Ulam Stability for a Class of Hill’s Equations

This paper deals with Ulam’s type stability for a class of Hill’s equations. In the two assertions of the main theorem, we obtain Ulam stability constants that are symmetrical to each other. By combining the obtained results, a necessary and sufficient condition for Ulam stability of a Hill’s equation is established. The results are generalized to nonhomogeneous Hill’s equations, and then application examples are presented. In particular, it is shown that if the approximate solution is unbounded, then there is an unbounded exact solution.


Introduction
The concept of Ulam's type stability was posed by Ulam, and its development is remarkable. Many researchers have studied this problem for functional equations. For an overview, see the book [1] written by Brzdęk, Popa, Raşa, and Xu. In 1998, Alsina and Ger [2] introduced this concept in the field of differential equations. After that, the study of Ulam's type stability for differential equations continued to grow (see, [3][4][5][6][7][8][9][10]). Recently, Fukutaka and Onitsuka [11,12] dealt with Ulam's type stability of the periodic linear differential equation y − p(x)y = 0 (1) on R, where p : R → R is a continuous periodic function. Throughout this paper, let I = (a, b), −∞ ≤ a < b ≤ ∞. We say that (1) has "Ulam stability (US)" on I if there exists a constant K > 0 with the following property: Let ε > 0 be a given arbitrary constant, and let η : I → R be a continuously differentiable function. If |η (x) − p(x)η(x)| ≤ ε holds for all x ∈ I, then there exists a solution y : R → R of (1) such that |η(x) − y(x)| ≤ Kε for all x ∈ I. We call such K a "US constant" for (1) on I. In 2020, Fukutaka and Onitsuka [12] established the following theorem of Ulam's type stability and a necessary and sufficient condition.
Theorem 1. Let P(x) and Q(x) be antiderivatives of p(x) and e −P(x) on R, respectively. Suppose that p(x) is a periodic function with period ω > 0 on R. Then the following hold: In recent years, Ulam stability has been actively studied not only for the first-order linear differential equations but also for the second-order linear differential equations. In 2010, Li [13] dealt with Ulam's type stability of the simple second-order linear differential equation where λ > 0. This study extends to more general equations with constant coefficient. For example see the works of Li and Huang [14], Li and Shen [15], and Xue [16]. On the other hand, there are many studies on the second-order linear differential equations with variable coefficients (see, [17][18][19][20][21][22][23][24]). It is well known that the most commonly encountered variable coefficient second order differential equation is Hill's equation where p(x) is a periodic function. This equation briefly describes the behavior of a large number of physical systems. For example, we can find a pendulum with moving support, electrons in a periodic potential, and beam stabilization in alternating gradient proton synchrotron (see, (Chapter 7 [25])). However, there are no studies on the Ulam stability of the second-order linear differential equations with periodic coefficients. So, this paper focuses on the stability of differential equations with periodic coefficients. The main equation in this paper is is also a periodic function with period ω > 0 on R, and thus, this equation is a member of Hill's equations. If λ(x) ≡ λ then the above equation is reduced to the equation y = λ 2 y.
The main purpose of this study is to establish a necessary and sufficient condition for Ulam stability of Hill's equations on R. In the second section, we will establish Ulam stability of nonhomogeneous equations. In Section 3, we will give the main theorem and its proof. Also, a instability theorem is given. By using the obtained results, a necessary and sufficient condition is established. In Section 4, we will disscus the minimal US constant for the case λ(x) ≡ λ. In addition, a comparison with previous results is also presented. In Section 5, the concept obtained in Section 3 is extended to nonhomogeneous Hill's equations. As an application example, we will discuss Ulam stability of Hill's equation whose coefficient is described by Fourier series. Finally, we will conclude that the unboundedness of the approximate solution implies the unboundedness of the exact solution.

Ulam Stability for Nonhomogeneous Equations
In this section, we consider the nonhomogeneous first-order linear differential equation on R, where p, f : R → R are continuous. We say that (2) has "Ulam stability (US)" on I if there exists a constant K > 0 with the following property: Let ε > 0 be a given arbitrary constant, and let η : for all x ∈ I, then there exists a solution y : R → R of (2) such that |η(x) − y(x)| ≤ Kε for all x ∈ I. We say such K a "US constant" for (2) on I. Using the previous result Theorem 1, we find the following theorem.
Theorem 3. Let P(x) and Q(x) be antiderivatives of p(x) and e −P(x) on R, respectively. Suppose that p(x) is a periodic function with period ω > 0 on R. Then the following hold: Proof. First, we will prove (i). Suppose that b = ∞ and ω 0 p(x)dx > 0. Let u(x) be a solution of (2) on R. Then we have on I. Using Theorem 1 with the above inequality, the periodicity of p(x) and ω 0 p(x)dx > 0, we see that there exists a solution v(x) of (1) such that on I. Now, we consider the function y(x) = u(x) + v(x) for all x ∈ R. Then This means that y(x) is a solution of (2) such that The proof of (ii) is omitted because it can be proved by the same way.

Main Results
We consider a Hill's equation of the form on R, where λ : R → R is a continuously differentiable, periodic function with period ω > 0. We say that (3) has "Ulam stability (US)" on I if there exists a constant K > 0 with the following property: Let ε > 0 be a given arbitrary constant, and let η : When I = R, we can establish the following theorem.
is a periodic function with period ω > 0 on R. Then the following hold: Since λ(x) and η (x) are continuously differentiable functions on R, we see that ψ : R → R is a continuously differentiable function. In addition, we have the inequality First, we will prove (i). From Theorem 1 (i) with b = ∞ and ω 0 λ(x)dx > 0, and |ψ (x) − λ(x)ψ(x)| ≤ ε on R, there exists a solution w(x) of the differential equation for all x ∈ R, and therefore, z(x) is a solution of (3). Next, we will prove (ii). From Theorem 1 (ii) with a = −∞ and for all x ∈ R. By the same calculation as (6), we conclude that z(x) is a solution of (3) on R. This completes the proof of Theorem 4.

Remark 2.
If we compare US constants in (i) and (ii), respectively, it can be seen that they have symmetry. Now we will show this fact below.
If λ 1 (x) := −λ 2 (x) then this assumption and US constant imply that ω 0 λ 2 (x)dx < 0 and where c 1 , c 2 and c 3 are real constants. That is, if λ 1 (x) := −λ 2 (x) then the assumption and US constant in (i) correspond to those in (ii). Therefore, it can be concluded that the assumptions and US constants in (i) and (ii) are symmetric with each other.
for all x ∈ R. Using this fact, we can prove this lemma. For the reference, see (Lemma 2.1 [12]).
Next, we will present an instability theorem.
This says that Λ(x) is also a periodic function with period ω, and thus, we have min x∈(0,ω] For any ε > 0, we define the function where c 1 , c 2 ∈ R. Since this function is a solution of the equation for all x ∈ R. Now, we consider the case b = ∞. By (7), we have for all x ≥ 0. From this, we see that for all x ≥ 0, and there exists a x 1 > max{0, a} such that for all x ≥ x 1 . Therefore, using (7) and (8), we obtain for all x ≥ x 1 , so that, lim x→∞ |η(x) − y(x)| = ∞. Hence, (3) does not have US on I = (a, ∞). The proof of the case a = −∞ is omitted as it can be proved in the same way.

Lemma 2. Let λ(x)
is a continuously differentiable function on R, and let b = ∞. If and (3) has US on I, then minimum US constant on I is at least 1/ sup x∈I λ 2 (x) − λ (x) .
Proof. Let ε > 0 be a given arbitrary constant, and let η(x) be a solution of the differential equation (3) has US, there exists a constant K 0 > 0 such that |η(x) − y(x)| ≤ K 0 ε on I. Now, let y(x) be any solution of (3) such that the above inequality holds. We will show that K 0 ≥ 1/ sup x∈I λ 2 (x) − λ (x) . By way of contradiction, we assume that K 0 < 1/ sup x∈I λ 2 (x) − λ (x) . From this assumption, we have and set x 0 ∈ I. Then we obtain for all x > x 0 . From this inequality, we have for all x > x 0 . Using this inequality, we see that lim x→∞ |η(x) − y(x)| = ∞. This contradicts the inequality |η(x) − y(x)| ≤ K 0 ε on I.

Remark 4.
The decisive difference between the previous study described in the introduction and this study is whether the interval is finite or not. In the previous studies [13][14][15][16], they assume that −∞ < a < b < ∞. If I = (a, b) = (−∞, ∞) = R, then Ulam stability cannot be judged by their results because the US constant in the proof of previous theorems depends on a or b, and, goes to ∞ when a → −∞ or b → ∞. However, our study established Ulam stability on an infinite interval, and obtained the minimum US constant.

Nonhomogeneous Hill's Equations and Applications
Consider the nonhomogeneous Hill's equation where f : R → R is continuous. Equation (9) has "Ulam stability (US)" on I if and only if there exists K > 0 such that the following holds: Let ε > 0 be given, and let η : ≤ ε on I, then there exists a solution y(x) of (9) such that |η(x) − y(x)| ≤ Kε on I. K is called a "US constant" for (9) on I. The following result is obtained by using Theorem 4.

Theorem 8.
Let Λ(x), Γ + (x) and Γ − (x) be antiderivatives of λ(x), e Λ(x) and e −Λ(x) on R, respectively. Suppose that λ(x) is a periodic function with period ω > 0 on R. Then the following hold: Proof. First, we will prove (i). Let u(x) be a solution of (9) on R. Suppose that on I. Using Theorem 4 with the above inequality, we see that there exists a solution v(x) of (3) such that Thus, y(x) is a solution of (9) such that The proof of (ii) is omitted because it can be proved by the same way.
Theorem 8 and Corollary 1 imply the following result.
Let η(x) be a solution of (12). Then it is an approximate solution of the nonhomogeneous Hill's equation where a 0 , a n and b n are the same as above, and ε > 0 is given. In fact, (a n cos 2nx holds, (13) has US on R if and only if α = 0 by Theorem 9.

Conclusions
In this paper, we have discussed the Ulam stability of periodic linear differential equations of the first order and second order, respectively. First, we have established a sufficient condition for Ulam stability of nonhomogeneous first order equations. Next, by using this result, Ulam stability of a class of Hill's equations is established. In particular, the exact US constant is given in the theorem. If the coefficient is a constant, it turns out to be the minimum US constant. On the other hand, the instability theorem is also given. By combining the obtained results, we have established a necessary and sufficient condition for Ulam stability of a class of Hill's equations. Using the idea of the results obtained first, this theory is extended to nonhomogeneous Hill's equations. Finally, we have discussed the Ulam stability of Hill's equations where the coefficient is described by a Fourier series. By using the obtained results, it can be concluded that the unboundedness of the approximate solution affects the unboundedness of the exact solution.