The Resolvent of Impulsive Singular Hahn–Sturm–Liouville Operators

Allahverdiev, Bilender P.; Tuna, Hüseyin; Isayev, Hamlet A.

doi:10.2478/amsil-2024-0001

Full Article

1.

Introduction

Impulsive differential equations are one of the interesting topics in the theory of differential equations. These equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states. These types of problems are especially encountered in heat and mass transfer problems ([17]). There are many studies on this subject in the literature [2, 6, 7, 8, 9, 12, 13, 14, 19, 23].

W. Hahn introduced the concept of the Hahn derivative to the literature in 1949 [10]. With this definition he made, he gathered two important operators under a single structure. These are the q-difference and forward difference operators. In 2018, Annaby et al. [4] using this definition instead of the classical derivative, investigated the fundamental properties of the Sturm–Liouville problems. In [5], the authors studied singular q-Sturm-Liouville equations. In [18], the author studied a q-analog of the singular Dirac problem. Recently in [21], the author proved a spectral expansion theorem by constructing the spectral function of the Hahn–Sturm–Liouville equation in the singular case under impulsive conditions.

In this paper, our aim is to consider Hahn–Sturm–Liouville problems under impulsive boundary conditions. The integral representation of the resolvent operator corresponding to this type of problem will be obtained using Weyl’s method [16, 22, 24].

2.

Preliminaries

Now, we provide a concise overview of the Hahn calculus [3, 4, 10, 11]. Let q ∈ (0, 1), ω₀ := ω/ (1 − q) , ω > 0, and let Ψ : J ⊂ ℝ ω ℝ be a function such that ω₀ ∈ J.

Definition 2.1 ([10], [11]).

The Hahn derivative D_ω,q Ψ is defined by $D_{ω, q} Ψ (ζ) = \begin{array}{l} \frac{Ψ (ω + q ζ) - Ψ (ζ)}{ω + (q - 1) ζ}, & ζ \neq ω_{0}, \\ Ψ^{'} (ω_{0}), & ζ = ω_{0}, \end{array}$ {D_{\omega ,q}}\Psi \left( \zeta \right) = \left\{ {\matrix{ {{{\Psi \left( {\omega + q\zeta } \right) - \Psi \left( \zeta \right)} \over {\omega + \left( {q - 1} \right)\zeta }},} \hfill & {\zeta \ne {\omega _0},} \hfill \cr {\Psi '\left( {{\omega _0}} \right)\;,} \hfill & {\zeta = {\omega _0},} \hfill \cr } } \right. where the expression Ψ^′ (ω₀) shows the ordinary derivative of Ψ at ω₀.

Definition 2.2 ([3]).

Let a, b, ω₀ ∈ J. The Hahn integral (ω, q-integral) is defined by $\int_{a}^{b} Ψ (ζ) d_{ω, q} ζ : = \int_{ω_{0}}^{b} Ψ (ζ) d_{ω, q} ζ - \int_{ω_{0}}^{a} Ψ (ζ) d_{ω, q} ζ,$ \int_a^b {\Psi \left( \zeta \right){d_{\omega ,q}}\zeta } : = \int_{{\omega _0}}^b {\Psi \left( \zeta \right){d_{\omega ,q}}\zeta } - \int_{{\omega _0}}^a {\Psi \left( \zeta \right){d_{\omega ,q}}\zeta } , where $\int_{ω_{0}}^{ζ} Ψ (t) d_{ω, q} t : = ((1 - q) ζ - ω) \sum_{n = 0}^{\infty} q^{n} Ψ (ω \frac{1 - q^{n}}{1 - q} + ζ q^{n}), ζ \in J,$ \int_{{\omega _0}}^\zeta {\Psi \left( t \right){d_{\omega ,q}}t: = \left( {\left( {1 - q} \right)\zeta - \omega } \right)} \sum\limits_{n = 0}^\infty {{q^n}\Psi \left( {\omega {{1 - {q^n}} \over {1 - q}} + \zeta {q^n}} \right)} ,\;\;\;\zeta \in J, provided that the series converges at ζ = a and ζ = b.

Definition 2.3.

The ω, q-Wronskian of Ψ₁ and Ψ₂ is defined by $W_{ω, q} (Ψ_{1}, Ψ_{2}) : = Ψ_{1} D_{ω, q} Ψ_{2} - Ψ_{2} D_{ω, q} Ψ_{1} .$ {W_{\omega ,q}}\left( {{\Psi _1},\;{\Psi _2}} \right): = {\Psi _1}{D_{\omega ,q}}{\Psi _2} - {\Psi _2}{D_{\omega ,q}}{\Psi _1}.

3.

Main results

Let us consider the following impulsive boundary-value problem (BVP) (3.1) $- \frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} y (ζ) + v (ζ) y (ζ) = λ y (ζ), ζ \in (ω_{0}, d) \cup (d, \frac{1}{q^{n}}),$ - {1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}y\left( \zeta \right) + v\left( \zeta \right)y\left( \zeta \right) = \lambda y\left( \zeta \right),\;\;\;\zeta \in \left( {{\omega _0},\;d} \right) \cup\left( {d,\;{1 \over {{q^n}}}} \right), (3.2) $y (ω_{0}, λ) \cos β + D_{- \frac{ω}{q}, \frac{1}{q}} y (ω_{0}, λ) \sin β = 0,$ y\left( {{\omega _0},\;\lambda } \right)\;\cos \;\beta + {D_{ - {\omega \over q},{1 \over q}}}y\left( {{\omega _0},\;\lambda } \right)\;\sin \;\beta = 0, (3.3) $y (d -) = η y (d +),$ y\left( {d - } \right) = \eta y\left( {d + } \right), (3.4) $D_{- \frac{ω}{q}, \frac{1}{q}} y (d -) = \frac{1}{η} D_{- \frac{ω}{q}, \frac{1}{q}} y (d +),$ {D_{ - {\omega \over q},{1 \over q}}}y\left( {d - } \right) = {1 \over \eta }{D_{ - {\omega \over q},{1 \over q}}}y\left( {d + } \right), (3.5) $y (\frac{1}{q^{n}}, λ) \cos γ + D_{- \frac{ω}{q}, \frac{1}{q}} y (\frac{1}{q^{n}}, λ) \sin γ = 0,$ y\left( {{1 \over {{q^n}}},\;\lambda } \right)\;\cos \;\gamma + {D_{ - {\omega \over q},{1 \over q}}}y\left( {{1 \over {{q^n}}},\;\lambda } \right)\;\sin \;\gamma = 0, where q ∈ (0, 1), ω₀ := ω/ (1 − q), ω > 0, γ, β ∈ ℝ, $\frac{1}{q^{n}} > d$ {1 \over {{q^n}}} > d , n ∈ ℕ := {1, 2, 3, . . .}, η > 0, λ ∈ ℂ, y(d±) := lim_ζd± y (ζ) , v is a real-valued continuous function on [ω₀, d) ∪ (d, ∞), and has finite limits v(d±).

A similar problem has been studied by the authors without impulsive boundary conditions ([1]).

$H_{n} = L_{ω, q}^{2} (ω_{0}, d) \dot{+} L_{ω, q}^{2} (d, \frac{1}{q^{n}})$ {H_n} = L_{\omega ,q}^2\left( {{\omega _0},\;d} \right) {^ \dot +} L_{\omega ,q}^2\left( {d,\;{1 \over {{q^n}}}} \right) , $\frac{1}{q^{n}} > d$ {1 \over {{q^n}}} > d , $n \in ℕ (H = L_{ω, q}^{2} (ω_{0}, d) \dot{+} L_{ω, q}^{2} (d, \infty))$ n \in {\mathbb N}\left( {H = L_{\omega ,q}^2 \left( {\omega _0 ,d} \right) {^ \dot +} L_{\omega ,q}^2 \left( {d,\infty } \right)} \right) is a Hilbert space endowed with the following inner product $\begin{array}{l} {〈 y, z 〉}_{n} : = \int_{ω_{0}}^{d} y^{(1)} \bar{z^{(1)}} d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} y^{(2)} \bar{z^{(2)}} d_{ω, q} ζ, \\ (〈 y, z 〉 : = \int_{ω_{0}}^{d} y^{(1)} \bar{z^{(1)}} d_{ω, q} ζ + \int_{d}^{\infty} y^{(2)} \bar{z^{(2)}} d_{ω, q} ζ) \end{array}$ \matrix{ {{{\langle y,\;z\rangle }_n}: = \int_{{\omega _0}}^d {{y^{\left( 1 \right)}}\overline {{z^{\left( 1 \right)}}} {d_{\omega ,q}}\zeta } + \int_d^{{1 \over {{q^n}}}} {{y^{\left( 2 \right)}}\overline {{z^{\left( 2 \right)}}} {d_{\omega ,q}}\zeta } ,} \hfill \cr {(\langle y,\;z\rangle : = \int_{{\omega _0}}^d {{y^{\left( 1 \right)}}\overline {{z^{\left( 1 \right)}}} {d_{\omega ,q}}\zeta } + \int_d^\infty {{y^{\left( 2 \right)}}\overline {{z^{\left( 2 \right)}}} {d_{\omega ,q}}\zeta )} } \hfill \cr } where $y (ζ) = \begin{array}{l} y^{(1)} (ζ), & ζ \in ω_{0}, d), \\ y^{(2)} (ζ), & ζ \in (d, \infty), \end{array}$ y\left( \zeta \right) = \left\{ {\matrix{ {{y^{\left( 1 \right)}}\left( \zeta \right)\;,\;} \hfill & {\zeta \in \left[ {{\omega _0},\;d} \right)\;,} \hfill \cr {{y^{\left( 2 \right)}}\left( \zeta \right)\;,\;} \hfill & {\zeta \in \left( {d,\;\infty } \right)\;,} \hfill \cr } } \right. and $z (ζ) = \begin{array}{l} z^{(1)} (ζ), & ζ \in ω_{0}, d), \\ z^{(2)} (ζ), & ζ \in (d, \infty) . \end{array}$ z\left( \zeta \right) = \left\{ {\matrix{ {{z^{\left( 1 \right)}}\left( \zeta \right),} \hfill & {\;\zeta \in \left[ {{\omega _0},\;d} \right),} \hfill \cr {{z^{\left( 2 \right)}}\left( \zeta \right),} \hfill & {\;\zeta \in \left( {d,\;\infty } \right).} \hfill \cr } } \right.

Let $ψ (ζ, λ) = \begin{array}{l} ψ^{(1)} (ζ, λ), & ζ \in ω_{0}, d), \\ ψ^{(2)} (ζ, λ), & ζ \in (d, \infty), \end{array}$ \psi \left( {\zeta ,\;\lambda } \right) = \left\{ {\matrix{ {{\psi ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right),} \hfill & {\;\zeta \in \left[ {{\omega _0},\;d} \right),} \hfill \cr {{\psi ^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right),\;} \hfill & {\zeta \in \left( {d,\;\infty } \right),} \hfill \cr } } \right. and $θ (ζ, λ) = \begin{array}{l} θ^{(1)} (ζ, λ), & ζ \in ω_{0}, d), \\ θ^{(2)} (ζ, λ), & ζ \in (d, \infty), \end{array}$ \theta \left( {\zeta ,\;\lambda } \right) = \left\{ {\matrix{ {{\theta ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right),} \hfill & {\;\zeta \in \left[ {{\omega _0},\;d} \right),} \hfill \cr {{\theta ^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right),} \hfill & {\;\zeta \in \left( {d,\;\infty } \right),} \hfill \cr } } \right. be solutions of Eq. (3.1) satisfying the following conditions $\begin{array}{l} ψ^{(1)} (ω_{0}, λ) = \cos β, D_{- \frac{ω}{q}, \frac{1}{q}} ψ^{(1)} (ω_{0}, λ) = \sin β, \\ θ^{(1)} (ω_{0}, λ) = \sin β, D_{- \frac{ω}{q}, \frac{1}{q}} θ^{(1)} (ω_{0}, λ) = - \cos β, \end{array}$ \eqalign{ & {\psi ^{\left( 1 \right)}}\left( {{\omega _0},\;\lambda } \right) = \;\cos \;\beta ,\;{D_{ - {\omega \over q},{1 \over q}}}{\psi ^{\left( 1 \right)}}\left( {{\omega _0},\;\lambda } \right) = \;\sin \;\beta , \cr & {\theta ^{\left( 1 \right)}}\left( {{\omega _0},\;\lambda } \right) = \;\sin \;\beta ,\;{D_{ - {\omega \over q},{1 \over q}}}{\theta ^{\left( 1 \right)}}\left( {{\omega _0},\;\lambda } \right) = - \;\cos \;\beta , \cr} and $\begin{array}{l} θ (d -, λ) & = & η θ (d +, λ), \\ D_{- \frac{ω}{q}, \frac{1}{q}} θ (d -, λ) & = & \frac{1}{η} D_{- \frac{ω}{q}, \frac{1}{q}} θ (d +, λ), \\ ψ (d -, λ) & = & η ψ (d +, λ), \\ D_{- \frac{ω}{q}, \frac{1}{q}} ψ (d -, λ) & = & \frac{1}{η} D_{- \frac{ω}{q}, \frac{1}{q}} ψ (d +, λ) . \end{array}$ \matrix{ {\theta \left( {d - ,\;\lambda } \right)} \hfill & = \hfill & {\eta \theta \left( {d + ,\;\lambda } \right),} \hfill \cr {{D_{ - {\omega \over q},{1 \over q}}}\theta \left( {d - ,\;\lambda } \right)} \hfill & = \hfill & {{1 \over \eta }{D_{ - {\omega \over q},{1 \over q}}}\theta \left( {d + ,\;\lambda } \right),} \hfill \cr {\psi \left( {d - ,\;\lambda } \right)} \hfill & = \hfill & {\eta \psi \left( {d + ,\;\lambda } \right),} \hfill \cr {{D_{ - {\omega \over q},{1 \over q}}}\psi \left( {d - ,\;\lambda } \right)} \hfill & = \hfill & {{1 \over \eta }{D_{ - {\omega \over q},{1 \over q}}}\psi \left( {d + ,\;\lambda } \right).} \hfill \cr }

Then the solution of Eq. (3.1) be represented $ψ (ζ, λ) + 𝓁 (λ, \frac{1}{q^{n}}) θ (ζ, λ)$ \psi \left( {\zeta ,\;\lambda } \right) + {\ell} \left( {\lambda ,\;{1 \over {{q^n}}}} \right)\theta \left( {\zeta ,\;\lambda } \right) which satisfies the boundary condition $\begin{array}{l} (D_{- \frac{ω}{q}, \frac{1}{q}} ψ^{(2)} (\frac{1}{q^{n}}, λ) + 𝓁 (λ, \frac{1}{q^{n}}) D_{- \frac{ω}{q}, \frac{1}{q}} θ^{(2)} (\frac{1}{q^{n}}, λ)) \sin γ \\ + (ψ^{(2)} (\frac{1}{q^{n}}, λ) + 𝓁 (λ, \frac{1}{q^{n}}) θ^{(2)} (\frac{1}{q^{n}}, λ)) \cos γ = 0. \end{array}$ \matrix{ {\left( {{D_{ - {\omega \over q},{1 \over q}}}{\psi ^{\left( 2 \right)}}\left( {{1 \over {{q^n}}},\;\lambda } \right)\; + \;{\ell}\left( {\lambda ,\;{1 \over {{q^n}}}} \right){D_{ - {\omega \over q},{1 \over q}}}{\theta ^{\left( 2 \right)}}\left( {{1 \over {{q^n}}},\;\lambda } \right)} \right)\;\sin \;\gamma } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \;\left( {{\psi ^{\left( 2 \right)}}\left( {{1 \over {{q^n}}},\;\lambda } \right) + {\ell} \left( {\lambda ,\;{1 \over {{q^n}}}} \right){\theta ^{\left( 2 \right)}}\left( {{1 \over {{q^n}}},\;\lambda } \right)} \right)\;\cos \;\gamma = 0.} \hfill \cr } Hence $𝓁 (λ, \frac{1}{q^{n}}) = - \frac{ψ^{(2)} (\frac{1}{q^{n}}, λ) cot γ + D_{- \frac{ω}{q}, \frac{1}{q}} ψ^{(2)} (\frac{1}{q^{n}}, λ)}{θ (2) (\frac{1}{q^{n}}, λ) cot γ + D_{- \frac{ω}{q}, \frac{1}{q}} θ ψ^{(2)} (\frac{1}{q^{n}}, λ)} .$ {\ell} \left( {\lambda ,\;{1 \over {{q^n}}}} \right) = - {{{\psi ^{\left( 2 \right)}}\left( {{1 \over {{q^n}}},\lambda } \right){\rm{\;cot\;}}\gamma + {D_{ - {\omega \over q},{1 \over q}}}{\psi ^{\left( 2 \right)}}\left( {{1 \over {{q^n}}},\lambda } \right)} \over {\theta \left( 2 \right)\left( {{1 \over {{q^n}}},\lambda } \right){\rm{\;cot\;}}\gamma + {D_{ - {\omega \over q},{1 \over q}}}\theta {\psi ^{\left( 2 \right)}}\left( {{1 \over {{q^n}}},\lambda } \right)}}.

Lemma 3.1.

Let $Z_{\frac{1}{q^{n}}} (ζ, λ) = ψ (ζ, λ) + 𝓁 (λ, \frac{1}{q^{n}}) θ (ζ, λ),$ {Z_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\lambda } \right) = \psi \left( {\zeta ,\;\lambda } \right) + {\ell} \left( {\lambda ,\;{1 \over {{q^n}}}} \right)\theta \left( {\zeta ,\;\lambda } \right), where $Z_{\frac{1}{q^{n}}} \in H_{n}$ {Z_{{1 \over {{q^n}}}}} \in {H_n} and $\frac{1}{q^{n}} > d$ {1 \over {{q^n}}} > d , n ∈ ℕ. Then, for each nonreal λ, the following relations hold: $\begin{array}{l} Z_{\frac{1}{q^{n}}} (ζ, λ) \to Z (ζ, λ), n \to \infty, \\ \int_{ω_{0}}^{d} {Z_{\frac{1}{q^{n}}} (ζ, λ)|}^{2} d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} {Z_{\frac{1}{q^{n}}} (ζ, λ)|}^{2} d_{ω, q} ζ \\ \to \int_{ω_{0}}^{d} {Z (ζ, λ)|}^{2} d_{ω, q} ζ + \int_{d}^{\infty} {(ζ, λ)|}^{2} d_{ω, q} ζ, n \to \infty . \end{array}$ \matrix{ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{Z_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\lambda } \right) \to Z\left( {\zeta ,\;\lambda } \right)\;,\;n \to \infty ,} \hfill \cr {\int_{{\omega _0}}^d {{{\left| {{Z_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } + \int_d^{{1 \over {{q^n}}}} {{{\left| {{Z_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\; \to \int_{{\omega _0}}^d {{{\left| {Z\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } + \int_d^\infty {{{\left| {\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } ,\;\;\;\;n \to \infty .} \hfill \cr }

Proof

It is immediate that $Z_{\frac{1}{q^{n}}} (ζ, λ) = Z (ζ, λ) + 𝓁 (λ, \frac{1}{q^{n}}) - m (λ)] θ (ζ, λ)$ {Z_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\lambda } \right) = Z\left( {\zeta ,\;\lambda } \right) + \left[ {\ell \left( {\lambda ,\;{1 \over {{q^n}}}} \right) - m\left( \lambda \right)} \right]\theta \left( {\zeta ,\;\lambda } \right) where Z(·, λ) ∈ H and m(λ) is the Titchmarsh–Weyl function. $𝓁 (λ, \frac{1}{q^{n}})$ \ell \left( {\lambda ,\;{1 \over {{q^n}}}} \right) varies on a circle with a finite radius $r_{\frac{1}{q^{n}}}$ {r_{{1 \over {{q^n}}}}} in the plane. In the limit-circle case, $𝓁 (λ, \frac{1}{q^{n}}) \to m (λ) (n \to \infty)$ \ell \left( {\lambda ,\;{1 \over {{q^n}}}} \right) \to m\left( \lambda \right)\left( {n \to \infty } \right) ; therefore $Z_{\frac{1}{q^{n}}} (ζ, λ) \to Z (ζ, λ) (n \to \infty) .$ {Z_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\lambda } \right) \to Z\left( {\zeta ,\;\lambda } \right)\;\;\;\;\left( {n \to \infty } \right).

Hence $\begin{array}{l} \int_{ω_{0}}^{d} {Z_{\frac{(1) 1}{q^{n}}} (ζ, λ)|}^{2} d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} {Z_{\frac{(2) 1}{q^{n}}} (ζ, λ)|}^{2} d_{ω, q} ζ \\ \to \int_{ω_{0}}^{d} {Z^{(1)} (ζ, λ)|}^{2} d_{ω, q} ζ + \int_{d}^{\infty} {Z^{(2)} (ζ, λ)|}^{2} d_{ω, q} ζ (n \to \infty), \end{array}$ \matrix{ {\int_{{\omega _0}}^d {{{\left| {{Z_{{{\left( 1 \right)1} \over {{q^n}}}}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } + \int_d^{{1 \over {{q^n}}}} {{{\left| {{Z_{{{\left( 2 \right)1} \over {{q^n}}}}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \to \int_{{\omega _0}}^d {{{\left| {{Z^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } + \int_d^\infty {{{\left| {{Z^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta \;\left( {n \to \infty } \right)} ,} \hfill \cr } due to Z(·, λ) ∈ H. In the limit-point case, we find $\begin{array}{l} 𝓁 (λ, \frac{1}{q^{n}}) - m (λ)| \leq r_{\frac{1}{q^{n}}} \\ = {(2 Im λ \int_{ω_{0}}^{d} {θ^{(1)} (ζ, λ)|}^{2} d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} {θ^{(2)} (ζ, λ)|}^{2} d_{ω, q} ζ])}^{- 1}, \end{array}$ \matrix{ {\left| {\ell \left( {\lambda ,\;{1 \over {{q^n}}}} \right) - m\left( \lambda \right)} \right| \le {r_{{1 \over {{q^n}}}}}} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {{\left( {2\;{\mathop{\rm Im}\nolimits} \;\;\lambda \left[ {\int_{{\omega _0}}^d {{{\left| {{\theta ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } + \int_d^{{1 \over {{q^n}}}} {{{\left| {{\theta ^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } } \right]} \right)}^{ - 1}},} \hfill \cr } where Im λ ≠ 0. As $r_{\frac{1}{q^{n}}} \to 0$ {r_{{1 \over {{q^n}}}}} \to 0 , $Z_{\frac{1}{q^{n}}} (ζ, λ) \to Z (ζ, λ) (n \to \infty)$ {Z_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\lambda } \right) \to Z\left( {\zeta ,\;\lambda } \right)\;\left( {n \to \infty } \right) . Moreover, we have $\begin{array}{l} \int_{ω_{0}}^{d} {| {ℓ (λ, \frac{1}{q^{n}}) - m (λ)} θ^{(1)} (ζ, λ) |}^{2} d_{ω, q} ζ \\ + \int_{d}^{\frac{1}{q^{n}}} {| {𝓁 (λ, \frac{1}{q^{n}}) - m (λ)} θ^{(2)} (ζ, λ) |}^{2} d_{ω, q} ζ \\ = {| 𝓁 (λ, \frac{1}{q^{n}}) - m (λ) |}^{2} (\int_{ω_{0}}^{d} {| θ^{(1)} (ζ, λ) |}^{2} d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} {| θ^{(2)} (ζ, λ) |}^{2} d_{ω, q} ζ) \\ \leq {(4 {(Im λ)}^{2} [\int_{ω_{0}}^{d} {| θ^{(1)} (ζ, λ) |}^{2} d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} {| θ^{(2)} (ζ, λ) |}^{2} d_{ω, q} ζ])}^{- 1}, \end{array}$ \matrix{ {\int_{{\omega _0}}^d {{{\left| {\left\{ {\ell \left( {\lambda ,\;{1 \over {{q^n}}}} \right) - m\left( \lambda \right)} \right\}{\theta ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \;\int_d^{{1 \over {{q^n}}}} {{{\left| {\left\{ {\ell\left( {\lambda ,\;{1 \over {{q^n}}}} \right) - m\left( \lambda \right)} \right\}{\theta ^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } } \hfill \cr {\;\; = \;{{\left| {\ell \left( {\lambda ,\;{1 \over {{q^n}}}} \right) - m\left( \lambda \right)} \right|}^2}\left( {\int_{{\omega _0}}^d {{{\left| {{\theta ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } + \int_d^{{1 \over {{q^n}}}} {{{\left| {{\theta ^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } } \right)} \hfill \cr {\;\;\;\;\;\;\;\;\;\; \le {{\left( {4{{({\mathop{\rm Im}\nolimits} \;\;\lambda )}^2}\left[ {\int_{{\omega _0}}^d {{{\left| {{\theta ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } + \int_d^{{1 \over {{q^n}}}} {{{\left| {{\theta ^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right)} \right|}^2}{d_{\omega ,q}}\zeta } } \right]} \right)}^{ - 1}},} \hfill \cr } which implies that $\begin{array}{l} \int_{ω_{0}}^{d} {| Z_{\frac{(1) 1}{q^{n}}} (ζ, λ) |}^{2} d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} {| Z_{\frac{(2) 1}{q^{n}}} (ζ, λ) |}^{2} d_{ω, q} ζ \\ \to \int_{ω_{0}}^{d} {| Z^{(1)} (ζ, λ) |}^{2} d_{ω, q} ζ + \int_{d}^{\infty} {| Z^{(2)} (ζ, λ) |}^{2} d_{ω, q} ζ . \end{array}$ \begin{array}{*{20}c} {\int_{\omega _0 }^d {\left| {Z_{_{\frac{1} {{q^n }}} }^{\left( 1 \right)} \left( {\zeta ,\lambda } \right)} \right|^2 d_{\omega ,q} \zeta } + \int_d^{\frac{1} {{q^n }}} {\left| {Z_{\frac{1} {{q^n }}}^{\left( 2 \right)} \left( {\zeta ,\lambda } \right)} \right|^2 d_{\omega ,q} \zeta } } \hfill \\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \to \int_{\omega _0 }^d {\left| {Z^{\left( 1 \right)} \left( {\zeta ,\lambda } \right)} \right|^2 d_{\omega ,q} \zeta } + \int_d^\infty {\left| {Z^{\left( 2 \right)} \left( {\zeta ,\lambda } \right)} \right|^2 d_{\omega ,q} \zeta } .} \hfill \\ \end{array}

Let $f \in H_{n} (\frac{1}{q^{n}} > d, n \in ℕ)$ f \in H_n \left( {\frac{1}{{q^n }} > d,n \in {\mathbb N}} \right) . Define (3.6) $\begin{array}{l} G_{\frac{1}{q^{n}}} (ζ, ς, λ) = \begin{array}{l} Z_{\frac{1}{q^{n}}} (ζ, λ) θ (ζ, λ), & ς \leq ζ, \\ θ (ζ, λ) Z_{\frac{1}{q^{n}}} (ς, λ), & ς > ζ, \end{array} \\ \begin{array}{l} (R_{\frac{1}{q^{n}}} f) (ζ, λ) = \int_{ω_{0}}^{d} G_{\frac{1}{q^{n}}} (ζ, ς, λ) f^{(1)} (ς) d_{ω, q} ς \\ + \int_{d}^{\frac{1}{q^{n}}} G_{\frac{1}{q^{n}}} (ζ, ς, λ) f^{(2)} (ς) d_{ω, q} ς, λ \in ℂ . \end{array} \end{array}$ \eqalign{ & {G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma ,\;\lambda } \right) = \left\{ {\matrix{ {{Z_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\lambda } \right)\theta \left( {\zeta ,\;\lambda } \right),\;} \hfill & {\varsigma \le \zeta ,} \hfill \cr {\theta \left( {\zeta ,\;\lambda } \right){Z_{{1 \over {{q^n}}}}}\left( {\varsigma ,\;\lambda } \right),\;} \hfill & {\varsigma > \zeta ,} \hfill \cr } } \right. \cr & \matrix{ {\left( {{R_{{1 \over {{q^n}}}}}f} \right)\left( {\zeta ,\;\lambda } \right) = \int_{{\omega _0}}^d {{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma ,\;\lambda } \right){f^{\left( 1 \right)}}\left( \varsigma \right){d_{\omega ,q}}\varsigma } } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \int_d^{{1 \over {{q^n}}}} {{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma ,\;\lambda } \right){f^{\left( 2 \right)}}\left( \varsigma \right){d_{\omega ,q}}\varsigma ,\;\lambda \in {\mathbb C}} .} \hfill \cr } \cr}

Without loss of generality, we can assume that λ = 0 is not an eigenvalue of the BVP (3.1)–(3.5). Now let us prove that the resolvent operator is compact.

Theorem 3.2.

$G_{\frac{1}{q^{n}}} (ζ, ς) (λ = 0) (\frac{1}{q^{n}} > d, n \in ℕ)$ {G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma } \right)\left( {\lambda = 0} \right)\;({1 \over {{q^n}}} > d,\;n \in {\mathbb N}) defined as (3.6) is a ω, q-Hilbert–Schmidt kernel, i.e., $\begin{array}{l} \int_{ω_{0}}^{d} \int_{ω_{0}}^{d} | G_{\frac{1}{q^{n}}} (ζ, ς) |^{2} d_{ω, q} ζ d_{ω, q} ς < + \infty, \\ \int_{d}^{\frac{1}{q^{n}}} \int_{d}^{\frac{1}{q^{n}}} | G_{\frac{1}{q^{n}}} (ζ, ς) |^{2} d_{ω, q} ζ d_{ω, q} ς < + \infty . \end{array}$ \eqalign{ & \mathop \smallint \nolimits_{{\omega _0}}^d \mathop \smallint \nolimits_{{\omega _0}}^d |{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma } \right){|^2}{d_{\omega ,q}}\zeta {d_{\omega ,q}}\varsigma< + \infty , \cr & \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} |{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma } \right){|^2}{d_{\omega ,q}}\zeta {d_{\omega ,q}}\varsigma< + \infty . \cr}

Proof

By (3.6), it is obvious that $\begin{array}{l} \int_{ω_{0}}^{d} d_{ω, q} ζ \int_{ω_{0}}^{d} | G_{\frac{1}{q^{n}}} (ζ, ς) |^{2} d_{ω, q} ς < + \infty, \\ \int_{d}^{\frac{1}{q^{n}}} d_{ω, q} ζ \int_{d}^{\frac{1}{q^{n}}} | G_{\frac{1}{q^{n}}} (ζ, ς) |^{2} d_{ω, q} ς < + \infty, \end{array}$ \eqalign{ & \mathop \smallint \nolimits_{{\omega _0}}^d {d_{\omega ,q}}\zeta \mathop \smallint \nolimits_{{\omega _0}}^d |{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma } \right){|^2}{d_{\omega ,q}}\varsigma< + \infty , \cr & \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {d_{\omega ,q}}\zeta \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} |{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma } \right){|^2}{d_{\omega ,q}}\varsigma< + \infty , \cr} due to $Z_{\frac{1}{q^{n}}}$ {Z_{{1 \over {{q^n}}}}} , (·, λ), θ (·, λ) ∈ H_n $(\frac{1}{q^{n}} > d, n \in ℕ)$ (\frac{1}{{q^n }} > d,n \in {\mathbb N}) . Hence (3.7) $\begin{array}{l} \int_{ω_{0}}^{d} \int_{ω_{0}}^{d} | G_{\frac{1}{q^{n}}} (ζ, ς) |^{2} d_{ω, q} ζ d_{ω, q} ς < + \infty, \\ \int_{d}^{\frac{1}{q^{n}}} \int_{d}^{\frac{1}{q^{n}}} | G_{\frac{1}{q^{n}}} (ζ, ς) |^{2} d_{ω, q} ζ d_{ω, q} ς < + \infty . \end{array}$ \eqalign{ & \mathop \smallint \nolimits_{{\omega _0}}^d \mathop \smallint \nolimits_{{\omega _0}}^d |{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma } \right){|^2}{d_{\omega ,q}}\zeta {d_\omega }_{,q}\varsigma< + \infty , \cr & \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} |{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma } \right){|^2}{d_{\omega ,q}}\zeta {d_{\omega ,q}}\varsigma< + \infty . \cr}

Theorem 3.3 ([20]).

Let A {t_i} = {x_i}, i ∈ ℕ, where (3.8) $x_{i} = \sum_{k = 1}^{\infty} η_{i k} t_{k}, i, k \in ℕ .$ {x_i} = \mathop \sum \nolimits_{k = 1}^\infty {\eta _{ik}}{t_k},\;\,\,i,\;k \in {\rm{\mathbb N}}.

If (3.9) $\sum_{i, k = 1}^{\infty} | η_{i k} |^{2} < + \infty,$ \sum\limits_{i,k = 1}^\infty {|{\eta _{ik}}{|^2}< + \infty } , then the operator A is compact in l².

Theorem 3.4.

Let 𝒯 be the ω, q-integral operator 𝒯: H_n → H_n ( $\frac{1}{q^{n}} > d, n \in ℕ$ \frac{1}{{q^n }} > d,\,n \in {\mathbb N} ), $(T f) (ζ) = \begin{array}{l} \int_{ω_{0}}^{d} G_{\frac{1}{q^{n}}} (ζ, ς) f^{(1)} (ς) d_{ω, q} ς, ζ \in ω_{0}, d), \\ \int_{d}^{\frac{1}{q^{n}}} G_{\frac{1}{q^{n}}} (ζ, ς) f^{(2)} (ς) d_{ω, q} ς, ζ \in d, \frac{1}{q^{n}}], \end{array}$ \left( {{\cal T}f} \right)\left( \zeta \right) = \left\{ {\matrix{ {\mathop \smallint \nolimits_{{\omega _0}}^d {G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma } \right){f^{\left( 1 \right)}}\left( \varsigma \right){d_{\omega ,q}}\varsigma ,\;\zeta \in \left[ {{\omega _0},\;d} \right)\;,} \hfill \cr {\mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma } \right){f^{\left( 2 \right)}}\left( \varsigma \right){d_{\omega ,q}}\varsigma ,\;\zeta \in \left( {d,\;{1 \over {{q^n}}}} \right],} \hfill \cr } } \right. where $f (ζ) = \begin{array}{l} f^{(1)} (ζ), ζ \in ω_{0}, d), \\ f^{(2)} (ζ), ζ \in d, \frac{1}{q^{n}}] . \end{array}$ f\left( \zeta \right) = \left\{ {\matrix{ {{f^{\left( 1 \right)}}\left( \zeta \right)\;,\;\zeta \in \left[ {{\omega _0},\;d} \right)\;,} \hfill \cr {{f^{\left( 2 \right)}}\left( \zeta \right)\;,\;\zeta \in \left( {d,\;{1 \over {{q^n}}}} \right].} \hfill \cr } } \right.

Then 𝒯 is a compact self-adjoint operator in space H_n.

Proof

Let $φ_{i} : = φ_{i} (ζ) = \begin{array}{l} φ_{i}^{(1)} (ζ), & ζ \in ω_{0}, d), \\ φ_{i}^{(2)} (ζ), & ζ \in d, \frac{1}{q^{n}}], \end{array} (i, n \in ℕ, \frac{1}{q^{n}} > d)$ \varphi _i : = \varphi _i \left( \zeta \right) = \left\{ {\begin{array}{*{20}c} {\varphi _i^{\left( 1 \right)} \left( \zeta \right),} \hfill & {\zeta \in \left[ {\omega _0 ,d} \right),} \hfill \\ {\varphi _i^{\left( 2 \right)} \left( \zeta \right),} \hfill & {\zeta \in \left( {d,\frac{1} {{q^n }}} \right],} \hfill \\ \end{array} } \right.\,\,\,\,\,(i,n \in {\mathbb N},\frac{1} {{q^n }} > d) be a complete, orthonormal basis of H_n. Let i, k, n ∈ ℕ, $\frac{1}{q^{n}} > d$ {1 \over {{q^n}}} > d . Write $\begin{array}{l} t_{i} = {〈 f, φ_{i} 〉}_{n} = \int_{ω_{0}}^{d} f^{(1)} (ζ) \bar{φ_{i}^{(1)} (ζ)} d_{ω, q} ζ \\ + \int_{d}^{\frac{1}{q^{n}}} f^{(2)} (ζ) \bar{φ_{i}^{(2)} (ζ)} d_{ω, q} ζ, \\ x_{i} = {〈 g, φ_{i} 〉}_{n} = \int_{ω_{0}}^{d} g^{(1)} (ζ) \bar{φ_{i}^{(1)} (ζ)} d_{ω, q} ζ \\ + \int_{d}^{\frac{1}{q^{n}}} g^{(2)} (ζ) \bar{φ_{i}^{(2)} (ζ)} d_{ω, q} ζ, \\ η_{i k} = \int_{ω_{0}}^{d} \int_{ω_{0}}^{d} G_{\frac{1}{q^{n}}} (ζ, ς) φ_{i}^{(1)} (ζ) \bar{φ_{k}^{(1)} (ς)} d_{ω, q} ζ d_{ω, q} ς \\ + \int_{d}^{\frac{1}{q^{n}}} \int_{d}^{\frac{1}{q^{n}}} G_{\frac{1}{q^{n}}} (ζ, ς) φ_{i}^{(2)} (ζ) \bar{φ_{k}^{(2)} (ς)} d_{ω, q} ζ d_{ω, q} ς . \end{array}$ \begin{gathered} t_i = \left\langle {f,\varphi _i } \right\rangle _n = \mathop \smallint \nolimits_{\omega _0 }^d f^{\left( 1 \right)} \left( \zeta \right)\overline {\varphi _i^{\left( 1 \right)} \left( \zeta \right)} d_{\omega ,q} \zeta \hfill \\ \,\,\,\,\,\,\,\, + \mathop \smallint \nolimits_d^{\frac{1} {{q^n }}} f^{\left( 2 \right)} \left( \zeta \right)\overline {\varphi _i^{\left( 2 \right)} \left( \zeta \right)} d_{\omega ,q} \zeta , \hfill \\ x_i = \langle g,\varphi _i \rangle _n = \mathop \smallint \nolimits_{\omega _0 }^d g^{\left( 1 \right)} \left( \zeta \right)\overline {\varphi _i^{\left( 1 \right)} \left( \zeta \right)} d_{\omega ,q} \zeta \hfill \\ \,\,\,\,\,\,\, + \mathop \smallint \nolimits_d^{\frac{1} {{q^n }}} g^{\left( 2 \right)} \left( \zeta \right)\overline {\varphi _i^{\left( 2 \right)} \left( \zeta \right)} d_{\omega ,q} \zeta , \hfill \\ \eta _{ik} = \mathop \smallint \nolimits_{\omega _0 }^d \mathop \smallint \nolimits_{\omega _0 }^d G_{\frac{1} {{q^n }}} \left( {\zeta ,\varsigma } \right)\varphi _i^{\left( 1 \right)} \left( \zeta \right)\overline {\varphi _k^{\left( 1 \right)} \left( \varsigma \right)} d_{\omega ,q} \zeta d_{\omega ,q} \varsigma \hfill \\ \,\,\,\,\,\,\, + \mathop \smallint \nolimits_d^{\frac{1} {{q^n }}} \mathop \smallint \nolimits_d^{\frac{1} {{q^n }}} G_{\frac{1} {{q^n }}} \left( {\zeta ,\varsigma } \right)\varphi _i^{\left( 2 \right)} \left( \zeta \right)\overline {\varphi _k^{\left( 2 \right)} \left( \varsigma \right)} d_{\omega ,q} \zeta d_{\omega ,q} \varsigma . \hfill \\ \end{gathered} H_n is mapped isometrically on to l². By this mapping, 𝒯 transforms into the operator A defined by (3.8) in l² and (3.7) is translated into (3.9). By Theorems 3.2 and 3.3, we see that A and 𝒯 are compact operators.

Let h, g ∈ H_n and $\frac{1}{q^{n}} > d$ {1 \over {{q^n}}} > d , n ∈ ℕ. Then we have $\begin{array}{l} {〈 𝒯 h, g 〉}_{n} & = \int_{d}^{ω_{0}} (𝒯 h^{(1)}) (ζ) \bar{g^{(1)} (ζ}) d_{ω, q} ζ + \int_{\frac{1}{q^{n}}}^{d} (𝒯 h^{(2)}) (ζ) \bar{g^{(2)} (ζ}) d_{ω, q} ζ \\ = \int_{d}^{ω_{0}} \int_{d}^{ω_{0}} G_{\frac{1}{q^{n}}} (ζ, ς) h^{(1)} (ς) d_{ω, q} ς \bar{g^{(1)} (ζ)} d_{ω, q} ζ \\ + \int_{\frac{1}{q^{n}}}^{d} \int_{\frac{1}{q^{n}}}^{d} G_{\frac{1}{q^{n}}} (ζ, ς) h^{(2)} (ς) d_{ω, q} ς \bar{g^{(2)} (ζ)} d_{ω, q} ζ \\ = \int_{d}^{ω_{0}} h^{(1)} (ς) (\int_{d}^{ω_{0}} G_{\frac{1}{q^{n}}} (ς, ζ) \bar{g^{(1)} (ζ)} d_{ω, q} ζ) d_{ω, q} ς \\ + \int_{\frac{1}{q^{n}}}^{d} h^{(2)} (ς) (\int_{\frac{1}{q^{n}}}^{d} G_{\frac{1}{q^{n}}} (ς, ζ) \bar{g^{(2)} (ζ)} d_{ω, q} ζ) d_{ω, q} ς \\ = {〈 h, T g 〉}_{n}, \end{array}$ \eqalign{ & \matrix{ {{{\left\langle {{\cal T}h,g} \right\rangle }_n}} \hfill & { = \mathop \smallint \nolimits_{{\omega _0}}^d ({\cal T}{h^{\left( 1 \right)}})\left( \zeta \right)\overline {{g^{\left( 1 \right)}}(\zeta } ){d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} ({\cal T}{h^{\left( 2 \right)}})\left( \zeta \right)\overline {{g^{\left( 2 \right)}}(\zeta } ){d_{\omega ,q}}\zeta } \hfill \cr {} \hfill & { = \mathop \smallint \nolimits_{{\omega _0}}^d \mathop \smallint \nolimits_{{\omega _0}}^d {G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma } \right){h^{\left( 1 \right)}}\left( \varsigma \right){d_{\omega ,q}}\varsigma \overline {{g^{\left( 1 \right)}}\left( \zeta \right)} {d_{\omega ,q}}\zeta } \hfill \cr {} \hfill & {\,\,\, + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma } \right){h^{\left( 2 \right)}}\left( \varsigma \right){d_{\omega ,q}}\varsigma \overline {{g^{\left( 2 \right)}}\left( \zeta \right)} {d_{\omega ,q}}\zeta } \hfill \cr {} \hfill & { = \mathop \smallint \nolimits_{{\omega _0}}^d {h^{\left( 1 \right)}}\left( \varsigma \right)\left( {\mathop \smallint \nolimits_{{\omega _0}}^d {G_{{1 \over {{q^n}}}}}\left( {\varsigma ,\;\zeta } \right)\overline {{g^{\left( 1 \right)}}\left( \zeta \right)} {d_{\omega ,q}}\zeta } \right){d_{\omega ,q}}\varsigma } \hfill \cr {} \hfill & {\,\,\, + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {h^{\left( 2 \right)}}\left( \varsigma \right)\left( {\mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {G_{{1 \over {{q^n}}}}}\left( {\varsigma ,\;\zeta } \right)\overline {{g^{\left( 2 \right)}}\left( \zeta \right)} {d_{\omega ,q}}\zeta } \right){d_{\omega ,q}}\varsigma } \hfill \cr {} \hfill & { = {{\left\langle {h,\;Tg} \right\rangle }_n},} \hfill \cr } \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \cr & \,\,\,\,\,\,\,\,\,\,\,\, \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \cr & \,\,\,\,\,\,\,\,\,\, \cr} since $G_{\frac{1}{q^{n}}} (ζ, γ)$ {G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\gamma } \right) is a symmetric function.

From Theorem 3.4, we conclude that 𝒯 has a discrete spectrum. Let $λ_{m, \frac{1}{q^{n}}}$ {\lambda _{m,{1 \over {{q^n}}}}} and $θ_{m, \frac{1}{q^{n}}} (ζ) : = \begin{array}{l} θ_{m, \frac{1}{q^{n}}}^{(1)} (ζ, λ_{m, \frac{1}{q^{n}}}), ζ \in ω_{0}, d), \\ θ_{m, \frac{1}{q^{n}}}^{(2)} (ζ, λ_{m, \frac{1}{q^{n}}}), ζ \in d, \frac{1}{q^{n}}], \end{array} (m, n \in ℕ, \frac{1}{q^{n}} > d)$ \theta _{m,\frac{1} {{q^n }}} \left( \zeta \right): = \left\{ {\begin{array}{*{20}c} {\theta _{m,\frac{1} {{q^n }}}^{\left( 1 \right)} \left( {\zeta ,\lambda _{m,\frac{1} {{q^n }}} } \right),\,\,\,\,\,\zeta \in \left[ {\omega _0 ,d} \right),} \hfill \\ {\theta _{m,\frac{1} {{q^n }}}^{\left( 2 \right)} \left( {\zeta ,\lambda _{m,\frac{1} {{q^n }}} } \right),\,\,\,\,\zeta \in \left( {d,\frac{1} {{q^n }}} \right],} \hfill \\ \end{array} } \right.\,\,\,\,\,\,\,(m,n \in \mathbb{N},\,\,\frac{1} {{q^n }} > d) be the eigenvalues and eigenfunctions of the BVP (3.1)–(3.5) and $α_{m, \frac{1}{q^{n}}}^{2} = \int_{ω_{0}}^{d} θ_{m, \frac{1}{q^{n}}}^{(1) 2} (ζ) d_{ω, q} ς + \int_{d}^{\frac{1}{q^{n}}} θ_{m, \frac{1}{q^{n}}}^{(2) 2} (ζ) d_{ω, q} ς .$ \alpha _{m,{1 \over {{q^n}}}}^2 = \mathop \smallint \nolimits_{{\omega _0}}^d \theta _{m,{1 \over {{q^n}}}}^{\left( 1 \right)2}\left( \zeta \right){d_{\omega ,q}}\varsigma + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} \theta _{m,{1 \over {{q^n}}}}^{\left( 2 \right)2}\left( \zeta \right){d_{\omega ,q}}\varsigma .

By Theorem 3.4 and the Hilbert–Schmidt theorem, we infer that (3.10) $\begin{array}{l} \int_{ω_{0}}^{d} | f^{(1)} (ζ) |^{2} d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} | f^{(2)} (ζ) |^{2} d_{ω, q} ζ \\ = \sum_{m = 1}^{\infty} \frac{1}{α_{m, \frac{1}{q^{n}}}^{2}} {\int_{ω_{0}}^{d} f^{(1)} (ζ) ϕ_{m, \frac{1}{q^{n}}}^{(1)} (ζ) d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} f^{(2)} (ζ) ϕ_{m, \frac{1}{q^{n}}}^{(2)} (ζ) d_{ω, q} ζ|}^{2} . \end{array}$ \eqalign{ & \mathop \smallint \nolimits_{{\omega _0}}^d |{f^{\left( 1 \right)}}\left( \zeta \right){|^2}{d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} |{f^{\left( 2 \right)}}\left( \zeta \right){|^2}{d_{\omega ,q}}\zeta \cr & = \mathop \sum \nolimits_{m = 1}^\infty {1 \over {\alpha _{m,{1 \over {{q^n}}}}^2}}{\left| {\mathop \smallint \nolimits_{{\omega _0}}^d {f^{\left( 1 \right)}}\left( \zeta \right)\phi _{m,{1 \over {{q^n}}}}^{\left( 1 \right)}\left( \zeta \right){d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {f^{\left( 2 \right)}}\left( \zeta \right)\phi _{m,{1 \over {{q^n}}}}^{\left( 2 \right)}\left( \zeta \right){d_{\omega ,q}}\zeta } \right|^2}. \cr}

Define $ϱ_{\frac{1}{q^{n}}} (λ) = \begin{array}{l} - \sum_{λ < λ_{m, \frac{1}{n}} < 0} \frac{1}{α_{m, \frac{1}{q^{n}}}^{2}}, & for λ \leq 0, \\ \sum_{0 \leq λ_{m, \frac{1}{q^{n}}} < λ} \frac{1}{α_{m, \frac{1}{q^{n}}}^{2}}, & for λ > 0. \end{array}$ {\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right) = \left\{ {\matrix{ { - \sum\limits_{\lambda< {\lambda _{m,{1 \over n}}}< 0} {{1 \over {\alpha _{m,{1 \over {{q^n}}}}^2}}\;,} } \hfill & {{\rm{for}}\;\lambda \le 0,} \hfill \cr {\sum\limits_{0 \le {\lambda _{m,{1 \over {{q^n}}}}}< \lambda } {{1 \over {\alpha _{m,{1 \over {{q^n}}}}^2}},} } \hfill & {{\rm{for}}\;\lambda > 0.} \hfill \cr } } \right. Then, (3.10) can be written as (3.11) $\int_{ω_{0}}^{d} {f^{(1)} (ζ)|}^{2} d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} {f^{(2)} (ζ)|}^{2} d_{ω, q} ζ = \int_{- \infty}^{\infty} | F (λ) |^{2} d ϱ_{\frac{1}{q^{n}}} (λ),$ \mathop \smallint \nolimits_{{\omega _0}}^d {\left| {{f^{\left( 1 \right)}}\left( \zeta \right)} \right|^2}{d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {\left| {{f^{\left( 2 \right)}}\left( \zeta \right)} \right|^2}{d_{\omega ,q}}\zeta = \mathop \smallint \nolimits_{ - \infty }^\infty |F\left( \lambda \right){|^2}d{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right), where $F (λ) = \int_{ω_{0}}^{d} f^{(1)} (ζ) ϕ_{m, \frac{1}{q^{n}}}^{(1)} (ζ) d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} f^{(2)} (ζ) ϕ_{m, \frac{1}{q^{n}}}^{(2)} (ζ) d_{ω, q} ζ .$ F\left( \lambda \right) = \mathop \smallint \nolimits_{{\omega _0}}^d {f^{\left( 1 \right)}}\left( \zeta \right)\phi _{m,{1 \over {{q^n}}}}^{\left( 1 \right)}\left( \zeta \right){d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {f^{\left( 2 \right)}}\left( \zeta \right)\phi _{m,{1 \over {{q^n}}}}^{\left( 2 \right)}\left( \zeta \right){d_{\omega ,q}}\zeta .

Lemma 3.5.

For any positive S, there is a positive number B = B (S) not depending on n so that $- ∨_{- S}^{S} ϱ_{\frac{1}{q^{n}}} (λ) = \sum_{- S \leq λ_{m, \frac{1}{q^{n}}} < S} \frac{1}{α_{m, \frac{1}{q^{n}}}^{2}} = ϱ_{\frac{1}{q^{n}}} (S) - ϱ_{\frac{1}{q^{n}}} (- S) < B .$ - \mathop \lor \limits_{{\text{ - }}S}^S \varrho _{\frac{1} {{q^n }}} \left( \lambda \right) = \sum\limits_{ - S \leqslant \lambda _{m,\frac{1} {{q^n }}}< S} {\frac{1} {{\alpha _{m,\frac{1} {{q^n }}}^2 }} = \varrho _{\frac{1} {{q^n }}} \left( S \right) - \varrho _{\frac{1} {{q^n }}} \left( { - S} \right)< B.}

Proof

Let sin β ≠ 0. Since θ(ζ, λ) is continuous in domain −S ≤ λ ≤ S, $ω_{0}, d) \cup (d, \frac{1}{q^{n}}]$ \left[ {{\omega _0},\;d} \right) \cup (d,\;{1 \over {{q^n}}}] , and the condition θ⁽¹⁾(ω₀, λ) = sin β, there exists a positive number h such that for |λ| < S, (3.12) $\frac{1}{h^{2}} {(\int_{ω_{0}}^{ω_{0} + h} θ^{(1)} (ζ, λ) d_{ω, q} ζ)}^{2} > \frac{1}{2} {sin}^{2} β .$ {1 \over {{h^2}}}{\left( {\mathop \smallint \nolimits_{{\omega _0}}^{{\omega _0} + h} {\theta ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta } \right)^2} > {1 \over 2}{\rm{si}}{{\rm{n}}^2}\beta .

Let $f_{h} (ζ) = \begin{array}{l} \frac{1}{h}, & ω_{0} \leq ζ \leq ω_{0} + h, \\ 0, & ζ > ω_{0} + h . \end{array}$ {f_h}\left( \zeta \right) = \left\{ {\matrix{ {{1 \over h},} \hfill & {{\omega _0} \le \zeta \le {\omega _0} + h,} \hfill \cr {0,} \hfill & {\zeta > {\omega _0} + h.} \hfill \cr } } \right.

From (3.12), we find $\begin{array}{l} \int_{ω_{0}}^{ω_{0} + h} f_{h}^{2} (ζ) d_{ω, q} ζ & = \frac{1}{h} = \int_{- \infty}^{\infty} {(\frac{1}{h} \int_{ω_{0}}^{ω_{0} + h} θ^{(1)} (ζ, λ) d_{ω, q} ζ)}^{2} d ϱ_{\frac{1}{q^{n}}} (λ) \\ \geq \int_{- S}^{S} {(\frac{1}{h} \int_{ω_{0}}^{ω_{0} + h} θ^{(1)} (ζ, λ) d_{ω, q} ζ)}^{2} d ϱ_{α} (λ) \\ > \frac{1}{2} {sin}^{2} β ϱ_{\frac{1}{q^{n}}} (S) - ϱ_{\frac{1}{q^{n}}} (- S)\} . \end{array}$ \matrix{ {\mathop \smallint \nolimits_{{\omega _0}}^{{\omega _0} + h} f_h^2\left( \zeta \right){d_{\omega ,q}}\zeta } \hfill & { = {1 \over h} = \mathop \smallint \nolimits_{ - \infty }^\infty {{\left( {{1 \over h}\mathop \smallint \nolimits_{{\omega _0}}^{{\omega _0} + h} {\theta ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta } \right)}^2}d{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right)} \hfill \cr {} \hfill & { \ge \mathop \smallint \nolimits_{ - S}^S {{\left( {{1 \over h}\mathop \smallint \nolimits_{{\omega _0}}^{{\omega _0} + h} {\theta ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta } \right)}^2}d{\varrho _\alpha }\left( \lambda \right)} \hfill \cr {} \hfill & { > {1 \over 2}{\rm{si}}{{\rm{n}}^2}\beta \left\{ {{\varrho _{{1 \over {{q^n}}}}}\left( S \right) - {\varrho _{{1 \over {{q^n}}}}}\left( { - S} \right)} \right\}.} \hfill \cr }

If sin β = 0, then we define f_h(ζ) as $\begin{array}{l} f_{h} (ζ) = \begin{array}{l} \frac{1}{h^{2}}, & ω_{0} \leq ζ \leq ω_{0} + h, \\ 0, & ζ > ω_{0} + h . \end{array} \\ \begin{array}{l} \end{array} \end{array}$ \eqalign{ & {f_h}\left( \zeta \right) = \left\{ {\matrix{ {{1 \over {{h^2}}},} \hfill & {\;{\omega _0} \le \zeta \le {\omega _0} + h,} \hfill \cr {0,} \hfill & {\zeta > {\omega _0} + h.} \hfill \cr } } \right. \cr & \matrix{ {} \hfill \cr \; \hfill \cr } \cr}

This proves the lemma.

Now, we will give an expansion into a Fourier series of resolvent. By ω, q-integration by parts, we obtain $\begin{array}{l} \int_{d}^{ω_{0}} [\frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} y^{(1)} (ζ, λ) - v (ζ) y^{(1)} (ζ, λ)] θ_{m, \frac{1}{q^{n}}}^{(1)} (ζ) d_{ω, q} ζ \\ + \int_{\frac{1}{q^{n}}}^{d} [\frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} y^{(2)} (ζ, λ) - v (ζ) y^{(2)} (ζ, λ)] θ_{m, \frac{1}{q^{n}}}^{(2)} (ζ) d_{ω, q} ζ \\ = \int_{d}^{ω_{0}} [\frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} ϕ_{m, \frac{1}{q^{n}}}^{(1)} (ζ) - v (ζ) θ_{m, \frac{1}{q^{n}}}^{(1)} (ζ)] y^{(1)} (ζ, λ) d_{ω, q} ζ \\ + \int_{\frac{1}{q^{n}}}^{d} [\frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} ϕ_{m, \frac{1}{q^{n}}}^{(2)} (ζ) - v (ζ) θ_{m, \frac{1}{q^{n}}}^{(2)} (ζ)] y^{(2)} (ζ, λ) d_{ω, q} ζ \\ = - λ_{m, \frac{1}{q^{n}}} \int_{d}^{ω_{0}} y^{(1)} (ζ, λ) θ_{m, \frac{1}{q^{n}}}^{(1)} (ζ) d_{ω, q} ζ - λ_{m, \frac{1}{q^{n}}} \int_{\frac{1}{q^{n}}}^{d} y^{(2)} (ζ, λ) θ_{m, \frac{1}{q^{n}}}^{(2)} (ζ) d_{ω, q} ζ \\ = - λ_{m, \frac{1}{q^{n}}} φ_{m} (λ), \end{array}$ \eqalign{ & \mathop \smallint \nolimits_{{\omega _0}}^d \left[ {{1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}{y^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right) - v\left( \zeta \right){y^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right)} \right]\theta _{m,{1 \over {{q^n}}}}^{\left( 1 \right)}\left( \zeta \right){d_{\omega ,q}}\zeta \cr & \,\,\,\, + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} \left[ {{1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}{y^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right) - v\left( \zeta \right){y^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right)} \right]\theta _{m,{1 \over {{q^n}}}}^{\left( 2 \right)}\left( \zeta \right){d_{\omega ,q}}\zeta \cr & = \mathop \smallint \nolimits_{{\omega _0}}^d \left[ {{1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}\phi _{m,{1 \over {{q^n}}}}^{\left( 1 \right)}\left( \zeta \right) - v\left( \zeta \right)\theta _{m,{1 \over {{q^n}}}}^{\left( 1 \right)}\left( \zeta \right)} \right]{y^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta \cr & \,\,\,\, + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} \left[ {{1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}\phi _{m,{1 \over {{q^n}}}}^{\left( 2 \right)}\left( \zeta \right) - v\left( \zeta \right)\theta _{m,{1 \over {{q^n}}}}^{\left( 2 \right)}\left( \zeta \right)} \right]{y^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta \cr & = - {\lambda _{m,{1 \over {{q^n}}}}}\mathop \smallint \nolimits_{{\omega _0}}^d {y^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right)\theta _{m,{1 \over {{q^n}}}}^{\left( 1 \right)}\left( \zeta \right){d_{\omega ,q}}\zeta - {\lambda _{m,{1 \over {{q^n}}}}}\mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {y^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right)\theta _{m,{1 \over {{q^n}}}}^{\left( 2 \right)}\left( \zeta \right){d_{\omega ,q}}\zeta \cr & = - {\lambda _{m,{1 \over {{q^n}}}}}{\varphi _m}\left( \lambda \right), \cr} where m ∈ ℕ. Let $\begin{array}{l} y (ζ, λ) = \sum_{m = 1}^{\infty} φ_{m} (λ) ψ_{m, \frac{1}{q^{n}}} (ζ), \\ a_{m} = \int_{ω_{0}}^{d} f (ζ) ψ_{m, \frac{1}{q^{n}}}^{(1)} (ζ) d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} f (ζ) ψ_{m, \frac{1}{q^{n}}}^{(2)} (ζ) d_{ω, q} ζ, \end{array}$ \eqalign{ & y\left( {\zeta ,\;\lambda } \right) = \sum\limits_{m = 1}^\infty {{\varphi _m}\left( \lambda \right){\psi _{m,{1 \over {{q^n}}}}}\left( \zeta \right),} \cr & {a_m} = \mathop \smallint \nolimits_{{\omega _0}}^d f\left( \zeta \right)\psi _{m,{1 \over {{q^n}}}}^{\left( 1 \right)}\left( \zeta \right){d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} f\left( \zeta \right)\psi _{m,{1 \over {{q^n}}}}^{\left( 2 \right)}\left( \zeta \right){d_{\omega ,q}}\zeta , \cr} where m ∈ ℕ. Since y(ζ, λ) satisfies the equation $- \frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} y (ζ, λ) + (v (ζ) - λ) y (ζ, λ) = f (ζ),$ - {1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}y\left( {\zeta ,\;\lambda } \right) + \left( {v\left( \zeta \right) - \lambda } \right)y\left( {\zeta ,\;\lambda } \right) = f\left( \zeta \right), we find $\begin{array}{l} a_{m} = \int_{ω_{0}}^{d} - \frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} y^{(1)} (ζ, λ) + (v (ζ) - λ) y^{(1)} (ζ, λ)] θ_{m, \frac{1}{q^{n}}}^{(1)} (ζ) d_{ω, q} ζ \\ + \int_{d}^{\frac{1}{q^{n}}} - \frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} y^{(2)} (ζ, λ) + (v (ζ) - λ) y^{(2)} (ζ, λ)] θ_{m, \frac{1}{q^{n}}}^{(2)} (ζ) d_{ω, q} ζ \\ = λ_{m, \frac{1}{q^{n}}} φ_{m} (λ) - λ φ_{m} (λ), m, n \in ℕ, \frac{1}{q^{n}} > d . \end{array}$ \eqalign{ & a_m = \mathop \smallint \nolimits_{\omega _0 }^d \left[ { - \frac{1} {q}D_{ - \frac{\omega } {q},\frac{1} {q}} D_{\omega ,q} y^{\left( 1 \right)} \left( {\zeta ,\lambda } \right) + \left( {v\left( \zeta \right) - \lambda } \right)y^{\left( 1 \right)} \left( {\zeta ,\lambda } \right)} \right]\theta _{m,\frac{1} {{q^n }}}^{\left( 1 \right)} \left( \zeta \right)d_{\omega ,q} \zeta \cr & \,\,\,\,\,\,\,\,\, + \mathop \smallint \nolimits_d^{\frac{1} {{q^n }}} \left[ { - \frac{1} {q}D_{ - \frac{\omega } {q},\frac{1} {q}} D_{\omega ,q} y^{\left( 2 \right)} \left( {\zeta ,\lambda } \right) + \left( {v\left( \zeta \right) - \lambda } \right)y^{\left( 2 \right)} \left( {\zeta ,\lambda } \right)} \right]\theta _{m,\frac{1} {{q^n }}}^{\left( 2 \right)} \left( \zeta \right)d_{\omega ,q} \zeta \cr & \,\,\,\,\,\,\,\, = \lambda _{m,\frac{1} {{q^n }}} \varphi _m \left( \lambda \right) - \lambda \varphi _m \left( \lambda \right)\,\,{\text{, }}m,n \in {\mathbb N},\frac{1} {{q^n }} > d. \cr}

Thus, we get $φ_{m} (λ) = \frac{a_{m}}{λ_{m, \frac{1}{q^{n}}} - λ} (m, n \in ℕ, \frac{1}{q^{n}} > d),$ \varphi _m \left( \lambda \right) = \frac{{a_m }} {{\lambda _{m,\frac{1} {{q^n }}} - \lambda }}\,\,\,\,\,\,(m,n \in {\mathbb N},\frac{1} {{q^n }} > d), and $y (ζ, λ) = {G_{\frac{1}{q^{n}}} (ζ, \cdot, λ), \bar{f (\cdot)}〉}_{n} = \sum_{m = 1}^{\infty} \frac{a_{m} θ_{m, \frac{1}{q^{n}}} (ζ)}{λ_{m, \frac{1}{q^{n}}} - λ} .$ y\left( {\zeta ,\;\lambda } \right) = {\left\langle {{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\; \cdot ,\;\lambda } \right),\;\overline {f\left( \cdot \right)} } \right\rangle _n} = \sum\limits_{m = 1}^\infty {{{{a_m}{\theta _{m,{1 \over {{q^n}}}}}\left( \zeta \right)} \over {{\lambda _{m,{1 \over {{q^n}}}}} - \lambda }}.}

Hence (3.13) $\begin{array}{l} (R_{\frac{1}{q^{n}}} f) (ζ, z) = \sum_{m = 1}^{\infty} \frac{θ_{m, \frac{1}{q^{n}}} (ζ)}{α_{m, \frac{1}{q^{n}}}^{2} (λ_{m, \frac{1}{q^{n}}} - z)} {f (\cdot), θ_{m, \frac{1}{q^{n}}} (\cdot)〉}_{n} \\ = \int_{- \infty}^{\infty} \frac{θ (ζ, λ)}{λ - z} {f (\cdot), θ_{m, \frac{1}{q^{n}}} (\cdot)〉}_{n}\} d ϱ_{\frac{1}{q^{n}}} (λ) . \end{array}$ \eqalign{ & \left( {{R_{{1 \over {{q^n}}}}}f} \right)\left( {\zeta ,\;z} \right) = \sum\limits_{m = 1}^\infty {{{{\theta _{m,{1 \over {{q^n}}}}}\left( \zeta \right)} \over {\alpha _{m,{1 \over {{q^n}}}}^2\left( {{\lambda _{m,{1 \over {{q^n}}}}} - z} \right)}}{{\left\langle {f\left( \cdot \right),\;{\theta _{m,{1 \over {{q^n}}}}}\left( \cdot \right)} \right\rangle }_n}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop \smallint \nolimits_{ - \infty }^\infty {{\theta \left( {\zeta ,\lambda } \right)} \over {\lambda - z}}\left\{ {{{\left\langle {f\left( \cdot \right),\;{\theta _{m,{1 \over {{q^n}}}}}\left( \cdot \right)} \right\rangle }_n}} \right\}d{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right). \cr}

Lemma 3.6.

For each nonreal z and fixed ζ, the following relation holds (3.14) $\int_{- \infty}^{\infty} {\frac{θ (ζ, λ)}{z - λ}|}^{2} d ϱ_{\frac{1}{q^{n}}} (λ) < S .$ \mathop \smallint \nolimits_{ - \infty }^\infty {\left| {{{\theta \left( {\zeta ,\lambda } \right)} \over {z - \lambda }}} \right|^2}d{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right)< S.

Proof

Writing $f (ς) = \frac{θ_{m, \frac{1}{q^{n}}} (ς)}{α_{m, \frac{1}{q^{n}}}}$ f\left( \varsigma \right) = {{{\theta _{m,{1 \over {{q^n}}}}}\left( \varsigma \right)} \over {{\alpha _{m,{1 \over {{q^n}}}}}}} yields (3.15) $\frac{1}{α_{m, \frac{1}{q^{n}}}} {G_{\frac{1}{q^{n}}} (ζ, \cdot, λ), θ_{m, \frac{1}{q^{n}}} (\cdot)〉}_{n} = \frac{θ_{m, \frac{1}{q^{n}}} (ζ)}{α_{m, \frac{1}{q^{n}}} (λ_{m, \frac{1}{q^{n}}} - z)},$ {1 \over {{\alpha _{m,{1 \over {{q^n}}}}}}}{\left\langle {{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\; \cdot ,\;\lambda } \right),\;{\theta _{m,{1 \over {{q^n}}}}}\left( \cdot \right)} \right\rangle _n} = {{{\theta _{m,{1 \over {{q^n}}}}}\left( \zeta \right)} \over {{\alpha _{m,{1 \over {{q^n}}}}}\left( {{\lambda _{m,{1 \over {{q^n}}}}} - z} \right)}}, due to the eigenfunctions $θ_{m, \frac{1}{q_{n}}} (ζ)$ {\theta _{m,{1 \over {{q_n}}}}}(\zeta ) are orthogonal. Combining (3.15) and (3.10), we see that $\begin{array}{l} \int_{ω_{0}}^{d} {G_{\frac{1}{q^{n}}} (ζ, ς, z)|}^{2} d_{ω, q} ς + \int_{d}^{\frac{1}{q^{n}}} {G_{\frac{1}{q^{n}}} (ζ, ς, z)|}^{2} d_{ω, q} ς \\ = \sum_{m = 1}^{\infty} \frac{{θ_{m, \frac{1}{q^{n}}} (ζ)|}^{2}}{α_{m, \frac{1}{q^{n}}}^{2} {λ_{m, \frac{1}{q^{n}}} - z|}^{2}} = \int_{- \infty}^{\infty} {\frac{θ (ζ, λ)}{λ - z}|}^{2} d ϱ_{\frac{1}{q^{n}}} (λ) . \end{array}$ \eqalign{ & \mathop \smallint \nolimits_{{\omega _0}}^d {\left| {{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma ,\;z} \right)} \right|^2}{d_{\omega ,q}}\varsigma + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {\left| {{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma ,\;z} \right)} \right|^2}{d_{\omega ,q}}\varsigma \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop \sum \nolimits_{m = 1}^\infty {{{{\left| {{\theta _{m,{1 \over {{q^n}}}}}\left( \zeta \right)} \right|}^2}} \over {\alpha _{m,{1 \over {{q^n}}}}^2{{\left| {{\lambda _{m,{1 \over {{q^n}}}}} - z} \right|}^2}}} = \mathop \smallint \nolimits_{ - \infty }^\infty {\left| {{{\theta \left( {\zeta ,\lambda } \right)} \over {\lambda - z}}} \right|^2}d{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right). \cr}

By Lemma 3.1, the integral on the left converges and the result is immediate.

It follows from Lemma 8 that the set $ϱ_{\frac{1}{q^{n}}} (λ)\}$ \left\{ {{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right)} \right\} is bounded. Using Helly’s theorems ([15]), one can find a sequence {1/q^nk} such that $ϱ_{\frac{1}{q^{n} k}} (λ)$ {\varrho _{{1 \over {{q^n}k}}}}\left( \lambda \right) converges to a monotone function ϱ(λ) (as n_k → ∞).

Lemma 3.7.

Let z be a nonreal number and ζ be a fixed number. Then we have (3.16) $\int_{- \infty}^{\infty} {\frac{θ (ζ, λ)}{z - λ}|}^{2} d ϱ (λ) \leq S .$ \mathop \int \nolimits_{ - \infty }^\infty {\left| {{{\theta \left( {\zeta ,\lambda } \right)} \over {z - \lambda }}} \right|^2}d\varrho \left( \lambda \right) \le S.

Proof

For arbitrary η > 0, it follows from (3.14) that $\int_{- η}^{η} {\frac{ϕ (ς, λ)}{z - λ}|}^{2} d ϱ_{\frac{1}{q^{n}}} (λ) < S .$ \mathop \int \nolimits_{ - \eta }^\eta {\left| {{{\phi \left( {\varsigma ,\lambda } \right)} \over {z - \lambda }}} \right|^2}d{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right)< S.

Letting η → ∞ and n → ∞, we get the desired result.

Lemma 3.8.

For arbitrary η > 0, we have $\int_{- \infty}^{- η} \frac{d ϱ (λ)}{| z - λ |^{2}} < \infty, \int_{η}^{\infty} \frac{d ϱ (λ)}{| z - λ |^{2}} < \infty .$ \mathop \int \nolimits_{ - \infty }^{ - \eta } {{d\varrho \left( \lambda \right)} \over {|z - \lambda {|^2}}}< \infty ,\;\mathop {\,\,\,\,\,\smallint }\nolimits_\eta ^\infty {{d\varrho \left( \lambda \right)} \over {|z - \lambda {|^2}}}< \infty .

Proof

Let sin β ≠ 0. Writing ζ = 0 in (3.16), we obtain $\int_{- \infty}^{\infty} \frac{d ϱ (λ)}{| z - λ |^{2}} < \infty .$ \mathop \int \nolimits_{ - \infty }^\infty {{d\varrho \left( \lambda \right)} \over {|z - \lambda {|^2}}}< \infty .

Let sin β = 0. Then $\frac{1}{α_{m, \frac{1}{q^{n}}}} {D_{q, ζ} G_{\frac{1}{q^{n}}} (ζ, \cdot, z), θ_{m, \frac{1}{q^{n}}} (\cdot)〉}_{n} = \frac{D_{q, ζ} θ_{m, \frac{1}{q^{n}}} (ζ)}{α_{m, \frac{1}{q^{n}}} (λ_{m, \frac{1}{q^{n}}} - z)} .$ {1 \over {{\alpha _{m,{1 \over {{q^n}}}}}}}{\left\langle {{D_{q,\zeta }}{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\; \cdot ,\;z} \right),\;{\theta _{m,{1 \over {{q^n}}}}}\left( \cdot \right)} \right\rangle _n} = {{{D_{q,\zeta }}{\theta _{m,{1 \over {{q^n}}}}}\left( \zeta \right)} \over {{\alpha _{m,{1 \over {{q^n}}}}}\left( {{\lambda _{m,{1 \over {{q^n}}}}} - z} \right)}}.

By (3.11), we find $\begin{array}{l} \int_{ω_{0}}^{d} {D_{q, ζ} G_{\frac{1}{q^{n}}} (ζ, ς, z)|}^{2} d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} {D_{q, ζ} G_{\frac{1}{q^{n}}} (ζ, ς, z)|}^{2} d_{ω, q} ζ \\ = \int_{- \infty}^{\infty} {\frac{D_{q, ζ} θ (ζ, λ)}{z - λ}|}^{2} d ϱ_{\frac{1}{q^{n}}} (λ) . \end{array}$ \eqalign{ & \mathop \smallint \nolimits_{{\omega _0}}^d {\left| {{D_{q,\zeta }}{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma ,\;z} \right)} \right|^2}{d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {\left| {{D_{q,\zeta }}{G_{{1 \over {{q^n}}}}}\left( {\zeta ,\;\varsigma ,\;z} \right)} \right|^2}{d_{\omega ,q}}\zeta \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop \smallint \nolimits_{ - \infty }^\infty {\left| {{{{D_{q,\zeta }}\theta \left( {\zeta ,\lambda } \right)} \over {z - \lambda }}} \right|^2}d{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right). \cr}

Lemma 3.9.

Let $(R f) (ζ, z) = \int_{ω_{0}}^{\infty} G (ζ, ς, z) f (ς) d_{ω, q} ς,$ \left( {Rf} \right)\left( {\zeta ,\;z} \right) = \mathop \smallint \nolimits_{{\omega _0}}^\infty G\left( {\zeta ,\;\varsigma ,\;z} \right)f\left( \varsigma \right){d_{\omega ,q}}\varsigma , where f ∈ H, and $G (ζ, ς, z) = \begin{array}{l} Z (ζ, z) θ (ς, z), & ς \leq ζ, ζ \neq d, ς \neq d, \\ θ (ζ, z) Z (ς, z), & ς > ζ, ζ \neq d, ς \neq d . \end{array}$ G\left( {\zeta ,\;\varsigma ,\;z} \right) = \left\{ {\matrix{ {Z\left( {\zeta ,\;z} \right)\theta \left( {\varsigma ,\;z} \right),} \hfill & {\;\varsigma \le \zeta ,\;\,\,\,\,\zeta \ne d,\;\,\,\,\varsigma \ne d,} \hfill \cr {\theta \left( {\zeta ,\;z} \right)Z\left( {\varsigma ,\;z} \right)\;,} \hfill & {\;\varsigma > \zeta ,\;\,\,\,\,\zeta \ne d,\;\,\,\varsigma \ne d.} \hfill \cr } } \right.

Then, we have $\begin{array}{l} \int_{ω_{0}}^{d} | (R f) (ζ, z) |^{2} d_{ω, q} ζ + \int_{d}^{\infty} | (R f) (ζ, z) |^{2} d_{ω, q} ζ \\ \leq \frac{1}{v^{2}} \int_{ω_{0}}^{d} {f^{(1)} (ζ)|}^{2} d_{ω, q} ζ + \int_{d}^{\infty} {f^{(2)} (ζ)|}^{2} d_{ω, q} ζ, \end{array}$ \eqalign{ & \mathop \smallint \nolimits_{{\omega _0}}^d |\left( {Rf} \right)\left( {\zeta ,\;z} \right){|^2}{d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^\infty |\left( {Rf} \right)\left( {\zeta ,\;z} \right){|^2}{d_{\omega ,q}}\zeta \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le {1 \over {{v^2}}}\mathop \smallint \nolimits_{{\omega _0}}^d {\left| {{f^{\left( 1 \right)}}\left( \zeta \right)} \right|^2}{d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^\infty {\left| {{f^{\left( 2 \right)}}\left( \zeta \right)} \right|^2}{d_{\omega ,q}}\zeta , \cr} where v = Im z.

Proof

Combining (3.13) and (3.10), for each $\frac{1}{q^{n}} > d$ {1 \over {{q^n}}} > d , n ∈ ℕ, we obtain $\begin{array}{l} \int_{ω_{0}}^{d} {(R_{\frac{1}{q^{n}}} f) (ζ, z)|}^{2} d_{ω, q} ζ + \int_{d}^{\frac{1}{q^{n}}} {(R_{\frac{1}{q^{n}}} f) (ζ, z)|}^{2} d_{ω, q} ζ \\ = \sum_{m = 1}^{\infty} \frac{{{〈 f (\cdot), θ_{m, \frac{1}{q^{n}}} (\cdot, z) 〉}_{n}|}^{2}}{α_{m, \frac{1}{q^{n}}}^{2} | λ_{m, \frac{1}{q^{n}}} - z |^{2}} \\ = \frac{1}{v^{2}} \int_{ω_{0}}^{d} {f^{(1)} (ς)|}^{2} d_{ω, q} ς + \frac{1}{v^{2}} \int_{d}^{\frac{1}{q^{n}}} {f^{(2)} (ς)|}^{2} d_{ω, q} ς . \end{array}$ \eqalign{ & \mathop \smallint \nolimits_{{\omega _0}}^d {\left| {\left( {{R_{{1 \over {{q^n}}}}}f} \right)\left( {\zeta ,\;z} \right)} \right|^2}{d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {\left| {\left( {{R_{{1 \over {{q^n}}}}}f} \right)\left( {\zeta ,\;z} \right)} \right|^2}{d_{\omega ,q}}\zeta \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop \sum \nolimits_{m = 1}^\infty {{{{\left| {{{\langle f\left( \cdot \right),{\theta _{m,{1 \over {{q^n}}}}}\left( { \cdot ,z} \right)\rangle }_n}} \right|}^2}} \over {\alpha _{m,{1 \over {{q^n}}}}^2|{\lambda _{m,{1 \over {{q^n}}}}} - z{|^2}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {1 \over {{v^2}}}\mathop \smallint \nolimits_{{\omega _0}}^d {\left| {{f^{\left( 1 \right)}}\left( \varsigma \right)} \right|^2}{d_{\omega ,q}}\varsigma + {1 \over {{v^2}}}\mathop \smallint \nolimits_d^{{1 \over {{q^n}}}} {\left| {{f^{\left( 2 \right)}}\left( \varsigma \right)} \right|^2}{d_{\omega ,q}}\varsigma . \cr}

Letting n → ∞, we get the desired result.

Theorem 3.10 (Integral Representation of the Resolvent).

For every non-real z and for each f ∈ H, we obtain $(R f) (ζ, z) = \int_{- \infty}^{\infty} \frac{θ (ζ, λ)}{λ - z} F (λ) d ϱ (λ),$ \left( {Rf} \right)\left( {\zeta ,\;z} \right) = \mathop \smallint \nolimits_{ - \infty }^\infty {{\theta \left( {\zeta ,\lambda } \right)} \over {\lambda - z}}F\left( \lambda \right)d\varrho \left( \lambda \right), where $F (λ) = \int_{ω_{0}}^{d} f^{(1)} (ζ) θ^{(1)} (ζ, λ) d_{ω, q} ζ + \lim_{σ \to \infty} \int_{d}^{σ} f^{(2)} (ζ) θ^{(2)} (ζ, λ) d_{ω, q} ζ .$ F\left( \lambda \right) = \mathop \smallint \nolimits_{{\omega _0}}^d {f^{\left( 1 \right)}}\left( \zeta \right){\theta ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta + \mathop {\lim }\limits_{\sigma \to \infty } \mathop \smallint \nolimits_d^\sigma {f^{\left( 2 \right)}}\left( \zeta \right){\theta ^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta .

Proof

Suppose that f(ζ) = f_σ(ζ) satisfies (3.2)–(3.4) and vanishes outside the set [ω₀, d) ∪ (d, σ], where $d < σ < \frac{1}{q^{n}}$ d< \sigma< {1 \over {{q^n}}} , n ∈ ℕ. Let $F_{σ} (λ) = \int_{ω_{0}}^{d} f_{σ}^{(1)} (ζ) θ^{(1)} (ζ, λ) d_{ω, q} ζ + \int_{d}^{σ} f_{σ}^{(2)} (ζ) θ^{(2)} (ζ, λ) d_{ω, q} ζ .$ {F_\sigma }\left( \lambda \right) = \mathop \smallint \nolimits_{{\omega _0}}^d f_\sigma ^{\left( 1 \right)}\left( \zeta \right){\theta ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^\sigma f_\sigma ^{\left( 2 \right)}\left( \zeta \right){\theta ^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta .

By (3.13), we see that (3.17) $\begin{array}{l} \begin{array}{l} (R_{\frac{1}{q^{n}}} f_{σ}) (ζ, z) & = \int_{- \infty}^{\infty} \frac{θ (ζ, λ)}{λ - z} F_{σ} (λ) d ϱ_{\frac{1}{q^{n}}} (λ) \\ = \int_{- \infty}^{- a} \frac{θ (ζ, λ)}{λ - z} F_{σ} (λ) d ϱ_{\frac{1}{q^{n}}} (λ) + \int_{- a}^{a} \frac{θ (ζ, λ)}{λ - z} F_{σ} (λ) d ϱ_{\frac{1}{q^{n}}} (λ) \\ + \int_{a}^{\infty} \frac{θ (ζ, λ)}{λ - z} F_{σ} (λ) d ϱ_{\frac{1}{q^{n}}} (λ) = I_{1} + I_{2} + I_{3} . \end{array} \end{array}$ \eqalign{ & \matrix{ {\left( {{R_{{1 \over {{q^n}}}}}{f_\sigma }} \right)\left( {\zeta ,\;z} \right)} \hfill & { = \mathop \smallint \nolimits_{ - \infty }^\infty {{\theta \left( {\zeta ,\lambda } \right)} \over {\lambda - z}}{F_\sigma }\left( \lambda \right)d{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right)} \hfill \cr {} \hfill & { = \mathop \smallint \nolimits_{ - \infty }^{ - a} {{\theta \left( {\zeta ,\lambda } \right)} \over {\lambda - z}}{F_\sigma }\left( \lambda \right)d{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right) + \mathop \smallint \nolimits_{ - a}^a {{\theta \left( {\zeta ,\lambda } \right)} \over {\lambda - z}}{F_\sigma }\left( \lambda \right)d{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right)} \hfill \cr {} \hfill & { + \mathop \smallint \nolimits_a^\infty {{\theta \left( {\zeta ,\lambda } \right)} \over {\lambda - z}}{F_\sigma }\left( \lambda \right)d{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right) = {I_1} + {I_2} + {I_3}.} \hfill \cr } \cr & \cr}

Firstly, we will estimate I₁. From (3.13), we deduce that $\begin{array}{l} I_{1} & = \int_{- \infty}^{- a} \frac{θ (ζ, λ)}{z - λ} F_{σ} (λ) d ϱ_{\frac{1}{q^{n}}} (λ) \\ = \sum_{λ_{k, \frac{1}{q^{n}}} < - a} \frac{θ_{k, \frac{1}{q^{n}}} (ζ)}{α_{k, \frac{1}{q^{n}}}^{2} (z - λ_{k, \frac{1}{q^{n}}})} {\int_{ω_{0}}^{d} f_{σ}^{(1)} (ζ) θ_{k, \frac{}{q^{n}}}^{(1)} (ζ) d_{ω, q} ζ + \int_{d}^{σ} f_{σ}^{(2)} (ζ) θ_{k, \frac{}{q^{n}}}^{(2)} (ζ) d_{ω, q} ζ} \\ \leq {(\sum_{λ_{k, \frac{1}{q^{n}}} < - a} \frac{θ_{k, \frac{1}{q^{n}}}^{2} (ζ)}{α_{k, \frac{1}{q^{n}}}^{2} {| z - λ_{k, \frac{1}{q^{n}}} |}^{2}})}^{1 / 2} \\ \times {(\sum_{λ_{k, \frac{1}{q^{n}}} < - a} \frac{1}{α_{k, \frac{1}{q^{n}}}^{2}} {| \int_{ω_{0}}^{d} f_{σ}^{(1)} (ζ) θ_{k, \frac{}{q^{n}}}^{(1)} (ζ) d_{ω, q} ζ + \int_{d}^{σ} f_{σ}^{(2)} (ζ) θ_{k, \frac{}{q^{n}}}^{(2)} (ζ) d_{ω, q} ζ |}^{2})}^{1 / 2} . \end{array}$ \matrix{ {{I_1}} \hfill & { = \mathop \smallint \nolimits_{ - \infty }^{ - a} {{\theta \left( {\zeta ,\lambda } \right)} \over {z - \lambda }}{F_\sigma }\left( \lambda \right)d{\varrho _{{1 \over {{q^n}}}}}\left( \lambda \right)} \hfill \cr {} \hfill & { = \sum\limits_{{\lambda _{k,{1 \over {{q^n}}}}}< - a} {{{{\theta _{k,{1 \over {{q^n}}}}}\left( \zeta \right)} \over {\alpha _{k,{1 \over {{q^n}}}}^2\left( {z - {\lambda _{k,{1 \over {{q^n}}}}}} \right)}}\left\{ {\mathop \smallint \nolimits_{{\omega _0}}^d f_\sigma ^{\left( 1 \right)}\left( \zeta \right)\theta _{k,{{_1} \over {{q^n}}}}^{(1)}\left( \zeta \right){d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^\sigma f_\sigma ^{\left( 2 \right)}\left( \zeta \right)\theta _{k,{{_1} \over {{q^n}}}}^{(2)}\left( \zeta \right){d_{\omega ,q}}\zeta } \right\}} } \hfill \cr {} \hfill & { \le {{\left( {\sum\limits_{{\lambda _{k,{1 \over {{q^n}}}}}< - a} {{{\theta _{k,{1 \over {{q^n}}}}^2\left( \zeta \right)} \over {\alpha _{k,{1 \over {{q^n}}}}^2{{\left| {z - {\lambda _{k,{1 \over {{q^n}}}}}} \right|}^2}}}} } \right)}^{1/2}}} \hfill \cr {} \hfill & { \times {{\left( {\sum\limits_{{\lambda _{k,{1 \over {{q^n}}}}}< - a} {{1 \over {\alpha _{k,{1 \over {{q^n}}}}^2}}{{\left| {\mathop \smallint \nolimits_{{\omega _0}}^d f_\sigma ^{\left( 1 \right)}\left( \zeta \right)\theta _{k,{{_1} \over {{q^n}}}}^{(1)}\left( \zeta \right){d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^\sigma f_\sigma ^{\left( 2 \right)}\left( \zeta \right)\theta _{k,{{_1} \over {{q^n}}}}^{(2)}\left( \zeta \right){d_{\omega ,q}}\zeta } \right|}^2}} } \right)}^{1/2}}.} \hfill \cr }

Integrating twice by parts, we find $\begin{array}{l} \int_{ω_{0}}^{d} f_{σ}^{(1)} (ζ) θ_{k, \frac{}{q^{n}}}^{(1)} (ζ) d_{ω, q} ζ + \int_{σ}^{d} f_{σ}^{(2)} (ζ) θ_{k, \frac{}{q^{n}}}^{(2)} (ζ) d_{ω, q} ζ \\ = - \frac{1}{λ_{k, \frac{1}{q^{n}}}} \int_{ω_{0}}^{d} f_{σ}^{(1)} (ζ) {\frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} θ_{k, \frac{}{q^{n}}}^{(1)} (ζ) - v (ζ) θ_{k, \frac{}{q^{n}}}^{(1)} (ζ)} d_{ω, q} ζ \\ - \frac{1}{λ_{k, \frac{1}{q^{n}}}} \int_{d}^{σ} f_{σ}^{(2)} (ζ) {\frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} θ_{k, \frac{}{q^{n}}}^{(2)} (ζ) - v (ζ) θ_{k, \frac{}{q^{n}}}^{(2)} (ζ)} d_{ω, q} ζ \\ = - \frac{1}{λ_{k, \frac{1}{q^{n}}}} \int_{ω_{0}}^{d} {\frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} f_{σ}^{(1)} (ζ) - v (ζ) f_{σ}^{(1)} (ζ)} θ_{k, \frac{}{q^{n}}}^{(1)} (ζ) d_{ω, q} ζ \\ - \frac{1}{λ_{k, \frac{1}{q^{n}}}} \int_{d}^{σ} {\frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} f_{σ}^{(2)} (ζ) - v (ζ) f_{σ}^{(2)} (ζ)} θ_{k, \frac{}{q^{n}}}^{(2)} (ζ) d_{ω, q} ζ . \end{array}$ \eqalign{ & \mathop \smallint \nolimits_{{\omega _0}}^d f_\sigma ^{\left( 1 \right)}\left( \zeta \right)\theta _{k,{{_1} \over {{q^n}}}}^{(1)}\left( \zeta \right){d_{\omega ,q}}\zeta + \mathop \smallint \nolimits_d^\sigma f_\sigma ^{\left( 2 \right)}\left( \zeta \right)\theta _{k,{{_1} \over {{q^n}}}}^{(2)}\left( \zeta \right){d_{\omega ,q}}\zeta \cr & = - {1 \over {{\lambda _{k,{1 \over {{q^n}}}}}}}\mathop \smallint \nolimits_{{\omega _0}}^d f_\sigma ^{\left( 1 \right)}\left( \zeta \right)\left\{ {{1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}\theta _{k,{{_1} \over {{q^n}}}}^{(1)}\left( \zeta \right) - v\left( \zeta \right)\theta _{k,{{_1} \over {{q^n}}}}^{(1)}\left( \zeta \right)} \right\}{d_{\omega ,q}}\zeta \cr & - {1 \over {{\lambda _{k,{1 \over {{q^n}}}}}}}\mathop \smallint \nolimits_d^\sigma f_\sigma ^{\left( 2 \right)}\left( \zeta \right)\left\{ {{1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}\theta _{k,{{_1} \over {{q^n}}}}^{(2)}\left( \zeta \right) - v\left( \zeta \right)\theta _{k,{{_1} \over {{q^n}}}}^{(2)}\left( \zeta \right)} \right\}{d_{\omega ,q}}\zeta \cr & = - {1 \over {{\lambda _{k,{1 \over {{q^n}}}}}}}\mathop \smallint \nolimits_{{\omega _0}}^d \left\{ {{1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}f_\sigma ^{\left( 1 \right)}\left( \zeta \right) - v\left( \zeta \right)f_\sigma ^{\left( 1 \right)}\left( \zeta \right)} \right\}\theta _{k,{{_1} \over {{q^n}}}}^{(1)}\left( \zeta \right){d_{\omega ,q}}\zeta \cr & - {1 \over {{\lambda _{k,{1 \over {{q^n}}}}}}}\mathop \smallint \nolimits_d^\sigma \left\{ {{1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}f_\sigma ^{\left( 2 \right)}\left( \zeta \right) - v\left( \zeta \right)f_\sigma ^{\left( 2 \right)}\left( \zeta \right)} \right\}\theta _{k,{{_1} \over {{q^n}}}}^{(2)}\left( \zeta \right){d_{\omega ,q}}\zeta . \cr}

By Lemma 3.6, we get $\begin{array}{l} I_{1} & \leq \frac{K^{1 / 2}}{a} \\ \times \sum_{λ_{k, \frac{1}{q^{n}}} < - a} \frac{1}{α_{k, \frac{1}{q^{n}}}^{2}} \int_{ω_{0}}^{d} {\frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} f_{σ}^{(1)} (ζ) - v (ζ) f_{σ}^{(1)} (ζ)} θ_{k, \frac{1}{q^{n}}}^{(1)} (ζ) d_{ω, q} ζ \\ {{+ \int_{d}^{σ} \frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} f_{σ}^{(2)} (ζ) - v (ζ) f_{σ}^{(2)} (ζ)\} θ_{k, \frac{1}{q^{n}}}^{(2)} (ζ) d_{ω, q} ζ|}^{2})}^{1 / 2} . \end{array}$ \matrix{ {{I_1}} \hfill & { \le {{{K^{1/2}}} \over a}} \hfill \cr {} \hfill & { \times \left( {\sum\limits_{{\lambda _{k,{1 \over {{q^n}}}}}< - a} {{1 \over {\alpha _{k,{1 \over {{q^n}}}}^2}}\mathop \smallint \nolimits_{{\omega _0}}^d \{ {1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}f_\sigma ^{\left( 1 \right)}\left( \zeta \right) - v\left( \zeta \right)f_\sigma ^{\left( 1 \right)}\left( \zeta \right)\} \theta _{k,{{{_1}} \over {{q^n}}}}^{(1)}\left( \zeta \right){d_{\omega ,q}}\zeta } } \right.} \hfill \cr {} \hfill & {\,\,\,{{\left. {{{\left. {\, + \mathop \smallint \nolimits_d^\sigma \left\{ {{1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}f_\sigma ^{\left( 2 \right)}\left( \zeta \right) - v\left( \zeta \right)f_\sigma ^{\left( 2 \right)}\left( \zeta \right)} \right\}\theta _{k,{{_1} \over {{q^n}}}}^{(2)}\left( \zeta \right){d_{\omega ,q}}\zeta } \right|}^2}} \right)}^{1/2}}.} \hfill \cr } Using Bessel inequality, we see that $\begin{array}{l} I_{1} \leq \frac{K^{1 / 2}}{a} \int_{ω_{0}}^{σ} {\frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} f_{σ}^{(1)} (ζ) - v (ζ) f_{σ}^{(1)} (ζ)|}^{2} d_{ω, q} ζ \\ + {\int_{d}^{σ} {\frac{1}{q} D_{- \frac{ω}{q}, \frac{1}{q}} D_{ω, q} f_{σ}^{(2)} (ζ) - v (ζ) f_{σ}^{(2)} (ζ)|}^{2} d_{ω, q} ζ]}^{1 / 2} = \frac{C}{a} . \end{array}$ \eqalign{ & {I_1} \le {{{K^{1/2}}} \over a}\left[ {\mathop \smallint \nolimits_{{\omega _0}}^\sigma {{\left| {{1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}f_\sigma ^{\left( 1 \right)}\left( \zeta \right) - v\left( \zeta \right)f_\sigma ^{\left( 1 \right)}\left( \zeta \right)} \right|}^2}{d_{\omega ,q}}\zeta } \right. \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + {\left. {\mathop \smallint \nolimits_d^\sigma {{\left| {{1 \over q}{D_{ - {\omega \over q},{1 \over q}}}{D_{\omega ,q}}f_\sigma ^{\left( 2 \right)}\left( \zeta \right) - v\left( \zeta \right)f_\sigma ^{\left( 2 \right)}\left( \zeta \right)} \right|}^2}{d_{\omega ,q}}\zeta } \right]^{1/2}} = {C \over a}. \cr}

It is proved similarly that $I_{3} \leq \frac{C}{a}$ {I_3} \le {C \over a} . Then I₁ and I₃ tend to zero as a → ∞, uniformly in $\frac{1}{q_{n}}$ {1 \over {{q_n}}} . It follows from the Helly selection theorem and (3.17) that (3.18) $(R f_{σ}) (ζ, z) = \int_{- \infty}^{\infty} \frac{θ (ζ, λ)}{z - λ} F_{σ} (λ) d ϱ (λ) .$ \left( {R{f_\sigma }} \right)\left( {\zeta ,\;z} \right) = \mathop \smallint \nolimits_{ - \infty }^\infty {{\theta \left( {\zeta ,\lambda } \right)} \over {z - \lambda }}{F_\sigma }\left( \lambda \right)d\varrho \left( \lambda \right). As is known, if f(·) ∈ H, then we find a sequence ${f_{σ} (ς)\}}_{σ = 1}^{\infty}$ \left\{ {{f_\sigma }\left( \varsigma \right)} \right\}_{\sigma = 1}^\infty that satisfies the previous conditions and tends to f(ζ) as σ → ∞. From (3.10), the sequence of Fourier transform converges to the transform of f(ζ). Using Lemmas 3.7 and 3.9, we can pass to the limit σ → ∞ in (3.18). Thus, we get the desired result.

Remark 3.11.

Using Theorem 3.10, we infer that $\begin{array}{l} \int_{ω_{0}}^{\infty} (R f^{(1)}) (ς, z) g^{(1)} (ς) d_{ω, q} ς + \int_{d}^{\infty} (R f^{(2)}) (ς, z) g^{(2)} (ς) d_{ω, q} ς \\ = \int_{- \infty}^{\infty} \frac{F (λ) G (λ)}{z - λ} d ϱ (λ), \end{array}$ \eqalign{ & \mathop \smallint \nolimits_{{\omega _0}}^\infty (R{f^{\left( 1 \right)}})\left( {\varsigma ,\;z} \right){g^{\left( 1 \right)}}\left( \varsigma \right){d_{\omega ,q}}\varsigma + \mathop \smallint \nolimits_d^\infty (R{f^{\left( 2 \right)}})\left( {\varsigma ,\;z} \right){g^{\left( 2 \right)}}\left( \varsigma \right){d_{\omega ,q}}\varsigma \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop \smallint \nolimits_{ - \infty }^\infty {{F\left( \lambda \right)G\left( \lambda \right)} \over {z - \lambda }}d\varrho \left( \lambda \right), \cr} where $F (λ) = \int_{ω_{0}}^{d} f^{(1)} (ζ) θ^{(1)} (ζ, λ) d_{ω, q} ζ + \lim_{σ \to \infty} \int_{d}^{σ} f^{(2)} (ζ) θ^{(2)} (ζ, λ) d_{ω, q} ζ,$ F\left( \lambda \right) = \mathop \smallint \nolimits_{{\omega _0}}^d {f^{\left( 1 \right)}}\left( \zeta \right){\theta ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta + \mathop {\lim }\limits_{\sigma \to \infty } \mathop \smallint \nolimits_d^\sigma {f^{\left( 2 \right)}}\left( \zeta \right){\theta ^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta , and $G (λ) = \int_{ω_{0}}^{d} g^{(1)} (ζ) θ^{(1)} (ζ, λ) d_{ω, q} ζ + \lim_{σ \to \infty} \int_{ω_{0}}^{σ} g^{(2)} (ζ) θ^{(2)} (ζ, λ) d_{ω, q} ζ .$ G\left( \lambda \right) = \mathop \smallint \nolimits_{{\omega _0}}^d {g^{\left( 1 \right)}}\left( \zeta \right){\theta ^{\left( 1 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta + \mathop {\lim }\limits_{\sigma \to \infty } \mathop \smallint \nolimits_{{\omega _0}}^\sigma {g^{\left( 2 \right)}}\left( \zeta \right){\theta ^{\left( 2 \right)}}\left( {\zeta ,\;\lambda } \right){d_{\omega ,q}}\zeta .

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