We rebuild the fundamental solutions of Sturm–Liouville problem (1.1)–(1.5) and get the asymptotic formulas for the eigenvalues and eigenfunctions in this section.
Lemma 3.1
(see [26])
Let the real-valued function
\(q(x) \)
be continuous on
\(I=[0,\xi)\cup(\xi,1] \), and
\(f(\lambda)\), \(g(\lambda) \)
be given entire functions. Then, for any
\(\lambda\in\mathbb{C} \), the following equation
$$ Ly:=-y''(x)+q(x)y(x)=\lambda y(x),\quad x\in[0, \xi) \cup(\xi,1] $$
has the unique solution
\(y=y(x, \lambda) \)
satisfying the initial conditions
$$ y(0,\lambda)=f(\lambda), \qquad y'(0,\lambda)=g(\lambda). $$
For each fixed
\(x\in[0, \xi)\cup(\xi,1] \), \(y(x, \lambda) \)
is an entire function of
λ.
Now, we define fundamental solutions \(\varphi(x,\lambda) \) and \(\chi (x,\lambda) \) of equation (1.1) by the following procedure, where
$$ \varphi(x,\lambda)= \textstyle\begin{cases} \varphi_{1}(x, \lambda),& x\in[0,\xi),\\ \varphi_{2}(x, \lambda),& x\in(\xi,1] \end{cases} $$
and
$$ \chi(x,\lambda)= \textstyle\begin{cases} \chi_{1}(x, \lambda),& x\in[0,\xi),\\ \chi_{2}(x, \lambda),& x\in(\xi,1]. \end{cases} $$
Set \(\varphi_{1}(x, \lambda) \) is the solution of equation (1.1) on the interval \([0, \xi) \), which satisfies the initial conditions
$$\begin{aligned}& \varphi_{1}(0,\lambda)=1, \end{aligned}$$
(3.1)
$$\begin{aligned}& \varphi'_{1}(0,\lambda)=f(\lambda). \end{aligned}$$
(3.2)
In accordance with Lemma 3.1, we can define the solution \(\varphi_{2}(x, \lambda) \) of equation (1.1) on \((\xi, 1] \) by the initial conditions
$$ \left ( \textstyle\begin{array}{@{}c@{}} \varphi_{2}(\xi+0) \\ \varphi'_{2}(\xi+0) \end{array}\displaystyle \right )=\left ( \textstyle\begin{array}{@{}c@{}} \delta_{1}\varphi_{1}(\xi-0) \\ \delta_{2}\varphi'_{1}(\xi-0) \end{array}\displaystyle \right ); $$
(3.3)
similarly, we define the solutions \(\chi_{2}(x, \lambda) \) and \(\chi _{1}(x, \lambda) \) of equation (1.1) by the initial conditions, respectively,
$$\begin{aligned}& \chi_{2}(1,\lambda)=1, \end{aligned}$$
(3.4)
$$\begin{aligned}& \chi'_{2}(1,\lambda)=g(\lambda), \end{aligned}$$
(3.5)
$$\begin{aligned}& \left ( \textstyle\begin{array}{@{}c@{}} \chi_{1}(\xi-0,\lambda) \\ \chi'_{1}(\xi-0,\lambda) \end{array}\displaystyle \right )=\left ( \textstyle\begin{array}{@{}c@{}} \frac{\chi_{2}(\xi+0,\lambda)}{\delta_{1}} \\ \frac{\chi'_{2}(\xi+0,\lambda)}{\delta_{2}} \end{array}\displaystyle \right ). \end{aligned}$$
(3.6)
Now we consider Wronskians
$$ W_{i}(\lambda):=W(\varphi_{i}, \chi_{i}; x) =\varphi_{i}(x, \lambda)\chi'_{i}(x, \lambda)-\varphi'_{i}(x, \lambda)\chi _{i}(x, \lambda)\quad (i=1, 2). $$
By the dependence of solutions of initial value problems on the parameter, we have \(W_{i}(\lambda)\) (\(i=1, 2\)) are independent of x.
Lemma 3.2
For every
\(\lambda\in\mathbb{C} \), \(W_{2}(\lambda)= \delta_{1}\delta_{2} W_{1}(\lambda)\).
Proof
By the definition of \(W_{i}(\lambda) \), we get
$$\begin{aligned}& W_{1}(\lambda)=\varphi_{1}(\xi-0,\lambda) \chi_{1}'(\xi-0,\lambda )-\varphi_{1}'( \xi-0,\lambda)\chi_{1}(\xi-0,\lambda), \\& W_{2}(\lambda)=\varphi_{2}(\xi+0,\lambda) \chi_{2}'(\xi+0,\lambda )-\varphi_{2}'( \xi+0,\lambda)\chi_{2}(\xi+0,\lambda). \end{aligned}$$
Using the transmission conditions (1.4)–(1.5), short calculation gives that
$$ W(\varphi_{2},\chi_{2};\xi+0)=\delta_{1} \delta_{2} W(\varphi_{1},\chi_{1};\xi-0). $$
Thus, for every \(\lambda\in\mathbb{C} \), we have \(W_{2}(\lambda)= \delta_{1}\delta_{2} W_{1}(\lambda) \), this completes the proof. □
Besides, we set \(W(\lambda):=W_{1}(\lambda)=\frac{1}{\delta_{1}\delta _{2}}W_{2}(\lambda) \).
Theorem 3.1
The eigenvalues of Sturm–Liouville problem (1.1)–(1.5) coincide with the roots of
\(W(\lambda)=0 \).
Proof
Let \(\nu_{0}(x,\lambda_{0}) \) be any eigenfunction corresponding to eigenvalue \(\lambda_{0} \), then the function \(\nu_{0}(x,\lambda_{0}) \) can be written as the form
$$ \nu_{0}(x,\lambda_{0})= \textstyle\begin{cases} m_{1}\varphi_{1}(x, \lambda_{0})+m_{2}\chi_{1}(x, \lambda_{0}),& x\in [0,\xi), \\ m_{3}\varphi_{2}(x, \lambda_{0})+m_{4}\chi_{2}(x, \lambda_{0}),& x\in (\xi,1], \end{cases} $$
where at least one of the constants \(m_{i}\) (\(i=1, 2, 3, 4\)) is nonzero. We should show that \(W(\lambda_{0})=0 \). Suppose to the contrary that there exists \(\lambda_{0}\in\mathbb{R} \) such that \(W(\lambda_{0})=W_{1}(\lambda_{0})=\frac{1}{\delta_{1}\delta _{2}}W_{2}(\lambda_{0})\neq0 \). Since the eigenfunction \(\nu_{0}(x, \lambda_{0}) \) satisfies both boundary and transmission conditions (1.2)–(1.5), we have \(L_{i}\nu_{0}(x, \lambda_{0})=0\) (\(i=1, 2, 3, 4\)). However, the determinant of coefficient matrix is nonzero, so we get \(m_{i}=0\) (\(i=1,2,3,4\)), which is a contradiction. Then \(W(\lambda_{0})=0 \). Conversely, set \(W(\lambda_{0})=0 \), then \(W(\lambda_{0})=W_{1}(\lambda_{0})=\frac{1}{\delta_{1}\delta _{2}}W_{2}(\lambda_{0})=0 \), therefore, \(\chi_{i}(x, \lambda _{0})=k\varphi_{i}(x, \lambda_{0})\) (\(i=1, 2\)) for some \(k\neq0 \). Since both \(\varphi_{2}(x, \lambda_{0}) \) and \(\chi_{2}(x, \lambda _{0}) \) satisfy the boundary condition (1.3), thus
$$ \varphi(x, \lambda_{0})= \textstyle\begin{cases} \varphi_{1}(x, \lambda_{0}),& x\in[0, \xi),\\ \varphi_{2}(x, \lambda_{0}),& x\in(\xi, 1] \end{cases} $$
satisfies problem (1.1)–(1.5). So the function \(\varphi (x,\lambda_{0}) \) is an eigenfunction of problem (1.1)–(1.5) corresponding to eigenvalue \(\lambda_{0} \). Our claim is established. □
Remark 3.1
We define \(y(x,\lambda) \) to be a non-trivial solution of (1.1), (1.4)–(1.5), write \(F(\lambda)=y'(0,\lambda)-f(\lambda )y(0,\lambda) \), \(G(\lambda)=y'(1,\lambda)-g(\lambda)y(1,\lambda) \), \(W(\lambda)=F(\lambda)G(\lambda) \). If \(W(\lambda)=0 \), then λ is an eigenvalue of (1.1)–(1.5). If, in addition, \(W_{\lambda}(\lambda)\neq0 \), then we call λ a simple eigenvalue, where the suffix denotes differentiation with respect to λ. To discuss the poles of \(f(\lambda) \), we use \(\Omega (\lambda)=(y(0,\lambda)-\frac{y'(0,\lambda)}{f(\lambda)})(y(1,\lambda )-\frac{y'(1,\lambda)}{g(\lambda)}) \) instead of \(W(\lambda) \). Particularly, if \(y(0,\lambda)=0 \), then \(\lambda=c_{i} \) is an eigenvalue. In addition, if \(y_{\lambda}(0,\lambda)\neq0 \), then \(\lambda=c_{i} \) is a simple eigenvalue. If \(y(1,\lambda)=0 \), then \(\lambda=e_{j} \) is an eigenvalue. In addition, if \(y_{\lambda}(1,\lambda )\neq0 \), then \(\lambda=e_{j} \) is a simple eigenvalue.
Theorem 3.2
All eigenvalues of Sturm–Liouville problem (1.1)–(1.5) are analytically single.
Proof
Set \(\lambda=s+it \), we use the following notations: \(\varphi= \varphi (x, \lambda) \), \(\varphi_{1\lambda}=\frac{\partial\varphi_{1}}{\partial \lambda} \), \(\varphi'_{1\lambda}=\frac{\partial\varphi_{1}^{'}}{\partial \lambda} \). Differentiating the equation \(A\chi=\lambda\chi\) with respect to λ, we obtain
$$ A \chi_{\lambda}=\chi+\lambda\chi_{\lambda}. $$
(3.7)
Then
$$ \langle\lambda\chi_{\lambda}, \varphi\rangle_{H_{1}}-\langle \chi_{\lambda}, \lambda\varphi\rangle_{H_{1}}=\langle\chi, \varphi \rangle_{H_{1}}. $$
Using integration by parts, we have
$$ \langle A\chi_{\lambda},\varphi\rangle_{1}-\langle \chi_{\lambda}, A\varphi \rangle_{1}=\delta_{1} \delta_{2}\bigl(\chi_{1\lambda}\overline{\varphi_{1}}'- \chi '_{1\lambda}\overline{\varphi_{1}} \bigr)|^{\xi}_{0}+\bigl(\chi_{2\lambda}\overline { \varphi_{2}}'-\chi'_{2\lambda}\overline{ \varphi_{2}}\bigr)|^{1}_{\xi}. $$
(3.8)
Further, by the initial conditions, we have
$$\begin{aligned} &\delta_{1}\delta_{2}\bigl(\chi_{1\lambda}\overline{ \varphi_{1}}'-\chi'_{1\lambda }\overline{ \varphi_{1}}\bigr)|^{\xi}_{0}+\bigl( \chi_{2\lambda}\overline{\varphi _{2}}'- \chi'_{2\lambda}\overline{\varphi_{2}} \bigr)|^{1}_{\xi} \\ &\quad =\delta_{1}\delta_{2}\chi'_{1\lambda}(0)- \overline{\varphi_{2}(1)}\Biggl[c+\sum^{N}_{j=1} \frac{d_{j}}{(e_{j}-\lambda)^{2}}\Biggr]-\delta_{1}\delta_{2} \chi_{1\lambda }(0) \Biggl(-a\lambda+b+\sum^{M}_{i=1} \frac{b_{i}}{\lambda-c_{i}}\Biggr). \end{aligned}$$
By virtue of the definition of \(W(\lambda) \), (3.1), and (3.2), we observe that
$$\begin{aligned} W(\lambda)&=\varphi_{1}(0,\lambda)\chi'_{1}(0, \lambda)-\varphi '_{1}(0,\lambda)\chi_{1}(0, \lambda) \\ &=\chi'_{1}(0,\lambda)-\Biggl(-a\lambda+b+\sum ^{M}_{i=1}\frac{b_{i}}{\lambda -c_{i}}\Biggr) \chi_{1}(0,\lambda). \end{aligned}$$
Differentiating it we get
$$ W'(\lambda)=\chi'_{1\lambda}(0,\lambda)- \chi_{1\lambda}(0,\lambda ) \Biggl(-a\lambda+b+\sum ^{M}_{i=1}\frac{b_{i}}{\lambda-c_{i}}\Biggr)+ \chi_{1}(0,\lambda )\Biggl[a+\sum^{M}_{i=1} \frac{b_{i}}{(\lambda-c_{i})^{2}}\Biggr]. $$
Next, let \(\lambda_{0} \) be an arbitrary root of \(W(\lambda)=0 \). Then we get \(\varphi_{i}(x, \lambda_{0})=k\chi_{i}(x, \lambda_{0})\) (\(i=1,2\)) (\(k\neq 0\)), \(k\in\mathbb{R} \). Noting that \(\lambda_{0} \in\mathbb{R} \), by a short calculation, (3.8) becomes
$$\begin{aligned} W'(\lambda_{0}) =&\overline{k} \int^{\xi}_{0} \bigl\vert \chi_{1}(x) \bigr\vert ^{2}\,dx+\frac {\overline{k}}{\delta_{1}\delta_{2}}\Biggl[ \int^{1}_{\xi} \bigl\vert \chi _{2}(x) \bigr\vert ^{2}\,dx+c+\sum^{N}_{j=1} \frac{d_{j}}{(e_{j}-\lambda_{0})^{2}}\Biggr] \\ &{}+\frac {1}{k}\Biggl[a+\sum^{M}_{i=1} \frac{b_{i}}{(\lambda_{0}-c_{i})^{2}}\Biggr]. \end{aligned}$$
Since \(c>0 \), \(a>0 \), \(b_{i}>0\) (\(i=1, 2, \ldots, M \)), \(d_{j}>0\) (\(j=1, 2, \ldots, N \)), \(\delta_{1}\delta_{2}>0 \), \(k\neq0 \), so we know \(W'(\lambda_{0})\neq0 \). Hence the eigenvalues of Sturm–Liouville problem (1.1)–(1.5) are analytically single. □
Lemma 3.3
Let
\(\lambda=s^{2}\), \(s=\sigma+it \). Then the following equalities hold for
\(k=0 \)
and
\(k=1 \):
$$\begin{aligned}& \begin{aligned}[b] \frac{d^{k}}{dx^{k}}\varphi_{1}(x, \lambda)&=\frac{d^{k}}{dx^{k}}\cos (sx)-\Biggl(-as+\frac{b}{s}+\sum ^{M}_{i=1}\frac{b_{i}}{s^{3}-c_{i}s}\Biggr)\frac {d^{k}}{dx^{k}} \sin(sx) \\ &\quad {}+\frac{1}{s} \int^{x}_{0}\frac{d^{k}}{dx^{k}}\sin\bigl[s(x-\tau) \bigr]q(\tau )\varphi_{1}(\tau)\,d\tau, \end{aligned} \end{aligned}$$
(3.9)
$$\begin{aligned}& \begin{aligned}[b] \frac{d^{k}}{dx^{k}}\varphi_{2}(x, \lambda)&=\delta_{1}\varphi_{1}(\xi -0)\frac{d^{k}}{dx^{k}} \cos\bigl[s(x-\xi)\bigr]+\frac{1}{s}\delta_{2} \varphi'_{1}(\xi -0)\frac{d^{k}}{dx^{k}}\sin\bigl[s(x-\xi) \bigr] \\ &\quad {}+\frac{1}{s} \int^{x}_{\xi}\frac{d^{k}}{dx^{k}}\sin\bigl[s(x-\tau) \bigr]q(\tau )\varphi_{2}(\tau)\,d\tau. \end{aligned} \end{aligned}$$
(3.10)
Proof
For the case of \(k=0 \), since \(-\varphi''_{1}+q\varphi_{1}=s^{2}\varphi_{1} \), we have \(q\varphi_{1}=\varphi''_{1}+s^{2}\varphi_{1} \), and
$$ \int^{x}_{0}\sin\bigl[s(x-\tau)\bigr]q(\tau) \varphi_{1}(\tau)\,d\tau=s^{2} \int ^{x}_{0}\varphi_{1}(\tau)\sin \bigl[s(x-\tau)\bigr]\,d\tau+ \int^{x}_{0}\sin\bigl[s(x-\tau )\bigr] \varphi''_{1}(\tau)\,d\tau. $$
Using integration by parts and noting the initial conditions
$$ \varphi_{1}(0)=1, \qquad \varphi'_{1}(0)=-as^{2}+b+ \sum^{M}_{i=1}\frac{b_{i}}{s^{2}-c_{i}}, $$
we obtain
$$ s\varphi_{1}(x,\lambda)=\Biggl(-as^{2}+b+\sum ^{M}_{i=1}\frac {b_{i}}{s^{2}-c_{i}}\Biggr)\sin(sx)+s\cos(sx)+ \int^{x}_{0}\sin\bigl[s(x-\tau)\bigr]q(\tau ) \varphi_{1}(\tau)\,d\tau. $$
Thus,
$$\begin{aligned} \varphi_{1}(x, \lambda) =&\Biggl(-as+\frac{b}{s}+\sum ^{M}_{i=1}\frac {b_{i}}{s^{3}-c_{i}s}\Biggr)\sin(sx)+\cos(sx) \\ &{}+\frac{1}{s} \int^{x}_{0}\sin\bigl[s(x-\tau )\bigr]q(\tau) \varphi_{1}(\tau)\,d\tau. \end{aligned}$$
(3.11)
Then (3.9) can be got by differentiating (3.11) with respect to x. The proof for (3.10) is similar. □
Similarly, we have the following lemma.
Lemma 3.4
Let
\(\lambda=s^{2}\), \(s=\sigma+it \). Then the following equalities hold for
\(k=0 \)
and
\(k=1 \):
$$\begin{aligned}& \begin{aligned}[b] \frac{d^{k}}{dx^{k}}\chi_{1}(x, \lambda)&=\frac{\chi_{2}(\xi+0)}{\delta _{1}}\frac{d^{k}}{dx^{k}}\cos\bigl[s(x-\xi)\bigr]+ \frac{\chi'_{2}(\xi+0)}{s\delta _{2}}\frac{d^{k}}{dx^{k}}\sin\bigl[s(x-\xi)\bigr] \\ &\quad {}-\frac{1}{s} \int^{\xi}_{x}\frac{d^{k}}{dx^{k}}\sin\bigl[s(x-\tau) \bigr]q(\tau )\chi_{1}(\tau)\,d\tau, \end{aligned} \end{aligned}$$
(3.12)
$$\begin{aligned}& \begin{aligned}[b] \frac{d^{k}}{dx^{k}}\chi_{2}(x, \lambda)&=\frac{d^{k}}{dx^{k}}\cos \bigl[s(x-1)\bigr]+\Biggl(cs+\frac{d}{s}- \sum^{N}_{j=1}\frac{d_{j}}{e_{j}s-s^{3}}\Biggr) \frac {d^{k}}{dx^{k}}\sin\bigl[s(x-1)\bigr] \\ &\quad {}-\frac{1}{s} \int^{1}_{x}\frac{d^{k}}{dx^{k}}\sin\bigl[s(x-\tau) \bigr]q(\tau)\chi _{2}(\tau)\,d\tau. \end{aligned} \end{aligned}$$
(3.13)
Lemma 3.5
Let
\(\lambda=s^{2}\), \(s=\sigma+it \). Then
\(\varphi(x,\lambda) \)
have the following asymptotic representations for
\(k=0, 1\):
$$\begin{aligned}& \frac{d^{k}}{dx^{k}}\varphi_{1}(x, \lambda)=-sa\frac{d^{k}}{dx^{k}}\sin (sx)+O\bigl( \vert s \vert ^{k}e^{ \vert t \vert x}\bigr), \\& \frac{d^{k}}{dx^{k}}\varphi_{2}(x, \lambda)=-sa\sin(s \xi)\frac {d^{k}}{dx^{k}}\cos\bigl[s(x-\xi)\bigr] \\& \hphantom{\frac{d^{k}}{dx^{k}}\varphi_{2}(x, \lambda)={}}{}-sa\delta_{2}\cos(s\xi) \frac {d^{k}}{dx^{k}}\sin\bigl[s(x-\xi)\bigr] +O\bigl( \vert s \vert ^{k}e^{ \vert t \vert x}\bigr). \end{aligned}$$
(3.14)
Each of these estimations holds uniformly for
x
as
\(|\lambda |\rightarrow\infty\).
Proof
The proof of asymptotic equalities for \(\varphi_{1}(x, \lambda) \) are similar to those of Titchmarsh’s proof for \(\varphi(x, \lambda) \) (see [25]), so we only prove (3.14), the other asymptotic equalities are similar.
For the case of \(k=0 \), by the estimations of \(\varphi_{1}(x, \lambda) \) and \(\varphi'_{1}(x, \lambda) \), we get
$$\begin{aligned}& \varphi_{1}(\xi-0, \lambda)=-sa\sin(s\xi)+O\bigl( \vert s \vert e^{ \vert t \vert \xi}\bigr), \\& \varphi'_{1}(\xi-0, \lambda)=-s^{2}a\cos(s \xi)+O\bigl( \vert s \vert ^{2}e^{ \vert t \vert \xi}\bigr). \end{aligned}$$
Substituting them into (3.10) and noting (3.3), we have
$$ \varphi_{2}(x, \lambda)=-sa\sin(s\xi)\cos\bigl[s(x-\xi)\bigr]-sa \delta_{2}\cos(s\xi )\sin\bigl[s(x-\xi)\bigr]+O\bigl(|s|e^{|t|x} \bigr). $$
Then, differentiating it with respect to x, we obtain (3.14). □
Lemma 3.6
Let
\(\lambda=s^{2}\), \(s=\sigma+it \). Then
\(\chi(x,\lambda) \)
have the following asymptotic representations for
\(k=0, 1\):
$$\begin{aligned}& \frac{d^{k}}{dx^{k}}\chi_{1}(x, \lambda)=-\frac{sc}{\delta_{1}}\sin \bigl[s(\xi -1)\bigr]\frac{d^{k}}{dx^{k}}\cos\bigl[s(x-\xi)\bigr] \\& \hphantom{\frac{d^{k}}{dx^{k}}\chi_{1}(x, \lambda)={}}{}+\frac{sc}{\delta_{2}} \cos\bigl[s(\xi -1)\bigr]\frac{d^{k}}{dx^{k}}\sin\bigl[s(x-\xi)\bigr]+O\bigl( \vert s \vert ^{k}e^{ \vert t \vert (x-1)}\bigr), \\& \frac{d^{k}}{dx^{k}}\chi_{2}(x, \lambda)=sc\frac{d^{k}}{dx^{k}}\sin \bigl[s(x-1)\bigr]+O\bigl( \vert s \vert ^{k}e^{ \vert t \vert (x-1)} \bigr). \end{aligned}$$
Each of these estimations holds uniformly for
x
as
\(|\lambda |\rightarrow\infty\).
Theorem 3.3
Let
\(\lambda=s^{2}\), \(s=\sigma+it \). The function
\(W_{1}(\lambda) \)
has the following estimations:
$$ W_{1}(\lambda)=\frac{acs^{3}\cos(s\xi)\sin[s(\xi-1)]}{\delta_{1}}-\frac {acs^{3}\sin(s\xi)\cos[s(\xi-1)]}{\delta_{2}}+O\bigl( \vert s \vert ^{2}e^{ \vert t \vert }\bigr). $$
Proof
By the definition of \(W_{1}(\lambda) \), we have
$$\begin{aligned} W_{1}(\lambda)&=\varphi_{1}(0,\lambda) \chi'_{1}(0,\lambda)-\varphi '_{1}(0, \lambda)\chi_{1}(0,\lambda) \\ &=\chi'_{1}(0,\lambda)-\Biggl(-sa+\frac{b}{s}+ \sum^{M}_{i=1}\frac {b_{i}}{s^{3}-c_{i}s}\Biggr)s \chi_{1}(0,\lambda). \end{aligned}$$
According to the estimations of \(\chi_{1}(0,\lambda) \) and \(\chi '_{1}(0,\lambda) \) in Lemma 3.6, we can obtain the asymptotic representations of \(W_{1}(\lambda) \). □
Corollary 3.1
The eigenvalues of Sturm–Liouville problem (1.1)–(1.5) are bounded below.
Proof
Setting \(s=it\) (\(t>0\)) in Theorem 3.3, we get \(W_{1}(\lambda )=W_{1}(-t^{2})\rightarrow\infty\) (\(t\rightarrow\infty\)). Then \(W_{1}(-t^{2})\neq0 \) for λ negative and sufficiently large. Our claim is established. □
Further, according to the asymptotic representations for \(W_{1}(\lambda) \), we have the following theorem. For the convenience, in the sequel, we assume \(\delta_{1}=\delta_{2}=\delta\).
Theorem 3.4
The eigenvalues
\(\lambda_{n}=s^{2}_{n}\) (\(n=0, 1, 2, \ldots\)) of discontinuous Sturm–Liouville problem (1.1)–(1.5) have the following estimations as
\(n\rightarrow\infty\):
$$ \sqrt{\lambda_{n}}=(n-1)\pi+O\biggl(\frac{1}{n}\biggr). $$
Proof
Using the well-known Rouché theorem in a closed curve removing \(c_{i} \) (\(i=1,2,\ldots,M\)) and \(e_{j} \) (\(j=1,2,\ldots,N\)), we can obtain this result (see [23], Theorem 2.3). □
Combining Theorem 3.4, Lemma 3.5 with Lemma 3.6, we can obtain the following asymptotic representations of the eigenfunctions \(\varphi(x, \lambda_{n}) \) and \(\chi(x,\lambda_{n}) \):
Theorem 3.5
The eigenfunctions
\(\varphi(x,\lambda_{n}) \)
and
\(\chi(x,\lambda_{n}) \) (\(n=0, 1, 2, \ldots\)) of Sturm–Liouville problem (1.1)–(1.5) have the following asymptotic representations as
\(n\rightarrow \infty\):
$$\begin{aligned}& \varphi(x,\lambda_{n})= \textstyle\begin{cases} -a(n-1)\pi\sin[(n-1)\pi x]+O(1),& x\in[0,\xi),\\ -a\delta(n-1)\pi\sin[(n-1)\pi x]+O(1),& x\in(\xi,1]. \end{cases}\displaystyle \\& \chi(x,\lambda_{n})= \textstyle\begin{cases} \pi c\frac{(n-1)}{\delta}\sin[(n-1)\pi(x-1)]+O(1), &x\in[0,\xi),\\ c(n-1)\pi\sin[(n-1)\pi(x-1)]+O(1),& x\in(\xi,1]. \end{cases}\displaystyle \end{aligned}$$