if and only if its characteristic equation has no roots on the imaginary axis [
5]. This result was generalized by D. Popa and I. Raşa who obtained a sharp estimate of Ulam constant [
6]. Recently, A.R. Baias and D. Popa obtained the best Ulam constant for a second order linear differential operator with constant coefficients [
7]. Recall also the result obtained by M. Onitsuka et al. on the best constant of linear differential equation of the first order [
8]. For more details on Ulam stability of differential equations and differential operators, we refer the reader to [
9,
10,
11,
12].
To this moment there are few results on Ulam stability for the linear differential operator of higher order with nonconstant coefficients. This is indeed the reason for which in this paper we deal with Ulam stability of a second order linear differential operator with nonconstant coefficients.
In what follows, let X be a Banach space over . Let be an interval, , . By , we denote as usual the set of all n-times continuously differentiable functions defined on I with values in the Banach space X.
Definition 1. [13] Let A be a linear space over the field.
A functionis called a gauge on A if: - 1.
if and only if;
- 2.
for all.
Let us consider the linear differential operator with nonconstant coefficients
given by
Let
and denote
Then is a gauge. We suppose also that , are endowed with the same gauge for
Definition 2. The operator D is called Ulam stable if there exists a nonnegative constant L such that for everyand everysatisfyingthere existssuch thatand A function
y satisfying (
2) for some
is called an
approximate solution of the equation
. The number
L in the relation (
3) is called an
Ulam constant of
D.
2. Main Result
The main result on Ulam stability for the operator
D defined by the relation (
1) is contained in the next two theorems.
Theorem 1. Suppose that there existssuch thatand the Riccati equationhas a solutionwith.
Then for everysatisfyingthere exists,
such thatand Proof. With the above
u satisfying (
4), let
and
Multiplying it by
, by using (
4) and having in mind that
, we get
which leads to
Multiplying the previous relation by
, we obtain
Since
, it follows that
and this yields
To resume, (
7) is equivalent to
In particular, the homogeneous equation
is equivalent to
Now let
and
satisfying (
5). Then
y satisfies also (
7) and (
8) with
. Let
, be a solution of (
9) with
and
. According to (
10),
and it satisfies the relation
. So it remains to prove (
6). Using (
8) and (
10), we obtain
Suppose that
. Then
If
, then by (
11) we obtain
To finish the proof it remains to show that
To this end, let us remark that
where
Since
we have
and using (
4) we get
Combined with , this yields for and , for respectively.
Accordingly,
for
and
for
, respectively. Since
, we conclude that
. Thus, (
12) is proved and this finishes the proof of the theorem. □
A similar result is presented in the following.
Theorem 2. Suppose that there existssuch thatand the equationhas a solution,
with.
Then the operator D is Ulam stable with the Ulam constant K. Proof. The proof is analogous to the proof of Theorem 1. Only (
13) should be replaced by
□
In what follows we obtain some direct consequences of the Theorems 1 and 2.
Corollary 1. Suppose that there existssuch thatand the equationhas a solutionwithfor some.
Then the operatoris Ulam stable with the Ulam constant K. Corollary 2. Suppose that there existssuch thatand the equationhas a solutionwithfor some.
Then the operatoris Ulam stable with the Ulam constant K. The proof of the previous corollaries follows immediately by taking in the Theorems 1 and 2, respectively.
Remark 1. The particular case,
being a periodic function, is studied in [14,15] and corresponds to the Hill’s equation. Remark 2. The stability of the operator D defined by (1) depends on the existence of a global solution for the Riccati Equation (4). In the following is given an example where some appropriate conditions on the coefficients of the Equation (4) lead to the existence of such a solution. For this we will use the following result contained in [16]. Theorem 3. Letandand.
Iffor all,
then the Cauchy problemhas a unique nonnegative solutionon.
Corollary 3. Letand suppose that:
- (i)
;
- (ii)
;
- (iii)
Then the operator D defined by (1) is Ulam stable with the Ulam constant K. Proof. The result is a simple consequence of the Theorems 1 and 3. □
Analogously, we obtain the following result.
Corollary 4. Letand suppose that:
- (i)
;
- (ii)
;
- (iii)
Then the operator D defined by (1) is Ulam stable with the Ulam constant K. Example 1. Letand suppose that.
Then the operator D defined by the relationis Ulam stable with the Ulam constant.
Proof. Let . The condition and the continuity of g implies , or (clearly, if g has two values of opposite signs, then g must take the value zero at some point in I).
If , then, according to Corollary 3, D is Ulam stable with the Ulam constant . If , then D is Ulam stable with the Ulam constant , in view of Corollary 4. □
3. Conclusions
In this paper, we studied the Ulam stability of a linear differential operator of second order acting on a Banach space. We proved the Ulam stability of the equation using the assumption that a certain Riccati differential equation has a global solution. In appropriate conditions of the coefficients of the operators, we obtained an explicit representation of its Ulam constant. Finally, we presented some stability results for particular cases of the operator.
Author Contributions
These authors contributed equally to this work. All authors read and agreed to the published version of the manuscript.
Funding
This work was partially supported by a grant of the Romanian Ministry of Research and Innovation, project number 10PFE/16.10.2018, Perform-Tech-UPT—The increasing of the institutional performance of the Polytechnic University of Timişoara by strengthening the research, development and technological transfer capacity in the field of “Energy, Environment and Climate Change, within Program 1—Development of the national system of Research and Development, Subprogram 1.2—Institutional Performance—Institutional Development Projects—Excellence Funding Projects in RDI, PNCDI III.
Acknowledgments
The authors would like to thank the referees for carefully reading of the manuscript and for giving useful comments and suggestions which improve the paper.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Hyers, D.H.; Isac, G.; Rassias, T. Stability of Functional Equations in Several Variables; Birkhäuser Boston Inc.: Basel, Switzerland, 1998. [Google Scholar]
- Brzdȩk, J.; Popa, D.; Raşa, I.; Xu, B. Ulam Stability of Operators; Academic Press: Cambridge, MA, USA, 2018. [Google Scholar]
- Obloza, M. Hyers stability of linear differential equation. Roczn. Nauk. Dydakt. Pr. Math. 1993, 13, 259–270. [Google Scholar]
- Alsina, C.; Ger, R. On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 1998, 2, 373–380. [Google Scholar] [CrossRef]
- Miura, T.; Miyajima, S.; Takahasi, S.E. Hyers–Ulam stability of linear differential operator with constant coeffcients. Math. Nachr. 2003, 258, 90–96. [Google Scholar] [CrossRef]
- Popa, D.; Raşa, I. On the Hyers–Ulam stability of the linear differential equation. J. Math. Anal. Appl. 2011, 381, 530–537. [Google Scholar] [CrossRef] [Green Version]
- Baias, A.B.; Popa, D. On the best Ulam constant of the second order linear differential operator. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 2020, 114, 23. [Google Scholar] [CrossRef]
- Onitsuka, M.; Shoji, T. Hyers–Ulam stability of first-order homogeneous linear differential equations with a real-valued coefficient. Appl. Math. Lett. 2017, 63, 102–108. [Google Scholar] [CrossRef] [Green Version]
- Jung, S.-M. Hyers–Ulam stability of a system of first order linear differential equations with constant coefficients. J. Math. Anal. Appl. 2006, 320, 549–561. [Google Scholar] [CrossRef] [Green Version]
- Miura, T.; Miyajima, S.; Takahasi, S.E. A characterization of Hyers–Ulam stability of first order linear differential operators. J. Math. Anal. Appl. 2003, 286, 136–146. [Google Scholar] [CrossRef] [Green Version]
- Popa, D.; Raşa, I. Hyers–Ulam stability of the linear differential operator with nonconstant coefficients. Appl. Math. Comput. 2012, 219, 1562–1568. [Google Scholar] [CrossRef]
- Takahasi, S.E.; Takagi, H.; Miura, T.; Miyajima, S. The Hyers–Ulam stability constants of first order linear differential operators. J. Math. Anal. Appl. 2004, 296, 403–409. [Google Scholar] [CrossRef] [Green Version]
- Brzdȩk, J.; Popa, D.; Raşa, I. Hyers–Ulam stability with respect to gauges. J. Math. Anal. Appl. 2017, 453, 620–628. [Google Scholar] [CrossRef]
- Fukutaka, R.; Onitsuka, M. Ulam Stability for a Class of Hill’s Equations. Symmetry 2019, 11, 1483. [Google Scholar] [CrossRef] [Green Version]
- Fukutaka, R.; Onitsuka, M. Best constant for Ulam stability of Hill’s equations. Bull. Sci. Math. 2020, 163, 102888. [Google Scholar] [CrossRef]
- Barbu, V. Differential Equations; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
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