Abstract
The article is focused on the development of a method allowing one to construct asymptotics for solutions to ODEs of arbitrary order with oscillating coefficients on the semiaxis. The idea of the method is presented on the example of studying the asymptotics of the Sturm–Liouville equation.
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1 INTRODUCTION
A substantial number of works are devoted to studying asymptotic properties of solutions to singular Sturm–Liouville equations and differential equations of arbitrary order, see [1–3] and the references cited there. However, in these works, the circumstance was essentially used that the coefficients of the equation have a regular growth at infinity.
In works [4–11] the asymptotic properties of solutions to ordinary differential equations were investigated for equations with coefficients from a broader classes, in particular, those not satisfying the Titchmarsh–Levitan conditions.
In work [11] we proposed a method for studying the asymptotic behavior of solutions to the Sturm–Liouville equation
in the case when \(q(x)\) is a rapidly oscillating function from the class \(\sigma\) (see [11]) that has the property \(\int_{x_{0}}^{\infty}q(x)dx<\infty\). This method allows constructing asymptotic formulas for solutions both in the case when \(q(x)\) affects the dominate term of the asymptotic and in the opposite case. Note that the above mentioned method does not allow studying the case when \(q(x)\) oscillates, but not falls into the class described in [11]. An example of such function is a function given by \(\sin x/x^{\alpha},\;\alpha>0\). The current work is aimed at developing this approach to construct asymptotics with perturbations of type \(p(x)/x^{\alpha},\;\alpha>0\), where \(p(x)\) is an almost periodic function.
2 CONSTRUCTION OF ASYMPTOTIC FORMULAS
Consider the equation
where \(0<\alpha\) and \(p(x)\) is an almost periodic function of the form
The main result of this paper is the following
Theorem. Let a function \(p(x)\) have form \((2)\) and
\((1)\) for any set of numbers \(\{c_{1},\ldots,c_{m}\}\), where \(c_{j}\in\{0\}\cup\mathbb{N}\), the condition is met:
\((2)\) for any \(p_{k}\), \(k=\overline{1,m}\), it is true that
Then for the fundamental system of solutions to Eq. \((1)\) the following asymptotic relations hold as \(x\to+\infty\):
Proof. The scheme of the proof is as follows. We reduce Eq. (1) to an equivalent system of equations. We introduce a vector function \(\mathbf{z}(x,\lambda)=(z_{1},z_{2})\): \(z_{1}=y\), \(z_{2}=y^{\prime}\).
Then Eq. (1) is rewritten as
Replacement
transits Eq. (6) to the system
where
We make one more replacement:
where the parameter \(k\) is such that \(\alpha\cdot(k+1)>1\). Below, we show that by the conditions of the theorem the matrices \(B_{i}\) are bounded as \(x\to\infty\). Replacement (8) leads to the system
Assuming \(x^{-\alpha}\) a small parameter as \(x\to\infty\), we seek the matrices \(B_{i}(x)\) from the following system of matrix equations:
From the first equation of system (10) we have
We present the formulas for solutions to the following equations:
from which we finally obtain a representation for \(B(x)\)
which also implies nondegeneracy of the matrix \(B(x)\).
At integration of system (9) we need to perform operations of matrix multiplication and finding the antiderivative. It is easy to prove that due to conditions (3), (4) the mean value of any product \(D\cdot B_{j},\;j=\overline{0,k},\) is zero, which results in the boundedness of \(B_{j}(x)\).
We denote
and due to the inequality \(\alpha\cdot(k+1)>1\) the matrix \(K(x)\) is summable on \((x_{0},\infty)\). Taking into account (10), we can write system (9) as
We proceed here to an equivalent system of integral equations and apply the method of successive approximations, thus obtaining the expressions for the dominate term of asymptotic of the fundamental system of solutions to the latter system:
Performing the inverse replacements (7) and (8), we finally obtain the required asymptotic formulas (5) for the fundamental system of solutions to Eq. (1). The theorem is proved.
Note that condition (3) of the theorem may be excessive. In view of this, we provide a more accurate formulation of the conditions ensuring the boundedness of the matrices \(B_{j}(x)\).
We introduce a matrix column \(\mathfrak{B}=(B_{0},B_{1},\ldots B_{k})^{T}\) and a matrix \(\mathfrak{A}=i\mu I\otimes L_{0}+J\otimes D\), where \(I\) is the identity matrix of order \(k+1\), \(J\) is the lower-shear matrix of order \(k+1\). Then system (10) is equivalent to the system
Note that the Floquet theory is applicable to system (11). Let \(\rho_{j}\), \(j=\overline{0,k},\) be multiplicators of system (11). It is clear that the condition
provides existence of a bounded solution to system (11) and, therefore, to system (10) equivalent to it. It is important to note that \(\rho_{j}\) can be computed for any \(\mu\in\mathbb{C}\); therefore, condition (12) can be checked.
Remark 1. From the proved theorem it follows that the perturbation \(p(x)/x^{\alpha}\) does not affect the dominate part of asymptotic of solutions to Eq. (1) if conditions (3) and (4) are met.
Remark 2. Formulas (8) allow refining the asymptotic formulas up to order \(1/x^{k\alpha}\) inclusively.
Remark 3. The proof of the theorem implies that, in the case of a periodic function \(p(x)\), despite the fact that condition (3) of the theorem is not met, the given algorithm of constructing the asymptotic of the fundamental system of solutions for the original equation can be executed under the condition \(\alpha>1/2\). Let us illustrate the above said by the following example.
3 EXAMPLE
Consider an equation
In this case \(p(x)=\frac{1}{2i}(e^{ix}-e^{-ix})\), \(p_{1}=1\), \(p_{2}=-1\), and, for instance, for \(c_{1}=c_{2}=1\) we have \(c_{1}p_{1}+c_{2}p_{2}=0\), that is, condition (3) of the theorem is violated. In (8) we put \(k=1\), then the matrix \(B(x)\) has the form
where
For this example we compute the matrices \(D_{0}\) and \(D_{1}\). We have
To compute the matrix \(D_{1}\), we need to integrate the elements of the matrix \(D_{0}\). The condition \(2\mu\neq\pm 1\) provides a nonzero imaginary part for all elements of the matrix \(D_{0}\), that is, absence of a resonance. We have
Note that, to make the matrix \(B(x)\) contain three and more terms, we need to compute the elements of the matrix \(D_{0}\cdot D_{1}\) and then integrate this product. Because condition (3) of the theorem is violated for this example, some elements of the matrix \(D_{0}\cdot D_{1}\) contain zero imaginary part, which results in unboundedness of the matrices \(B_{k},\;k>1\).
Thus, for this example we succeed in finding and explicitly computing the bounded matrices \(B_{0}(x)\), \(B_{1}(x)\) and obtaining the first two terms of the asymptotic of the fundamental system of solutions to Eq. (1) as \(x\to\infty\):
REFERENCES
A. Zettl, Sturm–Liouville Theory, Mathematical Surveys and Monographs, Vol. 121 (American Mathematical Society, Providence, R.I., 2005).
M. A. Naimark, Linear Differential Operators (Nauka, Moscow, 1969).
Ya. T. Sultanaev, ‘‘Asymptotics of solutions to ordinary differential equations in the degenerate case,’’ Tr. Seminara I.G. Petrovskogo 13, 36–55 (1988).
W. N. Everitt and L. Markus, in Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-Differential Operators (American Mathematical Society, Providence, R.I., 1999), pp. 1-60.
N. N. Konechnaya, K. A. Mirzoev, and A. A. Shkalikov, ‘‘On the asymptotic behavior of solutions to two-term differential equations with singular coefficients,’’ Math. Notes 104, 244–252 (2018). https://doi.org/10.1134/S0001434618070258
N. N. Konechnaya, K. A. Mirzoev, and Ya. T. Sultanaev, ‘‘On the asymptotics of solutions of some classes of linear differential equations,’’ Azerbaijan J. Math. 10, 162–171 (2020).
P. N. Nesterov, ‘‘Construction of the asymptotics of the solutions of the one-dimensional Schrodinger equation with rapidly oscillating potential,’’ 80, 233–243 (2006). https://doi.org/10.1007/s11006-006-0132-5
N. F. Valeev and Ya. T. Sultanaev, ‘‘On the deficiency indices of the singular Sturm–Liouville operator with rapidly oscillating perturbation,’’ Dokl. Math. 62, 271–273 (2000).
N. F. Valeev, E. A. Nazirova, and Ya. T. Sultanaev, ‘‘On a new approach for studying asymptotic behavior of solutions to singular differential equations,’’ Ufa Math. J. 7 (3), 9-14 (2015). https://doi.org/10.13108/2015-7-3-9
N. F. Valeev, O. V. Myakinova, and Ya. T. Sultanaev, ‘‘On the asymptotics of solutions of a singular \(n\)th-order differential equation with nonregular coefficients,’’ Math. Notes 104, 606–611 (2018). https://doi.org/10.1134/S0001434618090262
L. N. Valeeva, É. A. Nazirova, and Ya. T. Sultanaev, ‘‘On a method for studying the asymptotics of solutions of Sturm–Liouville differential equations with rapidly oscillating coefficients,’’ Math. Notes 112, 1059–1064 (2022). https://doi.org/10.1134/S0001434622110372
Funding
The studies of E.A. Nazirova and Ya.T. Sultanaev are supported by the Russian Science Foundation, project no. 23-21-00225.
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Valeev, N.F., Nazirova, E.A. & Sultanaev, Y.T. Constructing the Asymptotics of Solutions to Differential Sturm–Liouville Equations in Classes of Oscillating Coefficients. Moscow Univ. Math. Bull. 78, 253–257 (2023). https://doi.org/10.3103/S0027132223050066
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DOI: https://doi.org/10.3103/S0027132223050066