Asymptotic analysis of Sturm-Liouville problem with Robin and two-point boundary conditions

Artu¯ras Štikonas

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Abstract: Weanalyzetheinitialvalueproblemandgetasymptoticexpansionsforsolution. We investigate the characteristic equation for Sturm–Liouville problem with one classical Robin type boundary condition and another two-point nonlocal boundary condition. Finally, we obtain asymptotic expansions for eigenvalues and eigenfunctions.

Keywords: Sturm–Liouville problem, Robin condition, two-point nonlocal conditions, asymptotics of eigenvalues and eigenfunctions.

Summary: A. Štikonas

Mes analizuojame pradin˛i uždavini˛ ir gauname jo sprendinio asimptotinius skleidinius. Mes tiriame Sturmo ir Liuvilio uždavini˛ su su viena klasikine Robino tipo kraštine sąlyga ir kita dvitaške nelokalia kraštine sąlyga. Galiausiai gauname tikriniu˛ reikšmiu˛ ir tikriniu˛ funkciju˛ asimptotinius skleidinius.

Keywords: Šturmo ir Liuvilio uždavinys, Robino sąlyga, dvitaške˙s nelokaliosios sąlygos, tikriniu˛ reikšmiu˛ ir tikriniu˛ funkciju˛ asimptotika.

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Articles

Asymptotic analysis of Sturm-Liouville problem with Robin and two-point boundary conditions

Šturmo ir Liuvilio uždavinio su Robino ir dvitaške kraštine˙mis sąlygomis asimptotine˙ analize˙

Artu¯ras Štikonas arturas.stikonas@mif.vu.lt

Vilnius University, Lituania

Lietuvos matematikos rinkinys, vol. 63 Ser. A, pp. 9-18, 2022
Vilniaus Universitetas

Esta obra está bajo una Licencia Creative Commons Atribución 4.0 Internacional.

Recepción: 01 Julio 2022

Publicación: 10 Diciembre 2022

DOI: https://doi.org/10.15388/LMR.2022.29692

Introduction

Consider the following one-dimensional Sturm–Liouville equation

- u^{"} (t) + q (t) u (t) = λ u (t), t \in [0, 1]

where the real-valued function $q \in C [0, 1]; λ = s^{2}$ is a complex spectral parameter and $s = x + ι y; x, y \in ℝ$ . We will use the notation $Q (t) = \frac{1}{2} \int_{0}^{t} q (τ) d τ$ .

In this article $S \in ℂ_{s} : = ℝ_{s} U ℂ_{s}^{+} U ℂ_{s}^{-}$ , where $ℝ_{s} : = ℝ_{s}^{-} U ℝ_{s}^{+}, U ℝ_{s}^{0}$ $ℝ_{s}^{-} : = {s = x + ι y \in ℂ : x = 0,$ $y > 0}, ℝ_{s}^{+} : = {s = x + ι y \in ℂ : x > 0, y = 0},$ $ℝ_{s}^{0} : = {s = 0}, ℂ_{s}^{+} : = {s = x + ι y \in ℂ : x > 0, y > 0}$ and $ℂ_{s}^{-} : = {s = x + ι y \in ℂ : x > 0, y < 0}$ .

Then a map $λ = S^{2}$ is the bijection between $ℂ_{s}$ and $ℂ_{λ} : = ℂ$ .

We shall investigate Sturm–Liouville Problem (SLP) which consist of equation (1) on $[0, 1]$ with one classical (local) Robin type Boundary Condition (BC)

\cos α u (0) + \sin α u^{'} (0) = 0, α \in (0, π),

and another two-point Nonlocal Boundary Condition (NBC)

(C a s e 1) u^{'} (1) = γ u (ξ), ξ \in [0, 1],

(C a s e 2) u^{'} (1) = γ u^{'} (ξ), ξ \in [0, 1),

(C a s e 2) u (1) = γ u (ξ), ξ \in [0, 1),

where $γ \in ℝ$ . We consider the Dirichlet and the Neumann BC:

(C a s e d) u (0) = 0,

(C a s e n) u^{'} (0) = 0,

too. The Sturm–Liouville problem (1), (4_d), (3₃) was investigated in [2], the Sturm–Liouville problem (1), (4_n), (3) was investigated in [4].

1 Asymptotic expansions for Initial Value Problem

In this section we present some statements about solution of IVP. These statements were proved in [3]. We will use them for investigation asymptotic expansions for SLP (1)–(3). Additionally, we introduce some notation related to our asymptotical analysis of this problem.

Let $λ = s^{2}, s \in ℂ_{s}$ and $w_{α s} (t)$ be a solution of equation (1) satisfying the initial conditions

w_{α s} (0) = \sin α, w_{α s}^{'} (0) = - \cos α .

The function $w (t, s, α) = w_{α s} (t)$ is an analytic (holomorphic) function of s and this function satisfies boudary condition (2). We denote $φ s (t) = φ_{o s} (t) : = w (t, s, 0)$ and $ψ_{s} (t) = φ_{- 1, s} (t) = w (t, s, π / 2)$ .

Under the condition that $q \in ℂ^{r} [0, 1], r \in ℕ_{0} : = ℕ U {0}$ , asymptotic expansions may be obtained for $φ_{s} (t)$ [3] and $ψ_{s} (t)$ [4]. We will use recursive formula

p_{i + 1}^{0} (t) = - \frac{1}{2} \int_{0}^{t} q (τ) p_{i}^{0} (τ) d τ - \sum_{j = 2 + \partial}^{i} \frac{(q p_{j - 1}^{0}) (i - j) (t) + (- 1)^{i} (q p_{j - 1}^{0}) (i - j) (0)}{2^{i - j + 2}}

Lemma 1.(See [3, Lemma 7]) Let $s \in ℂ_{s}$ and $q \in C^{r} [0, 1]$ . Then for $| S | \geq q_{0}$ we have the asymptotic expansions

(φ_{s})_{s}^{(l)} (t, s) = - {\sum p_{j (t) \cos}^{l}}_{j = 1}^{r + 1} (s t + \frac{π}{2} (j - l)) s^{- j} + O (s^{- (r + 2) e (r + 2) | y | t),}

(φ_{s}^{'})_{s}^{(l)} (t, s) = - \sum_{j = 0}^{r} p_{j}^{- l} \cos (s t + \frac{π}{2} (j - l)) s^{- j} + O (s^{- (r + 1) e (r + 2) | y | t})

for $l \in ℕ_{0}$ , where $p_{1}^{k} (t) = - t p_{1}^{k - 1} (t), p_{i}^{k} (t) = (1 - i) p_{i - 1}^{k - 1} (t) - t p_{i}^{k - 1} (t), i = \overline{2, r + 1,}$ $p_{0}^{- k} (t) = - t p_{0}^{- k - 1} (t), p_{i}^{- k} (t) = (1 - i) p_{i - 1}^{- k - 1} (t) - t p_{i}^{- k - 1} (t), = \overline{1, r k \in ℕ,}$ $p_{i}^{- 0} (t) = p_{i}^{0'} (t) - p_{i + 1}^{0} (t), i = \overline{1, r,} p_{0}^{- 0} (t) = 1$ , and $p_{j}^{0} (t)$ is calculated by (6) $i = \overline{1, r}$ ,with $p_{1}^{0} (t) = - 1$ .

Lemma 2. (See [4, Lemma 9].) Let $S \in ℂ_{s}$ and $q \in ℂ^{r} [0, 1]$ . Then for $| s | \geq q_{0}$ we have the asymptotic expansions

(ψ_{s})_{s}^{(l)} (t, s) = - {\sum p_{j}^{l} (t) \cos (s t + \frac{π}{2} (j - l) s^{- j} + O (s^{- (r + 1) e (r + 2) | y | t})}_{j = 0}^{r}

(ψ_{s}^{'})_{s}^{(l)} (t, s) = - \sum_{j = - 1}^{r - 1} p_{j}^{- l} \cos (s t + \frac{π}{2} (j - l) s^{- j} + O (s^{- r} e^{(r + 2) | y | t})

for $l \in ℕ_{0}$ . Where $p_{0}^{k} (t) = - t p_{0}^{k - 1} (t), p_{i}^{k} (t) = (1 - i) p_{i - 1}^{k - 1} (t) - t p_{i}^{k - 1} (t), i = \overline{1, r,}$ $p_{- 1}^{- k} (t) = - t p_{- 1}^{- k - 1} (t), p_{i}^{- k} (t) = (1 - i) p_{i - 1}^{- k - 1} (t) - t p_{i}^{- k - 1} (t), i = \overline{0, r - 1,}$ $k \in ℕ, p_{i}^{- 0} (t) = p_{i}^{0'} (t) - p_{i + 1}^{0} (t), i = \overline{0, r - 1,} p_{- 1}^{- 0} (t) = 1$ and $p_{j}^{0} (t)$ is calculated by (6) for $i = \overline{0, r - 1}$ with $p_{0}^{0} (t) = - 1$ .

Remark 1. (See [3, Lemma 7], [4, Lemma 7].) In the case $q \in C [0, 1]$ and $l = 0$ we have the asymptotic expansions:

φ_{s} (t) = - \sin (s t) s^{- 1} + O (s^{- 2} e^{| y | t}), ψ_{s} (t) = \cos (s t) + O (s^{- 1} e^{| y | t}),

φ_{s}^{'} (t) = - \cos (s t) + O (s^{- 1} e^{| y | t}), ψ_{s}^{'} (t) = - s \sin (s t) + O (e^{| y | t}) .

Remark 2. In the case $q \in ℂ^{1} [0, 1]$ and $l = 0$ we have the asymptotic expansions

φ_{s} (t) = - \sin (s t) s^{- 1} + Q (t) \cos (s t) s^{2} + O (s^{- 3} e^{3 | y | t}),

ψ_{s} (t) = \cos (s t) + Q (t) \sin (s t) s^{- 1} + O (s^{- 2} e^{3 | y | t}),

φ_{s}^{'} (t) = - \cos (s t) - Q (t) \sin (s t) s^{- 1} + O (s^{- 2} e^{3 | y | t}),

ψ_{s}^{'} (t) = - s \sin (s t) + Q (t) \cos (s t) + O (s^{- 1} e^{3 | y | t}),

where $Q (t) = \frac{1}{2} \int_{0}^{t} q (τ) d τ .$

We can calculate the first functions $p_{j}^{l}$ and $p_{j}^{- l}$ in Case d (for function $φ$ ):

p_{1}^{0} = - 1, p_{2}^{0} = Q (t), p_{3}^{0} = - \frac{1}{2} (Q (t))^{2} \frac{1}{4} q (t) + \frac{1}{4} q (0),

p_{1}^{1} = t, p_{2}^{1} = 1 - t Q (t), p_{1}^{2} = - t^{2},

{\overline{p}}_{0}^{0} = 1, {\overline{p}}_{1}^{0} = - Q (t), {\overline{p}}_{2}^{0} = \frac{1}{2} (Q (t))^{2} + \frac{1}{4} q (t) - \frac{1}{4} q (0),

{\overline{p}}_{0}^{1} = - t, {\overline{p}}_{1}^{1} = t Q (t), {\overline{p}}_{0}^{2} = t^{2};

and in Case n (for function $ψ$ ):

p_{0}^{0} = - 1, p_{1}^{0} = Q (t), p_{2}^{0} = - \frac{1}{2} (Q (t))^{2} + \frac{1}{4} q (t) - \frac{1}{4} q (0),

p_{0}^{1} = t, p_{1}^{1} = - t Q (t), p_{0}^{2} = - t^{2},

{\overline{p}}_{- 1}^{0} = 1, {\overline{p}}_{0}^{0} = - Q (t), {\overline{p}}_{1}^{0} = \frac{1}{2} (Q (t))^{2} + \frac{1}{4} q (t) + \frac{1}{4} q (0),

{\overline{p}}_{- 1}^{1} = - t, {\overline{p}}_{0}^{1} = 1 + t Q (t), {\overline{p}}_{- 1}^{2} = t^{2} .

We will use an additional index to distinguish cases: $p_{j}^{d, l} (t), p_{j}^{- d, l} (t)$ (Case d), $p_{j}^{n, l} (t), p_{j}^{- n, l} (t)$ (in Case n).

The following integral equation holds [1,5]:

w_{α s} (t) - \frac{1}{s} \int_{0}^{t} q (τ) \sin (s (t - τ)) w_{α s} (τ) d τ = \sin α \cos (s t) - \cos α \frac{\sin (s t)}{s} .

If $α = 0$ , then the solution of this integral equation is $φ_{s}$ , if $α = π / 2$ , then the solution of this integral equation is $ψ_{s}$ . By superposition principle we have

w_{α s} (t) = \cos α . φ_{s} (t) + \sin α . ψ_{s} (t) .

So, we get asymptotic expansions for function $w_{α s}$ .

Lemma 3. Let $s \in ℂ_{s}$ and $q \in C^{r} [0, 1]$ . Then for $| s | \geq q_{0}$ we have the asymptotic expansions

(w_{α s})_{s}^{(l)} (t, s) = - \sum_{j = 0}^{r} p_{j}^{l} (t) \cos (s t + \frac{π}{2} (j - l)) s^{- j} + O (s^{- (r + 1)} e^{(r + 2) | y | t})

(w_{α s}^{'})_{s}^{(l)} (t, s) = - {\sum {\overline{p}}_{j}^{- l} (t) \cos (s t + \frac{π}{2} (j - l)) s^{- j}}_{j = - 1}^{r - 1} + O (s^{- r} e^{(r + 2) | y | t)})

for $l \in ℕ_{0}$ , where

Remark 3. Formulas (15)–(16) are valid for $α = 0$ , but in (7)–(8) we have more accurate the asymptotic expansions with

p_{j}^{d, l} (t) \cos (s t + \frac{π}{2} (r + 1 - l)) s^{- (r + 1)} + O (s^{- (r + 2)} e^{(r + 2) | y | t}),

p_{j}^{- d, l} (t) \cos (s t + \frac{π}{2} (r - l)) s^{- r} + O (s^{- (r + 1)} e^{(r + 1) | y | t}),

instead $O (s^{- (r + 1)} e^{(r + 2) | y | t})$ and $O (s^{- r} e^{r + 2) | y | t})$ in (13)–(14).

Corollary 1. If $q \in C [0, 1]$ and $α \in (0, π)$ , then we have asymptotic expansions:

w_{α s} (t) = \sin α \cdot \cos (s t) + O (s^{- 1} e^{| y | t}), w_{α s}^{'} (t) = - \sin α \cdot \sin (s t) s + O (e^{| y | t}) .

Corollary 2.If $q \in C^{1} [0, 1]$ and $α \in (0, π)$ , then we have asymptotic expansions:

w_{α s} (t) = \sin α . \cos (s t) + (- \cos α + \sin α Q (t)) \sin (s t) s^{- 1} + O (s^{- 2} e^{3 | y | t}),

w_{α s} (t) = - \sin α . \sin (s t) s + (- \cos α + \sin α Q (t)) \cos (s t) + O (s^{- 1} e^{3 | y | t}) .

Lemma 4.Let $x \in ℝ_{s}^{+}, δ \in ℝ, q \in C^{s} [0, 1], Q_{j} (x), j = \overline{1, r}$ , are bounded functions.

If $s = x + δ,$

δ = \sum_{j = 1}^{r} Q_{j} (x) x^{- j} + O (x^{- (r + 1)}),

then we have the following equations

w_{α s} (t) = {\sum R_{j} (t, x) x^{- j}}_{j = 0}^{r} + O (x^{- (r + 1)}), w_{α s}^{'} (t) = {\sum {\overline{R}}_{j} (t, x) x^{- j}}_{j = - 1}^{r - 1} + O (x^{- r}),

where

R_{0} (t, x) = \sin α \cdot R_{0}^{n} (t, x), R_{j} (t, x) = \cos α \cdot R_{j}^{d} (t, x) + \sin α \cdot R_{j}^{n} (t, x), j = \overline{1, r,}

{\overline{R}}_{- 1} (t, x) = \sin α \cdot {\overline{R}}_{- 1}^{n} (t, x), {\overline{R}}_{j} (t, x) = \cos α \cdot {\overline{R}}_{j}^{d} (t, x) + \sin α \cdot {\overline{R}}_{j}^{n} (t, x), j = \overline{0, r - 1,}

and functions

m = \overline{0, r .}

Proof. The proof follows from (12) and asymptotic expansions for $φ_{s} (t)$ [3,Corollary 2] and $ψ_{s} (t)$ [4,Corollary2].

Remark 4. In the case $α = 0$ we have more accurate the asymptotic expansions

φ_{s} (t) = {\sum R_{j}^{d} (t, x) x^{- j}}_{j = 1}^{r + 1} + O (x^{- (r + 2)}), φ_{s}^{'} (t) = {\sum {\overline{R}}_{j}^{d}}_{j = 0}^{r} (t, x) x^{- j} + O (x^{- (r + 1)}) .

Corollary 3. If $q \in C [0, 1]$ and $α \in (0, π)$ , then we have asymptotic expansions:

w_{α s} (t) = \sin α \cdot \cos (x t) + O (x^{- 1}), w_{α s}^{'} (t) = - \sin α \cdot \sin (x t) x + O (1) .

Corollary 4.If $q \in C^{1} [0, 1]$ and $α \in (0, π)$ , then we have asymptotic expansions:

w_{α s} (t) = \sin α . \cos (x t) + (- \cos α + \sin α (Q (t) - t Q_{1} (x))) \sin (x t) x^{- 1} + O (x^{- 2}),

w_{α s}^{'} (t) = - \sin α . \sin (x t) x + (- \cos α + \sin α (Q (t) - t Q_{1} (x))) \cos (x t) + O (x^{- 1}) .

2 Asymptotic expansions for characteristic equations

Substituting $w_{α s} (t)$ into (3) we get the characteristic equation

h_{α} (s) : = w_{α s}^{'} (1) - γ_{w α s} (ξ) = 0,

h_{α} (s) : = w_{α s}^{'} (1) - γ_{w_{α s}^{'}} (ξ) = 0,

h_{α} (s) : = w_{α s} (1) - γ_{w α s} (ξ) = 0 .

Let’s define functions:

h_{- 1}^{l} (s) : = - {\overline{p}}_{- 1}^{l} (1) \sin (s - \frac{π}{2} l) = (- 1)^{l - 1} \sin α \sin (s - \frac{π}{2} l),

h_{j}^{l} (s) : = γ p_{j}^{l} (ξ) \cos (ξ s + \frac{π}{2} (j - l)) - {\overline{p}}_{j}^{l} (1) \cos (s + \frac{π}{2} (j - l)), j = \overline{0, r - 1,}

h_{j}^{l} (s) : = γ {\overline{p}}_{j}^{l} (ξ) \cos (ξ s + \frac{π}{2} (j - l)) - {\overline{p}}_{j}^{l} (1) \cos (s + \frac{π}{2} (j - l)), j = \overline{- 1, r - 1,}

h_{j}^{l} (s) : = γ p_{j}^{l} (ξ) \cos (ξ s + \frac{π}{2} (j - l)) - p_{j}^{l} (1) \cos (s + \frac{π}{2} (j -)), j = \overline{0, r,}

where functions $p_{j}^{l}$ and ${\overline{p}}_{j}^{l}$ are defined by formulas (15) – (16). For example,

h_{- 1}^{0} (s) = - \sin α \sin s, h_{- 1}^{1} (s) = - \sin α \cos s,

h_{0}^{0} (s) = (\sin α Q (1) - \cos α) \cos s - \sin α γ \cos (ξ s),

h_{- 1}^{0} (s) = \sin α (\sin s - γ \sin (ξ s)), h_{- 1}^{1} (s) = - \sin α (\cos s - γ ξ \cos (ξ s)),

h_{0}^{0} (s) = (\sin α Q (1) - \cos α) \cos s - (\sin α Q (ξ) - \cos α) γ \cos (ξ s),

h_{0}^{0} (s) = \sin α (\cos s - γ \cos (ξ s)), h_{0}^{1} (s) = - \sin α (\sin s - γ ξ \sin (ξ s)),

h_{- 1}^{0} (s) = (\sin α Q (1) - \cos α) \sin s - (\sin α Q (ξ) - \cos α) γ \sin (ξ s) .

We will use the notation: $ρ = - 1, a_{k} : = (k - 1 / 2) π$ in Cases $1, 2; ρ = 0, a_{k} : = (k - 1) π$ in Cases $3, k \in ℕ$ .

Lemma 5.Suppose $| γ | < 1$ in Cases 2 and 3. Then $| h_{ρ}^{0} (a_{k} + ι y) | \geq {κe}^{| y |}, κ > 0 .$

Proof. In Case 1 we have

| h_{- 1}^{0} (a_{k} + ι y) | = \sin α | \sin (a_{k} + ι y) | = \sin α | \sin a_{k} \cos h y + ι \cos a_{k} \sin h y) | = \sin α \cos h y \geq \sin α e^{| y |} / 2 .

From inequalities

| \sin s - γ \sin (ξ s) | \geq (| \sin x | - | γ | | \sin (ξ s) |) \cos h y \geq (| \sin x | - | γ |) \cos h y,

| \cos s - γ \cos (ξ s) | \geq (| \cos x | - | γ | | \cos (ξ s) |) \cos h y \geq (| \cos x | - | γ |) \cos h y

we get (in Cases 2 and 3)

| h_{ρ}^{0} (a_{k} + ι y) | \geq \sin α (1 - | γ |) \cos h y \geq \sin α (1 - | γ |) e^{| y |} / 2 .

Lemma 6. Suppose $| γ | < 1$ in Cases 2 and 3. There exists $B > 0$ such that $| h_{ρ}^{0} (s) | \geq {κe}^{| y |}, k > 0$ for $| y | \geq B$ .

Proof. We estimate

| \sin s - γ \sin (ξ s) | \geq | \sin s | - | γ | | \sin (ξ s) | \geq \sin h | y | - | γ | \cos h (ξ y),

| \cos h s - γ \cos (ξ s) | \geq | \cos s | - | γ | | \cos (ξ s) | \geq \sin h | y | - | γ | \cos h (ξ y) .

For $| γ | < 1$ we have

\lim_{y \to + ∞} (\sin h y - | γ | \cos h (ξ y)) e^{- y} = \frac{1}{2} (1 - | γ | . ⌊ ξ ⌋ \geq \frac{1}{2} (1 - | γ |) > 0 .

So, in Cases 2 and 3 there exists $B > 0$ such that

| h_{ρ}^{0} (s) | \geq \frac{1}{4} \sin α (1 - | γ |) e^{| y |} .

for $| y | \geq B$ . The proof in Case 1 repeats the proof in Case 2 with $γ = 0$ .

Lemma 7.Let $s \in ℂ_{s}$ and $q \in C^{r} [0, 1]$ . Then for $| s | \geq q_{0}$ the asymptotic expansion

h_{α}^{(l)} (s) = {\sum h_{j}^{l} (s) s^{- j} + O (s^{- (r + 1 + ρ)} e^{(r + 2) | y |})}_{j = ρ}^{r + ρ}

is valid, $l ℕ \in ℕ_{0}$ .

Proof. The proof for $α \in (0, π)$ is the same as in case $α = π / 2$ [4,see Lemma11].

Remark 5. In the case $q \in C [0, 1]$ we have (see Corollary 1) the asymptotic expansion

h_{α} (s) = h_{ρ}^{0} (s) s^{- ρ} + O (s^{- (1 + ρ)} e^{| y |}) .

Remark 6. If $α = 0$ (the problem (1), (4_d), (3)) formula (21) is valid with $ρ = 0$ in Cases 1, 2; $ρ = 1$ in Case 3, and

h_{0}^{0} (s) = - \cos s, h_{0}^{1} (s) = - \sin s,

h_{0}^{0} (s) = γ \cos (ξ s) - \cos s, h_{0}^{1} (s) = \sin s - γ ξ \sin (ξ s),

h_{1}^{0} (s) = γ \sin (ξ s) - \sin s, h_{1}^{1} (s) = γ ξ \cos (ξ s) - \cos s .

So, if $α = 0$ , then Lemma 5 and Lemma 6 are valid for the problem (1), (4_d), (3) with $ρ = 0, α_{k} = k π, k \in ℕ$ , in Cases 1 and 2, $ρ = 0, α_{k} = (k - 1 / 2) π, k \in ℕ$ , in Case 3.

Let us consider positive $s = x > 0, q \in C^{r} [0, 1]$ . We investigate equation $h_{α} (x + δ) = 0, δ \in ℝ$ , with additional condition

| h_{ρ}^{1} (x) | \geq κ > 0

Lemma 8.Suppose $| γ | < 1$ in Cases 2 and 3. If $h_{ρ}^{0} (x) = 0$ , then (24) is valid. The constant κ is the same for all such x.

Proof. In Case 1 if $h_{- 1}^{0} (x) = - \sin α \sin x = 0,$ then $x = x_{k} = k π, k \in ℕ$ and $| h_{- 1}^{1} (x_{k}) | = | - \sin α \cos x_{k} | = \sin α > 0$ .

In Case 2 we have equation $h_{- 1}^{0} (x) = - \sin α (\sin x - γ \sin (ξ x))$ . If $x_{k}$ are the root of equation $x - γ \sin (ξ x) = 0$ , then we get [2, see Lemma 4 and Corollary 3] that $| h_{- 1}^{1} (x_{k}) | = | - \sin α (\cos x_{k} - γ ξ \cos (ξ x)) | \geq \sin α | \cos x_{k} - γ ξ \cos (ξ x_{k}) | \geq \sin α . k = κ \geq 0$ .

In Case 3 we have equation $h_{0}^{0} (x) = \sin α (\cos x - γ \cos (ξ x))$ . If $x_{k}$ are the root of equation $\cos x - γ \cos (ξ x) = 0$ , then we get [2, see Lemma 5] that $| h_{0}^{1} (x_{k}) | \geq \sin α | \sin x_{k} - γ ξ \sin (ξ_{x k}) | \geq \sin α (| \sin x_{k} | - | γ | . | \sin (ξ_{x k}) |) \geq \sin α . k = κ > 0$ .

Remark 7. Lemma 4 and Lemma 5 in [2] were proved for $ξ \in (0, 1)$ , but it is easy to see, that they are valid for $ξ = 0$ too. So, Lemma 8 is proved for all $ξ$ (see (3)).

Remark 8. If $α = 0$ then Lemma 8 is valid with $ρ = 0$ Cases 1 and 2, $ρ = 1$ in Case 3. The proof is the same.

Let’s denote the function

Q_{1} (x) = - h_{1 + ρ}^{0} (x) (h_{ρ}^{1} (x))^{- 1}

If functions $Q_{1}, . . ., Q_{k - 1}$ are defined, then we can find functions

$l = \overline{1, k - 1}$ and function

If $q \in C^{1} [0, 1]$ , then

Q_{1} (x) = \frac{(\sin α Q (1) - \cos α) \cos x - \sin α γ \cos (ξ x)}{\sin α \cos x}

Q_{1} (x) = \frac{(\sin α Q (1) - \cos α) \cos x - (\sin α Q (ξ) - \cos α) γ \cos (ξ x)}{\sin α (\cos x - γ ξ \cos (ξ x))}

Q_{1} (x) = \frac{(\sin α Q (1) - \cos α) \sin x - (\sin α Q (ξ) - \cos α) γ \sin (ξ x)}{\sin α (\sin x - γ ξ \sin (ξ x))}

Lemma 9. If $q \in C^{r} [0, 1]$ and $δ = o (1), h_{ρ}^{0} (x) = 0$ , then asymptotic expansion

δ = {\sum Q_{j} (x) x^{- j}}_{j = 1}^{r} + O (x^{- (r + 1)})

is valid, where $Q_{j} (x), j = \overline{1, r,}$ , are bounded functions.

Proof. The proof can be found in [4, see proof of Lemma 14].

3 Spectral asymptotics for eigenvalues and eigenfunctions

In this section we assume, that $| γ | < 1$ in Cases 2, 3. Let us denote domains $D_{k} = {s \in ℂ : | x | \leq a_{k}, | y | \leq a_{k}}, D_{s k} = ℂ_{s} \cap D_{k}, k \in ℕ$ ( $k > 1$ in Case 3), contours $T_{s k} = ℂ_{s} \cap \partial D_{k}$ , and intervals $I_{k} : = (a_{k}, a_{k + 1}) \subset D_{s, k + 1} ∖ D_{s k}, k \in ℕ$ .

Lemma 10.Suppose $| γ | < 1$ in Cases 2 and 3. If $q \in C [0, 1]$ , then it follows that the number of zeros of functions $h_{α} (s)$ and $h_{ρ}^{0} (s) s^{- ρ}$ is the same inside $T_{s k}$ for sufficiently large k.

Proof. We have (see (22)) $h_{α} (s) = h_{ρ}^{0} (s) s^{- ρ} + O (s^{- 1 - ρ} e^{| y |})$ . Using Lemma 5 and Lemma 6we estimate $| O (s^{- 1 - ρ} e^{| y |}) | \leq c_{1} | s |^{- 1 - ρ} e^{| y |} < m i n {k, κ} | s |^{- ρ} e^{| y |} \leq | h_{ρ}^{0} (s) s^{- ρ} |$ on the contours $T_{s k}$ for sufficiently large k. Therefore, by Rouché theorem it follows that the number of zeros of $h_{α} (s)$ and $h_{ρ}^{0} (s) s^{- ρ}$ are the same inside $T_{s k}$ for sufficiently large k.

Corollary 5.If $q \in C [0, 1]$ , then it follows that the number of zeros of functions $h_{α} (s)$ and $h_{ρ}^{0} (s)$ is the same between $T_{s k}$ and $T_{s, k + 1}$ for sufficiently large k.

Remark 9. If $α = 0$ , then Lemma 10 and Corollary 5 are valid. The proof is the same. In Case 3 $s = 0$ isn’t zero of the function $h_{1}^{0} (s) s^{- 1} = γ s^{- 1} \sin (ξ s) - s^{- 1} \sin s$ for $| γ | < 1$ .

From (19) (and (23) for $α = 0$ ) we have that function $h_{ρ}^{0}$ has only one positive root $x_{k} \in I_{k}, k \in ℕ$ . For example, $x_{k} = π k$ (and $x_{k} = π (k + 1 / 2$ ) for $α = 0$ ) in Case 1. In Cases 2 and 3 existence of such root follows from [4, Lemma 4 and Lemma 5]. Thus, function $h_{α} (s)$ has only one root sk between $T_{s k}$ and $T_{s, k + 1}$ for sufficiently large k.

In Case 2 $(\sin α \neq 0) a_{k} = (k - 1 / 2) π$ and

h_{α} (a_{k}) = - \sin α (\sin a_{k} - γ \sin (ξ a_{k})) a_{k} + O (1) = a_{k} \sin α ((- 1)^{k} + γ \sin (ξ a_{k}) + O (k^{- 2})) .

If $| γ | < 1$ , then sign $((- 1)^{k} + γ \sin (ξ a_{k}) + O (k^{- 1})) = (- 1)^{k}$ $s i g n (h_{α} (a_{k}) h_{α} (a_{k + 1})) = - 1$

for sufficiently large k. This formula is valid in Cases 1 and 3. Moreover, it is valid for $α = 0$ . Then from Intermediate Value Theorem at least one root of the function $h_{α} (s)$ lies in $I_{k}$ for sufficiently large k. So, $s_{k}$ is real root for such k.

We have $s_{k} \sim x_{k} \sim π k (a s k \to ∞)$ . Then $h_{α} (s_{k}) \cdot s_{k}^{ρ} = h_{ρ}^{0} (s_{k}) + O (k^{- 1}) = 0$ and $\lim_{k \to ∞} h_{ρ}^{0} (s_{k}) = 0$ . The function $s_{k} \to x_{k} a s k \to ∞$ or

s_{k} = x_{k} + o (1) (a s k \to ∞) .

Now we will investigate the distribution of these positive eigenvalues of problem (1)–(3), and we leave out the note about sufficiently large k.

Let us denote $δ_{k} = s_{k} - x_{k}$ . We have that $δ_{k} = o (1)$ .

Theorem 1.Let $q \in C^{r} [0, 1]$ . For eigenvalues $λ_{k} = s_{k}^{2}$ and eigenfunctions $u_{k}$ of problem (1)–(3), we have the asymptotic expansions

s_{k} = x_{k} + {\sum Q_{j} (x_{k}) x_{k}^{- j}}_{j = 1}^{r} + O (k^{- (r + 1)}),

u_{k} (t) = {\sum R_{j} (t, x_{k}) x_{k}^{- j}}_{j = 0}^{r} + O (k^{- (r + 1)})

for sufficiently large k.

Proof. We have $δ_{k} = o (1)$ . So, all conditions of Lemma 9 are valid, and it follows (27).Then we apply Corollary 1 and get (28).

Corollary 6.Let $q \in C [0, 1]$ . For eigenvalues $λ_{k} = s_{k}^{2}$ and eigenfunctions $u_{k}$ of problem (1)–(3), the asymptotic equations

s_{k} = x_{k} + O (k^{- 1}), u_{k} (t) = R_{0} (t, x_{k}) + O (k^{- 1})

are valid for sufficiently large k, where $R_{0} (t, x) = \sin α \cos (x t)$ .

Corollary 7.If $q \in C^{1} [0, 1]$ , then the asymptotic equations

s_{k} = x_{k} + Q_{1} (x_{k}) x_{k}^{- 1} + O (k^{- 2}),

u_{k} (t) = R_{0} (t, x_{k}) + R_{1} (t, x_{k}) x_{k}^{- 1} + O (k^{- 2})

are valid for sufficiently large k, where $Q_{1} (x)$ is defined by (25) $R_{0} (t, x) = \sin α \cos (x t), R_{1} (t, x) = (- \cos α + \sin α (Q (t) - t Q_{1} (x))) \sin (x t)$ .

Remark 10. (See [2].) If $α = 0$ , then formula (27) and asymptotic expansion

u_{k} (t) = {\sum R_{j} (t, x_{k}) x_{k}^{- j}}_{j = 1}^{r + 1} + O (k^{- (r + 2)})

are valid for sufficiently large k. If $q \in C [0, 1]$ , then $R_{1} (t, x) = - \sin (x t)$ . If $q \in C^{1} [0, 1]$ , then $R_{2} (t, x) = (Q (t) - t Q_{1} (x))) \cos (x t)$ and

Q_{1} (x) = \frac{Q (1) \sin x - γ \sin (ξ x)}{\sin x}

Q_{1} (x) = \frac{Q (1) \sin x - γ Q (ξ) \sin (ξ x)}{\sin x - γ ξ \sin (ξ x)}

Q_{1} (x) = \frac{Q (1) \cos x - γ Q (ξ) \cos (ξ x)}{\cos x - γ ξ \cos (ξ x)}

Material suplementario

Información adicional

AMS Subject Classification: : 34B24; 34L20; 35R10

References

[1] B.M. Levitan, I.S. Sargsjan. Sturm–Liouville and Dirac operators. Kluwer, Dordrecht, 1991.

[2] E. Şen, A. Štikonas. Asymptotic distribution of eigenvalues and eigenfunctions of a nonlocal boundary value problem. Math. Model. Anal., 26(2):253–266, 2021. https://doi.org/10.3846/mma.2021.13056.

[3] A. Štikonas, E. Şen. Asymptotic analysis of Sturm–Liouville problem with nonlocal integral-type boundary condition. Nonlinear Anal. Model. Control, 26(5):969–991, 2021. https://doi.org/10.15388/namc.2021.26.24299.

[4] A. Štikonas, E. Şen. Asymptotic analysis of Sturm–Liouville problem with Neumann and nonlocal two-point boundary conditions. Lith. Math. J, 62(4):519–541, 2022. https://doi.org/10.1007/s10986-022-09577-6.

[5] E.C. Titchmarsh. Eigenfunction expansions associated with second-order differential equations. Clarendon Press, Oxford, 1946.

Notas