Now we will derive asymptotic formulae of the eigenvalues and eigenfunctions in a way similar to the techniques of [9, 20] and [13, 15].
Lemma 2
Let
\(\phi_{\lambda} ( x )\)
be the solution of equation (1) defined in Section
3, and let
\(\lambda = s^{2}\). Then the following integral equations hold for
\(k = 0 \)
and
\(k = 1\):
$$\begin{aligned}& \phi_{ - \varepsilon,\lambda}^{ ( k )} ( x ) = \beta_{2} \bigl( \cos \bigl( s ( x - a ) \bigr) \bigr)^{ ( k )} - \frac{\beta_{1}}{s} \bigl( \sin \bigl( s ( x - a ) \bigr) \bigr)^{ ( k )} \\& \hphantom{\phi_{ - \varepsilon,\lambda}^{ ( k )} ( x ) =}{}+ \frac{1}{s} \int_{a}^{x} \sin \bigl( s ( x - t ) \bigr)^{ ( k )} q ( t )\phi_{ - \varepsilon,\lambda} ( t )\,dt, \end{aligned}$$
(29)
$$\begin{aligned}& \phi_{\varepsilon,\lambda}^{ ( k )} ( x ) = \bigl( \mu_{1} \phi_{ - \varepsilon,\lambda} ( \theta_{ - \varepsilon} - ) + \mu_{2} \phi_{ - \varepsilon,\lambda} ' ( \theta_{ - \varepsilon} - ) \bigr) \bigl( \cos \bigl( s ( x - \theta_{ - \varepsilon} ) \bigr) \bigr)^{ ( k )} \\& \hphantom{\phi_{\varepsilon,\lambda}^{ ( k )} ( x ) =}{}+ \frac{1}{s} \bigl( \mu_{1}'\phi_{ - \varepsilon,\lambda} ( \theta_{ - \varepsilon} - ) + \mu_{2}'\phi_{ - \varepsilon,\lambda} ' ( \theta_{ - \varepsilon} - ) \bigr) \bigl( \sin \bigl( s ( x - \theta_{ - \varepsilon} ) \bigr) \bigr)^{ ( k )} \\& \hphantom{\phi_{\varepsilon,\lambda}^{ ( k )} ( x ) =}{}+ \frac{1}{s} \int_{\theta_{ - \varepsilon}}^{x} \bigl( \sin \bigl( s ( x - t ) \bigr) \bigr)^{ ( k )}q ( t )\phi_{\varepsilon,\lambda} ( t )\,dt, \end{aligned}$$
(30)
$$\begin{aligned}& \phi_{ + \varepsilon,\lambda}^{ ( k )} ( x ) = \bigl( \eta_{1} \phi_{\varepsilon,\lambda} ( \theta_{ + \varepsilon} - ) + \eta_{2} \phi_{\varepsilon,\lambda} ' ( \theta_{ + \varepsilon} - ) \bigr) \bigl( \cos \bigl( s ( x - \theta_{ + \varepsilon} ) \bigr) \bigr)^{ ( k )} \\& \hphantom{\phi_{ + \varepsilon,\lambda}^{ ( k )} ( x ) =}{}+ \frac{1}{s} \bigl( \eta_{1}'\phi_{\varepsilon,\lambda} ( \theta_{ + \varepsilon} - ) + \eta_{2}'\phi_{\varepsilon,\lambda} ' ( \theta_{ + \varepsilon} - ) \bigr) \bigl( \sin \bigl( s ( x - \theta_{ + \varepsilon} ) \bigr) \bigr)^{ ( k )} \\& \hphantom{\phi_{ + \varepsilon,\lambda}^{ ( k )} ( x ) =}{}+ \frac{1}{s} \int_{\theta_{ + \varepsilon}}^{x} \bigl( \sin \bigl( s ( x - t ) \bigr) \bigr)^{ ( k )}q ( t )\phi_{ + \varepsilon,\lambda} ( t )\,dt. \end{aligned}$$
(31)
Proof
For proving it is enough to substitute \(s^{2}\phi_{ - \varepsilon,\lambda} ( t ) + \phi_{ - \varepsilon,\lambda}'' ( t )\), \(s^{2}\phi_{\varepsilon,\lambda} ( t ) + \phi_{\varepsilon,\lambda}'' ( t )\) and \(s^{2}\phi_{ + \varepsilon,\lambda} ( t ) + \phi_{ + \varepsilon,\lambda}'' ( t )\) instead of \(q ( t )\phi_{ - \varepsilon,\lambda} ( t )\), \(q ( t )\phi_{\varepsilon,\lambda} ( t )\) and \(q ( t )\phi_{ + \varepsilon,\lambda} ( t )\) in the integral terms of (29), (30) and (31), respectively, and integrate by parts twice. □
Lemma 3
Let
\(\lambda = s^{2}\). \(\operatorname{Im} s = \ell\). Then the function
\(\phi_{\lambda} ( x )\)
has the following asymptotic representations for
\(\vert \lambda \vert \to \infty\), which hold uniformly for
\(x \in I_{ i } \) (\(i = 1,2,3 \)):
$$\begin{aligned}& \phi_{ - \varepsilon,\lambda}^{ ( k )} ( x ) = \beta_{2} \bigl( \cos \bigl( s ( x - a ) \bigr) \bigr)^{ ( k )} + O \bigl( \vert s \vert ^{k - 1}e^{\vert \ell \vert ( x - a )} \bigr), \end{aligned}$$
(32)
$$\begin{aligned}& \phi_{\varepsilon,\lambda}^{ ( k )} ( x ) = - s\mu_{2}\beta_{2}\sin \bigl( s ( \theta_{ - \varepsilon} - a ) \bigr) \bigl( \cos \bigl( s ( x - \theta_{ - \varepsilon} ) \bigr) \bigr)^{ ( k )} + O \bigl( \vert s \vert ^{k}e^{\vert \ell \vert ( x - a )} \bigr), \end{aligned}$$
(33)
$$\begin{aligned}& \phi_{ + \varepsilon,\lambda}^{ ( k )} ( x ) = s^{2}\mu_{2} \eta_{2}\beta_{2}\sin \bigl( s ( \theta_{ - \varepsilon} - a ) \bigr)\sin \bigl( s ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} ) \bigr) \bigl( \cos \bigl( s ( x - \theta_{ + \varepsilon} ) \bigr) \bigr)^{ ( k )} \\& \hphantom{\phi_{ + \varepsilon,\lambda}^{ ( k )} ( x ) =}{}+ O \bigl( \vert s \vert ^{k + 1}e^{\vert \ell \vert ( x - a )} \bigr), \end{aligned}$$
(34)
for
\(\beta_{2} \ne 0\),
$$\begin{aligned}& \phi_{ - \varepsilon,\lambda}^{ ( k )} ( x ) = - \frac{\beta_{1}}{s} \bigl( \sin \bigl( s ( x - a ) \bigr) \bigr)^{ ( k )} + O \bigl( \vert s \vert ^{k - 2}e^{\vert \ell \vert ( x - a )} \bigr), \end{aligned}$$
(35)
$$\begin{aligned}& \phi_{\varepsilon,\lambda}^{ ( k )} ( x ) = - \mu_{2} \beta_{1}\cos \bigl( s ( \theta_{ - \varepsilon} - a ) \bigr) \bigl( \cos \bigl( s ( x - \theta_{ - \varepsilon} ) \bigr) \bigr)^{ ( k )} + O \bigl( \vert s \vert ^{k - 1}e^{\vert \ell \vert ( x - a )} \bigr), \end{aligned}$$
(36)
$$\begin{aligned}& \phi_{ + \varepsilon,\lambda}^{ ( k )} ( x ) = s\mu_{2}\eta_{2} \beta_{1}\cos \bigl( s ( \theta_{ - \varepsilon} - a ) \bigr)\sin \bigl( s ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} ) \bigr) \bigl( \cos \bigl( s ( x - \theta_{ + \varepsilon} ) \bigr) \bigr)^{ ( k )} \\& \hphantom{\phi_{ + \varepsilon,\lambda}^{ ( k )} ( x ) =}{}+ O \bigl( \vert s \vert ^{k}e^{\vert \ell \vert ( x - a )} \bigr), \end{aligned}$$
(37)
for
\(\beta_{2} = 0\).
Proof
These formulae can be proven similar to Titchmarsh’s proof [20] and also the techniques in [13, 15]. □
Lemma 4
Let
\(\lambda = s^{2}\). \(\operatorname{Im} s = \ell\). Then the characteristic function
\(\omega ( \lambda )\)
has the following asymptotic representations:
Case 1. If
\(\beta_{2} \ne 0\), \(\alpha_{2}' \ne 0\), then
$$\begin{aligned} \omega ( \lambda ) =& \frac{1}{D_{1}D_{2}}s^{5}\beta_{2} \alpha_{2}'\mu_{2}\eta_{2}\sin \bigl( s ( \theta_{ - \varepsilon} - a ) \bigr)\sin \bigl( s ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} ) \bigr)\sin \bigl( s ( b - \theta_{ + \varepsilon} ) \bigr) \\ &{}+ O \bigl( \vert s \vert ^{4}e^{\vert \ell \vert ( b - a )} \bigr). \end{aligned}$$
(38)
Case 2. If
\(\beta_{2} \ne 0\), \(\alpha_{2}' = 0\), then
$$\begin{aligned} \omega ( \lambda ) =& \frac{1}{D_{1}D_{2}}s^{4}\beta_{2} \alpha_{1}'\mu_{2}\eta_{2}\sin \bigl( s ( \theta_{ - \varepsilon} - a ) \bigr)\sin \bigl( s ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} ) \bigr)\cos \bigl( s ( b - \theta_{ + \varepsilon} ) \bigr) \\ &{}+ O \bigl( \vert s \vert ^{3}e^{\vert \ell \vert ( b - a )} \bigr). \end{aligned}$$
(39)
Case 3. If
\(\beta_{2} = 0\), \(\alpha_{2}' \ne 0\), then
$$\begin{aligned} \omega ( \lambda ) =& \frac{1}{D_{1}D_{2}}s^{4}\beta_{1} \alpha_{2}'\mu_{2}\eta_{2}\cos \bigl( s ( \theta_{ - \varepsilon} - a ) \bigr)\sin \bigl( s ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} ) \bigr)\sin \bigl( s ( b - \theta_{ + \varepsilon} ) \bigr) \\ &{}+ O \bigl( \vert s \vert ^{3}e^{\vert \ell \vert ( b - a )} \bigr). \end{aligned}$$
(40)
Case 4. If
\(\beta_{2} = 0\), \(\alpha_{2}' = 0\), then
$$\begin{aligned} \omega ( \lambda ) = &\frac{1}{D_{1}D_{2}}s^{3}\beta_{1} \alpha_{1}'\mu_{2}\eta_{2}\cos \bigl( s ( \theta_{ - \varepsilon} - a ) \bigr)\sin \bigl( s ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} ) \bigr)\cos \bigl( s ( b - \theta_{ + \varepsilon} ) \bigr) \\ &{}+ O \bigl( \vert s \vert ^{2}e^{\vert \ell \vert ( b - a )} \bigr). \end{aligned}$$
(41)
Proof
The proof is immediate by substituting (34) and (37) into the representation
$$\omega ( \lambda ) = \frac{1}{D_{1}D_{2}} \bigl\{ \bigl( \lambda \alpha_{1}' + \alpha_{1} \bigr) \phi_{ + \varepsilon,\lambda} ( b ) - \bigl( \lambda \alpha_{2}' + \alpha_{2} \bigr)\phi_{ + \varepsilon,\lambda} ' ( b ) \bigr\} . $$
□
Corollary 3
The eigenvalues of the problem (1)-(7) is bounded from below.
We are now ready to find the asymptotic approximation formulae for the eigenvalues of the problem (1)-(7). Since the eigenvalues coincide with the zeros of the entire functions \(\omega ( \lambda )\), it follows that they have no finite accumulation point. Moreover, all eigenvalues are real and bounded below by Corollaries 1 and 3. Therefore, we may renumber them as \(\lambda_{0} \le \lambda_{1} \le \lambda_{2} \le\cdots\), which are counted according to their multiplicity. Below we shall denote \(s_{n}^{2} = \lambda_{n}\).
Theorem 2
The problem (1)-(7) has a precisely denumerable number of real eigenvalues, whose behaviour may be expressed by the three sequences
\(\{ \lambda_{n}' \}\), \(\{ \lambda_{n}'' \} \)
and
\(\{ \lambda_{n} ''' \}\)
with the following asymptotics representations for
\(n \to \infty\):
Case 1. If
\(\beta_{2} \ne 0\), \(\alpha_{2}' \ne 0\), then
$$ \begin{aligned} &s_{n}' = \frac{ ( n - 1 )\pi}{ ( \theta_{ - \varepsilon} - a )} + O \biggl( \frac{1}{n} \biggr),\quad\quad s_{n}^{\prime\prime} = \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} + O \biggl( \frac{1}{n} \biggr), \\ &s_{n} ''' = \frac{ ( n - 2 )\pi}{ ( b - \theta_{ + \varepsilon} )} + O \biggl( \frac{1}{n} \biggr). \end{aligned} $$
(42)
Case 2. If
\(\beta_{2} \ne 0\), \(\alpha_{2}' = 0\), then
$$ \begin{aligned} &s_{n}' = \frac{ ( n - 1 )\pi}{ ( \theta_{ - \varepsilon} - a )} + O \biggl( \frac{1}{n} \biggr),\quad\quad s_{n}^{\prime\prime} = \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} + O \biggl( \frac{1}{n} \biggr),\\ & s_{n} ''' = \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} + O \biggl( \frac{1}{n} \biggr). \end{aligned} $$
(43)
Case 3. If
\(\beta_{2} = 0\), \(\alpha_{2}' \ne 0\), then
$$ \begin{aligned} &s_{n}' = \frac{ ( n - 1/2 )\pi}{ ( \theta_{ - \varepsilon} - a )} + O \biggl( \frac{1}{n} \biggr), \quad\quad s_{n}^{\prime\prime} = \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} + O \biggl( \frac{1}{n} \biggr),\\ & s_{n} ''' = \frac{ ( n - 1 )\pi}{ ( b - \theta_{ + \varepsilon} )} + O \biggl( \frac{1}{n} \biggr). \end{aligned} $$
(44)
Case 4. If
\(\beta_{2} = 0\), \(\alpha_{2}' = 0\), then
$$ \begin{aligned} &s_{n}' = \frac{ ( n - 1/2 )\pi}{ ( \theta_{ - \varepsilon} - a )} + O \biggl( \frac{1}{n} \biggr), \quad\quad s_{n}^{\prime\prime} = \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} + O \biggl( \frac{1}{n} \biggr), \\ &s_{n} ''' = \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} + O \biggl( \frac{1}{n} \biggr). \end{aligned} $$
(45)
Proof
We will only consider the first case. By applying the well-known Rouche theorem on a sufficiently large contour, it follows that \(\omega ( \lambda )\) has the same number of zeros inside the contour as the leading term in (38). Hence, if \(\lambda_{0}' \le \lambda_{1}' \le \lambda_{2}' \le\cdots\) are the zeros of \(\omega ( \lambda )\) and \(s_{n}^{2'} = \lambda_{n}'\) we have
$$ s_{n}' = \frac{ ( n - 1 )\pi}{ ( \theta_{ - \varepsilon} - a )} + \delta_{n}', $$
(46)
for sufficiently large n, where \(\vert \delta_{n}' \vert \le \frac{\pi}{2 ( \theta_{ - \varepsilon} - a )}\). By using (38) we have \(\delta_{n}' = O ( \frac{1}{n} )\), which completes the proof for the first formula of Case 1. The proof for the other cases are similar. □
Then from (32)-(37) (for \(k = 0\)) and the above theorem, the asymptotic behaviour of the eigenfunctions of the problem (1)-(7) is given by:
Case 1. If \(\beta_{2} \ne 0\), \(\alpha_{2}' \ne 0\), then
$$\begin{aligned}& \phi_{\lambda_{n}'} ( x ) = \textstyle\begin{cases} \beta_{2}\cos ( \frac{ ( n - 1 )\pi}{ ( \theta_{ - \varepsilon} - a )} ( x - a ) ) + O ( \frac{1}{n} ),& x \in [ a,\theta_{ - \varepsilon} ), \\ O ( \frac{1}{n} ),& x \in ( \theta_{ - \varepsilon},\theta_{ + \varepsilon} ), \\ O ( \frac{1}{n} ),& x \in ( \theta_{ + \varepsilon},b ], \end{cases}\displaystyle \\& \phi_{\lambda_{n}''} ( x ) = \textstyle\begin{cases} \beta_{2}\cos ( \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} ( x - a ) ) + O ( \frac{1}{n} ),& x \in [ a,\theta_{ - \varepsilon} ), \\ - \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )}\mu_{2}\beta_{2}\sin ( \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} ( \theta_{ - \varepsilon} - a ) )\cos ( \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} ( x - \theta_{ - \varepsilon} ) ) \\ \quad{}+ O ( 1 ),& x \in ( \theta_{ - \varepsilon},\theta_{ + \varepsilon} ), \\ O ( \frac{1}{n} ),& x \in ( \theta_{ + \varepsilon},b ], \end{cases}\displaystyle \\& \phi_{\lambda_{n} ''' } ( x ) = \textstyle\begin{cases} \beta_{2}\cos ( \frac{ ( n - 2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( x - a ) ) + O ( \frac{1}{n} ),& x \in [ a,\theta_{ - \varepsilon} ), \\ - \frac{ ( n - 2 )\pi}{ ( b - \theta_{ + \varepsilon} )}\mu_{2}\beta_{2}\sin ( \frac{ ( n - 2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( \theta_{ - \varepsilon} - a ) )\cos ( \frac{ ( n - 2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( x - \theta_{ - \varepsilon} ) ) \\ \quad{}+ O ( 1 ),& x \in ( \theta_{ - \varepsilon},\theta_{ + \varepsilon} ), \\ ( \frac{ ( n - 2 )\pi}{ ( b - \theta_{ + \varepsilon} )} )^{2}\mu_{2}\eta_{2}\beta_{2}\sin ( \frac{ ( n - 2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( \theta_{ - \varepsilon} - a ) )\sin ( \frac{ ( n - 2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} ) ) \\ \quad{}\times \cos ( \frac{ ( n - 2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( x - \theta_{ + \varepsilon} ) ) + O ( n ),& x \in ( \theta_{ + \varepsilon},b ]. \end{cases}\displaystyle \end{aligned}$$
Case 2. If \(\beta_{2} \ne 0\), \(\alpha_{2}' = 0\), then
$$\begin{aligned}& \phi_{\lambda_{n}'} ( x ) = \textstyle\begin{cases} \beta_{2}\cos ( \frac{ ( n - 1 )\pi}{ ( \theta_{ - \varepsilon} - a )} ( x - a ) ) + O ( \frac{1}{n} ),& x \in [ a,\theta_{ - \varepsilon} ), \\ O ( \frac{1}{n} ),& x \in ( \theta_{ - \varepsilon},\theta_{ + \varepsilon} ), \\ O ( \frac{1}{n} ),& x \in ( \theta_{ + \varepsilon},b ], \end{cases}\displaystyle \\& \phi_{\lambda_{n}''} ( x ) = \textstyle\begin{cases} \beta_{2}\cos ( \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} ( x - a ) ) + O ( \frac{1}{n} ),& x \in [ a,\theta_{ - \varepsilon} ), \\ - \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )}\mu_{2}\beta_{2}\sin ( \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} ( \theta_{ - \varepsilon} - a ) )\cos ( \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} ( x - \theta_{ - \varepsilon} ) ) \\ \quad{}+ O ( 1 ),& x \in ( \theta_{ - \varepsilon},\theta_{ + \varepsilon} ), \\ O ( \frac{1}{n} ),& x \in ( \theta_{ + \varepsilon},b ], \end{cases}\displaystyle \\& \phi_{\lambda_{n} ''' } ( x ) = \textstyle\begin{cases} \beta_{2}\cos ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( x - a ) ) + O ( \frac{1}{n} ),&\hspace{-3pt} x \in [ a,\theta_{ - \varepsilon} ), \\ - \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} \mu_{2}\beta_{2}\sin ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( \theta_{ - \varepsilon} - a ) )\cos ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( x - \theta_{ - \varepsilon} ) ) \\ \quad{}+ O ( 1 ),& \hspace{-3pt}x \in ( \theta_{ - \varepsilon},\theta_{ + \varepsilon} ), \\ ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} )^{2}\mu_{2}\eta_{2}\beta_{2}\sin ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( \theta_{ - \varepsilon} - a ) )\sin ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} ) )\\ \quad{}\times \cos ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( x - \theta_{ + \varepsilon} ) ) + O ( n ), &\hspace{-3pt} x \in ( \theta_{ + \varepsilon},b ]. \end{cases}\displaystyle \end{aligned}$$
Case 3. If \(\beta_{2} = 0\), \(\alpha_{2}' \ne 0\), then
$$\begin{aligned}& \phi_{\lambda_{n}'} ( x ) = \textstyle\begin{cases} - \frac{\beta_{1} ( \theta_{ - \varepsilon} - a )}{ ( n - 1/2 )\pi} \sin ( \frac{ ( n - 1/2 )\pi}{ ( \theta_{ - \varepsilon} - a )} ( x - a ) ) + O ( \frac{1}{n^{2}} ),& x \in [ a,\theta_{ - \varepsilon} ), \\ O ( \frac{1}{n} ), &x \in ( \theta_{ - \varepsilon},\theta_{ + \varepsilon} ), \\ O ( 1 ),& x \in ( b,\theta_{ + \varepsilon} ], \end{cases}\displaystyle \\& \phi_{\lambda_{n}''} ( x ) = \textstyle\begin{cases} - \frac{\beta_{1} ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )}{ ( n - 1 )\pi} \sin ( \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} ( x - a ) ) + O ( \frac{1}{n^{2}} ),& x \in [ a,\theta_{ - \varepsilon} ), \\ - \mu_{2}\beta_{1}\cos ( \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} ( \theta_{ - \varepsilon} - a ) )\cos ( \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} ( x - \theta_{ - \varepsilon} ) ) \\ \quad{}+ O ( \frac{1}{n} ), &x \in ( \theta_{ - \varepsilon},\theta_{ + \varepsilon} ), \\ O ( 1 ),& x \in ( b,\theta_{ + \varepsilon} ], \end{cases}\displaystyle \\& \phi_{\lambda_{n} ''' } ( x ) = \textstyle\begin{cases} - \frac{\beta_{1} ( b - \theta_{ + \varepsilon} )}{ ( n - 1 )\pi} \sin ( \frac{ ( n - 1 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( x - a ) ) + O ( \frac{1}{n^{2}} ),& x \in [ a,\theta_{ - \varepsilon} ), \\ - \mu_{2}\beta_{1}\cos ( \frac{ ( n - 1 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( \theta_{ - \varepsilon} - a ) )\cos ( \frac{ ( n - 1 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( x - \theta_{ - \varepsilon} ) ) \\ \quad{}+ O ( \frac{1}{n} ),& x \in ( \theta_{ - \varepsilon},\theta_{ + \varepsilon} ), \\ \frac{ ( n - 1 )\pi}{ ( b - \theta_{ + \varepsilon} )}\mu_{2}\beta_{1}\eta_{2}\cos ( \frac{ ( n - 1 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( \theta_{ - \varepsilon} - a ) )\sin ( \frac{ ( n - 1 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} ) ) \\ \quad{}\times \cos ( \frac{ ( n - 1 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( x - \theta_{ - \varepsilon} ) ) + O ( 1 ),& x \in ( b,\theta_{ + \varepsilon} ]. \end{cases}\displaystyle \end{aligned}$$
Case 4. If \(\beta_{2} = 0\), \(\alpha_{2}' = 0\), then
$$\begin{aligned}& \phi_{\lambda_{n}'} ( x ) = \textstyle\begin{cases} - \frac{\beta_{1} ( \theta_{ - \varepsilon} - a )}{ ( n - 1/2 )\pi} \sin ( \frac{ ( n - 1/2 )\pi}{ ( \theta_{ - \varepsilon} - a )} ( x - a ) ) + O ( \frac{1}{n^{2}} ), &x \in [ a,\theta_{ - \varepsilon} ), \\ O ( \frac{1}{n} ),& x \in ( \theta_{ - \varepsilon},\theta_{ + \varepsilon} ), \\ O ( 1 ),& x \in ( b,\theta_{ + \varepsilon} ], \end{cases}\displaystyle \\& \phi_{\lambda_{n}''} ( x ) = \textstyle\begin{cases} - \frac{\beta_{1} ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )}{ ( n - 1 )\pi} \sin ( \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} ( x - a ) ) + O ( \frac{1}{n^{2}} ),& x \in [ a,\theta_{ - \varepsilon} ), \\ - \mu_{2}\beta_{1}\cos ( \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} ( \theta_{ - \varepsilon} - a ) )\cos ( \frac{ ( n - 1 )\pi}{ ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} )} ( x - \theta_{ - \varepsilon} ) ) \\ \quad{}+ O ( \frac{1}{n} ),& x \in ( \theta_{ - \varepsilon},\theta_{ + \varepsilon} ), \\ O ( 1 ),& x \in ( b,\theta_{ + \varepsilon} ], \end{cases}\displaystyle \\& \phi_{\lambda_{n} ''' } ( x ) = \textstyle\begin{cases} - \frac{\beta_{1} ( b - \theta_{ + \varepsilon} )}{ ( n - 1/2 )\pi} \sin ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( x - a ) ) + O ( \frac{1}{n^{2}} ),& x \in [ a,\theta_{ - \varepsilon} ), \\ - \mu_{2}\beta_{1}\cos ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( \theta_{ - \varepsilon} - a ) )\cos ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( x - \theta_{ - \varepsilon} ) ) \\ \quad{}+ O ( \frac{1}{n} ),& x \in ( \theta_{ - \varepsilon},\theta_{ + \varepsilon} ), \\ \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )}\mu_{2}\beta_{1}\eta_{2}\cos ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( \theta_{ - \varepsilon} - a ) )\sin ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( \theta_{ + \varepsilon} - \theta_{ - \varepsilon} ) ) \\ \quad{}\times \cos ( \frac{ ( n - 1/2 )\pi}{ ( b - \theta_{ + \varepsilon} )} ( x - \theta_{ - \varepsilon} ) ) + O ( 1 ),& x \in ( b,\theta_{ + \varepsilon} ]. \end{cases}\displaystyle \end{aligned}$$
All these asymptotic formulae hold uniformly for \(x \in I\).