Solvability and coerciveness of multi-point Sturm–Liouville problems with abstract linear functionals

Abstract

In this paper, we consider a Sturm–Liouville problem with finite discontinuous points inside an interval and with abstract linear functionals in the boundary and transmission conditions. For such a problem, the properties such as isomorphism, Fredholmness, and coerciveness with respect to the spectral parameter are investigated.

1 Introduction

We will discuss the following differential equation:

$$ M(\lambda)y:=-\bigl(py'\bigr)'(x)+ \lambda^{2}y(x)+Ty(x)=f(x),\quad x\in I $$

(1.1)

with nonclassical boundary conditions

$$\begin{aligned} M_{k}y&:=a_{01}y^{(n_{k})}(\xi_{0})+\sum _{h=1}^{n}\bigl(a_{hk}y^{(n_{k})}( \xi _{h}-)+\tilde{a}_{hk}y^{(n_{k})}(\xi_{h}+) \bigr) \\ &\quad {}+a_{(n+1)k}y^{(n_{k})}(\xi_{n+1})+\sum _{j=1}^{n_{k}}\chi _{kj}y^{(n_{k})}(x_{kj})+F_{k}y=f_{k} , \end{aligned}$$

(1.2)

where $k=1,2,\ldots,2(n+1)$, $\xi_{0}=-1$, $\xi_{h}\in(-1,1)$, $\xi _{n+1}=1$, $\xi_{0}<\xi_{1}<\cdots<\xi_{n+1}$; set $I_{1}=[\xi_{0},\xi_{1})$, $I_{t}=(\xi_{t-1},\xi_{t})$, $I_{n+1}=(\xi_{n},\xi_{n+1}]$, $I=\bigcup_{i=1}^{n+1}I_{i}$, and $J_{i}=(\xi_{i-1},\xi_{i})$, $J=\bigcup_{i=1}^{n+1}J_{i}$ ($h=1,2,\ldots,n$; $t=2,3,\ldots,n$); $p(x)$ is a piecewise constant function, $p(x)=p_{i}$ for $x\in I_{i}$ ($i=1,2,\ldots,n+1$); T is a linear operator; $a_{0k}$, $a_{hk}$, $\tilde{a}_{hk}$, $a_{(n+1)k}$, $\chi_{kj}$, $p_{i}$ ($j=1,2,\ldots,n_{k}$) are complex coefficients, and assume that $p_{i}\neq0$, $|a_{0k}|+\sum_{h=1}^{n}(|a_{hk}|+|\tilde{a}_{hk}|)+|a_{(n+1)k}|\neq0$; λ is the complex parameter; $n_{k}$ are integers; $x_{kj}\in J$ are internal points; $F_{k}$ is a linear function in the space $L_{q}[-1,1]$ ($L_{q}[-1,1]$ is a set of qth order integrable functions on $[-1,1]$).

In recent years, the classical Sturm–Liouville problem has been generalized into various types for its new importance in physical sciences and applied mathematics. For example, theoretical investigations have become focused on the discontinuous Sturm–Liouville problems for their application in physics. The discontinuity of the coefficients of the equations in the Sturm–Liouville problems corresponds to the fact that the heterogeneous media consist of two different materials. Moreover, boundary value problems with discontinuities arise in many physical problems such as heat and mass transfer, electrostatics, and diffraction problem [1, 2]. It should be noted that some works on the spectral properties and coercive solvability of boundary value problems in Sobolev spaces can be found in [3, 4]. Some boundary value problems for differential equations with discontinuous coefficients were investigated by Rasulov [5]. Note that an abstract theory of the boundary value problems with continuous coefficients and an eigenvalue parameter in the boundary conditions have been constructed by Yakubov and Yakubov (see [4] and corresponding bibliography). Many authors have been devoted to the study of discontinuous problems [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. To deal with the discontinuity of the problem, transmission conditions are imposed on the discontinuous points. There are also other terminologies such as point interaction, interface condition, etc. [18, 23]. The properties of isomorphism, Fredholmness, and coerciveness of Sturm–Liouville problems with one discontinuous point were investigated by Mukhtarov and his coauthors in [6, 8, 9, 24].

In this paper, we investigate a Sturm–Liouville problem with discontinuities at finite points and with abstract linear functionals in the boundary-transmission conditions. We obtain the properties such as isomorphism, Fredholmness, and coerciveness of this problem.

2 Boundary value problems with nonhomogeneous transmission conditions

In this section, we consider the homogeneous differential equation

$$ M_{0}(\lambda)y:=-\bigl(py' \bigr)'(x)+\lambda^{2}y(x)=0,\quad x\in I $$

(2.1)

with the nonclassical boundary conditions

$$ M_{k0}y:=a_{0k}y^{(n_{k})}( \xi_{0})+\sum_{h=1}^{n} \bigl(a_{hk}y^{(n_{k})}(\xi _{h}-)+\tilde{a}_{hk}y^{(n_{k})}( \xi_{h}+)\bigr) +a_{(n+1)k}y^{(n_{k})}(\xi_{n+1})=f_{k} , $$

(2.2)

where $k=1,2,\ldots,2(n+1)$. We shall use the notations

$$\begin{array}{rl}& {\Gamma }_{k}y:=\sum _{j=1}^{n_k}{\chi }_{kj}{y}^{(n_k)}(x_{kj}),k=1,2,\dots ,2(n+1),\\ & w_{2i-1}:=p_i^{-1/2},w_{2i}:=-p_i^{-1/2},i=1,2,\dots ,n+1,\\ & \alpha :=\underset{1\le i\le n+1}{min}\{arg{p}_i\},\beta :=\underset{1\le i\le n+1}{max}\{arg{p}_i\},\\ & \omega =\left[\begin{array}{ccccc}{a}_{01}{w}_1^{n_1}& a_{11}{w}_2^{n_1}& {\tilde{a}}_{11}{w}_3^{n_1}& \cdots & a_{(n+1)1}{w}_{2(n+1)}^{n_1}\\ a_{02}{w}_1^{n_2}& a_{12}{w}_2^{n_2}& {\tilde{a}}_{12}{w}_3^{n_2}& \cdots & a_{(n+1)2}{w}_{2(n+1)}^{n_2}\\ a_{03}{w}_1^{n_3}& a_{13}{w}_2^{n_3}& {\tilde{a}}_{13}{w}_3^{n_3}& \cdots & a_{(n+1)3}{w}_{2(n+1)}^{n_3}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{0,2(n+1)}{w}_1^{n_{2(n+1)}}& a_{1,2(n+1)}{w}_2^{n_{2(n+1)}}& {\tilde{a}}_{1,2(n+1)}{w}_3^{n_{2(n+1)}}& \cdots & a_{(n+1),2(n+1)}{w}_{2(n+1)}^{n_{2(n+1)}}\end{array}\right]\end{array}$$

and

$$\varOmega_{\varepsilon}(\alpha,\beta):= \biggl\{ \lambda\in\mathbb{C}\Big| \frac {1}{2}(\pi+\beta+\varepsilon) < \arg \lambda< \frac{1}{2}(3\pi+ \alpha-\varepsilon) \biggr\} $$

for real $\varepsilon>0$ sufficiently small.

Note that the direct sum of Sobolev spaces $W_{q}^{m}=W_{q}^{m}(J_{1})\oplus W_{q}^{m}(J_{2})\oplus\cdots\oplus W_{q}^{m}(J_{n})$ (for an integer $m\geq0$ and real $q>1$) is defined as a Banach space of complex-valued function $y=y(x)$ on I which belongs to $W_{q}^{m}(J_{i})$ ($i=1,2,\ldots,(n+1)$) in intervals $J_{i}$ respectively, with the norm

$$\| y\|_{q,m}:=\sum_{i=1}^{n+1}\| y \| _{W_{q}^{m}(J_{i})}. $$

Here, as usual, $W_{q}^{m}(a,b)$ is the Sobolev space, i.e., the Banach space consisting of all measurable functions that have generalized derivatives up to mth order in the interval $({a,b})$ inclusive with the finite norm

$$\| y\|_{W_{q}^{m}(a,b)}:=\sum_{v=0}^{m} \biggl( \int _{a}^{b} \bigl\vert y^{(v)}(x) \bigr\vert ^{q}\,dx \biggr)^{1/q}. $$

Theorem 2.1

If $\omega\neq0$, then for any $\varepsilon>0$ there exists $\mu _{\varepsilon}>0$ such that, for all $\lambda\in\varOmega_{\varepsilon}(\alpha,\beta)$ for which $|\lambda|>\mu _{\varepsilon}$, problem (2.1)–(2.2) has a unique solution $y(x,\lambda )\in W_{q}^{\sigma}$ for arbitrary $\sigma\geq\max\{2,\max_{1\leq k\leq2(n+1)}\{n_{k}\}+1\}$ and λ has the following coercive estimate:

$$ \sum_{\ell=1}^{\sigma}\|y \|_{q,\ell}\leq C(\varepsilon)\sum_{k=1}^{2(n+1)}| \lambda|^{\sigma-n_{k}-q^{-1}}|f_{k}|. $$

(2.3)

Proof

Let $y_{k}(x,\lambda)$ ($k=1,2,\ldots,2(n+1)$) be the basic solution of equation (2.1), then $y_{k}(x,\lambda)$ can be represented as

$$ y_{k}(x,\lambda)= \textstyle\begin{cases} \exp(w_{k}\lambda(x-\tilde{\xi}_{k})), & \mbox{for }x\in\tilde{I}_{k}; \\ 0, & \mbox{for }x\notin\tilde{I}_{k}, \end{cases} $$

(2.4)

where $\tilde{I}_{2i-1}=\tilde{I}_{2i}=I_{i}$ for every i, $k=2i-1, 2i$ ($i=1,2,\ldots,n+1$), $\tilde{\xi}_{1}=\xi_{0}$, $\tilde{\xi}_{2h}=\tilde{\xi}_{2h+1}=\xi_{i}$, $\tilde{\xi}_{2(n+1)}=\xi_{n+1}$ ($h=1,2,\ldots,n$). It is clear that the general solution of (2.1) can be written as

$$ y(x,\lambda)=\sum_{k=1}^{2(n+1)}C_{k}y_{k}(x, \lambda). $$

(2.5)

Substituting equation (2.5) into boundary-transmission conditions (2.2), we obtain a linear system with respect to $C_{k}$ ($k=1,2,\ldots,2(n+1)$)

$$\begin{aligned} &(w_{1}\lambda)^{n_{k}} \bigl(a_{0k}+a_{1k}e^{w_{1}\lambda(\xi_{1}-\xi_{0})} \bigr)C_{1}+(w_{2}\lambda)^{n_{k}} \bigl(a_{1k}+a_{0k}e^{w_{2}\lambda(\xi_{0}-\xi _{1})} \bigr)C_{2} \\ &\quad {}+\sum_{t=2}^{n+1} \bigl[(w_{2t-1}\lambda)^{n_{k}} \bigl(\tilde{a}_{(t-1)k} +a_{tk}e^{w_{2t-1}\lambda(\xi_{t}-\xi_{t-1})} \bigr)C_{2t-1} \\ &\quad {}+(w_{2t}\lambda)^{n_{k}} \bigl(a_{tk} + \tilde{a}_{(t-1)k}e^{w_{2t}\lambda(\xi_{t-1}-\xi_{t})} \bigr)C_{2t} \bigr]=f_{k}, \end{aligned}$$

(2.6)

where $k=1,2,\ldots,2(n+1)$. It follows from $\lambda\in\varOmega_{\varepsilon}(\alpha,\beta)$ that

$$\begin{aligned}& \frac{1}{2}(\pi+\varepsilon)< \arg(w_{k}\lambda)< \frac{1}{2}(3\pi -\varepsilon),\quad k=1,3,\ldots,2 n+1; \\& -\frac{1}{2}(\pi-\varepsilon)< \arg(w_{k}\lambda)< \frac{1}{2}(\pi -\varepsilon),\quad k=2,4,\ldots,2( n+1). \end{aligned}$$

Therefore, for this λ and $\varepsilon>0$ (sufficiently small), we obtain

$$(-1)^{k+1}\operatorname{Re}(w_{k}\lambda)< -| \lambda||w_{k}|\sin\frac{\varepsilon}{2},\quad k=1,2,\ldots,2(n+1). $$

Thus, the determinant of system (2.6) has the form

$$A(\lambda)=\lambda^{\tilde{n}}\bigl(\omega+\theta(\lambda)\bigr), $$

where $\tilde{n}=n_{1}+n_{2}+\cdots+n_{2(n+1)}$,

$$\theta (\lambda )=e^{\kappa }\left[\begin{array}{cccc}{a}_{11}{w}_1^{n_1}& a_{01}{w}_2^{n_1}& \cdots & {\tilde{a}}_{n1}{w}_{2(n+1)}^{n_1}\\ a_{12}{w}_1^{n_2}& a_{02}{w}_2^{n_2}& \cdots & {\tilde{a}}_{n2}{w}_{2(n+1)}^{n_2}\\ ⋮& ⋮& ⋮& ⋮\\ a_{1,2(n+1)}{w}_1^{n_{2(n+1)}}& a_{0,2(n+1)}{w}_2^{n_{2(n+1)}}& \cdots & {\tilde{a}}_{n,2(n+1)}{w}_{2(n+1)}^{n_{2(n+1)}}\end{array}\right],$$

$\kappa=\lambda\sum_{i=1}^{n+1}(\xi_{i}-\xi_{i-1})(w_{2i-1}-w_{2i})$ and $\theta(\lambda)\rightarrow0$ as $|\lambda|\rightarrow\infty$ in the angle $\varOmega_{\varepsilon}(\alpha,\beta)$. Since $\omega\neq0$, there exists $\mu_{\varepsilon}>0$ such that, for all $\lambda\in\varOmega_{\varepsilon}(\alpha,\beta)$ and $|\lambda|>\mu_{\varepsilon}$, we have $A(\lambda)\neq0$. So, for these λ, the unique solution for the system of linear homogeneous equations (2.6) has the following representation:

$$C_{k}(\lambda)=\frac{1}{A(\lambda)}\sum_{\eta=1}^{2(n+1)}A_{\eta k}( \lambda)f_{\eta},\quad k=1,2,\ldots,2(n+1), $$

where $A_{\eta k}(\lambda)$ is an algebraic cofactor of $(\eta,k)$th element of the determinant $A(\lambda)$. It is obvious that each of the determinants $A_{\eta k}(\lambda)$ can be represented as

$$A_{\eta k}(\lambda)=\lambda^{\tilde{n}-n_{\eta}}\bigl(\omega_{\eta k}+ \theta _{\eta k}(\lambda)\bigr), $$

where $\omega_{\eta k}\in\mathbb{C}$ and $\theta(\lambda)\rightarrow0$ as $|\lambda|\rightarrow\infty$ in the angle $\varOmega_{\varepsilon}(\alpha,\beta)$. Hence, we have

$$C_{k}(\lambda)=\sum_{\eta=1}^{2(n+1)} \lambda^{-n_{\eta}}\frac{\omega_{\eta k}+\theta_{\eta k}(\lambda)}{\omega+\theta(\lambda)}f_{\eta},\quad k=1,2, \ldots,2(n+1). $$

So, the solution of (2.1)–(2.2) has the following representation:

$$y(x,\lambda)=\sum_{k=1}^{2(n+1)}\sum _{\eta=1}^{2(n+1)}\lambda^{-n_{\eta}} \frac{\omega_{\eta k}+ \theta_{\eta k}(\lambda)}{\omega+\theta(\lambda)}f_{\eta}y_{k}(x,\lambda). $$

From the expression of $y(x,\lambda)$ we obtain that, for each integer $\tau\geq0$ and $\lambda\in\varOmega_{\varepsilon}(\alpha,\beta)$, $|\lambda|\rightarrow \infty$, the estimate

$$ \bigl\Vert y^{(\tau)} \bigr\Vert _{L_{q}(-1,1)}\leq C \sum_{\eta =1}^{2(n+1)} \Biggl(| \lambda|^{\tau-n_{\eta}}|f_{\eta}| \sum_{k=1}^{2(n+1)} \bigl\Vert y_{k}(x,\lambda) \bigr\Vert _{Lq(\tilde {I}_{k})} \Biggr) $$

(2.7)

is valid. Further, the inequalities

$$\begin{aligned}& \begin{aligned}[b] \bigl\Vert y_{2i-1}(x,\lambda) \bigr\Vert _{L_{q}(\tilde{I}_{2i-1})}^{q} &= \int_{\xi_{i-1}}^{\xi_{i}}e^{q\operatorname{Re}(w_{2i-1}\lambda)(x-\xi _{i-1})}\,dx \\ &\leq \int_{\xi_{i-1}}^{\xi_{i}}e^{-q|w_{2i-1}||\lambda|(x-\xi_{i-1})\sin \frac{\varepsilon}{2}}\,dx \\ &={}^{t=|\lambda|(x-\xi_{i-1})} \int_{0}^{|\lambda|(\xi_{i}-\xi_{i-1})} e^{-q|w_{2i-1}|(\sin\frac{\varepsilon}{2})t}d \biggl( \frac{t}{|\lambda |}+\xi_{i-1} \biggr) \\ &\leq|\lambda|^{-1} \int_{0}^{+\infty}e^{-q|w_{2i-1}|(\sin\frac{\varepsilon }{2})t}dt \\ &=|\lambda|^{-1} \biggl(q|w_{2i-1}|\sin\frac{\varepsilon}{2} \biggr)^{-1} \\ &=C_{2i-1}(\varepsilon)|\lambda|^{-1}, \end{aligned} \end{aligned}$$

(2.8)

$$\begin{aligned}& \begin{aligned}[b] \bigl\Vert y_{2i}(x,\lambda) \bigr\Vert _{L_{q}(\tilde{I}_{2i})}^{q} &= \int_{\xi_{i-1}}^{\xi_{i}}e^{q\operatorname{Re}(w_{2i}\lambda)(x-\xi _{i})}\,dx \\ &\leq \int_{\xi_{i-1}}^{\xi_{i}}e^{-q|w_{2i}||\lambda|(\xi_{i}-x)\sin \frac{\varepsilon}{2}}\,dx \\ &={}^{t=|\lambda|(\xi_{i}-x)} \int_{|\lambda|(\xi_{i}-\xi_{i-1})}^{0} e^{-q|w_{2i}|(\sin\frac{\varepsilon}{2})t}d \biggl( \xi_{i}-\frac {t}{|\lambda|} \biggr) \\ &\leq|\lambda|^{-1} \int_{0}^{+\infty}e^{-q|w_{2i}|(\sin\frac{\varepsilon }{2})t}dt \\ &=|\lambda|^{-1} \biggl(q|w_{2i}|\sin\frac{\varepsilon}{2} \biggr)^{-1} \\ &=C_{2i}(\varepsilon)|\lambda|^{-1}, \end{aligned} \end{aligned}$$

(2.9)

where $i=1,2,\ldots,(n+1)$, hold by (2.4). Substituting (2.8), (2.9) into (2.7) yields

$$ \bigl\Vert y^{(\tau)}(x) \bigr\Vert _{L_{q}(-1,1)}\leq C( \varepsilon)\sum_{\eta=1}^{2(n+1)}| \lambda|^{\tau-n_{\eta}-q^{-1}}|f_{\eta}|, $$

which in turn gives us the needed estimation (2.3). The proof is completed. □

3 Fredholm property for multi-point boundary value problem with functional conditions

Let M̃ be the linear operator corresponding to problem (1.1)–(1.2). Suppose that $\sigma\geq\max\{2,\max_{1\leq k\leq2(n+1)}\{ n_{k}\}+1\}$ and define M̃ from $W_{q}^{\sigma}$ into $W_{q}^{\sigma-2}\oplus\mathbb{C}^{2(n+1)}$ by the rule

$$\widetilde{M}y=\bigl(M(\lambda), M_{1}y, M_{2}y,\ldots, M_{2(n+1)}y\bigr). $$

Theorem 3.1

Assume that the following conditions hold:

• (1) For $x\in I_{i}$, $p_{i}\neq0$;

• (2) $F_{k}$ ($k=1,2,\ldots,2(n+1)$) are continuous functionals in $W_{q}^{\sigma}$;

• (3) The operator T from $W_{q}^{\sigma}$ into $W_{q}^{\sigma -2}$ is compact.

Then M̃ is bounded and Fredholm operator.

Proof

The operator M̃ can be represented as

$$\begin{aligned} &\widetilde{M}_{0}y=\bigl(M_{0}(\lambda)y,M_{10}y,M_{20}y, \ldots,M_{2(n+1)0}y\bigr), \\ &\widetilde{M}_{1}y=(Ty,\varGamma_{1}y+F_{1}y, \varGamma_{2}y+F_{2}y,\ldots,\varGamma _{2(n+1)}y+F_{2(n+1)}y). \end{aligned}$$

The operator $\widetilde{M}_{0}$ is an isomorphism from $W_{q}^{\sigma}$ onto $W_{q}^{\sigma-2}\oplus\mathbb{C}^{2(n+1)}$ by Theorem 2.1. Furthermore, it follows from (2) and (3) that the operator $\widetilde {M}_{1}$ acts compactly from $W_{q}^{\sigma}$ onto $W_{q}^{\sigma-2}\oplus\mathbb{C}^{2(n+1)}$.

Therefore, by the definition of isomorphism and Theorem 1.2.8 in [3] (or [25, p. 238]), the operator $\widetilde{M}=\widetilde{M}_{0}+\widetilde{M}_{1}$ is Fredholm. Moreover, it is obvious that the operator M̃ is bounded. So, the desired results are obtained. □

4 Isomorphism and coerciveness of the principal part of the problem

We consider the principle part of main problem (1.1)–(1.2) without internal points, that is,

$$\begin{aligned}& M_{0}(\lambda)y:=-\bigl(py'\bigr)'(x)+ \lambda^{2}y(x)=f(x),\quad x\in I, \end{aligned}$$

(4.1)

$$\begin{aligned}& \begin{aligned}[b] M_{k0}y&:=a_{0k}y^{(n_{k})}( \xi_{0})+\sum_{h=1}^{n} \bigl(a_{hk}y^{(n_{k})}(\xi _{h}-)+\tilde{a}_{hk}y^{(n_{k})}( \xi_{h}+) \bigr) \\ &\quad {}+a_{(n+1)k}y^{(n_{k})}(\xi_{n+1})=f_{k} \end{aligned} \end{aligned}$$

(4.2)

for $k=1,2,\ldots,2(n+1)$. The corresponding operator is

$$\widehat{M}_{0}y:=\bigl(M_{0}(\lambda)y,M_{10}y,M_{20}y, \ldots,M_{2(n+1)0}y\bigr). $$

Theorem 4.1

Let the condition $\omega\neq0$ and $\sigma\geq \max\{2,\max_{1\leq k\leq2(n+1)}\{n_{k}\}+1\}$ be satisfied. Then, for each $\varepsilon>0$, there exists $\mu_{\varepsilon}>0$ such that, for all complex numbers $\lambda\in\varOmega_{\varepsilon}(\alpha,\beta)$, $|\lambda|>\mu _{\varepsilon}$, the operator $\widehat{M}_{0}$ from $W_{q}^{\sigma}$ onto $W_{q}^{\sigma -2}\oplus\mathbb{C}^{2(n+1)}$ is an isomorphism, and for these λ the inequality

$$ \sum_{\ell=0}^{\sigma}| \lambda|^{\sigma-\ell}\|y\|_{q,{\ell}}\leq C(\varepsilon) \Biggl(\|f \|_{q,{\sigma-2}}+ |\lambda|^{\sigma-2}\|f\|_{q,0}+\sum _{k=1}^{2(n+1)}|\lambda|^{\sigma -n_{k}-q^{-1}}|f_{k}| \Biggr) $$

(4.3)

holds for the solution of (4.1)–(4.2).

Proof

Obviously, the linear operator $\widehat{M}_{0}$ acts continuously from the space $W_{q}^{\sigma}$ into $W_{q}^{\sigma-2}\oplus\mathbb{C}^{2(n+1)}$. Let us prove that, for any $(f(x), f_{1}, f_{2},\ldots, f_{2(n+1)})\in W_{q}^{\sigma-2}\oplus\mathbb {C}^{2(n+1)}$ and $f_{i}$, problem (4.1)–(4.2) has a unique solution belonging to $W_{q}^{\sigma}$. Denote by $f_{i}(x)$ the restriction of $f(x)$ on the interval $J_{i}$. Let $\tilde{f}_{i}(x)\in W_{q}^{\sigma-2}(\mathbb{R})$ be an extension of $f_{i}(x)\in W_{q}^{\sigma-2}(J_{i})$. By Lemma 1.7.6 in [3] there exists an extension operator $T_{i}f_{i}:=\tilde{f}_{i}$ from $W_{q}^{\sigma-2}$ into $W_{q}^{\sigma-2}(\mathbb{R})$ is bounded for $i=1,2,\ldots,n+1$, where as usual $\mathbb{R}=(-\infty,+\infty)$. We shall find the solution $y(x,\lambda)$ of problem (4.1)–(4.2) in the form of $y(x,\lambda)=y_{1}(x,\lambda)+y_{2}(x,\lambda)$, where $y_{1}(x,\lambda)=(y_{1i}(x,\lambda))$, the function $y_{1i}(x,\lambda)$ is the restriction of the solution $\tilde{y}_{1i}(x,\lambda)$ on $J_{i}$ of the following equation:

$$-\bigl(p_{i}\tilde{y}^{\prime}\bigr)^{\prime}(x)+ \lambda^{2}\tilde{y}(x)=\tilde {f}_{i}(x),\quad x\in \mathbb{R} $$

for $i=1,2,\ldots,n+1$.

By virtue of Theorem 3.2.1 in [3], we get that this equation has a unique solution $\tilde{y}_{1i}=\tilde{y}_{1i}(x,\lambda)\in W_{q}^{\sigma}(\mathbb {R})$, and for $y_{1i}$, the estimate

$$ \sum_{\ell=0}^{\sigma}| \lambda|^{\sigma-\ell}\|y_{1i}\|_{W_{q}^{\ell }(J_{i})}\leq C(\varepsilon) \bigl(\|f\|_{W_{q}^{\sigma-2}(J_{i})}+|\lambda |^{\sigma-2}\|f\|_{L_{q}(J_{i})}\bigr), $$

(4.4)

where $i=1,2,\ldots,n+1$, holds for all $\lambda\in\varOmega_{\varepsilon}(\alpha,\beta)$ sufficiently large in modulus.

Hence, the function

$$ y_{1}(x,\lambda) = \textstyle\begin{cases} y_{11}(x,\lambda), & \mbox{for }x\in J_{1}; \\ y_{12}(x,\lambda), & \mbox{for }x\in J_{2}; \\ \quad \vdots \\ y_{1,n+1}(x,\lambda), & \mbox{for }x\in J_{n+1}, \end{cases} $$

(4.5)

satisfies equation (4.1), and from (4.4) the following estimate

$$ \sum_{\ell=0}^{\sigma}| \lambda|^{\sigma-\ell}\|y_{1}\|_{q,\ell}\leq C(\varepsilon) \bigl(\|f\|_{q,\sigma-2}+|\lambda|^{\sigma-2}\|f\|_{q,0}\bigr) $$

(4.6)

holds for all $\lambda\in\varOmega_{\varepsilon}(\alpha,\beta)$ sufficiently large in modulus. In light of solution (4.5), consider the following boundary value problem:

$$\begin{aligned}& -\bigl(py^{\prime}\bigr)^{\prime}(x)-\lambda^{2}y(x)=0, \quad x\in J, \\& M_{k0}y=f_{k}-M_{k0}y_{1}(x,\lambda), \quad k=1,2,\ldots,2(n+1). \end{aligned}$$

By Theorem 2.1, this problem has a unique solution $y_{2}=y_{2}(x,\lambda )\in W_{q}^{\sigma}$ for all complex numbers $\lambda\in\varOmega_{\varepsilon}(\alpha,\beta)$ sufficiently large in modulus, and for these λ the estimate

$$ \sum_{\ell=0}^{\sigma}| \lambda|^{\sigma-\ell}\|y_{2}\|_{q,\ell}\leq C(\varepsilon) \sum_{k=1}^{2(n+1)}|\lambda|^{\sigma -n_{k}-q^{-1}} \bigl( \vert f_{k} \vert +|M_{k0}y_{1}|\bigr) $$

(4.7)

is valid. Applying Theorem 1.7.7/2 in [3] and (2.3), one has that, for all $\lambda\in\varOmega_{\varepsilon}(\alpha,\beta)$ and $\sigma\geq \max\{2,\max_{1\leq k\leq2( n+1)}\{n_{k}\}+1\}$, the following estimates hold:

$$\begin{aligned} |\lambda|^{\sigma-n_{k}-q^{-1}}|M_{k0}y_{1}|&\leq C| \lambda|^{\sigma -n_{k}-q^{-1}}\sum_{i=1}^{n+1} \|y_{1}\|_{C^{n_{k}}(I_{i})} \\ &\leq C\sum_{i=1}^{n+1}\bigl(| \lambda|^{\sigma}\|y_{1i}\|_{L_{q}(J_{i})}+\| y_{1i} \|_{W_{q}^{\sigma}(J_{i})}\bigr) \\ &\leq C\bigl(|\lambda|^{\sigma}\|y_{1}\|_{q,0}+ \|y_{1}\|_{q,\sigma}\bigr) \\ &\leq C(\varepsilon) \bigl( \vert \lambda \vert _{\sigma-2}\|f \|_{q,0}+\|f\|_{q,\sigma-2}\bigr). \end{aligned}$$

(4.8)

Via (4.7) and (4.8) we have the inequality

$$ \sum_{\ell=0}^{\sigma}| \lambda|^{\sigma-\ell}\|y_{2}\|_{q,\ell}\leq C(\varepsilon) \Biggl(\|f\|_{q,\sigma-2}+|\lambda|^{\sigma-2}\|f\|_{q,0}+\sum _{k=1}^{2(n+1)}|\lambda|^{\sigma-n_{k}-q^{-1}}|f_{k}| \Biggr). $$

(4.9)

It is easy to see that $y(x,\lambda)=y_{1}(x,\lambda)+y_{2}(x,\lambda)$ is the solution of problem (4.1)–(4.2). Taking into account estimates (4.6) and (4.9), we see that for this solution the needed estimation (4.3) is valid. Moreover, from estimate (4.3) the uniqueness of the solution follows. On the other hand, by Theorem 3.1 the operator $\widehat{M}_{0}$ is Fredholm from $W_{q}^{\sigma}$ onto $W_{q}^{\sigma-2}\oplus\mathbb{C}^{2(n+1)}$. Now, an isomorphism of this operator follows from the fact that it is a Fredholm and one-to-one operator. So, the proof of the theorem is completed. □

5 Solvability and coerciveness of the main problem with nonclassical boundary conditions

Now, we will study the main problem (1.1)–(1.2).

Theorem 5.1

Assume that the following conditions hold:

• (1) $\omega\neq0$ and $\sigma\geq\max\{2,\max_{1\leq k\leq2( n+1)}\{n_{k}\}+1\}$;

• (2) The operator T from $W_{q}^{\sigma}$ into $W_{q}^{\sigma -2}$ is compact, and for all $\varepsilon>0$

  $$\begin{aligned}& \|Ty\|_{q,0}\leq\varepsilon\|y\|_{q,2}+C(\varepsilon)\|y \|_{q,0},\quad y\in W_{q}^{2}; \\& \|Ty\|_{q,\sigma-2}\leq\varepsilon\|y\|_{q,\sigma}+C(\varepsilon)\|y\| _{q,0}, \quad y\in W_{q}^{\sigma}; \end{aligned}$$

• (3) Functionals $F_{k}$ in $W_{q}^{n_{k}}$ ($k=1,2,\ldots,2(n+1)$) are continuous.

Then, for each $\varepsilon>0$, there exists $\mu_{\varepsilon}>0$ such that, for all $\lambda\in\varOmega_{\varepsilon}(\alpha,\beta)$ and $|\lambda|>\mu _{\varepsilon}$, the operator

$$\widehat{M}y=\bigl(M(\lambda)y,M_{1}y,M_{2}y,\ldots, M_{2(n+1)}y\bigr) $$

is an isomorphism from $W_{q}^{\sigma}$ onto $W_{q}^{\sigma-2}\oplus \mathbb{C}^{2(n+1)}$, and for these λ we have the following coercive estimate for the solution of problem (1.1)–(1.2):

$$ \sum_{\ell=0}^{\sigma}| \lambda|^{\sigma-\ell}\|y\|_{q,\ell}\leq C(\varepsilon) \Biggl(\|f \|_{q,\sigma-2}+|\lambda|^{\sigma-2}\|f\|_{q,0}+\sum _{k=1}^{2(n+1)}|\lambda|^{\sigma-n_{k}-q^{-1}}|f_{k}| \Biggr), $$

(5.1)

where $C(\varepsilon)$ is a constant which depends only on ε.

Proof

Let $(f(x),f_{1},f_{2},\ldots,f_{2(n+1)})$ be any element of $W_{q}^{\sigma -2}\oplus\mathbb{C}^{2(n+1)}$. Suppose that there exists a solution $y=y(x,\lambda)$ of problem (1.1)–(1.2) corresponding to this element. Then this solution satisfies the equalities

$$\begin{aligned}& M_{0}(\lambda)y:=-\bigl(py^{\prime}\bigr)^{\prime}+ \lambda^{2}y=M(\lambda)y-Ty, \end{aligned}$$

(5.2)

$$\begin{aligned}& \begin{aligned}[b] M_{k0}y&:=a_{0k}y^{(n_{k})}( \xi_{0})+\sum_{h=1}^{n} \bigl(a_{hk}y^{n_{k}}(\xi _{h}-)+\tilde{a}_{hk}y^{(n_{k})}( \xi_{h}+)\bigr) \\ &\quad {}+a_{(n+1)k}y^{(n_{k})}(\xi_{n+1})=M_{k}y- \varGamma_{k}y-F_{k}y, \end{aligned} \end{aligned}$$

(5.3)

where $k=1,2,\ldots,2(n+1)$. By Theorem 4.1 we have that for this solution the following a priori estimates hold:

$$\begin{aligned} \sum_{\ell=0}^{\sigma}|\lambda|^{\sigma-\ell} \Vert y \Vert _{q,\ell}&\leq C(\varepsilon) \Biggl( \Vert f \Vert _{q,\sigma-2}+|\lambda|^{\sigma-2} \Vert f \Vert _{q,0}+\sum _{k=1}^{2(n+1)}|\lambda|^{\sigma-n_{k}-q^{-1}}|f_{k}| \Biggr) \\ &=C(\varepsilon) \Biggl( \bigl\Vert M(\lambda)y-Ty \bigr\Vert _{q,\sigma-2}+|\lambda|^{\sigma-2} \bigl\Vert M(\lambda)y-Ty \bigr\Vert _{q,0} \\ &\quad {}+\sum_{k=1}^{2(n+1)}| \lambda|^{\sigma-n_{k}-q^{-1}}|M_{k}y-\varGamma _{k}y-F_{k}y| \Biggr) \\ &\leq C(\varepsilon) \Biggl( \Vert f \Vert _{q,\sigma-2}+|\lambda|^{\sigma-2} \Vert f \Vert _{q,0}+ \Vert Ty \Vert _{q,\sigma-2}+| \lambda|^{\sigma-2} \Vert Ty \Vert _{q,0} \\ &\quad {}+\sum_{k=1}^{2(n+1)}| \lambda|^{\sigma-n_{k}-q^{-1}}\bigl( \vert f_{k} \vert +| \varGamma_{k}y|+|F_{k}y|\bigr)\Biggr). \end{aligned}$$

(5.4)

Let ζ be any real number satisfying

$$\begin{aligned} 0&< \zeta \\ &< \min\biggl\{ \frac{\xi_{i}-\xi_{i-1}}{2},|\xi_{i}-x_{kj}|,|\xi _{0}-x_{kj}|: \\ &\quad i=1,2,\ldots,n+1; k=1,2,\ldots,2(n+1); j=1,2,\ldots,n_{k} \biggr\} . \end{aligned}$$

Using the same method in [15, Sect. 2.8.3], it is not difficult to construct a function $\phi(x)\in\mathbb{C}_{0}^{\infty}(\mathbb{R})$ such that

$$ \phi(x) = \textstyle\begin{cases} 1, & x\in\bigcup_{i=1}^{n+1}[\xi_{i-1}+\zeta,\xi_{i}-\zeta]; \\ 0, & x\in\bigcup_{i=1}^{n+1}[\xi_{0},\xi_{0}+\frac{\zeta }{2}]\cup(\bigcup_{i=1}^{n}[\xi_{i}-\frac{\zeta}{2},\xi_{i}+\frac{\zeta }{2}])\cup[\xi_{n+1}-\frac{\zeta}{2},\xi_{n+1}] \end{cases} $$

and $0\leq\phi(x)\leq1$ for all $x\in[-1,1]$. It is obvious that

$$ |\varGamma_{k}y|\leq C \bigl\Vert y^{(n_{k})} \bigr\Vert _{C(\bigcup_{i=1}^{n+1}[\xi_{i-1}+\zeta ,\xi_{i}-\zeta])}\leq C \bigl\Vert (\phi y)^{(n_{k})} \bigr\Vert _{C[-1,1]}. $$

(5.5)

By Theorem 3.10.4 in [15], for $y\in W_{q}^{\sigma}$, the estimate

$$ |\lambda|^{\sigma-n_{k}-q^{-1}} \bigl\Vert y^{(n_{k})} \bigr\Vert _{C[-1,1]}\leq C\bigl( \Vert y \Vert _{q,\sigma}+| \lambda|^{\sigma} \Vert y \Vert _{q,0}\bigr) $$

(5.6)

holds. By Theorem 4.1, from (5.4), (5.5) it follows that, for all $\lambda\in \varOmega_{\varepsilon}(\alpha,\beta)$ sufficiently large in modulus, the following estimates hold:

$$\begin{aligned} |\lambda|^{\sigma-n_{k}-q^{-1}}|\varGamma_{k}y|&\leq C|\lambda|^{\sigma -n_{k}-q^{-1}} \bigl\Vert (\phi y)^{(n_{k})} \bigr\Vert _{C[-1,1]} \\ &\leq C\bigl( \Vert \phi y \Vert _{q,\sigma}+|\lambda|^{\sigma} \Vert \phi y \Vert _{q,0}\bigr) \\ &\leq C(\varepsilon) \bigl( \bigl\Vert M_{0}(\lambda) (\phi y) \bigr\Vert _{q,\sigma-2}+|\lambda |^{\sigma-2} \bigl\Vert M_{0}(\lambda) (\phi y) \bigr\Vert _{q,0}\bigr) \\ &\leq C(\varepsilon) \Biggl( \bigl\Vert M_{0}(\lambda)y \bigr\Vert _{q,\sigma-2}+|\lambda|^{\sigma -2} \bigl\Vert M_{0}( \lambda)y \bigr\Vert _{q,0}+ \sum_{\ell=0}^{\sigma-1}| \lambda|^{\sigma-1-\ell} \Vert y \Vert _{q,\ell} \Biggr) \\ &\leq C(\varepsilon) \Biggl( \bigl\Vert M(\lambda)y \bigr\Vert _{q,\sigma-2}+|\lambda|^{\sigma-2} \bigl\Vert M(\lambda)y \bigr\Vert _{q,0}+ \Vert Ty \Vert _{q,\sigma-2} \\ &\quad {}+|\lambda|^{\sigma-2} \Vert Ty \Vert _{q,0}+\sum _{\ell=0}^{\sigma-1}|\lambda |^{\sigma-1-\ell} \Vert y \Vert _{q,\ell}\Biggr) \\ &\leq C(\varepsilon) \Biggl( \Vert f \Vert _{q,\sigma-2}+| \lambda|^{\sigma-2} \Vert f \Vert _{q,0}+ \Vert Ty \Vert _{q,\sigma-2} \\ &\quad {}+|\lambda|^{\sigma-2} \Vert Ty \Vert _{q,0}+\sum _{\ell=0}^{\sigma-1}|\lambda |^{\sigma-1-\ell} \Vert y \Vert _{q,\ell}\Biggr). \end{aligned}$$

(5.7)

By virtue of Theorem 1.7.7/2(b) and Remark 1.1.7/5 in [3], the following inequality

$$\|y\|_{q,\ell}\leq\delta\|y\|_{q,\ell+1}+C(\delta)\|y \|_{q,0} $$

holds for any $\delta>0$. Then, by virtue of (5.7),

$$\begin{aligned} |\lambda|^{\sigma-n_{k}-q^{-1}} \Vert \varGamma_{k}y \Vert &\leq C( \varepsilon) \Biggl( \bigl\Vert M(\lambda)y \bigr\Vert _{q,\sigma-2}+| \lambda|^{\sigma-2} \bigl\Vert M(\lambda)y \bigr\Vert _{q,0}+ \Vert Ty \Vert _{q,\sigma-2} \\ &\quad {}+|\lambda|^{\sigma-2} \Vert Ty \Vert _{q,0}+\sum _{\ell=0}^{\sigma-1}|\lambda |^{\sigma-1-\ell}\bigl( \delta \Vert y \Vert _{q,\ell+1}+C(\delta) \Vert y \Vert _{q,0}\bigr)\Biggr) \\ &\leq C(\varepsilon) \bigl( \bigl\Vert M(\lambda)y \bigr\Vert _{q,\sigma-2}+| \lambda|^{\sigma-2} \bigl\Vert M(\lambda)y \bigr\Vert _{q,0}+ \Vert Ty \Vert _{q,\sigma-2}\bigr) \\ &\quad {}+|\lambda|^{\sigma-2} \Vert Ty \Vert _{q,0}+C( \varepsilon) \bigl(\delta+C(\delta )|\lambda|^{-1}\bigr)\sum _{\ell=0}^{\sigma}|\lambda|^{\sigma-\ell} \Vert y \Vert _{q,\ell}. \end{aligned}$$

(5.8)

From conditions (2), (3), inequality (5.8), and Theorem 1.7.7/2 in [3], for any $\delta>0$, we have

$$\begin{aligned} & \Vert Ty \Vert _{q,\sigma-2}+|\lambda|^{\sigma-2} \Vert Ty \Vert _{q,0}+\sum_{k=1}^{2(n+1)}| \lambda|^{\sigma-n_{k}-q^{-1}}\bigl( \vert \varGamma_{k}y \vert +|F_{k}y|\bigr) \\ &\quad \leq C(\varepsilon) \bigl( \bigl\Vert M(\lambda)y \bigr\Vert _{q,\sigma-2}+|\lambda|^{\sigma-2} \bigl\Vert M(\lambda)y \bigr\Vert _{q,0}\bigr) +\delta\bigl( \Vert y \Vert _{q,\sigma}+| \lambda|^{\sigma-2} \Vert y \Vert _{q,2}\bigr) \\ &\qquad {}+C(\delta)|\lambda|^{\sigma-2} \Vert y \Vert _{q,0}+C(\varepsilon) \bigl(\delta +C(\delta)|\lambda|^{-1} \bigr) \sum_{\ell=0}^{\sigma}|\lambda|_{\sigma-\ell} \Vert y \Vert _{q,\ell} \\ &\qquad {}+\sum_{k=1}^{2(n+1)}| \lambda|^{\sigma-n_{k}-q^{-1}} \Vert y \Vert _{q,n_{k}} \\ &\quad \leq C(\varepsilon) \bigl( \Vert f \Vert _{q,\sigma-2}+| \lambda|^{\sigma-2} \Vert f \Vert _{q,0}\bigr) \\ &\qquad {}+C(\varepsilon) \bigl(\delta+C(\delta)|\lambda|^{-q^{-1}}\bigr) \sum_{\ell =0}^{\sigma}|\lambda|^{\sigma-\ell} \Vert y \Vert _{q,\ell}. \end{aligned}$$

(5.9)

Substituting (5.9) into (5.4) yields

$$\begin{aligned} \sum_{\ell=0}^{\sigma}|\lambda|^{\sigma-\ell}\|y \|_{q,\ell}&\leq C(\varepsilon) \Biggl(\|f\|_{q,\sigma-2}+| \lambda|^{\sigma-2}\|f\|_{q,0} +\sum_{k=1}^{2(n+1)}| \lambda|^{\sigma-n_{k}-q^{-1}}|f_{k}| \\ &\quad {}+\bigl(\delta+C(\delta)|\lambda|^{-q^{-1}}\bigr)\sum _{\ell=0}^{\sigma}|\lambda |^{\sigma-\ell}\|y \|_{q,\ell}\Biggr). \end{aligned}$$

It is obvious that for fixed $\varepsilon>0$ it is possible to choose $\delta>0$ so small and $|\lambda|$ so large that $C(\varepsilon)(\delta+C(\delta)|\lambda|^{-q^{-1}})<1$. Hence, for $\lambda\in\varOmega_{\varepsilon}(\underline{w},\overline{w})$ sufficiently large in modulus, we obtain a priori estimate (5.1).

It follows from estimate (5.1) that we can obtain the uniqueness property of solution of problem (1.1)–(1.2), i.e., the operator M̂ is a one-to-one operator. Moreover, by Theorem 3.1 the operator M̂ from $W_{q}^{\sigma }$ onto $W_{q}^{\sigma-2}\oplus\mathbb{C}^{2(n+1)}$ is Fredholm.

In view of condition (2), the operator T from $W_{q}^{\sigma}$ into $W_{q}^{\sigma-2}\oplus\mathbb{C}^{2(n+1)}$ is compact. From above, we get that the operator M̂ is an isomorphism from $W_{q}^{\sigma}$ onto $W_{q}^{\sigma-2}\oplus\mathbb{C}^{2(n+1)}$. The proof is completed. □

6 Conclusion

Sturm–Liouville problem with discontinuous points inside an interval has attracted extensive attention for its wide application in physical and mathematical fields. In this paper, we go into Sturm–Liouville problem with finite discontinuous points, and for such a problem, we establish the properties such as isomorphism, Fredholmness, and coerciveness with respect to the spectral parameter. These results are of both theoretical and practical significance.