Spectral Problems for Non-Linear Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions

2000 ◽  
Vol 52 (2) ◽  
pp. 248-264 ◽  
Author(s):  
Paul A. Binding ◽  
Patrick J. Browne ◽  
Bruce A. Watson
Keyword(s):  
Boundary Conditions ◽  
Liouville Equation ◽  
Spectral Problems ◽  
Non Linear ◽  
Sturm Liouville ◽  
Independent Variable

AbstractThe nonlinear Sturm-Liouville equation−(pyʹ)ʹ + qy = λ(1 − f)ry on [0, 1]is considered subject to the boundary conditions(ajλ + bj)y(j) = (cjλ + dj)(pyʹ)(j), j = 0, 1.Here a0 = 0 = c0 and p, r > 0 and q are functions depending on the independent variable x alone, while f depends on x, y and yʹ. Results are given on existence and location of sets of (λ, y) bifurcating from the linearized eigenvalues, and for which y has prescribed oscillation count, and on completeness of the y in an appropriate sense.

scholarly journals Geometric aspects of self-adjoint Sturm–Liouville problems

2017 ◽  
Vol 147 (6) ◽  
pp. 1279-1295
Author(s):  
Yicao Wang
Keyword(s):  
Boundary Conditions ◽  
Liouville Equation ◽  
Adjoint Action ◽  
Principal Type ◽  
Unitary Matrices ◽  
Adjoint Orbits ◽  
Sturm Liouville ◽  
Refined Classification ◽  
Adjoint Orbit

In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S2, and investigate the behaviour of the nth eigenvalue λnas a function on such orbits.

scholarly journals STURM–LIOUVILLE PROBLEMS WITH BOUNDARY CONDITIONS RATIONALLY DEPENDENT ON THE EIGENPARAMETER. I

2002 ◽  
Vol 45 (3) ◽  
pp. 631-645 ◽  
Author(s):  
Paul A. Binding ◽  
Patrick J. Browne ◽  
Bruce A. Watson
Keyword(s):  
Boundary Conditions ◽  
Liouville Equation ◽  
Subject Classification ◽  
Asymptotics Of Eigenvalues ◽  
Primary 34B24 ◽  
Sturm Liouville

AbstractWe consider the Sturm–Liouville equation$$ -y''+qy=\lambda y\quad\text{on }[0,1], $$subject to the boundary conditions$$ y(0)\cos\alpha=y'(0)\sin\alpha,\quad\alpha\in[0,\pi), $$and$$\frac{y'}{y}(1)=a\lambda+b-\sum_{k=1}^N\frac{b_k}{\lambda-c_k}. $$Topics treated include existence and asymptotics of eigenvalues, oscillation of eigenfunctions, and transformations between such problems.AMS 2000 Mathematics subject classification: Primary 34B24; 34L20

scholarly journals Isospectral sets for boundary value problems on the unit interval

1988 ◽  
Vol 8 (8) ◽  
pp. 301-358 ◽  
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Liouville Equation ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Unit Interval ◽  
Trace Function ◽  
Sturm Liouville ◽  
Real Self

AbstractWe analyse isospectral sets of potentials associated to a given ‘generalized periodic’ boundary condition in SL(2, R) for the Sturm-Liouville equation on the unit interval. This is done by first studying the larger manifold M of all pairs of boundary conditions and potentials with a given spectrum and characterizing the critical points of the map from M to the trace a + d Isospectral sets appear as slices of M whose geometry is determined by the critical point structure of the trace function. This paper completes the classification of isospectral sets for all real self-adjoint boundary conditions.

scholarly journals Inverse problems for the Sturm–Liouville equation with a spectral parameter in the boundary condition

2017 ◽  
Author(s):  
Namig J. Guliyev
Keyword(s):  
Boundary Condition ◽  
Inverse Problems ◽  
Boundary Conditions ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Eigenvalue Parameter ◽  
Sturm Liouville ◽  
Necessary And Sufficient

Inverse problems of recovering the coefficients of Sturm--Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

scholarly journals On Basis Property of Root Functions For a Class Second Order Differential Operator

2020 ◽  
Vol 5 (1) ◽  
pp. 361-368
Author(s):  
Volkan Ala ◽  
Khanlar R. Mamedov
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Basis Property ◽  
Second Order ◽  
Order Differential Operator ◽  
Root Functions ◽  
Sturm Liouville

AbstractIn this work we investigate the completeness, minimality and basis properties of the eigenfunctions of one class discontinuous Sturm-Liouville equation with a spectral parameter in boundary conditions.

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