Here, we will show the existence of the Floquet solutions of the equation \(\ell_{\lambda}(y)=0\), which plays an important role in the investigation of the spectrum of the operator \(L_{\lambda}\). If the characteristic polynomial has more than one multiple root, then there may arise various cases to obtain the fundamental system of solutions. Below, we consider the cases when there exist simple roots or one multiple root.
Case I. The characteristic polynomial \(\phi(z)\) has different simple roots \(\omega_{1},\omega_{2},\ldots,\omega_{m}\).
Theorem 1
If (1), (2) hold and
ω
is any root of
\(\phi(z)=0\)
then for each
\(\lambda\in \mathbb{C}\), \(\lambda\neq i\alpha_{n} ( \omega_{j}-\omega ) ^{-1}\), \(j=1,2,\ldots,m\), \(\omega_{j}\neq\omega \), \(n\in \mathbb{N}\), the differential equation
$$ y^{(m)}+\sum_{\gamma=1}^{m}p_{\gamma}(x, \lambda)y^{(m-\gamma)}=0 $$
(3)
has the solution
$$ f(x,\lambda)=e^{\omega\lambda x} \Biggl( 1+\sum_{n=1}^{\infty }U_{n}( \lambda )e^{i\alpha_{n}x} \Biggr) , $$
(4)
where
$$ U_{n}(\lambda)=U_{n}+ \sum_{k=1}^{n} \sum_{\substack{ j=1 \\ \omega_{j}\neq\omega}}^{m}\frac{U_{jkn}}{ [ i\alpha_{k}+(\omega -\omega_{j})\lambda ] },\quad \forall n\in \mathbb{N}, $$
with
\(U_{n},U_{jkn}\in \mathbb{C}\)
and series
$$ \sum_{n=1}^{\infty}\alpha_{n}^{\gamma} \bigl\vert U_{n}(\lambda )\bigr\vert ,\quad \gamma=0,1, \ldots,m, $$
(5)
is majorized in each compact set
\(S\subseteq \mathbb{C}\)
which does not contain the numbers
\(\lambda=i\alpha _{n} ( \omega_{j}-\omega ) ^{-1}\)
for
\(j=1,2,\ldots,m\), \(\omega _{j}\neq\omega\), \(n\in \mathbb{N}\).
Proof
Let ω be any root of the characteristic polynomial \(\phi(\omega)\). If we assume the existence of the solution of equation (3) represented as (4) with convergent series (5), then we can find the derivatives of \(f(x,\lambda)\) with respect to x as
$$ f^{(\gamma)}(x,\lambda)=e^{\omega\lambda x} \Biggl( (\omega\lambda )^{\gamma}+\sum_{n=1}^{\infty}(i \alpha_{n}+\omega\lambda)^{\gamma }U_{n}( \lambda)e^{i\alpha_{n}x} \Biggr) ,\quad \gamma=0,1,\ldots,m. $$
(6)
If we substitute these derivatives in (3) and divide both sides by \(e^{\omega \lambda x}\), then we get
$$\begin{aligned}& (\omega\lambda)^{m}+\sum_{n=1}^{\infty} (i\alpha_{n} +\omega\lambda)^{m}{U}_{n} ( \lambda)e^{i\alpha_{n}x}+\sum_{\gamma=1}^{m}(\omega \lambda)^{m-\gamma }p_{\gamma\gamma}\lambda^{\gamma} \\& \quad {}+\sum _{\gamma=1}^{m}p_{\gamma \gamma}\lambda^{\gamma}\sum _{n=1}^{\infty}(i\alpha_{n}+\omega \lambda)^{m-\gamma}U_{n}(\lambda)e^{i\alpha_{n}x} \\& \quad {}+\sum_{\gamma=1}^{m}(\omega \lambda)^{m-\gamma }\sum_{n=1}^{\infty} \tilde{p}_{\gamma n}(\lambda)e^{i\alpha _{n}x} \\& \quad {} +\sum _{\gamma=1}^{m}\sum_{n=1}^{\infty} \tilde{p}_{\gamma n}(\lambda)e^{i\alpha_{n}x}\sum _{n=1}^{\infty}(i\alpha _{n}+\omega \lambda)^{m-\gamma}U_{n}(\lambda)e^{i\alpha_{n}x}=0 \end{aligned}$$
or
$$\begin{aligned}& {\lambda}^{m}\phi(\omega)+\sum_{n=1}^{\infty} \Biggl[ (i\alpha_{n}+\omega\lambda)^{m}+\sum _{\gamma=1}^{m}p_{\gamma \gamma}\lambda^{\gamma}(i \alpha_{n}+\omega\lambda)^{m-\gamma } \Biggr] U_{n}( \lambda)e^{i\alpha_{n}x} \\& \quad {}+\sum_{n=1}^{\infty }\sum _{\gamma=1}^{m}(\omega\lambda)^{m-\gamma} \tilde{p}_{\gamma n}(\lambda)e^{i\alpha_{n}x} \\& \quad {}+\sum_{\gamma=1}^{m}\sum _{n=2}^{\infty} \biggl( \sum _{\alpha_{r}+\alpha_{s}=\alpha_{n}}\tilde{p}_{\gamma s}(\lambda) (i \alpha_{r}+\omega\lambda)^{m-\gamma}U_{r}(\lambda ) \biggr) e^{i\alpha_{n}x}=0 . \end{aligned}$$
Taking into account the uniqueness theorem for almost-periodic functions we have
$$\begin{aligned}& \Biggl[ (i\alpha_{n}+\omega\lambda)^{m}+\sum _{\gamma =1}^{m}p_{\gamma\gamma}\lambda^{\gamma}(i \alpha_{n}+\omega\lambda )^{m-\gamma} \Biggr] U_{n}( \lambda)+\sum_{\gamma=1}^{m}(\omega \lambda)^{m-\gamma}\tilde{p}_{\gamma n}(\lambda) \\& \quad {} +\sum_{\gamma=1}^{m} \sum _{\alpha_{r}+\alpha _{s}=\alpha_{n}}\tilde{p}_{\gamma s}(\lambda) (i\alpha _{r}+\omega \lambda)^{m-\gamma}U_{r}(\lambda)=0 , \quad n\in \mathbb{N} . \end{aligned}$$
Using the expansion
$$\begin{aligned}& (i\alpha_{n}+\omega\lambda)^{m}+\sum _{\gamma=1}^{m}p_{\gamma \gamma}\lambda^{\gamma}(i \alpha_{n}+\omega\lambda)^{m-\gamma} \\& \quad =\lambda^{m} \Biggl[ \biggl(\frac{i\alpha_{n}}{\lambda}+\omega \biggr)^{m}+\sum_{\gamma =1}^{m}p_{\gamma\gamma} \biggl(\frac{i\alpha_{n}}{\lambda}+\omega \biggr)^{m-\gamma} \Biggr]= \lambda^{m}\phi\biggl(\frac{i\alpha_{n}}{\lambda}+\omega\biggr) \\& \quad = \bigl[ i\alpha _{n}+\lambda ( \omega-\omega_{1} ) \bigr] \cdot \bigl[ i\alpha _{n}+\lambda ( \omega- \omega_{2} ) \bigr] \cdot\ldots\cdot \bigl[ i\alpha_{n}+ \lambda ( \omega-\omega_{m} ) \bigr] , \end{aligned}$$
we obtain
$$ U_{n}(\lambda)=- \frac{\sum_{\gamma=1}^{m}(\lambda { \omega})^{m-\gamma}\tilde{p}_{\gamma n}(\lambda )+\sum_{\gamma=1}^{m} \sum_{\alpha_{r}+\alpha _{s}=\alpha_{n}}(i\alpha_{r}+{ \omega}\lambda)^{m-\gamma}\tilde{p}_{\gamma s}(\lambda){U_{r}(\lambda)}}{[i\alpha _{n}+\lambda(\omega-\omega_{1})]\cdot[ i\alpha_{n}+\lambda (\omega-\omega_{2})]\cdot\ldots\cdot[ i\alpha_{n}+\lambda (\omega -\omega_{m})]} $$
(7)
for \(\lambda\in \mathbb{C}\), \(\lambda\neq i\alpha_{n} ( \omega_{j}-\omega ) ^{-1}\), \(j=1,2,\ldots,m\), \(\omega_{j}\neq\omega\), \(n\in \mathbb{N}\).
On the contrary, if \(\{ { U}_{n}{ (\lambda)} \} \) satisfies the system of equations (7) and the series (5) converges, then it can be shown that \(f(x,\lambda)\) determined by (4) is a solution of (3). Therefore, the solvability of (7) and the convergence of the series (5) are sufficient to prove the theorem.
From (7), \(\{ U_{n}(\lambda) \} \) is determined by the recurrent manner uniquely. It is possible to see that \(U_{n}(\lambda)\) is the rational function which can have simple poles \(\lambda=i\alpha _{k}(\omega _{j}-\omega)^{-1}\), \(j=1,2,\ldots,m\), \(\omega_{j}\neq\omega\), \(k=1,2,\ldots,n\), and therefore it can be uniquely written as
$$ U_{n}(\lambda)=U_{n}+ \sum_{k=1}^{n} \sum_{\substack{ j=1 \\ \omega_{j}\neq\omega}}^{m}\frac{U_{jkn}}{ [ i\alpha_{k}+(\omega -\omega_{j})\lambda ] },\quad \forall n\in \mathbb{N}, $$
where \(U_{n},U_{jkn}\in \mathbb{C}\). Let \(S\subseteq \mathbb{C}\) be a compact set which does not contain the points \(\lambda=i\alpha _{n}(\omega_{j}-\omega)^{-1}\) for \(j=1,2,\ldots,m\), \(\omega_{j}\neq \omega\), \(n\in\mathbb{N}\). Let us show that the series (5) is majorized in S for \(\{ { U}_{n}{ (\lambda)} \} \) which is determined from (7).
It is obvious that there exist \(c_{0}>0\), \(q>1\) such that
$$ c_{0}\alpha_{n}^{m}\leq\bigl\vert \bigl[ i \alpha_{n}+(\omega-\omega _{1})\lambda\bigr]\cdot\bigl[ i \alpha_{n}+(\omega-\omega_{2})\lambda \bigr]\cdot \ldots \cdot\bigl[ i\alpha_{n}+(\omega-\omega_{m})\lambda\bigr]\bigr\vert $$
and \(\vert \omega\lambda \vert \leq q\) for \(\forall n\in \mathbb{N}\), \(\forall\lambda\in S\). Then from (7), we have
$$\begin{aligned} c_{0}\alpha_{n}^{m}\bigl\vert U_{n}(\lambda)\bigr\vert \leq& \sum_{\gamma=1}^{m} \vert { \omega}\lambda \vert ^{m-\gamma}\bigl\vert \tilde{p}_{\gamma n}( \lambda)\bigr\vert +\sum_{\gamma=1}^{m} \sum _{\alpha_{r}+\alpha _{s}=\alpha_{n}}\bigl\vert U_{r}(\lambda)\bigr\vert \cdot \vert i{{\alpha _{r}+\omega\lambda}} \vert ^{m-\gamma}\bigl\vert \tilde{p}_{\gamma s}(\lambda)\bigr\vert \\ \leq&\sum_{\gamma=1}^{m}q^{m-\gamma} \tilde{p}_{\gamma n}+\sum_{\gamma=1}^{m} \sum _{\alpha_{r}+\alpha _{s}=\alpha_{n}}\bigl\vert U_{r}(\lambda)\bigr\vert \bigl(\vert {{\alpha _{r}}}\vert +q\bigr)^{m-\gamma} \tilde{p}_{\gamma s} \\ \leq& q^{m-1}\sum_{\gamma=1}^{m} \tilde{p}_{\gamma n}+\sum_{\gamma=1}^{m} \sum _{\alpha_{r}+\alpha _{s}=\alpha_{n}}\bigl\vert U_{r}(\lambda)\bigr\vert \alpha _{r}^{m-\gamma }\biggl(1+\frac{q}{{{\alpha}_{1}}} \biggr)^{m-\gamma}\tilde{p}_{\gamma s} \\ \leq& q^{m-1}\sum_{\gamma=1}^{m} \tilde{p}_{\gamma n}+\biggl(1+\frac{q}{{{\alpha}_{1}}}\biggr)^{m-1} \sum _{\alpha_{r}+\alpha_{s}=\alpha _{n}}\bigl\vert U_{r}(\lambda)\bigr\vert \alpha _{r}^{m-1}\cdot\sum _{\gamma=1}^{m}\tilde{p}_{\gamma s} \end{aligned}$$
for \(\forall\lambda\in S\), \(\forall n\in \mathbb{N}\) and \(\tilde{p}_{\gamma n}=\sup_{\lambda\in S}\vert \tilde{p}_{\gamma n}{ (\lambda)}\vert \), \(\gamma=1,2,\ldots,m\).
Let
$$\begin{aligned}& A_{n}= \frac{q^{m-1}}{c_{0}}\sum_{\gamma=1}^{m}\tilde{p}_{\gamma n} , \\& B_{n}= \frac{1}{c_{0}} \biggl( 1+\frac{q}{\alpha_{1}} \biggr) ^{m-1}\sum_{\gamma=1}^{m} \tilde{p}_{\gamma n} . \end{aligned}$$
Then from the last inequality we obtain
$$ \alpha_{n}^{m}\bigl\vert U_{n}(\lambda)\bigr\vert \leq A_{n}+\sum_{\alpha_{r}+\alpha_{s}=\alpha_{n}} \alpha _{r}^{m-1}\bigl\vert U_{r}(\lambda)\bigr\vert B_{s} ,\quad n\in \mathbb{N}. $$
If \(u_{n}= \sup_{\lambda\in S}\vert U_{n}(\lambda )\vert \), then we have
$$ \alpha_{n}^{m}u_{n}\leq A_{n}+ \sum _{\alpha_{r}+\alpha _{s}=\alpha_{n}}\alpha_{r}^{m-1}u_{r}B_{s} , \quad n\in \mathbb{N}$$
or
$$\begin{aligned} \sum_{n=1}^{t}\alpha_{n}^{m}u_{n} \leq& \sum_{n=1}^{t}A_{n}+\sum _{n=1}^{t}\sum_{\alpha_{r}+\alpha_{s}=\alpha _{n}} \alpha_{r}^{m-1}u_{r}B_{s} \\ \leq& A+\sum_{r=1}^{t-1}\alpha _{r}^{m-1}u_{r}\sum_{s=1}^{t-1}B_{s} \leq A+B\sum_{n=1}^{t-1}\alpha _{n}^{m-1}u_{n},\quad n\in \mathbb{N}. \end{aligned}$$
From (2) it is clear that \(A= \sum_{n=1}^{\infty}A_{n}<+\infty \) and \(B= \sum_{n=1}^{\infty}B_{n}<+\infty\).
Therefore, for all \(t\in \mathbb{N}\), \(\sum_{n=1}^{t}\alpha_{n}^{m}u_{n}\leq A+B\sum_{n=1}^{t-1}\alpha_{n}^{m-1}u_{n}\) is satisfied. By using this inequality and \(\alpha_{n}\rightarrow+\infty\), we can easily show the convergence of the series \(\sum_{n=1}^{+\infty}\alpha _{n}^{m}u_{n} \) according to the lemma in [12] (see [12], pp.21-22). In this case, \(\sum_{n=1}^{+\infty}\alpha_{n}^{m}| U_{n}(\lambda)|\) is a majorized series in S. According to the Weierstrass theorem, the series (5) is uniform convergent in S. Since \(S\subseteq \mathbb{C}\) is an arbitrarily chosen compact set, the series (5) is convergent for all \(\lambda\neq i\alpha_{n}(\omega_{j}-\omega)^{-1}\), \(j=1,2,\ldots,m\), \(\omega_{j}\neq\omega\), and \(n\in \mathbb{N}\). Thus \(f(x,\lambda)\) is a solution of equation (3). The theorem is proved. □
It is clear that \(\lambda_{jn}=i\alpha_{n}(\omega_{j}-\omega)^{-1}\) may be a singular point of \(f(x,\lambda)\) for any \(j=1,2,\ldots,m\), \(\omega _{j}\neq\omega\), \(n\in \mathbb{N}\). Actually, according to Theorem 1, the functional series in the representation
$$ \bigl[ i\alpha_{n}+(\omega-\omega_{j}) \bigr] f(x, \lambda)=e^{\omega \lambda x} \Biggl( 1+\sum_{r=1}^{\infty} \bigl[ i\alpha_{n}+(\omega -\omega _{j})\lambda \bigr] U_{r}(\lambda)e^{i\alpha_{r}x} \Biggr) $$
and the obtained series by m times term by term differentiation are absolutely and uniformly convergent with respect to λ in the closed disk with a small radius centered in point \(\lambda_{jn}\). Therefore, the finite limits
$$ \lim_{\lambda\rightarrow\lambda_{jn}} \bigl[ i\alpha _{n}+(\omega- \omega_{j})\lambda \bigr] \frac{\partial ^{s}f(x,\lambda)}{\partial x^{s}}=e^{\omega\lambda_{jn}x}{\sum _{r=n}^{\infty }U_{jnr}(\omega \lambda_{jn}+i\alpha_{r})^{s}}e^{i\alpha_{r}x}, \quad s=0,1,2,\ldots,m, $$
exist. Moreover, the series \({\sum_{r=n}^{\infty}} \vert {U_{jnr}}\vert {\alpha_{r}^{m}}\) is convergent. If this limit is not zero, then it means that the point \(\lambda_{jn}\) is a simple pole of the functions \(\frac{\partial^{s}f(x,\lambda)}{\partial x^{s}}\), \(s=0,1,\ldots,m\).
Corollary 1
For
\(\forall x\in R\), the functions
\(\frac {\partial ^{s}f(x,\lambda)}{\partial x^{s}}\), \(s=0,1,\ldots,m\), are meromorphic functions with respect to
λ
and they can have only simple poles
\({\lambda_{jn}}=i\alpha_{n}(\omega_{j}-\omega)^{-1}\), \(j=1,2,\ldots,m\), \(\omega_{j}\neq\omega\), \(n\in\mathbb{N}\). Moreover, these functions are also continuous functions of the pair
\((x,\lambda)\)
for all
\((x,\lambda)\in \mathbb{R}\times\mathbb{C}\), \(\lambda\neq i\alpha_{n}(\omega_{j}-\omega)^{-1}\), \(j=1,2,\ldots,m\), \(\omega_{j}\neq\omega\), \(n\in \mathbb{N}\).
Corollary 2
For
\(\forall\lambda\neq\lambda _{sjn}\), \(s,j=1,2,\ldots,m\), \(j\neq s\), \(n\in \mathbb{N}\), equation (3) has the Floquet solutions represented as
$$ f_{s}(x,\lambda)=e^{\omega_{s}\lambda x} \Biggl( {1+\sum _{n=1}^{\infty}U_{n}^{(s)}( \lambda)e^{i\alpha_{n}x}} \Biggr) , $$
where
$$ U_{n}^{(s)}(\lambda)=U_{n}^{(s)}+ \sum _{k=1}^{n}\sum _{\substack{ j=1 \\ j\neq s}}^{m}\frac{U_{jkn}^{(s)}}{i\alpha_{k}+(\omega _{s}-\omega _{j})\lambda}. $$
The Wronskian of the functions
\(f_{1}(x,\lambda ), f_{2}(x,\lambda),\ldots, f_{m}(x,\lambda)\)
for
\(\lambda\in\mathbb{C}\backslash\Lambda\)
is found as
$$ W[f_{1},f_{2},\ldots,f_{m}]=W(x,\lambda)= \lambda^{\frac{m(m-1)}{2}}W_{m}e^{-\lambda p_{11}x-\sum_{n=1}^{\infty}\frac{p_{10n}}{i\alpha _{n}}e^{i\alpha_{n}x}}. $$
(8)
Here
\(W_{m}\)
is the Vandermonde determinant of the numbers
\(\omega_{1},\omega_{2},\ldots,\omega_{m}\). The functions
\(f_{1}(x,\lambda), f_{2}(x,\lambda),\ldots, f_{m}(x,\lambda)\)
form the fundamental system of solutions of equation (3) in the interval
\(( -\infty,+\infty ) \)
for
\(\forall\lambda\in \mathbb{C}\backslash\Lambda_{0}\).
It is clear that the existence of the solutions \(f_{1}(x,\lambda), f_{2}(x,\lambda),\ldots, f_{m}(x,\lambda)\) follows from Theorem 1 for \(\omega=\omega_{s}\), \(s=1,2,\ldots,m\). Since every \(q(x)\in \mathit{AP}^{+}\) can be extended to the upper semi-plane as an analytic function of x and \(\lim_{\operatorname{Im}x\rightarrow+\infty}q(x)=0\), by passing to the limit as \(\operatorname{Im}x\rightarrow+\infty\) on both sides of the equation
$$ W(x,\lambda)e^{\lambda p_{11}x}=W(0,\lambda )e^{-\int_{0}^{x}p_{10}(t)\, dt} $$
we get equation (8). From (8) it follows that if \(\lambda\neq0\), then \(W(x,\lambda)\neq0\). Thus the system of functions is independent.
Note that the solutions of type \(f_{s}(x,\lambda )\), \(s=1,2,\ldots,m\), are obtained in [6–9] under the different conditions and in various forms of the representation.
According to Corollary 1, it is obvious that the function
$$ f_{sjn}(x)= {\lim_{\lambda\rightarrow\lambda _{sjn}}f_{s}(x,\lambda)} \bigl[ {i\alpha_{n}+(\omega_{s}-\omega _{j}) \lambda} \bigr] {=e^{{\omega_{s}}\lambda_{sjn}x}\sum_{k=n}^{\infty }U_{jnk}^{(s)}e^{i\alpha_{k}x}} $$
is a solution of equation (3) for \(\lambda=\lambda_{sjn}\), where the series \({\sum_{k=n}^{\infty}\alpha_{k}^{m}}\vert {U_{jnk}^{(s)}} \vert \) is convergent. As in [4], writing the equation which is satisfied by the coefficients \(U_{jnk}^{(s)}\) (see [4], p.778), it is seen easily that \(U_{jnk}^{(s)}=0\) for every \(k\geq n\) if \(U_{jnn}^{(s)}=0\). Therefore, \(f_{sjn}(x)\equiv0\) if and only if \(U_{jnn}^{(s)}=0\). In this case, \(f_{s}(x,\lambda)\) is regular at point \(\lambda=\lambda_{sjn}\) and \(f_{s}(x,\lambda_{sjn})\) is a solution of equation (3). It can be shown that the functions \(f_{sjn}(x)\) and \(f_{j}(x,\lambda_{sjn})\) are linearly dependent. Moreover, \(f_{sjn}(x)=U_{jnn}^{(s)}f_{j}(x,\lambda_{sjn})\) for any \(s=1,2,\ldots,m\), \(s\neq j=1,2,\ldots,m\), \(n\in \mathbb{N}\), is valid which is important for establishing the fundamental system of solutions of equation (3) for \(\lambda=\lambda_{sjn}\).
Let s, j, n be fixed and \(f_{s}(x,\lambda), f_{s_{1}}(x,\lambda ),f_{s_{2}}(x,\lambda),\ldots,f_{s_{\mu}}(x,\lambda)\) be all functions which have a pole at the point \(\lambda=\lambda_{sjn}\). It is only possible when the equality \(\lambda_{sjn}=\lambda_{s_{\beta}j_{\beta}{ n}_{\beta }}\), \(\beta=1,2,\ldots,\mu\), is valid for some different indices \(n_{1},n_{2},\ldots,n_{\mu}\in \mathbb{N}\), and \(1\leq j_{1},j_{2},\ldots,j_{\mu}\leq m\). Then all other functions \(f_{j}(x,\lambda), f_{j_{1}}(x,\lambda), f_{j_{2}}(x,\lambda ),\ldots,f_{j_{\mu}}(x,\lambda), f_{j_{\mu+1}}(x,\lambda ),\ldots,f_{j_{\nu }}(x,\lambda)\), \(\mu+\nu+2=m\), are regular at the point \(\lambda =\lambda _{sjn}\).
If we define the functions \(f_{kjn}(x)= {\lim_{\lambda \rightarrow\lambda_{sjn}}f_{k}(x,\lambda)} [ {i\alpha _{n}+(\omega _{s}-\omega_{j})\lambda} ]\), \(k=s_{1},s_{2},\ldots,s_{\mu}\), then it is obvious that the functions
$$ \begin{aligned} &f_{sjn}(x),\qquad f_{s_{1}jn}(x),\qquad f_{s_{2}jn}(x), \qquad \ldots, \\ &f_{s_{\mu }jn}(x),\qquad f_{k}(x, \lambda_{sjn}),\quad k=j,j_{1},j_{2}, \ldots,j_{\nu},\end{aligned} $$
(9)
are solutions of equation (3) for \(\lambda=\lambda_{sjn}\) and the functions of this system are linear dependent in \(( -\infty ,+\infty ) \), since their Wronskian is equal to zero. Moreover, any three of the numbers \(\operatorname{Re}(\lambda _{sjn}\omega _{k})\), \(k=1,2,\ldots,m\), can not be equal and there are some equal pairs between them. These equal pairs are \(\operatorname{Re}(\lambda _{sjn}\omega _{s})=\operatorname{Re}(\lambda _{sjn}\omega _{j})\), \(\operatorname{Re}(\lambda _{sjn}\omega _{s_{\beta }})= \operatorname{Re}(\lambda _{sjn}\omega _{j_{\beta }})\), \(\beta =1,2,\ldots,\mu \). Then taking into our account the behaviors as \(x\rightarrow \pm \infty \) of the functions belonging to the system (9), as it is shown in [12] (see pp.43-45), we have the existence of some constants \(b_{k}\), \(k=s\), \(s_{1},s_{2},\ldots,s_{\mu }\) such that
$$ f_{sjn}(x)=b_{s}f_{j}(x,\lambda _{sjn}),\qquad f_{s_{\beta }jn}(x)=b_{s_{\beta }}f_{j_{\beta }}(x,\lambda _{sjn}),\quad \beta =1,2,\ldots,\mu . $$
From the equality \(f_{sjn}(x)=b_{s}f_{j}(x,\lambda_{sjn})\), according to the uniqueness theorem for almost-periodic functions, it is seen that \(b_{s}=U_{jnn}^{(s)}\). Using the equalities
$$ f_{sjn}(x)=b_{s}f_{j}(x,\lambda_{sjn}), \qquad f_{s_{\beta }jn}(x)=b_{s_{\beta}}f_{j_{\beta}}(x, \lambda_{sjn}),\quad \beta =1,2,\ldots,\mu, $$
the system of linearly independent solutions of equation (3) corresponding to \(\lambda=\lambda_{sjn}\) can be established.
Since the functions \(f_{k}(x,\lambda)\), \(k=j,j_{1},j_{2},\ldots,j_{\mu }\), are regular at \(\lambda=\lambda_{sjn}\), the functions
$$\begin{aligned}& \tilde{f}_{sjn}(x)= \lim_{\lambda\rightarrow\lambda _{sjn}} \biggl( f_{s}(x,\lambda)- {\frac{b_{s}f_{j}(x,\lambda)}{i\alpha_{n}+(\omega_{s}-\omega_{j})\lambda}} \biggr) , \\& \tilde{f}_{s_{\beta}jn}(x)= \lim_{\lambda\rightarrow \lambda_{sjn}} \biggl( f_{s_{\beta}}(x,\lambda)- {\frac{b_{s_{\beta}}f_{j_{\beta}}(x,\lambda)}{i\alpha_{n}+(\omega _{s}-\omega _{j})\lambda}} \biggr) ,\quad \beta=1,2, \ldots,\mu, \end{aligned}$$
are also solutions of equation (3) corresponding to \(\lambda =\lambda_{sjn}\). According to the expressions of the functions \(f_{k}(x,\lambda )\), \(k=s, s_{1},s_{2},\ldots,s_{\mu},j,j_{1},j_{2},\ldots,j_{\mu}\), we conclude that \(\tilde{f}_{kjn}(x)=e^{i\omega_{k}\lambda_{sjn}x}(\psi _{kjn}(x)+x\phi _{kjn}(x))\), where \(\psi_{kjn}(x)\) and \(\phi_{kjn}(x)\) are Bohr almost-periodic functions for \(k=s,s_{1},s_{2},\ldots,s_{\mu}\). From the explicit form of the functions \(\tilde{f}_{sjn}(x),\tilde{f}_{s_{1}jn}(x), \tilde{f}_{s_{2}jn}(x),\ldots, \tilde{f}_{s_{\mu}jn}(x), f_{k}(x,\lambda_{sjn})\), \(k=j,j_{1},j_{2},\ldots,j_{\nu}\), it is seen that these functions are linearly independent on \((-\infty,+\infty)\). Therefore, these functions form a fundamental system of solutions of equation (3) for \(\lambda=\lambda_{sjn}\).
Now let us construct the linearly independent solutions of equation (3) for \(\lambda=0\).
Note that, since the Wronskian of the solutions \(f_{s}(x,\lambda)\), \(s=1,2,\ldots,m\), is equal to zero for \(\lambda=0\), they are linearly dependent. Linearly independent solutions of equation (3) corresponding to \(\lambda =0\) are established according to Theorem 1. It is clear that solutions of the equation
$$ y^{(m)}+\sum_{\gamma=1}^{m}p_{\gamma0}(x)y^{(m-\gamma)}= \lambda ^{m}y $$
(10)
corresponding to \(\lambda=0\) are also solutions of equation (3) for \(\lambda=0\). By Theorem 1, equation (10) has the solution
$$ \tilde{f}(x,\lambda)=e^{\lambda x} \Biggl( {1+\sum _{n=1}^{\infty}}\widetilde{{U}} {_{n}( \lambda)e^{i\alpha_{n}x}} \Biggr), $$
which is analytic with respect to λ in some small neighborhood of \(\lambda=0\). By putting \(\tilde{f}(x,\lambda)\) in (10) and by differentiating equation (10) with respect to λ, it is sure that functions \(\tilde{f}_{s}(x)= \frac{\partial^{s}\tilde{f}(x,\lambda)}{\partial\lambda^{s}}|_{\lambda=0}\), \(s=0,1,\ldots,m-1\), are also solutions of (10) and (3) corresponding to \(\lambda =0\). We can see easily that \(\tilde{f}_{0}(x)=\alpha_{00}(x)\), \(\tilde {f}_{1}(x)=x\alpha _{11}(x)+\alpha_{10}(x)\), … , \(\tilde{f}_{m-1}(x)=x^{m-1}\alpha _{m-1,m-1}(x)+x^{m-2}\alpha_{m-1,m-2}(x)+\cdots+\alpha_{m-1,0}(x)\), where \(\alpha_{sj}(x)\), \(s=0,1,\ldots,m-1\), \(j=0,1,\ldots,s\), are Bohr almost-periodic functions and \(\alpha_{ss}(x)\), \(s=0,1,\ldots,m-1\), are nonzero. The linear independence of \(\tilde{f}_{s}(x)\), \(s=0,1,\ldots,m-1\), in \((-\infty ,+\infty)\) is seen from their open form.
Case II. The characteristic polynomial has a unique multiple root \(\omega_{0}\), i.e.
\(\phi(z)=(z-\omega_{0})^{m}\).
In this case, to find the particular solutions of equation (3) we will use the following theorem.
Theorem 2
If the characteristic polynomial has a unique multiple root
\(\omega_{0}\), then for each function
\(g(x,\lambda )=e^{\omega_{0}\lambda x}\sum_{n=1}^{\infty}g_{n}(\lambda )e^{i\alpha_{n}x}\)
such that
\(g_{n}(\lambda)\), \(n\in \mathbb{N}\), are polynomials whose degree does not exceed
\(n(m-1)\)
and the series
\(\sum_{n=1}^{\infty} \vert g_{n}(\lambda )\vert \)
is majorized in any compact set
\(S\subseteq \mathbb{C}\), the equation
$$ y^{(m)}+\sum_{\gamma=1}^{m}p_{\gamma}(x, \lambda)y^{(m-\gamma )}=g(x,\lambda) $$
(11)
has a solution
$$ h(x,\lambda)=e^{\omega_{0}\lambda x} \Biggl( {1+\sum _{n=1}^{\infty}h_{n}(\lambda)e^{i\alpha_{n}x}} \Biggr) $$
(12)
in
\((-\infty,+\infty)\)
for every
\(\lambda\in \mathbb{C}\). Here the coefficients
\(h_{n}(\lambda)\), \(n\in \mathbb{N}\), are polynomials whose degrees do not exceed
\(n(m-1)\), and the series
\(\sum_{n=1}^{\infty} \vert {h_{n}(\lambda)} \vert \alpha _{n}^{m}\)
is majorized in each compact set
\(S\subseteq \mathbb{C}\).
Proof
If we substitute the function (12) in (11), to find the coefficients sequence \(\{ {h_{n}(\lambda)} \} \) as in the proof of Theorem 1, we obtain a system of equations,
$$\begin{aligned}& \Biggl[ (i\alpha_{n}+\omega\lambda)^{m}+\sum _{\gamma =1}^{m}p_{\gamma\gamma}\lambda^{\gamma}(i \alpha_{n}+\omega\lambda )^{m-\gamma} \Biggr] h_{n}( \lambda)+\sum_{\gamma=1}^{m}(\omega \lambda)^{m-\gamma}\tilde{p}_{\gamma n}(\lambda) \\ & \quad {}+\sum_{\gamma=1}^{m} \sum _{\alpha_{r}+\alpha _{s}=\alpha_{n}}\tilde{p}_{\gamma s}(\lambda) (i\alpha _{r}+\omega \lambda)^{m-\gamma}h_{r}( \lambda)=g_{n}(\lambda),\quad n\in \mathbb{N}, \lambda\in \mathbb{C} \end{aligned}$$
or
$$\begin{aligned}& ( i\alpha_{n} ) ^{m}h_{n}(\lambda)+\sum _{\gamma =1}^{m}(\omega\lambda)^{m-\gamma} \tilde{p}_{\gamma n}(\lambda) \\ & \quad {}+\sum_{\gamma=1}^{m} \sum _{\alpha_{r}+\alpha _{s}=\alpha_{n}}\tilde{p}_{\gamma s}(\lambda) (i\alpha _{r}+\omega \lambda)^{m-\gamma}h_{r}( \lambda)=g_{n}(\lambda), \quad n\in \mathbb{N}, \lambda\in \mathbb{C}, \end{aligned}$$
or
$$\begin{aligned}& h_{n}(\lambda)=- \frac{\sum_{\gamma=1}^{m}(\lambda { \omega})^{m-\gamma}\tilde{p}_{\gamma n}(\lambda )+\sum_{\gamma=1}^{m} {\sum_{\alpha_{r}+\alpha _{s}=\alpha_{n}}(i\alpha_{r}+{ \omega}\lambda)}^{m-\gamma}\tilde{p}_{\gamma s}(\lambda){h_{r}(\lambda)-}g_{n}(\lambda)}{ ( i\alpha_{n} ) ^{m}}, \\ & \quad n\in \mathbb{N}, \lambda \in \mathbb{C}. \end{aligned}$$
(13)
The coefficients \({h_{n}(\lambda)}\) are found uniquely from equation (13). In fact, the degree of the polynomial \(h_{1}(\lambda)=-\frac{\sum_{\gamma =1}^{m}(\lambda{ \omega})^{m-\gamma}\tilde{p}_{\gamma 1}(\lambda)-g_{1}(\lambda)}{ ( i\alpha_{1} ) ^{m}}\) for \(n=1\) does not exceed \(m-1\). Subsequently, for \(n=2\) the degree of the polynomial \(h_{2}(\lambda)\) does not exceed \(2 ( m-1 ) \) and, for each n, \(h_{n}(\lambda)\) is found as a polynomial whose degree does not exceed \(n ( m-1 ) \). If for the obtained coefficients \(h_{n}(\lambda )\), \(n\in\mathbb{N}\), \(h_{n}= \sup_{\lambda\in S}\vert h_{n}(\lambda )\vert \), then convergence of the series \(\sum_{n=1}^{+\infty }\alpha_{n}^{m}h_{n}\) and so majorization of the series \(\sum_{n=1}^{+\infty}\alpha_{n}^{m}\vert h_{n}(\lambda) \vert \) in the set \(S\subseteq \mathbb{C}\) easily can be shown as in the proof of Theorem 1. Therefore, for each \(\lambda\in S\) the function \(h(x,\lambda)=e^{\omega\lambda x} ( {1+\sum_{n=1}^{\infty}h_{n}(\lambda)e^{i\alpha_{n}x}} ) \) is the solution of equation (11) in \((-\infty,+\infty)\). The theorem is proved. □
Corollary 3
If the characteristic polynomial
\(\phi(z)\)
has a unique multiple root
\(\omega_{0}\), then equation (3) has a solution
\(\hat{f}(x,\lambda)=e^{\omega\lambda x}q(x,\lambda ) \)
in
\((-\infty,+\infty)\)
for every
\(\lambda\in \mathbb{C}\). Here
\(q(x,\lambda)=1+\sum_{n=1}^{\infty}q_{n}(\lambda )e^{i\alpha_{n}x}\)
is a Bohr almost-periodic function. The
\(q_{n}(\lambda)\), \(n\in \mathbb{N}\), are polynomials whose degree does not exceed
\(n(m-1)\), the series
\(\sum_{n=1}^{+\infty}\alpha_{n}^{m}\vert q_{n}(\lambda)\vert \)
is majorized in each compact set
\(S\subseteq\mathbb{C}\). \(\hat{f}(x,\lambda),\frac{\partial\hat {f}(x,\lambda)}{\partial x},\ldots,\frac{\partial^{m}\hat{f}(x,\lambda)}{\partial x^{m}} \)
are continuous functions in
\(\mathbb{R}\times \mathbb{C}\)
with respect to the ordered pair
\((x,\lambda)\)
and they are an entire function of
λ.
To prove Corollary 3, it is enough to take \(g(x,\lambda )=0\) in Theorem 2 and to see m times differentiability term by term of the series in the expression of the obtained solution \(\hat{f}(x,\lambda )\) with respect to x. Here the obtained series are uniformly convergent in each bounded set of the ordered pairs \((x,\lambda )\), therefore functions \(\hat{f}(x,\lambda ),\frac{\partial \hat{f}(x,\lambda )}{\partial x},\ldots,\frac{\partial ^{m}\hat{f}(x,\lambda )}{\partial x^{m}}\) are continuous functions of the ordered pairs \((x,\lambda )\) and they are entire functions of λ.
Theorem 3
If
\(\phi(z)=(z-\omega_{0})^{m}\), then equation (3) has Floquet solutions in the interval
\((-\infty ,+\infty)\)
as
$$\begin{aligned}& \hat{f}_{1}(x,\lambda) = e^{\omega\lambda x}q_{1}(x,\lambda ),\qquad \hat{f}_{2}(x,\lambda)=e^{\omega\lambda x} \bigl[ xq_{1}(x,\lambda )+q_{2}(x,\lambda) \bigr] ,\qquad \ldots, \\& \hat{f}_{m}(x,\lambda) = e^{\omega\lambda x} \biggl[ \frac {x^{m-1}}{(m-1)!}q_{1}(x,\lambda)+\cdots+xq_{m-1}(x, \lambda)+q_{m}(x,\lambda ) \biggr], \end{aligned}$$
where the functions
\(q_{1}(x,\lambda), q_{2}(x,\lambda ),\ldots,q_{m}(x,\lambda)\)
are almost-periodic function:
$$ q_{s}(x,\lambda)=1+ \sum_{n=1}^{\infty}q_{sn}( \lambda )e^{i\alpha_{n}x},\quad s=1,2,\ldots,m $$
for
\(\forall\lambda\in \mathbb{C}\). Here
\(q_{sn}(\lambda)\), \(s=1,2,\ldots,m\), \(n\in\mathbb{N}\), are polynomials whose degrees do not exceed
\(n(m-1)\)
and the series
\(\sum_{n=1}^{\infty}\alpha_{n}^{m}\vert q_{sn}(\lambda )\vert \)
are majorized in each compact set
\(S\subseteq \mathbb{C}\).
Proof
When \(\phi(z)= ( z-\omega_{0} ) ^{m}\), equation (3) has a solution \(\hat{f}_{1}(x,\lambda)=e^{\omega \lambda x}q_{1}(x,\lambda)\) according to Corollary 3. In order to obtain other solutions which form a fundamental system of solutions of equation (3) together with \(\hat{f}_{1}(x,\lambda)\), let us use the properties of the linear differential operator \(L:C^{m}( \mathbb{R} )\rightarrow C( \mathbb{R})\), which is defined as
$$ L(y)=p_{0}(x)y^{(m)}+p_{1}(x)y^{(m-1)}+p_{2}(x)y^{(m-2)}+ \cdots+p_{m-1}(x)y^{ \prime}+p_{m}(x)y, $$
(14)
where \(p_{j}(x)\in C( \mathbb{R})\), \(j=0,1,\ldots,m\). Let us define the operators \(L^{(k)}(y)=\sum_{ \gamma=0}^{m-k}A_{m}^{k}p_{\gamma}(x) y^{(m-\gamma-k)}\), \(L^{(k)}:C^{m}(\mathbb{R} )\rightarrow C( \mathbb{R})\), \(k=1,2,\ldots,m\). Here, \(A_{m}^{k}=m(m-1)\cdot\ldots\cdot(m-k+1)\), \(k=1,2,\ldots,m\), and \(A_{m}^{0}=1\). For any system of functions \(y_{1}(x),y_{2}(x),\ldots,y_{m}(x)\in C^{m}( \mathbb{R})\) it is not difficult to show that the identities
$$\begin{aligned}& L(xy_{1}+y_{2})=L(y_{2})+L^{(1)}(y_{1})+xL(y_{1}), \\& L\biggl(\frac {x^{2}}{2!} y_{1}+xy_{2}+y_{3} \biggr)= L(y_{3})+L^{(1)}(y_{2})+ \frac{1}{2!}L^{(2)}(y_{1})+x \bigl[ L(y_{2})+L^{(1)}(y_{1}) \bigr] + \frac{x^{2}}{2!}L(y_{1}), \\& \ldots, \\& L\biggl(\frac{x^{s-1}}{(s-1)!}y_{1}+\frac{x^{s-2}}{(s-2)!}y_{2}+ \cdots+\frac {x}{1!}y_{s-1}+y_{s}\biggr) \\& \quad =L(y_{s})+L^{(1)}(y_{s-1})+ \frac{1}{2!}L^{(2)}(y_{s-2})+\cdots+\frac{1}{(s-1)!}L^{(s-1)}(y_{1}) \\& \qquad {}+x \biggl[ L(y_{s-1})+\frac{1}{1!}L^{(1)}(y_{s-2})+ \cdots+\frac{1}{(s-2)!}L^{(s-2)}(y_{1}) \biggr] +\cdots \\& \qquad {}+\frac{x^{s-2}}{(s-2)!} \bigl[ L(y_{2})+L^{(1)}(y_{1}) \bigr] +\frac {x^{s-1}}{(s-1)!}L(y_{1}),\quad s=2,3,\ldots,m, \forall x\in \mathbb{R}, \end{aligned}$$
hold. Therefore, when the equations
$$\begin{aligned}& L(y_{1})=0, \\& L(y_{2})+L^{(1)}(y_{1})=0, \\& L(y_{3})+L^{(1)}(y_{2})+\frac{1}{2!}L^{(2)}(y_{1})=0, \\& \ldots, \\& L(y_{m})+L^{(1)}(y_{m-1})+\frac{1}{2!}L^{(2)}(y_{m-2})+ \cdots+\frac {1}{(m-1)!}L^{(m-1)}(y_{1})=0, \end{aligned}$$
(15)
are satisfied, the functions
$$ \tilde{y}_{1}=y_{1},\qquad \tilde{y}_{j}= \frac{x^{j-1}}{(j-1)!} y_{1}+\frac{x^{j-2}}{(j-2)!}y_{2}+\cdots+ \frac{x}{1!}y_{j-1}+y_{j},\quad j=2,3, \ldots,m, $$
(16)
are solutions of the equation \(L(y)=0\).
Let us show the existence of functions \(y_{s}=e^{\omega\lambda x}q_{s}(x,\lambda)\), \(s=1,2,\ldots,m\), satisfying the system of equations (15) for the operators \(L=L_{\lambda}\), \(L^{(k)}=L_{\lambda}^{(k)}\), \(k=1,2,\ldots,m\). If we set in these equations \(L=L_{\lambda}\) and \(y_{1}=\hat{f}_{1}(x,\lambda)\), the solution \(y_{2}=e^{\omega\lambda x}q_{2}(x,\lambda ) \), which satisfies the equation
$$ L_{\lambda}(y_{2})+L_{\lambda}^{(1)}(y_{1})=0, $$
exists according to Theorem 2 for \(g(x,\lambda)=-L_{\lambda}^{(1)}(y_{1})\). It is not difficult to verify that the conditions of Theorem 2 are satisfied. In the same manner, when the functions \(y_{s}=e^{\omega\lambda x}p_{s}(x,\lambda)\), \(s=1,2,\ldots,k-1\), were found, the existence of the function \(y_{k}=e^{\omega\lambda x}q_{k}(x,\lambda)\) which satisfies the equation
$$ L_{\lambda}(y_{k})=-L_{\lambda}^{(1)}(y_{k-1})- \frac{1}{2!}L_{\lambda }^{(2)}(y_{k-2})-\cdots- \frac{1}{(k-1)!}L_{\lambda}^{(k-1)}(y_{1}),\quad k=2,3, \ldots,m, $$
is obtained according to Theorem 2 by induction for \(g(x,\lambda )=-L_{\lambda}^{(1)}(y_{k-1})-\frac{1}{2!}L_{\lambda }^{(2)}(y_{k-2})-\cdots-\frac{1}{(k-1)!}L_{\lambda}^{(k-1)}(y_{1})\). Consequently, according to (15), (16) the functions
$$\begin{aligned}& \hat{f}_{1}(x,\lambda) = e^{\omega\lambda x}q_{1}(x,\lambda ), \qquad \hat{f}_{2}(x,\lambda)=e^{\omega\lambda x} \bigl[ xq_{1}(x,\lambda )+q_{2}(x,\lambda) \bigr] , \\& \ldots, \\& \hat{f}_{m}(x,\lambda) = e^{\omega\lambda x}\biggl[\frac {x^{m-1}}{(m-1)!}q_{1}(x, \lambda)+\cdots+xq_{m-1}(x,\lambda)+q_{m}(x,\lambda)\biggr] \end{aligned}$$
are solutions of equation (3) in \(( -\infty,+\infty ) \) for \(\lambda\in \mathbb{C}\). The theorem is proved. □
Note that the solutions of type \(\hat{f}_{s}(x,\lambda )\), \(s=1,2,\ldots,m \), are obtained in [7] under the different conditions and in various form of the representation.
Corollary 4
When
\(\phi(z)= ( z-\omega_{0} ) ^{m}\), for each
\(\lambda\in \mathbb{C}\), \(x\in \mathbb{R}\)
Wronskian of functions
\(\hat{f}_{1}(x,\lambda),\hat {f}_{2}(x,\lambda),\ldots,\hat{f}_{m}(x,\lambda)\)
is found as
$$ \widehat{W}(x,\lambda)=e^{m\omega_{0}\lambda x-\sum_{n=1}^{\infty}\frac{p_{10n}}{i\alpha_{n}}e^{i\alpha_{n}x}}\neq0 $$
(17)
and hence for each
\(\lambda\in \mathbb{C}\), the functions
\(\hat{f}_{1}(x,\lambda),\hat {f}_{2}(x,\lambda ),\ldots,\hat{f}_{m}(x,\lambda)\)
form the fundamental system of solutions of equation (3) in the interval
\(( -\infty ,+\infty ) \).