On the Asymptotic of Solutions of Odd-Order Two-Term Differential Equations

Abstract

This work is devoted to the development of methods for constructing asymptotic formulas as $x\to \infty$ of a fundamental system of solutions of linear differential equations generated by a symmetric two-term differential expression of odd order. The coefficients of the differential expression belong to classes of functions that allow oscillation (for example, those that do not satisfy the classical Titchmarsh–Levitan regularity conditions). As a model equation, the fifth-order equation $\frac{i}{2}\left[{\left(p\left(x\right)y^{‴}\right)}^{″}+{\left(p\left(x\right)y^{″}\right)}^{‴}\right]+q\left(x\right)y=\lambda y$, along with various behaviors of coefficients $p\left(x\right),q\left(x\right)$, is investigated. New asymptotic formulas are obtained for the case when the function $h\left(x\right)=-1+p^{-1/2}\left(x\right)\notin L_1[1,\infty )$ significantly influences the asymptotics of solutions to the equation. The case when the equation contains a nontrivial bifurcation parameter is studied.

1. Introduction

Analyzing the asymptotic behavior as $x\to \infty$ of a fundamental system of solutions of arbitrary-order singular differential equations, being of independent interest, is an effective method for studying qualitative spectral characteristics for corresponding differential operators [1,2,3]. As a rule, in these books, differential equations with regular coefficients with regular growth at infinity are investigated. Therefore, the study of the asymptotic behavior of solutions to equations with coefficients from other classes of functions is of particular interest. Such classes of functions were described by us in a previous paper [4]. Let us also note the works [5,6,7], where differential operators with distribution coefficients were studied.

For example, in the work [7], asymptotic formulas were obtained for the fundamental system of solutions of a two-term equation of even order:

$${(-1)}^n{\left(p\left(x\right)y^{\left(n\right)}\right)}^{\left(n\right)}+q\left(x\right)y=\lambda y,x\in [1,\infty ),$$

where the locally summable function p can be represented as $p\left(x\right)={(1+r\left(x\right))}^{-1},$$r\in L_1[1,\infty )$ and q is a generalized function representable for some fixed k, $0\le k\le n$, in the form $q={\sigma }^{\left(k\right)}$, where $\sigma \in L_1[1,\infty )$ if $k<n$, $\left|\sigma \right|(1+\left|r\right|\left)\right(1+\left|\sigma \right|)\in L_1[1,\infty ),$ if $k=n$.

Since 2014, we have been publishing a series of articles devoted to the study of the asymptotic behavior of solutions to singular ordinary differential equations with regularly oscillating coefficients [4,8,9,10,11]. In this case, a new approach was used for the study based on a sequence of matrix transformations and the use of Campbell’s identity [12].

The use of this approach made it possible to obtain new asymptotic formulas in different cases. For example, in [4,11], new asymptotic formulas were obtained for solutions of the Sturm–Liouville equation

$$y^{″}+\left({\mu }^2+\frac{sin\left(x^{\beta }\right)}{x^{\alpha }}\right)y=0,0<\alpha \le 1,\beta >\frac{\alpha }{2}+1$$

under some relations between $\alpha ,\beta ,\mu$. Note that $\mu$ has the meaning of a bifurcation parameter. By the way, this equation is one of the equations for testing new methods for constructing asymptotic formulas (see, for example, [13], p. 160).

Equations of odd order for irregular classes of coefficients (in the Titchmarsh–Levitan sense) have been studied less. In the works [7,10], the asymptotics of solutions of odd-order equations were studied in the case when the coefficient of the highest derivative is equal or equivalent to unity.

Here, we develop an approach that was proposed in [4,8,9,10,11] and can be implemented to study the asymptotic behavior as $x\to \infty$ of a fundamental system of solutions of two-term equations of arbitrary odd order of the form

$$ly=\frac{i}{2}\left[{\left(p\left(x\right)y^{\left(n\right)}\right)}^{(n+1)}+{\left(p\left(x\right)y^{(n+1)}\right)}^{\left(n\right)}\right]+q\left(x\right)y=\lambda y,x\ge 1$$

(1)

for various behaviors of coefficients $p\left(x\right),q\left(x\right)$.

This method allows us to significantly expand the classes of coefficients $p\left(x\right)$ and $q\left(x\right)$ for which we can write out the asymptotic behavior of solutions. In particular, new formulas are obtained in cases where $p\left(x\right)$ and $q\left(x\right)$ allow oscillations. Note that the new formulas obtained allow us to study the spectral properties of differential operators generated by the expression $ly$ (1).

2. Transition to the Ordinary System of Differential Equations Using Quasi-Derivatives

Let us write Equation (1) in the form of a system of ordinary differential equations of the first order. To do this, we use the apparatus of quasi-derivatives (for more detail, see [14,15,16]). Let us define the functions $q_n\left(x\right)\in L_{1,loc}[1,\infty )$ so that

$$q_n^{\left(n\right)}\left(x\right)=q\left(x\right)$$

(2)

and introduce into consideration quasi-derivatives defined by the following formulas:

$$\left\{\begin{array}{c}{z}_1=y,z_{n+2}=\sqrt{p}{z}_{n+1}^{\prime }-i{q}_n{z}_1\hfill \\ z_2=z_1^{\prime },z_{n+3}=z_{n+1}^{\prime }+i{C}_n^1{q}_n{z}_2\hfill \\ ........\hfill \\ z_n=z_{n-1}^{\prime },z_{2n}=z_{2n-1}^{\prime }+i{(-1)}^{n-1}{C}_n^{n-2}{q}_n{z}_{n-1}\hfill \\ z_{n+1}=\sqrt{p}{z}_n^{\prime },z_{2n+1}=z_{2n}^{\prime }+i{(-1)}^n{C}_n^{n-1}{q}_n{z}_n.\hfill \end{array}\right.$$

(3)

Then, Equation (1) is equivalent to the relation

$$z_{2n+1}^{\prime }=\lambda z_1-i{(-1)}^{n+1}\frac{q_n}{\sqrt{p}}{z}_{n+1}.$$

Let us introduce the column vector $\mathbf{z}=column(z_1,z_2,...,z_{2n+1})$ and write Equation (1) as a system of ordinary differential equations

$${\mathbf{z}}^{\prime }=S\mathbf{z},$$

where $S(x,\lambda )$ is the Shin–Zettl matrix [14].

$$S=\left(\begin{array}{cccccccccc}0& 1& 0& \dots & 0& 0& 0& 0& \dots & 0\\ 0& 0& 1& \dots & 0& 0& 0& 0& \dots & 0\\ & & & & & ..............\\ 0& 0& 0& \dots & 1& 0& 0& 0& \dots & 0\\ 0& 0& 0& \dots & 0& \frac{1}{\sqrt{p}}& 0& 0& \dots & 0\\ \frac{i{q}_n}{\sqrt{p}}& 0& 0& \dots & 0& 0& \frac{1}{\sqrt{p}}& 0& \dots & 0\\ 0& -in{q}_n& 0& \dots & 0& 0& 0& 1& \dots & 0\\ & & & & & ...............\\ 0& 0& 0& \dots & 0& {(-1)}^{n-1}in{q}_n& 0& 0& \dots & 1\\ -i\lambda & 0& 0& \dots & 0& 0& \frac{{(-1)}^{n}i{q}_n}{\sqrt{p}}& 0& \dots & 0\end{array}\right),$$

where the non-zero elements of the matrix $S(x,\lambda )$ are given by the formulas

$$s_{kj}=1,j=1+k,k=\overline{1,n-1},k=\overline{n+2,2n},s_{n,n+1}=s_{n+1,n+2}=\frac{1}{\sqrt{p}},$$

$$s_{n+1,1}=\frac{i{q}_n}{\sqrt{p}},s_{n+k,k}={(-1)}^{k-1}i{C}_n^{k-1}{q}_n,k=\overline{2,n},$$

$$s_{2n+1,2n-1}=\frac{{(-1)}^{n}i{q}_n}{\sqrt{p}},s_{2n+1,1}=-i\lambda .$$

Note that from the relation $q_n^{\left(n\right)}\left(x\right)=q\left(x\right)$, the function $q_n\left(x\right)$ is determined up to a polynomial of order $n-1$. However, the fundamental system of solutions of Equation (1) does not depend on the choice of integration constants, which follows directly from Formula (3). Conditions for choosing the coefficients of the polynomial are formulated for each case under study.

Further, in order to avoid complicated formulas, we limit ourselves to considering the 5th-order two-term equation

$$ly=\frac{i}{2}\left[{\left(p\left(x\right)y^{″}\right)}^{‴}+{\left(p\left(x\right)y^{‴}\right)}^{″}\right]+q\left(x\right)y=\lambda y,x\ge 1.$$

(4)

Using Formula (3), we introduce quasi-derivatives

$$\left\{\begin{array}{c}{z}_1=y\hfill \\ z_2=z_1^{\prime }\hfill \\ z_3=\sqrt{p}{z}_2^{\prime }\hfill \\ z_4=\sqrt{p}{z}_3^{\prime }-i{q}_2{z}_1\hfill \\ z_5=z_4^{\prime }+2i{q}_2{z}_2.\hfill \end{array}\right.$$

Then, Equation (4) is equivalent to the relation

$$z_5^{\prime }=-i\lambda z_1+\frac{i{q}_2}{\sqrt{p}}{z}_3$$

and can be written as a system of ordinary differential equations:

$${\mathbf{z}}^{\prime }=\left(\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 1/\sqrt{p}& 0& 0\\ {\mathit{iq}}_2/\sqrt{p}& 0& 0& 1/\sqrt{p}& 0\\ 0& -2{\mathit{iq}}_2& 0& 0& 1\\ -i\lambda & 0& {\mathit{iq}}_2/\sqrt{p}& 0& 0\end{array}\right)\mathbf{z},$$

where $\mathbf{z}=column(z_1,z_2,z_3,z_4,z_5).$

Let the function $p\left(x\right)$ admit the representation

$$\frac{1}{\sqrt{p\left(x\right)}}=1+h\left(x\right),h\left(x\right)\in L_{1,loc}[1,\infty ).$$

(5)

Let us write the last system of equations in the following form, taking into account (5):

$${\mathbf{z}}^{\prime }=\left(L_0+h\left(x\right)L_1+i{q}_2\left(x\right)D_0+ih\left(x\right)q_2\left(x\right))D_1\right)\mathbf{z},$$

(6)

$$L_0=\left(\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ -i\lambda & 0& 0& 0& 0\end{array}\right),L_1=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right),$$

$$D_0=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& -2& 0& 0& 0\\ 0& 0& 1& 0& 0\end{array}\right),D_1=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\end{array}\right).$$

3. Construction of Asymptotic Formulas

3.1. Case 1

Let us set

$$\tilde{h}\left(x\right)=q_2\left(x\right)h\left(x\right).$$

(7)

Let the following conditions be satisfied:

$$h\left(x\right),q_2\left(x\right)\in L_1[1,\infty ),\tilde{h}\left(x\right)\in L_{1,loc}[1,\infty ).$$

For example, these conditions are true for

$$h\left(x\right)=\frac{1}{x^{\gamma }},\gamma >1;q\left(x\right)=x^{\alpha }sin{x}^{\beta },\alpha >0,\beta >\frac{\alpha +3}{2}.$$

Let the constant matrix T reduce the matrix $L_0$ to diagonal form. Let us make a replacement:

$$\mathbf{z}=T\mathbf{u},T^{-1}{L}_{0}T=\mathsf{\Lambda },{\mu }_k^5=-i\lambda ,k=\overline{1,5},$$

$$\mathsf{\Lambda }=\left(\begin{array}{ccccc}{\mu }_1& 0& 0& 0& 0\\ 0& {\mu }_2& 0& 0& 0\\ 0& 0& {\mu }_3& 0& 0\\ 0& 0& 0& {\mu }_3& 0\\ 0& 0& 0& 0& {\mu }_3\end{array}\right),T=\left(\begin{array}{ccccc}1& 1& 1& 1& 1\\ {\mu }_1& {\mu }_2& {\mu }_3& {\mu }_4& {\mu }_5\\ {\mu }_1^2& {\mu }_2^2& {\mu }_3^2& {\mu }_4^2& {\mu }_5^2\\ {\mu }_1^3& {\mu }_2^3& {\mu }_3^3& {\mu }_4^3& {\mu }_5^3\\ {\mu }_1^4& {\mu }_2^4& {\mu }_3^4& {\mu }_4^4& {\mu }_5^4\end{array}\right).$$

Then, system (5) takes the form

$${\mathbf{u}}^{\prime }=\left(\mathsf{\Lambda }+h\left(x\right)T^{-1}{L}_{1}T+i{q}_2\left(x\right)T^{-1}{D}_{0}T+i{q}_2\left(x\right)h\left(x\right)T^{-1}{D}_{1}T\right)\mathbf{u}.$$

(8)

Obviously, because of the imposed conditions, System (8) satisfies the conditions of Lemma 1 in [3], p. 288, and is L-diagonal, which means we can write out asymptotic formulas as $x\to \infty$ for the fundamental system solutions of this system:

$${\mathbf{z}}_k(x,\lambda )=T·{\mathbf{u}}_k(x,\lambda )=e^{{\mu }_{k}x}·T·({\mathbf{e}}_k+o\left(1\right)),k=\overline{1,5},$$

where ${\mathbf{e}}_k$ are unit basis vectors.

3.2. Case 2

Let the following conditions be satisfied:

$$q_2\left(x\right)\notin L_1[1,\infty ),h\left(x\right),q_3\left(x\right),\tilde{h}\left(x\right)\in L_1[1,\infty ).$$

(9)

These conditions are true for

$$h\left(x\right)=\frac{1}{x^{\gamma }},\gamma >1;q\left(x\right)=x^{\alpha }sin{x}^{\beta },\alpha >0,\frac{\alpha +3}{2}\ge \beta >\frac{\alpha +4}{3}.$$

Following the approach outlined in the paper [8], we make a replacement in System (6):

$$\mathbf{z}=e^{i{q}_3\left(x\right)D_0}\mathbf{u}.$$

(10)

We obtain

$${\mathbf{u}}^{\prime }=e^{-i{q}_3\left(x\right)D_0}\left(L_0+h\left(x\right)L_1+i\tilde{h}\left(x\right)D_1\right)e^{i{q}_3\left(x\right)D_0}\mathbf{u}.$$

(11)

Let us apply Campbell’s identity to transform the right-hand side of (11) to

$$e^{-i{q}_3\left(x\right)D_0}{L}_0{e}^{i{q}_2\left(x\right)D_0}=L_0-i{q}_3\left(x\right)[D_0,L_0]+\frac{i^2{q}_3^2\left(x\right)}{2!}[D_0,[D_0,L_0]]-$$

$$-\frac{i^3{q}_3^3\left(x\right)}{3!}[D_0,[D_0,[D_0,L_0]]]+\dots ,$$

where $[A,B]=AB-BA$ is a matrix commutator.

Below, we use the following obvious consideration: if the matrix A is nilpotent, then a nonzero sequence of matrix commutators of the form $[A,[A,\dots ,[A,B]]\dots ]$ is finite.

Note that the matrix $D_0$ is nilpotent. By sequentially calculating the commutators on the right side of the last relation, we obtain that all terms, starting from the fourth, are equal to zero, and non-zero terms can be calculated:

$$[D_0,L_0]=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ -1& 0& 0& 0& 0\\ 0& 3& 0& 0& 0\\ 0& 0& -3& 0& 0\\ 0& 0& 0& 1& 0\end{array}\right),[D_0,[D_0,L_0]]=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 5& 0& 0& 0& 0\\ 0& 5& 0& 0& 0\end{array}\right).$$

Similar calculations can be carried out for the remaining terms on the right side of (11):

$$e^{-i{q}_3\left(x\right)D_0}h\left(x\right)L_1{e}^{i{q}_3\left(x\right)D_0}=h\left(x\right)L_1-i{q}_{3}h\left(x\right)[D_0,L_1]+\frac{h\left(x\right)i^2{q}_3^2\left(x\right)}{2!}[D_0,[D_0,L_1]]+....,$$

$$[D_0,L_1]=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ -1& 0& 0& 0& 0\\ 0& 2& 0& 0& 0\\ 0& 0& -2& 0& 0\\ 0& 0& 0& 1& 0\end{array}\right),[D_0,[D_0,L_1]]=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 4& 0& 0& 0& 0\\ 0& 4& 0& 0& 0\end{array}\right).$$

Because $[D_0,D_1]=0$, the following representation is true:

$$e^{-i{q}_3\left(x\right)D_0}\tilde{h}\left(x\right)D_1{e}^{i{q}_3\left(x\right)D_0}=\tilde{h}\left(x\right)D_1.$$

Then, Equation (11) can be rewritten as

$${\mathbf{u}}^{\prime }=(L_0+h{L}_1+\tilde{h}{D}_1-i{q}_3[D_0,L_0]+\frac{i^2{q}_3^2}{2!}[D_0,[D_0,L_0]]-i{q}_{3}h[D_0,L_1]+\frac{i^{2}h{q}_3^2}{2!}[D_0,[D_0,L_1]])\mathbf{u}.$$

(12)

Because of the imposed conditions on the functions $h\left(x\right),q\left(x\right)$, the last system can be written as

$${\mathbf{u}}^{\prime }=(L_0+D\left(x\right))\mathbf{u},$$

where $D\left(x\right)$ is a matrix whose elements belong to $L_1[1,\infty )$. Just as in Case 1, let us make the replacement $\mathbf{u}=T\mathbf{v};$ then,

$${\mathbf{v}}^{\prime }=\left(\mathsf{\Lambda }+T^{-1}D\left(x\right)T\right)\mathbf{v}.$$

(13)

System (13) satisfies the conditions of Lemma 1 in [3] and is L-diagonal, which means, taking into account (10), we can write asymptotic formulas for $x\to \infty$ for its fundamental system of solutions:

$${\mathbf{z}}_k(x,\lambda )=e^{{\mu }_{k}x}·e^{i{q}_3\left(x\right)D_0}·T·({\mathbf{e}}_k+o\left(1\right)),k=\overline{1,5}.$$

where ${\mathbf{e}}_k$ are unit vectors.

Remark 1.

Let us note the importance of resulting Equation (12). Imposing various conditions on the coefficients of this equation, $h\left(x\right),q_3\left(x\right)h\left(x\right),q_3^2\left(x\right)h\left(x\right),\tilde{h}\left(x\right),q_3\left(x\right)$, and $q_2^3\left(x\right)$, different from the conditions in (9), one can obtain different asymptotics of the fundamental system of solutions with nontrivial properties.

3.3. Case 3

Let us define the function $h_1\left(x\right)$ so that

$$h_1^{\prime }\left(x\right)=h\left(x\right).$$

(14)

Let us now consider the case when

$$h\left(x\right),q_2\left(x\right)\notin L_1[1,\infty ),q_3\left(x\right),h_1\left(x\right),\tilde{h}\left(x\right)\in L_1[1,\infty ).$$

For example, these conditions are true for

$$h\left(x\right)=\frac{1}{x^{\gamma }},0<\gamma <1;q\left(x\right)=x^{\alpha }sin{x}^{\beta },2>\alpha >0,\frac{\alpha +3}{2}\ge \beta >\frac{\alpha +3-\gamma }{2}.$$

Just as in Case 2, let us make a replacement in System (6)

$$\mathbf{z}=e^{h_1\left(x\right)L_1}\mathbf{u}.$$

(15)

Then, System (6) takes the form

$${\mathbf{u}}^{\prime }=e^{-h_1\left(x\right)L_1}\left(L_0+i{q}_2\left(x\right)D_0+i\tilde{h}\left(x\right)D_1\right)e^{h_1\left(x\right)L_1}\mathbf{u}.$$

(16)

Let us apply Campbell’s identity to transform the right-hand side of (16):

$$e^{-h_1\left(x\right)L_1}{L}_0{e}^{h_1\left(x\right)L_1}=L_0-h_1\left(x\right)[L_1,L_0]+\frac{h_1^2\left(x\right)}{2!}[L_1,[L_1,L_0]]-\frac{h_1^3\left(x\right)}{3!}[L_1,[L_1,[L_1,L_0]]]+\dots$$

Note that the matrix $L_1$ is nilpotent. By sequentially calculating the commutators on the right side of the last relation, we obtain that all terms, starting from the fourth, are equal to zero, and non-zero terms can be calculated:

$$[L_1,L_0]=\left(\begin{array}{ccccc}0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right),[L_1,[L_1,L_0]]=\left(\begin{array}{ccccc}0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right).$$

Similar calculations can be carried out for the remaining terms on the right side of (16):

$$e^{-h_1\left(x\right)L_1}i{q}_2\left(x\right)D_0{e}^{h_1\left(x\right)L_1}=i{q}_2\left(x\right)D_0-i{q}_2\left(x\right)h_1\left(x\right)[L_1,D_0]+\frac{i{q}_2\left(x\right)h_1^2\left(x\right)}{2!}[L_1,[L_1,D_0]]-$$

$$-\frac{i{q}_2\left(x\right)h_1^3\left(x\right)}{3!}[L_1,[L_1,[L_1,D_0]]]+\frac{i{q}_2\left(x\right)h_1^4\left(x\right)}{4!}[L_1,[L_1,[L_1,[L_1,D_0]]]].$$

$$[L_1,D_0]=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ -1& 0& 0& 0& 0\\ 0& 2& 0& 0& 0\\ 0& 0& -2& 0& 0\\ 0& 0& 0& 1& 0\end{array}\right),[L_1,[L_1,D_0]]=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& -2& 0& 0& 0\\ 0& 0& 4& 0& 0\\ 0& 0& 0& -2& 0\\ 0& 0& 0& 0& 0\end{array}\right),$$

$$[L_1,[L_1,[L_1,D_0]]]=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 6& 0& 0\\ 0& 0& 0& -6& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right),[L_1,[L_1,[L_1,[L_1,D_0]]]]=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& -12& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right),$$

$$e^{-h_1\left(x\right)L_1}i\tilde{h}\left(x\right)D_1{e}^{h_1\left(x\right)L_1}=i\tilde{h}\left(x\right)D_1-i{h}_1\left(x\right)\tilde{h}\left(x\right)[L_1,D_1],$$

$$[L_1,D_1]=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 0\end{array}\right).$$

Because of the imposed conditions on the functions $h\left(x\right),q\left(x\right)$, the last system can be written as

$${\mathbf{u}}^{\prime }=(L_0+i{q}_2\left(x\right)D_0+D\left(x\right))\mathbf{u}.$$

where $D\left(x\right)$ is a matrix whose elements belong to $L_1[1,\infty )$. Unlike Case 2, the resulting system is not yet L-diagonal. Let us make one more transformation:

$$\mathbf{u}=e^{i{q}_3\left(x\right)D_0}\mathbf{v}.$$

(17)

Then, the last system will take the form

$${\mathbf{v}}^{\prime }=e^{-i{q}_3\left(x\right)D_0}\left(L_0+i{q}_2\left(x\right)D_0+D\left(x\right)\right)e^{i{q}_3\left(x\right)D_0}\mathbf{v}.$$

(18)

Let us apply Campbell’s identity to transform the right-hand side of (18). Just as in Case 2, taking into account the nilpotency of the matrix $D_0$ and sequentially calculating all the necessary matrix commutators, we obtain the following form of System (18):

$${\mathbf{v}}^{\prime }=(L_0+\tilde{D}\left(x\right))\mathbf{v}.$$

Here, the matrix $\tilde{D}\left(x\right)$ is defined by the expression

$$\tilde{D}\left(x\right)=-i{q}_3\left(x\right)[D_0,L_0]+\frac{i^2{q}_3^2}{2!}[D_0,[D_0,L_0]]+e^{-i{q}_3\left(x\right)D_0}D\left(x\right)e^{i{q}_3\left(x\right)D_0},$$

which because of the conditions imposed above on the functions $h\left(x\right),q\left(x\right)$, is obviously a matrix with elements summable over $[1,\infty )$.

Next, we make the replacement $\mathbf{v}=T\mathbf{s}.$ Then,

$${\mathbf{s}}^{\prime }=\left(\mathsf{\Lambda }+T^{-1}\tilde{D}\left(x\right)T\right)\mathbf{s}.$$

(19)

System (19) satisfies the conditions of Lemma 1 in [3] and is L-diagonal, which means, taking into account (15) and (17), we can write out asymptotic formulas as $x\to \infty$ for its fundamental system of solutions:

$${\mathbf{z}}_k(x,\lambda )=e^{{\mu }_{k}x}·e^{h_1\left(x\right)L_1}·e^{i{q}_3\left(x\right)D_0}·T·({\mathbf{e}}_k+o\left(1\right)),k=\overline{1,5},$$

where ${\mathbf{e}}_k$ are unit vectors.

Summarizing Cases 1–3, we find that we have proven the following theorem:

Theorem 1.

Let functions $q_2\left(x\right),q_3\left(x\right),h\left(x\right),\tilde{h}\left(x\right),h_1\left(x\right)$ be defined by Formulas (2), (5), (7), and (14) correspondingly and one of the following conditions be satisfied:

(1) 

$h\left(x\right),q_2\left(x\right)\in L_1[1,\infty ),$

(2) 

$h\left(x\right),q_3\left(x\right),\tilde{h}\left(x\right)\in L_1[1,\infty )$,

(3) 

$q_3\left(x\right),h_1\left(x\right),\tilde{h}\left(x\right)\in L_1[1,\infty )$.

Then, the asymptotic formulas as $x\to \infty$ for the fundamental system of solutions of Equation (4) are valid:

$$y_j(x,\lambda )=e^{{\mu }_{j}x}·(1+o\left(1\right)),j=\overline{1,5}.$$

In fact, we obtain the asymptotic formulas as $x\to \infty$ for vector function; we may also write down the asymptotic formulas for quasi-derivatives of solutions.

3.4. Counterexample

Let us show that the conditions of Theorem 1 are essential.

Let

$$h\left(x\right),q_2\left(x\right)\notin L_1[1,\infty ),q_3\left(x\right),h_1\left(x\right)\in L_1[1,\infty ),\tilde{h}\left(x\right)\in L_{1,loc}[1,\infty ).$$

In the same way as in Case 3, we make sequential transformations

$$\mathbf{z}=e^{h_1\left(x\right)L_1}\mathbf{u},\mathbf{u}=e^{i{q}_3\left(x\right)D_0}\mathbf{w},$$

which brings Equation (6) to the form

$${\mathbf{w}}^{\prime }=(L_0+ih\left(x\right)q_2\left(x\right))D_1)\mathbf{w}+F\left(x\right)\mathbf{w},$$

(20)

where $F\left(x\right)\in L_1[1,\infty )$.

The last system of equations allows for a large variety in the asymptotic behavior as $x\to +\infty$ and can be the subject of a separate study.

We limit ourselves to considering a model example on which we demonstrate an unusual property of equations with oscillating coefficients, namely, the influence of the algebraic structure of the coefficients of the equation on the asymptotic behavior of the solutions.

Let

$$h\left(x\right)=asin\left(e^x\right),q_2\left(x\right)=sin\left(k{e}^x\right),$$

from which

$$\tilde{h}\left(x\right)=h\left(x\right)q_2\left(x\right)=asin\left(e^x\right)sin\left(k{e}^x\right)=\frac{1}{2}a[cos\left((k-1)e^x\right)+cos\left((k+1)e^x\right)].$$

Consider two cases: $k=\pm 1$ and $k\ne \pm 1$. Let $k\ne \pm 1$. Define the function ${\tilde{h}}_1\left(x\right)$ so that ${\tilde{h}}_1^{\prime }\left(x\right)=\tilde{h}\left(x\right)$. Note that, in this case, ${\tilde{h}}_1\left(x\right)\in L_1[1,\infty )$.

In System (20), we set

$$\mathbf{w}=e^{i{\tilde{h}}_1\left(x\right)D_1}\mathbf{v};$$

then, for $\mathbf{v}$, we obtain the system

$${\mathbf{v}}^{\prime }=(L_0+F_1\left(x\right))\mathbf{v},$$

(21)

where $F_1\left(x\right)\in L_1[1,\infty )$ and which can easily be reduced to an L-diagonal system. Consequently, the main term of the asymptotics of the fundamental system of solutions (21), as above, are determined:

$${\mathbf{z}}_j(x,\lambda )=e^{{\mu }_{j}x}·T·({\mathbf{e}}_j+o\left(1\right)),j=\overline{1,5}.$$

Let $k=\pm 1$. Note that now

$$h\left(x\right)q_2\left(x\right)=asin\left(e^x\right)sin\left(k{e}^x\right)=$$

$$\frac{1}{2}a[cos\left((k-1)e^x\right)+cos\left((k+1)e^x\right)]=\frac{1}{2}a(1+cos\left(2e^x\right))$$

whence it follows that $h\left(x\right)q_2\left(x\right)\notin L_1[1,\infty )$. Let us denote $\sigma \left(x\right)=\frac{i}{2}acos\left(2e^x\right)$ and represent the system (21) in the following form:

$${\mathbf{w}}^{\prime }=\left(L_0+\frac{i}{2}a{D}_1+\sigma \left(x\right)D_1\right)\mathbf{w}+F\left(x\right)\mathbf{w},$$

(22)

where $F\left(x\right)\in L_1[1,\infty ),$ and the matrix $L_0+\frac{i}{2}a{D}_1$ is constant. Considering

$${\sigma }_1\left(x\right)=-\frac{i}{2}a{\int }_x^{+\infty }cos\left(2e^{\xi }\right)d\xi \in L_1[1,\infty ),$$

we make a replacement in System (22)

$$\mathbf{w}=e^{{\sigma }_1\left(x\right)D_1}\mathbf{v},$$

and again using the technique described above, we obtain the system

$${\mathbf{v}}^{\prime }=(L_0+\frac{i}{2}a{D}_1)\mathbf{w}+F_1\left(x\right)\mathbf{w}.$$

(23)

Here, taking into account that ${\sigma }_1\left(x\right)\in L_1[1,\infty )$, we have $F_1\left(x\right)\in L_1[1,\infty )$. Let matrix $\widehat{T}$ reduce matrix $L_0+\frac{i}{2}a{D}_1$ to diagonal form, ${\widehat{\mu }}_j,werej=\overline{1,5}$-eigenvalues of the matrix $L_0+\frac{i}{2}a{D}_1$. Let us make a replacement:

$$\mathbf{v}=\widehat{T}\mathbf{s},{\widehat{T}}^{-1}(L_0+\frac{i}{2}a{D}_1)\widehat{T}=\widehat{\mathsf{\Lambda }}.$$

Then, System (22) takes the form

$${\mathbf{s}}^{\prime }=(\widehat{\mathsf{\Lambda }}+{\widehat{T}}^{-1}{F}_1\left(x\right)\widehat{T})\mathbf{s}.$$

The resulting system is equivalent to the L-diagonal system. Then, as above, the fundamental system of solutions of Equation (6) can be represented as

$${\mathbf{z}}_j(x,\lambda )=e^{{\widehat{\mu }}_{j}x}·\widehat{T}·({\mathbf{e}}_j+o\left(1\right)),j=\overline{1,5}.$$

Thus, when the numerical coefficient k passes through the points $k=\pm 1$, the asymptotics of the fundamental system of solutions of Equation (4) undergoes a qualitative change. In other words, the points $k=\pm 1$ are bifurcation points for System (6) and corresponding Equation (4).

Such bifurcation points for differential equations and systems of equations with regularly oscillating coefficients are typical and were recently noted by us in works devoted to the study of the asymptotic behavior of the Sturm–Liouville equation with an oscillating potential [4,11].

4. Discussion

The results obtained have important applications in the spectral theory of differential operators generated by the left side of Equation (1). In particular, they make it possible to calculate the deficiency indices of the corresponding minimal differential operator.

The authors intend to investigate this issue in the future. In addition, we will be interested in the qualitative nature of the spectrum of such operators.

5. Conclusions

In this paper, we present a new approach to studying asymptotic behavior for $x\to \infty$ solutions of singular binomial differential equations of odd order for new classes of coefficients the corresponding differential expression.

This approach is based on the transition to a first-order system using quasi-derivatives and sequences of matrix transformations of this system. The key point of our approach is using the features of the algebraic matrix structure of the resulting system of ordinary differential equations.

This makes it possible to construct asymptotic formulas for solutions of new classes of equations, for example, for equations with oscillating coefficients. Note that similar results for such equations were not previously known.

Of particular interest is the counterexample we constructed, showing the phenomenon of “resonance” of oscillating coefficients: the existence of certain values of the numerical parameters of the coefficients of the equation at which the qualitative change asymptotic behavior of solutions. This property is typical, in general, for differential equations with oscillating coefficients of arbitrary order and was previously shown by us for the Sturm–Liouville equation.