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Sturm-Liouville problems with a boundary condition depending bilinearly on an eigenparameter

Yagub N. Aliyev, Narmin N. Aliyeva

Abstract

This paper studies a Sturm--Liouville boundary value problem in which one of the boundary conditions depends bilinearly on the spectral parameter. The differential equation is considered on the interval $(0,1)$ with a classical boundary condition at one endpoint and an eigenparameter--dependent boundary condition at the other. Explicit formulas for the inner products and norms of eigenfunctions are obtained. These relations make it possible to analyze the structure of the system of root functions and the corresponding biorthogonal system. Using these results, the minimality of the system of root functions in $L_2(0,1)$ is established. Furthermore, the basis properties of the system of root functions in the spaces $L_p(0,1)$, $1<p<\infty$, are investigated. Necessary and sufficient conditions under which the system forms a basis are derived. Special attention is given to the cases of multiple eigenvalues and the case when the eigenvalue coincides with the critical value $-d/c$. The obtained results reveal a symmetry between different spectral cases and provide a more elementary approach that avoids the use of the characteristic function. Several examples are presented to illustrate the theoretical results.

Sturm-Liouville problems with a boundary condition depending bilinearly on an eigenparameter

Abstract

This paper studies a Sturm--Liouville boundary value problem in which one of the boundary conditions depends bilinearly on the spectral parameter. The differential equation is considered on the interval with a classical boundary condition at one endpoint and an eigenparameter--dependent boundary condition at the other. Explicit formulas for the inner products and norms of eigenfunctions are obtained. These relations make it possible to analyze the structure of the system of root functions and the corresponding biorthogonal system. Using these results, the minimality of the system of root functions in is established. Furthermore, the basis properties of the system of root functions in the spaces , , are investigated. Necessary and sufficient conditions under which the system forms a basis are derived. Special attention is given to the cases of multiple eigenvalues and the case when the eigenvalue coincides with the critical value . The obtained results reveal a symmetry between different spectral cases and provide a more elementary approach that avoids the use of the characteristic function. Several examples are presented to illustrate the theoretical results.
Paper Structure (14 sections, 25 theorems, 195 equations)

This paper contains 14 sections, 25 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Inner products and norms of eigenfunctions.
  3. Inner products with the first associated function.
  4. Inner products with the second associated function.
  5. Existence of special associated functions $y_{k+1}^*$ and $y_{k+1}^\#$.
  6. Existence of special associated functions $y_{k+2}^*$ and $y_{k+2}^\#$.
  7. Basis properties of the root functions in the cases (i) and (ii).
  8. Case (i).
  9. Case (ii).
  10. Basis properties of the root functions in the case (iii) and (iv).
  11. Case (iii).
  12. Case (iv).
  13. Examples.
  14. Declarations

Key Result

Lemma 2.1

Let $y_{n}, y_{m}$ be eigenfunctions corresponding to the eigenvalues $\lambda _{n}$ and $\lambda _{m}$ such that $\lambda _{n}\ne \overline{\lambda _{m}}$. (a) If $\lambda_{n},\lambda_{m}\neq - \frac{d}{c}$, then (b) If $\lambda_{n}= -\frac{d}{c} \neq \lambda_{m}$, then

Theorems & Definitions (50)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 40 more